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Essays on beliefs, networks and spatial modeling
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Essays on beliefs, networks and spatial modeling
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ESSAYS ON BELIEFS, NETWORKS AND SPATIAL MODELING Ida Johnsson A dissertation submitted for the degree of Doctor of Philosophy in Economics FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA May 2018 A Jorge, Kiki, Madde y Nika. Ett s arskilt tack till de tv asistn amnda f or deras itiga tummh allande igenom aren. Tack till mamma och pappa f or allt. I'd like to thank Professor Hyungsik Roger Moon and Professor Hashem Pesaran for their continued guidance and support during my PhD. Also special thanks to Dr. Morgan Ponder for his invaluable help. Contents 0 Introduction 7 1 Estimation of Peer Eects in Endogenous Social Networks: Control Func- tion Approach 9 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2 Model of Peer Eects with an Endogenous Network . . . . . . . . . . . . . . 13 1.3 Endogenous Network Formation and Identication of peer eects using a con- trol function approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.1 Model of Network Formation . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.2 Control Function of Network Endogeneity . . . . . . . . . . . . . . . 17 1.3.3 Identication of Peer Eects . . . . . . . . . . . . . . . . . . . . . . . 18 1.4 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.4.1 With a i as Control Function . . . . . . . . . . . . . . . . . . . . . . . 23 1.4.2 With (x 2i ;a i ) as Control Function . . . . . . . . . . . . . . . . . . . . 25 1.4.3 Sieve Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.5 Limit Distribution and Standard Error . . . . . . . . . . . . . . . . . . . . . 28 1.5.1 Limiting Distribution and Standard Error of ^ 2SLS . . . . . . . . . . 29 1.5.2 Limiting Distribution and Standard Error of 2SLS . . . . . . . . . . 31 1.6 Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 Appendices 45 A.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 A.2 Outline of the proof of Theorem 1.5.1 . . . . . . . . . . . . . . . . . . . . . . 49 S.1 Supplementary Appendix - Introduction . . . . . . . . . . . . . . . . . . . . 51 S.2 For ^ 2SLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 S.2.1 Controlling the Sampling Error ^ a i a i in Sieve Estimation . . . . . . 53 S.2.2 Controlling the Series Approximation Error . . . . . . . . . . . . . . 59 S.2.3 Limiting Distribution of ^ 2SLS . . . . . . . . . . . . . . . . . . . . . . 61 S.2.4 Further Supporting Lemmas . . . . . . . . . . . . . . . . . . . . . . . 70 S.3 For 2SLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 S.3.1 Limiting distribution of 2SLS . . . . . . . . . . . . . . . . . . . . . . 77 S.3.2 Controlling the Sampling Error d deg i deg i in Sieve Estimation . . . . 78 S.3.3 Controlling the Series Approximation Error for r K (x 2i ; deg i ) . . . . . 79 S.3.4 Limiting distribution of 2SLS . . . . . . . . . . . . . . . . . . . . . . 79 S.4 Supplementary Monte Carlo results . . . . . . . . . . . . . . . . . . . . . . . 81 2 Double-question Survey Measures for the Analysis of Financial Bubbles and Crashes 89 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 2.2 Valuation and expected price changes . . . . . . . . . . . . . . . . . . . . . . 95 2.3 Double-question surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 2.3.1 Survey waves and respondent characteristics . . . . . . . . . . . . . . 106 2.3.2 Filters applied to survey responses . . . . . . . . . . . . . . . . . . . 107 2.3.3 Socio-demographic characteristics of respondents: . . . . . . . . . . . 109 2.3.4 Geographic location of respondents . . . . . . . . . . . . . . . . . . . 110 2.4 Price change expectations and valuation indicators . . . . . . . . . . . . . . 111 2.4.1 Eects of individual-specic characteristics on price expectations . . . 114 2.5 Constructing leading indicators of bubbles and crashes from DQ surveys . . 118 2.6 Bubble and crash indicators and realized house price changes across MSAs . 122 4 2.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Appendices 131 S1 Supplementary Appendix - Introduction . . . . . . . . . . . . . . . . . . . . 131 S2 Relationship between expected price changes and the valuation indicator for higher order horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 S3 Fixed eects-time eects (FE-TE) estimators for unbalanced panels . . . . . 134 S4 FE-TE Filtered estimators of the time-invariant eects for unbalanced panels 136 S5 Dynamic panel regressions with bubble and crash indicators . . . . . . . . . 139 S6 American Life Panel Surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 S6.1 Recruitment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 S6.2 Demographics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 S6.3 Response Rates and Attrition . . . . . . . . . . . . . . . . . . . . . . 142 S7 Survey questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 S8 Truncation lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 S9 Respondent location and respondent characteristics . . . . . . . . . . . . . . 151 S10 Denition of US regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 S11 Spatial weight matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 S12 Data sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 S12.1 Survey data downloaded from the RAND ALP website . . . . . . . . 156 S12.2 Data and codes for replicating results . . . . . . . . . . . . . . . . . . 158 S13 Selected MSA summary statistics . . . . . . . . . . . . . . . . . . . . . . . . 159 S14 Estimates for males and females . . . . . . . . . . . . . . . . . . . . . . . . . 162 S15 Random eect estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 S16 Comparison of FEF and RE estimates of the price expectation equations . . 171 S17 FE-TE Filtered estimates of the price expectation equations . . . . . . . . . 175 S18 Comparison of alternative estimates of (h) and implied interest rate r . . . . 182 S19 Regression results controlling for home-ownership . . . . . . . . . . . . . . . 184 5 3 Spatial Equilibrium and Search Frictions - an Application to the NYC Taxi Market 193 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 3.1.1 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 3.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 3.3 Benchmark Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 3.3.1 Model Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . 202 3.3.2 Demand-Driven Pricing . . . . . . . . . . . . . . . . . . . . . . . . . 204 3.4 Matching with Frictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 3.4.1 Calculating ^ it . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 3.4.2 Market Tightness and Frictions . . . . . . . . . . . . . . . . . . . . . 212 3.4.3 Local Market Tightness . . . . . . . . . . . . . . . . . . . . . . . . . 217 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Appendices 221 A1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 A1.1 Neighborhood Tabulation Areas . . . . . . . . . . . . . . . . . . . . . 224 6 Chapter 0 Introduction The essays in this dissertation focus on three topics: social networks, beliefs and asset prices, and spatio-temporal relationships. In the rst chapter (joint with Hyungsik Roger Moon) we propose a method of estimat- ing the linear-in-means model of peer eects in which the peer group, dened by a social network, is endogenous in the outcome equation for peer eects. Endogeneity is due to un- observable individual characteristics that in uence both link formation in the network and the outcome of interest. We propose two estimators of the peer eect equation that control for the endogeneity of the social connections using a control function approach. We leave the functional form of the control function unspecied and treat it as unknown. To estimate the model, we use a sieve semiparametric approach, and we establish asymptotics of the semiparametric estimator. In the second chapter (joint with Hashem Pesaran) we propose a new double-question survey whereby an individual is presented with two sets of questions; one on beliefs about current asset values and another on price expectations. A theoretical asset pricing model with heterogeneous agents is advanced and the existence of a negative relationship between price expectations and asset valuations is established, which is tested using survey results on equity, gold and house prices. Leading indicators of bubbles and crashes are proposed and their potential value is illustrated in the context of a dynamic panel regression of realized 7 house price changes across key MSAs in the US. In the nal chapter, I employ a dynamic spatial equilibrium model to analyze the eect of matching frictions and pricing policy on the spatial allocation of taxicabs and the aggregate number of taxi-passenger meetings. A spatial equilibrium model, in which meetings are frictionless but aggregate matching frictions can arise endogenously for certain parameter values, is calibrated using data on more than 45 million taxi rides in New York. It is shown how the set of equilibria changes for dierent pricing rules and dierent levels of aggregate market tightness, dened as the ratio of total supply to total demand. Finally, a novel data- driven algorithm for inferring unobserved demand from the data is proposed, and is applied to analyze how the relationship between demand and supply in a market with frictions compares to the frictionless equilibrium outcome. 8 Chapter 1 Estimation of Peer Eects in Endogenous Social Networks: Control Function Approach 1 Ida Johnsson y , Hyungsik Roger Moon z We propose a method of estimating the linear-in-means model of peer eects in which the peer group, dened by a social network, is endogenous in the outcome equation for peer eects. Endogeneity is due to unobservable individual characteristics that in uence both link formation in the network and the outcome of interest. We propose two estimators of the peer eect equation that control for the endogeneity of the social connections using a control function approach. We leave the functional form of the control function unspecied and treat 1 We thank Bryan Graham and three referees for their helpful and valuable comments and suggestions. We are particularly grateful to one of the referees for suggesting the idea that is presented in Section 1.4.2 of the paper. We also appreciate the comments and discussions of the participants at the 2015 USC Dornsife INET Conference on Networks, University of Southern California, 2016 North American Summer Meeting of the Econometric Society, 2016 California Econometrics Conference, the 2017 Asian Meeting of Econometric Society, the 2017 IAAE conference. The rst draft of the paper was written while Johnsson was a graduate fellow of USC Dornsife INET and Moon was the associate director of USC Dornsife INET. Moon acknowledges that this work was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2017S1A5A2A01023679). Address for correspondence: Hyungsik Roger Moon, Department of Economics, University of Southern California, 3620 S. Vermont Ave., Los Angeles, CA 90089, USA. E-mail: moonr@usc.edu. y Department of Economics, University of Southern California z Department of Economics, University of Southern California, and School of Economics, Yonsei Univer- sity 9 it as unknown. To estimate the model, we use a sieve semiparametric approach, and we establish asymptotics of the semiparametric estimator. Keywords: peer effects, endogenous network, sieve estimation, control function JEL Classification: C14, C21 1.1 Introduction The ways in which interconnected individuals in uence each other are usually referred to as peer eects. One of the rst to formally model peer eects is Manski (1993). He proposes the linear-in-means model, in which an individual's action depends on the average action of other individuals and possibly also on their average characteristics. Manski (1993) assumes that all individuals within a given group are connected. Later literature allows for more complex patterns of connections, in which an individual might be directly in uenced by a subset of the group. Examples are Bramoull e et al. (2009), Lee et al. (2010), Lee (2007b) among others. Models of peer eects have been applied in various areas, such as education, health and development, and various application examples are found in recent review papers such as Blume et al. (2010), Manski (2000), Epple and Romano (2011), and Graham (2011). Many models considered in earlier literature assume that connections between individuals are independent of unobserved individual characteristics that in uence outcomes. However, the exogeneity of the network or peer group might be restrictive in many applications. For example, consider the following, widely studied, empirical application of peer eects: peer in uence on scholarly achievement. The assumption that friendships are exogenous in the outcome equation for scholarly achievement means that there are no unobservables that in- uence both friendship formation and individual grades. However, even if the researchers control for observable individual characteristics such as gender, age, race and parents' edu- cation, it is likely they leave out factors that in uence both students' choice of friends and their GPA, for example parents' expectations, psychological disorders or substance use. For more examples of endogenous peer groups and networks, see Weinberg (2007), Shalizi (2012) 10 and Hsieh and Lee (2016), among others. In this paper we investigate estimating a linear-in-means model of peer eects, where the peer group is dened by a network that is endogenous to the outcome equation. Our model allows correlation between the unobserved individual heterogeneity that impacts network formation and the unobserved characteristics of the outcome. For this, we use a dyadic network formation model that allows the unobserved individual attributes of two dierent agents to in uence link formation, in which links are pairwise independent conditional on the observed and unobserved individual attributes. The network formation we consider in the paper is dense and nonparametric. The main contributions of the paper are methodological. First, given the endogenous peer group formation, we show that we can identify the peer eects by controlling the un- observed individual heterogeneity of the network formation equation. Secondly, we propose an empirically tractable implementation of the control function whose functional form is not parametrically specied. For this, we propose two approaches, one based on the estimator of the unobserved individual heterogeneity and the other one based on the node degrees of the network. 2 Our estimation method is semiparametric because we do not restrict the functional form of the control function. Thirdly, we derive the limiting distributions of the estimators within a large single network. The main challenge of the asymptotics is to handle the strong dependence of observables caused by the dense network. Closely related papers that adopt a control function approach include Goldsmith-Pinkham and Imbens (2013), Hsieh and Lee (2016), Qu and Lee (2015), and Arduini et al. (2015). Our paper adopts a frequentist approach based on nonparametric specication of the network formation, while Goldsmith-Pinkham and Imbens (2013) and Hsieh and Lee (2016) use the Bayesian method based on a full parametric specication of the network formation and the outcome equation. Like our paper, Qu and Lee (2015) assume the network (spatial weights in their model) to be endogenous through the unobserved individual heterogeneity. How- ever, our paper is dierent from Qu and Lee (2015) in many aspects. They consider sparse 2 We acknowledge that this approach is developed based on the idea provided by one of the referees. We thank the referee. 11 network formation models while we consider a dense network. They restrict the functional form of the control function to be linear, while we impose no restriction on the functional form. The two papers propose dierent implementations of the control function. Our paper is dierent from Arduini et al. (2015) regarding the main source of the endogeneity of the network and the form of the control function. Arduini et al. (2015) assume that the endo- geneity of the network is allowed through dependence between the outcome equation error and the idiosyncratic network formation error, like the conventional sample selection model. Arduini et al. (2015) consider a control function (both parametric and semiparametric) to deal with the selection bias problem. Regarding asymptotics, in both Qu and Lee (2015) and Arduini et al. (2015) the asymptotics are derived using near-epoch dependence and are based on the assumption that the number of connections does not increase at the same rate as the square of the network size. After the completion of the rst draft of our paper we were made aware of the paper by Auerbach (2016). His network formation model is the same as ours. To identify and estimate peer eects, he proposes a pairwise matching method which resembles Powell (1987), Heckman et al. (1998), and Abadie and Imbens (2006). The remainder of the paper is organized as follows. In Section 1.2 we formally present our model. In Section 1.3, we show how to identify the peer eects using control functions. Estimation is discussed in Section 1.4, and in Section 1.5 we discuss the limiting distribution of the estimator and propose standard errors. In Section 1.6 we present results of Monte Carlo simulations. There we compare the nite sample performance of our two semipara- metric estimators against an estimator that assumes unobserved characteristics enter in a linear way, as well as an instrumental variable (IV) estimator that does not control for net- work endogeneity. We investigate both high degree and low degree networks. Section 1.7 concludes. A word on notation. In what follows we denote scalars by lowercase letters, vectors by lowercase bold letters, and matrices by uppercase bold letters. 12 1.2 Model of Peer Eects with an Endogenous Net- work Suppose thatd ij;N are the observed links among individualsi2f1;:::;Ng, such thatd ij;N = 1 if i and j are directly connected and 0 otherwise. We assume that individual outcomes, y i , are given by the linear-in-means model of peer eects y i = 0 1 N X j=1 j6=i g ij;N y j + x 0 1i 0 2 + 0 B @ N X j=1 j6=i g ij;N x 1j 1 C A 0 0 3 + i ; (1.2.1) where x 1i are observed individual characteristics that aect the outcomey i ,v i are unobserved individual characteristics, and g ij;N = 8 > < > : 0 if i =j d ij;N P j6=i d ij;N otherwise: is the weight of the peer eects. Using the terminology of Manski (1993), 0 1 captures the endogenous social eect, and 0 3 measures the exogenous social eect. We let 0 := ( 0 1 ; 0 0 2 ; 0 0 3 ) 0 and denote = ( 1 ; 0 2 ; 0 3 ) 0 . Throughout the paper, we assume thatj 0 1 j< 1, and so the peer eect model has a unique solution. We let D N be the (NN) adjacency matrix of the network whose (i;j) th element is d ij;N . We letd ii;N = 0 for alli, following the convention. Let G N be the matrix whose (i;j) th element isg ij;N , and G N is obtained by row-normalizing D N . Denote X 1N = (x 0 11 ;:::; x 0 1N ) 0 , y N = (y 1 ;:::;y N ) 0 and N = ( 1 ;:::; N ) 0 . In the standard linear-in-means model of peer eects, the main focus has been identi- cation and estimation of peer eects, assuming that the peer group (or the network) is exogenous, that is,E[ i jX 1N ; G N ] = 0. For example, see Manski (1993) and Bramoull e et al. (2009), Lee (2007b), and Blume et al. (2015). To identify and estimate the linear-in-means model of peer eects when the peer group is exogenous it is necessary to take into account the fact that the regressor P N i=1 g ij;N y j is correlated with the error term i . For example, if 13 i i:i:d:(0; 2 ), it is true that E[(G N y N ) 0 N ] = [(G N (I N 1 G N ) 1 (X 1N 2 + G N X 1N 3 + N )) 0 N ] =E[(G N (I N 1 G N ) 1 N ) 0 N ] = 0 tr(G N (I N 1 G N ) 1 )6= 0: (1.2.2) To solve this endogeneity problem dierent estimators have been proposed in the liter- ature, see for example Kelejian and Prucha (1998), Lee (2003) and Lee (2007a). One of the widely used estimation methods is the Instrumental Variable (IV) approach based on using G 2 N X 1N as the IV of the endogenous regressor G N y N (see for example Kelejian and Prucha (1998), Lee (2003), and Bramoull e et al. (2009)). Then, the natural estimator is the Two-Stage Least Squares (2SLS) estimator ^ 2SLS N = (W 0 N Z N (Z 0 N Z N ) 1 Z N W N ) 1 W 0 N Z N (Z 0 N Z N ) 1 Z 0 N y N ; (1.2.3) where W N = [G N y N ; X 1N ; G N X 1N ] and Z N = [X 1N ; G N X 1N ; G 2 N X 1N ] is the matrix of instruments. For the IVs Z N to be strong, we assume that 0 2 6= 0. When the network matrix is endogenous, E[G N N ]6= 0, and the procedure used by Kelejian and Prucha (1998), Lee (2003), Bramoull e et al. (2009) and others is no longer valid since the IV matrix Z N = [X 1N ; G N X 1N ; G 2 N X 1N ] is correlated with the error term N . Specically, the validity of the 2SLS estimator depends on the orthogonality condition E[ N jZ N ] = 0, which is implied if E[ N jX 1N ; D N ] = 0. However, it does not hold if the network D N (or equivalently, the network G N ) is correlated with N , which is true if un- observed individual characteristics of the network D N (or G N ) directly in uence both link formation and individual outcomes. In this paper, we consider the case where it may be that E[ N jX 1N ; D N ]6= 0, so that unobserved characteristics that in uence link formation can also have a direct eect on individual outcomes. This is an important consideration in many common applications, for example the impact of school friendships on scholarly achievement or substance use. 14 1.3 Endogenous Network Formation and Identication of peer eects using a control function approach In this section, we introduce a network formation model ford ij;N and discuss the assumptions and implication of the model. Then we provide an identication argument for the peer eect equation when the network is endogenous. 1.3.1 Model of Network Formation Let x 2i be a vector of observable characteristics of individuali, and let x i = x 1i [x 2i . Dene X 2N analogously to X 1N and let X N = X 1N [ X 2N . We introduce a i , a scalar unobserved characteristic of individuali, which is treated as an individual xed-eect, and hence, might be correlated with x i . We denote the vector of individual unobserved characteristics by a N = (a 1 ;a 2 ;:::;a N ) 0 . Individuals are connected by an undirected network D N , with the (i;j) th element d ij;N = 1 if i and j are directly connected and 0 otherwise. We assume the network to be undirected 3 , d ij;N = d ji;N , and assume d ii;N = 0 for all i, following the convention. In this case, there are n = N 2 dyads. Let t ij denote an l T 1 vector of dyad-specic characteristics of dyad ij, and we assume that t ij =t(x 2i ; x 2j ). Agents form links according to d ij;N =I(g(t(x 2i ; x 2j );a i ;a j )u ij 0); (1.3.1) whereI() is an indicator function. In this setup, link surplus is transferable across directly linked agents and consists of three components: t ij :=t(x 2i ; x 2j ) is a systematic component that varies with observed dyad attributes and accounts for homophily, a i anda j account for unobserved dyad attributes (degree heterogeneity), and u ij is an idiosyncratic shock that is i.i.d. across dyads and independent of t ij and a i for all i;j. Since links are undirected, the surplus of link d ij;N must be the same for individual i and j. Hence, we assume that the function t ij is symmetric in i and j, and the function g is symmetric in a i and a j . 3 Our analysis can be extended to the directed network case, but we do not pursue it in this paper. 15 In the literature, various parametric versions of the network formation in (1.3.1) are used (see Graham (2015) and the references therein.). An important example of a parametric specication is the one in Graham (2017), d ij;N =I(t(x 2i ; x 2j ) 0 +a i +a j u ij > 0): (1.3.2) For the purpose of the paper, particularly in constructing the estimators that we introduce in Section 1.4, we do not need a parametric specication. Instead, we need the following two restrictions. First, we assume that the network formed by (1.3.1) is dense in the sense that the expected number of connections is proportional to the square of the network size. For this, we assume that the error u ij is drawn randomly from a distribution with full support, while g(t ij ;a i ;a j ) is bounded (see Assumption 11 in the Appendix). In this case, the probability of any two individuals forming a link is bounded away from zero. The dense network model is appropriate for scenarios where any two individuals can plausibly form a link. Secondly, we assume that the net link surplus is a monotonic function ofa i (anda j ), that is,g(t(x 2i ; x 2j );a i ;a j ) is a strictly monotonic function of a i (anda j ) (see Assumption 11(vi) in the Appendix.). Obviously, this condition is satised in the parametric model (1.3.2). This monotonic condition is crutial in constructing the estimator in Section 1.4.2. Regarding the network formation model (1.3.1), it is important to notice that the net- work formation model (1.3.1) rules out interdependent link preferences, and it assumes that links are formed independently conditional on observed and unobserved characteristics. As discussed in Graham (2017), this assumption is appropriate for settings where link formation is driven predominantly by bilateral concerns, such as certain types of friendship networks, trade networks and some models of con ict between nation-states. The model in (1.3.1) is not a good choice when important strategic aspects in uence link formation, for example when the identity of the nodes to which j is linked in uences i's return from forming a link with j. A discussion of networks with interdependent links can be found in Graham (2015) and De Paula (2016). Also, when network externalities are present, the additional 16 complication of multiple equilibria has to be considered, see for example Sheng (2012) for more details. 1.3.2 Control Function of Network Endogeneity In this subsection we discuss how to control the endogeneity of the peer group dened by the network formed in equation (1.3.1). First we introduce a basic assumption that we will maintain throughout the paper. Assumption 1 () (i) (x i ;a i ; i ) are i.i.d. for all i, i = 1;:::;N. (ii)fu ij g i;j=1;:::;N are independent of (X N ; a N ; N ) and i.i.d. across (i;j) with cdf (). (iii) E(v i jx i ;a i ) =E(v i ja i ): Assumption 1(i) implies that the observables x i and the unobservable characteristics, (a i ; i ), are randomly drawn. This is a standard assumption in the peer eects literature. Assumption 1(ii) assumes that the link formation error u ij is orthogonal to all other observables and unobservables in the model. This means that the dyad-specic unobservable shock u ij from the link formation process does not in uence outcomes y 1 ;:::;y N . However, we allow for endogeneity of the social interaction group through dependence between the two unobserved components a i and i . This means that the unobserved error i in the outcome equation can be correlated with unobserved individual characteristicsa i that are determinants of link formation. We also allow that the observed characteristics, x i , of the outcome equation and the network formation be correlated with the unobserved components, ( i ;a i ), so that the regressor x 1i can be endogenous in the outcome equation, and the network formation observables x 2i can be arbitrarily correlated with the unobserved individual characteristica i . In Assumption 1(iii), we assume that the dependence between x i and i exists only through a i . That is, a i is the xed eect of individual i and controls the endogeneity of x i with respect to i . Notice that the network D N dened in (1.3.1) and the (row normalized) network G N are measurable functions of (x 2i ; x 2;i ;a i ; a i ;fu ij g i;j=1;:::;N ); 17 where x 2;i = (x 2;1 ;:::; x 2;i1 ; x 2;i+1 ;:::; x 2N ) and a i is dened analogously. Under As- sumption 1 we have E[ i jX N ; D N ;a i ] = E[ i jx i ; D N (x 2;i ; a i ;fu ij g i;j=1;:::;N ; x 2i ;a i ); x i ;a i ] = E[ i jx i ;a i ] = E[ i ja i ]; (1.3.3) where the second equality holds because (x i ; a i ;fu ij g i;j=1;:::;N ) and (x i ;a i ; i ) are indepen- dent under Assumptions 1 (i) and (ii). This showsv i and (x i ; D N (x 2;i ; a i ;fu ij g i;j=1;:::;N ; x 2i ;a i )) are mean-independent conditioning on (x i ;a i ). The last line follows by the xed eect as- sumption, Assumption 1 (iii). The result (1.3.3) shows conditional on the unobserved heterogeneity a i in the network formation, the unobserved characteristic i that aects the outcomey i becomes uncorrelated with the network D N (and the observables X N ). This implies that the network endogeneity can be controlled by a i . We summarize the discussion above in the following lemma: Lemma 1 (Control Function of Peer Group Endogeneity) Suppose that Assumption 1 holds. Then, E[ i jX N ; D N ;a i ] =E[ i ja i ]: 1.3.3 Identication of Peer Eects In this section we show how to identify the peer eects in the outcome question when the endogenous network is formed by (1.3.1). We provide two identication methods depending on whether we control the network (peer group) endogeneity witha i or together with x 2i , in the case when x 2i and x 1i do not overlap. First notice that regardless of the possible endogeneity of the network G N , we need to control for the endogeneity of the term P N j=1 j6=i g ij;N y j that represents so-called the endogenous peer eects. When the peer group G N is exogenous and uncorrelated with N , G 2 N X 1N is often used as IVs for the endogenous peer eects term G N y N (See, for example, Kelejian 18 and Prucha (1998), Lee (2003), Bramoull e et al. (2009).). Let Z N = [X 1N ; G N X 1N ; G 2 N X 1N ] be the usual IV matrix used in 2SLS estimation of the peer eects equation. Note that Z N is not a valid IV matrix anymore in our framework because the peer group dened by the network G N is correlated with N due to potential cor- relation between two unobserved characteristics i anda i , Let W N = [G N y N ; X 1N ; G N X 1N ]. Further, denote the transpose of the ith row of Z N and W N by z i and w i , respectively. Suppose that Assumption 1 holds and so a i controls the network endogeneity. Then, E [ (z i E[z i ja i ]) ( i E( i ja i ))ja i ] = E[z i i ja i ]E[z i ja i ]E[ i ja i ] = E [E[z i i ja i ; X 1N ; G N ]ja i ]E[z i ja i ]E[ i ja i ] = E [z i E[ i ja i ; X 1N ; G N ]ja i ]E[z i ja i ]E[ i ja i ] (1) = E [z i E[ i ja i ]ja i ]E[z i ja i ]E[ i ja i ] = 0; (1.3.4) where equality (1) holds by Lemma 1(a). This shows that the instrumental variables z i or z i E[z i ja i ] become orthogonal to i E[ i ja i ]; the residual of i after projecting out a i . Furthermore, ifE (z i E[z i ja i ]) (w i E[w i ja i ]) 0 has full rank, then we can identify the peer eect coecients 0 as 0 = E [(z i E[z i ja i ]) (y i w 0 i E[y i w 0 i ja i ])] = E[(z i E[z i ja i ]) (w i E[w i ja i ]) 0 ]( 0 ) +E[(z i E[z i ja i ]) ( i E[ i ja i ])] (1) = E[(z i E[z i ja i ]) (w i E[w i ja i ]) 0 ]( 0 ) (2) , = 0 ; where equality (1) follows by the orthogonality result in (1.3.4) and equality (2) follows from the full rank condition. Assumption 2 (Rank condition) E (z i E[z i ja i ]) (w i E[w i ja i ]) 0 has full rank. For the full rank condition in Assumption 2, it is necessary that the IVs z i and the 19 regressors w i have additional variation after projecting out the control function a i . As shown in the Supplementary Appendix S.2.3, whenN is large, both z i and w i become close to functions that depend only on (x i ;a i ). In this case, for the full rank condition to be satised, it is necessary that there are additional random components in x i that are dierent from a i , so that the limits of z i and w i are not linearly dependent. As a summary, we have the following rst identication theorem. Theorem 1.3.1 (Identication) Under Assumptions 1 and 2, the parameter 0 is identi- ed by the moment condition E[(z i E(z i ja i )) (y i E(y i ja i ) (w i E(w i ja i )) 0 0 )] = 0: E[(z i E(z i ja i )) (y i E(y i ja i ) (w i E(w i ja i )) 0 )] = 0 () = 0 : Theorem 1.3.1 shows that we can identify parameter 0 by controlling the unobserved network heterogeneity a i in the outcome equation and taking the residuals y i E(y i ja i ) (w i E(w i ja i )) 0 , and then by using the instrumental variables z i E[z i ja i ]. Alternative Identication In view of the derivation of the control function in (1.3.3) under Assumption 1, it is possible to use any regressors in x i in addition to the unobserved heterogeneitya i . In this section, we discuss identication of the peer eects when we use (x 2i ;a i ) as control function. The reason to consider this particular control function is that we can implement this control function dierently from the control function that uses a i only, which will be discussed in detail in Section 1.4. First, suppose that there is no overlap between the regressors in the outcome equation, x 1i , and the network formation equation, x 2i and assume the conditions in Assumption 1. Assumption 3 () Assume that the conditions (i),(ii), and (iii) of Assumption 1 hold. Also, assume that (iv) x 1i \ x 2i =;. 20 Later in this section, we will discuss a more general case where x 1i and x 2i intersect. Then, under Assumption 1 and (1.3.3), it follows that E[ i jX N ; D N ;a i ] =E[ i ja i ] =E[ i jx 2i ;a i ]; (1.3.5) where the last line holds by Assumption 1(iii). Then, similar to (1.3.4), we can show that E [ (z i E[z i jx 2i ;a i ]) ( i E( i jx 2i ;a i ))j x 2i ;a i ] = E[z i i jx 2i ;a i ]E[z i jx 2i ;a i ]E[ i jx 2i ;a i ] = E [z i E[ i ja i ; X 1N ; G N ]jx 2i ;a i ]E[z i jx 2i ;a i ]E[ i jx 2i ;a i ] (1) = E [z i E[ i jx 2i ;a i ]jx 2i ;a i ]E[z i jx 2i ;a i ]E[ i jx 2i ;a i ] = 0; (1.3.6) where equality (1) holds by (1.3.5). Furthermore, suppose that the following full rank as- sumption is satised: Assumption 4 (Rank condition) E (z i E[z i jx 2i ;a i ]) (w i E[w i jx 2i ;a i ]) 0 has full rank. Notice that if x 1i and x 2i are overlapped, then the full rank condition in Assumption 4 does not hold. Using similar arguments that lead to Theorem 1.3.1, we can identify the peer eect coecients 0 as 0 = E [(z i E[z i jx 2i ;a i ]) (y i w 0 i E[y i w 0 i jx 2i ;a i ])] = E[(z i E[z i jx 2i ;a i ]) (w i E[w i jx 2i ;a i ]) 0 ]( 0 ) +E[(z i E[z i jx 2i ;a i ]) ( i E[ i jx 2i ;a i ])] (1) = E[(z i E[z i jx 2i ;a i ]) (w i E[w i jx 2i ;a i ]) 0 ]( 0 ) (2) , = 0 ; (1.3.7) where equality (1) follows by the orthogonality result in (1.3.6) and equality (2) follows from the full rank condition in Assumption 4. This is summarized in the following theorem. 21 Theorem 1.3.2 (Alternative Identication) Under Assumptions 1, 3, and 4, the pa- rameter 0 is identied by the moment condition E[(z i E(z i jx 2i ;a i )) ((y i E(y i jx 2i ;a i ) (w 0 i E(w i jx 2i ;a i )) 0 0 ] = 0: E[(z i E(z i jx 2i ;a i )) ((y i E(y i jx 2i ;a i ) (w 0 i E(w i jx 2i ;a i )) 0 ] = 0 () = 0 : So far, we have considered the case where the regressors x i1 and x 2i do not intersect. A more general case is when the regressors x 1i consist of two components, where one component is dierent from the observed control function x 2i and the other is a part of x 2i . That is, x 1i = (x 11i ; x 12i ), where x 11i does not share any elements with x 2i and x 11i is nonempty, and x 12i x 2i . Let 0 2 = ( 0 21 ; 0 22 ); 0 3 = ( 0 31 ; 0 32 ) conformable to the dimensions of (x 11i ; x 12i ). Similarly let 2 = ( 21 ; 22 ); 3 = ( 31 ; 32 ): In this case, with a properly modied rank condition of z (2);i and w (2);i that excludes the variables associated with x 12;i and P N j=1;6=i g ij;N x 12;j , we can identify the coecients 0 (2) := ( 0 1 ; 0 21 ; 0 31 ) using the same argument that leads to the identication in (1.3.7). However, we cannot identify the coecients that correspond to the variable x 12;i and P N j=1;6=i g ij;N x 12;j . The reason is that the controlling the network endogeneity with the control variable (x 2i ;a i ) wipes out the information in (x 12;i ; P N j=1;6=i g ij;N x 12;j ): x 12;i E[x 12;i jx 2i ;a i ] = 0 N X j=1;6=i g ij;N x 12;j E " N X j=1;6=i g ij;N x 12;j jx 2i ;a i # ! p 0; where the second convergence holds because P N j=1;6=i g ij;N x 12;j converges to a function that depends only on x 2i ;a i (see Section S.2.3 in the Supplementary Appendix.). Through the rest of the paper, when we consider (x 2i ;a i ) as control function, we will without loss of generality apply the restriction in Assumption 3 that x 1i and x 2i do not overlap. 22 1.4 Estimation In this section we present two estimation methods. In subsections 1.4.1 and 1.4.2 we discuss estimation using a i and (x 2i ;a i ) as control function, respectively. 1.4.1 With a i as Control Function The identication scheme of Theorem 1.3.1 identies the parameter of interest 0 with the two step procedure: (i) control a i in the outcome equation and yield y i E(y i ja i ) = (w i E(w i ja i )) 0 0 + i E( i ), and then (ii) use z i E[z i ja i ] as IVs for w i E(w i ja i ). Let h(a i ) = (h y (a i ); h w (a i ); h z (a i )) := (E[y i ja i ];E[w i ja i ];E[z i ja i ]): Let f W N = (w 1 h w (a 1 );:::; w N h w (a N )) 0 . Similarly we dene e Z N ; ~ y N . Suppose that we observe h(a i ). In view of the identication scheme of Theorem 1.3.1, we can estimate 0 by ^ inf 2SLS = f W 0 N e Z N e Z 0 N e Z N 1 e Z 0 N f W N 1 f W 0 N e Z N e Z 0 N e Z N 1 e Z 0 N ~ y N : However, since the individual heterogeneitya i is, not observed and the functions h() are not known, the estimator ^ inf 2SLS is not feasible. A natural implementation of the infeasible estimator ^ inf 2SLS is to replace h(a i ) in f W N ; e Z N , and ~ y N with its estimate, say ^ h(^ a i ). Notice that ifa i is observed, we can estimate h() using various nonparametric methods. In this paper we consider a sieve estimation method. 4 Suppose that h l (a) is the l th element in h(a) forl = 1;:::;L, whereL is the dimension of (y i ; w 0 i ; z 0 i ) 0 . The sieve estimation method assumes that each function h l (a), l = 1;:::;L is well approximated by a linear combination of base functions (q 1 (a);:::;q K N (a)): h l (a) = K N X k=1 q k (a) l k ; (1.4.1) as the truncation parameter K N !1. Let q K (a) = (q 1 (a);:::;q K (a)) 0 , Q N := Q N (a N ) = (q K (a 1 );:::;q K (a N )) 0 , h l (a N ) = 4 In principle we can use other nonparametric estimation methods such as kernel smoothing or local polynomial methods. 23 (h l (a 1 );:::;h l (a N )) 0 , and l N = ( l 1 ;:::; l K N ) 0 . Let b l i be the l th element in (y i ; w 0 i ; z 0 i ) 0 and denote b l N = (b l 1 ;:::;b l N ). If a N = (a 1 ;:::;a N ) 0 is observed, in view of (1.4.1), we can estimate the unknown function h l (a N ) by the OLS of b l i on q K (a i ): for l = 1;:::;L, ^ h l (a N ) = P Q N b l N ; (1.4.2) where P Q N = Q N (Q 0 N Q N ) Q 0 N . Here denotes any symmetric generalized inverse. Given this, we suggest to estimate h l (a N ) as follows: (i) rst, we estimate the unobserved individual heterogeneity and then (ii) plug the estimate in ^ h l (a N ) of (1.4.2). To be more specic, suppose ^ a N = (^ a 1 ;:::; ^ a N ) 0 is an estimator of a N = (a 1 ;:::;a N ) 0 . Denote ^ Q N := Q N (^ a N ) = (q K N (^ a 1 );:::; q K N (^ a N )) 0 . Then the rst estimator of h l (a N ) is dened by ^ h l := ^ h l (^ a N ) = P ^ Q N b l N (1.4.3) for l = 1;:::;L, and this leads the following estimator of 0 : ^ 2SLS = W 0 N M b Q N Z N Z 0 N M b Q N Z N 1 Z 0 N M b Q N W N 1 W 0 N M b Q N Z N Z 0 M b Q N Z N 1 Z 0 N M b Q N y N ; (1.4.4) where M b Q N =I N P b Q N . A desired estimator of a i should satisfy the following high level condition. Assumption 5 (Estimation of a i ) We assume that we can estimate a i with ^ a i such that max i j^ a i a i j = O p ( a (N) 1 ), where a (N)!1 as N!1, satisfying Assumption 8 in the Appendix. For the purpose of our paper, any estimation method that yields the estimator ^ a i satisfy- ing the restriction in Assumption 5 can be adopted. For example, assuming the parametric specication (1.3.2), d ij;N =I(t(x 2i ; x 2j ) 0 +a i +a j u ij ); 24 with regularity conditions of Assumption 6 in the Appendix, Graham (2017) shows that the joint maximum likelihood estimator that solves (^ a 1 ;:::; ^ a N ) := argmax ;a N N X i=1 X j<i d ij;N exp (t(x 2i ; x 2j ) 0 +a i +a j ) ln [1 + exp(t(x 2i ; x 2j ) 0 +a i +a j )] ! (1.4.5) satises sup 1iN j^ a i a i j<O r lnN N ! : with probability 1O(N 2 ). In this case we have a (N) = q N lnN . Examples of other estimation methods include Fern andez-Val and Weidner (2013), Jochmans (2016), Dzemski (2017), and Jochmans (2017). 1.4.2 With (x 2i ;a i ) as Control Function Suppose that x 1i and x 2i do not intersect as assumed in Assumption 3. The idea of the second approach is to implement a control function with the node degree of the network together with the regressors x 2i . The degree of node (or individual)i is the number of connections with node (individual) i in the network. Let d deg i be the degree of node i scaled by the network size: d deg i := 1 N 1 N X j=1;6=i d ij;N : Recall that the link d ij;N is formed by d ij;N =I(g(t(x 2i ; x 2j );a i ;a j )u ij 0): Recall that the unobserved link-specic error terms u ij are assumed to be independent of all the other variables and randomly drawn. Let () be the cdf of u ij . Also let (x 2 ;a) be the joint density funciton of (x 2i ;a i ). Then, for each (x 2i ;a i ), by the WLLN conditioning on 25 (x 2i ;a i ), we have d deg i := 1 N 1 N X j=1;6=i I(g(t(x 2i ; x 2j );a i ;a j )u ij 0) ! p Z (g(t(x 2i ; x 2 );a i ;a))(x 2 ;a)dx 2 da =: deg(x 2i ;a i ) =: deg i (1.4.6) as the network sizeN grows to innity. Furthermore, we can show that under the regularity conditions in Assumption 11 in the Appendix, sup i E[( p N( d deg i deg i )) 2B ] <1 for any nite integer B 2, from which we can deduce that max 1iN j d deg i deg i j =O p deg (N) 1 ; (1.4.7) where deg (N) :=o(1)N B1 2B : This corresponds to the condition in Assumption 5. Suppose that u ij is a continuous random variable so that () is a strictly increasing function. Then, since the function g is a strictly monotonic function of a i , the function deg(x 2i ;a i ) is a strictly monotonic function of a i , too, for each x 2i . Let b l i be the l th element in (y i ; w 0 i ; z 0 i ) 0 . The strict monotonicity of deg i = deg(x 2i ;a i ) in a i for each x 2i implies that (x 2i ;a i ) and (x 2i ; deg i ) have an one-to-one relation, and therefore we have h l (x 2i ;a i ) :=E[b l i jx 2i ;a i ] =E[b l i jx 2i ; deg i ] =:h l (x 2i ; deg i ): Suppose that the function h l (x 2i ; deg i );l = 1;:::;L is well approximated by a linear combination of base functions (r 1 (x 2 ; deg i );:::;r K (x 2 ; deg i )) : h l (x 2i ; deg i ) = K N X k=1 r k (x 2 ; deg i ) l k 26 as the truncation parameter K N !1. Let Deg N = (deg 1 ;:::; deg N ) 0 . Let r K (x 2i ; deg i ) = (r 1 (x 2i ; deg i );:::;r K (x 2i ; deg i )) 0 , R N := R N (X 2N ; Deg N ) = (r K (x 21 ; deg 1 );:::; r K (x 2N ; deg N )) 0 , and l = ( l 1 ;:::; l K N ) 0 . Let b l N = (b l 1 ;:::;b l N ). In the case where (x 2i ; deg i ) are observed, we can estimate h l (X 2N ; Deg N ) = (h l (x 2;1 ; deg 1 );:::;h l (x 2;N ; deg N ) for l = 1;::;L with b h l (X 2N ; Deg N ) := P R N b l N ; (1.4.8) where P R N = R N (R 0 N R N ) R 0 N . Here denotes any symmetric generalized inverse. In view of (1.4.6), the natural estimator of deg i is d deg i . This suggests that we estimate ^ h l (x 2i ; deg i ) by using d deg i in place of deg i . To be more specic, suppose that d Deg N = ( d deg 1 ;:::; d deg N ). Denote b R N := R N (X 2N ; d Deg N ) = (r K (x 21 ; d deg 1 );:::; r K (x 2N ; d deg N )) 0 . The estimator of h l (x 2i ;a i ) =h l (x 2i ; deg i ) is dened by the i th element of b h l (X 2N ; a N ) := b h l (X 2N ; d Deg N ) = P b R N b l N : Then, it leads the following second estimator of 0 : 2SLS := W 0 N M b R N Z N Z 0 N M b R N Z N 1 Z 0 N M b R N W N 1 W 0 N M b R N Z N Z 0 M b R N Z N 1 Z 0 N M b R N y N ; (1.4.9) where M b R N =I N P b R N . 1.4.3 Sieve Estimation In this subsection we discuss the sieve estimators used in estimating ^ 2SLS and 2SLS . For the regularity conditions of the sieve basis Q N and R N , we impose standard conditions such as those proposed by Newey (1997) and Li and Racine (2007). These assumptions ensure that Q 0 N Q N is asymptotically non-singular and control the rate of approximation of the sieve estimator. These assumptions are formally stated in Assumptions 7 and 9 of the Appendix. 27 Additionally, we require that the sieve basis satisfy a Lipschitz condition, which allows to control for the error introduced by the estimation ofa i with ^ a i in the estimation of ^ 2SLS , and the estimation of deg i with d deg i in the estimation of 2SLS 5 (see Assumptions 8 and 10). As an example, dene the polynomial sieve as follows. Let Pol(K N ) denote the space of polynomials on [1; 1] of degree K N , Pol(K N ) = ( 0 + K N X k=1 k a k ; a2 [1; 1]; k 2R ) : For any k we have a k 1 a k 2 =kj~ a k jja 1 a 2 jMkja 1 a 2 j; where ~ a2 [1; 1] and M is a nite constant. In sieve estimations an important issue is how to choose the truncation parameter K N . Well-known procedures for selecting K N are Mallows' C L , generalized cross-validation and leave-one-out cross-validation. For more on these methods see Chapter 15.2 in Li and Racine (2007), Li (1987), Wahba (1985), Andrews (1991) and Hansen (2014). However, these meth- ods are applicable mostly when the observations are cross-sectionally independent, which is not true in our case, especially when the network is dense, as we assume. Developing a data-driven choice of K N is beyond the scope of this paper and we leave it for future work. 1.5 Limit Distribution and Standard Error In this section we present the asymptotic distributions of the two 2SLS estimators ^ 2SLS and 2SLS , and show how to estimate standard errors. We also discuss key technical issues in deriving the limits. All details of the technical derivations and proofs can be found in the Appendix. 5 This issue is similar to the two step series estimation problem in Newey (2009). Other papers that investigated the problem of nonparametric or semiparametric analysis with generated regressors include Ahn and Powell (1993), Mammen et al. (2012), Hahn and Ridder (2013), and Escanciano et al. (2014), for example. 28 1.5.1 Limiting Distribution and Standard Error of ^ 2SLS Recall the denitionsh y (a i ) :=E[y i ja i ]; h (a i ) :=E[ i ja i ]; h w (a i ) :=E(w i ja i ); h z (a i ) := E(z i ja i ): Dene y i :=y i h y (a i ); i := i h (a i ); w i = w i h w (a i ); z i = z i h z (a i ): Let N = ( 1 ;:::; N ) 0 and H N (a N ) = (h (a 1 );:::;h (a N )) 0 . Let ^ h (a i ), ^ h w (a i ), and ^ h z (a i ) denote the sieve estimators of h (a i ), h w (a i ) and h z (a i ), respectively. The derivation of the asymptotic distribution of ^ 2SLS consists of three steps. First, we show that the sampling error caused by the use of ^ a N instead of a N is asymptotically negligible (see Lemma 2 of the Supplementary Appendix S.2.1.). Next, we control the error introduced by the non-parametric estimation of h l (a i ), where l2f; w; zg. In Lemma 7 of Supplementary Appendix S.2.2 we show that under the regularity conditions, the estimation error in ^ h l (a i ) vanishes at a suitable rate. Combining these two, we deduce p N( ^ 2SLS ^ inf 2SLS ) =o p (1): The last step is to derive the limiting distribution of the infeasible estimator p N( ^ inf 2SLS 0 ). In the Supplementary Appendix S.2.3 we show the following: 1 N N X i=1 (w i h w (a i ))(z i h z (a i )) 0 p ! S wz (1.5.1) 1 N N X i=1 (z i h z (a i ))(z i h z (a i )) 0 p ! S zz (1.5.2) 1 p N N X i=1 (z i h z (a i )) i )N (0; S zz ); (1.5.3) where the closed forms of the limits S wz and S zz are found in Lemma 11 and S zz in Lemma 12 of Supplementary Appendix. Notice that the derivation of the limiting distribution in (1.5.3) allows i = i E( i ja i ) to be conditionally heteroskedastic, and so 2 (x i ;a i ) :=E[( i E[ i ja i ]) 2 jx i ;a i ] is allowed to depend on (x i ;a i ). Combining all the limit results deduce the following theorem. 29 Theorem 1.5.1 (Limiting Distribution) Suppose that Assumptions 1, 2, 5, 7, 8, and 11(i)-(v) in the Appendix hold. Then, we have p N( ^ 2SLS 0 ))N (0; ); where = S wz (S zz ) 1 (S wz ) 0 1 S wz (S zz ) 1 S zz (S zz ) 1 (S wz ) 0 S wz (S zz ) 1 (S wz ) 0 1 : (1.5.4) The theorem requires several regularity conditions. Assumption 1 requires that (y i ; x i ;a i ) be randomly drawn and Assumption 2 is a full rank condition. Assumptions 5, 7 and 8 ensure thata i can be consistently estimated, and that the error between h(a i ) and ^ h l (^ a i ) converges to zero at a suitable rate. Assumption 11 imposes further restrictions on the outcome model (1.2.1) and the network formation model (1.3.1). It requires thatj 0 1 j be bounded below 1 so that the spillover eect has a unique solution, and thatk 0 2 k be bounded above 0 so that the IVs are strong. It also assumes that observables (y i ; x i ) and t ij are bounded, and a i has a compact support in [1; 1]. These boundedness conditions are required as a technical regularity condition in deriving the limits in (1.5.1), (1.5.2), and (1.5.3), which involves some uniformity in the limit. The asymptotic variance can be consistently estimated by b = b S wz b S zz 1 ( b S wz ) 0 1 b S wz b S zz 1 b S zz b S zz 1 ( b S wz ) 0 b S wz b S zz 1 ( b S wz ) 0 1 ; (1.5.5) 30 where ^ S wz = 1 N N X i=1 w i ^ h w (^ a i ) z i ^ h z (^ a i ) 0 ^ S zz = 1 N N X i=1 z i ^ h z (^ a i ) z i ^ h z (^ a i ) 0 ^ S ZZ 2 = 1 N N X i=1 z i ^ h z (^ a i ) z i ^ h z (^ a i ) 0 (^ i ) 2 ; and ^ i =y i ^ h y (^ a i ) (w i ^ h w (^ a i )) 0 ^ 2SLS : 1.5.2 Limiting Distribution and Standard Error of 2SLS The process is analogous to the one presented in the previous section. Again, let b l i be the l th element in (y i ; w 0 i ; z 0 i ) 0 . Recall the denition that h l (x 2i ;a i ) =E[b l i jx 2i ;a i ] =E[b l i jx 2i ; deg i ] =:h l (x 2i ; deg i ): Further, let l i = b l i h l (x 2i ;a i ) = b l h l (x 2i ; deg i ), and let ^ h l (x 2i ; deg i ) denote a sieve estimator of h l (x 2i ; deg i ). As in the previous section, we derive the asymptotic distribution of 2SLS in three steps. First, we show that the error that stems from the use of the estimate d deg i for deg i , ^ h l (x 2i ; d deg i ) ^ h l (x 2i ; deg i ), is asymptotically negligible. In the second step, we con- trol the error introduced by the non-parametric estimation of h l (x 2i ; deg i ), ^ h l (x 2i ; deg i ) h l (x 2i ; deg i ). This implies p N( 2SLS inf 2SLS ) =o p (1); where inf 2SLS = f W 0 N e Z N e Z 0 N e Z N 1 e Z 0 N f W N 1 f W 0 N e Z N e Z 0 N e Z N 1 e Z 0 N ~ y N and f W N = (w 1 h w (x 21 ;a 1 );:::; w N h w (x 2N ;a N )) 0 and e Z N ; ~ y N are dened analogously. 31 The last step is to derive the limiting distribution of the infeasible estimator p N( inf 2SLS 0 ) by showing 1 N N X i=1 (w i h w (x 2i ;a i ))(z i h z (x 2i ;a i )) 0 p ! S wz 1 N N X i=1 (z i h z (x 2i ;a i ))(z i h z (x 2i ;a i ) 0 p ! S zz 1 p N N X i=1 (z i h z (x 2i ;a i )) i )N (0; S zz ); Combining all the limit results we have the following theorem. Theorem 1.5.2 (Limiting Distribution) Suppose that Assumptions 1, 3, 4, 9, 10, and 11 hold. Then, we have p N( 2SLS 0 ))N 0; ; where = S wz S zz 1 ( S wz ) 0 1 S wz S zz 1 S zz S zz 1 ( S wz ) 0 S wz S zz 1 ( S wz ) 0 1 : Assumption 3 assumes that the regressors in the outcome equation, x 1i and the observ- ables in the network formation x 2i do not overlap. Assumption 4 is a full rank condition for 2SLS . Assumptions 9 and 10 assume the sieve used in constructing the estimator 2SLS . Comparing the assumptions assumed in Theorem 1.5.1, Theorem 1.5.2 does not require the high level condition of Assumption 5 because we do not use an estimator of a i . Instead it requires an additional restriction that the net surplus of link formation be strictly monotonic in a i , as explained in Section 1.4.2. Like in the case of ^ 2SLS , we allow i = i E( i jx 2i ;a i ) to be conditionally heteroskedas- tic, and 2 (x i ;a i ) :=E[( i E[ i jx 2i ;a i ]) 2 jx i ;a i ] is allowed to depend on (x i ;a i ). 32 The asymptotic variance can be consistently estimated by b = b S wz b S zz 1 ( b S wz ) 0 ) 1 b S wz b S zz 1 b S zz b S zz 1 ( b S wz ) 0 ) b S wz b S zz 1 ( b S wz ) 0 ) 1 ; (1.5.6) where ^ S wz = 1 N N X i=1 w i ^ h w (x 2i ; d deg i ) z i ^ h z (x 2i ; d deg i ) 0 ^ S zz = 1 N N X i=1 z i ^ h z (x 2i ; d deg i ) z i ^ h z (x 2i ; d deg i ) 0 ^ S zz 2 = 1 N N X i=1 z i ^ h z (x 2i ; d deg i ) z i ^ h z (x 2i ; d deg i ) 0 (^ i ) 2 ; and ^ i =y i ^ h y (x 2i ; d deg i ) (w i ^ h w (x 2i ; d deg i )) 0 2SLS : 1.6 Monte Carlo The Monte Carlo design of the network formation process follows Graham (2017). Links are formed according to d ij;N =Ifx 2i x 2j +a i +a j u ij 0g; wherex 2i 2f1; 1g, = 1 andu ij follows a logistic distribution. This link rule implies that agents have a strong taste for homophilic matching since x 2i x 2j = 1 when x 2i = x 2j and x 2i x 2j =1 when x 2i 6= x 2j . Individual-level degree heterogeneity is generated according to a i ='( L Ifx 2i =1g + H Ifx 2i = 1g + i ); with L H and i a centered Beta random variable i jx 2i n Beta( 0 ; 1 ) 0 0 + 1 o so that a i 2 ' h L 0 0 + 1 ; H + 1 0 + 1 i . ' is a scaling factor that assures thatja i j 1 in the designs that so require. 33 We set the parameter values L =3=2, H = 1, 0 = 1=4 and 1 = 3=4. This design involves degree heterogeneity distributions that are correlated with x 2i and right skewed, which mimics distributions observed in real world networks. We have explored other speci- cations of the network formation parameters and the results of the Monte Carlo simulations are not notably dierent. These results are available upon request. Individual outcomes are generated according to y i = 1 N X j=1 j6=i g ij;N y j + 2 x 1i + 3 N X j=1 j6=i g ij;N x 1j +h(a i ) +" i : In the simulations, we set 2 = 3 = 1, 1 2f0:2; 0:5; 0:8g, x 1i = 3q 1 + cos(q 2 )=0:8 + i , where q 1 ;q 2 N (x 2i ; 1), and " i ; i N (0; 1). For h(a i ) we use the following functional forms: h(a i ) = exp(a i ), and h(a i ) = sin(a i ) with = 3. We estimate the outcome equation coecients ( 1 ; 2 ; 3 ) using the standard 2SLS es- timator for peer eects without controlling for the network endogeneity, using a control function linear in ^ a i , ^ h(^ a i ), ^ h( d deg i ;x 2i ) 6 , andh(a i ) with polynomial and Hermite polynomial sieve bases. We perform simulations with network sizeN = 100; 250 with a dense and sparse network design. The average number of links for the dense design is 24 for N = 100 and 60 for N = 250. The corresponding numbers for the sparse design are 2 and 5, respectively. In the paper, we present Monte Carlo results with the Hermite polynomial sieve and h(a i ) = sin(3a i ) for selected values of K N . Specically, Tables 1.1 and 1.2 include results for the dense network specication for K N = 4 and K N = 6, respectively. Results for the sparse network for K N = 3 are presented in Table 1.3. Results for the other specications are not notably dierent and are included in the Supplementary Appendix. For the sparse design, we present results with the control function ^ h( d deg i ;x 2i ). 6 Note that since x 2i is a discrete with a nite support x 2i 2 fx 1 ;:::;x M g we have r(x 2i ;deg i ) = P M m=1 r(x m ;deg i )Ifx 2i = x m g: We can then approximate r(x 2i ;deg i ) ' P K N k=1 n P M m=1 m;k q d k (deg)Ifx 2i =x m g o : 34 We also perform the conventional leave-one-out cross validation to nd data-dependent K N (chosen as theK N that minimizes the Root Mean Square Error (RMSE) of the prediction based on the leave-one-out estimator, see for example Andrews (1991), Hansen (2014)). We report the data-dependent K N , K N; , for each design in the footnotes of the tables in the case of control function ^ h( d deg i ;x 2i ). The dierences in RMSE are very small between the dierent values of K N . Analyzing Monte Carlo results we can see that the order ofK N does not have a signicant impact on the estimates for K N between 3 and 8. The conventional data-driven method for the choice of K N gives us K N in this range. In almost all specications, as expected from our asymptotic theories, the control functions ^ h(^ a i ) and ^ h( d deg i ;x 2i ) perform better than the estimator with a linear control function, as well as the estimator that does not control for the endogeneity of the network, both in terms of mean bias and size. Also, as expected, the true control function h(a i ) performs best of all specications. An interesting observation is that our estimator continues to show good nite sample performances even in the specication with the sparse network presented in Table 1.3. 35 Table 1.1: Hermite Polynomial Sieve: Parameter values across 1000 Monte Carlo replications with h(a) = sin(3a i ) and K N = 4 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0:200 -1.681 -0.398 -0.101 0.007 -0.010 -0.445 -0.630 -0.091 -0.023 -0.019 mean bias (26.702 ) (0.325 ) (0.243 ) (0.349 ) (0.246 ) (160.512 ) (0.352 ) (0.232 ) (0.372 ) (0.297 ) std 0.597 0.301 0.078 0.069 0.018 0.668 0.563 0.078 0.063 0.014 size 2 = 1 0.128 0.022 0.002 0.001 0.001 -0.043 0.019 0.001 0.000 0.001 mean bias (2.379 ) (0.041 ) (0.036 ) (0.037 ) (0.035 ) (6.620 ) (0.024 ) (0.021 ) (0.021 ) (0.022 ) std 0.372 0.109 0.076 0.086 0.016 0.557 0.137 0.057 0.061 0.018 size 3 = 1 2.674 0.520 0.094 -0.002 0.015 0.627 0.937 0.097 0.029 0.033 mean bias (41.110 ) (0.530 ) (0.411 ) (0.473 ) (0.425 ) (277.957 ) (0.605 ) (0.407 ) (0.499 ) (0.532 ) std 0.544 0.203 0.062 0.062 0.017 0.668 0.430 0.069 0.058 0.013 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0:5 -1.095 -0.211 -0.062 0.003 -0.005 -0.462 -0.321 -0.056 -0.015 -0.009 mean bias (14.839 ) (0.170 ) (0.151 ) (0.219 ) (0.140 ) (25.186 ) (0.171 ) (0.142 ) (0.230 ) (0.147 ) std 0.771 0.313 0.078 0.069 0.023 0.787 0.576 0.079 0.063 0.015 size 2 = 1 0.148 0.015 0.001 0.001 0.000 0.023 0.013 0.001 0.000 0.000 mean bias (2.399 ) (0.037 ) (0.035 ) (0.036 ) (0.033 ) (1.468 ) (0.023 ) (0.021 ) (0.021 ) (0.021 ) std 0.316 0.100 0.075 0.083 0.021 0.529 0.096 0.057 0.063 0.020 size 3 = 1 2.154 0.317 0.060 0.002 0.011 0.910 0.571 0.063 0.022 0.022 mean bias (30.259 ) (0.364 ) (0.329 ) (0.363 ) (0.310 ) (55.722 ) (0.383 ) (0.321 ) (0.369 ) (0.336 ) std 0.663 0.174 0.061 0.064 0.018 0.786 0.378 0.066 0.055 0.015 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0:8 -0.168 -0.093 -0.036 0.002 -0.002 -0.320 -0.136 -0.033 -0.009 -0.004 mean bias (0.285 ) (0.079 ) (0.088 ) (0.130 ) (0.070 ) (5.928 ) (0.075 ) (0.083 ) (0.136 ) (0.065 ) std 0.802 0.321 0.078 0.069 0.025 0.897 0.592 0.079 0.064 0.022 size 2 = 1 0.026 0.008 0.000 0.001 -0.000 0.028 0.008 0.001 0.000 0.000 mean bias (0.072 ) (0.034 ) (0.035 ) (0.035 ) (0.031 ) (0.470 ) (0.021 ) (0.021 ) (0.021 ) (0.020 ) std 0.135 0.084 0.077 0.082 0.021 0.227 0.072 0.057 0.065 0.020 size 3 = 1 0.401 0.162 0.037 0.004 0.007 0.879 0.301 0.041 0.017 0.013 mean bias (0.817 ) (0.251 ) (0.274 ) (0.292 ) (0.219 ) (18.665 ) (0.244 ) (0.265 ) (0.288 ) (0.203 ) std 0.479 0.127 0.062 0.068 0.024 0.849 0.279 0.064 0.053 0.020 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 CF - control function. (0) - none, (1) - ^ a i , (2) - ^ h(^ a i ), (3) - ^ h( d deg i ;x 2i ), (4) - h(a i ). Average number of links for N = 100 is 24:1, for N = 250 it is 60:2. The bias of ^ a i is calculated as a i ^ a i K 100;0:2 = 7, K 100;0:5 = 3, K 100;0:8 = 3 K 250;0:2 = 8, K 250;0:5 = 6, K 250;0:8 = 4 36 Table 1.2: Hermite Polynomial Sieve: Parameter values across 1000 Monte Carlo replications with h(a) = sin(3a i ) and K N = 6 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0:200 -1.681 -0.398 -0.102 0.005 -0.010 -0.445 -0.630 -0.090 -0.023 -0.019 mean bias (26.702 ) (0.325 ) (0.245 ) (0.350 ) (0.246 ) (160.512 ) (0.352 ) (0.233 ) (0.372 ) (0.297 ) std 0.597 0.301 0.082 0.065 0.018 0.668 0.563 0.078 0.065 0.014 size 2 = 1 0.128 0.022 0.002 0.001 0.001 -0.043 0.019 0.001 0.000 0.001 mean bias (2.379 ) (0.041 ) (0.037 ) (0.037 ) (0.035 ) (6.620 ) (0.024 ) (0.021 ) (0.021 ) (0.022 ) std 0.372 0.109 0.077 0.085 0.016 0.557 0.137 0.057 0.063 0.018 size 3 = 1 2.674 0.520 0.096 0.000 0.015 0.627 0.937 0.095 0.029 0.033 mean bias (41.110 ) (0.530 ) (0.415 ) (0.475 ) (0.425 ) (277.957 ) (0.605 ) (0.409 ) (0.499 ) (0.532 ) std 0.544 0.203 0.062 0.061 0.017 0.668 0.430 0.073 0.056 0.013 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0:5 -1.095 -0.211 -0.063 0.002 -0.005 -0.462 -0.321 -0.055 -0.015 -0.009 mean bias (14.839 ) (0.170 ) (0.152 ) (0.220 ) (0.140 ) (25.186 ) (0.171 ) (0.143 ) (0.230 ) (0.147 ) std 0.771 0.313 0.080 0.066 0.023 0.787 0.576 0.078 0.066 0.015 size 2 = 1 0.148 0.015 0.001 0.001 0.000 0.023 0.013 0.001 0.000 0.000 mean bias (2.399 ) (0.037 ) (0.036 ) (0.036 ) (0.033 ) (1.468 ) (0.023 ) (0.021 ) (0.021 ) (0.021 ) std 0.316 0.100 0.078 0.089 0.021 0.529 0.096 0.055 0.064 0.020 size 3 = 1 2.154 0.317 0.061 0.004 0.011 0.910 0.571 0.062 0.022 0.022 mean bias (30.259 ) (0.364 ) (0.332 ) (0.364 ) (0.310 ) (55.722 ) (0.383 ) (0.323 ) (0.370 ) (0.336 ) std 0.663 0.174 0.063 0.060 0.018 0.786 0.378 0.070 0.056 0.015 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0:8 -0.168 -0.093 -0.037 0.001 -0.002 -0.320 -0.136 -0.032 -0.009 -0.004 mean bias (0.285 ) (0.079 ) (0.089 ) (0.130 ) (0.070 ) (5.928 ) (0.075 ) (0.084 ) (0.136 ) (0.065 ) std 0.802 0.321 0.080 0.065 0.025 0.897 0.592 0.078 0.066 0.022 size 2 = 1 0.026 0.008 0.000 0.001 -0.000 0.028 0.008 0.001 0.000 0.000 mean bias (0.072 ) (0.034 ) (0.035 ) (0.036 ) (0.031 ) (0.470 ) (0.021 ) (0.021 ) (0.021 ) (0.020 ) std 0.135 0.084 0.078 0.090 0.021 0.227 0.072 0.057 0.064 0.020 size 3 = 1 0.401 0.162 0.038 0.006 0.007 0.879 0.301 0.040 0.017 0.013 mean bias (0.817 ) (0.251 ) (0.277 ) (0.293 ) (0.219 ) (18.665 ) (0.244 ) (0.267 ) (0.289 ) (0.203 ) std 0.479 0.127 0.061 0.069 0.024 0.849 0.279 0.068 0.054 0.020 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 CF - control function. (0) - none, (1) - ^ a i , (2) - ^ h(^ a i ), (3) - ^ h( d deg i ;x 2i ), (4) - h(a i ). Average number of links for N = 100 is 24:1, for N = 250 it is 60:2. The bias of ^ a i is calculated as a i ^ a i K 100;0:2 = 7, K 100;0:5 = 3, K 100;0:8 = 3 K 250;0:2 = 8, K 250;0:5 = 6, K 250;0:8 = 4 37 Table 1.3: Hermite Polynomial Sieve: Parameter values across 1000 MC replications with h(a) = sin(3a i ) and K N = 3 N 100 250 CF (0) (1) (2) (0) (1) (2) 1 = 0:2 0.011 -0.003 -0.002 0.034 -0.001 -0.001 mean bias (0.048 ) (0.055 ) (0.049 ) (0.042 ) (0.051 ) (0.045 ) std 0.329 0.075 0.050 0.366 0.052 0.048 size 2 = 1 0.003 0.001 0.001 -0.005 0.001 0.001 mean bias (0.035 ) (0.040 ) (0.035 ) (0.022 ) (0.024 ) (0.022 ) std 0.320 0.066 0.054 0.290 0.054 0.053 size 3 = 1 -0.003 0.004 0.003 -0.034 0.000 0.001 mean bias (0.078 ) (0.081 ) (0.077 ) (0.069 ) (0.074 ) (0.071 ) std 0.306 0.065 0.044 0.324 0.048 0.042 size N 100 250 CF (0) (1) (2) (0) (1) (2) 1 = 0:5 0.007 -0.002 -0.001 0.019 -0.001 -0.001 mean bias (0.029 ) (0.033 ) (0.030 ) (0.025 ) (0.032 ) (0.027 ) std 0.313 0.071 0.054 0.377 0.042 0.038 size 2 = 1 0.003 -0.001 -0.000 -0.002 0.001 0.001 mean bias (0.034 ) (0.038 ) (0.034 ) (0.021 ) (0.022 ) (0.021 ) std 0.305 0.057 0.047 0.293 0.067 0.059 size 3 = 1 -0.003 0.001 0.002 -0.021 0.003 0.002 mean bias (0.071 ) (0.074 ) (0.071 ) (0.058 ) (0.063 ) (0.059 ) std 0.313 0.067 0.052 0.321 0.049 0.039 size N 100 250 CF (0) (1) (2) (0) (1) (2) 1 = 0:8 0.003 -0.000 -0.000 0.008 -0.001 -0.001 mean bias (0.012 ) (0.014 ) (0.012 ) (0.011 ) (0.017 ) (0.014 ) std 0.295 0.050 0.036 0.384 0.051 0.035 size 2 = 1 0.004 -0.001 -0.000 -0.001 -0.000 -0.000 mean bias (0.031 ) (0.037 ) (0.031 ) (0.019 ) (0.022 ) (0.019 ) std 0.288 0.057 0.049 0.329 0.061 0.048 size 3 = 1 -0.001 0.000 0.001 -0.011 0.002 0.002 mean bias (0.063 ) (0.065 ) (0.063 ) (0.048 ) (0.056 ) (0.051 ) std 0.289 0.065 0.043 0.313 0.049 0.038 size CF - control function. (0) - none, (1) - ^ a i , (2) - ^ h( d deg i ;x 2i ). Average number of links for N = 100 is 1:9, for N = 250 it is 4:9. The bias of ^ a i is calculated as a i ^ a i . K 100;0:2 = 8, K 100;0:5 = 5, K 100;0:8 = 6 K 250;0:2 = 4, K 250;0:5 = 6, K 250;0:8 = 5 38 1.7 Conclusions In this paper we show that, whenever the network is likely endogenous, it is important to control for this endogeneity when estimating peer eects. Failing to control for the endogeneity of the connections matrix in general leads to biased estimates of peer eects. We show that under specic assumptions, we can use the control function approach to deal with the endogeneity problem. We assume that unobserved individual characteristics directly aect link formation and individual outcomes. We leave the functional form through which unobserved individual characteristics enter the outcome equation unspecied and estimate it using a non-parametric approach. 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A.1 Assumptions In this section we introduce the assumptions used in the proof of Theorem 1.5.1. First, we introduce a set of sucient conditions under which we can estimatea i satisfying the conditions in Assumption 5. This assumption corresponds to Assumptions 1, 2, 3 and 5 of Graham (2017). Assumption 6 (Sucient Conditions for Assumption 5) (i) t ij = t ji . (ii) u ij i:i:d: for all ij a logistic distribution. (iii) The supports of , t ij , a i are compact. The next four assumptions are about the sieves used in the semiparametric estimators. The rst two are for ^ 2SLS and the next two are for 2SLS . Assumption 7 (Sieve) For every K N there is a non-singular matrix of constants B such that for ~ q K N (a) = Bq K N (a), (i) The smallest eigenvalue of E[~ q K N (a i )~ q K N (a i ) 0 ] is bounded away from zero uniformly in K N . 45 (ii) There exists a sequence of constants 0 (K N ) that satisfy the condition sup a2A k~ q K N (a)k 0 (K N ); where K N satises 0 (K N ) 2 K N =N ! 0 as N!1. (iii) For f(a) being an element of h(a) = (E[y i ja i = a];E[z i ja i = a];E[w i ja i = a]), there exists a sequence of f K N and a number > 0 such that sup a2A kf(a) q K N (a) 0 f K N k =O(K N ) as K N !1. (iv) As N ! 1; K N ! 1 with p NK N ! 0 and K N =N ! 0. Assumption 8 (Lipschitz condition) The sieve basis satises the following condition: there exists a positive number 1 (k) such that kq k (a) q k (a 0 )k 1 (k)kaa 0 k8 k = 1;:::;K N with 1 a (N) 2 K N X k=1 2 1 (k) =o(1) and 0 (K N ) 6 1 a (N) 2 K N X k=1 2 1 (k) ! =o(1): In our paper, we use the following sieves for the Monte Carlo simulations. (i) Polynomial: Forjaj 1, dene Pol(K N ) = ( 0 + K N X k=1 k a k ; a2 [1; 1] k 2R ) 46 (ii) The Hermite Polynomial sieve: HPol(K N ) = ( K N +1 X k=1 k H k (a) exp a 2 2 ; a2 [1; 1]; k 2R ) ; where H k (a) = (1) k e a 2 d k da k e a 2 . For polynomial sieve, it is known that 0 = O(K N ) (e.g., Newey (1997)). Then, since 1 (k) = O(k), P K N k=1 2 1 (k) = O(K 3 N ). Hence, the conditions that must be satised for the polynomial sieve areK 3 N =N ! 0 and p NK N ! 0. Further, when a (N) 2 = N lnN , we need a (N) 2 O(K 9 N ) =o(1): The next two assumptions are for the sieves used in 2SLS . These assumptions modify Assumption 7 and Assumption 8. Assumption 9 (Sieve) For every K N there is a non-singular matrix of constants B such that for ~ r K N (x 2i ;deg i ) = Br K N (x 2i ;deg i ), (i) The smallest eigenvalue of E[~ r K N (x 2i ;deg i )~ r K N (x 2i ;deg i ) 0 ] is bounded away from zero uniformly in K N . (ii) There exists a sequence of constants 0 (K N ) that satisfy the condition sup (x 2i ;deg i )2S k~ r K N (x 2i ;deg i )k 0 (K N ); where K N satises 0 (K N ) 2 K N =N ! 0 as N!1, andS is the domain of (x 2i ; deg i ). (iii) Forf(x 2i ; deg i ) being an element of h (x 2i ; deg i ) = (E[y i jx 2i ; deg i ];E[z i jx 2i ; deg i ];E[w i jx 2i ; deg i ]), there exists a sequence of f K N and a number > 0 such that sup (x 2i ;deg i )2S kf r K N 0 f K N k =O(K N ) as K N !1. (iv) As N ! 1; K N ! 1 with p NK N ! 0 and K N =N ! 0. 47 Recall from (17) that sup i j d deg i deg i j = O( deg (N) 1 ) with deg (N) = o(1)N B1 2B for some integer B 2. Assumption 10 (Lipschitz) For 0 (K N ) being the constant from Assumption 10, there exists a positive number 1 (k) such that kr k (x 2i ;deg i ) r k (x 2i ;deg 0 i )k 1 (k)kdeg i deg 0 i k8 k = 1;:::;K N with deg (N) 2 K N X k=1 2 1 (k) =o(1) and 0 (K N ) 6 deg (N) 2 K N X k=1 2 1 (k) ! =o(1): The next assumptions restrict the models of the outcome in (1.2.1) and the network formation of (1.3.1). We need Assumption 11 to derive the limiting distribution of ^ 2SLS in Theorem 1.5.1. Assumption 11 We assume the following: (i) The true coecients satisesj 0 1 j 1 andk 0 2 k> for some small . (ii) The parameter set B for is bounded. (iii) The observables (y i ; x i ) are bounded. The unobserved characteristic a i has a compact support in [1; 1]. (iv) The network formation error u ij has an unbounded full support R. (v) The net surplus of the network g(t ij ;a i ;a j ) is bounded by a nite constant. (vi) The net surplus function g(t ij ;a i ;a j ) is strictly monotonic in a i and a j for all t ij . Condition (i) is standard in the linear-in-means peer eect literature. The condition j 0 1 j 1 is required for the unique solution of the spillover eect. We need the restriction 48 k 0 2 k> for the IVs to be strong. The boundedness conditions in (ii) and (iii) are important technical assumptions for asymptotics which require some uniform convergence. Also, these conditions imply key regularity conditions for the CLT. Conditions (vi) and (v) assume that the network is dense andE[d ij;N = 1]> 0. Finally, notice that Assumption 11 allows i E( i ja i ) to be conditionally heteroskedastic, and so 2 (x i ;a i ) := E[( i E[ i ja i ]) 2 jx i ;a i ] depends on (x i ;a i ). This is also true for i E( i ja i ) A.2 Outline of the proof of Theorem 1.5.1 By denition, we have ^ 2SLS 0 = W 0 N M b Q N Z N Z 0 N M b Q N Z N 1 Z 0 N M b Q N W N 1 W 0 N M b Q N Z N Z 0 M b Q N Z N 1 Z 0 N M b Q N N h (a N ) b Q N K N : The derivation of the asymptotic distribution of ^ 2SLS consists of three steps. Step 1. First, we control the sampling error coming from the fact that we do not observe a N and approximate it with ^ a N . Under suitable assumptions (see Supplementary Appendix S.2.1), we show that the error that stems from the estimation of a N by ^ a N is asymptotically negligible: p N ^ 2SLS 0 = 1 N W 0 N M Q N Z N 1 N Z 0 N M Q N Z N 1 1 N Z 0 N M Q N W N ! 1 1 N W 0 N M Q N Z N 1 N Z 0 N M Q N Z N 1 1 p N Z 0 N M Q N N +o p (1): (See Lemma 2 in Supplementary Appendix S.2.1 ) Step 2. Next, we consider the error introduced by the non-parametric estimation of h(a i ). Let h w (a i ) = E(w i ja i ); w i = w i h w (a i ), h z (a i ) = E(z i ja i ) and z i = z i h z (a i ). Let 49 ^ h w (a i ) and ^ h z (a i ) denote the series approximation of h w (a i ) and h z (a i ), respectively. In Lemma 7 in Supplementary Appendix S.2.2 we show that under the regularity conditions (see Supplementary Appendix S.2.2), the error from estimating h(a i ) with ^ h(a i ) converges to zero at a suitable rate and we have 1 N W 0 N M Q Z N = 1 N N X i=1 w i ^ h w (a i ) z i ^ h z (a i ) 0 = 1 N N X i=1 (w i h w (a i )) (z i h z (a i )) 0 +o p (1) 1 N Z 0 N M Q Z N = 1 N N X i=1 z i ^ h z (a i ) z i ^ h z (a i ) 0 = 1 N N X i=1 (z i h z (a i )) (z i h z (a i )) 0 +o p (1); 1 p N Z 0 N M Q v N = 1 p N N X i=1 z i ^ h z (a i ) v i = 1 p N N X i=1 (z i h z (a i )) i +o p (1): Step 3. The consequence of these two approximations is that p N( ^ 2SLS ^ inf 2SLS ) =o p (1). Fi- nally in Step 3, we derive the limiting distribution of the infeasible estimator p N( ^ inf 2SLS 0 ). 50 S.1 Supplementary Appendix - Introduction We use the following notation. M denotes a nite generic constant and a? b means that a and b are orthogonal to each other. For an NN matrix A, we dene matrix norms as follows: kAk = P i;j ja ij j 2 1=2 denotes the Frobenius norm,kAk o denotes the operator norm of matrix A, that is,kAk o = max (A 0 A) 1=2 , min (A) denotes the minimum eigenvalue of A. Notice that kAk o kAkkAk o rank(A): (S.1.1) Further, for matrix A, [a] i denotes the i'th row of A. Denote [GX 1 ] i by X 1;G;i , [G 2 X 1 ] i by X 1;G 2 ;i , [Gy] i by Y G;i . The ith row of the instrument matrix Z N is given by z 0 i = [X 0 2;i ; X 1;G;i ; X 1;G 2 ;i ], z i is (3l x ) 1. Similarly, w 0 i = [Y G;i ; X 0 1;i ; X 1;G;i ]. We denote matrices by uppercase bold letters and vectors by lowercase bold letters, Z N = (Z 0 1 ;:::; Z 0 N ) 0 , W N = (W 0 1 ;:::; W 0 N ) 0 and a N = (a 1 ;:::;a N ) 0 . S.2 For ^ 2SLS Outline of the proof of Theorem 1.5.1: By denition, we have ^ 2SLS 0 = W 0 N M b Q N Z N Z 0 N M b Q N Z N 1 Z 0 N M b Q N W N 1 W 0 N M b Q N Z N Z 0 M b Q N Z N 1 Z 0 N M b Q N N h (a N ) b Q N K N : The derivation of the asymptotic distribution of ^ 2SLS consists of three steps. Step 1. First, we control the sampling error coming from the fact that we do not observe a N and approximate it with ^ a N . Under suitable assumptions (see Appendix S.2.1), we show that the error that stems from the estimation of a N by ^ a N is asymptotically 51 negligible: p N ^ 2SLS 0 = 1 N W 0 N M Q N Z N 1 N Z 0 N M Q N Z N 1 1 N Z 0 N M Q N W N ! 1 1 N W 0 N M Q N Z N 1 N Z 0 N M Q N Z N 1 1 p N Z 0 N M Q N N +o p (1): (See Lemma 2 in Appendix S.2.1 ) Step 2. Next, we consider the error introduced by the non-parametric estimation of h(a i ). Let h w (a i ) = E(w i ja i ); w i = w i h w (a i ), h z (a i ) = E(z i ja i ) and z i = z i h z (a i ). Let ^ h w (a i ) and ^ h z (a i ) denote the series approximation of h w (a i ) and h z (a i ), respectively. In Lemma 7 in Appendix S.2.2 we show that under the regularity conditions (see Appendix S.2.2), the error from estimating h(a i ) with ^ h(a i ) converges to zero at a suitable rate and we have 1 N W 0 N M Q Z N = 1 N N X i=1 w i ^ h w (a i ) z i ^ h z (a i ) 0 = 1 N N X i=1 (w i h w (a i )) (z i h z (a i )) 0 +o p (1) 1 N Z 0 N M Q Z N = 1 N N X i=1 z i ^ h z (a i ) z i ^ h z (a i ) 0 = 1 N N X i=1 (z i h z (a i )) (z i h z (a i )) 0 +o p (1); 1 p N Z 0 N M Q v N = 1 p N N X i=1 z i ^ h z (a i ) v i = 1 p N N X i=1 (z i h z (a i )) i +o p (1): Step 3. The consequence of these two approximation is that p N( ^ 2SLS ^ inf 2SLS ) =o p (1). Fi- nally in Step 3, we derive the limiting distribution of the infeasible estimator p N( ^ inf 2SLS 0 ). 52 S.2.1 Controlling the Sampling Error ^ a i a i in Sieve Estimation In this section, we show that the error coming from the estimation of a i by ^ a i is of order o p (1). All supporting Lemmas can be found in Appendix S.2.1. Lemma 2 Assume Assumptions 1 2, 7, 8, and 11. Then the following hold. (a) 1 N (Z 0 N P b Q N W N Z 0 N P Q N W N ) =o p (1). (b) 1 N (Z 0 N P b Q N Z N Z 0 N P Q N Z N ) =o p (1). (c) 1 p N (Z 0 N P b Q N N Z 0 N P Q N N ) =o p (1). (d) 1 p N (Z 0 M b Q N (h (a N ) b Q N K N )) =o p (1). Proof 1 Part (a). 1 N (Z 0 N P b Q N W N Z 0 N P Q N W N ) = Z 0 N b Q N Q N N b Q 0 N b Q N N ! 1 b Q 0 N W N N Z 0 N Q N N 8 < : Q 0 N Q N N 1 b Q 0 N b Q N N ! 1 9 = ; Q 0 N W N N + Z 0 N Q N N b Q 0 N b Q N N ! 1 b Q N Q N 0 W N N = Z 0 N b Q N Q N N b Q 0 N b Q N N ! 1 ( b Q N Q N ) 0 W N N + Z 0 N b Q N Q N N b Q 0 N b Q N N ! 1 Q 0 N W N N Z 0 N Q N N 8 < : Q 0 N Q N N 1 b Q 0 N b Q N N ! 1 9 = ; Q 0 N W N N + Z 0 N Q N N b Q 0 N b Q N N ! 1 b Q N Q N 0 W N N = I 1 +I 2 I 3 +I 4 ;say: For the desired result, by (S.1.1) we show that 1 N (Z 0 N P b Q N W N Z 0 N P Q N W N ) o =o p (1); which follows by triangular inequality if we show kI 1 k o ;kI 2 k o ;kI 3 k o ;kI 4 k o =o p (1): 53 For term I 1 , kI 1 k o Z N p N b Q N Q N p N 2 b Q 0 N b Q N N ! 1 o W N p N = O p (1) 1 a (N) 2 K N X k=1 1 (k) 2 ! O P (1)O(1) =o p (1); where the last line holds by (S.2.1), Lemmas 4 and 6, and by Assumption 8. For term I 2 , kI 2 k o Z N p N b Q N Q N p N b Q 0 N b Q N N ! 1 o Q N p N W N p N = O p (1) 1 a (N) 2 K N X k=1 1 (k) 2 ! 1=2 O P (1) 0 (K N )O(1) =o p (1); where the last line holds by (S.2.1), Lemmas 4 and 6, and by Assumption 8. For term I 3 , write I 3 = Z 0 N Q N N b Q 0 N b Q N N ! 1 ( b Q 0 N b Q N N ! Q 0 N Q N N ) Q 0 N Q N N 1 Q 0 N W N N = Z 0 N Q N N b Q 0 N b Q N N ! 1 b Q 0 N ( b Q N Q N ) N ! Q 0 N Q N N 1 Q 0 N W N N + Z 0 N Q N N b Q 0 N b Q N N ! 1 ( b Q N Q N ) 0 Q N N ! Q 0 N Q N N 1 Q 0 N W N N : Then, kI 3 k o O p (1) 0 (K N )O p (1) 0 (K N ) 1 a (N) 2 K N X k=1 1 (k) 2 ! 1=2 O p (1) 0 (K N )O p (1) =o p (1); where the last equality follows by Assumption 8. The desired result of term I 4 follows by similar argument used for term I 2 . Part (b) can be shown in a similar way as Part (a). 54 Part (c). 1 p N (Z 0 N P b Q N N Z 0 N P Q N N ) = Z 0 N b Q N Q N N b Q 0 N b Q N N ! 1 ( b Q N Q N ) 0 N p N + Z 0 N b Q N Q N N b Q 0 N b Q N N ! 1 Q 0 N N p N Z 0 N Q N N 8 < : Q 0 N Q N N 1 b Q 0 N b Q N N ! 1 9 = ; Q 0 N N p N + Z 0 N Q N N b Q 0 N b Q N N ! 1 b Q N Q N 0 N p N = III 1 +III 2 III 3 +III 4 ;say; and the desired result of Part (c) follows if we show that for j = 1;:::; 4, kIII j k =o p (1): First, for term III 1 , we have kIII 1 k Z N p N b Q N Q N p N b Q 0 N b Q N N ! 1 ( b Q N Q N ) 0 N p N = O p (1) 1 a (N) 2 K N X k=1 1 (k) 2 ! 1=2 O p (1) ( b Q N Q N ) 0 N p N ; where the last line holds by (S.2.1), Lemmas 4 and 6. Under Assumption we can show that E 2 4 ( b Q N Q N ) 0 N p N 2 X 1N ; G N ; a N 3 5 = 1 N b Q N Q N 2 : Then, by Lemma 4 and Assumption 8, we have the required result for term III 1 . The rest of the required results follow by similar fashion and we omit the proof. Part (d). 55 Notice that 1 p N (Z 0 N M b Q N (h (a N ) b Q N K N )) = 1 p N Z 0 N M b Q N h (a N ) = 1 p N Z 0 N M b Q N M Q N h (a N ) + 1 p N Z 0 N M Q N h (a N ) Q N K N = IV 1 +IV 2 ;say: We can show IV 1 =o p (1) by applying similar arguments used in the proof of Part (a). For term IV 2 , notice that kIV 2 k = kIV 2 k o 1 p N Z N o kM Q N k o h (a N ) Q N K N o = 1 p N Z N h (a N ) Q N K N = O p (1) p NO(K N ) =o p (1) by Assumption 7 (iii) and (iv). Supporting Lemmas First notice that by the boundedness condition (ii) and (iii) in Assumption 11, we have 1 N kZ N k 2 =O p (1); 1 N kW N k 2 =O p (1): (S.2.1) Lemma 3 Under Assumption 7, we have 1 N kQ N k 2 M 2 0 (K N ): Proof 2 1 N kQ N k 2 = 1 N N X i=1 kq K (a i )k 2 sup i kq K (a i )k 2 = 2 0 (K N ) 56 by Assumption 7 (ii). Lemma 4 Under Assumptions 1, 5, 7, and 8, we have 1 N k b Q N Q N k 2 =M 1 a (N) 2 K N X k=1 1 (k) 2 : Proof 3 1 N k b Q N Q N k 2 = 1 N N X i=1 K N X k=1 kq k (^ a i )q k (a i )k 2 1 N N X i=1 K N X k=1 1 (k) 2 k^ a i a i k 2 1 N N X i=1 K N X k=1 1 (k) 2 1 a (N) 2 = 1 a (N) 2 K N X k=1 1 (k) 2 ; where the rst inequality follows from Assumption 8 and the second inequality follows from Assumption 5. Lemma 5 For symmetric matrices A and B it is true that j min (A) min (B)jkA Bk Proof 4 Let x A be the eigenvector associated with the minimum eigenvalue of A. Dene xx B analogously. First we showj min (A) min (B)jkA Bk: min (A) min (B) = x 0 A Ax A x 0 B Bx B x 0 B (A B)x B jx 0 B (A B)x B jkA Bk: Also, we can prove the other direction. Notice that min (A) min (B) = x 0 A Ax A x 0 B Bx B x 0 A (A B)x A jx 0 B (A B)x B jkA Bk: 57 Then, we have the required result. Lemma 6 Under 1, 5, 7, and 8, W.p.a.1, there exists a positive constant C > 0 such that 1 C min Q 0 N Q N N ; min b Q 0 N b Q N N ! : Proof 5 First we show that there exists a positive constant C such that 1 C min Q 0 N Q N N ; which follows by Assumption 7(i) if we show min Q 0 N Q N N E[q K N (a i )q K N (a i ) 0 ] =o p (1): For this, by Lemma 5, we have min Q 0 N Q N N E[q K N (a i )q K N (a i ) 0 ] Q 0 N Q N N E[q K N (a i )q K N (a i ) 0 ] = 1 N N X i=1 q K N (a i )q K N (a i ) 0 E[q K N (a i )q K N (a i ) 0 ] : Then, by Assumption 7(ii), we have E 1 N N X i=1 q K N (a i )q K N (a i ) 0 E[q K N (a i )q K N (a i ) 0 ] 2 = K N X k=1 K N X l=1 E 1 N N X i=1 (q k (a i )q l (a i )E[q k (a i )q l (a i )]) ! 2 1 N K N X k=1 K N X l=1 E[q k (a i )q l (a i )] 2 1 N sup a K N X k=1 q k (a) 2 ! 2 0 (K N ) 4 N =o(1); where the last line holds by Assumptions 7(ii) and 8. Next, given the rst part of the lemma, the second claim of the lemma follows if we show min b Q 0 N b Q N N ! min Q 0 N Q N N =o p (1): 58 Notice by Lemma 5, for symmetric matrices p A and B, we have k min (A) min (B)kkA Bk: Then, min b Q 0 N b Q N N ! min Q 0 N Q N N b Q 0 N b Q N N Q 0 N Q N N ( ^ Q N Q N ) 0 p N Q N p N + Q 0 N p N ( ^ Q N Q N ) p N + ( ^ Q N Q N ) 0 p N ( ^ Q N Q N ) p N : Then, by lemmas 3 and 4 and by Assumption 8, we have min b Q 0 N b Q N N ! min Q 0 N Q N N M 0 @ 0 (K N ) v u u t 1 a (N) 2 K N X k=1 2 1 (k) + 1 a (N) 2 K N X k=1 2 1 (k) 1 A =o p (1); as desired. S.2.2 Controlling the Series Approximation Error Lemma 7 (Series Approximation) Assume the assumptions in Lemma 2. Then, we have (a) 1 N P N i=1 w i ^ h w (a i ) z i ^ h z (a i ) 0 = 1 N P N i=1 (w i h w (a i )) (z i h z (a i )) 0 +o p (1), (b) 1 N P N i=1 z i ^ h z (a i ) z i ^ h z (a i ) 0 = 1 N P N i=1 (z i h z (a i )) (z i h z (a i )) 0 +o p (1), (c) 1 p N P N i=1 z i ^ h z (a i ) i = 1 p N P N i=1 (z i h z (a i )) i +o p (1). Proof 6 Lemma 7 follows if we show (i) 1 N P N i=1 ^ h w (a i ) h w (a i ) ^ h w (a i ) h w (a i ) 0 =o p (1): (ii) 1 N P N i=1 ^ h z (a i ) h z (a i ) ^ h z (a i ) h z (a i ) 0 =o p (1). 59 (iii) 1 p N P N i=1 ^ h z (a i ) h z (a i ) i =o p (1): Lemma 7 (i) and (ii) is true by Lemma 10 and Lemma 7 (iii) follows from (ii). See the remainder of this section. Following Newey (1997), we assume B = I in Assumption 7, hence, ~ q K (a) =q K (a). Also, we assume P =E[q K (a i )(q K (a i )) 0 ] =I. 7 Lemma 8 Assume Assumption 7. Then, E[k ~ P Ik 2 ] = O( 0 (K N ) 2 K N =N), where ~ P = (Q 0 N Q N )=N. Proof 7 For proof see Li and Racine (2007) page 481. Note that this Lemma implies thatk ~ P Ik = O p ( 0 (K N ) p K N =N = o p (1). Also, since the smallest eigenvalue of ~ P I is bounded byk ~ P Ik, this implies that the smallest eigenvalue of ~ P converges to one in probability. Letting 1 N be the indicator function for the smallest eigenvalue of ~ P being greater than 1=2, we have Pr(1 N = 1)! 1. Lemma 9 Assume Assumption 7. Then,k~ f f k =O p (K N ), where ~ f = (Q 0 N Q N ) 1 Q 0 N f, where (f) satises Assumption 7 and f(a)2fh y (a);h z (a);h w (a)g. Proof 8 1 N k~ (f) (f) k = 1 N k(Q 0 N Q N ) 1 Q 0 N (f Q N f )k = 1 N f(f Q N f ) 0 Q N (Q 0 N Q N ) 1 (Q 0 N Q N =N) 1 Q 0 N (f Q N f )=Ng 1=2 = 1 N O P (1)f(f Q N (f) ) 0 Q N (Q 0 N Q N ) 1 Q 0 N (f Q N f )=Ng 1=2 O p (1)f(f Q N f ) 0 (f Q N f )=Ng 1=2 =O p (K N ) by Lemma 8, Assumption 7(iii), the fact that Q N (Q 0 N Q N ) 1 Q 0 N is idempotent and Pr(1 N = 1)! 1. 7 The Lemmas in this section follow Section 15.6 in Li and Racine (2007). 60 Lemma 10 Assume Assumption 7. Let f(a)2 (h y (a); h z z(a); h w (a)) and ~ f = Q N ~ f N . Then, 1 N kf ~ fk 2 =O p (K 2 N ) =o p (N 1=2 ). Proof 9 The required result for the lemma follows because 1 N kf ~ fk 2 1 N fkf Q N f N k 2 +kQ N ( (f) N ~ f N )k 2 g = O(K 2 N ) + ( f N ~ f N ) 0 (Q 0 N Q N =N)( f N ~ f N ) = O(K 2 N ) +O p (1)k f N ~ f N k 2 =O p (K 2 N ) by Assumption 7(iii), Lemma 8 and Lemma 9. S.2.3 Limiting Distribution of ^ 2SLS In this section we derive the distribution of the infeasible estimator ^ inf 2SLS . All supporting lemmas can be found in Section S.2.4. We introduce the following notation. Let s 0 (x i ;a i ) be a function of (x i ;a i ) such that s 0 (;) is bounded over the support of (x i ;a i ). We denote an N vector-valued function that stacks s 0 (x i ;a i ) over i = 1;:::;N as S 0;N = (s 0 (x 1 ;a 1 );:::;s 0 (x N ;a N )) 0 : Dene s 0;N;i :=s 0 (x i ;a i ): (S.2.2) Next, for m = 1; 2;:::; we dene recursively s m;N;i := N X j=1;6=i g ij;N s m1;N;i = [G N S m1;N ] i ; (S.2.3) where S m1;N := (s m1;N;1 ;:::;s m1;N;N ) 0 : For m = 0; 1; 2;:::, we dene s x 1 m;N;i and S x 1 m;N with initial function s 0;N;i = s 0 (x i ;a i ) = x 1i , and dene s a m;N;i and S a m;N with initial function s 0;N;i =s 0 (x i ;a i ) =h (a i ). Next, we dene recursively the probability limit ofs m;N;i dened with the initial function 61 s 0;N;i =s 0 (x i ;a i ) for each i as N!1. For this, let ~ s 0 (x i ;a i ) =s 0 (x i ;a i ) =s 0;N;i : Note that for xed i, s 1;N;i has the following limit as N!1: s 1;N;i = [G N S 0;N ] i = 1 N X j6=i d ij;N ! 1 1 N X j6=i d ij;N s 0 (x j ;a j ) = 1 N X j6=i Ifg(t(x 2i ; x 2j );a i ;a j )u ij g ! 1 1 N X j6=i Ifg(t(x 2i ; x 2j );a i ;a j )u ij gs 0 (x j ;a j ) p ! RRR p(g(t(x 2i ; x 2 );a i ;a)s 0 (x;a)(x;a)dxda RR p(g(t(x 2i ; x 2 );a i ;a)(x 2 ;a)dx 2 da = E[d ij;N s 0 (x j ;a j )jx i ;a i ] E[d ij;N jx i ;a i ] =: ~ s 1 (x i ;a i ); (S.2.4) where (x;a) with x = (x 1 ; x 2 ) is the joint density of x i = (x 1i ; x 2i ) and a i , and (x 2 ;a) is the joint density of (x 2i ;a i ). Here note that the limit ~ s 1 (x i ;a i ) depends only on (x i ;a i ), not on (x i ;a i ), while s 1;N;i depends on both (x i ;a i ) and (x i ;a i ). We dene the following recursively for m = 2; 3; as follows: ~ s m (x i ;a i ) := E[d ij;N ~ s m1 (x j ;a j )jx i ;a i ] E[d ij;N jx i ;a i ] (S.2.5) = RR p(g(t(x 2i ; x 2 );a i ;a)~ s m1 (x;a)(x;a)dxda RR p(t(x 2i ; x 2 );a i ;a)(x 2 ;a)dx 2 da = plim N!1 1 N X j6=i d ij;N ! 1 1 N X j6=i d ij;N ~ s m1 (x j ;a j ) = plim N!1 [G N ~ S m1 ] i ; where ~ S m = (~ s m (x 1 ;a 1 );:::; ~ s m (x N ;a N )): Using this general denitions of (S.2.4) and (S.2.5), with ~ s x 1 0 (x i ;a i ) = s x 1 0 (x i ;a i ) = x 1i 62 and ~ s a 0 (x i ;a i ) = s a 0 (x i ;a i ) = h(a i ), we dene ~ s x 1 m (x i ;a i ) and ~ s a m (x i ;a i ), respectively, for m = 1; 2;:::. Let ~ S x 1 m = (~ s x 1 m (x 1 ;a 1 );::: ~ s x 1 m (x N ;a N )) 0 : and ~ S a m = (~ s a m (x 1 ;a 1 );::: ~ s a m (x N ;a N )) 0 . Next, with the initial function s 0;N;i = i and S 0;N := (s 0;N;1 ;:::;s 0;N;N ) 0 , we dene recursively s m;N;i := [G N S m1;N ] i = N X j=1;6=i g ij;N s m1;N;i ; (S.2.6) and S m;N := (s m;N;1 ;:::;s m;N;N ) 0 for m = 1; 2;:::. Lemma 11 Under Assumptions 1 and 11, as N!1, we have (a) 1 N N X i=1 (w i h w (a i ))(z i h z (a i )) 0 =: 0 B B B B @ 1 N P N i=1 GY i ( x 1 i ) 0 1 N P N i=1 GY i ( Gx 1 i ) 0 1 N P N i=1 GY i ( G 2 x 1 i ) 0 1 N P N i=1 x 1 i ( x 1 i ) 0 1 N P N i=1 x 1 i ( Gx 1 i ) 0 1 N P N i=1 x 1 i ( G 2 x 1 i ) 0 1 N P N i=1 Gx 1 i ( x 1 i ) 0 1 N P N i=1 Gx 1 i ( Gx 1 i ) 0 1 N P N i=1 Gx 1 i ( G 2 x 1 i ) 0 1 C C C C A p ! 0 B B B B @ S GY;x 1 S GY;Gx 1 S GY;G 2 x 1 S x 1 ;x 1 S x 1 ;Gx 1 S x 1 ;G 2 x 1 S Gx 1 ;x 1 S Gx 1 ;Gx 1 S Gx 1 ;G 2 x 1 1 C C C C A =:S wz ; (b) 1 N N X i=1 (z i h z (a i ))(z i h z (a i )) 0 =: 0 B B B B @ 1 N P N i=1 x 1 i ( x 1 i ) 0 1 N P N i=1 x 1 i ( Gx 1 i ) 0 1 N P N i=1 x 1 i ( G 2 x 1 i ) 0 1 N P N i=1 Gx 1 i ( x 1 i ) 0 1 N P N i=1 Gx 1 i ( x 1 i ) 0 1 N P N i=1 Gx 1 i ( Gx 1 i ) 0 1 N P N i=1 G 2 x 1 i ( G 2 x 1 i ) 0 1 N P N i=1 G 2 x 1 i ( x 1 i ) 0 1 N P N i=1 G 2 x 1 i ( G 2 x 1 i ) 0 1 C C C C A p ! 0 B B B B @ S x 1 ;x 1 S x 1 ;Gx 1 S x 1 ;G 2 x 1 S Gx 1 ;x 1 S Gx 1 ;Gx 1 S Gx 1 G 2 x 1 S G 2 x 1 ;x 1 S G 2 x 1 ;Gx 1 S G 2 x 1 ;G 2 x 1 1 C C C C A =:S zz ; 63 where S GY;G r x 1 =E " 1 X m=0 0 0 2 ~ ~ s x 1 m (x i ;a i ) + 0 0 3 ~ ~ s x 1 m+1 (x i ;a i ) + ~ ~ s a m (x i ;a i ) ! ~ ~ s x 1 r (x i ;a i ) 0 # ; r = 0; 1; 2 S G r x 1 ;G s x 1 =E h ~ ~ s x 1 r (x i ;a i )) ~ ~ s x 1 s (x i ;a i ) 0 i ; r;s = 0; 1; 2 ~ ~ s x 1 m (x i ;a i ) = ~ s x 1 m (x i ;a i )E[~ s x 1 m (x i ;a i )ja i ]) with ~ s x 1 0 (x i ;a i ) = x 1i ~ ~ s a m (x i ;a i ) = ~ s a m (x i ;a i )E[~ s a m (x i ;a i )ja i ]) with ~ s a 0 (x i ;a i ) =h (a i ): and ~ ~ s x 1 m (x i ;a i ) and ~ ~ s a m (x i ;a i ) are dened recursively as in (S.2.5). Proof We take the element 1 N P N i=1 GY i ( G 2 x 1 i ) 0 as an example. The proofs of the rest are similar and we omit them. Whenj 0 1 j< 1, G N y N = 1 X m=0 ( 0 1 ) m G m N (X 1N 0 2 + G N X 1N 0 3 + h (a N ) + N ); and [G N y N ] i = 00 2 " 1 X m=0 ( 0 1 ) m G m N X 1N # i + 00 3 " 1 X m=0 ( 0 1 ) m G m+1 N X 1N # i + " 1 X m=0 ( 0 1 ) m G m N h(a N ) # i + " 1 X m=0 ( 0 1 ) m G m N v N # i : 64 Set s x 1 0 (x i ;a i ) = ~ s x 1 0 (x i ;a i ) = x 1i . We have 1 N N X i=1 GY i ( G 2 x 1 i ) 0 = 1 N N X i=1 ([G N y N ] i Ef[G N y N ] i ja i g) [G 2 N X 1N ] i Ef[G 2 N X 1N ] i ja i g 0 = 1 N N X i=1 0 0 2 1 X m=0 ( 0 1 ) m s x 1 m;N;i E[s x 1 m;N;i ja i ] ! s x 1 2;N;i E[s x 1 2;N;i ja i ] 0 + 1 N N X i=1 0 0 3 1 X m=0 ( 0 1 ) m s x 1 m+1;N;i E[s x 1 m+1;N;i ja i ] ! s x 1 2;N;i E[s x 1 2;N;i ja i ] 0 + 1 N N X i=1 0 0 2 1 X m=0 ( 0 1 ) m s a m;N;i E[s a m;N;i ja i ] ! s x 1 2;N;i E[s x 1 2;N;i ja i ] 0 + 1 N N X i=1 0 0 2 1 X m=0 ( 0 1 ) m s m;N;i E[s m;N;i ja i ] ! s x 1 2;N;i E[s x 1 2;N;i ja i ] 0 =I +II +III +IV; say: Consider term I, 1 N N X i=1 0 0 2 1 X m=0 ( 0 1 ) m s x 1 m;N;i E[s x 1 m;N;i ja i ] ! s x 1 2;N;i E[s x 1 2;N;i ja i ] 0 : Denote A 1i := 0 0 2 1 X m=0 ( 0 1 ) m s x 1 m;N;i E[s x 1 m;N;i ja i ] A 2i :=s x 1 2;N;i E[s x 1 2;N;i ja i ] A 3i := 1 X m=0 ( 0 1 ) m s m;N;i E[s m;N;i ja i ] B 1i := 0 0 2 1 X m=0 ( 0 1 ) m f~ s x 1 m (x i ;a i )E[~ s x 1 m (x i ;a i )ja i ]g B 2i := ~ s x 1 2 (x i ;a i )E[~ s x 1 2 (x i ;a i )ja i ] B 3i := i = i E[ja i ]: 65 First, notice that 1 N N X i=1 A 1i A 0 2i 1 N N X i=1 B 1i B 0 2i = 1 N N X i=1 (A 1i B 1i )A 0 2i + 1 N N X i=1 B 1i (A 2i B 2i ) 0 1 N N X i=1 (A 1i B 1i )A 0 2i + 1 N N X i=1 B 1i (A 2i B 2i ) 0 sup i kA 1i B 1i k sup i kA 2i k + sup i kB 1i k sup i kA 2i B 2i k (S.2.7) According to Lemma 16 and Lemma 14, we have sup i kA 1i B 1i k =o p (1); sup i kA 2i B 2i k =o p (1): Also, under Assumption 11, sup i kA 2i k and sup i kB 1i k are bounded by a nite constant. Therefore, we deduce that I = 1 N N X i=1 B 1i B 0 2i +o p (1): Then, we apply the WLLN to 1 N P N i=1 B 1i B 0 2i and deduce 1 N N X i=1 B 1i B 0 2i p !E [B 1i B 0 2i ] =E " 0 0 2 1 X m=0 ( 0 1 ) m f~ s x 1 m (x i ;a i )E[~ s x 1 m (x i ;a i )ja i ]g ! (~ s x 1 2 (x i ;a i )E[s x 1 2 (x i ;a i )ja i ]) # =E " 0 0 2 1 X m=0 ( 0 1 ) m ~ ~ s x 1 m (x i ;a i ) ! ~ ~ s x 1 2 (x i ;a i ) # We can derive the probability limits of terms II and III by similar fashion. For term IV , rst notice that for each m = 0; 1; 2;:::, E[s m;N;i ja i ] =E ([G m N N ] i ja i ) =EfE ([G m N N ] i jX N ; D N ;a i )ja i g =Ef[G m N E( N jX N ; D N ;a i )] i ja i g = 0; 66 where the last equality holds by Lemma 1. Then, A 3i := P 1 m=0 ( 0 1 ) m s m;N;i : Similar to the bound in (S.2.7), notice that 1 N N X i=1 A 3i A 0 2i 1 N N X i=1 B 3i B 0 2i sup i kA 3i B 3i k sup i kA 2i k + sup i kB 3i k sup i kA 2i B 2i k: According to Lemma 16 and Lemma 14, sup i kA 3i B 3i k =o p (1); sup i kA 2i B 2i k =o p (1): Also, under Assumption 11, sup i kA 2i k and sup i kB 3i k are bounded by a nite constant. Therefore, we deduce that IV = 1 N N X i=1 B 3i B 0 2i +o p (1): Then, we apply the WLLN to 1 N P N i=1 B 3i B 0 2i and deduce 1 N N X i=1 B 3i B 0 2i p !E [B 3i B 0 2i ] =E [ a i (~ s x 1 2 (x i ;a i )E[s x 1 2 (x i ;a i )ja i ])] =E ( i E[ i ja i ]) ~ ~ s x 1 2 (x i ;a i ) =E E ( i E[ i ja i ]jx i ;a i ) ~ ~ s x 1 2 (x i ;a i ) = 0: Let 2 (x i ;a i ) :=E[( i ) 2 jx i ;a i ] =E[( i E[ i ja i ]) 2 jx i ;a i ]: Lemma 12 Under Assumptions 1 and 11, as N!1, we have 1 N N X i=1 (z i h z (a i ))(z i h z (a i )) 0 2 (x i ;a i ) p ! S zz ; where the limit variance S zz is dened in Lemma 13. Proof 67 The proof is similar to that of the results in Lemma 11 and we omit it. Lemma 13 Under Assumptions 1 and 11, as N!1, we have 1 p N N X i=1 (z i h z (a i )) i )N (0; S zz ); where S zz = 0 B B B B @ S x 1 x 1 S x 1 Gx 1 S x 1 G 2 x 1 S Gx 1 x 1 S Gx 1 Gx 1 S Gx 1 G 2 x 1 S G 2 x 1 x 1 S G 2 x 1 Gx 1 S G 2 x 1 G 2 x 1 1 C C C C A and S G r x 1 G s x 1 =E h ~ ~ s x 1 r (x i ;a i )) ~ ~ s x 1 s (x i ;a i ) 0 2 (x i ;a i ) i ; r;s = 0; 1; 2 ~ ~ s x 1 m (x i ;a i ) = ~ s x 1 m (x i ;a i )E[~ s x 1 m (x i ;a i )ja i ]) with ~ s x 1 0 (x i ;a i ) = x 1i 2 (x i ;a i ) :=E[( i ) 2 jx i ;a i ] =E[( i E[ i ja i ]) 2 jx i ;a i ]; where ~ ~ s x 1 m (x i ;a i ) is dened recursively as in (S.2.5). Proof LetF i = (X 1N ; D N ;a i ; v 1 ;:::; v i1 ). Conditional on (X 1N ; D N ;a i ), E[(z i h z (a i )) v i jF i ] = (z i h z (a i ))E[ v i jF i ] = 0; and sof(z i h z (a i )) v i ;F i g is a martingale dierence sequence. Since i = i E[ i ja i ] is bounded by a constant under Assumption 11, E[( i ) 4 jF i1 ]<M (S.2.8) for some nite constant M. 68 Also notice under Assumptions 1, we have E[( i ) 2 jF i ] = E[( i E(ja i )) 2 jx i ;a i ; x i ; a i ; D N (x i ; a i ;fu ij g i;j=1;:::;N ; x i ;a i );f j g j<i ] = E[( i E(ja i )) 2 jx i ;a i ] =: 2 (x i ;a i ): Let ` be a nonzero vector whose dimension is the same as the IVs z i . Then, E[` 0 Z i ( Z i ) 0 `( v i ) 2 jF i ] = [` 0 (z i h z (a i ))(z i h z (a i )) 0 `]E[( i ) 2 jF i ] = [` 0 (z i h z (a i ))(z i h z (a i )) 0 `] 2 (x i ;a i ): Let s 2 N := 1 N N X i=1 E[` 0 (z i h z (a i ))(z i h z (a i )) 0 `( i ) 2 jF i ] = 1 N n X i=1 [` 0 (z i h z (a i ))(z i h z (a i )) 0 `] 2 (x i ;a i ): According to Lemma 12, s 2 N p ! S zz : Also, since ` 0 (z i h z (a i )) i =` 0 (z i h z (a i ))( i E[ i ja i ]) is bounded by a constant, under Assumption 11 the Lindeberg-Feller condition is satised, that is, for any > 0, 1 N N X i=1 E h [` 0 (z i h z (a i ))(z i h z (a i )) 0 `]( i ) 2 I n j` 0 (z i h z (a i )) i j> p N o jF i i N X i=1 1 2 N 2 E [` 0 (z i h z (a i ))(z i h z (a i )) 0 `] 2 ( i ) 4 jF i M N ! 0 as N!1: Then, by the Martingale Central Limit Theorem (e.g., see Corollary 3.1 Hall and Heyde 69 (2014)), we have the desired result for theorem: 1 p N N X i=1 (z i h z (a i )) i )N (0; S zz ): Proof of Theorem 1.5.1. Theorem 1.5.1 follows from Lemma 2, Lemma 7, Lemma 11, and Lemma 13. S.2.4 Further Supporting Lemmas Lemma 14 (Uniform Convergence of s m;N;i in i) Assume Assumptions 1, 5, 7, 8 and 11. Suppose that s 0 (x i ;a i ) is a bounded function of x i and a i . Suppose that we dene s m;N;i as in (S.2.3) and consider its probability limit ~ s m (x i ;a i ) in equation (S.2.5) for eachi. Then, for each m = 0; 1; 2; (a) sup 1iN js m;N;i ~ s m (x i ;a i )j =o p (1) (b) sup 1iN jE[s m;N;i ja i ]E[~ s m (x i ;a i )ja i ]j =o p (1): Proof Part (a). For m = 0: The required result for the lemma holds trivially because of the denition that s 0;N;i = ~ s 0 (x i ;a i ). Next we show the required result form = 1 and then use mathematical induction for the rest m = 2; 3;:::. For m = 1: The claim for the case m = 1 is proved in three steps. 70 Step 1. Notice that s 1;N;i = 1 N X j6=i d ij;N ! 1 1 N X j6=i d ij;N s 1;N;j = 1 N X j6=i Ifg(t(x 2i ; x 2j );a i ;a j )u ij g ! 1 1 N X j6=i Ifg(t(x 2i ; x 2j );a i ;a j )u ij gs 0 (x j ;a j ): Then, by the WLLN, for each i, 1 N X j6=i Ifg(t(x 2i ; x 2j );a i ;a j )u ij g p ! Z Z ((t(x 2i ; x 2 );a i ;a)(x 2 ;a)dx 2 da = E[d ij;N jx i ;a i ] (S.2.9) 1 N X j6=i Ifg(t(x 2i ; x 2j );a i ;a j )u ij gs 0 (x j ;a j ) p ! Z Z (t(x 2i ; x 2 ) 0 0 +a i +a)s 0 (x;a)(x;a)dxda = E[d ij;N s 0 (x j ;a j )jx i ;a i ]: (S.2.10) SinceE[d ij;N jx i ;a i ]> 0 uniformly ini;j under Assumption 11 (vi),(v), and (vi) for eachi as N!1, we have s 1;N;i ! p ~ s 1 (x i ;a i ) = RR (g(t(x 2i ; x 2 );a i ;a)s 0 (x;a)(x;a)dxda RR (g(t(x 2i ; x 2 );a i ;a)(x 2 ;a)dx 2 da Step 2. In this step, we show that the convergences in (S.2.9) and (S.2.10) hold uniformly in i. For this, we introduce the following notation. Let i;N;1 = 1 N N X j=1;6=i (d ij;N E[d ij;N jx i ;a i ]) and i;N;2 = 1 N N X j=1;6=i (d ij;N s 0 (x j ;a j )E[d ij;N s 0 (x j ;a j )jx i ;a i ]): 71 Notice that conditional on (x i ;a i ), d ij;N and d ij;N s 0 (x j ;a j ) are iid with conditional mean zero and bounded by a constant across j = 1;:::;N;6=i: Then, there exists a nite constant M 1 such that sup i E k p N i;N;k k 4 jx i ;a i M 1 ; and we can deduce the desired result sup i k i;N;k k =O p (N 1=4 ) =o p (1) because for any >, we choose M 2 = M 1 and then Pfsup i k i;N;k kN 1=4 M 1=4 2 jx i ;a i g =Pfsup i N 1=4 k p N i;N;k kM 1=4 2 jx i ;a i g =Pfsup i N 1 k p N i;N;k k 4 M 2 jx i ;a i g P ( 1 N N X i=1 k p N i;N;k k 4 M 2 jx i ;a i ) 1 M 2 1 N N X i=1 E k p N i;N;k k 4 jx i ;a i M 1 M 2 =: Step 3. Now we prove the desired result for the case m = 1. Dene i;N;1 = 1 N P j6=i d ij;N and i;N;2 = 1 N P j6=i d ij;N s 0 (x j ;a j ). Then, s 1;N;i = i;N;1 i;N;2 : 72 Let i;1 = 1 N P N j=1;6=i E[d ij;N jx i ;a i ] and i;2 = 1 N P N j=1;6=i E[d ij;N s 0 (x j ;a j )jx i ;a i ]. Notice that sup i ks 1;N;i k = sup i i;N;2 i;N;1 i;2 i;1 sup i i;N;2 i;2 i;N;1 + sup i i;2 ( i;N;1 i;1 ) i;N;1 i;1 =o p (1); where the last line holds becausek i;N;k i;k k =o p (1) by Step 2, and i;1 > 0 andk i;2 k is bounded by a constant. This shows the required result sup i ks 1;N;i ~ s 1 (x i ;a i )k =o p (1): For m 2. Given that we show the required result of the lemma with m = 1, we show the rest by mathematical induction. For this, suppose that sup 1iN ks m;N;i ~ s m (x i ;a i )k =o p (1): Then, we have sup 1iN ks m+1;N;i ~ s m+1 (x i ;a i )k = sup 1iN 1 N P N j=1;6=i d ij;N s m;N;i 1 N P N j=1;6=i d ij;N E[d ij;N ~ s m (x j ;a j )jx i ;a i ] E[d ij;N jx i ;a i ] sup 1iN 1 N P N j=1;6=i d ij;N (s m;N;i E[d ij;N ~ s m (x j ;a j )jx i ;a i ]) 1 N P N j=1;6=i d ij;N + sup 1iN kE[d ij;N ~ s m (x j ;a j )jx i ;a i ]k sup 1iN 1 1 N P N j=1;6=i d ij;N 1 E[d ij;N jx i ;a i ] : For the rst term, we have by the denition ofg ij;N = d ij;N P N j=1;6=i d ij;N and since P j=1;6=i g ij;N = 1, 73 we have sup 1iN 1 N P N j=1;6=i d ij;N (s m;N;i E[d ij;N ~ s m (x j ;a j )jx i ;a i ]) 1 N P N j=1;6=i d ij;N = sup 1iN 1 N N X j=1;6=i g ij;N (s m;N;i E[d ij;N ~ s m (x j ;a j )jx i ;a i ]) sup 1iN ks m;N;i E[d ij;N ~ s m (x j ;a j )jx i ;a i ]k =o p (1); where the last line holds by the assumption of mathematical induction. We can show the second term sup 1iN kE[d ij;N ~ s m (x j ;a j )jx i ;a i ]k sup 1iN 1 1 N P N j=1;6=i d ij;N 1 E[d ij;N jx i ;a i ] =o p (1) by using similar argument used in the proof of Step 3 of the case m = 1. Part (b). Notice that under Assumption 11,E[s m;N;i ja i ] andE[~ s m (x i ;a i )ja i ] are bounded by a nite constant. The required argument follows by similar arguments used in the proof of Part (a). Lemma 15 (Uniform Convergence of s m;N;i in i) Assume Assumptions Assumptions 1, 5, 7, 8 and 11. Suppose that we dene s m;N;i as in (S.2.6). Then, for each m = 1; 2; sup 1iN js m;N;i j =o p (1): Proof The proof is similar to that of Lemma 14. First, we show that for eachi andm = 1; 2;::: the probability limit of s m;N;i dened with s 0;i = i = i E[ i ja i ] recursively as (S.2.6) is 74 zero as N!1. To verify this, let ~ s 0;i = i = i E[ i ja i ]: For m = 1, s 1;N;i = 1 N X j6=i d ij;N ! 1 1 N X j6=i d ij;N s 0;j : Consider the numerator. Notice by denition that d ij;N s 0;j =Ifg(t(x 2i ; x 2j ;a i ;a j )u ij g ( j E[ j ja j ]) are i.i.d. acrossj conditioning on (x 2i ;a i ) and bounded by a nite constant under Assumption 11. Then, by the WLLN conditioning on (x 2i ;a i ), we have 1 N X j6=i d ij;N s 0;j p !E [d ij;N ( j E[ j ja j ])jx 2i ;a i ] =E [d ij;N E ( j E[ j ja j ]jX N ; D N ;a i )jx 2i ;a i ] = 0; where the last equality holds by Lemma 1. The denominator converges to 1 N X j6=i Ifg(t(x 2i ; x 2j );a i ;a j )u ij g! p Z Z (g(t(x 2i ; x 2 );a i ;a)(x 2 ;a)dx 2 da> 0; where the last inequality holds under Assumption 11. This shows that as N!1 1 N X j6=i g ij;N s 0;j p ! 0 =: ~ s 1;i for each i. 75 Then, using similar argument in Step 2 of the proof of Lemma 14, we deduce sup 1iN 1 N X j6=i g ij;N s 0;j =o p (1): Also, form = 2;:::, we follow the same mathematical induction argument in Steps 3 and 4 of the proof of Lemma 14 and deduce that sup 1iN 1 N X j6=i g ij;N s m;N;j =o p (1): Lemma 16 Assume Assumptions 1, 5, 7, 8 and 11. Suppose that s 0 (x i ;a i ) is a bounded function of x i and a i . Suppose that we dene s m;N;i as in equation (S.2.3) and consider its probability limit ~ s m (x i ;a i ) in equation (S.2.5) for each i. Then, (a) sup 1iN 1 X m=0 ( 0 1 ) m (s m;N;i ~ s m (x i ;a i )) =o p (1) (b) sup 1iN 1 X m=0 ( 0 1 ) m (E[s m;N;i ja i ]E[~ s m (x i ;a i )ja i ]) =o p (1): Also, suppose that we dene s m;N;i as in equation (S.2.6). Let ~ s 0;i = a i and ~ s m;i = 0 for m = 1; 2;:::. Then, (c) sup 1iN 1 X m=0 ( 0 1 ) m s m;N;i ~ s m =o p (1): Proof Part (a). Notice from Assumption 11 thatj 0 1 j< 1 ands m;N;i ; ~ s m (x i ;a i );E[s m;N;i ja i ];E[~ s m (x i ;a i )ja i ] are bounded by a nite constant, say, M. For given > 0, we choose m such that 2M P 1 m=m +1 ( 0 1 ) m . Then, by denition, we have sup 1iN 1 X m=m +1 ( 0 1 ) m (s m;N;i ~ s m (x i ;a i )) 2M 1 X m=m +1 ( 0 1 ) m : 76 Notice that sup 1iN 1 X m=0 ( 0 1 ) m (s m;N;i ~ s m (x i ;a i )) sup 1iN m X m=0 ( 0 1 ) m (s m;N;i ~ s m (x i ;a i )) + m sup 1iN js m;N;i ~ s m (x i ;a i )j + =o p (1) +; where the last inequality holds since m is nite and by Lemma 16. Since is arbitrary, we have the desired result for Part (a). Parts (b) and (c). Under Assumption 11,E[s m;N;i ja i ];E[~ s m (x i ;a i )ja i ], and i = i h (a i ) are bounded by a constant. Apply the same argument used in the proof of Part (a), then we deduce the required result of Parts (b) and (c). S.3 For 2SLS S.3.1 Limiting distribution of 2SLS Recall the denition that for any variable b l i being an element of (y i ; w i ; w i ) and i , l i :=b l i h l (x 2i ;a i ) =b l i h l (x 2i ; deg i ); i := i h (x 2i ;a i ) i h (x 2i ; deg i ): Let N = ( 1 ;:::; N ) 0 . Outline: Step 1 Show that p N( 2SLS 0 ) = W 0 N M R N Z N (Z 0 N M R N Z N ) 1 Z 0 N M R N W N 1 W 0 N M R N Z N (Z 0 M R N Z N ) 1 Z 0 N M R N N +o p (1): (S.3.1) 77 Step 2 Show 1 N N X i=1 b l i ^ h l (x 2i ; deg i ) b l i ^ h l (x 2i ; deg i ) 0 1 N N X i=1 b l i h l (x 2i ; deg i ) b l i h l (x 2i ; deg i ) 0 +o p (1) and 1 p N N X i=1 b l i ^ h l (x 2i ; deg i ) i = 1 p N N X i=1 b l i h l (x 2i ; deg i ) i +o p (1): Step 3 Derive the limits of 1 N N X i=1 b l i h l (x 2i ; deg i ) b l i h l (x 2i ; deg i ) 0 = 1 N N X i=1 b l i h l (x 2i ;a i ) b l i h l (x 2i ;a i 0 and 1 p N N X i=1 b l i h l (x 2i ; deg i ) i = 1 p N N X i=1 b l i h l (x 2i ;a i ) i S.3.2 Controlling the Sampling Error d deg i deg i in Sieve Estima- tion Equation (S.3.1) holds if the following Lemma is true. Lemma 17 Assume Assumptions Assumptions 1, 3, 4, 9, 10 and 11. Then the following holds. (a) 1 N (Z 0 N P b R N W N Z 0 N P R N W N ) =o p (1). (b) 1 N (Z 0 N P b R N Z N Z 0 N P R N Z N ) =o p (1). (c) 1 p N (Z 0 N P b R N v N Z 0 N P R N v N ) =o p (1). (d) 1 p N (Z 0 M b R N (H(a N ) b R N )) =o p (1). Proof 10 We can apply a similar argument as in Lemma 2 and derive the desired result. 78 S.3.3 Controlling the Series Approximation Error for r K (x 2i ; deg i ) Lemma 18 (Series Approximation) Assume the assumptions in Lemma 17. Then, we have (a) 1 N P N i=1 (w i ^ h w (x 2i ; deg i ))(z i ^ h z (x 2i ; deg i )) 0 = 1 N P N i=1 (w i h w (x 2i ; deg i ))(z i h z (x 2i ; deg i )) 0 +o p (1), (b) 1 N P N i=1 (z i ^ h z (x 2i ; deg i ))(z i ^ h z (x 2i ; deg i )) 0 = 1 N P N i=1 (z i h z (x 2i ; deg i ))(z i h z (x 2i ; deg i )) 0 + o p (1), (c) 1 p N P N i=1 (z i ^ h z (x 2i ; deg i )) i = 1 p N P N i=1 (z i h z (x 2i ; deg i )) i +o p (1). Then the proofs are analogous to the proofs presented in Section S.2.2 and we omit them. S.3.4 Limiting distribution of 2SLS Note that h l (x 2i ;deg i ) = h l (x 2i ;a i ). Using this relationship we can state the following Lemmas. Lemma 19 Under Assumption 1, 3, and 11, we have 1 N N X i=1 (w i h w (x 2i ;a i ))(z i h z (x 2i ;a i )) 0 p ! 0 B B B B @ S GY;x 1 S GY;Gx 1 S GY;G 2 x 1 S x 1 ;x 1 S x 1 ;Gx 1 S x 1 ;G 2 x 1 S Gx 1 ;x 1 S Gx 1 ;Gx 1 S Gx 1 ;G 2 x 1 1 C C C C A =: S wz ; and 1 N N X i=1 (z i h z (x 2i ;a i ))(z i h z (x 2i ;a i )) 0 p ! 0 B B B B @ S x 1 ;x 1 S x 1 ;Gx 1 S x 1 ;G 2 x 1 S Gx 1 ;x 1 S Gx 1 ;Gx 1 S Gx 1 G 2 x 1 S G 2 x 1 ;x 1 S G 2 x 1 ;Gx 1 S G 2 x 1 ;G 2 x 1 1 C C C C A =: S zz ; 79 where S GY;G r x 1 =E " 1 X m=0 0 0 2 ~ ~ s x 1 m (x i ;a i ) + 0 0 3 ~ ~ s x 1 ;m+1 (x i ;a i ) + ~ ~ s a m (x i ;a i ) ! ~ ~ s x 1 r (x i ;a i ) 0 # ; r = 0; 1; 2 S G r x 1 ;G s x 1 =E h ~ ~ s x 1 r (x i ;a i )) ~ ~ s x 1 s (x i ;a i ) 0 i ; r;s = 0; 1; 2 ~ ~ s x 1 m (x i ;a i ) = ~ s x 1 m (x i ;a i )E[~ s x 1 m (x i ;a i )jx 2i ;a i ]) with ~ s x 1 0 (x i ;a i ) = x 1i ~ ~ s a m (x i ;a i ) = ~ s a m (x i ;a i )E[~ s a m (x i ;a i )jx 2i ;a i ]) with ~ s a 0 (x i ;a i ) =h (x 2i ;a i ); where ~ ~ s x 1 m (x i ;a i ) and ~ ~ s a m (x i ;a i ) are dened recursively as in (S.2.5). Lemma 20 Under Assumption 1, 3, and 11, 1 N N X i=1 (z i h z (x 2i ;a i ))(z i h z (x 2i ;a i )) 0 2 (x i ;a i )! p S zz ; where the limit variance S zz is dened in Lemma 21. Lemma 21 Under Assumption 1, 3, and 11, 1 p N N X i=1 (z i h z (x 2i ;a i )) i )N (0; S zz ); where S zz = 0 B B B B @ S x 1 x 1 S x 1 Gx 1 S x 1 G 2 x 1 S Gx 1 x 1 S Gx 1 Gx 1 S Gx 1 G 2 x 1 S G 2 x 1 x 1 S G 2 x 1 Gx 1 S G 2 x 1 G 2 x 1 1 C C C C A 80 and S G r x 1 G s x 1 =E h ~ ~ s x 1 r (x i ;a i )) ~ ~ s x 1 s (x i ;a i ) 0 2 (x i ;a i ) i ; r;s = 0; 1; 2 ~ ~ s x 1 m (x i ;a i ) = ~ s x 1 m (x i ;a i )E[~ s x 1 m (x i ;a i )jx 2i ;a i ]) with ~ s x 1 0 (x i ;a i ) = x 1i 2 (x i ;a i ) :=E[( i ) 2 jx i ;a i ] =E[( i E[ i jx 2i ;a i ]) 2 jx i ;a i ]; where ~ ~ s x 1 m (x i ;a i ) is dened recursively as in (S.2.5). S.4 Supplementary Monte Carlo results 81 Table S.4.1: Polynomial Sieve: Parameter values across 1000 Monte Carlo replications with h(a) = exp(3a i ) and K N = 4 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0 0.081 0.074 0.011 0.007 -0.007 0.126 0.115 0.002 -0.018 -0.008 mean bias (0.140 ) (0.178 ) (0.214 ) (0.308 ) (0.165 ) (0.109 ) (0.163 ) (0.205 ) (0.331 ) (0.135 ) std 0.383 0.089 0.057 0.061 0.048 0.537 0.109 0.060 0.068 0.044 size 2 = 1 -0.005 -0.004 0.000 0.001 0.000 -0.004 -0.003 0.000 0.000 0.000 mean bias (0.031 ) (0.033 ) (0.035 ) (0.036 ) (0.031 ) (0.020 ) (0.021 ) (0.021 ) (0.021 ) (0.020 ) std 0.312 0.061 0.069 0.077 0.047 0.334 0.072 0.061 0.063 0.056 size 3 = 1 -0.113 -0.100 -0.006 -0.003 0.012 -0.201 -0.181 0.007 0.024 0.016 mean bias (0.268 ) (0.335 ) (0.384 ) (0.437 ) (0.298 ) (0.214 ) (0.315 ) (0.379 ) (0.458 ) (0.247 ) std 0.348 0.076 0.057 0.064 0.051 0.465 0.092 0.057 0.057 0.050 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0:5 0.042 0.042 0.007 0.004 -0.004 0.062 0.064 0.001 -0.012 -0.004 mean bias (0.076 ) (0.104 ) (0.135 ) (0.197 ) (0.095 ) (0.059 ) (0.093 ) (0.129 ) (0.209 ) (0.073 ) std 0.385 0.091 0.059 0.062 0.046 0.540 0.110 0.060 0.068 0.043 size 2 = 1 -0.003 -0.003 0.000 0.001 0.000 -0.003 -0.003 0.000 0.000 0.000 mean bias (0.031 ) (0.032 ) (0.034 ) (0.035 ) (0.031 ) (0.020 ) (0.020 ) (0.021 ) (0.021 ) (0.020 ) std 0.315 0.060 0.070 0.077 0.049 0.335 0.071 0.064 0.064 0.058 size 3 = 1 -0.067 -0.065 -0.001 0.002 0.008 -0.116 -0.122 0.008 0.019 0.011 mean bias (0.196 ) (0.259 ) (0.313 ) (0.343 ) (0.223 ) (0.153 ) (0.237 ) (0.306 ) (0.348 ) (0.172 ) std 0.338 0.078 0.057 0.070 0.051 0.425 0.083 0.058 0.062 0.046 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0:8 0.015 0.018 0.004 0.002 -0.001 0.019 0.027 0.001 -0.007 -0.001 mean bias (0.033 ) (0.049 ) (0.080 ) (0.118 ) (0.045 ) (0.025 ) (0.042 ) (0.076 ) (0.125 ) (0.031 ) std 0.388 0.090 0.058 0.062 0.041 0.539 0.110 0.060 0.068 0.042 size 2 = 1 -0.001 -0.002 0.000 0.001 -0.000 -0.001 -0.001 0.000 0.000 -0.000 mean bias (0.030 ) (0.032 ) (0.034 ) (0.035 ) (0.030 ) (0.020 ) (0.020 ) (0.020 ) (0.021 ) (0.020 ) std 0.307 0.060 0.071 0.077 0.046 0.331 0.066 0.063 0.063 0.054 size 3 = 1 -0.021 -0.032 0.001 0.004 0.004 -0.035 -0.065 0.009 0.015 0.006 mean bias (0.135 ) (0.192 ) (0.264 ) (0.281 ) (0.154 ) (0.099 ) (0.164 ) (0.256 ) (0.276 ) (0.105 ) std 0.314 0.072 0.058 0.065 0.041 0.361 0.070 0.058 0.060 0.041 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 CF - control function. (0) - none, (1) - ^ a i , (2) - ^ h(^ a i ), (3) - ^ h( d deg i ;x 2i ), (4) - h(a i ). Average number of links for N = 100 is 24:1, for N = 250 it is 60:2. The bias of ^ a i is calculated as a i ^ a i K 100;0:2 = 6, K 100;0:5 = 3, K 100;0:8 = 4 K 250;0:2 = 8, K 250;0:5 = 3, K 250;0:8 = 7 82 Table S.4.2: Polynomial Sieve: Parameter values across 1000 Monte Carlo replications with h(a) = exp(3a i ) and K N = 6 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0 0.081 0.074 0.010 0.006 -0.007 0.126 0.115 0.003 -0.018 -0.008 mean bias (0.140 ) (0.178 ) (0.216 ) (0.314 ) (0.165 ) (0.109 ) (0.163 ) (0.206 ) (0.332 ) (0.135 ) std 0.383 0.089 0.059 0.067 0.048 0.537 0.109 0.060 0.069 0.044 size 2 = 1 -0.005 -0.004 0.001 0.001 0.000 -0.004 -0.003 0.000 0.000 0.000 mean bias (0.031 ) (0.033 ) (0.035 ) (0.036 ) (0.031 ) (0.020 ) (0.021 ) (0.021 ) (0.021 ) (0.020 ) std 0.312 0.061 0.072 0.078 0.047 0.334 0.072 0.065 0.065 0.056 size 3 = 1 -0.113 -0.100 -0.005 -0.000 0.012 -0.201 -0.181 0.005 0.023 0.016 mean bias (0.268 ) (0.335 ) (0.389 ) (0.443 ) (0.298 ) (0.214 ) (0.315 ) (0.381 ) (0.460 ) (0.247 ) std 0.348 0.076 0.061 0.067 0.051 0.465 0.092 0.058 0.058 0.050 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0:5 0.042 0.042 0.006 0.003 -0.004 0.062 0.064 0.002 -0.012 -0.004 mean bias (0.076 ) (0.104 ) (0.137 ) (0.200 ) (0.095 ) (0.059 ) (0.093 ) (0.129 ) (0.210 ) (0.073 ) std 0.385 0.091 0.060 0.067 0.046 0.540 0.110 0.060 0.069 0.043 size 2 = 1 -0.003 -0.003 0.000 0.001 0.000 -0.003 -0.003 0.000 0.000 0.000 mean bias (0.031 ) (0.032 ) (0.035 ) (0.036 ) (0.031 ) (0.020 ) (0.020 ) (0.021 ) (0.021 ) (0.020 ) std 0.315 0.060 0.072 0.072 0.049 0.335 0.071 0.065 0.069 0.058 size 3 = 1 -0.067 -0.065 -0.000 0.004 0.008 -0.116 -0.122 0.006 0.018 0.011 mean bias (0.196 ) (0.259 ) (0.317 ) (0.348 ) (0.223 ) (0.153 ) (0.237 ) (0.307 ) (0.350 ) (0.172 ) std 0.338 0.078 0.059 0.067 0.051 0.425 0.083 0.059 0.066 0.046 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0:8 0.015 0.018 0.004 0.002 -0.001 0.019 0.027 0.001 -0.007 -0.001 mean bias (0.033 ) (0.049 ) (0.081 ) (0.121 ) (0.045 ) (0.025 ) (0.042 ) (0.077 ) (0.126 ) (0.031 ) std 0.388 0.090 0.060 0.068 0.041 0.539 0.110 0.060 0.069 0.042 size 2 = 1 -0.001 -0.002 0.000 0.000 -0.000 -0.001 -0.001 0.000 0.000 -0.000 mean bias (0.030 ) (0.032 ) (0.034 ) (0.035 ) (0.030 ) (0.020 ) (0.020 ) (0.021 ) (0.021 ) (0.020 ) std 0.307 0.060 0.072 0.070 0.046 0.331 0.066 0.068 0.068 0.054 size 3 = 1 -0.021 -0.032 0.002 0.006 0.004 -0.035 -0.065 0.007 0.014 0.006 mean bias (0.135 ) (0.192 ) (0.268 ) (0.285 ) (0.154 ) (0.099 ) (0.164 ) (0.257 ) (0.278 ) (0.105 ) std 0.314 0.072 0.063 0.069 0.041 0.361 0.070 0.057 0.065 0.041 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 CF - control function. (0) - none, (1) - ^ a i , (2) - ^ h(^ a i ), (3) - ^ h( d deg i ;x 2i ), (4) - h(a i ). Average number of links for N = 100 is 24:1, for N = 250 it is 60:2. The bias of ^ a i is calculated as a i ^ a i K 100;0:2 = 6, K 100;0:5 = 3, K 100;0:8 = 4 K 250;0:2 = 8, K 250;0:5 = 3, K 250;0:8 = 7 83 Table S.4.3: Polynomial Sieve: Parameter values across 1000 Monte Carlo replications with h(a) = exp(3a i ) and K N = 8 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0 0.081 0.074 0.009 0.007 -0.007 0.126 0.115 0.003 -0.017 -0.008 mean bias (0.140 ) (0.178 ) (0.219 ) (0.314 ) (0.165 ) (0.109 ) (0.163 ) (0.207 ) (0.332 ) (0.135 ) std 0.383 0.089 0.058 0.069 0.048 0.537 0.109 0.060 0.066 0.044 size 2 = 1 -0.005 -0.004 0.000 0.001 0.000 -0.004 -0.003 0.001 0.000 0.000 mean bias (0.031 ) (0.033 ) (0.036 ) (0.036 ) (0.031 ) (0.020 ) (0.021 ) (0.021 ) (0.021 ) (0.020 ) std 0.312 0.061 0.074 0.074 0.047 0.334 0.072 0.063 0.067 0.056 size 3 = 1 -0.113 -0.100 -0.003 -0.001 0.012 -0.201 -0.181 0.004 0.023 0.016 mean bias (0.268 ) (0.335 ) (0.394 ) (0.443 ) (0.298 ) (0.214 ) (0.315 ) (0.382 ) (0.461 ) (0.247 ) std 0.348 0.076 0.058 0.064 0.051 0.465 0.092 0.060 0.059 0.050 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0:5 0.042 0.042 0.005 0.003 -0.004 0.062 0.064 0.002 -0.011 -0.004 mean bias (0.076 ) (0.104 ) (0.138 ) (0.201 ) (0.095 ) (0.059 ) (0.093 ) (0.130 ) (0.210 ) (0.073 ) std 0.385 0.091 0.059 0.071 0.046 0.540 0.110 0.060 0.066 0.043 size 2 = 1 -0.003 -0.003 0.000 0.000 0.000 -0.003 -0.003 0.000 0.000 0.000 mean bias (0.031 ) (0.032 ) (0.035 ) (0.035 ) (0.031 ) (0.020 ) (0.020 ) (0.021 ) (0.021 ) (0.020 ) std 0.315 0.060 0.074 0.067 0.049 0.335 0.071 0.065 0.071 0.058 size 3 = 1 -0.067 -0.065 0.001 0.003 0.008 -0.116 -0.122 0.006 0.018 0.011 mean bias (0.196 ) (0.259 ) (0.321 ) (0.348 ) (0.223 ) (0.153 ) (0.237 ) (0.308 ) (0.350 ) (0.172 ) std 0.338 0.078 0.062 0.069 0.051 0.425 0.083 0.063 0.065 0.046 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0:8 0.015 0.018 0.003 0.002 -0.001 0.019 0.027 0.001 -0.007 -0.001 mean bias (0.033 ) (0.049 ) (0.082 ) (0.121 ) (0.045 ) (0.025 ) (0.042 ) (0.077 ) (0.126 ) (0.031 ) std 0.388 0.090 0.060 0.071 0.041 0.539 0.110 0.060 0.066 0.042 size 2 = 1 -0.001 -0.002 -0.000 0.000 -0.000 -0.001 -0.001 0.000 0.000 -0.000 mean bias (0.030 ) (0.032 ) (0.034 ) (0.035 ) (0.030 ) (0.020 ) (0.020 ) (0.021 ) (0.021 ) (0.020 ) std 0.307 0.060 0.074 0.067 0.046 0.331 0.066 0.067 0.070 0.054 size 3 = 1 -0.021 -0.032 0.004 0.006 0.004 -0.035 -0.065 0.007 0.014 0.006 mean bias (0.135 ) (0.192 ) (0.272 ) (0.285 ) (0.154 ) (0.099 ) (0.164 ) (0.258 ) (0.278 ) (0.105 ) std 0.314 0.072 0.067 0.072 0.041 0.361 0.070 0.063 0.064 0.041 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 CF - control function. (0) - none, (1) - ^ a i , (2) - ^ h(^ a i ), (3) - ^ h( d deg i ;x 2i ), (4) - h(a i ). Average number of links for N = 100 is 24:1, for N = 250 it is 60:2. The bias of ^ a i is calculated as a i ^ a i K 100;0:2 = 6, K 100;0:5 = 3, K 100;0:8 = 4 K 250;0:2 = 8, K 250;0:5 = 3, K 250;0:8 = 7 84 Table S.4.4: Polynomial Sieve: Parameter values across 1000 Monte Carlo replications with h(a) = sin(3a i ) and K N = 4 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0:200 -1.681 -0.398 -0.101 0.006 -0.010 -0.445 -0.630 -0.091 -0.024 -0.019 mean bias (26.702 ) (0.325 ) (0.243 ) (0.349 ) (0.246 ) (160.512 ) (0.352 ) (0.232 ) (0.372 ) (0.297 ) std 0.638 0.301 0.075 0.066 0.018 0.698 0.563 0.080 0.064 0.014 size 2 = 1 0.128 0.022 0.002 0.001 0.001 -0.043 0.019 0.001 0.000 0.001 mean bias (2.379 ) (0.041 ) (0.036 ) (0.037 ) (0.035 ) (6.620 ) (0.024 ) (0.021 ) (0.021 ) (0.022 ) std 0.511 0.109 0.073 0.088 0.016 0.616 0.137 0.056 0.062 0.018 size 3 = 1 2.674 0.520 0.094 -0.002 0.015 0.627 0.937 0.097 0.030 0.033 mean bias (41.110 ) (0.530 ) (0.411 ) (0.473 ) (0.425 ) (277.957 ) (0.605 ) (0.407 ) (0.499 ) (0.532 ) std 0.597 0.203 0.062 0.062 0.017 0.705 0.430 0.071 0.054 0.013 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0:5 -1.095 -0.211 -0.062 0.003 -0.005 -0.462 -0.321 -0.056 -0.015 -0.009 mean bias (14.839 ) (0.170 ) (0.151 ) (0.219 ) (0.140 ) (25.186 ) (0.171 ) (0.142 ) (0.230 ) (0.147 ) std 0.779 0.313 0.075 0.066 0.023 0.810 0.576 0.080 0.064 0.015 size 2 = 1 0.148 0.015 0.001 0.001 0.000 0.023 0.013 0.001 0.000 0.000 mean bias (2.399 ) (0.037 ) (0.035 ) (0.036 ) (0.033 ) (1.468 ) (0.023 ) (0.021 ) (0.021 ) (0.021 ) std 0.497 0.100 0.074 0.085 0.021 0.599 0.096 0.058 0.065 0.020 size 3 = 1 2.154 0.317 0.060 0.003 0.011 0.910 0.571 0.063 0.023 0.022 mean bias (30.259 ) (0.364 ) (0.329 ) (0.363 ) (0.310 ) (55.722 ) (0.383 ) (0.321 ) (0.369 ) (0.336 ) std 0.701 0.174 0.059 0.065 0.018 0.804 0.378 0.067 0.056 0.015 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0:8 -0.168 -0.093 -0.036 0.001 -0.002 -0.320 -0.136 -0.033 -0.009 -0.004 mean bias (0.285 ) (0.079 ) (0.088 ) (0.130 ) (0.070 ) (5.928 ) (0.075 ) (0.083 ) (0.136 ) (0.065 ) std 0.813 0.321 0.075 0.067 0.025 0.906 0.592 0.080 0.066 0.022 size 2 = 1 0.026 0.008 0.000 0.001 -0.000 0.028 0.008 0.001 0.000 0.000 mean bias (0.072 ) (0.034 ) (0.035 ) (0.035 ) (0.031 ) (0.470 ) (0.021 ) (0.021 ) (0.021 ) (0.020 ) std 0.364 0.084 0.075 0.084 0.021 0.461 0.072 0.058 0.066 0.020 size 3 = 1 0.401 0.162 0.037 0.005 0.007 0.879 0.301 0.041 0.018 0.013 mean bias (0.817 ) (0.251 ) (0.274 ) (0.293 ) (0.219 ) (18.665 ) (0.244 ) (0.265 ) (0.288 ) (0.203 ) std 0.585 0.127 0.060 0.069 0.024 0.846 0.279 0.064 0.054 0.020 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 CF - control function. (0) - none, (1) - ^ a i , (2) - ^ h(^ a i ), (3) - ^ h( d deg i ;x 2i ), (4) - h(a i ). Average number of links for N = 100 is 24:1, for N = 250 it is 60:2. The bias of ^ a i is calculated as a i ^ a i K 100;0:2 = 8, K 100;0:5 = 7, K 100;0:8 = 6 K 250;0:2 = 7, K 250;0:5 = 5, K 250;0:8 = 7 85 Table S.4.5: Polynomial Sieve: Parameter values across 1000 Monte Carlo replications with h(a) = sin(3a i ) and K N = 6 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0:200 -1.681 -0.398 -0.102 0.005 -0.010 -0.445 -0.630 -0.089 -0.024 -0.019 mean bias (26.702 ) (0.325 ) (0.246 ) (0.354 ) (0.246 ) (160.512 ) (0.352 ) (0.233 ) (0.373 ) (0.297 ) std 0.638 0.301 0.081 0.071 0.018 0.698 0.563 0.076 0.071 0.014 size 2 = 1 0.128 0.022 0.002 0.001 0.001 -0.043 0.019 0.001 0.000 0.001 mean bias (2.379 ) (0.041 ) (0.037 ) (0.037 ) (0.035 ) (6.620 ) (0.024 ) (0.021 ) (0.021 ) (0.022 ) std 0.511 0.109 0.079 0.081 0.016 0.616 0.137 0.058 0.065 0.018 size 3 = 1 2.674 0.520 0.096 0.001 0.015 0.627 0.937 0.094 0.029 0.033 mean bias (41.110 ) (0.530 ) (0.416 ) (0.479 ) (0.425 ) (277.957 ) (0.605 ) (0.409 ) (0.501 ) (0.532 ) std 0.597 0.203 0.060 0.062 0.017 0.705 0.430 0.072 0.059 0.013 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0:5 -1.095 -0.211 -0.063 0.002 -0.005 -0.462 -0.321 -0.055 -0.015 -0.009 mean bias (14.839 ) (0.170 ) (0.152 ) (0.222 ) (0.140 ) (25.186 ) (0.171 ) (0.143 ) (0.231 ) (0.147 ) std 0.779 0.313 0.079 0.073 0.023 0.810 0.576 0.076 0.072 0.015 size 2 = 1 0.148 0.015 0.001 0.001 0.000 0.023 0.013 0.001 0.000 0.000 mean bias (2.399 ) (0.037 ) (0.036 ) (0.036 ) (0.033 ) (1.468 ) (0.023 ) (0.021 ) (0.021 ) (0.021 ) std 0.497 0.100 0.079 0.080 0.021 0.599 0.096 0.057 0.066 0.020 size 3 = 1 2.154 0.317 0.062 0.005 0.011 0.910 0.571 0.061 0.022 0.022 mean bias (30.259 ) (0.364 ) (0.333 ) (0.368 ) (0.310 ) (55.722 ) (0.383 ) (0.323 ) (0.371 ) (0.336 ) std 0.701 0.174 0.059 0.061 0.018 0.804 0.378 0.069 0.060 0.015 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0:8 -0.168 -0.093 -0.037 0.001 -0.002 -0.320 -0.136 -0.032 -0.009 -0.004 mean bias (0.285 ) (0.079 ) (0.089 ) (0.132 ) (0.070 ) (5.928 ) (0.075 ) (0.084 ) (0.136 ) (0.065 ) std 0.813 0.321 0.079 0.073 0.025 0.906 0.592 0.077 0.072 0.022 size 2 = 1 0.026 0.008 0.000 0.001 -0.000 0.028 0.008 0.001 -0.000 0.000 mean bias (0.072 ) (0.034 ) (0.035 ) (0.036 ) (0.031 ) (0.470 ) (0.021 ) (0.021 ) (0.021 ) (0.020 ) std 0.364 0.084 0.079 0.084 0.021 0.461 0.072 0.057 0.067 0.020 size 3 = 1 0.401 0.162 0.038 0.007 0.007 0.879 0.301 0.040 0.017 0.013 mean bias (0.817 ) (0.251 ) (0.278 ) (0.296 ) (0.219 ) (18.665 ) (0.244 ) (0.266 ) (0.289 ) (0.203 ) std 0.585 0.127 0.058 0.072 0.024 0.846 0.279 0.066 0.059 0.020 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 CF - control function. (0) - none, (1) - ^ a i , (2) - ^ h(^ a i ), (3) - ^ h( d deg i ;x 2i ), (4) - h(a i ). Average number of links for N = 100 is 24:1, for N = 250 it is 60:2. The bias of ^ a i is calculated as a i ^ a i K 100;0:2 = 8, K 100;0:5 = 7, K 100;0:8 = 6 K 250;0:2 = 7, K 250;0:5 = 5, K 250;0:8 = 7 86 Table S.4.6: Polynomial Sieve: Parameter values across 1000 Monte Carlo replications with h(a) = sin(3a i ) and K N = 8 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0:200 -1.681 -0.398 -0.103 0.005 -0.010 -0.445 -0.630 -0.089 -0.023 -0.019 mean bias (26.702 ) (0.325 ) (0.249 ) (0.354 ) (0.246 ) (160.512 ) (0.352 ) (0.234 ) (0.374 ) (0.297 ) std 0.638 0.301 0.081 0.073 0.018 0.698 0.563 0.078 0.068 0.014 size 2 = 1 0.128 0.022 0.002 0.001 0.001 -0.043 0.019 0.001 0.000 0.001 mean bias (2.379 ) (0.041 ) (0.037 ) (0.037 ) (0.035 ) (6.620 ) (0.024 ) (0.021 ) (0.021 ) (0.022 ) std 0.511 0.109 0.081 0.078 0.016 0.616 0.137 0.055 0.065 0.018 size 3 = 1 2.674 0.520 0.099 0.000 0.015 0.627 0.937 0.093 0.029 0.033 mean bias (41.110 ) (0.530 ) (0.422 ) (0.480 ) (0.425 ) (277.957 ) (0.605 ) (0.411 ) (0.501 ) (0.532 ) std 0.597 0.203 0.062 0.065 0.017 0.705 0.430 0.073 0.060 0.013 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0:5 -1.095 -0.211 -0.064 0.002 -0.005 -0.462 -0.321 -0.055 -0.015 -0.009 mean bias (14.839 ) (0.170 ) (0.154 ) (0.222 ) (0.140 ) (25.186 ) (0.171 ) (0.144 ) (0.231 ) (0.147 ) std 0.779 0.313 0.081 0.074 0.023 0.810 0.576 0.078 0.070 0.015 size 2 = 1 0.148 0.015 0.001 0.001 0.000 0.023 0.013 0.001 0.000 0.000 mean bias (2.399 ) (0.037 ) (0.036 ) (0.036 ) (0.033 ) (1.468 ) (0.023 ) (0.021 ) (0.021 ) (0.021 ) std 0.497 0.100 0.076 0.078 0.021 0.599 0.096 0.058 0.064 0.020 size 3 = 1 2.154 0.317 0.064 0.004 0.011 0.910 0.571 0.061 0.022 0.022 mean bias (30.259 ) (0.364 ) (0.338 ) (0.368 ) (0.310 ) (55.722 ) (0.383 ) (0.324 ) (0.371 ) (0.336 ) std 0.701 0.174 0.061 0.061 0.018 0.804 0.378 0.068 0.061 0.015 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 N 100 250 CF (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) 1 = 0:8 -0.168 -0.093 -0.037 0.001 -0.002 -0.320 -0.136 -0.032 -0.009 -0.004 mean bias (0.285 ) (0.079 ) (0.090 ) (0.132 ) (0.070 ) (5.928 ) (0.075 ) (0.084 ) (0.136 ) (0.065 ) std 0.813 0.321 0.081 0.074 0.025 0.906 0.592 0.078 0.071 0.022 size 2 = 1 0.026 0.008 -0.000 0.000 -0.000 0.028 0.008 0.001 -0.000 0.000 mean bias (0.072 ) (0.034 ) (0.035 ) (0.036 ) (0.031 ) (0.470 ) (0.021 ) (0.021 ) (0.021 ) (0.020 ) std 0.364 0.084 0.075 0.085 0.021 0.461 0.072 0.057 0.065 0.020 size 3 = 1 0.401 0.162 0.040 0.006 0.007 0.879 0.301 0.039 0.016 0.013 mean bias (0.817 ) (0.251 ) (0.281 ) (0.296 ) (0.219 ) (18.665 ) (0.244 ) (0.268 ) (0.290 ) (0.203 ) std 0.585 0.127 0.057 0.073 0.024 0.846 0.279 0.062 0.058 0.020 size ^ a - mean bias=0.016, median bias=0.007 ^ a - mean bias=0.006, median bias=0.003 CF - control function. (0) - none, (1) - ^ a i , (2) - ^ h(^ a i ), (3) - ^ h( d deg i ;x 2i ), (4) - h(a i ). Average number of links for N = 100 is 24:1, for N = 250 it is 60:2. The bias of ^ a i is calculated as a i ^ a i K 100;0:2 = 8, K 100;0:5 = 7, K 100;0:8 = 6 K 250;0:2 = 7, K 250;0:5 = 5, K 250;0:8 = 7 87 88 Chapter 2 Double-question Survey Measures for the Analysis of Financial Bubbles and Crashes 1 Hashem Pesaran y , Ida Johnsson z This paper proposes a new double-question survey whereby an individual is presented with two sets of questions; one on beliefs about current asset values and another on price expectations. A theoretical asset pricing model with heterogeneous agents is advanced and the existence of a negative relationship between price expectations and asset valuations is established, which is tested using survey results on equity, gold and house prices. Leading indicators of bubbles and crashes are proposed and their potential value is illustrated in the 1 Earlier versions of this paper have been presented at the Reserve Bank of India, Queen's University, Canada, York University, England, the Federal Reserve Bank of San Francisco, and at the Rady School of Management at UCSD. The survey questions have been designed jointly with Je Dominitz and Charles Manski. We are grateful to Arie Kapteyn (now at USC but previously at RAND) for his generous support of this project, and to Julie Newell, Angela Hung, and Tania Gutsche for overseeing the conduct of the surveys at RAND American Life Panel. Qiankun Zhou helped with carrying out some of the panel data regressions and Jorge Tarras o helped with the initial analysis of the survey data. We have also received helpful comments from Ron Smith. Address for correspondence: Hashem Pesaran, Department of Economics, University of Southern California, 3620 S. Vermont Ave., Los Angeles, CA 90089, USA. E-mail: pesaran@usc.edu. y Department of Economics & USC Dornsife INET, University of Southern California, and Trinity College, Cambridge, UK z Department of Economics, University of Southern California (Ida Johnsson was a Graduate Fellow of USC Dornsife INET while working on this project during 2014-2016.) 89 context of a dynamic panel regression of realized house price changes across key MSAs in the US. Keywords: Price expectations, bubbles and crashes, house prices, belief valuations. JEL Classification: C83, D84, G12, G14. 2.1 Introduction Expectations formation is an integral part of the decision making process, yet little is known about the way individuals actually form expectations. At the theoretical level and in the context of representative agent models, the rational expectations hypothesis (REH) has gained general acceptance as the dominant model of expectations formation. But in reality markets are populated with agents that dier in a priori beliefs, information, knowledge, cognitive and processing abilities, and there is no reason to believe that such heterogeneities will be eliminated by market interactions alone. As argued in the seminal work of Grossman and Stiglitz (1980), the price revelation cannot be perfect and heterogeneity is likely to be a prevalent feature of expectations across individuals. Allowing for heterogeneity of expectations is particularly important for a better understanding of bubble and crashes in asset prices. This is apparent in the theoretical literature on price bubbles where most recent contributions consider dierent types of traders, variously refereed to as \fundamental" and \noise" traders, or \behavioral" traders. See, for example, Allen et al. (1993), Daniel et al. (1998), Hirshleifer (2001), Odean (1998), Thaler (1991), Shiller (2000), Shleifer (2000), and Abreu and Brunnermeier (2003). There is also a related literature on higher-order beliefs in asset pricing, inspired from Keynes's example of the beauty contest, that focus on the departure of asset prices from the average expectations of the fundamentals across agents. See, for example, Allen et al. (2006), Bacchetta and Van Wincoop (2006), and Bacchetta and Van Wincoop (2008). This literature provides a formal framework for the analysis of market psychology and the possibility of bubbles and crashes arising when market expectations of the fundamentals deviate from realized asset prices. 90 Furthermore, it has proved dicult to develop tests of bubbles/crashes based on rep- resentative agent models, as was recognized early on by Blanchard (1979), who concluded that \...Detecting their [bubbles] presence or rejecting their existence is likely to prove very hard." There is also a large econometrics literature on tests of asset price bubbles based on long historical time series of asset returns. 2 But the outcomes of such tests are generally inconclusive. For example, G urkaynak (2008) after surveying a large number of studies con- cludes that \We are still unable to distinguish bubbles from time-varying or regime switching fundamentals, while many small sample econometrics problems of bubble tests remain un- resolved." Recent recursive time series tests proposed in a series of papers by Phillips and Yu provide more powerful tests, but these tests are purely statistical in nature and do not allow us to infer if structural breaks detected in the time series processes of asset prices are evidence of bubbles or are due to breaks in the underlying (unobserved) fundamentals. See Phillips et al. (2011) and Phillips et al. (2015). Also see Homm and Breitung (2012). Analysis of aggregate time series observations can provide historical information about price reversals and some of their proximate causes. But it is unlikely that such aggregate time series observations on their own could provide timely evidence of building up of bubbles and their subsequent collapse. In this paper we consider an alternative survey-based strategy and propose indicators of bubbles and crashes that exploit the heterogeneity of expectations across individuals and the disparities that exist between individual subjective asset valuations and their expected price changes. We show that in a heterogeneous agent model with bubble-free equilibrium outcomes, we would expect a negative association between valuation and expected price changes, and use this theoretical result as a bench-mark for categorizing individual respon- dents as belonging to bubble, crash and normal states. The proportions of respondents in bubble and crash states can be used as leading indicators in forecasting or policy analysis. The heterogeneity of expectations is a key feature of our analysis and has been well documented in the literature. For example, Ito (1990) considers expectations of foreign 2 There are a few empirical studies that use panel data regressions, but such studies face the additional challenge of allowing for bubbles at dierent times in dierent markets and possible bubble spill-overs across markets. 91 exchange rates in Japan, and nds that exporters tend to anticipate a yen depreciation while importers anticipate an appreciation, a kind of `wishful thinking'. Dominitz and Manski (2011) and Branch (2004) study the heterogeneity of equity price expectations using the Michigan Surveys, and nd that there is a large degree of heterogeneity in expectation formation. Similar patterns of expectations heterogeneity are documented for house prices. See, for example, Case and Shiller (1988), Case and Shiller (2003), Case et al. (2012), Niu and Van Soest (2014), Kuchler and Zafar (2015), and Bover (2015). 3 However, all surveys of price expectations focus on individual expectations of future price movements either qualitatively (whether the prices are expected to rise, fall or stay the same) or quantitatively in the form of predictive densities. The outcomes of such surveys are used in disaggregated or aggregated forms in tests of rationality of expectations and for forecasting of aggregate trends. Typically, such survey questions are not placed in particular decision contexts. However, for the analysis of many economic problems more information about the nature of individual beliefs and expectations is required. This is particularly the case when individual decisions depend not only on their own expectations of future outcomes, but also on their beliefs about the expectations of other market participants. But elicitation of individual expectations of others can be quite dicult. It is also likely to be unreliable since the reference group might not be known and could be changeable over time. In this paper we approach the problem indirectly and present an individual respondent with two sets of questions, one that asks about the individual's subjective belief regarding valuations (whether the prevailing asset price is "fairly valued"), and another regarding the individual's expectations of the future price of that asset. 4 Responses to these two questions are then used to measure the extent to which prices are likely to move towards or away from the subjectively perceived fundamental values. These questions do not require that the notation of a fundamental value is commonly understood or agreed upon. We report the results of such double-question surveys for gold, equity and house prices 3 A review of the literature on survey expectations can be found in Pesaran and Weale (2006). 4 The double-question surveys proposed in this paper are to be distinguished from other double-questions considered in the survey literature, such as the "double-barreled" questions that ask a respondent two questions but require one answer, and questions with anchoring vignettes, introduced by King et al. (2004), which are aimed at enhancing cross-respondent comparability of survey measures. 92 conducted with US households using RAND American Life Panel (ALP). 5 The ALP covers over 6,000 members with ages 18 and over, and is nationally representative, drawing from respondents recruited from several sources, including University of Michigan Phone-Panel and Internet-Panel Cohorts, and National Survey Project Cohort. We started with two pilot surveys, and introduced the double-question surveys as a new module starting in January 2012 and ended January 2013 (13 waves altogether). The number of survey participants ranged form a low of 4,477 in January 2012, to a high of 5,911 in January 2013. All re- spondents provided demographic information, but were not compelled to respond to our questions. Nevertheless, as it turned out the response rate was around 72%, and we ended up with a panel of around 4,000 individuals who completed our survey questions over the period January 2012 to January 2013. The survey responses provide information on individuals' price expectations as well as their valuation beliefs. It is the two questions together that allow us to construct bubble and crash indicators. To our knowledge this has not been done before. Under standard representative agent rational expectations models there is no relationship between asset valuations and future expected price changes, and the question on valuation will not be necessary. But as we shall show, this is not the case when we consider hetero- geneous agent rational expectations models where agents dier in their beliefs about future dividend processes. Under certain conditions on how individuals form expectations of others in the market place, we shall show that individual expectations of price changes are nega- tively related to their market valuation. In the absence of price bubbles/crashes, individuals who believe market prices are too high tend to have lower price expectations, whilst those who believe market prices are too low tend to have higher price expectations. However, such an error-correcting process need not hold at times of bubbles (or crashes) when individuals could believe the prices to be too high (low), and yet expect higher (lower) prices. 6 The 5 For details of ALP see http://www.rand.org/pubs/corporate pubs/CP508-2015-05.html. The survey questions have been designed jointly with Je Dominitz (Resolution Economics) and Charles Manski (North- western University). 6 This pattern of expectations formation is in line with theories of speculative behavior and bubbles and crashes, which argue that rational traders understand that market prices might be over-valued, but continue to expect higher prices as they believe they can ride the bubble and exit just before the crash. See, for 93 theoretical relationship between expected future price changes and valuation allows us to dene bubbles and crash indicators by considering responses that contradict the predictions of the theory under rational expectations. We provide estimates of the relationship between expected price changes and a valuation indicator using an unbalanced panel of responses from the double-question surveys. We nd statistically signicant relationships between expected price changes (at one, three and twelve months ahead) and asset valuations (under or over) for all the three asset classes. But these relationships are error correcting (in the sense discussed above) for equity price expectations at longer horizons and for house price expectations at all three horizons being considered. Gold price expectations do not seem to be equilibrating. The eects of demographic factors, such as sex, age, education, ethnicity, and income are also investigated. It is shown that for house price expectations such demographic factors cease to be statistically signicant once we condition on the respondents' location and their asset valuation indicator. Finally, using the double-question survey responses we propose bubble and crash indica- tors for use as early warning signals of bubbles and crashes in the economy as a whole or in a particular region. There is also the issue of how to evaluate the usefulness of such indicators. One approach would be to investigate their contribution in modeling and forecasting real- ized price changes in a given region or nationally. A pure time series approach would require suciently long time series data and is not possible in the case of the present survey (which covers a very short time period). But it is possible to exploit the panel dimension of our data and see if crash and bubble indicators can signicantly contribute to the explanation of realized house price changes across dierent metropolitan statistical areas (MSAs). To this end we begin with a dynamic xed eects panel data model in monthly realized house price changes and then add expected house price changes and crash and bubble indicators at dierent horizons to see if such survey based indicators can help in cross-sectional expla- nation of realized house price changes. We employ dynamic panel data models with xed and time eects and include MSA-specic crash and bubble indicators together with similar indicators constructed for the neighboring MSAs. We nd such indicators to have signicant example, Abreu and Brunnermeier (2003). 94 explanatory power for realized house price changes over and above past price changes. All estimated coecients have the correct signs, predicting expected price changes to rise with bubble indicators and to fall with the crash indicators. The remainder of the paper is organized as follows: Section 2.2 sets out the theoretical as- set pricing model with heterogeneous agents and derives the relationship between individual expected price changes and their asset valuations at dierent horizons. Section 2.3 describes the survey design, provides summary statistics of survey responses, and presents some pre- liminary data analyses. Section 2.4 gives the panel regressions of respondents' expected price changes on their valuation indicator, and discusses the eects of location, socio-demographic characteristics and other factors on the expectations formation process. Section 2.5 intro- duces the bubble and crash leading indicators. Section 2.6 investigates the importance of such leading indicators for the analysis of realized house price changes across MSAs. Section 2.7 ends with some concluding remarks. The exact survey questions and the ltering rules used to clean the survey data for panel regression analyses are given in an Online Supple- ment. Additional results and descriptions are provided in the Online Supplement which is available from the authors on request. 2.2 Valuation and expected price changes In this section we consider a rational expectations model with heterogeneous agents and show that in a bubble-free equilibrium agents' expected future price changes and their valuations belief are consistent, in the sense that if they believe assets are over-valued then they expect prices to fall and vice versa. This theoretical result is new and will form the basis of our development of bubble and crash indicators in Section 2.5. The importance of heterogeneity for speculative behavior and over-valuation has been emphasized by Miller (1977). Miller was the rst to show that in markets with heteroge- neous agents and short-sales constraints, security prices are likely to be over-valued, since short-sales restrictions deter the pessimists from trading without a commensurate eect on the optimists. The quantitative importance of this eect is investigated by Chen et al. (2002). 95 Miller's result is obtained in a static framework, but similar outcomes are also obtained in a dynamic setting. Harrison and Kreps (1978) show that in the presence of short-sales restric- tions, and when agents dier in their beliefs about the probability distributions of dividend streams, then over-valuation can arise since agents believe that in the future they will nd a buyer willing to pay more than their asset's current worth. In a related paper, Scheinkman and Xiong (2003) argue that such speculative behavior can generate important bubble com- ponents even for small dierences in beliefs. As noted earlier Allen et al. (2006), Bacchetta and Van Wincoop (2006), and Bacchetta and Van Wincoop (2008) have also emphasized the importance of high-order beliefs for under- and over-valuation of asset prices. In particular, Bacchetta and Van Wincoop (2008) investigate the impact of higher-order expectations on the equilibrium price and establish the existence of a gap between the equilibrium price and the average expectations of the fundamentals, which they refer to as the "higher order wedge". They show that such a non-zero wedge is compatible with rationality and arises purely due to persistent heterogeneity across agents. These and other theoretical models of asset price over-valuation in the literature provide important insights into interactions of trader heterogeneity and other market features such as short-sales constraints. However, they are silent on the way over-valuation (or under- valuation) can aect price expectations. In what follows, building on the contributions of Allen et al. (2006), and Bacchetta and Van Wincoop (2008) we consider a multi-period asset pricing model with heterogeneous traders, and show that the model has a unique bubble- free solution when traders are anonymous and individual traders base their expectations of others only on publicly available information. Our model solution strategy diers from the one adopted in the literature on higher-order beliefs and does not aim to provide an explicit solution for the equilibrium asset price. Instead we make use anonamity of traders in the network to derive an explicit relationship between expected price changes and a valuation indicator. Specically, we show that individual traders' expected price changes are related to their asset valuation, as measured by the gap between market prices and traders' own valuation. This relationship is shown to be error correcting in expectations formation, with 96 traders who believe the market to be over-valued (under-valued) expecting prices to fall (rise). This result holds for expectations formed for longer horizons, with the weight attached to the asset valuation variable declining with the horizon. By implication, it also follows that the error correcting mechanism could become perverse if cross-agent expectations are likely to lead to indeterminate outcomes, possibly resulting in the build-up of forces for bubbles or crashes. In such situations, it is possible for traders to believe the market is over-valued (under-valued), and yet continue to expect prices to rise (fall). More formally, suppose there aren traders withn suciently large. Let it = it [ t ;i = 1; 2;:::;n, denote traderi th information set composed of his/her private information, it , and the public information t that contains at least current and past prices. Each trader decides on how many units, q it , of a particular asset to hold by maximizing E i [U i (W t+1;i )j it ], where U i (W t+1;i ) represents the constant absolute risk aversion utility function with i as the absolute risk aversion coecient of the i th trader, and E i (j it ) is the expectations operator for trader i conditional on his/her information set, it . Under this set up and assuming normally distributed asset returns and no transaction costs, it is easily established that asset demand for trader i is given by P t q d it = E i (R t+1 j it )r t i Var i (R t+1 j it ) ; where R t+1 = (P t+1 P t +D t+1 )=P t , is the rate of return on holding the asset over the period t to t + 1, P t is the asset price at t, D t+1 is the dividend paid on holding the asset over periodt tot + 1;r t is the risk free rate of return, andVar i (R t+1 j it ) is thei th trader's conditional variance of asset returns. Assuming no new shares are issued, the market clearing condition is given by P n i=1 q d it = 0; and we have 7 P t = 1 1 +r t " n X i=1 w it E i (P t+1 j it ) + n X i=1 w it E i (D t+1 j it ) # ; (2.2.1) where w it = [ i Var i (R t+1 j it )] 1 = P n j=1 [ j Var j (R t+1 j jt )] 1 . This is a generalization of 7 This assumption can be relaxed and replaced by P n i=1 q d it = Q, where Q is the net addition to the supply of shares. In this case, our results hold if it is assumed that Q=n! 0 as n!1. 97 the standard asset pricing model and allows for the possible eects of information hetero- geneity across traders on the determination of asset prices. 8 The weights w it satisfy the adding up condition, P N i=1 w it = 1, and capture the relative importance of the traders in the market. When information and priori beliefs are the same across traders,E i (P t+1 j it ) =E (P t+1 j t ) and E i (D t+1 j it ) =E (D t+1 j t ), and the price equation reduces to P t = 1 1 +r t [E (P t+1 j t ) +E (D t+1 j t )]; with homogeneous expected price changes given by e i;t+h =E ( t+h j t ) =r t E D t+1 P t j t ; for all i, where t+h = (P t+h P t )=hP t . However, in the presence of information heterogeneity the solution will be subject to the "innite regress" problem. 9 Each trader needs to form ex- pectations of other traders' price and dividend expectations for all future dates, which is a multi-period version of Keynes' well known beauty contest. In general, the solution is inde- terminate even if we impose transversality conditions on all traders, individually. There are many possible solutions. In what follows we consider a set of simplifying assumptions that allow for heterogeneity but lead to a unique bubble-free market solution. In this way we are able to model the cross section heterogeneity of expectations in an equilibrium context, so that bubble and crash states can be dened as deviations from the equilibrium benchmark. Specically, we make the following assumptions: Assumption 12 (Risk free rate) Risk free rate, r t , is time-invariant, namely r t = r, Var (R t+1 j it ) = 2 i for all t, and 0 < c < i 2 i < C <1, for some strictly nite pos- itive constants, c<C. 8 See also Eq. (3) in Bacchetta and Van Wincoop (2008), and note that we allow for the eects of individual risk premia in the weights, whilst in Bacchetta and Van Wincoop (2008) average price and dividend expectations and risk premia are shown separately. 9 For an early discussion of the innite regress problem see Phelps (1983), Townsend (1983) and Pesaran (1987) Ch. 4. 98 Assumption 13 (Network anonymity) The traders i = 1; 2;:::;n belong to an anonymous network and each trader i th expectations of other traders' price expectations are given by E i [E j ( t+h j j;t+h1 )j it ] =E i ( t+h j it ) + (h) it ; (2.2.2) for all i and j = 1; 2;:::;n, and h = 1; 2;:::, where (h) it is the idiosyncratic part of trader i th expectations of trader j th price change expectations at horizon h, and satisfy the following E i (h) jt j it = (h) it , for j =i (2.2.3) = 0, for j6=i. Assumption 14 (Dividend processes) Traders commonly believe that the dividend process, fD t g, follows a geometric random walk, but dier in their beliefs about the drift and volatility of the dividend process. Specically, trader i th dividend process is given by model M i M i :D t =D t1 exp( i + i " t ); for i = 1; 2;:::;n; (2.2.4) where " t is i:i:d:N(0; 1). The true dividend process is given by DGP :D t =D t1 exp( +" t ); (2.2.5) Remark 1 Conditional expectations taken under model M i and under the DGP will be de- noted by E i (j) and E (j), respectively. Assumption 15 (Market pooling condition) Market expectations of individual traders' price expectations are given by E [E i (P t+1 j t )j t ] =E (P t+1 j t ); (2.2.6) the transversality condition lim H!1 (1 +r) H E (P t+H j t ) = 0 holds, and exp(g) < 1 +r, where g = + (1=2) 2 ; with and 2 dened by (2.2.5). 99 Remark 2 Assumption 15 ensures the existence of a representative agent model associated with the underlying multi-agent set up. To allow for market pooling of traders' disparate beliefs regarding the dividend growth process, we introduce the following assumption: Assumption 16 (Distribution of trader disparities) Trader-specic belief regarding his/her steady state growth rate of dividends, g i , dened by (2.2.8), are distributed independently across i as N(g;! 2 g ). Under Assumption 12 the price equation (2.2.1) simplies to P t = 1 1 +r " n X s=1 w s E s (P t+1 j st ) + n X s=1 w s E s (D t+1 j st ) # : Also, under Assumption 14 it is easily seen that E s (D t+h j st ) =D t exp(hg s ), (2.2.7) where g s = s + (1=2) 2 s : (2.2.8) Hence P t = 1 1 +r n X s=1 w s E s (P t+1 j st ) + n 1 +r D t ; (2.2.9) where n = n X s=1 w s exp(g s ): (2.2.10) Now suppose that the asset pricing equation (2.2.9) is common knowledge, and is therefore used by all traders to form their price expectations and asset price valuations. In cases where expectations are homogeneous across all traders or when dierences in expectations are common knowledge then applying the conditional expectations operator for thei th trader, E i (j it ) to both sides of (2.2.9) will yield the same result, namely P t . However, this is not the case in the more realistic scenario where dierences in expectations are not common 100 knowledge. Clearly, for the left hand side of (2.2.9) we have E i (P t j it ) = P t since P t is included in it . But application of E i (j it ) to the right hand side of (2.2.9) need not be equal to P t since exact expressions for terms such as E i [E s (P t+1 j st )j it ] are not known to traderi, and he/she has no choice but to use some form of an approximation, such as the one proposed in Assumption 13. Accordingly, we dene thei th trader's asset valuation at timet,P it , by applyingE i (j it ) to the right hand side of (2.2.9), namely P it = 1 1 +r n X s=1 w s E i [E s (P t+1 j st )j it ] + E i ( n ) 1 +r D t : Now under Assumption 13, and using the conditionE i [E s (P t+1 j st )j it ] =E i (P t+1 j it ) + (1) it P t ; we have P it = 1 1 +r h E i (P t+1 j it ) + (1) it P t i + E i ( n ) 1 +r D t : (2.2.11) Subtracting P t from both sides of (2.2.11) and after some re-arrangements we obtain E i (P t+1 j it )P t P t =(1 +r) P t P it P t + r E i ( n ) 1 +r D t P t (1) it ; which we write as 10 e i;t+1 =(1 +r)V it + r E i ( n ) 1 +r D t P t (1) it ; (2.2.12) where e i;t+1 =E i ( i;t+1 j it ) , and V it = P t P it P t : (2.2.13) Equation (2.2.12) relates the i th trader's expected rate of price change to his/her over- or under-valuation as dened by V it , which measures the degree to which trader i th asset valuation, P it , diers from the commonly observed prevail price, P t . In equilibrium the realized price dividend-ratio, P t =D t , is determined by taking expecta- tions of the asset pricing equation (2.2.9) conditional on the publicly available information, 10 Note that n is not known to trader i and E i ( n ) represents the i th trader's expectations of n . 101 t , across all traders. Specically, we have E (P t j t ) = P t = 1 1 +r n X i=1 w i E [E i (P t+1 j it )j t ] + E ( n ) 1 +r D t ; = 1 1 +r n X i=1 w i E [E i (P t+1 j t )j t ] + E ( n ) 1 +r D t : Further by Assumption 15 we have (recall that n i=1 w i = 1) P t = 1 1 +r E (P t+1 j t ) + E ( n ) 1 +r D t : This is a standard asset pricing model for a representative risk neutral agent with the divi- dend process given by (2.2.5). Under thr standard transversality condition applied to P t , it has the following unique solution: P t = E ( n ) 1 +r 1 X j=0 1 1 +r j E (D t+j j t ); which in view of (2.2.5) yields (recall that exp(g)< 1 +r ) P t =D t = E ( n ) 1 +re g = P n s=1 w s E [exp(g s )] 1 +re g : (2.2.14) Using this result in (2.2.12) now gives the following relationship between expectations and valuations e i;t+1 = i (1 +r)V it +u it ; (2.2.15) where e i;t+1 =E i ( t+1 j it ), V it = (P t P it )=P t , and i =r E i ( n ) (1 +re g ) E ( n ) , and u it = (1) it . (2.2.16) It is easily seen that in the homogeneous information case where, it = t , andg i =g; then we also have P it = P t , and E i ( n ) = E ( n )=D t , for all i. Furthermore, (2.2.15) reduces to e i;t+1 =e g 1, for all i. 102 The above solution also relates to the over-valuation results obtained in the literature. We rst note that the equilibrium price-dividend ratio under heterogeneous information, given by (2.2.14), tends to e g+0:5! 2 g = (1 +re g ), as n!1. 11 However, under homogeneity the equilibrium price-dividend ratio is given by e g D t = (1 +re g ) which is strictly less than the solution for the heterogenous case. This nding mirrors the over-valuation results due to Miller (1977) and Harrison and Kreps (1978), discussed above, but holds more generally even in the absence of short-sales constraints. The extent of over-valuation under heterogeneity depends on the degree of dispersion of opinion across traders about g i . Our result is also consistent with that the existence of the higher-order wedge identied by Bacchetta and van Wincoop (2008). In terms of our simplied set up the rst-order wedge is given by E (D t+1 j t ) P n i=1 w i E i (D t+1 j it ) = (e g n )D t , which tends to (1 e 0:5! 2 g )e g D t , as n ! 1. In this case the wedge is negative for ! 2 g > 0, which is consistent with asset over-valuation. Finally, the error-correction specication (2.2.15) can be generalized to price expectations for higher-order horizons. In general, for a nite h we have e i;t+h = (h) i (1 +r) h h V it +u (h) it ; (2.2.17) where e i;t+h = E i ( t+h j it ). Exact expressions for (h) i and u (h) it for h = 2 is given in Section S2 of the Online Supplement, and can be obtained similarly for a general h. But for the empirical analysis to follow, it is sucient to note that the asset valuation coecient, (1 +r) h =h; tends to fall with h for small values of r and so long as h is not too large. Empirically we model (h) i as individual xed eects and consider a general time series process for u (h) it . But rst we need to provide further details of the double-question surveys. 11 Recall that under Assumption 16, g i is IIDN(g;! 2 g ), with 1 +r>e g and ! 2 g > 0. 103 2.3 Double-question surveys To our knowledge the use of double-question surveys to elicit a respondent asset valuation along with her/his price expectations is new. Whilst there is a large and expanding literature on surveys of price expectations, there is no attempt at direct measurement of individual's subjective valuation of asset prices. We needed to carry out a fresh survey that simulta- neously included both questions on expectations and valuations. With this in mind and in collaboration with Je Dominitz and Charles Manski, we designed survey questions on expectations and valuations for US households, using RAND American Life Panel (ALP). 12 The ALP has a modular form, which allowed us to combine demographic, education and income data with the results from our double-question surveys. The double-question surveys on belief and expectations added to the ALP surveys covered equity, gold, and house prices. The two questions for equity prices were as follows: 13 12 We are particularly grateful to Arie Kapteyn for his generous support of this project. The sampling frame of ALP surveys, and other details can be found from the following link http://www.rand.org/pubs/ corporate_pubs/CP508-2016-04.html. 13 We also asked the respondents a third question regarding the chance of $1,000 investment to fall in three dierent ranges. Further details can be found in the Online Supplement. A similar set of questions was asked about gold prices. 104 Question 1 (equity) We have some questions about the price of publicly traded stocks. Do you believe the US stock market (as measured by S&P 500 index) to be currently: 1 Overvalued 2 Fairly valued (in the sense that the general level of stock prices is in line with what you personally regard to be fair) 3 Undervalued Note: The S&P 500 is an index of 500 common stocks actively traded in the United States. It provides one measure of the general level of stock prices. Question 2 (equity) Bearing in mind your response to the previous question, suppose now that today someone were to invest 1000 dollars in a mutual fund that tracks the movement of S&P 500 very closely. That is, this \index fund" invests in shares of the companies that comprise the S&P 500 Index. What do you expect the $1000 investment in the fund to be worth - in one month from now, - in three months from now, - in one year from now. For house prices respondents were also provided with the median price of a single family home in the area close to their place of residence. We used quarterly house prices disaggre- gated by 180 MSAs from the National Association of Realtors. 14 This turned out to be an important consideration given the large disparity of house prices across the US. Although, due to privacy considerations APL does not provide ZIP code information on respondents, we were able to match respondents to MSAs using their self-reported city and state of res- idence. Respondents who resided further than 500 miles away from a major metropolitan area were instead asked about the median US house price. The survey questions on house prices for respondents who resided closer than 500 miles away from a major metropolitan area are presented below. The exact wording of the survey questions can be found in the 14 All areas are metropolitan statistical areas (MSA) as dened by the US Oce of Management and Budget though in some areas an exact match is not possible from the available data. For further details see http://www.realtor.org/topics/existing-home-sales. 105 Online Supplement. Question 1 (house prices) We now have some questions about housing prices. The median price of a single family home in the [ll for city nearest to R zip code] cosmopolitan area is currently around [converted ll for median housing price in R zip code area] (Half of all single family homes in the area cost less than the median, and the other half cost more than the median.). Do you believe that current housing prices are: 1 just right (in the sense that housing prices are in line with what you personally regard to be fair), 2 too high, 3 too low as compared to the fair value? Question 2 (house prices) Bearing in mind your response to the previous question, suppose now that someone were to purchase a single family home in [ll for city nearest to R zip code] area for the price of [ . . . ] What do you expect the house to be worth (Please enter a numeric answer only, with no commas or punctuation) - 1 month from now, - 3 months from now, - 1 year from now. It is important to note that the survey design does not require that the notion of "fairly valued" to be commonly agreed on. What is important is the consistency in measurement of what a respondent considers an asset to be fairly valued and his/her expectations of future price change. Also, we do not ask respondents about percentage price changes but about future price itself, and we ask no direct questions on in ation expectations. 2.3.1 Survey waves and respondent characteristics The American Life Panel (ALP) consists of over 6,000 panel members aged 18 and older. Participants are recruited from various sources, such as the University of Michigan phone- panel and internet-panel and cohorts, mailing experiments, phone experiments and vulner- able population cohorts. The panel is representative of the nation, and panel members are 106 provided with equipment that allows them to respond any survey programmed by RAND. The attrition rate of ALP participants is relatively low, between 2006 and 2013 the annual attrition rates were between 6 and 13 per cent. Panel members who have answered a non- household information survey within the last year are considered active and are invited to surveys. Each survey, in addition to the specic survey questions, contains a \Demographics" module, which elicits demographic and socio-economic information about the respondent. The double-question (DQ) surveys were carried out over the period January 2012 to January 2013. But the rst two waves were dropped due to incomplete house price informa- tion provided to respondents residing more than 500 miles from major metropolitan areas. For the remaining survey waves (March 2012 to January 2013), we ended up with 5,480 respondents. ALP members were oered the opportunity to respond to our DQ surveys, but their participation was not made mandatory. Table 2.1 provides the number of ALP members who participated in the surveys and the fraction of those who completed the DQ surveys. The response rates were quite high and averaged around 72 per cent of the survey participants, and varied little across the 13 survey waves. This is a very high response rate as compared to other surveys of house prices conducted in the literature. For example, the average response rate of the home-buyers surveys conducted by Case and Shiller was around 22.7% over the years 1988, and 2003-2012. See Table 1 in Case et al. (2012). We found no signicant demographic dierences between the respondents and non-respondents of our DQ surveys. 2.3.2 Filters applied to survey responses To reduce the impact of extreme outlier responses on our analysis a number of lters were applied to the responses. We also dropped waves 1 and 2 since, as was noted above, in the case of these waves respondents residing more than 500 miles from major metropolitan areas were not provided with house price data. This shortcoming was rectied in the subsequent waves (3-11), by providing such respondents with US median house prices. For these remain- ing survey waves (March 2012 to January 2013), we ended up with 5,480 respondents. We 107 Table 2.1: Survey waves and response rates Waves Months All ALP Completed Filtered Samples participants DQ Surveys per cent (1) per cent (2) 1 January 2012 4477 3371 75 2707 80 2 February 2012 4864 3685 75 2727 74 3 March 2012 5015 3721 74 2991 80 4 April 2012 5260 3723 71 2967 80 5 May 2012 5464 3706 68 2982 80 6 June 2012 5568 4179 75 3379 81 7 July 2012 5674 4135 73 3363 81 8 August 2012 5713 4208 74 3445 82 9 September 2012 5762 4162 72 3425 82 10 October 2012 5772 4180 72 3421 82 11 November 2012 5847 3926 67 3169 81 12 December 2012 5894 4083 69 3404 83 13 January 2013 5911 4209 71 3415 81 The surveys were elded on the third Monday of the month (1) - Respondents who completed the DQ Surveys as a percentage of all ALP participants (2) - Filtered respondents as percentage of all respondents who completed the DQ Surveys applied the following truncation lters to the data. First, we dropped all respondents with missing responses to the survey questions or missing demographic characteristics. We also dropped respondents whose demographic characteristics were incomplete or contained incon- sistent entries over time. 15 Finally, for all expectations horizons (one month, three months and one year) and for all asset prices (equity, gold, housing) we remove respondents from our analysis if they report an expected price equal to zero for any of the survey questions, or report any expected price rises for equity or gold which are in excess of 400 per cent, or report expected price rises for equity or gold for all horizons in excess of 200 per cent, or report expected price falls of more than 90 per cent for all expectations horizons, or report expected house price rises in excess of 200 per cent, or if they report expected house price falls of more than 50 per cent for any expectation horizon. Around 20 per cent of the responses were ltered in any given survey wave, leaving us with 35,961 responses and 4,971 respondents. A comparison of the demographic characteristics of the ltered and unltered samples is provided in Table S1 in the Online Supplement and shows only minor dierences between the two. The frequency distribution of monthly 15 Detailed descriptions are provided in Section S8 of the Online Supplement. 108 participation of the respondents in the ltered sample is shown in Table 2.2. Just over a quarter of respondents (1,268) answered the DQ surveys for all the 11 waves (3 to 13), 50 per cent (2,453) answered 9 waves, suggesting a high degree of over-time participation of the respondents in the DQ surveys. Table 2.2: Empirical frequency distribution of participants by months Months 11 10 9 8 7 6 5 4 3 2 1 No. 1268 1933 2453 2779 3088 3331 3597 3860 4161 4520 4971 Per cent 25.51 38.89 49.35 55.90 62.12 67.01 72.36 77.65 83.71 90.93 100 The average and median number of months participated are 7:23 and 6, respectively. The distribution is based on respondents who remained in the sample after the truncation lter is applied. 2.3.3 Socio-demographic characteristics of respondents: For the purposes of the econometric analysis, we calculate respondent-specic time averages of the variables age, income and education. A summary of selected socio-demographic char- acteristics of the respondent sample is presented in Table 2.3. A detailed comparison of the Table 2.3: Summary statistics of respondent-specic time invariant characteris- tics Statistic Mean St. Dev. Min Median Max Age 47.80 15.50 16 49 94 Family income 1 ($) 52,470 36,627 5,000 45,000 200,000 Female (%) 0.59 0.49 0 1 1 Asian (%) 0.02 0.14 0 0 1 Black (%) 0.11 0.31 0 0 1 Hispanic/Latino (%) 0.19 0.39 0 0 1 Education Index 2 1.33 0.57 0 1 2 All statistics are based on the sample of 4,971 respondents. 1 - note that incomes higher than 200,000 were coded as equal to 200,000 2 - respondent's education averaged over the time period the respondent participated in the survey, where education is equal to 0 if the respondent has no high school diploma, 1 if the respondent is a high school graduate with a diploma, some college but no degree, an associate degree in college occupational/vocational or academic program, and 2 if the respondent has a Bachelor's degree or higher. socio-demographic characteristics of the respondents remaining in our sample and the US population are provided in the Online Supplement. The main dierences are as follows: Female respondents are over-represented at 59 per cent as compared to 51 per cent for the entire US population. 109 The age group 50 to 70 years old constitute a higher fraction of the ALP respondent sample compared to the US population. Roughly 2 per cent of the respondents identify as Asian or Pacic Islanders, the cor- responding number for the entire US population is 5.4 per cent. ALP respondents have a higher educational level than the US population. Households with an annual income higher than $125,000 are under-represented in the ALP respondent sample. 2.3.4 Geographic location of respondents Around 20 per cent of the respondents in any given survey wave resided further than 500 miles away from a major metropolitan area, and were thus given the median US house price instead of the local house price in the survey section on house prices. From the sample of 4,971 respondents, we could match exactly 4,000 to a Metropolitan Statistical Area. We achieved this using the information about the respondent's city and state of residence, provided in the survey. Information on the geographical distribution of the respondents as compared to the population density of the US are provided in the Online Supplement. Overall, we nd that the geographical distribution of the respondents over time is relatively stable and match closely the national distribution for the six out of the eight regions. The exceptions are South East and South West. Survey respondents are underrepresented in the South East region and over-represented in the South West region. Overall, the above comparative analysis suggests that the DQ sample of respondents are fairly typical of the US population and provide a reasonable mix of individuals with dierent demographic and location characteristics. Furthermore, to allow for unobserved characteristics of individual respondents (such as their optimistic or pessimistic disposition) we focus primarily on the xed eects estimates and report the full set of random eect estimates in the Online Supplement. 110 2.4 Price change expectations and valuation indicators We are now in a position to provide empirical evidence on the importance of individual asset valuations, V it , on expected prices changes, as set out in (2.2.17). Bearing in mind the survey questions, for equity and gold prices the expected rate of price change is dened by ^ e i;t+hjt = 100(P e i;t+hjt 1000)=(1000h), and for house prices it is computed as ^ e i;t+hjt = 100(P e i;t+hjt P 0 it )=(hP 0 it ), where P e i;t+hjt is the i th respondent's price expectation formed at time t for h months ahead, and P 0 it is the house price provided to the respondent i at time t. We assume that e i;t+hjt = ^ e i;t+hjt + i;t+h ; (2.4.1) where i;t+h is the error associated with the measurement of e i;t+hjt . Using responses to the rst question of the surveys we measuresign (V it ), byx it withx it = 1 if respondenti at time t believes the asset is over-valued (i:e: V it > 0), x it =1 if respondent i at time t believes the asset is under-valued (V it < 0), and x it = 0, otherwise. We then approximate V it by i x it , with i > 0, is a scalar constant. Using (2.4.1) in (2.2.17) and setting V it = i x it we obtain the following interactive xed-eects panel data model with individual eects, (h) i , heterogeneous slopes, (h) i =h 1 (1 +r) h i ^ e i;t+hjt = (h) i (h) i x it +u (h) it i;t+h : (2.4.2) Since the time dimension of the panel is short we can not identify the individual slope eects, (h) i . Instead we focus on estimation of the mean eect of x it on ^ e i;t+hjt by assuming the following random eects specication for i i = + i ; (2.4.3) 111 where i is assumed to be distributed independently ofx it and the composite erroru (h) it i;t+h . Substituting (2.4.3) in (2.4.2) we now obtain ^ e i;t+hjt = (h) i + (h) x it +" i;t+h ; (2.4.4) where (h) = (1 +r) h h , and " i;t+h =u (h) it (1 +r) h h i x it i;t+h : (2.4.5) Under the above assumptions x it and " i;t+h are uncorrelated, and (h) can be estimated consistently using xed eects estimation that allows for arbitrary correlations between the individual eects, (h) i , x it and the error term, " i;t+h . We also allow for common (economy- wide) eects on individual expectations by including a time eect in (2.4.4), which gives the following xed-eects, time-eects (FE-TE) panel regression ^ e i;t+hjt = (h) i + (h) t + (h) x it +" i;t+h : (2.4.6) This is a reasonably general framework that allows for random errors in measurement of expectations, random heterogeneity in the scale parameters i , and possible time eects. We also use robust standard errors for the FE-TE estimates of (h) , that allow for serial correlation in the errors, " i;t+h , and cross-sectional heteroskedasticity. We provide estimates of (h) for the three dierent asset classes, and for all the three horizons, h = 1; 3; and 12, separately. We use the full set of responses which yields an unbalanced panel and estimate (2.4.6) with and without time eects, allowing the individual eects, (h) i , to be correlated with " i;t+h (and hence with its components, i x it , u (h) it , and i;t+h ). We report FE and FE-TE estimates of (h) , together with standard errors robust to serially correlated and heteroskedastic errors in Table 2.1. 112 Table 2.1: Estimates of (h) in the panel regressions of individual ex- pected price changes on their belief valuation indicators for dierent assets (equation (2.4.6)) Equity Gold Housing Horizons FE FE-TE FE FE-TE FE FE-TE One Month -0.0991 -0.126 0.602*** 0.581*** -0.292*** -0.303*** Ahead (h = 1) (0.127) (0.128) (0.197) (0.198) (0.0643) (0.0642) Three Months -0.0905 -0.0995 0.222** 0.203* -0.106*** -0.109*** Ahead (h = 3) (0.0760) (0.0760) (0.108) (0.109) (0.0273) (0.0274) One Year -0.115*** -0.117*** -0.0226 -0.0316 -0.0481*** -0.0479*** Ahead (h = 12) (0.0365) (0.0364) (0.0488) (0.0489) (0.0102) (0.0102) Dependent variable: ^ e i;t+hjt . FE and FE-TE estimates are computed based on equation ^ e i;t+hjt = (h) i + (h) x it +u (h) it with an unbalanced panel of 4,971 respondents over 11 months, March 2012 to January 2013. N = 35; 961,T min = 1, T = 7:23,Tmax = 11 Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. Standard errors are robust to heteroskedastic- ity and residual serial correlation. The FE estimates of (h) for equity price expectations are statistically insignicant for h = 1 and 3, but become statistically signicant and negative for h = 12. These results are in line with our theoretical ndings and suggest that over the sample under consideration equity price expectations and belief valuations are consistently related. However, the same is not true of the results for gold prices, where (h) is estimated to be positive and statistically signicant for h = 1 and 3, and suggest that respondents might view gold prices to be over-valued and still expect gold prices to rise. Interestingly enough, even for gold prices (h) stops being statistically signicant for h = 12, suggesting the short term nature of the misalignment between expectations and valuations. By contrast, the estimates of (h) for house prices are much more coherent across h and are all negative and statistically highly signicant. Also, FE estimates of (h) for house prices fall withh, as predicted by the theory. Similar conclusions are obtained if the FE-TE estimates are considered. Although, the scaling parameter is not identied, an estimate of r, the discount rate 113 can be obtained using any two of the estimates ^ (h 1 ) and ^ (h 2 ) , so long asj ^ (h 1 ) j>j ^ (h 2 ) j. 16 For example, using the FE-TE estimates for one and three months ahead expectations, ^ (1) and ^ (3) , we obtain ^ r = 3:9%, which seems quite reasonable. Estimates of r based on other combinations of ^ (h 1 ) and ^ (h 2 ) yield similar but higher estimates of r. 17 Overall, the panel estimates support the predictions of the heterogeneous agent model developed in Section 2.2, and suggest a strong relationship between respondent's housing price expectations and their valuations which is shown to be equilibrating, at least over the period under consideration. The same cannot, however, be said about the gold price expectations. This could be due to the fact that respondents are likely to have more rst hand knowledge and experience about house prices as compared to international gold prices. The results for equity prices are ambiguous; there are no statistically signicant relationship between equity price expectations and valuations at one month and three months horizons, which is in line with the prediction of a representative agent model. Nevertheless, for one year horizons asset valuations seem to play a signicant role in respondent's price expectations formation process. 18 2.4.1 Eects of individual-specic characteristics on price expec- tations So far we have focused on the eects of valuations on price expectations, and by using inter- active xed eects panel data set up, we have shown our results to be robust to individual- specic heterogeneity. But it is also of interest to investigate possible eects of individual- specic characteristics of respondents on their price expectations. For example, Niu and Van Soest (2014) explore the relationship between house price expectations, local economic conditions, and individual household characteristics. Bover (2015) uses house price expecta- tions data from the Spanish Survey of Household Finances, and nds important dierences in 16 Specically, using (h) =h 1 (1 +r) h we have ^ r(h 1 ;h 2 ) = h1 h2 ^ (h 1 ) ^ (h 2 ) 1 h 1 h 2 1:: 17 See Table S1 in the Online Supplement for further details. 18 In the Online Supplement we also provide estimates of (h) across dierent sub-groups such as male and female, home-owners and renters, and nd that our main conclusion continues to hold. See Sections S14 and S19 of the Online Supplement. 114 expectations across gender and occupation. Kuchler and Zafar (2015) use data from Survey of Consumer Expectations and focus on how personal experiences aect expectations at the national level. They nd that experiencing a house price fall leads respondents to be more pessimistic about future US house prices. The above studies all point to important systematic dierences in price expectations across respondents. Similar disparities in expectations are also present in our surveys. Using the information in demographic modules of ALP, we considered the eects of sex, age, income, ethnicity and education on price expectations. Given the time-invariant nature of the demographic variables, there are two ways that this can be done. One possibility would be to augment the panel regressions in (2.4.6) with the observed individual-specic eects, and then treat (h) i as random eects, distributed independently of x it . Setting (h) i = (h) + z 0 i (h) + (h) i , where z i is the vector of time-invariant observed characteristics of the i th respondent, (h) i is the unobserved random component of (h) i assumed to be distributed independently of z i andx it . The associated random eects panel data model can now be written as ^ e i;t+hjt = (h) + (h) t + z 0 i (h) + (h) x it +" i;t+h + (h) i : (2.4.7) We consider model (2.4.7) both with and without time eects (h) t . For the elements of z i = (z i1 ;z i2 ;:::;z i7 ) 0 , we consider z i1 = 1 if the respondent identies as female, and 0 otherwise, z i2 = lnage i , z i3 measures the education level of respondent i, z i4 = ln income i , and z i5 to z i7 are dummy variables that take the value of 1 if the respondent identies her/himself as Asian, Black and Hispanic/Latino, respectively. For a detailed description of how the time-invariant variables are constructed see the Online Supplement. We allow " i;t+h + (h) i to be serially correlated and heteroskedastic. An alternative approach, that does not require (h) i and x it to be independently dis- tributed, is to employ the two-stage approach proposed recently in Pesaran and Zhou (2016), whereby in the rst stage FE (or FE-TE) estimates of (h) are used to lter out the eects ofx it , and in the second stage a pure cross section regression of ^ u i is run on an intercept and 115 z i , for i = 1; 2;:::;N, where ^ u i = P T t=1 s it ^ e i;t+hjt ^ (h) FETE x it P T t=1 s it ; ands it is an indicator variable which takes the value of 1 if respondent i is included in wave t of the survey and 0 otherwise. This estimator is referred to as the FE ltered estimator and denoted by ^ (h) FEF (or ^ (h) FEFTE ). Pesaran and Zhou (2016) provide standard errors for ^ (h) FEF that allow for the sampling uncertainty of ^ (h) FE (or ^ (h) FETE ), and possible error heteroskedasticity. The FE ltered and RE estimates of (h) and their robust standard errors are summarized for equity, gold and house price expectations in the Online Supplement in Tables S1, S2 and S3, respectively. For completeness we also report the estimates of (h) , although, as noted earlier, the RE estimates are not robust to possible correlations between i and x it . The FE estimates of (h) in Tables S1-S3 are the same as those already reported in Table 2.1. Inclusion of time dummies had little impact on the RE or FE estimates (the FE-TE estimates are reported in the Online Supplement). But we nd it matters a great deal, particularly to the regressions for house price expectations, if we did include a location (MSA) dummy in the regressions. As noted earlier, we have been able to identify the MSA within which a respondent resides from the demographic module of the survey and the house price information that was provided to the respondents. This additional information (often absent in other survey expectations) allows us to separate the location-specic nature of house price changes from respondent-specic characteristics. Comparing RE and FE estimates of (h) we note that they are generally quite close, although the RE estimates tend to be larger in absolute magnitude, and more statistically signicant. Judging by the implied estimates ofr, and the fact that FE estimates are robust to possible correlations betweenx it and i , the FE estimates are clearly to be preferred. But it is worth noting that our main conclusion that the valuation indicator plays a signicant role in price expectations formation holds irrespective of whether RE or FE estimates are 116 used. Also, RE estimates of (h) are robust to the inclusion of location dummies. 19 Regarding the eects of individual-specic characteristics on price expectations, we nd important dierences across assets. For equity prices sex, age and education are statistically signicant at all three horizons and irrespective of whether RE or FE ltered estimates are considered. Ethnicity also features signicantly for 3 and 12 months horizons. Females tend to have higher equity price expectations, whilst older respondents, and those with a higher level of income, tend to have lower equity price expectations. But it is interesting that the estimates and their statistical signicance are hardly aected by the inclusion of location and/or time dummies (the latter results reported in the Online Supplement). Similar results are obtained for gold price expectations where in addition to sex, age, income and ethnicity, education is also statistically signicant, with higher educated respondents having lower price expectations of gold prices. The picture is very dierent when we consider regressions for house price expectations (in Table S3). Generally speaking, the respondent-specic characteristics are not as signicant as compared to the equity and gold price regressions, and the test outcomes critically depend on the estimator and whether the regressions include location dummies. Using the preferred FE ltered estimates and considering the regressions with MSA dummies, we nd that only income is statistically signicant (with a positive sign) in the case of regressions for one month ahead, and ethnicity for the one year expectations. The heterogeneity of house price expectations across respondents seem to be largely explained by the location dummy once we condition on the valuation indicator, and all other respondent-specic characteristics loose their statistical signicance. 20 19 Note that the FE estimates are unaected by respondent-specic characteristics, including their location. 20 A similar result is also reported in Bover (2015) who shows that most of the observed heterogeneity in house price expectations can be explained by a location dummy at the postal code level. 117 2.5 Constructing leading indicators of bubbles and crashes from DQ surveys The equilibrium relation between expected price changes and the valuation indicator in (2.2.17) can also be used to construct time series indicators of bubbles and crashes at the level of individual respondents, that can then be aggregated at regional or national levels. Such indicators are likely to provide valuable information about the possibility of bubbles or crashes building up, and could prove useful as predictors of realized price changes. In what follows we suggest such indicators. We begin with respondent-specic indicators and for each horizon h consider individual i th responses to the DQ surveys that contradict the theoretical relations between ^ e i;t+hjt and x it , namely when respondent's valuation belief and price change expectations do not match the pattern predicted by (2.2.17), which is derived assuming an equilibrating mechanism. Accordingly, we dene the bubble indicator for respondent i at time t for h periods ahead by B i;t+hjt = I[(x it > 0)\ (^ e i;t+hjt 0)], and the crash indicator by C i;t+hjt = I[(x it < 0)\ (^ e i;t+hjt 0)]. Specically, a respondent is said to be in a bubble (crash) state if he/she believes the asset under consideration is overvalued (undervalued) but at the same time expects prices to rise (fall) or stay the same. Therefore, B i;t+hjt = 1 (or C i;t+hjt = 1) if respondent i is in bubble (crash) state and 0 otherwise. The proportion of respondents with non-zero bubble and crash indicators are summarized in Table 2.1. The results are summarized for all respondents and by gender. The proportion of respondents in bubble and crash states are relatively small for equity and house prices, but not for gold. The proportion of respondents who believe gold prices are over-valued and nevertheless expect gold prices to rise over the next month is around 47 per cent, as compared to 24 per cent for equity prices and 16 per cent for house prices. In all cases the proportion of respondents in bubble state falls with horizon, and beliefs and expectations are more likely to be aligned with our theoretical prediction when expectations are considered over longer horizons. These results are in line with the regression estimates reported in Table 118 2.1, where we nd positive and statistically signicant estimates of (h) only for gold prices and only at one month and three months horizons. Finally, the proportion of respondents in bubble and crash states do not dier much by gender, which is interesting considering the statistically signicant gender eect observed on expectations in the case of equity and gold prices. 21 Table 2.1: Respondents in bubble and crash states by gender (a) Equity One Month Three Months One Year Total Female Male Total Female Male Total Female Male Bubble 8700 4804 3896 8084 4542 3542 7949 4519 3430 (%) 24.19 23.32 25.37 22.48 22.05 23.06 22.10 21.93 22.33 Crash 3549 2422 1127 2168 1523 645 1177 836 341 (%) 9.87 11.76 7.34 6.03 7.39 4.20 3.27 4.06 2.22 Neither 23712 13376 10336 25709 14537 11172 26835 15247 11588 (%) 65.94 64.93 67.30 71.49 70.56 72.74 74.62 74.01 75.45 (b) Gold One Month Three Months One Year Total Female Male Total Female Male Total Female Male Bubble 16891 9561 7330 15437 8884 6553 13971 8224 5747 (%) 46.97 46.41 47.72 42.93 43.12 42.67 38.85 39.92 37.42 Crash 1116 799 317 699 533 166 473 369 104 (%) 3.10 3.88 2.06 1.94 2.59 1.08 1.32 1.79 0.68 Neither 17954 10242 7712 19825 11185 8640 21517 12009 9508 (%) 49.93 49.71 50.21 55.13 54.29 56.25 59.83 58.29 61.91 (c) Housing One Month Three Months One Year Total Female Male Total Female Male Total Female Male Bubble 5720 3370 2350 5147 3037 2110 5189 3077 2112 (%) 15.91 16.36 15.30 14.31 14.74 13.74 14.43 14.94 13.75 Crash 6322 3954 2368 4861 3053 1808 3000 1896 1104 (%) 17.58 19.19 15.42 13.52 14.82 11.77 8.34 9.20 7.19 Neither 23919 13278 10641 25953 14512 11441 27772 15629 12143 (%) 66.51 64.45 69.28 72.17 70.44 74.49 77.23 75.86 79.06 The statistics are calculated using a sample of 35,961 responses, with 15,359 male and 20,602 female responses. Male and female responses represent 43% and 57% of the sample, respec- tively. The percentages in the table are column percentages and sum to 100 % for each column. The time proles of bubble and crash indicators can be aggregated across respondents 21 Females tend to have higher price expectations as compared to male respondents. See the estimates reported in Section S17 of the Online Supplement. 119 and related to realized price changes. But since the survey results are available only over a very short time period, a time series evaluation of the usefulness of such indicators is not possible. Instead we consider a related question of whether spatially disaggregated bubble and crash indicators can help explain the cross-section variations of realized house price changes across ve US regions, and more formally across 48 Metropolitan Statistical Areas (MSAs). We begin by illustrating the evolution of the bubble and crash indicators along with realized house price changes across the US mainland regions Northeast, Southeast, Midwest, Southwest and West, as dened by the National Geographic Society. 22 Region-specic bubble and crash indicators are dened by simple averages of the individual responses averaged over the respondents that reside in region r, namely B r;t+hjt = P i2rt B i;t+hjt # rt ; C r;t+hjr = P i2rt C i;t+hjt # rt (2.5.1) where rt denotes the set of respondents in region r at time t. The regional bubble and crash indicators can then be related to realized house prices changes in these regions. In what follows we rst show how the balance of these regional indicators lagged three months, dened by BC r;t+h3jt3 = B r;t+h3jt3 C r;t+h3jt3 , can be viewed as leading indicators of future realized house price changes, rt . For illustrative purposes we also average the balance statistics over the horizons h = 1; 3 and 12, and focus on the relationship between BC r;t3 = (1=3) P h=1;3;12 B r;t+h3jt3 C r;t+h3jt3 and realized house price changes rt for the US as a whole and the ve regions. Figure 2.1 shows the plots of BC r;t3 and rt over the 11 months from July 2012 to May 2013 for the US as a whole and the ve regions. As can be seen the balance statistics, BC r;t3 , track reasonably well the evolution of house price changes three months ahead for all ve regions. 22 https://www.nationalgeographic.org/maps/united-states-regions/ See Section S10 in the On- line Supplement for an exact specication of the regions. 120 Figure 2.1: Realized house price changes and three months lagged values of balanced bubble-crash indicators by regions 121 2.6 Bubble and crash indicators and realized house price changes across MSAs Given the promising graphical results in the previous section, we develop a dynamic panel data model of realized house price changes and bubble and crash indicators across 48 MSAs. Specically, we dene thee bubble and crash indicators for MSA s at time t for h periods ahead as B s;t+hjt = P it2st B i;t+hjt # st ; and C s;t+hjt = P it2st C i;t+hjt # st : where st denotes the set of respondents in MSAs at timet. For each MSAs, we also dene bubble and crash indicators of neighboring areas as follows. Let W =fw ss 0g s;s 0 =1;2;:::;N denote an NN matrix with w ss 0 = 1 if MSAs s and s 0 lie in neighboring areas, and w ss 0 = 0; otherwise. w ss 0 is determined based on the Haversine distance between the geographic centers of MSAs s and s 0 . See Section S11 in the Online Supplement for further details. The neighboring area bubble and crash indicators for MSA s in month t are dened by B s;t+hjt = P N s 0 =1 w ss 0B s;t+hjt P N s 0 =1 w ss 0 ; and C s;t+hjt = P N s 0 =1 w ss 0C s;t+hjt P N s 0 =1 w ss 0 : We now consider the statistical signicance of the above indicators for explanation of realized house price changes across the 48 MSAs over the 11 survey waves. As a bench mark model we consider the following standard dynamic panel regression model for expectation horizons h = 1; 3; 12 months. M 1 : s;t+1 = (h) s + (h) 0 st + (h) 1 ^ e s;t+hjt +u s;t+1;h ; for h = 1; 3; 12; (2.6.1) where s;t+1 = 300 [ln(P s;t+1 ) ln(P st )] is the one month ahead realized house price change in MSAs (expressed in per cent per quarter), and ^ e s;t+hjt is the expected house price change formed in month t for h months ahead, and averaged across the respondents in MSA s. 122 Specically ^ e s;t+hjt = P it2st ^ e i;t+hjt # st : Given the importance of location in the formation of house price expectations discussed above, we also allow for MSA-specic xed eects, (h) s , in the benchmark model. We then augment the benchmark model (2.6.1), with the MSA-specic bubble and crash indicators. We consider the following specication M 2 : s;t+1 = (h) s + (h) 0 st + (h) 1 ^ e s;t+hjt + (h) 1 B s;t+hjt + (h) 2 C s;t+hjt (2.6.2) + (h) 1 B s;t+hjt + (h) 2 C s;t+hjt +u s;t+1;h : To isolate the importance of the bubble and crash indicators from the price expectations we also estimate (2.6.2), without the expectations variable, ^ e s;t+hjt , which we denote as model M 3 . All three specications are estimated using a balanced panel of observations overN = 48 MSAs, and T = 9 months, namely for s = 1; 2;:::; 48, and t = June 2012 - February 2013. First-dierencing is applied to eliminate the MSA-specic eects. Note that standard FE estimation of dynamic panel regressions will not be appropriate since T is small relative to N, and FE estimates can lead to signicant bias due to the presence of the lagged dependent variable in the panel regressions. After rst-dierencing we estimate the parameters by the two-step Generalized Method of Moments (GMM) method due to Arellano and Bond (1991), using the following moment conditions: 23 E (u s;t+1;h z s;j ) = 0, for j =t 2;t 1;t = 5(June 2012); 6;:::; 13(February 2013); (2.6.3) where we set z s;j = s;j ; ^ e s;j+hjj 0 , for the baseline model M 1 , z s;j = s;j ; ^ e s;j+hjj ;B s;j+hjj ;C s;j+hjj ;B s;j+hjj ;C s;j+hjj 0 , for model M 2 ; 23 Note that we do not use all available moment conditions suggested by Arellano and Bond (1991), to avoid the weak instrument problem. 123 and z s;j = s;j ;B s;j+hjj ;C s;j+hjj ;B s;j+hjj ;C s;j+hjj 0 , for model M 3 : The estimation results are summarized in Table 2.1. Note that we are primarily interested in the explanatory power of house price in ation expectations, ^ e s;t+hjt , and the crash and bubble indicators B s;t+hjt , C s;t+hjt , B s;t+hjt , and C s;t+hjt . The lagged value of realized house price changes, st , is included in the analysis to take account of the high degree of known persistence in realized price changes. Consider rst the estimates for the baseline model, M 1 . As expected, (h) 0 which measure the degree of persistence in the rate of house price changes, is estimated to be quite high and lies in the range 0:70 0:80, and is statistically signicant at all horizons. The coecient of house price expectations formed at t, (h) 1 , is also statistically signicant but its magnitude is disappointingly low, and in fact becomes negative forh = 12. In contrast, the bubble and crash indicators, included in modelM 2 , are statistically signicant and have the correct signs for all horizons,h = 1; 3; and 12. Forh = 1, the panel regressions predict that MSAs with a higher bubble indicator tend to experience a higher degree of house price changes, and MSAs with a higher crash indicator tend to experience a lower degree of house price changes. 24 It is also most interesting that similar eects are observed from spillover bubble and crash indicators, in the sense that MSAs that are surrounded by neighboring MSAs with a high (low) value of the bubble (crash) indicator also tend to show a higher (lower) degree of house price changes. The eects of changes in bubble and crash indicators on future house price changes get accentuated due to the fact that in general the bubble and crash indicators move in opposite directions. Finally, these results continue to hold even if the price expectations variable is dropped from the analysis. See the estimates under columns M 2 and M 3 in Table 2.1 The estimates clearly show that bubble and crash indicators and the associated neigh- boring indicators play an important role in future movements of realized house price changes across MSAs. For example, the estimates of model M 2 for the one month expectation hori- 24 It is also interesting to note that estimated coecients of crash indicators tend to be larger than those of the bubble indicators. But this could partly re ect the fact that over the survey period the proportion of respondents in the crash state is generally smaller than the proportion of respondents in the bubble state. 124 Table 2.1: Dynamic panel regressions of realized house prices by MSAs (Across 48 MSAs and months June 2012 to February 2013) One Month (h = 1) Three Months (h = 3) One Year (h = 12) M 1 M 2 M 3 M 1 M 2 M 3 M 1 M 2 M 3 st 0.712*** 0.765*** 0.771*** 0.704*** 0.736*** 0.741*** 0.721*** 0.792*** 0.798*** (0.00872) (0.00555) (0.00564) (0.00772) (0.00732) (0.00346) (0.00528) (0.00521) (0.00675) ^ e s;t+hjt 0.0159*** -0.0118** 0.0513*** -0.0115 -0.0924*** -0.247*** (0.00231) (0.00521) (0.00697) (0.0123) (0.0217) (0.0490) B s;t+hjt 2.018*** 1.669*** 2.921*** 2.841*** 1.825 2.174*** (0.637) (0.504) (1.020) (0.971) (1.158) (0.663) C s;t+hjt -8.623*** -8.836*** -8.395*** -8.638*** -14.36*** -13.02*** (0.736) (0.680) (0.622) (0.593) (1.659) (1.583) B s;t+hjt 3.529*** 3.742*** 8.410*** 8.401*** 3.452*** 3.564*** (0.650) (0.874) (0.991) (0.927) (0.543) (0.696) C s;t+hjt -11.84*** -11.99*** -9.669*** -10.04*** -16.83*** -18.84*** (0.874) (0.656) (1.245) (1.198) (1.470) (2.270) Dependent variable: s;t+1 (in per cent per quarter). The panel regression is estimated using a two-step GMM estimator (Arellano and Bond (1991)) using the moment conditions specied in Section S5 with heteroskedasticity- robust standard errors. Observations from the rst two survey waves April to May 2012 are used to initialize moment conditions. The estimates are based on a balanced panel with N = 48 and T = 9. Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels. zon imply that an increase in the bubble indicator from 0:2 to 0:5 leads to a 0:87 percentage point increase in the quarterly growth rate of house prices. A rise in crash indicators has the opposite eect and depresses future house prices. Finally, the explanatory value of bubble and crash indicators seems to be robust to averaging the indicators across the three horizons and/or introducing a longer lag between when the indicators are observed and the target date of house price changes. Table 2.2 provides estimates based on the following dynamic panel regressions M 4 : s;t+1 = (h) s + (h) 0 st + (h) 1 ^ e st + (h) 1 B s;t2 + (h) 2 C s;t2 (2.6.4) + (h) 1 B s;t2 + (h) 2 C s;t2 +u s;t+1;h ; where ^ e st = 1 3 (^ e s;t+1jt + ^ e s;t+3jt + ^ e s;t+12jt ); B st = 1 3 (B s;t+1jt + B s;t+3jt + B s;t+12jt ), C st = 1 3 (C s;t+1jt +C s;t+3jt +C s;t+12jt ); and so on. 25 The results are in fact stronger and more robust as compared to those reported in Table 2.1. The coecients of the average indicator variables are all statistically signicant with the a priori expected signs. Most importantly, lagging the indicators by two months has not reduced their explanatory power for future changes in house prices across MSAs. 25 See Section S5 in the Online Supplement for further details. 125 Table 2.2: Dynamic panel regressions of realized house prices by MSAs (Across 48 MSAs and months August 2012 to February 2013) st 0.765*** 0.923*** 0.913*** (0.0141) (0.0168) (0.0124) ^ e st 0.0318*** 0.0904*** (0.00723) (0.00664) B s;t2 4.088*** 4.071*** (1.239) (0.527) C s;t2 -11.51*** -11.36*** (1.128) (0.864) B s;t2 10.64*** 11.73*** (1.146) (0.578) C s;t2 -9.897*** -10.54*** (1.425) (1.138) Dependent variable: s;t+1 (in per cent per quarter). See notes to Table 2.1 and Section S5 in the Online Supplement. 2.7 Concluding remarks In this paper we have introduced a new type of survey which combines standard surveys of price expectations with questions regarding the respondents' subjective belief about asset values. Using a theoretical asset pricing model with heterogenous agents we show that there exists a negative relationship between the agents expectations of price changes and their asset valuation, a relationship that holds under dierent horizons. DQ surveys provide evidence in support of such relationships, particularly for house prices for which survey respondents are more likely to have a rst-hand knowledge as compared to other assets such as equities or gold prices which might not be of concern to many respondents in the survey. We also investigate the eects of demographic factors, such as sex, age, education, ethnicity, and income on price expectations, and nd important dierences in price expectations. But, interestingly enough, for house price expectations demographic factors stop being statistically signicant once we condition on the respondent's location and his/her valuation indicator. 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Section S3 gives the mathematical details of the FE-TE estimators and their standard errors, and Section S4 generalizes the FE-TE ltered esti- mators of the time-invariant variables proposed in Pesaran and Zhou (2016) to unbalanced panels. Section S5 describes the GMM estimators used for the dynamic panel regressions of realized house price changes across MSAs reported in Section 2.6 of the paper. Section S6 provides further details of the RAND American Life Panel (ALP) surveys discussed in Sec- tion 2.3 of the paper. Section S7 provides the survey questions, Section S8 gives the details of the truncation lters applied to the responses. Section S9 compares the socio-demographic characteristics and geographic location of the survey respondents and the US population. Section S10 denes US mainland regions referred to in Section 2.5 of the paper, and Section S11 describes the spatial weight matrix used in the construction of neighboring crash and bubble indicators used in the regressions. Section S12 contains a brief description of Data Sources as well as the les that replicate the results reported in paper and this supplement. Section S13 provides summary statistics for selected MSA level variables. Section S14 pro- vides estimates of the price expectation-valuation panel regressions, estimated separately for male and female respondents. Section S15 provides the random eect estimates of the model specications discussed in Section 2.4 of the paper, and Section S16 provides a comparison of FE and RE estimates. Section S17 gives the FE-TE ltered estimates. Section S18 provides 131 a comparison of the estimates of (h) (dened in Section 2.4 of the paper) obtained for dif- ferent model specications, as well as the corresponding interest rate estimates. Section S19 reports panel regression results including home-ownership dummies, obtained by matching the DQ Surveys with the \Eects of the Financial Crisis" survey, also carried out by RAND. By matching the two surveys we are able to control for the eects of home-ownership on expectations formation. S2 Relationship between expected price changes and the valuation indicator for higher order horizons Advancing both sides of equation (2.2.9) in the paper one period ahead we rst note that; P t+1 = 1 1 +r n X s=1 w s E s (P t+2 j s;t+1 ) + n 1 +r D t+1 ; and applying the conditional expectations operator, E i (j it ) we have E i (P t+1 j it ) = 1 1 +r n X s=1 w s E i f[E s (P t+2 j s;t+1 )]j it g + E i ( n ) 1 +r D t e g i : But by (2.2.2) in the paper, we haveE i [E s (P t+2 j s;t+1 )j it ] =E i (P t+2 j it ) + 2 (2) it P t ; and hence E i (P t+1 j it ) = 1 1 +r h E i (P t+2 j it ) + 2 (2) it P t i + E i ( n ) 1 +r D t e g i : Substituting this result in (2.2.9) in the paper yields P t = 1 1 +r n X i=1 w i 1 1 +r h E i (P t+2 j it ) + 2 (2) it P t i + E i ( n ) 1 +r D t e g i + n 1 +r D t ; and after some simplication we have P t = 1 1 +r 2 n X s=1 w s E s (P t+2 j st ) + 1 1 +r 2 n X s=1 2w s (2) st ! P t + n D t ; (S.1) 132 where n = n 1 +r + 1 1 +r 2 n X s=1 w s E s ( n )e gs ! : (S.2) As before P it is dened by applying the expectations operator E i (P t j it ) to the right hand side of (S.1), namely P it = 1 1 +r 2 n X s=1 w s E i [E s (P t+2 j st )j it ] + 1 1 +r 2 " n X s=1 w s E 2 (2) st j it # P t +E i ( n )D t : Now using (2.2.2) and (2.2.3) from the paper in the above equation yields P it = 1 1 +r 2h E i (P t+2 j it ) + 2 (2) it P t i +2w i 1 1 +r 2 (2) it P t +E i ( n )D t : SubtractingP t from both sides, using (2.2.14) from the paper, and after some simplications, and obtain e i;t+2 = (2) i (1 +r) 2 2 V it +u (2) it ; where e i;t+2 = E i ( t+2 j it ) , t+2 = P t+2 P t 2P t = P t+2 + P t+1 2P t ; (2) i = (1 +r) 2 1 2 (1 +r) 2 (1 +re g )E i ( n ) 2E ( n ) ; u (2) it =(1 +w i ) (2) it : Following similar derivations for h = 3; 4;:::; the general result given by equation (2.2.17) in the paper follows. 133 S3 Fixed eects-time eects (FE-TE) estimators for unbalanced panels Consider the panel data model y it = i + t +x it +u it ; (S.1) where i = 1; 2;:::;H and t = 1; 2;:::;T i for respondent i, and let T = max i T i . Let N t be the number of respondents observed in periodt and letN t be the set of respondents observed in period t. Let s it be a binary variable which takes the value of 1 if a response is recorded for respondent i at time period t, and equal to 0; otherwise. Finally, let N = P t N t . S1 Denote the available observations on respondents at time t by the N t 1 vector, y :t;Nt , whose elements are members of the set N t . Specically, N t = #N t . x :t;Nt is dened analo- gously. Stack y :t;Nt and x :t;Nt over t = 1; 2;:::;T to obtain y = 8 > > > > > > > < > > > > > > > : y :1;N 1 y :2;N 2 . . . y :T;N T 9 > > > > > > > = > > > > > > > ; ; and x = 8 > > > > > > > < > > > > > > > : x :1;N 1 x :2;N 2 . . . x :T;N T 9 > > > > > > > = > > > > > > > ; : Next, following the procedure described in Wansbeek and Kapteyn (1989), let D t be the N t H matrix obtained from theHH identity matrix from which the rows corresponding to the respondents not observed in period t have been omitted, and let H be the H 1 S1 In terms of paper's notation, y it corresponds to ^ e i;t+hjt in equation (2.4.6) of the paper. 134 vector of ones. Dene Z 1 = 8 > > > > > > > < > > > > > > > : D 1 D 1 H D 2 D 2 H . . . . . . D T D T H 9 > > > > > > > = > > > > > > > ; ; and Z 2 = 8 > > > > > > > < > > > > > > > : D 1 H D 2 H . . . D T H 9 > > > > > > > = > > > > > > > ; ; and set Z = (Z 1 ; Z 2 ). Also let Z = Z 2 Z 1 (Z 0 1 Z 1 ) 1 Z 0 1 Z 2 ; Q = Z 0 2 Z 2 Z 0 2 Z 1 (Z 0 1 Z 1 ) 1 Z 0 1 Z 0 2 ; and P = I N Z 1 (Z 0 1 Z 1 ) 1 Z 0 1 ZQ Z 0 ; where I N is theNN identity matrix, and Q is a generalized inverse of Q. The resultant P matrix does not depend on the choice of the generalized inverse ( see Wansbeek and Kapteyn (1989)). Now dene the transformed variables ~ y = Py and ~ v = Px and consider the transformed panel regression ~ y it =~ x it +" it : We estimate by ^ FETE = " H X i=1 T X t=1 s it (~ x it ~ x) 2 # 1 " H X h=1 T X t=1 s it (~ x it ~ x)(~ y it ~ y) # ; (S.2) 135 where ~ x = 1 N P H i=1 P T t=1 s it ~ x it , and ~ y is dened analogously. Let ^ " it;FETE = ~ y it ~ y(~ x it ~ x) ^ FETE , and ^ " i;FETE = (^ " it 1;i ;FETE ; ^ " it 2;i ;FETE ;:::; ^ " iT i ;FETE ) 0 , where t 1;i is the rst time period in which respondent i is observed. Also, dene x i: = 8 > > > > > > > < > > > > > > > : ~ x it 1;i ~ x ~ x it 2;i ~ x . . . ~ x iT i ~ x; 9 > > > > > > > = > > > > > > > ; : The variance of ^ FETE is computed as d Var( ^ FETE ) = H X i=1 x 0 i: x i: ! 1 H X i=1 x 0 i: ^ " i;FETE ^ " 0 i;FETE x i: ! H X i=1 x 0 i: x i: ! 1 (S.3) S4 FE-TE Filtered estimators of the time-invariant ef- fects for unbalanced panels The parameters of interest is the k 1 vector of time-invariant eects, , y it =a + 0 z i + t +x it +u it +" i ; obtained from (S.1), by replacing i with a + 0 z i +" i , where z i is the k 1 vector of time- invariant characteristics of respondent i. To estimate , we assume that z i is distributed independently of " i + u i , where u i = P T t=1 s it u it = P T t=1 s it , and s it = 1 if respondent i is in the sample, and 0 otherwise. Note that P T t=1 s it =T i , where T i denotes the number of time periods that respondent i is observed. To estimate we extend the method proposed in Pesaran and Zhou (2016) to unbalanced panels with time eects, and adopt a two-stage pro- cedure where in the rst-step the eects ofx it are ltered out, by considering the individual specic residuals after estimation of by application of FE-TE procedure to (S.1). In this 136 way we allow x it and u it to be correlated. Let ^ u it =y it ^ FETE x it ; and note that for a xed T and N large ^ u it =a + 0 z i + t +" i +O p (N 1=2 ): Then for each respondent averaging ^ u it overt, taking into account the unbalanced nature of the panel, we have ^ u i =a + 0 z i + s i +" i +O p (N 1=2 ); (S.1) where s i = P T t=1 s it t P T t=1 s it ! ; and ^ u i = T X t=1 s it ^ u it ! = T X t=1 s it ! : We note that s i = s i 0, if respondents i and i 0 have the same participation pattern, as represented by s i = (s i1 ;s i2 ;:::;s iT ) 0 . As Table 2.2 in the paper shows the frequency of participation across the survey waves has been quite high, and there is a good chance that many respondents have the same participation pattern, s i . Accordingly, we use a dummy variable to identify the set of respondents with the same participation pattern. Specically, letS be the set of unique response patterns in the data, S =f2f0; 1g T j = s i for at least one i = 1; 2;:::;Hg: Denote the cardinality ofS byjSj =m and assume that the elements ofS are ordered, with l denoting the l th element ofS. Note that m 2 T 1. Let d i = (d i1 ;d i2 ;:::;d i;m ) (S.2) 137 be the vector of time eects of respondent i, with d il = 1 if s i = l , and d il equal to zero, otherwise. In eect, respondents with the same participation pattern are grouped together and assigned a dummy variable which takes the value of unity if a respondent belong to the group and zero otherwise. With these additional dummy variables, (S.1) can be written as ^ u i =a + 0 z i + 0 d i +" i +O p (N 1=2 ); (S.3) or more compactly as ^ u i = 0 q i +" i +O p (N 1=2 ); where = (a; 0 ; 0 ) 0 and q i = (1; z 0 i ; d 0 i ) 0 . Then the FE-TE ltered (FE-TE-F) estimator of is computed as ^ FETEF = " H X i=1 (q i q)(q i q) 0 # 1 H X i=1 (q i q)( ^ u i ^ u); (S.4) where ^ u =H 1 P H i=1 ^ u i , and H is the total number of respondents in the sample The variance of ^ FETEF is estimated by (see also Proposition 2 of Pesaran and Zhou (2016)), d Var( ^ FETEF ) =H 1 Q 1 qq;H h ^ V qq;H + Q q x;H H d Var( ^ FETE ) Q 0 q x;H i Q 1 qq;H ; (S.5) where d Var( ^ FETE ) is given by (S.3), and Q qq;H = 1 H H X i=1 (q i q)(q i q) 0 ; Q q v;H = 1 H H X i=1 (q i q)(x i x) 0 ; x = H X i=1 x i =H; ^ V qq;H = 1 H H X i=1 (^ & i ^ &) 2 (q i q)(q i q) 0 ; 138 and ^ & i ^ & = y i y (x i x) ^ FETE (q i q) 0 ^ FETEF : S5 Dynamic panel regressions with bubble and crash indicators In this section we provide additional information on estimation of the dynamic panel re- gressions of realized house price changes. Note that the DQ surveys were conducted from the middle of one month to the middle of the following month. For example, indicators calculated using survey results conducted from mid-June to mid-July are used as predictors of the realized house price change in August. We follow the procedure described by Arellano and Bond (1991) with some modications. Consider the model s;t+1 = s + st + 0 x st +u s;t+1 (S.1) and x st includes the predictors that vary depending on the specication of Models M1 to M4 considered in Section 2.6 of the paper. For each h = 1; 3; and 12, x st = ^ e s;t+hjt ; for model M 1 ; x st = ^ e s;t+hjt ;B s;t+hjt ;C s;t+hjt ;B s;t+hjt ;C s;t+hjt ; , for model M 2 ; x st = B s;t+hjt ;C s;t+hjt ;B s;t+hjt ;C s;t+hjt ; , for model M 3 : Models M 1 to M 3 are estimated over s = 1; 2;:::;N (= 48 MSA), and t = 3; 4;:::;T (= 11 months) (June 2012-February 2013). Model M 4 is estimated with MSAs s = 1; 2;:::;N (= 48), over the months August 2012-February 2013 (T = 7), with x st set to ^ e st ; B s;t2 ; C s;t2 ; B s;t2 ; C s;t2 , where ^ e st = 1 3 (^ e s;t+1jt + ^ e s;t+3jt + ^ e s;t+12jt ); 139 B st = 1 3 (B s;t+1jt +B s;t+3jt +B s;t+12jt ); C st = 1 3 (C s;t+1jt +C s;t+3jt +C s;t+12jt ); B st and C st are dened analogously. The GMM estimation is carried out by rst dierencing equation (S.1) to eliminate the MSA xed eects, s , namely s;t+1 = st + 0 x st + u s;t+1 ; for s = 1; 2;:::;N, and t = 3; 4;:::;T . Then the T 2 available observations are stacked as s;+1 = ( s;3 ; s;4 ;:::; s;T ) 0 ; u s;+1 = (u s;3 ; u s;4 ;:::; u s;T ) 0 ; s; = ( s;2 ; s;3 ;:::; s;T1 ) 0 ; X s = (x 0 s;2 ; x 0 s;3 ;;:::; x 0 s;T1 ) 0 : x st is treated as predetermined and the following instrumental variable matrix is used W s = 0 B B B @ ( s1 ; s2 ;x s1 ;x s2 ) 0 ::: ::: 0 0 ( s2 ; s3 ;x s2 ;x s3 ) ::: ::: 0 . . . . . . . . . . . . 0 0 ::: ::: ::: ( s;T2 ; s;T1 ;x s;T2 ;x s;T1 ) 1 C C C A : The moment conditions can now be expressed as E (W 0 s u s;+1 ) = 0; for s = 1; 2;::::;N: where u +1 = +1 X; 140 with +1 = 0 B B B B B B B @ 1;+1 2;+1 . . . N;+1 1 C C C C C C C A ; = 0 B B B B B B B @ 1 2 . . . N 1 C C C C C C C A ; X = 0 B B B B B B B @ X 1 X 2 . . . X N 1 C C C C C C C A ; u +1 = 0 B B B B B B B @ u 1;+1 u 2;+1 . . . u N;+1 1 C C C C C C C A : The two-step Arellano-Bond estimator is given by ^ AB;2step = (G 0 ZS N Z 0 G) 1 G 0 ZS N Z 0 ; (S.2) where ^ AB;2step = ( ^ AB;2step ; ^ 0 AB;2step ), G = (; X), Z = (W; X), W = (W 1 ; W 2 ;:::; W N ) 0 ; S N = N X s=1 Z 0 s ^ u s ^ u 0 s Z s ! 1 ; Z s = (W s ; x s ) and ^ u s = G^ AB;1step , are the residuals using the rst-stage estimates ^ AB;1step = h G 0 Z (Z 0 Z) 1 Z 0 G i 1 G 0 Z (Z 0 Z) 1 Z 0 ; (S.3) with = (I N A), and A = 0 B B B B B B B B B B @ 2 1 ::: 0 0 1 2 ::: 0 0 . . . . . . . . . . . . . . . 0 0 . . . 2 1 0 0 . . . 1 2 1 C C C C C C C C C C A : See also Section 27.4.2 in Pesaran (2015). 141 S6 American Life Panel Surveys The American Life Panel (ALP) consists of over 6,000 panel members aged 18 and older. Detailed information about the panel can be found at https://alpdata.rand.org/index.php?page=panel. In what follows we provide selected information about the ALP surveys that we deem relevant to the DQs surveys. S6.1 Recruitment ALP participants are recruited through a number of sources, including the University of Michigan Monthly Surveys, both internet-panel cohort and phone-panel (CATI) cohort, the National Survey Project cohort, Snowball cohort, phone and mailing experiment cohort, vul- nerable population cohort, and ALP Intergenerational Cohort. The origin of each household in the survey is indicated by the \recruitment type" variable in the excel sheet survey result les. The ALP invites adult members of participating households to join the panel. Members of the same household can be identied in the panel, which allows for intra-household com- parisons. Currently, approximately 17 per cent of surveyed households have more than one panel member. S6.2 Demographics Each ALP survey contains a \Demographics" module, which by default contains informa- tion on "gender, date of birth, place of birth, US citizenship, household income, household members, employment, state of residence, ethnicity. and education. S6.3 Response Rates and Attrition The attrition rate of ALP participants is relatively low. Between 2006 and 2013 the annual attrition rate has been between 6 and 13 percent. Since panel members do not always give formal notication about their decision to leave the panel, in order to avoid retention of non- 142 responding panel members, RAND contacts members who have not been active for at least one year and asks them about their continued interest in participating. The ALP removes all those for whom such contact attempts fail, as well as all those who were not active in the previous year. Response rates for ALP surveys are calculated by dividing the number of completed interviews by the size of the associated underlying sample. Most selected panel members who complete the interview respond within one week of the elding of the survey, and almost all do so within two weeks. Response rates for the ALP survey typically average around 70 percent, but can vary signicantly by subgroups, how long the survey is kept in the eld, and the number of reminders sent. S7 Survey questions We are interested in learning your views about prices of houses, stocks and shares, and gold, and appreciate your responses to the following questions. H1 rate current housing prices We now have some questions about housing prices. The median price of a single family home in the [ll for city nearest to R zip code] cosmopolitan area is currently around [converted ll for median housing price in R zip code area] (Half of all single family homes in the area cost less than the median, and the other half cost more than the median.). Do you believe that current housing prices are: 1 just right (in the sense that housing prices are in line with what you personally regard to be fair), 2 too high, 3 too low as compared to the fair value? H2 intro 143 Bearing in mind your response to the previous question, suppose now that someone were to purchase a single family home in [ll for city nearest to R zip code] area for the price of [ . . . ] What do you expect the house to be worth (Please enter a numeric answer only, with no commas or punctuation) H2 1month 1 month from now, H2 3month 3 months from now, H2 1year 1 year from now. Respondents who reside further than 500 miles away from a major metropolitan area were provided with H1 alternate and H2 intro alternate instead of H1 and H2 intro. H1 alternate rate current housing prices We now have some questions about housing prices. The median price of a single family home in the USA is currently around $163,500 (Half of all single family homes in the area cost less than the median, and the other half cost more than the median.). Do you believe that current housing prices are: 1 just right (in the sense that housing prices are in line with what you personally regard to be fair), 2 too high, 3 too low as compared to the fair value? H2 intro alternate Bearing in mind your response to the previous question, suppose now that someone were to purchase a single family home in the USA for the price of $163,500. What do you expect the house to be worth (Please enter a numeric answer only, with no commas or punctuation) H2 1month 1 month from now, H2 3month 3 months from now, H2 1year 1 year from now. 144 H3 intro Will you please elaborate by providing responses to the following: What do you think is the per cent chance that one year from now the house will be worth H3 percent1 amount minus or plus 5 per cent. Between [ calculated low house value] and [calculated high house value] dollars? H3 percent2 amount less 5 per cent. Less than [calculated low house value] dollars? H3 percent3 amount more than 5 per cent. More than [ calculated high house value] dol- lars? Your responses should add up to 100 per cent. E1 rate stock price level We have some questions about the price of publicly traded stocks. Do you believe the US stock market (as measured by S&P 500 index) to be currently: 1 Overvalued 2 Fairly valued (in the sense that the general level of stock prices is in line with what you personally regard to be fair) 3 Undervalued E1 note explain stock index Note: The S&P 500 is an index of 500 common stocks actively traded in the United States. It provides one measure of the general level of stock prices. E2 intro estimate 1000 investment Bearing in mind your response to the previous question, suppose now that today someone were to invest 1000 dollars in a mutual fund that tracks the movement of S&P 500 very closely. That is, this \index fund" invests in shares of the companies that comprise the S&P 500 Index. What do you expect the $1000 investment in the fund to be worth 145 E2 1month in one month from now, E2 3month in three months from now, E2 1year in one year from now. E3 intro intro to per cent change Will you please elaborate by providing responses to the following: What do you think is the per cent chance that a year from today the investment will be worth E3 percent1 minus 5 to plus 5 per cent. Between [calculated low stock value] and [calculated high stock value] dollars? E3 percent2 minus 5 per cent. Less than [calculated low stock value] dollars? E3 percent3 plus 5 per cent. More than [calculated high stock value] dollars? Your responses should add up to 100 per cent. G1 rate current gold prices We now have some questions about the price of gold bullion traded internationally. Given the current price of gold, do you believe gold prices to be: 1 Overvalued 2 Fairly valued (in the sense that the general level of stock prices is in line with what you personally regard to be fair) 3 Undervalued G2 intro intro to G2 Bearing in mind your response to the previous question, suppose now that today someone were to invest 1000 dollars in gold bullion. What do you expect the $1000 investment in gold to be worth G2 1month 1 month from now, G2 3month 3 months from now, G2 1year 1 year from now. 146 G3 intro intro to G3 Will you please elaborate by providing responses to the following: What do you think is the per cent change that a year from today the investment in gold will be worth G3 percent1 minus 10 to plus 10 per cent. Between [calculated low gold value] and [calcu- lated high gold value] dollars? G3 percent2 minus 10 per cent. Less than [calculated low gold value] dollars? G3 percent3 plus 10 per cent. More than [calculated high gold value] dollars? Your responses should add up to 100 per cent. S8 Truncation lters Denote the price of asseta, witha =eq;gd;hs (equity, gold, house), provided to respondent i at time t by P (a) it . Note that P (eq) it = 1000 and P (gd) `t = 1000, for all t. The price of asset a expected by the i th respondent in month t for h months ahead is denoted by P e;(a) i;t+hjt . Re- spondent i's subjective valuation of asset a in period t is denoted by x (a) it , with x (a) it = 1 if the respondent believes that the asset is over-valued, x (a) it =1 if the respondent believes that the asset is under-valued, and x (a) it = 0, otherwise. z i is a 7 1 vector of time-invariant characteristics of the i th respondent. LetT i be the set of time periods (months) in which respondent i takes part in the survey. The elements of z i are z i1 = 1 if female, 0 otherwise. z i2 = 1 #T i P t2T i logage it ; average log age of respondent i. z i3 = 1 #T i P t2T i edu it respondent's education averaged over the time period the re- spondent participated in the survey, where edu it = 0 if the respondent has no high school diploma, edu it = 1 if the respondent is a high school graduate with a diploma, 147 some college but no degree, an associate degree in college occupational/vocational or academic program, andedu it = 2 if the respondent has a Bachelor's degree or higher. S2 z i4 = 1 #T i P t2T i logincome it ; average log income of respondent i. z i5 = 1 if Asian, 0 otherwise. z i6 = 1 if Black, 0 otherwise. z i7 = 1 if Hispanic/Latino, 0 otherwise. We came across a few cases where responses to gender and ethnicity questions did not remain invariant over the dierent survey waves. In such cases we used the following rule. Let d it be the binary variable that denotes the gender or ethnicity. (Asian, Black, His- panic/Latino) of respondent i in month t, and letT i denote the set of months during which respondent i participated in the surveys. Let d i = 1 #T i P t2T i d it . If d it varies over time, we consider the following cases. If d i 2=3, we set d it = 1 for all t2T i . If d i 1=3, we set d it = 0 for all t2T i . If 1=3< d i < 2=3, we remove respondent i from the data. Truncation lter criteria For respondent i in period t, x (a) it , P e;(a) i;t+hjt for a = eq;gd;hs; and h = 1; 3; 12, are removed from the data set if any of the following criteria apply: (a) Missing responses: x (a) it or P e;(a) i;t+hjt is missing for any a =eq;gd;hs or any h = 1; 3; 12, z 1;i , z 2;i , z 3;i , z 4;i , age it , income it or edu it are missing, S2 z 5;i z 6;i andz 7;i are constructed after all steps of the truncation lter described in Section S8 have been applied. 148 (b) Equity prices: P e;(eq) i;t+hjt > 4000 or P e;(eq) i;t+hjt = 0 for any h = 1; 3; 12; P e;(eq) i;t+hjt < 100 for all h, or P e;(eq) i;t+hjt > 2000 for all h, S3 (c) Gold prices: P e;(gd) i;t+hjt > 4000 or P e;(gd) i;t+hjt = 0 for any h = 1; 3; 12 P e;(gd) i;t+hjt < 100 for all h, or P e;(gd) i;t+hjt > 2000 for all h, and (d) House prices: P e;(hs) i;t+hjt < 0:5P (hs) it or P e;(hs) i;t+hjt > 2P (hs) it or P e;(hs) i;t+hjt = 0 for any h = 1; 3; 12. Table S1 provides a comparison of the characteristics of ltered and unltered respon- dents. S3 Examples of responses (P e;(eq) i;t+1jt , P e;(eq) i;t+3jt , P e;(eq) i;t+12jt ) that would be truncated are: (4020; 1030; 1020), (90; 80; 99), (2020; 2010; 3000). Examples of responses that would not be truncated are (90; 1020; 1010), (2030; 2020; 1050). 149 Table S1: Comparison of original and ltered respondent samples Wave Age Income Female Asian Black Hispanic/Latino Education average average per cent per cent per cent per cent average unltered ltered unltered ltered unltered ltered unltered ltered unltered ltered unltered ltered unltered ltered 3 49:05 49:77 52; 887 56; 683 59:62 57:81 2:13 2:31 10:76 8:29 17:39 14:61 1:33 1:39 4 49:04 49:84 51; 965 56; 507 59:17 56:99 2:05 2:33 11:01 8:70 18:27 14:32 1:31 1:38 5 49:09 49:86 51; 285 55; 962 58:96 56:84 1:78 1:98 11:10 8:72 17:84 14:52 1:30 1:37 6 48:90 49:47 51; 736 56; 039 58:78 56:76 1:89 2:13 11:52 8:97 18:53 15:54 1:31 1:37 7 48:70 49:33 51; 518 55; 240 59:51 57:57 1:86 1:96 11:84 9:43 18:58 15:73 1:31 1:37 8 48:86 49:50 51; 967 55; 444 59:62 57:53 1:95 2:03 11:19 9:20 18:99 15:82 1:31 1:37 9 48:99 49:66 51; 423 54; 983 59:35 57:93 1:78 1:87 11:57 9:69 18:50 16:00 1:30 1:36 10 49:11 49:69 51; 900 55; 689 59:09 56:85 1:87 1:99 11:59 9:68 19:07 15:81 1:31 1:37 11 49:02 49:90 52; 003 56; 105 59:20 57:43 1:94 1:99 11:19 9:02 18:11 14:52 1:31 1:38 12 49:33 49:93 51; 423 54; 992 59:10 57:26 1:84 2:00 11:53 9:37 19:40 16:75 1:30 1:36 13 48:78 49:44 51; 659 55; 565 58:98 57:19 1:93 2:17 11:70 9:22 18:50 15:26 1:31 1:38 Average 48:99 49:67 51; 797 55; 746 59:22 57:29 1:91 2:07 11:36 9:12 18:47 15:35 1:31 1:37 (1) - original sample of 5,480 respondents (2) - ltered sample of 4,971 respondents Education is equal to 0 if the respondent has no high school diploma, 1 if the respondent is a high school graduate with a diploma, some college but no degree, an associate degree in college occupational/vocational or academic program, and 2 if the respondent has a Bachelor's degree or higher. 150 S9 Respondent location and respondent characteris- tics Figure S1: Age distribution of ALP respondents and US population The ALP distributions are based on the sample of 4,971 respondents. The data on US population is obtained from the following sources: http://www.census.gov/population/age/data/2012comp.html https://www.census.gov/popest/data/historical/2010s/vintage_2012/national.html 151 Figure S2: Ethnicity of ALP respondents and US population The ALP distributions are based on the sample of 4,971 respondents. The data on US population is obtained from the following sources: http://www.census.gov/population/age/data/2012comp.html https://www.census.gov/popest/data/historical/2010s/vintage_2012/national.html Figure S3: Educational attainment of ALP respondents and US population The ALP education distribution is based on 4,968 (out of 4,971) respondents who are aged 18 or older. The data on US population is obtained from the following sources: http://www.census.gov/hhes/socdemo/education/data/cps/2012/tables.html http://www.census.gov/data/tables/time-series/demo/income-poverty/cps-hinc/hinc-06.2012. html. 152 Figure S4: Income distribution of ALP respondents and US population The ALP income distribution is based on the sample of 4,971 respondents. The data on US population is obtained from the following sources: http://www.census.gov/hhes/socdemo/education/data/cps/2012/tables.html http://www.census.gov/data/tables/time-series/demo/income-poverty/cps-hinc/hinc-06.2012. html. Figure S5: Location of Respondents in the DQ Surveys 153 Figure S6: US population density 154 S10 Denition of US regions Table S1: Categorization of regions Region States NorthEast CT, ME, MA, NH, NJ, NY, PA, RI, VT SouthEast Al, AR, DE, DC, FL, GA, KY, LA, MD, MS, NC, SC, TN, VA, WV MidWest IL, IN, IA, KS, MI, MN, MO, NE, ND, OH, SD, WI SouthWest AZ, NM, OK, TX West CA, CO, ID, MT, NV, OR, UT, WA, WY National Geographic Society proposes this region categorization according to their geographic position on the continent. According to its denition, a region is de- ned by natural or articial features, for example language, government, religion, forests, wildlife or climate. S11 Spatial weight matrix Consider MSAs s = 1; 2;:::;S. Let G (d) denote the SS geodesic based spatial matrix calculated using the Haversine distance between MSAs. Specically, we say that MSA s ands 0 ared-neighbors if the Haversine distance between their geographic centers is less than or equal to d miles. Then G (d) (s;s 0 ) = 1 if s and s 0 are d-neighbors, and G (d) (s;s 0 ) = 0 otherwise. Also, G (d) (s;s) = 0 for all s = 1; 2;:::;S. Denote the s th row of a matrix A by [A] s and let a ss 0 denote the (s;s 0 ) element of A, and let 0 S be a 1S vector of zeros, and dene W = (w ss 0) as follows. For s = 1; 2;:::;S, [W] s = [G (100) ] s if [G (100) ] s 6= 0 S . If [G (100) ] s = 0 S and [G (200) ] s 6= 0 S , [W] s = [G (200) ] s . If [G (200) ] s = 0 S , w ss 0 = 1 for s 0 = 1; 2;:::;S; s 0 6=s and w ss = 0. 155 S12 Data sources The survey data can be accessed from the link https://alpdata.rand.org/index.php?page=data. The survey is labeled \Asset Price Expectations" [W01]-[W15]. The house price data used in the MSA level analysis is sourced from the National Association of Realtors. The house prices are disaggregated by 180 MSAs as dened by the US Oce of Management and Budget. For further details see http://www.realtor.org/topics/existing-home-sales. In Section S12.1 we describe the survey data as released by RAND, and in Section S12.2 we describe how to replicate our results. S12.1 Survey data downloaded from the RAND ALP website The folder \DQ Survey data Aug 2012-Jan 2013" contains all survey data for the DQ Survey as available on the RAND ALP website. The results of each survey wave is included a separate csv le, and contains the following modules: Demographics - demographic information about the respondent, such as age, gender, education, employment etc. Base Module - information about the exact time when the respondent lled out the survey. Housing Prices - DQ survey module about house prices. Stock Prices - DQ survey module about stock prices. Gold Value - DQ survey module about gold prices. Closing - assessment of the interview experience. A list of the variables available in each survey wave can be found in the les \List of variables in each survey wave.xlsx". An overview of the modules can be accessed by clicking 156 on the survey name on the RAND website. An example for survey wave 13 is shown in Figure S1. Information about the non-respondents of the survey can also be found on this page. Further information about the questions contained in the module can be accessed by clicking on the name of the module. See Figure S2 for an example, where some of the variables in the Demographics module are displayed. Finally, more information about a variable can be obtained by clicking on the variable name. Figure S3 shows the information displayed if we click on the variable name, \ms318 gender" in survey wave 13. Figure S1: Screenshot of Asset Price Expectations Survey Wave 13 Figure S2: Screenshot of Accessing Demographic Variables 157 Figure S3: Screenshot of Question about Gender S12.2 Data and codes for replicating results All data and codes necessary to replicate the results are provided in the zipped le called \DQ Survey Replication". When this le is unzipped you should see the folder and le structure displayed in Figure 4. This Figure shows the structure of the folders in which the codes are organized. Folders are marked with a blue color. Files that recreate the data sets used in the estimation are marked in yellow, and the numbers next to the yellow boxes indicate the order in which the les should be executed. Finally, green boxes indicate les that replicate the estimation results. These can be executed in an arbitrary order. All les necessary to replicate the estimation results are also provided in the \Data" folder. Hence, it is possible to run the estimation scripts marked with green color without previously re- creating the data sets. All estimates are saved in tex tables, which are automatically placed in the folder called \tex". The zipped le \DQ Survey Replication" contains a folder with the same name. To run the replication les on a PC, place the zipped folder in a directory of your choice and unzip it. Then change the path names in the les accordingly. For example, if the le is unzipped in the root directory \C:n", add \C:n" directly before the words \DQ Survey Replication" in the le path, so that the path begins with \C:nDQ Survey Replication". Additionally, /" in the path denitions need to be changed to \n". Similarly, on a Mac or Linux computer, unzip the folder in a directory of your choice. Suppose the folder \DQ Survey Replication" is unzipped in the directory \/Users/home/Desktop/". 158 Then change the path names in the replication les so that they begin with \/Users/home/Desktop/DQ Survey Replication". The data sets used in the empirical analysis can be found in the folder \DQ Sur- vey Replication/Data/csv/". The data les are \panel ind.csv", \panel fef loc.csv" and \panel fetef.csv". These are the data sets containing all individual level variables such as valuation and price expectation as well as demographics. The latter two les also contain location and response pattern dummies, respectively. The panel data of 48 MSAs used in the MSA level analysis is contained in the le \Panel 48 MSAs.xlsx" in the same folder. For convenience, all the survey data les covering the period August 2012 to January 2013 are also available in the zipped le "DQ survey data Aug 2012-Jan 2013". Figure S4: Structure of Replication Directory S13 Selected MSA summary statistics 159 Table S1: Summary statistics of variables used in the realized house price change regressions Mean St. Dev. Min Pctl(25) Median Pctl(75) Max st 1.726 2.565 3.408 0.251 1.401 3.464 10.084 ^ e s;t+1jt 2.181 5.462 55.552 2.869 1.264 0.159 6.543 ^ e s;t+3jt 0.678 1.991 18.744 1.173 0.391 0.166 5.391 ^ e s;t+12jt 0.063 0.682 5.041 0.207 0.145 0.426 2.525 B s;t+1jt 0.177 0.112 0.000 0.088 0.164 0.250 0.591 C s;t+1jt 0.186 0.117 0.000 0.089 0.174 0.265 0.527 B s;t+1jt 0.167 0.091 0.000 0.104 0.165 0.199 0.552 C s;t+1jt 0.193 0.098 0.000 0.146 0.187 0.250 0.475 B s;t+3jt 0.160 0.104 0.000 0.076 0.148 0.231 0.591 C s;t+3jt 0.134 0.099 0.000 0.051 0.117 0.193 0.473 B s;t+3jt 0.153 0.086 0.000 0.095 0.153 0.184 0.515 C s;t+3jt 0.141 0.082 0.000 0.097 0.136 0.180 0.409 B s;t+12jt 0.159 0.105 0.000 0.076 0.148 0.227 0.591 C s;t+12jt 0.073 0.070 0.000 0.022 0.052 0.108 0.350 B s;t+12jt 0.155 0.088 0.000 0.093 0.149 0.182 0.539 C s;t+12jt 0.079 0.057 0.000 0.041 0.074 0.100 0.350 The statistics are based on the sample of 48 MSAs and 11 months: April 2012 to February 2013. st and ^ e s;t+hjt for h = 1; 3; 12 are expressed in per cent per quarter. The indicators B s;t+hjt ;C s;t+hjt ;B s;t+hjt ;C s;t+hjt for h = 1; 3; 12 are fractions between 0 and 1. 160 Table S2: Summary statistics of selected variables by MSA for 48 MSAs Average value during the period April 2012-February 2013 Nst st B s;t+1jt C s;t+1jt B s;t+3jt C s;t+3jt B s;t+12jt C s;t+12jt Albuquerque, NM 27:82 0:55 0:19 0:12 0:17 0:09 0:20 0:07 Amarillo, TX 20:18 0:40 0:32 0:06 0:30 0:03 0:31 0:02 Atlanta-Sandy Springs-Roswell, GA 49:36 3:17 0:06 0:34 0:04 0:27 0:04 0:15 Austin-Round Rock, TX 45:27 2:12 0:33 0:01 0:32 0:004 0:33 0:004 Boise City, ID 22:64 4:02 0:12 0:20 0:09 0:16 0:09 0:09 Chattanooga, TN-GA 29:45 0:89 0:08 0:26 0:07 0:19 0:07 0:07 Chicago-Naperville-Elgin, IL-IN-WI 68 0:43 0:09 0:39 0:07 0:28 0:07 0:13 Cleveland-Elyria, OH 41:55 0:26 0:06 0:40 0:04 0:34 0:03 0:25 Columbus, OH 22:36 0:67 0:08 0:31 0:06 0:25 0:07 0:10 Corpus Christi, TX 59:09 1:54 0:31 0:03 0:29 0:02 0:29 0:01 Cumberland, MD-WV 29:55 0:07 0:15 0:16 0:11 0:10 0:10 0:03 Dallas-Fort Worth-Arlington, TX 63:64 1:48 0:19 0:18 0:17 0:12 0:18 0:07 Denver-Aurora-Lakewood, CO 27:64 2:82 0:31 0:11 0:28 0:07 0:29 0:04 Detroit-Warren-Dearborn, MI 54:91 3:74 0:06 0:42 0:05 0:35 0:04 0:20 Dover, DE 20:45 0:33 0:10 0:22 0:08 0:15 0:09 0:05 El Paso, TX 51:09 0:13 0:27 0:05 0:21 0:04 0:20 0:04 Fort Wayne, IN 36:27 0:67 0:14 0:30 0:08 0:28 0:07 0:23 Grand Rapids-Wyoming, MI 34 2:11 0:07 0:27 0:07 0:22 0:07 0:13 Green Bay, WI 26:73 0:13 0:23 0:15 0:20 0:10 0:15 0:03 Greensboro-High Point, NC 30:82 0:56 0:11 0:21 0:11 0:14 0:09 0:07 Houston-The Woodlands-Sugar Land, TX 46:82 1:83 0:16 0:04 0:14 0:04 0:15 0:03 Indianapolis-Carmel-Anderson, IN 27:45 0:85 0:07 0:25 0:06 0:20 0:06 0:11 Kansas City, MO-KS 26:55 0:93 0:19 0:19 0:19 0:15 0:19 0:09 Lansing-East Lansing, MI 21:82 1:79 0:12 0:26 0:11 0:13 0:12 0:08 Los Angeles-Long Beach-Anaheim, CA 176:18 3:22 0:35 0:06 0:32 0:04 0:30 0:02 Miami-Fort Lauderdale-West Palm Beach, FL 43:09 3:18 0:13 0:14 0:11 0:06 0:11 0:02 Milwaukee-Waukesha-West Allis, WI 24:91 0:15 0:24 0:21 0:22 0:15 0:20 0:12 Minneapolis-St. Paul-Bloomington, MN-WI 36:91 2:32 0:11 0:26 0:11 0:15 0:10 0:10 New York-Newark-Jersey City, NY-NJ-PA 136:36 0:26 0:30 0:08 0:27 0:05 0:26 0:03 Philadelphia-Camden-Wilmington, PA-NJ-DE-MD 35:55 0:44 0:14 0:15 0:15 0:10 0:16 0:03 Phoenix-Mesa-Scottsdale, AZ 42:55 5:77 0:07 0:18 0:07 0:14 0:07 0:07 Raleigh, NC 24:82 0:72 0:19 0:12 0:18 0:10 0:17 0:07 Reading, PA 21:27 0:46 0:12 0:32 0:12 0:26 0:13 0:16 Riverside-San Bernardino-Ontario, CA 44:82 3:98 0:21 0:15 0:20 0:10 0:18 0:06 Sacramento{Roseville{Arden-Arcade, CA 64:18 4:83 0:17 0:20 0:15 0:14 0:15 0:07 Salt Lake City, UT 61:64 2:51 0:17 0:22 0:15 0:16 0:15 0:12 San Antonio-New Braunfels, TX 45:09 1:03 0:19 0:05 0:20 0:04 0:21 0:03 San Diego-Carlsbad, CA 36:27 3:55 0:33 0:03 0:32 0:02 0:32 0:002 San Francisco-Oakland-Hayward, CA 21:45 4:52 0:37 0:10 0:36 0:04 0:38 0:004 San Jose-Sunnyvale-Santa Clara, CA 39:64 4:18 0:40 0:05 0:29 0:03 0:32 0:02 Seattle-Tacoma-Bellevue, WA 43:55 3:16 0:24 0:12 0:22 0:07 0:23 0:04 Spartanburg, SC 24:27 0:40 0:02 0:31 0:02 0:20 0:02 0:09 St. Louis, MO-IL 21:36 0:45 0:12 0:32 0:11 0:22 0:10 0:06 Tallahassee, FL 20:45 0:62 0:22 0:10 0:22 0:07 0:20 0:03 Tucson, AZ 26:09 2:44 0:12 0:26 0:14 0:19 0:15 0:10 Tulsa, OK 33 0:65 0:25 0:16 0:19 0:12 0:17 0:01 Washington-Arlington-Alexandria, DC-VA-MD-WV 43:27 1:76 0:22 0:15 0:22 0:09 0:21 0:03 Youngstown-Warren-Boardman, OH-PA 24:91 0:75 0:02 0:22 0:02 0:17 0:04 0:11 Nst - number of respondents in month t and MSA s. st - realized price change in MSA s and month t, expressed in per cent per quarter. The data on house prices is sourced from the National Association of Realtors.The house prices are disaggregated by 180 MSAs as dened by the US Oce of Management and Budget. For further details see http://www.realtor.org/topics/existing-home-sales 161 S14 Estimates for males and females While not central to our paper, we also analyze how estimates of (h) in model (2.4.6) vary in terms of socio-economic characteristics. Specically, note that our estimates in Table 2.1 allow for random variation in (h) i across respondents. In this section we estimate equation (2.4.6) separately for male and female respondents. The estimates are summarized in Table S1. For equity prices, we nd no statistically signicant relationship between expected price changes and the valuation indicators for female respondents at any of the three expectations horizons. But for male respondents we nd the relationship to be statistically signicant and negative (thus equilibrating) for all three expectations horizons. Similar dierences between female and male respondents are also observed in the case of gold prices, with female respondents showing a positive and statistically signicant relationship between expected price changes and valuation indicators, whereas for male respondents we nd the relationship to be negative at three and twelve month expectations horizons. Finally, in terms of house prices, the valuation-expectation relationship is negative for both males and females. For females the results are statistically signicant for all expectation horizons, whilst for males they are statistically signicant only at the 12 month expectations horizon. 162 Table S1: Estimates of (h) in the panel regressions of individual ex- pected price changes on their belief valuation indicators for dierent assets by gender Dependent variable: ^ e i;t+hjt Female Respondents Equity Gold Housing Horizons FE FE-TE FE FE-TE FE FE-TE One Month 0.192 0.186 1.178*** 1.168*** -0.354*** -0.367*** Ahead (h = 1) (1.15) (1.11) (4.05) (4.01) (-4.85) (-5.02) Three Months 0.0895 0.0916 0.593*** 0.583*** -0.126*** -0.131*** Ahead (h = 1) (0.88) (0.90) (3.80) (3.74) (-3.70) (-3.82) One Year 0.00299 0.00489 0.181** 0.175** -0.0402** -0.0400** Ahead (h = 1) (0.06) (0.10) (2.74) (2.66) (-2.98) (-2.95) Male Respondents Equity Gold Housing Horizons FE FE-TE FE FE-TE FE FE-TE One Month -0.554** -0.617** -0.196 -0.236 -0.202 -0.211 Ahead (h = 1) (-2.83) (-3.14) (-0.82) (-0.99) (-1.74) (-1.81) Three Months -0.372*** -0.401*** -0.291* -0.323* -0.0767 -0.0782 Ahead (h = 1) (-3.33) (-3.58) (-2.10) (-2.32) (-1.69) (-1.73) One Year -0.300*** -0.308*** -0.304*** -0.319*** -0.0596*** -0.0594*** Ahead (h = 1) (-6.00) (-6.12) (-4.32) (-4.52) (-3.89) (-3.87) Fixed eect (FE) estimates of (h) in the panel regression ^ e i;t+hjt = (h) i + (h) x it +u (h) it are obtained with and without time eects (FE-TE) using an unbalanced panel of respondents over 11 months, March 2012 to January 2013. The regressions for females are estimated using 2,910 respondents and 20,602 responses. The regressions for males are estimated using 2,061 respondents and 15,359 responses. Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. Standard errors are robust to heteroskedasticity and residual serial correlation. 163 S15 Random eect estimates In what follows, we provide estimates of the panel data model ^ e i;t+hjt = (h) + z 0 i (h) + (h) x it + (h) t +" i;t+h + (h) i ; (S.1) which corresponds to equation (28) in the paper. We provide estimates both with and without time eects, and with and without MSA dummies. For the elements of z i = (z i1 ;z i2 ;:::;z i7 ) 0 , we consider z i1 = lnage i , z i2 = lnincome i , z i3 to z i6 are dummy variables that take the value of 1 if the respondenti identies her/himself as female, Asian, Black and Hispanic/Latino, respectively. Finally, z i7 measures the education level of the respondent. For a detailed description of how the time-invariant variables are constructed see Appendix A.2 of the paper. We allow " i;t+h + (h) i to be serially correlated and heteroskedastic. Ran- dom eects estimates of model (S.1) are presented in Tables S1-S3. We also consider the following model ^ e i;t+hjt = (h) + z 0 i (h) + (h) t +" i;t+h + (h) i ; (S.2) which we estimate with and without time eects and MSA dummies. These estimates are presented in Tables S4-S6. The estimates for equity and gold prices are similar across all model specications. It is interesting to note that for house prices, time-invariant character- istics cease to be statistically signicant once MSA (location) dummies are included. 164 Table S1: Random Eect Estimates of (h) and (h) in the Panel Regressions of Individual Expected Price Changes on Belief Valuation Indicators for Equity Dependent variable: ^ e i;t+hjt One Month Ahead Three Months Ahead One Year Ahead x it -0.124 -0.146 -0.133 -0.156 -0.118* -0.126* -0.120* -0.128* -0.138*** -0.140*** -0.139*** -0.140*** (0.116) (0.116) (0.116) (0.117) (0.0700) (0.0700) (0.0703) (0.0704) (0.0337) (0.0337) (0.0339) (0.0339) Female 0.654** 0.649** 0.684*** 0.680*** 0.825*** 0.822*** 0.829*** 0.826*** 0.553*** 0.551*** 0.553*** 0.551*** (0.261) (0.261) (0.264) (0.264) (0.155) (0.155) (0.156) (0.156) (0.0778) (0.0779) (0.0783) (0.0783) lnage -2.464*** -2.465*** -2.345*** -2.344*** -2.417*** -2.426*** -2.358*** -2.361*** -1.580*** -1.588*** -1.545*** -1.549*** (0.441) (0.441) (0.441) (0.442) (0.264) (0.264) (0.265) (0.265) (0.130) (0.131) (0.133) (0.133) Education -0.305 -0.309 -0.416 -0.420 -0.597*** -0.602*** -0.680*** -0.684*** -0.454*** -0.457*** -0.494*** -0.497*** (0.272) (0.272) (0.280) (0.280) (0.163) (0.163) (0.166) (0.166) (0.0800) (0.0802) (0.0819) (0.0820) lnincome -0.686*** -0.688*** -0.657*** -0.659*** -0.793*** -0.794*** -0.777*** -0.777*** -0.468*** -0.470*** -0.455*** -0.455*** (0.207) (0.207) (0.213) (0.214) (0.133) (0.133) (0.135) (0.136) (0.0646) (0.0647) (0.0657) (0.0657) Asian -1.254 -1.270 -1.191 -1.207 -0.133 -0.153 -0.0571 -0.0821 -0.137 -0.148 -0.115 -0.131 (0.931) (0.931) (0.948) (0.948) (0.552) (0.552) (0.568) (0.569) (0.248) (0.249) (0.258) (0.258) Black 0.998* 1.006* 0.772 0.779 1.523*** 1.533*** 1.369*** 1.375*** 1.149*** 1.155*** 1.051*** 1.053*** (0.586) (0.586) (0.599) (0.599) (0.335) (0.335) (0.343) (0.343) (0.162) (0.162) (0.168) (0.168) Hispanic/Latino 0.280 0.273 -0.234 -0.245 1.284*** 1.281*** 1.006*** 0.995*** 0.916*** 0.915*** 0.781*** 0.774*** (0.505) (0.505) (0.559) (0.559) (0.282) (0.282) (0.311) (0.311) (0.133) (0.133) (0.146) (0.146) Time Dummies No Yes No Yes No Yes No Yes No Yes No Yes MSA Dummies No No Yes Yes No No Yes Yes No No Yes Yes The estimates reported refer to the panel regressions ^ e i;t+hjt = (h) x it +z 0 i (h) + (h) i +" i;t+h using an unbalanced panel of 4,971 respondents over 11 months, March 2012 to January 2013. ^ e i;t+hjt is expressed in per cent per quarter for all h. N = 35; 961, T min = 1, T p25 = 4, T p50 = 6, T = 7:23, T p75 = 9, Tmax = 11 Random eect estimates with standard errors clustered at individual level. Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. 165 Table S2: Random Eect Estimates of (h) and (h) in the Panel Regressions of Individual Expected Price Changes on Belief Valuation Indicators for Gold Dependent variable: ^ e i;t+hjt One Month Ahead Three Months Ahead One Year Ahead x it 0.581*** 0.562*** 0.591*** 0.572*** 0.208** 0.190* 0.213** 0.195* -0.0565 -0.0646 -0.0524 -0.0606 (0.177) (0.177) (0.178) (0.178) (0.0993) (0.0994) (0.0997) (0.0998) (0.0455) (0.0456) (0.0457) (0.0458) Female 1.143*** 1.136*** 1.131*** 1.125*** 1.047*** 1.041*** 1.056*** 1.051*** 0.626*** 0.623*** 0.639*** 0.637*** (0.321) (0.321) (0.323) (0.323) (0.190) (0.190) (0.190) (0.190) (0.0943) (0.0944) (0.0942) (0.0943) lnage -3.925*** -3.936*** -3.840*** -3.842*** -3.072*** -3.082*** -3.052*** -3.055*** -1.707*** -1.712*** -1.708*** -1.710*** (0.539) (0.539) (0.542) (0.542) (0.313) (0.314) (0.314) (0.314) (0.152) (0.152) (0.152) (0.152) Education -1.294*** -1.299*** -1.266*** -1.273*** -1.269*** -1.274*** -1.226*** -1.231*** -0.783*** -0.785*** -0.768*** -0.770*** (0.319) (0.318) (0.330) (0.330) (0.198) (0.198) (0.200) (0.200) (0.0965) (0.0965) (0.0979) (0.0979) lnincome -1.381*** -1.385*** -1.325*** -1.326*** -1.140*** -1.142*** -1.092*** -1.092*** -0.663*** -0.665*** -0.632*** -0.633*** (0.265) (0.265) (0.271) (0.271) (0.158) (0.158) (0.161) (0.161) (0.0750) (0.0750) (0.0751) (0.0752) Asian 0.758 0.736 0.833 0.801 0.492 0.475 0.590 0.563 0.274 0.265 0.365 0.352 (1.212) (1.208) (1.232) (1.228) (0.651) (0.648) (0.666) (0.662) (0.318) (0.317) (0.327) (0.327) Black 2.071*** 2.078*** 1.742** 1.742** 1.898*** 1.904*** 1.768*** 1.767*** 1.297*** 1.300*** 1.240*** 1.240*** (0.695) (0.695) (0.711) (0.710) (0.388) (0.388) (0.394) (0.393) (0.189) (0.189) (0.193) (0.193) Hispanic/Latino 1.452** 1.449** 0.856 0.840 1.674*** 1.673*** 1.355*** 1.341*** 0.954*** 0.953*** 0.800*** 0.794*** (0.573) (0.573) (0.645) (0.645) (0.327) (0.327) (0.367) (0.367) (0.153) (0.153) (0.170) (0.170) Time Dummies No Yes No Yes No Yes No Yes No Yes No Yes MSA Dummies No No Yes Yes No No Yes Yes No No Yes Yes The estimates reported refer to the panel regressions ^ e i;t+hjt = (h) x it +z 0 i (h) + (h) i +" i;t+h using an unbalanced panel of 4,971 respondents over 11 months, March 2012 to January 2013. ^ e i;t+hjt is expressed in per cent per quarter for all h. N = 35; 961, T min = 1, T p25 = 4, T p50 = 6, T = 7:23, T p75 = 9, Tmax = 11 Random eect estimates with standard errors clustered at individual level. Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. 166 Table S3: Random Eect Estimates of (h) and (h) in the Panel Regressions of Individual Expected Price Changes on Belief Valuation Indicators for Housing Dependent variable: ^ e i;t+hjt One Month Ahead Three Months Ahead One Year Ahead x it -0.363*** -0.374*** -0.382*** -0.389*** -0.144*** -0.147*** -0.149*** -0.151*** -0.0638*** -0.0637*** -0.0634*** -0.0632*** (0.0604) (0.0602) (0.0547) (0.0546) (0.0254) (0.0254) (0.0242) (0.0242) (0.00945) (0.00947) (0.00940) (0.00942) Female 0.123 0.119 -0.0138 -0.0169 0.0236 0.0226 -0.0240 -0.0246 0.0356 0.0355 0.0216 0.0215 (0.180) (0.180) (0.0920) (0.0919) (0.0666) (0.0665) (0.0426) (0.0426) (0.0250) (0.0250) (0.0216) (0.0216) lnage 2.007*** 1.993*** -0.0146 -0.0207 0.663*** 0.659*** -0.00791 -0.00980 0.187*** 0.187*** 0.0227 0.0226 (0.280) (0.279) (0.134) (0.134) (0.103) (0.103) (0.0632) (0.0632) (0.0380) (0.0380) (0.0334) (0.0334) Education 0.384** 0.382** 0.114 0.113 0.166** 0.165** 0.0610 0.0607 0.0428* 0.0428* 0.00841 0.00852 (0.181) (0.180) (0.102) (0.102) (0.0652) (0.0651) (0.0439) (0.0439) (0.0237) (0.0237) (0.0211) (0.0211) lnincome 0.546*** 0.547*** 0.247*** 0.249*** 0.145*** 0.145*** 0.0497 0.0502 0.0137 0.0139 -0.00911 -0.00894 (0.132) (0.131) (0.0733) (0.0731) (0.0502) (0.0501) (0.0353) (0.0352) (0.0196) (0.0196) (0.0179) (0.0179) Asian -1.332* -1.343* -0.233 -0.247 -0.443* -0.446* -0.0547 -0.0584 -0.00995 -0.0103 0.0673 0.0668 (0.716) (0.713) (0.393) (0.390) (0.242) (0.242) (0.154) (0.154) (0.0879) (0.0878) (0.0753) (0.0752) Black -1.429*** -1.418*** -0.210 -0.204 -0.414*** -0.411*** -0.0211 -0.0193 -0.0362 -0.0358 0.0548 0.0550 (0.345) (0.344) (0.192) (0.192) (0.128) (0.128) (0.0889) (0.0889) (0.0500) (0.0500) (0.0456) (0.0457) Hispanic/Latino -2.074*** -2.076*** -0.196 -0.202 -0.521*** -0.522*** 0.100 0.0987 -0.0114 -0.0115 0.130*** 0.129*** (0.290) (0.289) (0.159) (0.159) (0.111) (0.111) (0.0783) (0.0783) (0.0412) (0.0412) (0.0390) (0.0390) Time Dummies No Yes No Yes No Yes No Yes No Yes No Yes MSA Dummies No No Yes Yes No No Yes Yes No No Yes Yes The estimates reported refer to the panel regressions ^ e i;t+hjt = (h) x it +z 0 i (h) + (h) i +" i;t+h using an unbalanced panel of 4,971 respondents over 11 months, March 2012 to January 2013. ^ e i;t+hjt is expressed in per cent per quarter for all h. N = 35; 961, T min = 1, T p25 = 4, T p50 = 6, T = 7:23, T p75 = 9, Tmax = 11 Random eect estimates with standard errors clustered at individual level. Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. 167 Table S4: Random Eect Estimates of Price Expectations and Individual Time-Invariant Characteristics for Equity Dependent variable: ^ e i;t+hjt One Month Ahead Three Months Ahead One Year Ahead Female 0.665** 0.663** 0.696*** 0.693*** 0.836*** 0.834*** 0.839*** 0.837*** 0.566*** 0.565*** 0.565*** 0.563*** (0.260) (0.260) (0.263) (0.263) (0.154) (0.155) (0.156) (0.156) (0.0778) (0.0779) (0.0783) (0.0784) lnage -2.467*** -2.469*** -2.350*** -2.349*** -2.420*** -2.430*** -2.362*** -2.366*** -1.584*** -1.592*** -1.550*** -1.554*** (0.441) (0.441) (0.441) (0.442) (0.263) (0.264) (0.265) (0.265) (0.130) (0.131) (0.133) (0.133) Education -0.304 -0.308 -0.414 -0.418 -0.596*** -0.600*** -0.678*** -0.683*** -0.453*** -0.455*** -0.492*** -0.495*** (0.272) (0.272) (0.280) (0.280) (0.163) (0.163) (0.166) (0.166) (0.0802) (0.0803) (0.0821) (0.0822) lnincome -0.670*** -0.670*** -0.642*** -0.641*** -0.778*** -0.778*** -0.763*** -0.762*** -0.451*** -0.452*** -0.439*** -0.439*** (0.207) (0.207) (0.214) (0.214) (0.133) (0.133) (0.136) (0.136) (0.0647) (0.0648) (0.0657) (0.0658) Asian -1.256 -1.272 -1.189 -1.205 -0.135 -0.155 -0.0555 -0.0805 -0.139 -0.150 -0.114 -0.129 (0.931) (0.932) (0.948) (0.948) (0.552) (0.552) (0.569) (0.569) (0.249) (0.249) (0.258) (0.259) Black 0.983* 0.988* 0.757 0.762 1.509*** 1.517*** 1.356*** 1.361*** 1.133*** 1.138*** 1.036*** 1.038*** (0.585) (0.585) (0.598) (0.598) (0.335) (0.335) (0.343) (0.343) (0.162) (0.162) (0.168) (0.168) Hispanic/Latino 0.274 0.266 -0.233 -0.243 1.279*** 1.275*** 1.007*** 0.996*** 0.909*** 0.908*** 0.782*** 0.775*** (0.505) (0.505) (0.559) (0.559) (0.282) (0.282) (0.311) (0.311) (0.133) (0.133) (0.146) (0.147) Time Dummies No Yes No Yes No Yes No Yes No Yes No Yes MSA Dummies No No Yes Yes No No Yes Yes No No Yes Yes The estimates reported refer to the panel regressions ^ e i;t+hjt =z 0 i (h) + (h) i +" i;t+h using an unbalanced panel of 4,971 respondents over 11 months, March 2012 to January 2013. ^ e i;t+hjt is expressed in per cent per quarter for all h. N = 35; 961, T min = 1, T p25 = 4, T p50 = 6, T = 7:23, T p75 = 9, Tmax = 11 Random eect estimates with standard errors clustered at individual level. Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. 168 Table S5: Random Eect Estimates of Price Expectations and Individual Time-Invariant Characteristics for Gold Dependent variable: ^ e i;t+hjt One Month Ahead Three Months Ahead One Year Ahead Female 1.118*** 1.111*** 1.105*** 1.099*** 1.038*** 1.033*** 1.047*** 1.043*** 0.629*** 0.626*** 0.641*** 0.639*** (0.320) (0.320) (0.323) (0.323) (0.190) (0.190) (0.190) (0.190) (0.0942) (0.0943) (0.0941) (0.0942) lnage -3.781*** -3.798*** -3.696*** -3.703*** -3.021*** -3.036*** -3.001*** -3.007*** -1.721*** -1.728*** -1.721*** -1.724*** (0.536) (0.536) (0.538) (0.538) (0.311) (0.312) (0.312) (0.312) (0.151) (0.151) (0.151) (0.152) Education -1.264*** -1.270*** -1.239*** -1.247*** -1.259*** -1.264*** -1.216*** -1.222*** -0.786*** -0.788*** -0.770*** -0.773*** (0.319) (0.319) (0.331) (0.330) (0.198) (0.198) (0.201) (0.201) (0.0965) (0.0966) (0.0979) (0.0980) lnincome -1.356*** -1.360*** -1.301*** -1.303*** -1.131*** -1.134*** -1.083*** -1.084*** -0.666*** -0.668*** -0.634*** -0.635*** (0.264) (0.264) (0.270) (0.270) (0.158) (0.158) (0.160) (0.160) (0.0750) (0.0750) (0.0751) (0.0752) Asian 0.781 0.758 0.852 0.819 0.500 0.482 0.597 0.570 0.272 0.263 0.363 0.350 (1.211) (1.207) (1.231) (1.226) (0.650) (0.647) (0.665) (0.662) (0.318) (0.317) (0.328) (0.327) Black 2.033*** 2.042*** 1.705** 1.706** 1.885*** 1.892*** 1.754*** 1.755*** 1.301*** 1.304*** 1.243*** 1.244*** (0.696) (0.695) (0.711) (0.711) (0.388) (0.388) (0.394) (0.394) (0.189) (0.189) (0.193) (0.193) Hispanic/Latino 1.434** 1.432** 0.842 0.826 1.667*** 1.667*** 1.350*** 1.337*** 0.956*** 0.955*** 0.801*** 0.795*** (0.574) (0.573) (0.645) (0.645) (0.327) (0.327) (0.367) (0.367) (0.153) (0.153) (0.170) (0.170) Time Dummies No Yes No Yes No Yes No Yes No Yes No Yes MSA Dummies No No Yes Yes No No Yes Yes No No Yes Yes The estimates reported refer to the panel regressions ^ e i;t+hjt =z 0 i (h) + (h) i +" i;t+h using an unbalanced panel of 4,971 respondents over 11 months, March 2012 to January 2013. ^ e i;t+hjt is expressed in per cent per quarter for all h. N = 35; 961, T min = 1, T p25 = 4, T p50 = 6, T = 7:23, T p75 = 9, Tmax = 11 Random eect estimates with standard errors clustered at individual level. Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. 169 Table S6: Random Eect Estimates of Price Expectations and Individual Time-Invariant Characteristics for Housing Dependent variable: ^ e i;t+hjt One Month Ahead Three Months Ahead One Year Ahead Female 0.144 0.140 0.00295 0.000163 0.0320 0.0311 -0.0175 -0.0180 0.0393 0.0391 0.0243 0.0243 (0.180) (0.180) (0.0926) (0.0925) (0.0669) (0.0667) (0.0429) (0.0429) (0.0251) (0.0251) (0.0217) (0.0217) lnage 2.085*** 2.072*** 0.0594 0.0546 0.694*** 0.690*** 0.0207 0.0192 0.201*** 0.200*** 0.0348 0.0346 (0.280) (0.280) (0.134) (0.134) (0.103) (0.103) (0.0631) (0.0631) (0.0381) (0.0381) (0.0336) (0.0335) Education 0.406** 0.405** 0.145 0.144 0.175*** 0.175*** 0.0729* 0.0728* 0.0467** 0.0467** 0.0134 0.0135 (0.182) (0.181) (0.102) (0.102) (0.0654) (0.0653) (0.0441) (0.0441) (0.0238) (0.0238) (0.0211) (0.0212) lnincome 0.607*** 0.609*** 0.307*** 0.309*** 0.169*** 0.170*** 0.0727** 0.0736** 0.0244 0.0245 0.000657 0.000778 (0.131) (0.131) (0.0731) (0.0729) (0.0502) (0.0501) (0.0353) (0.0352) (0.0197) (0.0197) (0.0180) (0.0180) Asian -1.370* -1.383* -0.241 -0.256 -0.459* -0.462* -0.0579 -0.0617 -0.0171 -0.0174 0.0663 0.0658 (0.716) (0.713) (0.395) (0.392) (0.242) (0.242) (0.155) (0.155) (0.0883) (0.0882) (0.0758) (0.0758) Black -1.531*** -1.523*** -0.298 -0.294 -0.455*** -0.452*** -0.0553 -0.0541 -0.0542 -0.0537 0.0403 0.0405 (0.345) (0.345) (0.194) (0.194) (0.128) (0.128) (0.0894) (0.0894) (0.0500) (0.0500) (0.0458) (0.0458) Hispanic/Latino -2.143*** -2.148*** -0.224 -0.230 -0.549*** -0.550*** 0.0896 0.0879 -0.0237 -0.0238 0.125*** 0.125*** (0.290) (0.290) (0.160) (0.160) (0.111) (0.111) (0.0786) (0.0786) (0.0413) (0.0413) (0.0392) (0.0391) Time Dummies No Yes No Yes No Yes No Yes No Yes No Yes MSA Dummies No No Yes Yes No No Yes Yes No No Yes Yes The estimates reported refer to the panel regressions ^ e i;t+hjt =z 0 i (h) + (h) i +" i;t+h using an unbalanced panel of 4,971 respondents over 11 months, March 2012 to January 2013. ^ e i;t+hjt is expressed in per cent per quarter for all h. N = 35; 961, T min = 1, T p25 = 4, T p50 = 6, T = 7:23, T p75 = 9, Tmax = 11 Random eect estimates with standard errors clustered at individual level. Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. 170 S16 Comparison of FEF and RE estimates of the price expectation equations In tables S1 to S3 we present the xed eects ltered and random eects estimates for the panel regressions discussed in Section 2.4.1 of the paper. Specically, we consider the panel data model ^ e i;t+hjt = (h) + (h) t + z 0 i (h) + (h) x it +" i;t+h + (h) i : (S.1) For the RE estimates we assume that (h) i and x it are independently distributed, and we allow " i;t+h + (h) i to be serially correlated and heteroskedastic. For the FEF estimates we allow (h) i andx it to be correlated, and employ the two-stage approach proposed by Pesaran and Zhou (2016). For a detailed discussion of the estimators and estimates see Section 2.4.1 of the paper. The FEF and RE estimates are similar across all model specications. As noted earlier, time-invariant respondent characteristics cease to be signicant predictors of the respondent's expected house price changes once we condition on the respondent's location. This is true for FEF and RE estimates. 171 Table S1: Fixed Eect Filtered and Random Eect Estimates of Price Expectation Equations for Equity Dependent variable: ^ e i;t+hjt One Month Ahead Three Months Ahead One Year Ahead FEF FEF RE RE FEF FEF RE RE FEF FEF RE RE x it (1) -0.099 -0.099 -0.124 -0.133 -0.090 -0.090 -0.118* -0.120* -0.115*** -0.115*** -0.138*** -0.139*** (0.127) (0.127) (0.116) (0.116) (0.076) (0.076) (0.0700) (0.0703) (0.036) (0.036) (0.0337) (0.0339) Female 0.767** 0.793*** 0.654** 0.684*** 0.898*** 0.896*** 0.825*** 0.829*** 0.570*** 0.567*** 0.553*** 0.553*** (0.301) (0.302) (0.261) (0.264) (0.171) (0.172) (0.155) (0.156) (0.084) (0.084) (0.0778) (0.0783) lnage -2.845*** -2.718*** -2.464*** -2.345*** -2.633*** -2.568*** -2.417*** -2.358*** -1.668*** -1.628*** -1.580*** -1.545*** (0.511) (0.508) (0.441) (0.441) (0.296) (0.297) (0.264) (0.265) (0.142) (0.144) (0.130) (0.133) Education -0.343 -0.424 -0.305 -0.416 -0.622*** -0.690*** -0.597*** -0.680*** -0.467*** -0.502*** -0.454*** -0.494*** (0.329) (0.337) (0.272) (0.280) (0.184) (0.187) (0.163) (0.166) (0.087) (0.089) (0.0800) (0.0819) lnincome -0.663*** -0.624** -0.686*** -0.657*** -0.816*** -0.790*** -0.793*** -0.777*** -0.479*** -0.461*** -0.468*** -0.455*** (0.241) (0.247) (0.207) (0.213) (0.148) (0.151) (0.133) (0.135) (0.070) (0.071) (0.0646) (0.0657) Asian -1.577 -1.430 -1.254 -1.191 -0.210 -0.082 -0.133 -0.0571 -0.170 -0.133 -0.137 -0.115 (1.056) (1.061) (0.931) (0.948) (0.619) (0.629) (0.552) (0.568) (0.274) (0.280) (0.248) (0.258) Black 0.956 0.787 0.998* 0.772 1.501*** 1.377*** 1.523*** 1.369*** 1.135*** 1.046*** 1.149*** 1.051*** (0.665) (0.681) (0.586) (0.599) (0.368) (0.376) (0.335) (0.343) (0.173) (0.178) (0.162) (0.168) Hispanic/Latino -0.098 -0.612 0.280 -0.234 1.205*** 0.928*** 1.284*** 1.006*** 0.895*** 0.759*** 0.916*** 0.781*** (0.586) (0.640) (0.505) (0.559) (0.310) (0.341) (0.282) (0.311) (0.142) (0.157) (0.133) (0.146) MSA dummies No Yes No Yes No Yes No Yes No Yes No Yes The estimates reported refer to the panel regressions ^ e i;t+hjt = (h) x it +z 0 i (h) + (h) i +" i;t+h using an unbalanced panel of 4,971 respondents over 11 months, March 2012 to January 2013. N = 35; 961, T min = 1, T p25 = 4, T p50 = 6, T = 7:23, T p75 = 9, Tmax = 11 FEF - estimator of Pesaran and Zhou (2016). Standard errors are robust to heteroskedasticity and serial correlation. RE - random eect estimates with standard errors clustered at individual level. Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. (1) - The FE estimates of (h) reported in the table are the same as those summarized in Table 2.1. 172 Table S2: Fixed Eect Filtered and Random Eect Estimates of Price Expectation Equations for Gold Dependent variable: ^ e i;t+hjt One Month Ahead Three Months Ahead One Year Ahead FE FE RE RE FE FE RE RE FE FE RE RE x it (1) 0.602*** 0.602*** 0.581*** 0.591*** 0.222** 0.222** 0.208** 0.213** -0.023 -0.023 -0.0565 -0.0524 (0.197) (0.197) (0.177) (0.178) (0.108) (0.108) (0.0993) (0.0997) (0.049) (0.049) (0.0455) (0.0457) Female 1.174*** 1.166*** 1.143*** 1.131*** 1.062*** 1.072*** 1.047*** 1.056*** 0.622*** 0.634*** 0.626*** 0.639*** (0.356) (0.358) (0.321) (0.323) (0.203) (0.202) (0.190) (0.190) (0.098) (0.098) (0.0943) (0.0942) lnage -4.132*** -4.039*** -3.925*** -3.840*** -3.140*** -3.117*** -3.072*** -3.052*** -1.725*** -1.722*** -1.707*** -1.708*** (0.591) (0.595) (0.539) (0.542) (0.332) (0.334) (0.313) (0.314) (0.158) (0.159) (0.152) (0.152) Education -1.312*** -1.277*** -1.294*** -1.266*** -1.280*** -1.235*** -1.269*** -1.226*** -0.798*** -0.780*** -0.783*** -0.768*** (0.370) (0.387) (0.319) (0.330) (0.220) (0.223) (0.198) (0.200) (0.103) (0.104) (0.0965) (0.0979) lnincome -1.362*** -1.302*** -1.381*** -1.325*** -1.119*** -1.070*** -1.140*** -1.092*** -0.656*** -0.624*** -0.663*** -0.632*** (0.302) (0.309) (0.265) (0.271) (0.172) (0.175) (0.158) (0.161) (0.080) (0.080) (0.0750) (0.0751) Asian 0.981 1.046 0.758 0.833 0.556 0.651 0.492 0.590 0.299 0.389 0.274 0.365 (1.407) (1.421) (1.212) (1.232) (0.687) (0.706) (0.651) (0.666) (0.325) (0.338) (0.318) (0.327) Black 1.977*** 1.639** 2.071*** 1.742** 1.886*** 1.756*** 1.898*** 1.768*** 1.309*** 1.254*** 1.297*** 1.240*** (0.751) (0.769) (0.695) (0.711) (0.409) (0.414) (0.388) (0.394) (0.198) (0.201) (0.189) (0.193) Hispanic/Latino 1.280** 0.645 1.452** 0.856 1.682*** 1.339*** 1.674*** 1.355*** 0.967*** 0.802*** 0.954*** 0.800*** (0.640) (0.717) (0.573) (0.645) (0.353) (0.395) (0.327) (0.367) (0.161) (0.179) (0.153) (0.170) MSA dummies No Yes No Yes No Yes No Yes No Yes No Yes The estimates reported refer to the panel regressions ^ e i;t+hjt = (h) x it +z 0 i (h) + (h) i +" i;t+h using an unbalanced panel of 4,971 respondents over 11 months, March 2012 to January 2013. N = 35; 961, T min = 1, T p25 = 4, T p50 = 6, T = 7:23, T p75 = 9, Tmax = 11 FEF - estimator of Pesaran and Zhou (2016). Standard errors are robust to heteroskedasticity and serial correlation. RE - random eect estimates with standard errors clustered at individual level. Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. (1) - The FE estimates of (h) reported in the table are the same as those summarized in Table 2.1. 173 Table S3: Fixed Eect Filtered and Random Eect Estimates of Price Expectation Equations for Housing Dependent variable: ^ e i;t+hjt One Month Ahead Three Months Ahead One Year Ahead FE FE RE RE FE FE RE RE FE FE RE RE x it (1) -0.292*** -0.292*** -0.363*** -0.382*** -0.106*** -0.106*** -0.144*** -0.149*** -0.048*** -0.048*** -0.0638*** -0.0634*** (0.064) (0.064) (0.0604) (0.0547) (0.027) (0.027) (0.0254) (0.0242) (0.010) (0.010) (0.00945) (0.00940) Female 0.139 -0.000 0.123 -0.0138 0.032 -0.016 0.0236 -0.0240 0.037 0.023 0.0356 0.0216 (0.185) (0.107) (0.180) (0.092) (0.069) (0.047) (0.0666) (0.0426) (0.026) (0.023) (0.0250) (0.0216) lnage 2.041*** 0.046 2.007*** -0.0146 0.673*** 0.002 0.663*** -0.00791 0.189*** 0.024 0.187*** 0.0227 (0.287) (0.162) (0.280) (0.134) (0.107) (0.073) (0.103) (0.0632) (0.040) (0.037) (0.0380) (0.0334) Education 0.389** 0.086 0.384** 0.114 0.165** 0.044 0.166** 0.061 0.042* 0.006 0.0428* 0.00841 (0.187) (0.121) (0.181) (0.102) (0.068) (0.049) (0.0652) (0.0439) (0.025) (0.022) (0.0237) (0.0211) lnincome 0.549*** 0.248*** 0.546*** 0.247*** 0.145*** 0.047 0.145*** 0.0497 0.011 -0.012 0.0137 -0.00911 (0.135) (0.087) (0.132) (0.0733) (0.052) (0.040) (0.0502) (0.0353) (0.021) (0.019) (0.0196) (0.0179) Asian -1.300* -0.097 -1.332* -0.233 -0.448* -0.041 -0.443* -0.0547 -0.016 0.067 -0.00995 0.0673 (0.738) (0.443) (0.716) (0.393) (0.246) (0.159) (0.242) (0.154) (0.089) (0.076) (0.0879) (0.0753) Black -1.490*** -0.289 -1.429*** -0.210 -0.433*** -0.035 -0.414*** -0.0211 -0.040 0.054 -0.0362 0.0548 (0.354) (0.222) (0.345) (0.192) (0.133) (0.100) (0.128) (0.0889) (0.053) (0.049) (0.0500) (0.0456) Hispanic/Latino -2.033*** -0.165 -2.074*** -0.196 -0.500*** 0.119 -0.521*** 0.100 -0.004 0.135*** -0.0114 0.130*** (0.295) (0.185) (0.290) (0.159) (0.114) (0.088) (0.111) (0.0783) (0.043) (0.042) (0.0412) (0.0390) MSA dummies No Yes No Yes No Yes No Yes No Yes No Yes The estimates reported refer to the panel regressions ^ e i;t+hjt = (h) x it +z 0 i (h) + (h) i +" i;t+h using an unbalanced panel of 4,971 respondents over 11 months, March 2012 to January 2013. N = 35; 961, T min = 1, T p25 = 4, T p50 = 6, T = 7:23, T p75 = 9, Tmax = 11 FEF - estimator of Pesaran and Zhou (2016). Standard errors are robust to heteroskedasticity and serial correlation. RE - random eect estimates with standard errors clustered at individual level. Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. (1) - The FE estimates of (h) reported in the table are the same as those summarized in Table 2.1. 2 174 S17 FE-TE Filtered estimates of the price expectation equations We consider the following model. ^ e i;t+hjt = (h) + z 0 i (h) + (h) x it + d 0 i (h) +" i;t+h + (h) i ; (S.1) with d i as specied in equation (S.2). There are m = 943 unique response patterns in our data, 456 of which belong to at least two respondents. We estimate two specications of the model. In the rst one we introduce dummies for each response pattern, i.e. d i 2R 942 (we leave out one dummy). Second, we estimate a model with time dummies for response patterns shared by at least two respondents, d i 2R 456 . Finally, as a benchmark, we estimate a model with no response pattern eects. Estimates of these models, with and without MSA dummies, are presented in Tables S1 -S6. As before, inclusion of location dummies have little eects on the estimates for equity and gold price equations across all specications. For house prices, however, the estimates dier signicantly depending on whether MSA xed eects are included or not. Specically, respondent characteristics cease to be statistically signicant once a location (MSA) dummy is included. 175 Table S1: FE-TE Filtered Estimates of (h) in the Panel Regressions of Individual Expected Price Changes on Belief Valuation Indicators for Dierent Assets (with 942 Response Pattern Dummies) Dependent variable: ^ e i;t+hjt One Month Ahead Three Months Ahead One Year Ahead Equity Gold Housing Equity Gold Housing Equity Gold Housing Female 0.817*** 1.067*** -0.018 0.951*** 1.016*** -0.009 0.582*** 0.610*** 0.025 (0.310) (0.377) (0.192) (0.178) (0.215) (0.071) (0.087) (0.103) (0.028) lnage -2.880*** -3.586*** 1.386*** -2.536*** -2.737*** 0.489*** -1.536*** -1.492*** 0.180*** (0.597) (0.658) (0.302) (0.337) (0.369) (0.114) (0.157) (0.171) (0.044) Education -0.220 -1.161*** 0.044 -0.410** -1.051*** 0.041 -0.328*** -0.643*** 0.024 (0.346) (0.397) (0.194) (0.191) (0.234) (0.070) (0.090) (0.107) (0.025) lnincome -0.595** -1.145*** 0.430*** -0.771*** -0.892*** 0.109* -0.422*** -0.536*** 0.010 (0.272) (0.328) (0.145) (0.161) (0.188) (0.056) (0.076) (0.087) (0.022) Asian -1.637 2.232 -1.575** -0.008 1.379** -0.550** -0.199 0.578* -0.065 (1.054) (1.445) (0.764) (0.641) (0.686) (0.251) (0.312) (0.334) (0.089) Black 1.217* 1.948** -1.355*** 1.333*** 1.507*** -0.403*** 0.963*** 1.043*** -0.071 (0.727) (0.795) (0.390) (0.401) (0.436) (0.146) (0.188) (0.210) (0.057) Hispanic/Latino -0.330 0.883 -1.706*** 0.912*** 1.380*** -0.415*** 0.693*** 0.818*** -0.002 (0.621) (0.680) (0.321) (0.330) (0.376) (0.123) (0.152) (0.171) (0.047) MSA Dummies No No No No No No No No The estimates reported refer to the panel regressions ^ e i;t+hjt = (h) +z 0 i (h) + (h) x it +d 0 i (h) +" i;t+h + (h) i using an unbalanced panel of 4,971 respondents over 11 months, March 2012 to January 2013. ^ e i;t+hjt is expressed in per cent per quarter for all h. N = 35; 961, T min = 1, T p25 = 4, T p50 = 6, T = 7:23, T p75 = 9, Tmax = 11 FE-TE Filtered estimates with standard errors robust to heteroskedasticity and serial correlation. Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. Filtered estimates are computed using the estimator of Pesaran and Zhou (2016). 176 Table S2: FE-TE Filtered Estimates of (h) in the Panel Regressions of Individual Expected Price Changes on Belief Valuation Indicators for Dierent Assets (with 942 Response Pattern Dummies and MSA Dummies) Dependent variable: ^ e i;t+hjt One Month Ahead Three Months Ahead One Year Ahead Equity Gold Housing Equity Gold Housing Equity Gold Housing Female 0.817*** 1.118*** -0.034 0.929*** 1.043*** -0.016 0.575*** 0.633*** 0.018 (0.315) (0.383) (0.120) (0.179) (0.214) (0.051) (0.088) (0.102) (0.025) lnage -2.852*** -3.603*** -0.025 -2.517*** -2.786*** 0.003 -1.530*** -1.519*** 0.055 (0.596) (0.668) (0.192) (0.338) (0.374) (0.084) (0.161) (0.174) (0.041) Education -0.335 -1.155*** 0.007 -0.477** -1.007*** 0.009 -0.357*** -0.627*** 0.010 (0.359) (0.418) (0.135) (0.198) (0.240) (0.053) (0.093) (0.109) (0.023) lnincome -0.577** -1.045*** 0.190* -0.770*** -0.836*** 0.030 -0.417*** -0.506*** -0.010 (0.274) (0.335) (0.097) (0.163) (0.190) (0.044) (0.076) (0.086) (0.020) Asian -1.470 2.131 0.161 0.102 1.394** 0.019 -0.165 0.635* 0.055 (1.035) (1.455) (0.405) (0.647) (0.702) (0.151) (0.315) (0.344) (0.077) Black 1.205 1.815** -0.309 1.309*** 1.507*** -0.058 0.927*** 1.033*** 0.014 (0.737) (0.811) (0.265) (0.410) (0.444) (0.114) (0.192) (0.213) (0.053) Hispanic/Latino -0.645 0.351 -0.264 0.754** 1.072** 0.040 0.621*** 0.676*** 0.096** (0.675) (0.754) (0.209) (0.365) (0.416) (0.094) (0.170) (0.188) (0.045) MSA Dummies Yes Yes Yes Yes Yes Yes Yes Yes Yes The estimates reported refer to the panel regressions ^ e i;t+hjt = (h) +z 0 i (h) + (h) x it +d 0 i (h) +" i;t+h + (h) i using an unbalanced panel of 4,971 respondents over 11 months, March 2012 to January 2013. ^ e i;t+hjt is expressed in per cent per quarter for all h. N = 35; 961, T min = 1, T p25 = 4, T p50 = 6, T = 7:23, T p75 = 9, Tmax = 11 FE-TE Filtered estimates with standard errors robust to heteroskedasticity and serial correlation. Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. Filtered estimates are computed using the estimator of Pesaran and Zhou (2016). 177 Table S3: FE-TE Filtered Estimates of (h) in the Panel Regressions of Individual Expected Price Changes on Belief Valuation Indicators for Dierent Assets (with 456 Response Pattern Dummies) Dependent variable: ^ e i;t+hjt One Month Ahead Three Months Ahead One Year Ahead Equity Gold Housing Equity Gold Housing Equity Gold Housing Female 0.792*** 1.181*** 0.071 0.888*** 1.075*** 0.003 0.567*** 0.646*** 0.029 (0.297) (0.363) (0.181) (0.171) (0.208) (0.068) (0.084) (0.099) (0.026) lnage -2.895*** -3.781*** 1.297*** -2.506*** -2.871*** 0.458*** -1.523*** -1.513*** 0.172*** (0.550) (0.616) (0.283) (0.314) (0.349) (0.107) (0.146) (0.160) (0.041) Education -0.171 -1.144*** 0.137 -0.463** -1.084*** 0.089 -0.360*** -0.676*** 0.042* (0.327) (0.375) (0.182) (0.181) (0.223) (0.066) (0.085) (0.102) (0.024) lnincome -0.695*** -1.292*** 0.417*** -0.773*** -0.980*** 0.105** -0.426*** -0.565*** 0.006 (0.257) (0.310) (0.135) (0.154) (0.178) (0.053) (0.072) (0.082) (0.021) Asian -2.073* 1.029 -1.653** -0.121 0.808 -0.535** -0.200 0.412 -0.035 (1.181) (1.523) (0.761) (0.647) (0.704) (0.246) (0.295) (0.330) (0.084) Black 1.107 1.664** -1.214*** 1.387*** 1.630*** -0.341*** 1.019*** 1.117*** -0.032 (0.679) (0.747) (0.352) (0.376) (0.414) (0.132) (0.176) (0.198) (0.054) Hispanic/Latino -0.139 1.161* -1.750*** 1.027*** 1.559*** -0.402*** 0.786*** 0.902*** 0.019 (0.564) (0.628) (0.298) (0.308) (0.352) (0.114) (0.142) (0.161) (0.044) MSA Dummies No No No No No No No No The estimates reported refer to the panel regressions ^ e i;t+hjt = (h) +z 0 i (h) + (h) x it +d 0 i (h) +" i;t+h + (h) i using an unbalanced panel of 4,971 respondents over 11 months, March 2012 to January 2013. ^ e i;t+hjt is expressed in per cent per quarter for all h. N = 35; 961, T min = 1, T p25 = 4, T p50 = 6, T = 7:23, T p75 = 9, Tmax = 11 FE-TE Filtered estimates with standard errors robust to heteroskedasticity and serial correlation. Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. Filtered estimates are computed using the estimator of Pesaran and Zhou (2016). 178 Table S4: FE-TE Filtered Estimates of (h) in the Panel Regressions of Individual Expected Price Changes on Belief Valuation Indicators for Dierent Assets (with 456 Response Pattern Dummies and MSA Dummies) Dependent variable: ^ e i;t+hjt One Month Ahead Three Months Ahead One Year Ahead Equity Gold Housing Equity Gold Housing Equity Gold Housing Female 0.801*** 1.208*** -0.027 0.877*** 1.099*** -0.031 0.564*** 0.668*** 0.017 (0.301) (0.365) (0.112) (0.173) (0.207) (0.048) (0.084) (0.099) (0.023) lnage -2.849*** -3.740*** -0.031 -2.476*** -2.883*** 0.002 -1.511*** -1.532*** 0.053 (0.551) (0.624) (0.173) (0.315) (0.352) (0.077) (0.149) (0.163) (0.038) Education -0.322 -1.161*** 0.068 -0.548*** -1.060*** 0.044 -0.399*** -0.669*** 0.024 (0.337) (0.391) (0.123) (0.187) (0.227) (0.049) (0.087) (0.104) (0.022) lnincome -0.672*** -1.227*** 0.183** -0.767*** -0.940*** 0.029 -0.417*** -0.541*** -0.011 (0.260) (0.316) (0.089) (0.156) (0.180) (0.041) (0.073) (0.082) (0.019) Asian -1.859 1.058 -0.113 0.022 0.874 -0.015 -0.166 0.495 0.078 (1.167) (1.529) (0.476) (0.658) (0.719) (0.161) (0.300) (0.340) (0.072) Black 1.042 1.390* -0.243 1.315*** 1.521*** -0.015 0.955*** 1.066*** 0.049 (0.690) (0.764) (0.233) (0.383) (0.421) (0.102) (0.181) (0.202) (0.050) Hispanic/Latino -0.428 0.749 -0.129 0.870** 1.324*** 0.124 0.714*** 0.790*** 0.137*** (0.617) (0.698) (0.190) (0.340) (0.389) (0.087) (0.158) (0.177) (0.042) MSA Dummies Yes Yes Yes Yes Yes Yes Yes Yes Yes The estimates reported refer to the panel regressions ^ e i;t+hjt = (h) +z 0 i (h) + (h) x it +d 0 i (h) +" i;t+h + (h) i using an unbalanced panel of 4,971 respondents over 11 months, March 2012 to January 2013. ^ e i;t+hjt is expressed in per cent per quarter for all h. N = 35; 961, T min = 1, T p25 = 4, T p50 = 6, T = 7:23, T p75 = 9, Tmax = 11 FE-TE Filtered estimates with standard errors robust to heteroskedasticity and serial correlation. Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. Filtered estimates are computed using the estimator of Pesaran and Zhou (2016). 179 Table S5: FE-TE Filtered Estimates of (h) in the Panel Regressions of Individual Expected Price Changes on Belief Valuation Indicators for Dierent Assets (with 0 Response Pattern Dummies) Dependent variable: ^ e i;t+hjt One Month Ahead Three Months Ahead One Year Ahead Equity Gold Housing Equity Gold Housing Equity Gold Housing Female 0.764** 1.173*** 0.139 0.898*** 1.061*** 0.031 0.570*** 0.622*** 0.037 (0.301) (0.356) (0.185) (0.171) (0.203) (0.069) (0.084) (0.098) (0.026) lnage -2.844*** -4.127*** 2.039*** -2.633*** -3.135*** 0.672*** -1.668*** -1.723*** 0.189*** (0.511) (0.591) (0.287) (0.296) (0.332) (0.107) (0.142) (0.158) (0.040) Education -0.343 -1.311*** 0.389** -0.622*** -1.279*** 0.165** -0.467*** -0.797*** 0.042* (0.329) (0.370) (0.187) (0.184) (0.220) (0.068) (0.087) (0.103) (0.025) lnincome -0.666*** -1.361*** 0.547*** -0.817*** -1.118*** 0.144*** -0.479*** -0.656*** 0.011 (0.241) (0.302) (0.135) (0.148) (0.172) (0.052) (0.070) (0.080) (0.021) Asian -1.577 0.982 -1.299* -0.210 0.557 -0.448* -0.170 0.299 -0.016 (1.056) (1.407) (0.738) (0.619) (0.687) (0.246) (0.274) (0.325) (0.089) Black 0.959 1.976*** -1.487*** 1.502*** 1.885*** -0.433*** 1.136*** 1.308*** -0.040 (0.665) (0.751) (0.354) (0.368) (0.409) (0.133) (0.173) (0.198) (0.053) Hispanic/Latino -0.097 1.279** -2.031*** 1.206*** 1.681*** -0.499*** 0.895*** 0.966*** -0.004 (0.586) (0.640) (0.295) (0.310) (0.353) (0.114) (0.142) (0.161) (0.043) MSA Dummies No No No No No No No No The estimates reported refer to the panel regressions ^ e i;t+hjt = (h) +z 0 i (h) + (h) x it +d 0 i (h) +" i;t+h + (h) i using an unbalanced panel of 4,971 respondents over 11 months, March 2012 to January 2013. ^ e i;t+hjt is expressed in per cent per quarter for all h. N = 35; 961, T min = 1, T p25 = 4, T p50 = 6, T = 7:23, T p75 = 9, Tmax = 11 FE-TE Filtered estimates with standard errors robust to heteroskedasticity and serial correlation. Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. Filtered estimates are computed using the estimator of Pesaran and Zhou (2016). 180 Table S6: FE-TE Filtered Estimates of (h) in the Panel Regressions of Individual Expected Price Changes on Belief Valuation Indicators for Dierent Assets (with 0 Response Pattern Dummies and MSA Dummies) Dependent variable: ^ e i;t+hjt One Month Ahead Three Months Ahead One Year Ahead Equity Gold Housing Equity Gold Housing Equity Gold Housing Female 0.791*** 1.165*** -0.001 0.895*** 1.071*** -0.016 0.567*** 0.634*** 0.023 (0.302) (0.358) (0.107) (0.172) (0.202) (0.047) (0.084) (0.098) (0.023) lnage -2.717*** -4.034*** 0.044 -2.568*** -3.113*** 0.001 -1.627*** -1.720*** 0.024 (0.508) (0.595) (0.162) (0.297) (0.334) (0.073) (0.144) (0.159) (0.037) Education -0.424 -1.276*** 0.085 -0.690*** -1.234*** 0.044 -0.502*** -0.780*** 0.006 (0.337) (0.387) (0.121) (0.187) (0.223) (0.049) (0.089) (0.104) (0.022) lnincome -0.627** -1.301*** 0.246*** -0.791*** -1.069*** 0.046 -0.461*** -0.624*** -0.012 (0.247) (0.309) (0.087) (0.151) (0.175) (0.040) (0.071) (0.080) (0.019) Asian -1.430 1.047 -0.097 -0.082 0.651 -0.041 -0.133 0.390 0.067 (1.061) (1.421) (0.443) (0.629) (0.706) (0.159) (0.280) (0.338) (0.076) Black 0.790 1.638** -0.286 1.378*** 1.755*** -0.034 1.046*** 1.253*** 0.054 (0.681) (0.769) (0.222) (0.376) (0.414) (0.100) (0.178) (0.201) (0.049) Hispanic/Latino -0.612 0.645 -0.164 0.928*** 1.338*** 0.119 0.759*** 0.802*** 0.135*** (0.640) (0.717) (0.185) (0.341) (0.395) (0.088) (0.157) (0.179) (0.042) MSA Dummies Yes Yes Yes Yes Yes Yes Yes Yes Yes The estimates reported refer to the panel regressions ^ e i;t+hjt = (h) +z 0 i (h) + (h) x it +d 0 i (h) +" i;t+h + (h) i using an unbalanced panel of 4,971 respondents over 11 months, March 2012 to January 2013. ^ e i;t+hjt is expressed in per cent per quarter for all h. N = 35; 961, T min = 1, T p25 = 4, T p50 = 6, T = 7:23, T p75 = 9, Tmax = 11 FE-TE Filtered estimates with standard errors robust to heteroskedasticity and serial correlation. Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. Filtered estimates are computed using the estimator of Pesaran and Zhou (2016). 181 S18 Comparison of alternative estimates of (h) and im- plied interest rate r In Table S2 we present a comparison of the estimates of (h) in the equation ^ e i;t+hjt = (h) + z 0 i (h) + (h) x it +" i;t+h + (h) i (S.1) for dierent model specications. We consider FE and FE-TE estimates of (h) . We also consider a model where (h) i is treated as random. We estimate the RE model with and without the time-invariant characteristics z i , and with/without time and MSA dummies. Then, using the estimates of (h) for the housing market, we calculate the estimated interest rate, ^ r. Given the estimates ^ (h 1 ) and ^ (h 2 ) , compute the interest rate estimates as follows: ^ r h 1 ;h 2 = h 1 h 2 (h 1 ) (h 2 ) 1 h 1 h 2 1; for cases wherej ^ (h 1 ) j<j ^ (h 2 ) j. The interest rate estimates are presented in Table S1. Table S1: Alternative estimates of the discount rate r, using FE, FE-TE and RE estimates of (h) for house prices FE FE-TE RE ^ r 3;1 0:044 0:039 0:082 0:082 0:055 0:057 0:091 0:086 0:082 0:079 ^ r 12;1 0:064 0:060 0:058 0:055 0:055 0:053 0:070 0:067 0:065 0:063 ^ r 12;3 0:069 0:065 0:053 0:049 0:055 0:052 0:066 0:063 0:061 0:059 Time Dummies No Yes No Yes No Yes No Yes MSA Dummies No No Yes Yes No No Yes Yes Demographics No No No No Yes Yes Yes Yes 182 Table S2: Estimates of (h) in equation (26) for dierent model specications horizon FE FE-TE RE equity One Month -0.0991 -0.126 -0.0849 -0.107 -0.108 -0.131 -0.124 -0.146 -0.133 -0.156 Ahead (0.127) (0.128) (0.116) (0.116) (0.116) (0.117) (0.116) (0.116) (0.116) (0.117) Three Months -0.0905 -0.0995 -0.0719 -0.0798 -0.0908 -0.0988 -0.118* -0.126* -0.120* -0.128* Ahead (0.0760) (0.0760) (0.0703) (0.0703) (0.0705) (0.0705) (0.0700) (0.0700) (0.0703) (0.0704) One Year -0.115*** -0.117*** -0.111*** -0.112*** -0.121*** -0.122*** -0.138*** -0.140*** -0.139*** -0.140*** Ahead (0.0365) (0.0364) (0.0339) (0.0339) (0.0340) (0.0340) (0.0337) (0.0337) (0.0339) (0.0339) gold One Month 0.602*** 0.581*** 0.409 ** 0.389 ** 0.455 *** 0.435 ** 0.581 *** 0.562 *** 0.591 *** 0.572 *** Ahead (0.197) (0.198) (0.175) (0.176) (0.176) (0.177) (0.177) (0.177) (0.178) (0.178) Three Months 0.222** 0.203* 0.0850 0.0678 0.113 0.0960 0.208 ** 0.190 * 0.213 ** 0.195 * Ahead (0.108) (0.109) (0.0986) (0.0987) (0.0990) (0.0992) (0.0993) (0.0994) (0.0997) (0.0998) One Year -0.0226 -0.0316 -0.114 ** -0.122 *** -0.0996 ** -0.108 ** -0.0565 -0.0646 -0.0524 -0.0606 Ahead (0.0488) (0.0489) (0.0453) (0.0454) (0.0455) (0.0456) (0.0455) (0.0456) (0.0457) (0.0458) housing One Month -0.292*** -0.303*** -0.443 *** -0.456 *** -0.412 *** -0.419 *** -0.363 *** -0.374 *** -0.382 *** -0.389 *** Ahead (0.0643) (0.0642) (0.0602) (0.0601) (0.0545) (0.0544) (0.0604) (0.0602) (0.0547) (0.0546) Three Months -0.106*** -0.109*** -0.173 *** -0.178 *** -0.153 *** -0.156 *** -0.144 *** -0.147 *** -0.149 *** -0.151 *** Ahead (0.0273) (0.0274) (0.0252) (0.0252) (0.0239) (0.0239) (0.0254) (0.0254) (0.0242) (0.0242) One Year -0.0481*** -0.0479*** -0.0687 *** -0.0686 *** -0.0618 *** -0.0616 *** -0.0638 *** -0.0637 *** -0.0634 *** -0.0632 *** Ahead (0.0102) (0.0102) (0.00941) (0.00943) (0.00937) (0.00939) (0.00945) (0.00947) (0.00940) (0.00942) Time Dummies No Yes No Yes No Yes No Yes MSA Dummies No No Yes Yes No No Yes Yes Demographics No No No No Yes Yes Yes Yes The equation ^ e i;t+hjt = (h) x it +z 0 i (h) + i +t +" i;t+h is estimated using an unbalanced panel of 4,971 respondents over 11 months, March 2012 to January 2013. ^ e i;t+hjt is expressed in per cent per quarter for all h. N = 35; 961, T min = 1, T p25 = 4, T p50 = 6, T = 7:23, T p75 = 9, Tmax = 11 Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. Standard errors are robust to heteroskedasticity and serial correlation. y Female= 1 183 S19 Regression results controlling for home-ownership In this section we present results obtained by matching the data from the DQ Survey with another survey carried out by RAND ALP - the Eects of the Financial Crisis Survey. The Financial Crisis Survey was elded during November 2008 - January 2016, and the survey data can be accessed at https://alpdata.rand.org/index.php?page=data. The survey is of interest to us since it contains information on home-ownership. To match the respondents form the two surveys, we used the fact that the respondent identier variable, \prim key", is uniquely assigned to a respondent across all surveys. For each month from March 2012 through January 2013, we kept those respondents of the Double Question Survey who had also participated in the Financial Crisis Survey in the same month. We also applied analo- gous lters to the one used for gender and race, which eliminates respondents who provides information that is not consistent over time with respect to the home-ownership variable. We ended up with a sample of 3,325 respondents who had participated in both surveys, and for whom we knew whether they were homeowners or not. The fraction of homeowners in this sample is 29%. This is signicantly lower than the national rate of home-ownership, which was around 65% during the survey period. We then estimate the model introduced in equation (2.4.6) in the paper separately for home- owners and non-homeowners. Specically, we consider ^ e i;t+hjt = (h) i + (h) 1 x it + (h) t +" i;t+h for i2 1 ; (S.1) and ^ e i;t+hjt = (h) i + (h) 2 x it + (h) t +" i;t+h for i2 2 ; (S.2) where 1 and 2 is the set of homeowners and non-homeowners, respectively. The estimates of ( (h) 1 ; (h) 2 ) for the three dierent asset classes, and for all the three horizons,h = 1; 3; and 12, are summarized in Table S1. 184 Table S1: Estimates of (h) in the panel regressions of individual expected price changes on their belief valuation indicators for dif- ferent assets by homeownership Dependent variable: ^ e i;t+hjt Homeowners Equity Gold Housing Horizons FE FE-TE FE FE-TE FE FE-TE One Month -0.259 -0.236 0.656 0.725 -0.170 -0.164 Ahead (h = 1) (-0.71) (-0.66) (1.38) (1.52) (-1.43) (-1.38) Three Months -0.133 -0.142 0.0932 0.128 -0.0364 -0.0301 Ahead (h = 1) (-0.66) (-0.72) (0.33) (0.46) (-0.59) (-0.49) One Year -0.0636 -0.0665 -0.0305 -0.0258 -0.0526 -0.0494 Ahead (h = 1) (-0.62) (-0.65) (-0.22) (-0.19) (-1.93) (-1.81) Non-Homeowners Equity Gold Housing Horizons FE FE-TE FE FE-TE FE FE-TE One Month -0.112 -0.141 0.0965 0.0604 -0.203 -0.223 Ahead (h = 1) (-0.68) (-0.86) (0.44) (0.27) (-1.86) (-2.06) Three Months -0.179 -0.198* -0.0729 -0.0996 -0.0818* -0.0897* Ahead (h = 1) (-1.83) (-2.04) (-0.59) (-0.81) (-2.05) (-2.27) One Year -0.202*** -0.210*** -0.185** -0.190** -0.0493*** -0.0507*** Ahead (h = 1) (-4.59) (-4.76) (-3.06) (-3.14) (-3.69) (-3.80) Fixed eect (FE) estimates of (h) in the panel regression ^ e i;t+hjt = (h) i + (h) x it +u (h) it are obtained with and without time eects (FE-TE) using an unbalanced panel of respondents over 11 months, March 2012 to January 2013. The regressions for homeowners are estimated using 2,910 respondents and 20,602 responses. The regressions for non-homeowners are estimated using 2,061 respondents and 15,359 responses. Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. Standard errors are robust to heteroskedasticity and residual serial correlation. 185 Then we estimate the panel data model ^ e i;t+hjt = (h) + ~ z 0 i (h) + + (h) x it +" i;t+h + (h) i ; (S.3) where the variables are the same as previously dened, except for ~ z i , which now includes a home-ownership dummy in addition to the previously considered time-invariant individual characteristics. FEF and RE estimates of the model are presented in tables S5-S4. Looking at the RE estimates in Tables S2-S4, we see that homeowners form slightly higher equity price expectations that non-homeowners for the three month and one year expectation horizons. There are no signicant eects for gold expectations, and the eects for housing are positive after controlling for MSA xed eects. Looking at the FEF estimates in Table S5, we see that the equity price expectations for three month and one year horizons are higher for homeowners, there are no signicant eects for gold, and the one month house price expectations for homeowners are lower. 186 Table S2: Random Eect Estimates of (h) and (h) in the Panel Regressions of Individual Expected Price Changes on Belief Valuation Indicators for Equity for Samples with Home-ownership Indicators Dependent variable:^ e i;t+hjt One Month Ahead Three Months Ahead One Year Ahead x it -0.197 -0.222* -0.180 -0.205 -0.205** -0.220*** -0.192** -0.207*** -0.190*** -0.196*** -0.184*** -0.189*** (0.135) (0.134) (0.136) (0.135) (0.0799) (0.0795) (0.0805) (0.0802) (0.0388) (0.0389) (0.0392) (0.0392) Female 0.121 0.118 0.147 0.144 0.448*** 0.442*** 0.488*** 0.482*** 0.431*** 0.427*** 0.447*** 0.444*** (0.255) (0.255) (0.260) (0.260) (0.159) (0.159) (0.163) (0.163) (0.0836) (0.0836) (0.0852) (0.0852) lnage -0.894* -0.877* -0.829* -0.813 -1.157*** -1.167*** -1.113*** -1.118*** -0.842*** -0.854*** -0.792*** -0.800*** (0.505) (0.506) (0.503) (0.504) (0.295) (0.296) (0.294) (0.295) (0.152) (0.152) (0.153) (0.153) Education -0.331 -0.332 -0.575** -0.575** -0.489*** -0.495*** -0.620*** -0.627*** -0.360*** -0.363*** -0.418*** -0.422*** (0.268) (0.269) (0.270) (0.270) (0.163) (0.163) (0.166) (0.166) (0.0833) (0.0834) (0.0859) (0.0860) lnincome -0.505** -0.505** -0.510** -0.511** -0.453*** -0.458*** -0.469*** -0.474*** -0.284*** -0.290*** -0.294*** -0.299*** (0.211) (0.212) (0.216) (0.216) (0.141) (0.142) (0.144) (0.144) (0.0744) (0.0744) (0.0749) (0.0749) Asian -0.309 -0.320 -0.462 -0.470 0.333 0.336 0.264 0.261 0.105 0.115 0.120 0.122 (0.976) (0.976) (1.008) (1.007) (0.601) (0.601) (0.636) (0.634) (0.286) (0.285) (0.303) (0.302) Black 0.983 0.984 0.648 0.652 1.628*** 1.647*** 1.444*** 1.461*** 1.210*** 1.228*** 1.118*** 1.132*** (0.639) (0.640) (0.624) (0.624) (0.381) (0.382) (0.377) (0.378) (0.200) (0.200) (0.204) (0.204) Hispanic/Latino 0.886 0.879 0.408 0.404 1.137*** 1.159*** 0.877** 0.880** 0.703*** 0.731*** 0.574*** 0.581*** (0.587) (0.584) (0.617) (0.616) (0.335) (0.334) (0.362) (0.362) (0.159) (0.159) (0.174) (0.174) Homeowner 0.233 0.236 0.256 0.260 0.527** 0.540** 0.554** 0.562** 0.373*** 0.384*** 0.410*** 0.415*** (0.376) (0.376) (0.393) (0.393) (0.220) (0.220) (0.227) (0.227) (0.112) (0.113) (0.115) (0.115) Time Dummies No Yes No Yes No Yes No Yes No Yes No Yes MSA Dummies No No Yes Yes No No Yes Yes No No Yes Yes The estimates reported refer to the panel regressions ^ e i;t+hjt = (h) x it +z 0 i (h) + (h) i +" i;t+h using an unbalanced panel of 3,325 respondents over 11 months, March 2012 to January 2013. ^ e i;t+hjt is expressed in per cent per quarter for all h. N = 20; 663, T min = 1, T p25 = 3, T p50 = 5, T p75 = 10, Tmax = 11 Random eect estimates with standard errors clustered at individual level. Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. 187 Table S3: Random Eect Estimates of (h) and (h) in the Panel Regressions of Individual Expected Price Changes on Belief Valuation Indicators for Gold for Samples with Home-ownership Indicators Dependent variable:^ e i;t+hjt One Month Ahead Three Months Ahead One Year Ahead x it 0.410** 0.391** 0.425** 0.406** 0.0352 0.0178 0.0474 0.0301 -0.150*** -0.155*** -0.144*** -0.149*** (0.184) (0.185) (0.186) (0.187) (0.109) (0.110) (0.110) (0.111) (0.0552) (0.0553) (0.0554) (0.0555) Female 0.681** 0.673** 0.648* 0.641* 0.791*** 0.787*** 0.819*** 0.816*** 0.517*** 0.515*** 0.546*** 0.545*** (0.330) (0.330) (0.336) (0.336) (0.212) (0.212) (0.214) (0.214) (0.108) (0.108) (0.108) (0.109) lnage -3.306*** -3.319*** -3.391*** -3.396*** -2.700*** -2.703*** -2.691*** -2.688*** -1.513*** -1.512*** -1.520*** -1.517*** (0.639) (0.640) (0.640) (0.640) (0.399) (0.399) (0.400) (0.401) (0.198) (0.198) (0.198) (0.198) Education -1.400*** -1.399*** -1.384*** -1.386*** -1.306*** -1.307*** -1.275*** -1.277*** -0.774*** -0.774*** -0.770*** -0.771*** (0.332) (0.332) (0.336) (0.335) (0.210) (0.210) (0.212) (0.212) (0.106) (0.106) (0.108) (0.108) lnincome -1.672*** -1.685*** -1.741*** -1.753*** -1.233*** -1.240*** -1.221*** -1.228*** -0.671*** -0.673*** -0.656*** -0.657*** (0.302) (0.302) (0.303) (0.302) (0.193) (0.193) (0.192) (0.192) (0.0944) (0.0944) (0.0923) (0.0923) Asian 0.750 0.786 0.839 0.857 0.582 0.598 0.724 0.732 0.414 0.419 0.516 0.519 (1.073) (1.070) (1.093) (1.089) (0.686) (0.684) (0.699) (0.696) (0.363) (0.363) (0.366) (0.365) Black 2.103** 2.141** 1.872** 1.901** 1.851*** 1.870*** 1.707*** 1.722*** 1.084*** 1.091*** 1.031*** 1.037*** (0.845) (0.846) (0.831) (0.831) (0.506) (0.507) (0.502) (0.502) (0.231) (0.231) (0.233) (0.234) Hispanic/Latino 0.988 1.061 0.451 0.476 0.793* 0.830** 0.411 0.422 0.497** 0.507*** 0.321 0.323 (0.670) (0.671) (0.753) (0.752) (0.407) (0.407) (0.462) (0.461) (0.194) (0.195) (0.217) (0.217) Homeowner -0.119 -0.0992 -0.195 -0.187 0.163 0.173 0.228 0.233 0.135 0.138 0.197 0.199 (0.475) (0.474) (0.486) (0.486) (0.288) (0.288) (0.294) (0.294) (0.142) (0.142) (0.145) (0.145) Time Dummies No Yes No Yes No Yes No Yes No Yes No Yes MSA Dummies No No Yes Yes No No Yes Yes No No Yes Yes The estimates reported refer to the panel regressions ^ e i;t+hjt = (h) x it +z 0 i (h) + (h) i +" i;t+h using an unbalanced panel of 3,325 respondents over 11 months, March 2012 to January 2013. ^ e i;t+hjt is expressed in per cent per quarter for all h. N = 20; 663, T min = 1, T p25 = 3, T p50 = 5, T p75 = 10, Tmax = 11 Random eect estimates with standard errors clustered at individual level. Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. 188 Table S4: Random Eect Estimates of (h) and (h) in the Panel Regressions of Individual Expected Price Changes on Belief Valuation Indicators for Housing for Samples with Home-ownership Indicators Dependent variable:^ e i;t+hjt One Month Ahead Three Months Ahead One Year Ahead x it -0.256*** -0.276*** -0.321*** -0.338*** -0.106*** -0.113*** -0.128*** -0.135*** -0.0658*** -0.0664*** -0.0696*** -0.0703*** (0.0777) (0.0773) (0.0672) (0.0671) (0.0304) (0.0305) (0.0284) (0.0286) (0.0112) (0.0113) (0.0111) (0.0112) Female 0.0887 0.0864 -0.0358 -0.0334 0.0289 0.0283 -0.0178 -0.0167 0.0357 0.0356 0.0205 0.0208 (0.200) (0.200) (0.105) (0.106) (0.0751) (0.0750) (0.0497) (0.0497) (0.0273) (0.0273) (0.0238) (0.0238) lnage 2.285*** 2.286*** 0.116 0.137 0.812*** 0.812*** 0.0949 0.101 0.259*** 0.261*** 0.0847** 0.0875** (0.354) (0.353) (0.163) (0.163) (0.132) (0.131) (0.0811) (0.0811) (0.0485) (0.0485) (0.0416) (0.0416) Education 0.124 0.123 -0.00314 -0.000636 0.114 0.114 0.0497 0.0509 0.0353 0.0353 0.00878 0.00915 (0.202) (0.201) (0.108) (0.108) (0.0746) (0.0745) (0.0499) (0.0499) (0.0269) (0.0269) (0.0241) (0.0241) lnincome 0.712*** 0.712*** 0.433*** 0.441*** 0.252*** 0.253*** 0.170*** 0.173*** 0.0695*** 0.0703*** 0.0532** 0.0544** (0.169) (0.169) (0.0933) (0.0931) (0.0649) (0.0647) (0.0448) (0.0448) (0.0249) (0.0249) (0.0223) (0.0224) Asian -1.708** -1.727** -0.601 -0.617 -0.466 -0.473 -0.121 -0.127 -0.0167 -0.0187 0.0509 0.0491 (0.836) (0.834) (0.407) (0.409) (0.290) (0.290) (0.171) (0.172) (0.102) (0.103) (0.0876) (0.0877) Black -0.946** -0.948** -0.275 -0.287 -0.260 -0.262 -0.0424 -0.0478 -0.0195 -0.0201 0.0382 0.0370 (0.440) (0.440) (0.262) (0.262) (0.164) (0.164) (0.120) (0.120) (0.0591) (0.0591) (0.0553) (0.0553) Hispanic/Latino -2.671*** -2.685*** -0.608*** -0.627*** -0.826*** -0.832*** -0.157 -0.165* -0.115** -0.118** 0.0273 0.0249 (0.393) (0.393) (0.197) (0.197) (0.149) (0.149) (0.0968) (0.0969) (0.0533) (0.0534) (0.0488) (0.0488) Homeowner -0.529* -0.524* 0.283* 0.283* -0.104 -0.103 0.179** 0.179** 0.0510 0.0505 0.127*** 0.127*** (0.277) (0.276) (0.151) (0.150) (0.104) (0.103) (0.0710) (0.0709) (0.0381) (0.0381) (0.0334) (0.0334) Time Dummies No Yes No Yes No Yes No Yes No Yes No Yes MSA Dummies No No Yes Yes No No Yes Yes No No Yes Yes The estimates reported refer to the panel regressions ^ e i;t+hjt = (h) x it +z 0 i (h) + (h) i +" i;t+h using an unbalanced panel of 3,325 respondents over 11 months, March 2012 to January 2013. ^ e i;t+hjt is expressed in per cent per quarter for all h. N = 20; 663, T min = 1, T p25 = 3, T p50 = 5, T p75 = 10, Tmax = 11 Random eect estimates with standard errors clustered at individual level. Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. 189 Table S5: Fixed Eects Filtered Estimates of Price Expectation Equations for Samples with Home-ownership Indi- cators Dependent variable: ^ e i;t+hjt One Month Ahead Three Months Ahead One Year Ahead Equity Gold Housing Equity Gold Housing Equity Gold Housing (h) -0.153 0.271 -0.194** -0.166* -0.021 -0.069** -0.163*** -0.137** -0.050*** (0.156) (0.213) (0.085) (0.090) (0.121) (0.033) (0.043) (0.060) (0.012) Female ( (h) 3 ) 0.056 0.621* 0.108 0.437** 0.771*** 0.038 0.427*** 0.511*** 0.039 (0.297) (0.359) (0.207) (0.177) (0.222) (0.078) (0.090) (0.112) (0.029) lnage ( (h) 1 ) -1.171** -3.546*** 2.315*** -1.308*** -2.774*** 0.821*** -0.870*** -1.516*** 0.260*** (0.549) (0.698) (0.364) (0.316) (0.416) (0.137) (0.159) (0.202) (0.051) Education ( (h) 7 ) -0.369 -1.410*** 0.126 -0.534*** -1.320*** 0.117 -0.389*** -0.784*** 0.032 (0.316) (0.368) (0.211) (0.183) (0.223) (0.078) (0.089) (0.110) (0.028) lnincome ( (h) 2 ) -0.568** -1.709*** 0.733*** -0.483*** -1.236*** 0.260*** -0.291*** -0.669*** 0.069*** (0.240) (0.333) (0.175) (0.153) (0.202) (0.068) (0.078) (0.097) (0.026) Asian ( (h) 4 ) -0.639 0.331 -1.676* 0.229 0.470 -0.443 0.070 0.373 -0.007 (1.197) (1.271) (0.863) (0.661) (0.736) (0.298) (0.299) (0.378) (0.104) Black ( (h) 5 ) 0.811 1.930** -0.934** 1.601*** 1.822*** -0.255 1.190*** 1.066*** -0.020 (0.706) (0.895) (0.448) (0.404) (0.527) (0.169) (0.208) (0.238) (0.061) Hispanic/Latino ( (h) 6 ) 0.812 0.894 -2.663*** 1.105*** 0.784* -0.829*** 0.711*** 0.501** -0.117** (0.691) (0.738) (0.400) (0.369) (0.427) (0.153) (0.168) (0.201) (0.055) Homeowner ( (h) 8 ) 0.207 -0.143 -0.551* 0.525** 0.160 -0.118 0.389*** 0.153 0.049 (0.418) (0.522) (0.287) (0.238) (0.305) (0.109) (0.119) (0.148) (0.040) The estimates reported refer to the panel regressions ^ e i;t+hjt = (h) x it +z i 0 (h) + (h) i +" i;t+h using an unbalanced panel of 3,325 respondents over 11 months, March 2012 to January 2013. ^ e i;t+hjt is expressed in per cent per quarter for all h. N = 20; 663, T min = 1, T p25 = 3, T p50 = 5, T p75 = 10, Tmax = 11 Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. Estimates of (h) are obtained using the FE Filtered estimator or Pesaran and Zhou (2016). 190 Table S6: Fixed Eects Filtered Estimates of Price Expectation Equations for Samples with Home-Ownership Indicators and MSA Dumies Dependent variable: ^ e i;t+hjt One Month Ahead Three Months Ahead One Year Ahead Equity Gold Housing Equity Gold Housing Equity Gold Housing (h) -0.153 0.271 -0.194** -0.166* -0.021 -0.069** -0.163*** -0.137** -0.050*** (0.156) (0.213) (0.085) (0.090) (0.121) (0.033) (0.043) (0.060) (0.012) Female ( (h) 3 ) 0.074 0.579 -0.017 0.478*** 0.799*** -0.007 0.442*** 0.540*** 0.025 (0.299) (0.364) (0.121) (0.180) (0.224) (0.055) (0.091) (0.111) (0.025) lnage ( (h) 1 ) -1.119** -3.660*** 0.111 -1.266*** -2.779*** 0.085 -0.815*** -1.527*** 0.082* (0.535) (0.692) (0.191) (0.311) (0.414) (0.090) (0.159) (0.200) (0.044) Education ( (h) 7 ) -0.600* -1.399*** -0.036 -0.662*** -1.295*** 0.043 -0.444*** -0.784*** 0.003 (0.316) (0.371) (0.125) (0.185) (0.224) (0.055) (0.092) (0.111) (0.025) lnincome ( (h) 2 ) -0.594** -1.793*** 0.501*** -0.509*** -1.232*** 0.188*** -0.305*** -0.658*** 0.054** (0.241) (0.330) (0.111) (0.154) (0.199) (0.050) (0.078) (0.094) (0.023) Asian ( (h) 4 ) -0.788 0.438 -0.728 0.158 0.617 -0.131 0.083 0.476 0.055 (1.208) (1.265) (0.524) (0.691) (0.746) (0.194) (0.315) (0.381) (0.090) Black ( (h) 5 ) 0.542 1.763** -0.298 1.439*** 1.697*** -0.045 1.107*** 1.021*** 0.038 (0.681) (0.870) (0.293) (0.394) (0.518) (0.129) (0.209) (0.237) (0.058) Hispanic/Latino ( (h) 6 ) 0.358 0.345 -0.695*** 0.855** 0.382 -0.193* 0.585*** 0.315 0.017 (0.716) (0.834) (0.223) (0.400) (0.488) (0.106) (0.185) (0.224) (0.051) Homeowner ( (h) 8 ) 0.219 -0.244 0.263 0.539** 0.212 0.167** 0.415*** 0.210 0.128*** (0.429) (0.529) (0.178) (0.244) (0.309) (0.080) (0.122) (0.150) (0.036) MSA Dummies Yes Yes Yes Yes Yes Yes Yes Yes Yes The estimates reported refer to the panel regressions ^ e i;t+hjt = (h) x it +z i 0 (h) + (h) i +" i;t+h using an unbalanced panel of 3,325 respondents over 11 months, March 2012 to January 2013. ^ e i;t+hjt is expressed in per cent per quarter for all h. N = 20; 663, T min = 1, T p25 = 3, T p50 = 5, T p75 = 10, Tmax = 11 Standard errors are in parentheses, *, ** and *** denote statistical signicance at 10%, 5% and 1% levels, respectively. Estimates of (h) are obtained using the FE Filtered estimator or Pesaran and Zhou (2016). 191 References Arellano, M. and S. Bond (1991). Some tests of specication for panel data: Monte carlo evidence and an application to employment equations. The Review of Economic Studies 58 (2) 277-297. Pesaran, M. H. (2015). Time Series and Panel Data Econometrics. Oxford: Oxford University Press. Pesaran, M. H. and Q. Zhou (2016). Estimation of time-invariant eects in static panel data models. Econometric Reviews (forthcoming). Wansbeek, T. and A. Kapteyn (1989) Estimation of the error-components model with incomplete panels. Journal of Econometrics 41 (3), 341-361. 192 Chapter 3 Spatial Equilibrium and Search Frictions - an Application to the NYC Taxi Market 1 Ida Johnsson y This paper uses a dynamic spatial equilibrium model to analyze the eect of matching frictions and pricing policy on the spatial allocation of taxicabs and the aggregate number of taxi-passenger meetings. A spatial equilibrium model, in which meetings are frictionless but aggregate matching frictions can arise endogenously for certain parameter values, is cal- ibrated using data on more than 45 million taxi rides in New York. It is shown how the set of equilibria changes for dierent pricing rules and dierent levels of aggregate market tightness, dened as the ratio of total supply to total demand. Finally, a novel data-driven algorithm for inferring unobserved demand from the data is proposed, and is applied to ana- lyze how the relationship between demand and supply in a market with frictions compares to the frictionless equilibrium outcome. Keywords: patial equilibrium, matching, industry dynamics, taxicabs valua- 1 I would like to thank Professor Hashem Pesaran and Professor Hyungsik Roger Moon for their guidance and support. y Department of Economics, University of Southern California 193 tions. 3.1 Introduction The distribution of taxicabs in big cities tends to be imbalanced, in some areas passengers have a hard time nding a taxi, in others taxis can't easily nd a passenger. In the taxi mar- ket, trades occur bilaterally between agents, and the equilibrium outcome depends on the nature of the meeting process. For example, the matching mechanism between yellow cabs in New York and passengers, who have to hail the cab from the street, is dierent than for ride-hailing services such as Uber or Lyft that match passengers and cabs using an algorithm. As prices are typically xed, the key equilibrium variables in the taxi market are meeting probabilities and aggregate market tightness (AMT), dened as the ratio of total supply to total demand. In this paper I rst consider the spatial equilibrium model proposed by Lagos (2000), in which the taxi-passenger matching process is frictionless. Frictions in the model can arise endogenously if taxis choose to locate in such a way that there is excess supply in some locations and excess demand in others. Frictions depend on aggregate market tightness and location heterogeneity. I calibrate the model using data on over 45 million taxi rides and show that an equilibrium with frictionless matching yields excess supply of cabs in all locations, and that aggregate matches are thus determined by total demand. I then extend the model of Lagos (2000) to analyze how varying the pricing policy impacts the ratio of supply to demand in each location, as well as the aggregate number of matches. I consider both the current two-part tari as well as a demand-dependent pricing policy. Finally, I consider a taxi-passenger matching process that involves local frictions in the sense that the number of rides originating from a neighborhood with x empty taxis looking for passengers and y customers in looking for a taxi might be smaller than minfx;yg. This type of matching process is considered by for example Frechette et al. (2015) and Buchholz 194 (2016). Similar to Frechette et al. (2015), Buchholz (2016) and Aan et al. (2015), I propose a method of inferring unobserved taxi demand in the presence of local matching frictions. My method is based on dening a local measure of taxi availability, which can be approximated using data on taxi rides. I than compute time-specic local market tightness (tLMT), dened as the ratio of supply to demand in a specic location during a certain time interval, and compare the empirical outcomes with the results implied by the equilibrium of the frictionless matching model. 3.1.1 Related Literature Lagos (2000) proposes a spatial equilibrium model in which taxi drivers choose their location depending on demand and policy parameters. Matching is assumed to be frictionless, and it is shown that frictions arise endogenously in equilibria that correspond to certain parameter values. Lagos (2000) shows that an aggregate matching function exists, and characterizes it's properties with respect to distances between locations, the size of agent populations, as well as policy parameters. He shows that the spatial equilibrium depends on the demand transition matrix, the full set of bilateral distances, demand and policy parameters that determine fare rate and the number of medallions. Lagos (2003) uses the model of Lagos (2000) to quantify the impact of increasing taxi fares and the medallion cap on the medallion prices and the process that rules meetings between passengers and taxicabs in New York. I also use the model of Lagos (2000). The dierences between my work and that of Lagos (2003) are as follows. Lagos (2003) assumes that all medallions are active throughout the day and uses data on 22,604 trips in 1988 to cali- brate the model, characterize all equilibria, and quantify the meeting process generated by Manhattan's market for taxicab rides. I use data from January-March 2013, which includes comprehensive information on all taxi rides during that period - around 45 million trips. In- stead of assuming that all medallions are active throughout the day, I consider two dierent time periods - day and night - and calculate the number of active medallions from my data. Further, Lagos (2003) simplies the map of Manhattan by dividing it into seven locations, 195 whereas I consider forty dierent neighborhoods, both in and outside of Manhattan. Similar to Lagos (2003), I calibrate the frictionless spatial equilibrium model and characterize the no-frictions frontier. Further, I extend the model to include demand-driven pricing rules and calculate the equilibrium allocations for selected pricing rules. Another strand of literature considers equilibrium models of the NYC Taxi market in the presence of matching frictions. Frechette et al. (2015) propose a dynamic general equilibrium model of the NYC Taxi market, and back out unobserved demand through a matching simu- lation. They assume that taxi search times and passenger wait times (the latter unobserved) are an unknown function of the unobserved number of waiting passengers, searching taxis and other exogenous time-varying variables. They simulate the matching process of waiting passengers and searching taxis on a grid that represents a simplied version of Manhattan to back out the unknown function and the unobserved inputs. The authors do not consider location-specic heterogeneity. Buchholz (2016) models the driver's choice problem using a state dependent value function. He assumes that demand for rides in each location comes from a location-specic Poisson distribution, and estimates the parameters of these distributions as well as the supply of taxis from the equilibrium of the model. My work is similar to Buchholz (2016) in the sense that I also consider location-specic supply and demand. However, rather than assuming a model for the equilibrium behavior of taxis and passengers and backing out unobserved supply and demand from the model equi- librium, I take advantage of the fact the we observe passenger pick-up and drop-o locations. I dene a measure of taxi availability in each neighborhood and time period, and estimate taxi and passenger arrival rates directly from the data. This approach is similar to the one taken by Aan et al. (2015). The rest of the paper is structured as follows. Section 3.2 describes the data. In Sec- tion 3.3 I present the benchmark model with frictionless matching, and analyze how the 196 no-frictions frontier changes for dierent pricing policies. In Section 3.4 I introduce the matching process with local frictions and show estimates of local market tightness for dier- ent matching eciency parameters. Section 3.5 concludes. I what follows I use lowercase letters to denote scalars, lowercase bold letters for vectors and uppercase bold letters for matrices. 3.2 Data I use a data set obtained by Mr. Chris Whong through a Freedom of Information Law (FOIL) request submitted to the Taxi and Limousine Commission (TLC) 2 . The data contains information on more than 173 million yellow cab trips in 2013. Each record contains complete information about a taxi ride. I use the following variables (a specication of all variables available in the data can be found in Appendix A1). Medallion - a unique identier of the taxi cab. Pickup and drop-o time and geographical coordinates Note that data on all yellow cab rides from 2009 till the present date, as well as green cab rides from mid-2013 onward and Uber rides for selected months of 2014 and 2015 is freely available in Google Big Query. However, that data does not contain information on the taxi or driver identiers, which is crucial for inferring unmet demand. Green cabs were introduced in the summer of 2013. Hence, I restrict my analysis of the yellow cab market to the period before the summer of 2013. Specically, I consider data from January-March, which contains information on approximately 45 million taxi rides. I restrict my analysis to this period due to the computational costs of geofencing the data. Using the geographical coordinates of the pickup and drop-o locations for each trip, I map 2 Mr. Whong made the data set available on his website http://chriswhong.com/open-data/foil_ nyc_taxi 197 the trip start and end points to a neighborhood. I use the neighborhood denitions of the NYC Neighborhood Tabulation Areas (NTAs) 3 , which are shown in Figure A1 in Appendix A1. Mapping pickup and dropo coordinates to NTAs for January-March 2013 took approxi- mately 72 hours running on an Amazon Web Services (AWS) memory-optimized PostgreSQL instance of type db.r3.8xlarge with 32 vCPUs and 244GiB Memory. I consider 15-minute time intervals during the time period January-March 2013. Plots of pickups and drop-os, aggregated into 15-minute time intervals, during a randomly chosen week in March for selected neighborhoods are shown below in Figure 3.1. As we can see, the data follows a pattern where pickups/drop-os are generally higher during the period 6am- 11pm, although the pattern varies by neighborhood. For example, pickups and Midtown- Midtown South are lower on weekends, whereas the opposite is true for Chinatown. This suggests that the taxi-passenger meeting dynamics vary according to the time of the day. I consider 40 neighborhoods for which the average number of pickups in a time interval during the time period considered is at least 5. These neighborhoods are listed in Table 3.1. Further, I divide my data into two time periods: (a) - from 6:00am to 11:00pm, and (b) from 11:00pm to 6:00am. Hence, the rst observation in (a) each day is the number of pickups between 6:00am and 6:15am, and the last observation is the number of pickups between 10:45pm and 11:00pm, and so on. There are 6,210 intervals in (a) and 2,430 intervals in (b). Remark 3 (Time Intervals) The choice of time intervals (a) and (b) is tentative, in con- tinued work I plan to explore the sensitivity of the results with respect to the choice of time intervals. Let y it and z it denote the pickups and drop-os in neighborhood i in time period t, re- spectively. Summary statistics for y it and z it for all time periods, as well as for a and b separately, are presented in Table 3.1. Note that out of the total of 345,600 observations, y it = 0 in 2.4% of the cases andz it = 0 in 0.6% of the cases. As we can see there is substantial location heterogeneity. 3 See https://www1.nyc.gov/site/planning/data-maps/open-data/dwn-nynta.page 198 Figure 3.1: Pickups and drop-os for selected neighborhoods during a selected week in March 199 Table 3.1: Summary statistics for pickups y it and drop-os z it mean std min median max interval a +b a b a +b a b a +b a b a +b a b a +b a b Pickups y it Astoria 12.1 10.4 17 8 6 11 0 0 0 10 9 13 86 48 86 Battery Park City-Lower Manhattan 120.6 151.3 42 69 52 40 0 4 0 135 156 26 334 334 215 Brooklyn Heights-Cobble Hill 9.4 11.1 5 7 6 6 0 0 0 8 10 3 41 38 41 Carroll Gardens-Columbia Street-Red Hook 11.6 13.2 8 9 9 9 0 0 0 9 11 4 58 56 58 Central Harlem North-Polo Grounds 4.8 5.2 4 4 4 4 0 0 0 4 4 3 47 36 47 Central Harlem South 15.3 17.5 10 9 8 9 0 1 0 14 16 7 71 64 71 Central Park 62.5 82.7 11 46 38 12 0 1 0 61 85 7 203 203 170 Chinatown 92.0 71.4 145 89 45 138 0 8 0 59 54 88 520 295 520 Clinton 213.6 239.4 148 96 74 114 0 26 0 214 232 102 513 483 513 DUMBO-Vinegar Hill-Downtown Brooklyn-Boerum Hill 20.4 21.6 17 13 11 17 0 0 0 18 20 10 84 73 84 East Harlem North 17.2 20.2 10 10 9 7 0 3 0 16 18 8 91 91 59 East Harlem South 37.2 46.0 15 20 15 11 0 4 0 39 45 11 145 145 85 East Village 172.1 158.1 208 133 93 196 0 17 0 128 130 120 742 638 742 East Williamsburg 6.4 2.8 15 11 4 16 0 0 0 2 1 10 96 55 96 Fort Greene 6.2 5.8 7 6 5 8 0 0 0 4 4 4 58 58 49 Gramercy 138.2 158.6 86 76 60 86 0 14 0 137 150 47 377 377 353 Greenpoint 5.8 3.6 12 9 4 13 0 0 0 3 2 6 56 38 56 Hamilton Heights 4.9 5.4 4 4 4 4 0 0 0 4 5 3 42 42 40 Hudson Yards-Chelsea-Flatiron-Union Square 481.9 558.6 286 242 198 233 0 28 0 527 569 192 1081 1081 929 Hunters Point-Sunnyside-West Maspeth 18.6 20.1 15 9 8 10 0 3 0 17 19 12 101 101 79 JFK Airport 74.8 89.7 37 46 39 42 0 0 0 76 90 16 216 216 188 La Guardia Airport 95.3 122.7 25 74 64 50 0 0 0 102 129 2 328 315 328 Lenox Hill-Roosevelt Island 206.6 260.9 68 118 87 61 0 15 0 233 262 46 563 563 396 Lincoln Square 196.3 247.1 66 112 78 75 0 9 0 228 252 34 546 546 440 Lower East Side 28.6 26.7 34 23 14 37 0 3 0 23 24 17 188 105 188 Manhattanville 4.7 5.3 3 3 3 3 0 0 0 4 5 2 62 62 24 Midtown-Midtown South 848.2 1028.6 387 440 342 308 0 52 0 938 1065 274 1982 1982 1437 Morningside Heights 51.5 62.8 23 29 24 18 0 2 0 53 62 17 244 244 129 Murray Hill-Kips Bay 234.1 281.6 113 125 93 113 0 18 0 254 286 64 531 531 483 North Side-South Side 33.8 20.2 68 45 24 64 0 0 0 15 9 45 267 149 267 Park Slope-Gowanus 11.6 9.4 17 14 10 21 0 0 0 6 6 8 143 101 143 Queensbridge-Ravenswood-Long Island City 7.6 8.3 6 4 4 4 0 0 0 7 8 5 37 37 30 SoHo-TriBeCa-Civic Center-Little Italy 262.1 292.9 184 143 119 167 0 13 0 267 284 112 667 596 667 Stuyvesant Town-Cooper Village 18.1 21.5 9 12 11 10 0 1 0 16 19 6 64 64 63 Turtle Bay-East Midtown 348.2 429.2 141 200 162 126 0 21 0 369 422 91 894 894 619 Upper East Side-Carnegie Hill 354.6 467.8 65 239 179 70 0 18 0 396 493 37 1108 1108 510 Upper West Side 237.9 300.5 78 132 89 76 0 17 0 275 306 51 677 677 574 Washington Heights South 5.0 5.7 3 4 3 4 0 0 0 5 5 2 48 32 48 West Village 364.9 382.4 320 208 167 282 0 16 0 335 353 202 1063 997 1063 Yorkville 144.8 179.5 56 80 59 54 0 18 0 154 175 36 421 421 417 Drop-os z it Astoria 24.8 16.9 45 24 14 30 0 0 0 15 12 39 141 114 141 Battery Park City-Lower Manhattan 132.7 152.2 83 73 65 68 0 3 0 135 145 54 437 437 261 Brooklyn Heights-Cobble Hill 15.0 15.2 15 13 12 14 0 0 0 11 11 9 73 73 60 Carroll Gardens-Columbia Street-Red Hook 14.6 13.7 17 13 12 15 0 0 0 9 9 11 71 71 63 Central Harlem North-Polo Grounds 16.2 14.7 20 11 9 14 0 0 0 13 12 17 81 61 81 Central Harlem South 24.8 25.8 22 15 13 17 0 1 0 22 23 17 89 84 89 Central Park 68.6 88.1 19 43 34 17 0 3 0 73 92 12 208 208 88 Chinatown 76.1 75.3 78 68 57 89 0 1 0 53 54 44 396 362 396 Clinton 192.1 218.4 125 99 84 104 0 13 0 195 209 83 541 541 466 DUMBO-Vinegar Hill-Downtown Brooklyn-Boerum Hill 26.8 26.2 28 19 17 24 0 0 0 21 21 20 110 110 104 East Harlem North 30.8 33.3 24 14 13 16 0 1 0 30 32 20 143 143 85 East Harlem South 46.4 51.7 33 20 16 23 0 8 0 48 51 26 154 154 118 East Village 142.7 146.6 133 124 119 135 0 4 0 94 99 78 580 580 577 East Williamsburg 14.0 8.8 27 16 10 20 0 0 0 7 5 23 87 67 87 Fort Greene 9.6 8.3 13 9 8 11 0 0 0 6 5 10 53 53 49 Gramercy 121.2 138.0 78 70 60 76 0 8 0 117 126 45 320 320 317 Greenpoint 17.8 13.4 29 18 14 22 0 0 0 9 7 23 96 77 96 Hamilton Heights 11.1 9.1 16 8 6 10 0 0 0 9 7 15 71 47 71 Hudson Yards-Chelsea-Flatiron-Union Square 455.4 546.6 222 241 188 200 0 27 0 507 563 136 1269 1269 886 Hunters Point-Sunnyside-West Maspeth 28.9 25.6 37 17 12 24 0 3 0 23 22 31 151 80 151 JFK Airport 34.6 44.0 11 26 24 16 0 0 0 34 41 4 137 137 118 La Guardia Airport 56.6 71.2 19 49 46 33 0 0 0 58 75 2 217 217 187 Lenox Hill-Roosevelt Island 200.8 241.2 97 105 80 90 0 19 0 221 240 59 490 490 425 Lincoln Square 180.6 224.5 68 112 96 63 0 9 0 192 224 41 659 659 286 Lower East Side 45.6 46.1 44 32 30 37 0 1 0 37 38 32 174 158 174 Manhattanville 7.4 7.0 9 5 4 6 0 0 0 7 7 7 39 38 39 Midtown-Midtown South 833.4 1049.2 282 477 362 225 0 46 0 896 1044 204 2164 2164 1067 Morningside Heights 51.9 59.2 33 27 22 28 0 2 0 56 61 23 200 200 131 Murray Hill-Kips Bay 236.8 279.6 128 121 90 121 0 11 0 265 289 73 497 487 497 North Side-South Side 40.0 32.6 59 39 31 48 0 0 0 24 19 44 199 174 199 Park Slope-Gowanus 24.5 20.8 34 25 22 29 0 0 0 13 12 23 140 140 118 Queensbridge-Ravenswood-Long Island City 8.5 8.3 9 4 4 6 0 0 0 8 8 8 30 26 30 SoHo-TriBeCa-Civic Center-Little Italy 236.4 279.5 126 130 104 126 0 7 0 255 283 72 596 588 596 Stuyvesant Town-Cooper Village 20.6 21.4 18 16 16 16 0 0 0 15 16 12 80 80 80 Turtle Bay-East Midtown 315.7 388.8 129 180 144 118 0 19 0 342 390 80 924 924 504 Upper East Side-Carnegie Hill 336.8 438.2 78 217 164 74 0 13 0 373 463 48 866 866 371 Upper West Side 234.3 284.7 106 142 124 98 0 12 0 244 290 66 800 800 480 Washington Heights South 16.7 15.5 20 9 7 13 0 0 0 15 14 17 119 119 71 West Village 315.1 362.1 195 217 201 210 0 9 0 295 320 106 1039 1039 894 Yorkville 145.8 168.0 89 94 88 85 0 9 0 136 158 50 462 462 432 y it - pickups in neighborhood i in the 15-minute time period t. z it - drop-os in neighborhood i in the 15-minute time period t. Summary statistics are based on observations from January-March 2013. Interval a contains 6,210 15 minute time periods between 6am and 11pm. The rst interval each day is calculated as the number of pickups that occured between 6:00am and 6:15am, the last interval is the number of pickups between 10:45pm and 11:00pm. Other time intervals are dened analogously. Interval b contains 8,640 15 minute time periods between 11pm and 6am. 200 3.3 Benchmark Model In this section I introduce the model proposed by Lagos (2000) and analyzed by Lagos (2003). First I describe the model and main result as shown in Lagos (2000). I then consider equilibrium characteristics under alternative pricing rules. Denote locations by i = 1; 2;:::;N. Assume that there are large populations of taxicabs and passengers, denoted by v and u, respectively. I focus on the steady-state outcome of an innite-horizon discrete time game. In each period, passengers wish to transition from locationi toj with probability a ij , with P j a ij = 1. The transition probability of passengers is captured by the Markov matrix A, and its steady-state distribution is denoted by = ( 1 ; 2 ;:::; N ) 4 . The meeting process between passengers and cabs in locationi is frictionless, and meetings occur according to m i = minfu i ;v i g; (3.3.1) with m i being the number of meetings in location i. Note that the failure of cabs and passengers to contact each other can occur only because of the location choices of the cab drivers. Assuming that meetings are random, the probability that a cab meets a passenger is given by p i = minf1= i ; 1g and the probability that a passenger is matched with a cab is given by p i i , where i =v i =u i . Assumption 17 (Prot Under Two-Part Tari) The passenger is charged a two part tari: an initial " ag-drop" rate b and a per-mile charge c. Taxis incur a xed per-mile cost c. Let = cc. Then a taxi taking a passenger from i to j earns a prot ij = b + ij , where ij is the distance (in miles) between locations i and j, In every period there is a meeting session. Let V i denote the value function of a cab in location i before the meeting session. If the cab does not match with a passenger it may relocate to any other location for the next meeting session. Let be the discount factor and 4 For simplicity, assume the time-invariant distribution is unique. 201 U i = max j fV j g j=1;2;:::;N be the value function of an unmatched cab. Then V i =p i X j a ij maxf ij +V j ;U i g + (1p i )U i : (3.3.2) A steady-state equilibrium is a time-invariant distribution of passengers and cabs such that the cabs have no incentive to relocate. Denition 1 (Steady-State Equilibrium) A steady-state equilibrium is a time-invariant distribution of passengers and cabs across locationsfu i ;v i g n i=1 such that P i u i =u, P i v i = v, V i =V n for all i and P j a ij m i = P j a ji m i . Proposition 1 (Equilibrium Outcomes) Let i = i =( P n j=1 j j ), where i = P j a ij ij , and dene = minf i g n i=1 . Let = v=u be the aggregate market tightness. Label locations so that 1 2 ::: k =::: = N : (3.3.3) Hence, locations labeled k and higher have the lowest conditional prot from rides. Then Lagos (2000) shows that the following holds in equilibrium. If > 1=, then there is a unique equilibrium in which all locations exhibit excess supply. If = 1= there is a unique equilibrium in which locations 1; 2;:::;k 1 exhibit excess supply, and the market clears in locations k;:::;N. If < 1= and k<n, there exists a continuum of equilibria with excess supply only in locations 1;:::;k 1 and excess demand in at least one of the remaining locations. If k = N the equilibrium is unique and location N exhibits excess demand and all other locations exhibit excess supply. 3.3.1 Model Parametrization I calculate the transition matrices for each time period, A (a) and A (b) directly from the data following the method described in Lagos (2003). Distances between neighborhoods are 202 computed as the average of the distances of the rides that occurred between the two neigh- borhoods. According to the New York City Taxi and Limousine Commission 2012 Fact Book the average fuel economy of the taxi eet was 29 mpg. The average per gallon gas price in New York City in January 2013 was $3.70 5 . Hence, I assume that the cost of driving one mile is 12 cents. Lagos (2000) uses data on 22,604 trips from 1988 to calibrate the model assumes all medallions are active throughout the day. I use the data to calculate the average number of active medallions for each of the time intervals (a): 6am-11pm and (b): 11pm-6am. On an average day, 380 trips per minute occurred during time interval (a), with the average trip duration being 11.44 minutes. The corresponding numbers for interval (b) are 183 trips per minute with an average trip duration of 11.02 minutes. 12,921 medallions were observed on average during time period (a), and 12,534 during time period (b). Using these numbers we can calculate taxi availability during an average minute in intervals (a) and (b). For time interval (a) we havev (a) = 1292111:44380 = 8574, and for interval (b) v (b) = 12534 11:02 183 = 10517. Let = b=. The current fare rates of New York yellow cabs yield b = 2:50 and c = 2, which yields = 1:05. We can then write (a) i = + P j a(a) ij ij + P i;j i a(a) ij ij . Remembering that (int) = minf (int) i g N i=1 , we have for interval (a), (a) = 0:71 and corresponds to Upper East Side-Carnegie Hill. For interval (b), (b) = 0:75 and corresponds to Gramercy. Since the aggregate number of meetings given the frictionless matching is minfu;vg, it follows that u (a) = 380 and u (b)= 183. This yields the following estimates of aggregate market tightness for periods (a) and (b): (a) 16 and (b) 43. Even assuming that not all observed taxi medallions are active throughout the time interval, this parametrization puts market tightness above the no-frictions frontier for both time intervals and implies excess supply in all locations. We have (a) () = +1:81 +2:40 and (b) () = +2:31 +2:93 . Remark 4 (Changing Two-Part Tari) We have ((0) (a) ) 1 = 1:33 and ((0) (b) ) 1 = 1:26. Also, lim !1 ((0) (a) ) 1 = lim !1 ((0) (b) ) 1 = 1 Since v=u > is equal to 16 and 5 source: https://www.nyserda.ny.gov/Researchers-and-Policymakers/Energy-Prices/ Motor-Gasoline/Monthly-Average-Motor-Gasoline-Prices 203 43 for time periods (a) and (b), respectively, changes in the two-part tari don't aect the equilibrium number of matches. Figure 3.1: No-Frictions Frontier with Two-Part Tari 3.3.2 Demand-Driven Pricing Now suppose that pricing is demand-driven. Specically, suppose that the two-part tari dened by the parameters b and c now potentially varies by location and depends on the relative volume of demand in the location as compared to other locations. Assumption 18 (Demand-Sensitive Pricing) Let the two-part tari for rides originat- ing in location i be given by b i = bf b ( i ), c i = cf c ( i ), where i is the steady state fraction of demand in location i. We can then write i = f b ( i ) +f c ( i ) P j a ij ij P i i f b ( i ) + P i;j f c ( i )a ij ij ; (3.3.4) 204 Table 3.1: i and 1 i for intervals (a) and (b) i 1 i interval (a) (b) (a) (b) Astoria 11.08 10.50 1.10 1.10 Battery Park City-Lower Manhattan 12.09 12.47 1.33 1.27 Brooklyn Heights-Cobble Hill 12.54 12.09 1.31 1.26 Carroll Gardens-Columbia Street-Red Hook 12.32 11.84 1.29 1.25 Central Harlem North-Polo Grounds 10.02 9.74 1.28 1.24 Central Harlem South 9.35 10.07 1.27 1.23 Central Park 7.48 8.89 1.26 1.22 Chinatown 8.64 9.28 1.24 1.21 Clinton 7.58 9.00 1.24 1.21 DUMBO-Vinegar Hill-Downtown Brooklyn-Boerum Hill 11.77 11.12 1.23 1.20 East Harlem North 9.22 9.50 1.22 1.19 East Harlem South 8.44 9.41 1.21 1.19 East Village 7.73 8.76 1.20 1.18 East Williamsburg 10.01 10.20 1.20 1.18 Fort Greene 11.65 11.75 1.19 1.17 Gramercy 7.22 8.17 1.18 1.17 Greenpoint 9.51 9.68 1.18 1.16 Hamilton Heights 12.40 11.67 1.17 1.16 Hudson Yards-Chelsea-Flatiron-Union Square 7.26 8.51 1.17 1.15 Hunters Point-Sunnyside-West Maspeth 10.43 10.68 1.16 1.15 JFK Airport 42.36 42.93 1.16 1.15 La Guardia Airport 26.76 25.36 1.15 1.14 Lenox Hill-Roosevelt Island 7.69 8.99 1.15 1.14 Lincoln Square 7.41 8.92 1.15 1.14 Lower East Side 8.61 9.48 1.14 1.13 Manhattanville 10.17 9.94 1.14 1.13 Midtown-Midtown South 7.51 8.53 1.14 1.13 Morningside Heights 9.27 9.54 1.13 1.13 Murray Hill-Kips Bay 7.23 8.20 1.13 1.12 North Side-South Side 9.86 10.10 1.13 1.12 Park Slope-Gowanus 11.65 12.26 1.13 1.12 Queensbridge-Ravenswood-Long Island City 8.72 9.93 1.12 1.12 SoHo-TriBeCa-Civic Center-Little Italy 8.61 9.19 1.12 1.11 Stuyvesant Town-Cooper Village 7.98 8.89 1.12 1.11 Turtle Bay-East Midtown 7.55 8.49 1.12 1.11 Upper East Side-Carnegie Hill 6.90 8.44 1.11 1.11 Upper West Side 7.71 9.41 1.11 1.11 Washington Heights South 15.61 13.19 1.11 1.10 West Village 7.43 8.71 1.11 1.10 Yorkville 8.25 9.55 1.11 1.10 205 where = b= c. Note that unlike the case with a xed two-part tari, where the location i with the lowest value i does not vary with , this location is potentially dierent for dierent values of the policy parameters b, c, f b and f c . Consider for example the following policies: f b ( i ) = i - short rides are more incentivized in locations with high demand. f c ( i ) = i - long rides are more incentivized in locations with high demand. f b ( i ) = f c ( i ) = i - both long and short rides are incentivized more in locations with high demand. As seen in Figures 3.2 (c) and (d), in these cases the no frictions frontier is substantially higher and there is excess demand in at least one location. As argued by Lagos (2003), evidence suggests that v and u do not respond much to fare increases. The calibrated ratio of market tightness puts the city above the no-frictions frontier, thus conrming the results of Lagos (2003). Changes in the two-part tari have no eect on the aggregate number of matches in a setting with frictionless matching, and the equilibrium of oversupply of taxis in each location does not change. Demand dependent pricing has an eect on the aggregate number of meetings if the no-frictions frontier rises above the current ratio of aggregate market tightness. 206 Figure 3.2: No-frictions frontier for dierent pricing mechanisms (a) Interval (a) (b) Interval (b) (c) Interval (a) (d) Interval (b) 207 3.4 Matching with Frictions The model presented and calibrated in the previous section is idealized - it is reasonable to assume that matching in the New York yellow cab market is not frictionless, as done by for example Frechette et al. (2015) and Buchholz (2016). In this section I argue that matching with frictions better describes that taxi-passenger meeting process. I then propose a novel way of inferring taxi availability in a given neighborhood and time period. Then, based on taxi arrival rates and matching mechanisms proposed in the literature, I estimate customer arrival rates and and time-specic local market tightness, i.e. the ratio of supply to demand in each neighborhood and time period. I analyze the variability of this measure throughout the day for dierent neighborhoods and compare my ndings with existing literature. Frechette et al. (2015) show that the fraction of time taxis spend searching is almost never lower than thirty percent and shows substantial variation throughout the day, ranging up to 65%. Under the current system of essentially xed fares, most inter-temporal variation in driver and customer welfare comes from varying search/wait times. Frechette et al. (2015) show that a simple linear model reveals that waiting time for passengers explains about 60% of the variation in hourly wages for drivers, which supports the theory that matching is not frictionless. In this section I propose an algorithm to estimate market tightness in the presence of matching frictions. Unlike Frechette et al. (2015) and Buchholz (2016), I do not assume a general equilibrium model. Instead I allow customer and taxi arrival rates to be dierent in each observed time period, and estimate them using observed data. As previously, I consider neighborhoods i = 1; 2;:::;N and time periods t = 1; 2;:::;T . Denote the set of all neighborhoods byI. Let d it denote demand for taxi rides in neighbor- hoodi and time periodt, and lety it denote the number of rides (taxi-passenger matches) in location i and time period t. Assumption 19 (Customer and Taxi Arrival Rates) Customers and empty taxis ar- rive deterministically at location i during time period t with rates C it and T it , respectively. 208 Note that vacant taxis in location i and time period t are composed of vacant taxis that transition from other neighborhoods and taxis that drop a passenger in i during t Assumption 20 (Matching Rate) Customers are matched to taxis at a rate it =( T it ; C it ), hence y it = C it it . Following Buchholz (2016), I assume the following functional form of the matching function, y it = T " 1 1 1 T it C it # : (3.4.1) As discussed by Buchholz (2016), the function exhibits several properties desirable in the context of the taxi market. For small values of T it and C it , the function is bounded below by y it = minf T it ; C it g, which means that matching becomes frictionless. This implies that smaller values of supply and demand occur when the matching area is small so that taxis and customers can easily nd each other. Further, the function exhibits constant returns to scale for larger values of supply and demand, which will prove important in inferring unmet demand as I show below. Matching eciency is represented by the parameter , with larger values of generating fewer matches, everything else constant. Note that the notion of supply of taxis in a neighborhoodi during a time intervalt is not straightforward. For example, would a supply of 1 mean that there was 1 taxi at some point during t, or that there was a taxi throughout the entire interval of t? Instead, I introduce a measure of taxi availability in location i throughout time period t. Denition 2 (Taxi Slack) Taxi availability in neighborhoodi during time periodt is mea- sured by taxi slack it , with one unit of it representing the state of a taxi that is vacant and in neighborhood i during the entire time period t. For example, suppose that there was one vacant taxi circulating in i for the entire time period t. Then it = 1. If the taxi was vacant for only one third or t, and there were no other taxis in the neighborhood, then it = 1=3, and so on. 209 Given the customer matching rate, vacant taxis accumulate at a rate T it it C it , and the total slack in i and time period t is given by the area of a triangle with base 1 and height T it it C it , it = 0:5( T it it C it ): (3.4.2) We can now derive the vacant taxi arrival rate. Remark 5 The vacant taxi arrival rate in locationi and time periodt is given by 2 it +y it = T it . Further, note that 1 2 it =y it + 1 =( T it ; C it ) C it T it : (3.4.3) Given the matching function in (3.4.1), the customer matching rate is ( T it ; C it ) = T C it " 1 1 1 T it C it # : (3.4.4) Since this function is approximately constant for a given ratio C it T it , we can write( T it ; C it ) p( it ), where it = C it T it is the relative demand intensity andp() is monotonous withp 0 ()< 0. Then 1 2 it =y it + 1 =p( it ) it : (3.4.5) Lemma 22 (Inferring Demand) We can infer customer arrival rates C it using equations (3.4.3), (3.4.5) and the fact that p() is strictly monotonous. 3.4.1 Calculating ^ it In this section I describe the algorithm I use to approximate it . I assume that the taxi takes the shortest route from one neighborhood to another. Specically, I represent neighborhoods as an undirected graph G =fg ij g i;j2I , where g ij =g ji = 1 if neighborhoods i and j share a border or a corner, and g ij = 0, otherwise. Let ij be the set of shortest paths from i to j, where the length of a path is equivalent to the number of vertices of G it contains, and let 210 ij be the cardinality of ij . I consider all cases where the time between a taxi dropping o a passenger and picking up the next one is shorter than 1 hour for day shifts and 1 hour 30 minutes for night shifts. I calculate ^ it in 15-minute intervals starting at each full hour, and I assume that each vacant taxi contributes 1 to the supply during a 15-minute interval. For example, suppose a taxi drops o a passenger in i at pm and picks up another passenger in i at 3:16pm. I then assume the taxi spent the time 3-3:15pm ini, and thus, the taxi contributes 1 to the supply of vacant taxis in i in this time period. If the taxi pick up the passenger at 3:06pm instead, it contributes 1/3 to the supply of vacant taxis in i during 3-3:15pm and so on. Now consider the case when a taxi drops o a passenger in i at 3:05pm and pick sup another passenger in j at 3:35pm. Suppose there are 2 shortest paths from i to j, so that ij = 2 and suppose the routes are (i;k;j) and (i;l;j). I calculate the supply contribution of the taxi for each area for each of the routes and weight it by ij as described below. Let a i denote the area of neighborhood i. Consider the path (i;k;j). The taxi was vacant during 30min, and I assume that the time the driver spend in each neighborhood in the path is equal to (a i ;a k ;a j )=(a i +a k +a j ) 30 min, i.e. proportional to the are of the neighborhood relative ot the other areas. Suppose this equals to (7; 13; 10) min, so the taxi is assumed to bin in i from 3:05pm to 3:12pm, in k from 3:12pm to 3:25pm and in j from 3:25pm to 3:35pm. The taxi then contributes (7=15)= ij to the supply in i for the time period 3-3:15pm, (3=15) ij to the supply ink calculated for the time period 3-3:15pm, and (10=15) ij to the supply for k in the time period 3:15-3:30pm, (5=15) ij to the supply in j for the time period 3:15-3:30pm, and nally, (5=15) ij to the supply in j for the time period 3:30-3:45pm. I repeat these calculations for the other route, and proceed analogously for all observations where the time between a drop-o and a subsequent pickup satises the above-mentioned criteria. 211 Remark 6 (Acknowledgment of shortcomings) Assuming that the taxi driver took one of the shortest routes according to the number of vertices in G is a simplication and might not be optimal in some circumstances. An improved version of the algorithm would use Google Maps API to calculate the optimal route between any given pair of locations. However, due to the quota on Google Maps API calls, I do not consider this approach at the moment. 3.4.2 Market Tightness and Frictions I use the results from Section 3.4 to infer the taxi and customer arrival rates T it and C it . I approximate it by ^ it , calculated as described in Section 3.4.1. Then I obtain the estimates ^ T it and ^ C it . The estimates of customer arrival rates depend on the eciency parameter of the matching function. Buchholz (2016) estimates the parameter of the matching function to be 1:3. My neighborhood denitions are similar in size to those used by Buchholz (2016), hence, assuming a similar value of seems reasonable. In what follows I present values of ^ C it obtained by assuming = (1; 1:3; 1:6). Figure 3.1 shows how the number of taxi-passenger matches varies for dierent levels of matching eciency. Higher values of imply lower matching eciency and lead to higher estimates of demand arrival rates, as illustrated in Figure 3.2. Below I present preliminary summary statistics and compare my results to those of Buchholz (2016). I estimate taxi and customer arrival rates for each time period and each value of . Buch- holz (2016) assumes that taxi and customer arrival rates are the same for any 5-minute time interval on weekdays, and estimates arrival rates for for example 3-3.05pm on Tuesdays etc. Since I estimate date and time-specic arrival rates, averaging them across dates allows me to compare my results to those of Buchholz (2016). Let 1 ; 2 ;:::; 96 denote unique 15-minute time intervals during 24 hours, with 1 being 12:00-12:15am and so on. Remembering that t = 1; 2;:::;T denote dated 15-minute time intervals, for example, 12:00-12:15am January 1st 2013, let t denote the time portion of 212 Figure 3.1: Number of matches for taxi arrival rate T it = 100 and dierent levels of matching eciency t, for example 1 is 12:00-12:15am. Let T k denote the set of all date-time intervals t with t = k . Finally, dene the average taxi arrival rate for neighborhood i for the time of day k as T i k = 1 jT k j P t2T k it , and dene C i k analogously. In Table 3.1 I present summary statistics of the estimated values of customer and taxi arrival rates for each time of the day. In Figure 3.2 I present estimates of taxi and customer arrival rates for dierent values of . Let ^ ( ) it = ^ T it = ^ C it . Figure 3.3 shows a plot of ^ (1:3) it for Midtown-Midtown South and West Village for a representative week. Similar to Buchholz (2016), I nd that there is an oversupply of taxis in Midtown during morning rush hours, and a slight undersupply of taxis in the afternoon. The opposite is true for West Village, which is also in line with the results of Buchholz (2016). This is an example of spatial misallocation. This evidence is anecdotal and should not be treated as conclusive evidence. There is uncertainty involved both in the parameter values and the estimates of demand and supply arrival rates. The main point I want to make is to show my results corroborate the ndings of Buchholz (2016) and point to frictions that result in misallocation. 213 Table 3.1: Estimated values of taxi and customer arrival rates for dierent pa- rameter values of the matching function matching function parameter = 1 = 1:3 = 1:6 ^ T i k ^ C i k ^ C i k = ^ T i k ^ C i k ^ C i k = ^ T i k ^ C i k ^ C i k = ^ T i k 6am-11pm mean 294.5 200.1 0.7 260.2 0.9 320.3 1.1 std 442.0 295.6 0.2 384.3 0.3 473.0 0.4 min 4.4 1.5 0.1 2.0 0.1 2.5 0.2 25% 26.5 11.6 0.5 15.2 0.7 18.7 0.8 50% 121.4 56.1 0.7 73.0 0.9 90.0 1.1 75% 395.0 307.0 0.8 399.2 1.1 491.4 1.3 max 2741.2 2119.0 1.8 2754.9 2.4 3390.7 3.0 11pm-6am mean 151.0 94.1 0.7 122.4 0.9 150.7 1.0 std 240.4 153.5 0.2 199.5 0.3 245.6 0.4 min 3.1 1.3 0.2 1.8 0.2 2.3 0.3 25% 21.1 8.9 0.5 11.7 0.7 14.4 0.8 50% 57.4 25.2 0.6 32.8 0.8 40.5 1.0 75% 173.3 116.5 0.8 151.5 1.0 186.6 1.2 max 2187.1 1346.2 1.8 1750.2 2.4 2154.1 3.0 The summary statistics for the time period 6am-11pm are based on 2,720 observations of 40 neighborhoods over 68 15-minute time intervals. The summary statistics for the time period 11pm-6am are based on 1,120 observations of 40 neighborhoods over 28 15-minute time intervals. 214 Figure 3.2: Estimated taxi and customer arrival rates for dierent values of the matching function parameter 215 Figure 3.3: Spatial misallocation 216 3.4.3 Local Market Tightness Finally, I consider a simple dynamic model. ^ ( ) it = i + P X p=1 p ^ ( ) i;tp + 0 z t + t + it ; (3.4.6) where t are time eects andz t are weather conditions,i = 1; 2;:::; 40 andt = 1; 2;:::; 2976 (all 15-minute time intervals in January 2013). Note that during the considered time period weather conditions do not vary notably as shown in Table 3.2. Parameter estimates for model (3.4.6) are presented in Table 3.3. The results do not vary notably if dierent lag orders of ^ ( ) it are included. The results indicate that demand pressure increases in the mornings and during periods with increased precipitation. The latter is interesting as it is in line with earlier ndings, see for example Farber (2015) and Kamga et al. (2015). Farber (2015) suggests that rain does not only drive up the demand for taxis, it also drives down the supply of drivers, making the problem worse from both angles. Table 3.2: Summary statistics Statistic Mean St. Dev. Min Pctl(25) Median Pctl(75) Max y it 123.49 202.27 1 8 31 163 1,834 ^ it 68.11 114.04 0.004 7.47 25.28 81.07 1,043.67 ^ T it 259.70 418.88 1.01 24.28 89.48 327.33 3,453.54 ^ C it , = 1 173.95 288.65 1.00 10.03 41.07 231.61 2,804.00 ^ C it , = 1:3 226.23 375.25 1.30 13.11 53.50 301.20 3,645.35 ^ C it , = 1:6 278.51 461.86 1.60 16.20 65.90 370.80 4,486.71 ^ ( ) it , = 1 2.53 2.74 0.13 1.33 1.76 2.62 84.60 ^ ( ) it , = 1:3 1.93 2.10 0.10 1.02 1.35 2.00 64.99 ^ ( ) it , = 1:6 1.57 1.70 0.08 0.83 1.10 1.62 52.76 precipitation 0.003 0.02 0.00 0.00 0.00 0.00 0.33 wind speed (m/s) 5.99 3.84 0 3 6 8 23 temperature (F) 34.50 11.54 0 28 36 43 64 Precipitation is measured in total accumulation in units of millimeters at the Central Park station during a 60-minute period. 217 Table 3.3: Parameter estimates for model (3.4.6) Dependent variable: ^ ( ) it matching eciency = 1 = 1:3 = 1:6 ^ ( ) i;t1 0.279 0.280 0.263 (0.024) (0.024) (0.029) ^ ( ) i;t2 0.222 0.222 0.196 (0.015) (0.015) (0.018) ^ ( ) i;t3 0.151 0.151 0.121 (0.009) (0.009) (0.014) 7am-10am 0.262 0.200 0.151 (0.065) (0.050) (0.039) 5pm-9pm 0.028 0.022 0.026 (0.034) (0.026) (0.021) 12am-6am 0.051 0.036 0.023 (0.043) (0.033) (0.022) temperature (F) 0.0003 0.0002 0.0001 (0.001) (0.0004) (0.0003) wind speed (m/s) 0.0003 0.0003 0.0002 (0.001) (0.001) (0.001) precipitationy 0.400 0.306 0.304 (0.219) (0.167) (0.132) Standard errors in parentheses., and denote statistical signicance at the 10%, 5% and 1% levels, respectively. y - precipitation is measured in total accumulation in units of millimeters at the Central Park station during a 60-minute period. Neighborhood xed eects are included in all models, balanced panel with N = 40, T = 2; 976 The results do not change notably if dierent lag orders of ^ are included. 3.5 Conclusions In a market with frictionless matching and the current two-part tari there should be ex- cess supply in all locations. This is still true if the two-part tari is modied so that the ratio of ag-drop rate to per-mile fare lies in the range of (0;1). Demand-driven pricing implemented in such a way that high-demand locations have higher fares than low demand locations can result in excess demand in at least one location depending on the degree of price discrimination. In this paper specic functional forms of demand-driven pricing are considered. A formal characterization of equilibria for general forms of the pricing function is beyond the scope of this paper and left for future research. 218 Data-driven estimation of taxi availability provides evidence for spatial misallocation, which implies excess demand or supply depending on the time of the day and location. Market tightness decreases in the mornings and during periods of higher precipitation, which is in line with earlier ndings in the literature. In continued research I plan to consider the above results in the framework of a spatial general equilibrium model and analyze the implications of a demand-driven pricing algorithm in the context of matching with frictions. Bibliography Aan, A., A. Odoni, and D. Rus (2015). Inferring unmet demand from taxi probe data. In Intelligent Transportation Systems (ITSC), 2015 IEEE 18th International Conference on, pp. 861{868. IEEE. Buchholz, N. (2016). Spatial Equilibrium, Search Frictions and Ecient Regulation in the Taxi Industry. Working paper, 1{64. Farber, H. S. (2015). Why you cant nd a taxi in the rain and other labor supply lessons from cab drivers. The Quarterly Journal of Economics 130 (4), 1975{2026. Frechette, G. R., A. Lizzeri, and T. Salz (2015). Frictions in a Competitive, Regulated Market Evidence from Taxis. SSRN Electronic Journal. Kamga, C., M. A. Yazici, and A. Singhal (2015). Analysis of taxi demand and supply in new york city: implications of recent taxi regulations. Transportation Planning and Technology 38 (6), 601{625. Lagos, R. (2000). An alternative approach to search frictions. Journal of Political Econ- omy 108 (5), 851{873. Lagos, R. (2003). An analysis of the market for taxicab rides in new york city. International Economic Review 44 (2), 423{434. 219 220 Appendices 221 A1 Data The data set contains the following information for each taxi ride. Medallion id Anonymized hack license Vendor ID - code indicating the TPEP provider that provided the record (Creative Mobile Technologies or VeriFone) Pickup date/time - date and time when the meter was engaged Drop-o date/time - date and time when the meter was disengaged Passenger count Trip distance in miles reported by the taximeter Pickup longitude and latitude Drop-o longitude and latitude RateCodeID - Standard rate, JKF, Newark, Nassau or Westchester, Negotiated fare or group ride. Store and forward ag - ag that indicates whether the trip record was held in vehicle memory before sending to the vendor, aka \store and forward", because the vehicle did not have a connection to the server. Payment type - credit card, cash, no charge, dispute, voided or unknown Time-and-distance fare calculated by the meter Extras and surcharges. Currently, this only includes the $0.50 and $1 rush hour and overnight charges. 222 MTA tax - $0.50 MTA tax that is automatically triggered based on the metered rate in use. item Improvement surcharge - $0.30 improvement surcharge assessed trips at the ag drop. The improvement surcharge began being levied in 2015. Tip amount - only takes into account card tips, not cash. Tolls - amount of tolls paid in the trip Total amount charged to passengers (does not include cash tips) 223 A1.1 Neighborhood Tabulation Areas Figure A1: Neighborhood Tabulation Areas 224
Abstract (if available)
Abstract
The essays in this dissertation focus on three topics: social networks, beliefs and asset prices, and spatio-temporal relationships. ❧ In the first chapter (joint with Hyungsik Roger Moon) we propose a method of estimating the linear-in-means model of peer effects in which the peer group, defined by a social network, is endogenous in the outcome equation for peer effects. Endogeneity is due to unobservable individual characteristics that influence both link formation in the network and the outcome of interest. We propose two estimators of the peer effect equation that control for the endogeneity of the social connections using a control function approach. We leave the functional form of the control function unspecified and treat it as unknown. To estimate the model, we use a sieve semiparametric approach, and we establish asymptotics of the semiparametric estimator. ❧ In the second chapter (joint with Hashem Pesaran) we propose a new double-question survey whereby an individual is presented with two sets of questions
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Johnsson, Ida Brigitta
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Core Title
Essays on beliefs, networks and spatial modeling
School
College of Letters, Arts and Sciences
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Doctor of Philosophy
Degree Program
Economics
Publication Date
04/10/2018
Defense Date
03/22/2018
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asset prices,beliefs,networks,OAI-PMH Harvest,social networks,spatial modeling
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Moon, Hyungsik Roger (
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ida.b.johnsson@gmail.com,johnsson@usc.edu
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asset prices
beliefs
networks
social networks
spatial modeling