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Individual heterogeneity and program evaluation
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Individual heterogeneity and program evaluation
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INDIVIDUAL HETEROGENEITY AND PROGRAM EVALUATION by Yan Shen A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHEN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ECONOMICS) August 2003 Copyright 2003 Yan Shen Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3116785 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3116785 Copyright 2004 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY O F SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES, CALIFORNIA 90089-1695 This dissertation, written by Yan Shen under the direction o f h er dissertation committee, and approved by all its members, has been presented to and accepted by the Director o f Graduate and Professional Programs, in partial fulfillment o f the requirements fo r the degree o f DOCTOR OF PHILOSOPHY Director Date Mftlj ft ; 2 o0 3 Dissertation Commit Chair Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Dedication To My Parents Jianhua Shen and Junxiang Yan, and My Husband Jian Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. iii Acknowledgements First and foremost, I would like to thank my advisor and chair of my Ph.D. committee, Dr. Cheng Hsiao, for giving every draft of this dissertation, in every stage of its development, as much attention and care as if it had been his own. I have benefited enormously from his dedication, guidance, inspiration, and patience when undertaking the doctoral program at University of Southern California. He will be my lifetime model in not only in commitment to scholarship, from whom I learn “Persistence and diligence are the only way to do research”, but also in style of teaching and guidance of students. I am also very grateful for having an exceptional doctoral committee and wish to thank Dr. Kathleen Johnson, Dr. Isabelle Perrigne, Dr. Geert Ridder, and Dr. Quang Vuong for their insightful comments and suggestions. My indebtedness is also to the College of Letters, Arts, & Sciences o f USC for the dissertation fellowship, and to the Economics Department in particular, for the teaching and research opportunities and for excellent academic and administrative help from Dr. Jeffery Nugent, Dr. Caroline Betts, Young Miller, Shiella Williams and Sena Schlessinger. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. iv I would like to thank workshop participants at China Center for Economic Research, State University of New York - Buffalo, Chinese University of Hong Kong, University of Hong Kong, University of Southern California, participants at the 10th International Panel Data Conference for comments and suggestions. I also would like to thank Dr. Wang Boqing, Dr. Greg Weeks of Washington State for providing me with the WorkFirst data, Employment Security Department of Washington State for partial financial assistance, and Michelle Petritz for kindly answering my numerous questions with regard to the data set. My final thanks go to God, for the marvelous ways he has brought people into my life who have helped me both in academics and in everyday life. In particular, I thank him for giving me wonderful parents, who cultivate and educate me with their endless love, and for blessing me by giving me Cai Jian as my husband. Jian’s support, encouragement, and companionship have turned the demanding journey of pursuing a Ph.D. degree into a real pleasure. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V Table of Contents Dedication................................................................................................................................ii Acknowledgements.............................................................................................................. iii List of Tables.......................................................................................................................viii List of Figures........................................................................................................................ x Abstract..................................................................................................................................xi 1 Introduction.................................................................................................................... 1 2 Aggregate vs. Disaggregate Analysis of Japan's Money Demand Function Under the Low Interest Rate Policy...........................................................................5 2.1 Introduction...........................................................................................................5 2.2 The Basic Formation............................................................................................ 7 2.3 Aggregate Time Series Analysis......................................................................... 8 2.4 The Disaggregate Time Series Analysis.........................................................21 2.4.1 Data Measurement................................................................................... 25 2.4.2 Statistical Modeling................................................................................. 26 2.4.3 Empirical Results....................................................................................27 2.5 Possible Sources of Discrepancy......................................................................35 2.5.1 Simultaneity............................................................................................ 35 2.5.2 Definitions of Money..............................................................................38 2.5.3 Aggregation Over Heterogeneous Units.................................................41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vi 2.6 A Simulation Study of Aggregate and Disaggregate Behavior.................. 46 2.6.1 The Simulation Design............................................................................47 2.6.2 Results Based on Simulated Data............................................................48 2.7 Choice between Aggregate Model or Disaggregate M odels........................54 2.8 Conclusion............................................................................................................ 64 3 Controlling Individual Heterogeneity in Evaluating the Effectiveness of Repeated Job Search Services................................................................................... 68 3.1 Introduction.........................................................................................................68 3.2 The Model............................................................................................................ 72 3.2.1 A Transitional Probability Model for the Outcomes...............................72 3.2.2 Evaluation of the Conditional and Unconditional Impacts.....................82 3.3 Findings................................................................................................................ 85 3.3.1 Impacts on the Job-seeker Group............................................................85 3.3.2 Impacts on the Job-holder Group............................................................92 3.3.3 The Equilibrium Impacts.........................................................................94 3.4 Diagnostic Checking...........................................................................................95 3.4.1 Controlling for the Selection on Unobservables.....................................96 3.4.2 Controlling for Individual specific Effects..............................................99 3.4.3 Controlling for both Selection Bias and Individual Specific Effects ... 110 3.5 Conclusion..........................................................................................................112 4 Concluding Remarks................................................................................................ 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vii References............................................................................................................................117 Appendices..........................................................................................................................122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables Table 1 ADF Tests of Unit Roots"’ 6 ......................................................................................11 Table 2 Johansen Likelihood Ratio Test When Ml (M2), GDP and Bond Rate are I (1).....12 Table 3 Johansen (1991) Normalized Cointegration Vector.................................................. 13 Table 4 Anderson’s (2001) Reduced Rank Estimation.......................................................... 15 Table 5 Unrestricted Error Correction Estimation: Ml (M2), GDP and Bond Rate Are 1(1) 16 Table 6 Least Squares Estimation of Money Demand...........................................................18 Table 7 ARIMA Modeling of M2 and M l............................................................................. 19 Table 8 Root Mean Square Error Comparison For One-Period-ahead Forecasting...............20 Table 9 Hierarchical Bayes Estimated Random Coefficients................................................29 Table 10 Hierarchical Bayes Estimated Mixed and Fixed and Random Coefficients...........30 Table 11 Hausman Specification Test of Measurement Error*.............................................31 Table 12 Fixed Effects (FE) and Random Effects (RE) Estimation......................................33 Table 13 Income Elasticity and Semi-Interest Rate Elasticity Based On Random Coefficient and Mixed Fixed and Random Coefficient Models.................................................35 Table 14 Two Stage Least Squares Estimation For Ml and M2............................................37 Table 15 Least Squares Estimation of Money Demand Ml (M2)-Currency.........................39 Table 16 Two Stage Least Squares Estimation For Ml (M2)- Currency...............................40 Table 17 Least Squares Estimation for the Simulated Ml and M2........................................49 Table 18 Translog Estimation for the Aggregate Data"’ 6 ......................................................55 Table 19 Box-Cox Transformation for the Aggregate Data...................................................56 Table 20 Estimating Money Demand Using the Inverse of Bond Rate.................................57 with permission of the copyright owner. Further reproduction prohibited without permission. ix Table 21 Error Sum of Square (ESS) and Predicted Error Sum of Squares (PES) for Disaggregate and Aggregate Data...........................................................................59 Table 22 Frequency Distribution of All Observations Over Time.........................................70 Table 23 Descriptive Statistics For Initially Unemployed Individuals*................................87 Table 24 Descriptive Statistics for Initially Employed Individuals * ....................................88 Table 25 MLE and NL2S Estimations for Initially Unemployed Clients..............................89 Table 26 MLE and NL2S Estimations for Initially Employed Clients..................................91 Table 27 Mean Group Impacts and Equilibrium Impacts*....................................................93 Table 28 With or Without Individual Heterogeneity*.........................................................110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. X List of Figures Figure 1 Plotting Real M2 for 47 Prefectures........................................................................23 Figure 2 Plotting Real Ml For 47 Prefectures.......................................................................24 Figure 3 Plotting Prefecture Income For 47 Prefectures........................................................24 Figure 4 The Distribution of Lagged MF1 Coefficients........................................................44 Figure 5 The Distribution of lagged MF2 Coefficients..........................................................44 Figure 6 Prefecture Income Weight Distributions Over 1992 - 1997....................................45 Figure 7 Histogram for Simulated M l(-l) Coefficients.........................................................50 Figure 8 Histogram for Simulated M2(-l) Coefficients.........................................................51 Figure 9 Histogram of Simulated M l(-l) Coefficients When Disaggregate Coefficients are Randomly Assigned.................................................................................................52 Figure 10 Histogram for Simulated M2(-l) Coefficients When Disaggregate Coefficients Are Randomly Assigned..........................................................................................53 Figure 11 Simulated Responses of the “True Aggregates” and “Simulated Aggregates” from Disaggregate Equations to Interest Rate Shock.......................................................62 Figure 12 “True Aggregates” and The Simulated Responses from the Aggregate and Disaggregate Equations...........................................................................................63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xi Abstract Through two empirical studies we focus on how to detect and control for unobserved individual heterogeneity to make inferences about the population characteristics for dynamic linear and nonlinear models. In the first study we use Japanese aggregate and disaggregate money demand data to show that conflicting inferences can arise when individual heterogeneity is ignored. The aggregate data appears to support the contention that there was no stable money demand function. The disaggregate data shows that there was a stable money demand function. Neither was there any indication of the presence of a liquidity trap. Possible sources of discrepancy are explored and the diametrically opposite results between the aggregate and disaggregate analysis are attributed to the neglected heterogeneity among micro units. We provide necessary and sufficient conditions for the existence of cointegrating relations among aggregate variables when heterogeneous cointegration relations among micro units exist. We also conduct simulation analysis to show that when such conditions are violated, it is possible to observe stable micro relations, but unit root phenomenon among macro variables. Moreover, the prediction of aggregate outcomes using aggregate data is less accurate than the prediction based on micro equations. A generalization of sophisticated individual behavior to aggregate behavior under the guise of representative agent can lead to seriously misleading aggregate model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xii In the second study we investigate the role of unobserved individual heterogeneity in evaluating the impacts of repeated job search services on the employment rate of female welfare recipients. We propose a transition probability model to take account issues of sample attrition, sample refreshment and duration dependence. We also propose a nonlinear two-stage least squares estimator to allow for selection on unobservables and generalize Honore and Kyriazidou’s (2000) conditional maximum likelihood estimator to allow for state-dependent individual specific effects and slope coefficients. We then provide a conditional nonlinear two-stage least squares estimator that allows for both unobserved individual heterogeneity and selection on unobservables. The specification tests indicate that the conditional independence assumption and the no individual specific heterogeneity assumption are not violated. We find that job search services do have positive and significant impacts on the employment rate of those who are initially unemployed, but their impacts are insignificant for those who are employed. Furthermore, there are significant experience-enhancing effects. These findings suggest that providing job search services to unemployed individuals to help them find jobs quickly may have a lasting impact on raising their employment rate. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 Introduction The availability of panel data allows us to check if there exists heterogeneity among micro units. This dissertation deals with the issue of testing the presence of unobserved heterogeneity and procedures for controlling them to obtain valid inference using linear and nonlinear panel data models. In chapter 2 we look at issues of misleading inference in aggregate time series analysis when homogeneity among micro units are violated. The consequences of ignoring unobserved individual heterogeneity have not been fully appreciated in empirical studies even though many studies have shown that the dynamics of aggregate variables and disaggregate variables can be very different. In p articular, aggregate data is often constructed from the disaggregate data under the representative agent assumption, which is usually justified by arguing that heterogeneities among cross-sectional units does not matter when the primary interest is in obtaining an unbiased estimator o f the average effect o f exogenous variables (Zellner (1969)). However, this result does not extend to dynamic models (Pesaran and Smith (1995), Pesaran, Smith and Im (1996)). The availability o f panel data allows an investigator to check the homogeneity assumption. We use Japan's aggregate and disaggregate data to investigate whether Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 there was a stable money demand function for Japan in the 1990's under the low interest rate policy using both aggregate and disaggregate time series data. The aggregate data appears to support the contention that there was no stable money demand function. In the disaggregate data analysis, we use random coefficient models to allow for heterogeneous slope coefficients among cross-sectional units. The disaggregate data results show the existence of a stable money demand function. Neither was there any indication of the presence of liquidity trap. The possible sources of discrepancy are then explored. The diametrically opposite results between the aggregate and disaggregate analysis are attributed to the neglected heterogeneity among micro units. We derive conditions for the presence of stable macro relations under heterogeneity and demonstrate that it is possible to have stable micro relations but unstable macro relations when these conditions are violated. Simulation analysis is conducted to show that when heterogeneities among micro units are present, the prediction of aggregate outcomes based on aggregate data is less accurate than the prediction based on disaggregate data. In short, caution is needed to generalize sophisticated individual behavior to the aggregate behavior under the representative agent assumption. The focus o f chapter 3 is to examine the importance of controlling for unobserved individual specific effects in evaluating effectiveness of repeated treatments when dependent variables are discrete. Controlling for individual specific effects in dynamic nonlinear models is particularly difficult since in general there is no simple Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 transformation to eliminate the individual specific effects (Hsiao (2003)). If we take a specific parametric approach and treat individual specific effects as fixed, a conditional maximum likelihood estimator can be constructed for logit models (Honore and Kyriazidou (2000)). However, as this type of method requires the differences of explanatory variables across two different time periods to have support around zero, the conditional maximum likelihood estimator suffers from two drawbacks. First, tim e dummies are excluded as explanatory variables. Second, as individual specific effects cannot be estimated, it is not possible to compute elasticities for individual agents at specified values of the explanatory variables (Hsiao (2003)). If individual specific effects are not present, however, we can have more data points for analysis and we can answer more policy-related questions. In Chapter 3 we evaluate how repeated job search services (JSS) and personal characteristics affect the employment rate of the prime-age female welfare recipients in the State of Washington. We propose a transition probability model to take account issues o f sample attrition, sample refreshment and duration dependence. We also propose a nonlinear two-stage least square estimator to allow for selection on unobservables and generalize Honore and Kyriazidou’s (2000) conditional maximum likelihood estimator to allow for state-dependent individual specific effects and slope coefficients. We then provide a conditional nonlinear two-stage least square estimator that allows for both unobserved individual heterogeneity and selection on unobservables. The specification tests indicate that the selection-only-due-to- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 observables assumption and the no-unobserved-individual-heterogeneity assumption are not violated. Therefore, we focus on the transition probability model for the empirical analysis. We find that job search services do have positive and significant impacts on the employment rate of those who are initially unemployed, with the first job search services increases the probability of being employed by 8.45%, the second job search services increases it by a further 2.15%, and the third job search services increases it by an additional 0.8%. But the impacts of JSS are insignificant for those who already have jobs. Furthermore, there are significant experience-enhancing effects. These findings suggest that providing job search services to unemployed individuals to help them find jobs quickly may have a lasting impact on raising their employment rate. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 2 Aggregate vs. Disaggregate Analysis of Japan's Money Demand Function Under the Low Interest Rate Policy 2.1 Introduction Many economists have relied on the “representative agent” argument to derive macro restrictions from sophisticated individual choice-theoretical analysis. However, as noted by L. Klein in his classical econometrics textbook published in 1953, such practice can lead to misleading inference in macro-econometric modeling and policy evaluations. Over the years, many economists have shown that the dynamics of aggregate variables can be very different from the dynamics o f micro variables (e.g. Fomi and Lippi (1997, 1999), Granger (1980), Lewbel (1992, 1994), Pesaran (1999), Stoker (1993), Theil (1954), Trivedi (1985)). In this chapter we use Japan's aggregate and disaggregate money demand data to show that ignoring heterogeneity in micro units can yield diametrically opposite results with regard to the issue of whether there exists a stable money demand function or the presence of a “liquidity trap” for Japan. Despite very aggressive fiscal and monetary policies, Japan's economy largely stagnated in the 90's. The dependence of national budget (ippan kaikei) on the issuance of the bonds on an ongoing basis has reached 38.5% in the fiscal year 2000 budget, a dramatic increase from 10.6% in 1990 (Highlights of the Budget for fiscal Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 year 2001 (April 2001)). On a stock basis, the government gross debt to GDP is approximately 135.3% in 2000, the worst level among industrialized countries (Fujiki, Okina and Shirutsuka (2001)). Doi (2000), based on the criteria of Bohn (1995), found that Japan's fiscal position had deteriorated to an unsustainable level. Given that the sustainability of fiscal debt is uncertain, it is natural that one might wonder if monetary policy could play a more important role in stimulating Japanese economy. However, the effectiveness o f monetary policy depends critically on the existence o f a stable money demand function and the nonexistence of liquidity trap. Nakashima and Saito (2000) analyze monthly aggregate time series data from 1985 to 1999 and conclude that (i) there was a structural break in January 1995, (ii) there was no stable relation between money demand and income; and (iii) there was evidence of liquidity trap after 1995. On the other hand, Fujiki, Hsiao and Shen (2001) use panel information of 47 prefectures of Japan and finds that there is indeed a stable money demand function at the disaggregate level. In this chapter we wish to investigate further whether Japan has a stable money demand equation under the low interest rate policy by relying on the evidence of aggregate and disaggregate tim e series data and explore the source of discrepancy between the information of the aggregate and disaggregate data. Section 2.2 presents the basic model for money demand equation. Section 2.3 presents the evidence of aggregate quarterly time series data from 1980.IV to 2000.IV, which basically supports the contention that there was no stable money Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 demand function and money demand is better modeled as a random walk with a drift. Section 2.4 uses a random coefficients framework to analyze d isaggregate annual time series data of 47 Japanese prefectures from 1992 to 1997. The evidence appears to support the contention that money demand is a stable function o f real income and nominal interest rate. In section 2.5 we explore possible sources of discrepancy between the evidence of aggregate and disaggregate time series data and provide arguments in favor of disaggregate data analysis. Section 2.6 provides simulation results of the relationship between aggregate and disaggregate data. Section 2.7 uses simulated data to illustrate the importance of relying on disaggregate data to predict the aggregate outcome and perform simulation analysis when heterogeneity is present in micro units. Conclusions are in section 2.8. 2.2 The Basic Formation Following Goldfeld and Sichel (1990), we assume that the logarithm of the desired real money balance, m* t , for region i at time t, is a linear function o f the logarithm of real income, y it, and nominal interest rate, rt ,' m it ~ a i + y it + c i rt> (221 ) 1 There are a lot of financial innovations in the 1980's and 1990's, which are considered as possible sources of yielding unstable money demand functions. Here, we are using data from 1992 - 1997, a relatively short time period with most of the financial innovations already introduced. Our aim is not to investigate the effect of financial innovations, but to see if there still exists a stable relation o f the form (2.2.3). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 where a *represents the regional specific effects that approximates regional rental costs of inputs to the production function o f output and financial services, as well as other regional specific effects. The actual money demand follows a stock adjustment principle in which the changes in money demand is proportional to the deviation between the desired and actual money holding (Nerlove (1958)), (™ it ~ mu _i) = y*(m * it - mijtA) + sit, (2.2.2) ) || j| c where yt denotes the speed of adjustment, typically assumed to be 0<y; <1, and eu denotes the noise. Substituting (2.2.1) into (2.2.2) yields a regional money demand equation of the form mit = Yimi,t-\ + btyit + C trt + a t + sit, (2.2.3) where yt = 1 - y *, ct = y*c*, at = y*a*. If | y |< 1 , equation (2.2.2) implies a long-run equilibrium relation between money and income o f the form b; Cj < 2 ; m, = — * - * + ^ r + - . (2.2.4) i -Y i i-Y i i-r,- 2.3 Aggregate Time Series Analysis In this subsection we report the results of using Japan's quarterly real M l (RM1), real M2 (RM2), real GDP (RGDP), and interest rate from 1980.IV to 2000.IV. The interest rates used by Nakashima and Saito (2000) or Fujiki, Hsiao and Shen (2001) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 are the Bank o f Japan overnight call rates. However, as pointed out by L. Klein • 23 that the call rate may be too short, we use five-year bond rate (rt ) instead. In the aggregate data analysis, we impose the assumption of homogeneity across micro units, namely, y, = yj = y , bt = bj - b ,c t = c j= c ,a t = a j = a , for all i, j. In other words, we estimate a money demand function of the form mt = ymt_i + byt + crt + a + st (2.3.1) where mt and yt denote the logarithm o f p er capita real m oney balance and real income, respectively. If jy |<1, then (2.3.1) yields the long run equilibrium relation between mt , yt and rt as — b c a « . . . mt =- y ,+ - rt +- + v„ (2.3.2) 1 - y 1 - y 1 - y where vt denotes the error term. Inference on (2.3.1) critically depends on the time series properties of mt , yt and rt . We use Augmented Dicky-Fuller statistic (ADF) to test if the logarithm of RM1, RM2, RGDP and rt are stationary or integrated of certain order. Because there is an issue of whether there is a structural break after the 2 The use o f overnight call rate yields more or less the same general conclusions. 3 Since our analysis uses Japanese GDP data based on 1993 System of National Account (Benchmark y ear = 1995), we can only use data from 1980 to 2000. See details of the revision of Japanese GDP statistics, http://www.esri.cao.go.jp/en/sna/menu.html. Thus our results are not comparable with literature using 1968 System of National Account, such as Miyao (1996). In addition, most Japanese research use M2+CD instead of M2, however, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 bubble burst in 1990, Table 1 reports the results of ADF using the complete sample and the sub-sample period 1992.1 - 2000.IV. Both the complete sample and sub-sample results overwhelmingly favor the hypothesis that they are integrated or order 1 ,1 (1). since our results will be compared with those using prefecture deposit that do not include CD, we will use M2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 Table 1 ADF Tests of Unit Roots"’ 4 Sample r c Real GDP Real M2 Real Ml Bond Rate Period ADF SBC ADF SBC ADF SBC ADF SBC Test of 1(0) vs. KD 1980.IV 0 -2.59 -9.3298* -3.33 -9.442 4.03 -8.5996 -0.95 -1.29* - 1 -2.76 -9.2884 -2.31 -9.882 1.76 -8.7956* -0.87 -1.24 2000.IV 2 -2.87 -9.2435 -2.24 -9.898* 1.66 -8.7414 -0.78 -1.21 3 -2.21 -9.3273 -2.19 -9.831 1.78 -8.6796 -0.89 -1.15 4 -1.76 -9.2902 -1.64 -9.796 1.41 -8.6301 -0.86 -1.08 5 -1.71 -9.2903 -3.33 -9.759 1.10 -8.5667 -0.90 -1.01 1992.1 0 -0.86 -8.957 3.85 -10.48 2.39 -8.731 -2.17 -1.58* - 1 -0.69 -8.9218 2.03 -10.49* 0.51 -9.057* -2.15 -1.48 2000.IV 2 -0.45 -8.9594* 1.09 -10.475 0.56 -8.96 -2.31 -1.43 3 -0.49 -8.8735 1.39 -10.411 0.78 -8.878 -2.60 -1.42 4 -0.48 -8.7777 0.63 -10.385 0.52 -8.78 -2.84 -1.37 The * indicates the optimally selected lag order. The 95% critical values are -2.897 and -2.95 for the 1980.IV - 2000.IV and 1992.1 - 2000.IV periods respectively. L denotes the lagged difference order. Under the assumption that mt , y t and rt are 1(1), then the issue of whether there exists a stable money demand equation is an issue of whether there exists a cointegrating relation between mt, y t and rt . Table 2 reports Johansen (1988, 1991) likelihood ratio test results when mt, y t and rt are treated as 1(1). For the complete sample period based on either AIC (Akaike (1973)) or Schawarz (1978) criterion selected order o f autoregressive process, there is one cointegration between real GDP, interest rate, and real M l or real M2. Focusing on the sub-sample period of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 1992.1 - 2000.IV there is one cointegrating relation between real M2, real GDP and interest rate but no cointegrating relation between RM1, RGDP and interest rate. Table 2 Johansen Likelihood Ratio Test When M l (M2), GDP and Bond Rate are 1 (1) Sample Period Hypothesis Lag Order Information Criterion Likelihood Ratio 5% critical value 1% critical value Number of cointegrated relation 1980.IV M l, r=0 1 SBC 62.44008 34.91 41.07 One - M l, r<=l 21.50105 19.96 24.60 2000.IV M l, r<=2 2.678774 9.24 12.97 2 AIC 63.36653 34.91 41.07 One 22.18303 19.96 24.60 2.393634 9.24 12.97 M2, r=0 2 SBC (AIC) 54.55278 34.91 41.07 One M2, r<=l 21.48478 19.96 24.60 M2, r<=2 5.156355 9.24 12.97 1992.1 M l, r=0 1 SBC (AIC) 33.29918 34.91 41.07 No - M l, r<=l 12.60056 19.96 24.60 2000.IV M l, r<=2 4.643863 9.24 12.97 M2, r=0 1 SBC 39.44755 34.91 41.07 One M2, r<=l 16.55820 19.96 24.60 M2, r<=2 4.706671 9.24 12.97 2 AIC 35.24054 34.91 41.07 One 16.12073 19.96 24.60 4.100181 9.24 12.97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 Table 3 presents the Johansen (1991) normalized cointegrating estimates between mt , yt and V f for both the complete sample and sub-sample period. The complete sample estimated income coefficient for the M l equation is exactly the opposite of what the economic theory predicts. The income coefficient of M2 although has the correct sign, the interest rate coefficient has the wrong sign. For the sub-sample Table 3 Johansen (1991) Normalized Cointegration Vector 1980.IV- 2000.IV 1992.1 - 2000.IV M1 M2 M l M2 Based on SBC GDP -0.72 3.83 75.23 8.225 (-1.337) (3.34) (258) (8.27) Bond Rate -0.13 0.108 2.666 0.316 (0.079) (0.17) (9.92) (0.41) Intercept -26.43 -44.29 -1156 -112.7 (21.22) (53.54) (4017) (129.3) Based on AIC GDP 0.183 11.036 (0.674) (12.33) Bond Rate -0.05 Same as Same as 0.404 (0.049) SBC SBC (0.552) Intercept -17.62 -156.4 (10.59) (192.4) * Standard Error in parentheses. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 period, although estimates of the income coefficients yield correct signs, they are of this unbelievable magnitude with income elasticity of 75.23 for M l based on SBC and 11.036 for M2 based on AIC. Moreover, the interest rate coefficients in these equations are of the wrong signs. The Johansen (1991) MLE of the cointegrating relations is based on the decomposition of the coefficient matrix of the level variables, I I , into the product of II = a/3' , in an error correction representation where /?' is referred as the cointegrating relations and a is referred as the adjustment coefficients of the deviations from the long-run equilibrium. However, as argued in Hsiao (2001a) that it is difficult to give a structural interpretation o f the cointegrating vectors J3' and it is preferable to view the corresponding row of II directly as the implied long-run relation of the equation of interest. Anderson (2001) has proposed an efficient method o f directly estimating II given the rank of cointegration. Table 4 presents Andersen (2001) reduced rank estimation of II assuming that there is one cointegrating relation for b o th th e bivariate and trivariate system. Again the result would imply a negative relation between m and y. Moreover, the estimated coefficients are close to zero and appear to support the contention that there is no stable relation between money demand, real GDP and interest rate. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 Table 4 Anderson’s (2001) Reduced Rank Estimation Sample Period 1980.1 V -2000.IV 1992.1 -2000.IV Monetary Aggregates Ml M2 Ml M2 The Long Rim Coefficient II -0.0004 -0.0002 0.0645 -0.0010 -0.0006 0.1523 -0.3178 -0.1775 48.0654 -0.0007 -0.0004 0.0645 -0.0013 -0.0008 0.1195 -0.5535 -0.3415 49.697 0.0026 0.0004 0.0376 0.0193 0.0031 0.2741 2.1893 0.3468 31.1393 0.0000 0.0000 0.0378 -0.0001 -0.0001 0.1061 -0.0348 -0.0156 31.3812 * II is the long run coefficient matrix in 'Ain {RGDPt) c," " * 1 1 7 1 1 2 * 1 3 ~\n(RGDPt_ x) Ain (M2 i,) = c2 + * 2 1 *22 * 2 3 ln(^2,,-l) A t ; -C 3 _ .*31 * 3 2 *33. r > - 1 'BnP BnP V Ain {RGDPt_p)~ uu BnP B2 2 p B2 3 P A \n(M2t_ p ) + u2 l _ B3 \p B3 2 p B3 3 p &,-P _«3*. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 Table 5 Unrestricted Error Correction Estimation: M l (M2), GDP and Bond Rate Are I (1) Sample Period Estimated Coefficients _ _ 0.36 (021) -0.01 (0.01) -0.01 (0.02) 0.004 (0.001) 0.39 (0.17) + -0.013 (0.01) -0.01 (001) -0.001 (-0.001) -26 (9.06) 1.56 (0.54) 0.18 (o.68) 0.24 (0.06) 1980.IV - 2000.IV A In [Mu ) A\n (RGDP,) A r, 0.32 0.06 -0.001 (0.09) (0.14) (0.002) 0.008 0.13 0.002 (0.08) (0.11) (0.002) 5.73 6.42 -0.18 (4.01) (5.96) (0.11) to K m ) ' In (RGDP,_,) Ain (M, Ain (RGDP,_{) Ain (M2i) Ain (RGDP,) A r, r _ 0.97 (0.84) 0.007 (0.02) -0.06 (0.06) -0.003 (0.001) = 5.37 (188) + 0.07 (0.05) -0.41 (0.15) -0.006 (-0.003) -218 (77.9) 1.78 _ (213) 15.87 (6.35) 0.35 (0.13) 0.093 (0.17) 0.009 (0.08) 0.002 (-0.001) + 0.007 (0.38) 0.011 (0.18) 0.003 (0.004) 38.88 (15.89) -13.62 (5.37) -0.2 (0.16) J 3.57 (2.04) -0.01 (0.02) -0.22 (0.15) -0.01 (0.002) = 7 (2.08) + 0.04 (0.02) -0.49 (0.14) -0.005 (0.003) -136 (96.9) 0.3 . (0 9 6 ) 8.51 (6.9) 0.21 (0.12) In (M2 ) In (RGDP,_,) A In (M , ) Ain (RGDP'_X ) A r,_ , 1992.1-2000. IV ' A In (M „) Ain {RGDP,) A r, 1 ,1 K m ) In (RGDP'_{) A In (M2i) A In (RGDP,) Ar. 0.83 (0.79) 0.01 (002) -0.06 (0.06) -0.003 (0.001) lnK ,_ ,) 5.54 (1.70) + 0.08 (0.04) 0.44 (0.13) -0.006 (0.003) In {RGDP^) -157 (78.7) _ -0.02 _ (2.13) 10.17 (6.03) 0.21 (0.12) r‘-' Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 The efficiency gain of Johansen (1991) MLE o f jB' or Anderson (2001) reduced rank estimation of II is based on prior knowledge of known rank of cointegration. However, it is well known that tests for cointegration have very poor finite sample performance. Therefore, in Table 5 we present the unrestricted error- correction estimation of the long-run relations between m, y and r4. The results again confirms the early findings that either the estimated long-run relation is opposite to what economic theory would predict or it is not unreasonable to assume n =0 after 1992, hence implying that there is no stable relation between money demand, real GDP and interest rate. The multivariate time series analysis is very sensitive to the time period covered and to the choice of the order o f autoregressive process, p. We find that either there is no cointegrating relation between money demand, real GDP and interest rate, or if there is one, the estimated relation is opposite of what economic theories predict. Therefore, we turn to the single equation modeling. Table 6 presents the least squares estimates of the money demand equation (2.3.1) for the period 1980.IV - 2000.IV and 1992.1 - 2000.IV. The estimated lagged dependent variable coefficient is almost exactly equal to one. The income coefficient is either insignificantly different from zero or has the wrong sign. The interest rate coefficient is statistically significant and has the correct sign. 4 The unrestricted error correction estimates are identical to the unrestricted vector autoregressive m odel estimates with proper linear transformation (e.g., H siao (2001a)). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 Table 6 Least Squares Estimation of Money Demand Dependent Variable Sample Variable Parameter Standard Period Estimate Error M2 1980.IV Intercept 1.30462 0.28975 - Real GDP -0.15425 0.04538 2000.IV RM2(-1) 1.07022 0.02790 Bond Rate -0.00186 0.00069 1992.1 Intercept -0.16272 0.85081 - Real GDP 0.00847 0.06772 2000.IV RM2 (-1) 1.00295 0.02248 Bond Rate -0.00250 0.00140 M l 1980.IV Intercept 0.46907 0.21852 - Real GDP -0.01857 0.01700 2000.IV RM1 (-1) 0.98964 0.01249 Bond Rate -0.00566 0.00135 1992.1 Intercept -0.68783 2.10228 - Real GDP 0.08414 0.14898 2000.IV RM1 (-1) 0.96038 0.01999 Bond Rate -0.01005 0.00283 Since neither the multivariate, nor the single equation estimates give economically meaningful interpretation of the money demand equation, we use Box and Jenkins (1970) procedure to model the money demand. Table 7 presents the ARIMA modeling o f RM1 and RM2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 Table 7 ARIMA Modeling of M2 and M l Sample Period Dependent Estimated equation* Variable 1980.IV- 2000.IV Amu Amu = 0 .0 1 1 5 5 + 0.56A w 1 ( , + e, (3'52) (6 .1 1 ) A m2l A m2t = 0.01+ 0.5Aw2 ,_, + 0.246Am2,_2 + et (3 .6 6 ) (459) (2 2 6 ) 1992.1- 2000.IV Amu Amu = 0.0042+ 0.558Am„ , + e, (1 .9 4 ) (2 .9 7 ) Aw2, Am7 f = - 0 .0 0 7 + 0 .5 17Am2. . + 0.479A m2. , + e . 2 1 2‘ (-1.49) ( 3 . 2 8) 2‘~ ' (3 .0 0 ) 2'~2 ' * Model selection based on SBC. To check whether the money demand equation o f the form (2.3.1) or the univariate ARIMA model better describes Japan’s aggregate money demand, we use the prediction principle. As remarked by Friedman and Schawarz (1991), “the real proof of the pudding is whether it produces a satisfactory explanation of data not used in baking it—data for subsequent or earlier years”, we therefore split the sample into two periods. We use data o f 1980.IV - 1997.IV to estimate various models. We then use the estimated coefficients to generate one period ahead prediction for the period 1998.1 — 2000.IV. The first column of Table 8 presents their root mean square prediction errors. The univariate ARIMA model predicts better than the models that also use real GDP and interest rate as explanatory variables. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 Table 8 Root Mean Square Error Comparison For One-Period-ahead Forecasting Dependent Model RMSE** RMSE** Variable 1998.1-2000.IV 1998.III-2000.IV M l 1 ARIMA (1,1,0) 0.0109* 0.011* 2 OLS 0.012 0.019 3 Unrestricted ECM 0.018 0.012 4 Two Stage Least Squares 0.015 0.015 M2 1 ARIMA (2,1,0) 0.0084* 0.004* 2 OLS 0.011 0.0047 3 Unrestricted ECM 0.011 0.007 4 Two Stage Least Squares 0.014 0.006 Smallest Root Mean Square Error. ** The RMSE for the 1998.1 - 2000.IV column is calculated by estimation using data from 1980.1 to 1997.IV and predict 1998.1 to 2000.IV; the RMSE for the 1998.III -2000.IV column is calculated by estimation using data from 1991.Ill to 1998.11 and predict 1998.Ill to 2000.IV. The above prediction comparison is based on data from 1980. IV - 2000.IV. If there is indeed a structural break in the early 90’s, then the complete sample comparison in general favors a random walk model to a vector autoregressive or an error correction model. To gauge against unfavorable treatment of structural break models, we also compare the prediction performance by using the data o f 1991.Ill to 1998.11 for estimation and generate one period ahead prediction for the period 1998.Ill to 2000.IV. The root mean square prediction error comparisons are reported at the second column of Table 8. The univariate ARIMA model again dominates the others. The fact that both the prediction results and the estimation o f the form (2.3.1) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 having coefficients of lagged dependent variable almost exactly equal to one appears to support the contention that Japan’s aggregate money demand is better modeled in terms of random walk with a drift (with a drift parameter possible changing for period after 1990). 2.4 The Disaggregate Time Series Analysis The fact that Japan's aggregate money demand is better modeled by a random walk with a drift is disconcerting to economists. It basically says that the empirical evidence has refuted theories of money demand (e.g., Laidler (1969)). However, the lack of stable relation between real money demand, real GDP and interest rate in aggregate time series analysis could be due to the shortages of degree o f freedom. But more importantly, it could be due to the lack of sample variability. For the period 1980.IV - 2000.IV, the minimum and maximum values of the logarithm o f real GDP, real M l, real M2 are (14.943, 15.4925), (13.609, 14.707) and (14.738, 15.708) respectively, and the corresponding standard deviations for real GDP, real M l and real M2 are 0.18, 0.31 and 0.29. For the period o f 1992.1 - 2000.IV, the minimum and maximum values of the logarithm of real GDP, real M l and real M2 are (15.3878,1 5.4925), (14.014, 14.707) and (15.4296,1 5.7089) respectively, and the corresponding standard deviations are 0.036, 0.23,0.09. When all sample points are clustered together, technically any slope coefficient estimates may be obtained by slightly changing the time period covered. To get more robust estimate of the relationship between money, real GDP and interest rate, we need data to vary over a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 wider range. One possible way to obtain sample of greater variability is to use disaggregate time series. Japan is divided into 47 prefectures. Annual data on prefecture employee income (the counter part to national GDP), prefecture price deflator, population statistics from 1985 - 1997 are available from the home page of Economic and Social Research Institute (former Economic Planning Agency of Japan). Data on Demand Deposits (MF1) and MF1 plus Savings Deposits (MF2) are available from Monthly Economic Statistics o f the Bank of Japan. MF1 and MF2 may be viewed as the prefecture counterpart of the national M l and M2 without currency held by individuals. There are several advantages o f using disaggregate time series. First, there are more degrees of freedom, more sample variability and less multicollinearity. Figures 1 - 3 plot the 47 per capita prefecture real MF1, real MF2 and income over the period of 1989 - 1997. The minimum and maximum prefecture logarithm o f per capital real income, real MF1 and real MF2 are (2.96, 3.79), (0.99, 3.44) and (2.65, 4.84) respectively for the sample period o f 1989 - 1997. The corresponding standard deviations for the same period are 0.15, 0.37 and 0.34 respectively. For period 1992 - 1997, the standard deviations are 0.145, 0.345 and 0.319 respectively. Secondly, the disaggregate tim e series data allow more accurate estimate of dynamic adjustment behavior even with a short time series. Third, it provides the possibility to control the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 impact of omitted variables. Forth, it provides means to get around structural tests which are based on large sample theory with dubious finite sample property (e.g., Hsiao (2001b)). Figure 1 Plotting Real M2 for 47 Prefectures plotting the prefecture M2 o ve rt ime 5 4 .5 4 3 .5 3 2 .5 1 9 tim e from 1 989 to 199 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 Figure 2 Plotting Real M l For 47 Prefectures 3 .5 2.5 1 .5 0.5 p lo tting the p re fe cture M 1 ov er tim e 198 9 19 90 1991 1 992 19 93 19 94 time from 1 989 to 19 97 19 95 1996 1997 Figure 3 Plotting Prefecture Income For 47 Prefectures plotting the prefecture in c o m e o v e r t i m e 3 .8 3 .7 3 .6 3 .5 3 .4 3 .3 3 .2 3.1 3 2 .9 I---- 1989 1990 199 1 1992 1993 199 4 1995 1996 19 9 7 time from 1 989 to 199 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 However, there are also a number of sample and statistical issues o f using prefecture data. We shall first discuss issues of data measurement, then statistical modeling, and finally present the empirical results. 2.4.1 Data Measurement There are three issues involved in using prefecture deposit data in domestically licensed banks. First, there is an issue of consistency of coverage of banks. There was a change in the definition o f banks surveyed in the deposit statistics in 1989. Due to an extension o f the coverage of regional II banks in the deposit statistics in that year, the Monthly Economic Statistics of the Bank o f Japan data show an unusual increase in 1989. Secondly, there is an issue of whether the bubble burst in 1990 created a noise that can no longer be viewed as random noise from the same population. Third, there is an issue o f people living in one prefecture but working in big metropolitan prefectures, Tokyo, Osaka and Kyoto, had bank accounts near where they worked. With the availability of panel data, it is fairly easy to check if there exist systematic measurement errors on data of some prefectures. To avoid the possibility of obtaining biased results because of inconsistent data measurements in 1989 and 1990, one may just fit a money demand equation for the year 1992 - 1997. To avoid the problem o f people living in one prefecture but having bank accounts in another prefecture, we can exclude the data of Tokyo, Osaka, Kyoto and their neighboring Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 prefectures Chiba, Saitama, Kanagawa and Hyogo from considerations and use the remaining 40 prefectures data to fit (2.3.1). 2.4.2 Statistical Modeling Disaggregate data focus on individual outcomes. Factors affecting individual outcomes are numerous. It is neither feasible, nor desirable to construct a model that includes all these factors in the specification. A model is not a mimic of reality, but a simplification of reality, designed to capture the essential factors affecting the outcomes. To allow for the possibility of heterogeneity among disaggregate units, we can let Oi = (yi,bi,ciy in (2.3.1). The question of whether the coefficient 0. ={yi,bi,ci)' should be assumed fixed and different or random and different depends on whether can be viewed as random draws from heterogeneous population or from a common population. In this chapter we shall favor a random coefficients approach for two reasons. First, the short time series dimension does not allow accurate estimation o f individual 0t . Second, if 0; are indeed fixed and different, the representative agent argument so frequently used by economists no longer appears relevant. Neither is it possible to make inference about population relationship between money demand, income and interest rate. Therefore, we assume that the coefficients > < are randomly distributed with mean 0i =(/,b,c)' and covariance matrix A. W hen the regional specific effects, a , , is treated as a fixed constant, we have a mixed fixed and random effects models (Hsiao and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 Tahmiscioglu (1997)). When a t is treated as a random variable that is independent o f (yit,rt ) and is independently distributed across / w ith m ean a and variance <r2 a , we have random effects model (e.g. Hsiao (2003a)). It is well known that when regressors are exogenous the estimators based on the sampling approach such as fixed and random effects yield consistent estimates o f the mean coefficients when the number of the cross-sectional units approaches infinity. However, the same results do not carry over to dynamic models. The negligence of coefficient heterogeneity in dynamic models creates correlation between the regressors and the error term as well as serial correlation in the residuals. Neither the least squares, nor the within estimator is consistent (Pesaran and Smith (1995)). Because there does not appear having any consistent estimator o f the mean coefficients when T is finite, we shall adopt a hierarchical Bayes estimator which is shown to have very good properties even when T=5 and N=50 (Hsiao, Pesaran and Tahmiscioglu (1999)). 2.4.3 Empirical Results We report the findings based on the disaggregate data analysis. The first thing we wish to check is that whether 0t - are identical across i as assumed by Fujiki, Hsiao and S hen (2001). The testing o f homogeneity a cross i yields an F-value of 69 for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 MF1 and 4.07 for MF2. Both are statistically significant at 1% level with 40 and 117 degrees o f freedom hence rejecting the homogeneity assumption. To check whether the issue of people living in one prefecture but having bank accounts in another prefecture might systematically bias the estimation results, we can compare the difference between the estimates using all 47 prefectures and the estimates based on prefecture data that are relatively free from this issue. If the systematic measurement errors are not serious, two estimates are expected to be close. Otherwise, they are expected to be different. Columns 2 and 3 o f Tables 9 present the estimated coefficients for the MF1 and MF2 using 47 prefectures (complete sample, measurement error not adjusted) and 40 prefectures respectively. Columns 2 and 3 of Table 10 present the mixed fixed and random effects estimated coefficients for the MF1 and MF2 equations using 47 and 40 prefectures respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 Table 9 Hierarchical Bayes Estimated Random Coefficients Complete Sample Measurement error not Adjusted 40 Prefectures Complete Sample Measurement Error Adjusted Variables Coefficients Standard Deviation Coefficients Standard Deviation Coefficients Standard Deviation M F l(-l) 0.6507019 0.0316268 0.6555294 0.0344682 0.6351 0.0404 Income 0.927945 0.1090934 0.8812118 0.1137414 0.7935 0.1101 Bond Rate -0.05055 0.006016 -0.047598 0.0061865 -0.0542 0.0069 Constant -2.272464 0.3708626 -2.125453 0.382336 -1.7913 0.3727 MF2(-1) 0.5248475 0.06054 0.5337534 0.0689148 0.5264 0.0691 Income 0.4776609 0.0563903 0.4725214 0.0642067 0.4821 0.0691 Bond Rate -0.009717 0.0032794 -0.008941 0.0034183 -0.0087 0.0035 Constant 0.0604931 0.2244695 0.0425582 0.2396531 0.0368 0.2449 Variance Covariance matrix of thetai for MF1 (40 Prefectures): 0.015 -0.0001 0.0017 -0.024 -0.001 0.177 0.059 -0.588 0.0017 -0.0559 0.0005 -0.023 -0.024 -0.588 -0.023 2.017 Variance Covariance matrix of thetai for MF2 (40 Prefectures): 0.068 -0.031 0.0023 -0.13 -0.0031 0.0062 0.0003 -0 .107 0.0023 0.0003 0.0014 -0 .009 -0.13 -0.107 -0.009 0.8385 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 Table 10 Hierarchical Bayes Estimated Mixed and Fixed and Random Coefficients Complete Sample Measurement error not Adjusted 40 Prefectures Complete Sample Measurement Error Adjusted Variables Coefficients Standard Deviation Coefficients Standard Deviation Coefficients Standard Deviation M F l(-l) 0.6389765 0.0409127 0.6533347 0.0339363 0.6324 0.0396 Income 0.2988121 0.0848417 0.8809016 0.1154574 0.7846 0.1093 Bond Rate -0.071933 0.007101 -0.048035 0.0062235 -0.0546 0.0067 Constant -0.116582 0.2817165 -2.119956 0.3889621 -1.7556 0.3662 MF2(-1) 0.4770931 0.0579894 0.5378975 0.0689838 0.5182 0.0696 Income 0.4318347 0.0565629 0.4742547 0.0631223 0.4760 0.0680 Bond Rate -0.013406 0.0028535 -0.008728 0.0034719 -0.0095 0.0034 Constant 0.3876087 0.192636 0.0223581 0.2436108 0.0860 0.2470 Variance Covariance matrix of thetai for MF1 (40 Prefectures): 0.014 -0.001 0.0017 -0.024 -0.001 0.0176 0.058 -0.584 0.0017 0.058 0.0005 -0.022 -0.024 -0 .584 -0 .022 2.000 Variance Covariance matrix of thetai for MF2 (47 Prefectures): 0.069 -0.032 0.0024 -0 .1 3 -0.0032 0.0063 0.0003 -0 .107 0.0024 0.0003 0.0015 -0 .009 -0 .1 3 -0 .107 -0 .009 0.8448 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 Table 11 presents the Hausman (1978) specification test of the presence of systematic measurement errors. The chi-square statistics reject the null hypothesis of no measurement error at 1% significance level in all cases. To avoid contamination due to measurement errors, we can rely on the results using 40 prefectures. Table 11 Hausman Specification Test of Measurement Error* Variables RE Models FE Models MF1 24.68* * 45.92 MF2 84.17*** ~ * Since we use Gibbs sampler to estimate the random coefficients models and the mixed fixed and random effects models, the calculated Hausman statistic can be slightly different for different simulations. **: Deleting Bond rate to avoid singularity problem. ***: Dropping constant to avoid singularity problem. — : Hausman Test statistics is negative. An alternative procedure to avoid measurement errors issue is to aggregate the data of Tokyo and its neighboring prefectures, Chiba, Saitama, Kanagawa, into one region and the data of Osaka, Kyoto and Hyogo into another region. Columns 3 of Tables 9 and 10 also present the estimates using 40 prefectures plus these two aggregated regions (in other words, all 47 prefecture data are used). The results are very similar to the results o f only using data of 40 prefectures (All the differences in the estimated coefficients are less than one standard error). Because o f the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 similarities o f t hese r esults, w e s hall o nly r eport the results of disaggregate data analysis using 40 prefectures for ease of comparison between the aggregate and disaggregate data analysis. There are three notable features arising from the disaggregate time series analysis. First, while aggregate time series analysis shows that there is no stable relation between money and income, the disaggregate time series analysis shows that there exists a stable money demand function. There is no indication that money demand follows a random walk with a drift. The estimated mean lagged dependent variable coefficient is significantly below one. The mean income coefficient is positive and statistically significant. The estimated mean interest rate coefficient is negative and statistically significant. The estimated mean short - run income coefficients for MF1 and MF2 are both less than 1 with the magnitude of 0.88 and 0.47 respectively. The estimated mean short-run interest rate coefficients for MF1 and MF2 are -0.047 and - 0.009 respectively. Secondly, the estimated mean relationship is very similar to those obtained using the formulation of Fujiki, Hsiao and Shen (2001) mn = Wi,-x + by i, + crt + a, + s it (2.4.1) where at is assumed either random or fixed (see Table 12, where ai fixed or random is referred to as fixed or random effects estimator, respectively (FE or RE)). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 Table 12 Fixed Effects (FE) and Random Effects (RE) Estimation Fixed Effects Random Effects Variables Coefficients t-value Coefficients t-value 40 Prefectures M lF (-l) Income Bond rate Constant 0.7223413 0.6500417 -0.035468 30.418895 8.9506568 -7.502673 0.6958649 0.7945655 -0.045742 -1.915043 25.69604 10.378702 -9.398455 -0.2925 M2F(-1) Income Bond rate Constant 0.6526298 0.2571169 -0.00954 14.061279 5.2103729 -3.320842 0.4982692 0.3673084 -0.014314 0.5143107 11.419157 7.3975925 -5.290189 0.2486889 The estimated coefficients from the random effects and fixed effects models for 40 prefectures are quite close to those presented in Table 9 and Table 10 for 40 prefectures. Furthermore, the variance-covariance matrix o f 0t as summarized at the bottom o f Tables 9 and 10 are fairly small in absolute magnitude. In other words, no matter which models we use for the analysis of the disaggregate data, they appear to support the contention that there is indeed a stable relation between money demand, income and interest rate as predicted by economic theory. b c Third, using the formula of - ~ ■ ■ or — =— the estimated long-run income elasticity 1- y 1 - y for MF1 is 2.175 and for MF2 is 1.018 respectively for the random coefficients Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 model and 2.13 and 0.988 respectively for the mixed random and fixed coefficients models. The long-run semi-interest rate for MF1 is -0.149 and for MF2 is -0.0184 for the random coefficients model, and -0.149 and -0.0197, respectively for the mixed fixed and random coefficients models. Table 13 presents the estimated income and semi-interest rate elasticities based on the random coefficients models and mixed fixed and random coefficients models, where the long-run elasticities are calculated using the formula V Hi— ! — w, or x~Yi V — Wj, with W : to be the average weight of the relative importance o f the ith prefecture money demand. The implied average long-run income elasticities for the MF1 and MF2 for the random coefficients model are 2.302 and 1.041, respectively, and are 2.296 and 1.02, respectively for the mixed fixed and random coefficients models. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 Table 13 Income Elasticity and Semi-Interest Rate Elasticity Based On Random Coefficient and Mixed Fixed and Random Coefficient Models Elasticities of Interest MF1 MF2 Short Run Long Rim Short Run Long Run Income Elasticity Random Coefficients Mixed Fixed 0.7846 2.302 0.476 1.041 and Random Coefficients 0.7935 2.296 0.4821 1.020 Semi-interest Elasticity Random Coefficients Mixed Fixed -0.0546 -0.145 -0.0095 -0.0185 and Random Coefficients -0.054 -0.146 -0.0087 -0.0189 2.5 Possible Sources of Discrepancy The diametrically opposite evidence obtained from the aggregate and disaggregate data series analysis makes one wonder what constitutes a well grounded level of analytical framework and what are the interesting interactions between disaggregate and aggregate models. In this section we explore whether the issues of simultaneity, different definitions of money, and aggregation o f heterogeneous units are the reasons behind these conflicting implications. 2.5.1 Simultaneity In a disaggregate framework it may be plausible to assume that income and interest rate are exogenous, but the same cannot be said for the aggregate data. When money Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 demand, income and interest rate are jointly determined, least squares regressions of the form (2.3.1) yields biased and inconsistent estimates even though regressors are I (1) (Hsiao (1997a,b). On the other hand, two-stage least squares estimator (2SLS) is consistent. Moreover, as the Monte Carlo studies conducted by Hsiao and Wang (2003) have shown, despite the limiting distribution o f the 2SLS of a structural vector autoregressive model is not standard normal or mixed normal where there does not exist strictly exogenous driving force, it still dominates modified estimators that possess desirable large sample properties in terms of bias, root mean square error and closeness o f the actual size to the nominal size in finite sample. We reestimate models of the form (2.3.1) by the two-stage least squares and present the results in Table 14. Again, the implication is similar to the least square results. The income coefficients have the wrong signs and the sums of the lagged dependent variable coefficients are almost exactly equal to one. In other words, even after taking account of simultaneity, we still cannot find a stable money demand function in the 1990's. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 Table 14 Two Stage Least Squares Estimation For M l and M2 Dependent Variable Sample Period Variable Parameter Estimate Standard Error t value M l 1980.IV Intercept 0.375692 0.207400 1.81 - M l(-l) 1.319778 0.093789 14.07 2000.IV M l (-2) -0.32958 0.093070 -3.54 GDP(-l) -0.01374 0.016063 -0.86 Bond Rate -0.00402 0.001347 -2.99 1992.1 Intercept 4.311913 3.230759 1.33 - M l(-l) 1.489444 0.166541 8.94 2000.IV M l (-2) -0.49015 0.156715 -3.13 GDP(-l) -0.27628 0.231358 -1.19 Bond Rate -0.01024 0.003025 -3.39 M2 1980.IV Intercept 0.931084 0.332153 2.80 - M 2(-l) 1.502049 0.087207 17.22 2000.IV M2(-2) -0.45290 0.098063 -4.62 GDP(-l) -0.10920 0.050970 -2.14 Bond Rate -0.00147 0.000604 -2.44 1992.1 Intercept -0.10242 1.140174 -0.09 - M 2(-l) 1.208455 0.174600 6.92 2000.IV M2(-2) -0.20761 0.173027 -1.20 GDP(-l) 0.006507 0.094242 0.07 Bond Rate -0.00202 0.001584 -1.28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 2.5.2 Definitions o f Money Disaggregate time series analysis uses demand deposit (MF1) and demand deposit plus time deposit (MF2) while aggregate time series analysis uses M l (currency and demand deposit) and M2 (M l + time deposit). To see if it is the component of currency holding that contributes to the lack o f relationship between money demand and income, we redo the analysis for the (M l- currency) and (M2 - currency). The results of this analysis are the same as those of M l and M2 as can be seen from the least squares and 2SLS estimates presented in Tables 15 and 16. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 Table 15 Least Squares Estimation of Money Demand M l (M2)-Currency Dependent Variable Sample Period Variable Parameter Estimate Standard Error t value M l 1980.IV Intercept 0.40465 0.27360 1.48 - Real GDP -0.01656 0.01870 -0.89 2000.IV M l(-l) 0.99184 0.01379 71.93 Bond Rate -0.00561 0.00168 -3.34 1992.1 Intercept -0.13807 2.55447 -0.05 - Real GDP 0.04922 0.17968 0.27 2000.IV M l(-l) 0.95927 0.02226 43.09 Bond Rate -0.01217 0.00335 -3.63 M2 1980.IV Intercept 1.42314 0.30912 4.60 - Real GDP -0.17217 0.04797 -3.59 2000.IV M 2(-l) 1.08074 0.02928 36.91 Bond Rate -0.00182 0.00069043 -2.64 1992.1 Intercept -0.05223 0.86170 -0.06 - Real GDP -0.00191 0.06930 -0.03 2000.IV M 2(-l) 1.00613 0.02477 40.62 Bond Rate -0.00246 0.00143 -1.72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 Table 16 Two Stage Least Squares Estimation For M l (M2)- Currency Dependent variable Sample Period Variable Parameter Estimate Standard Error t value M l 1980.IV Intercept 0.309286 0.252302 1.23 - M l(-l) 1.374958 0.094940 14.48 2000.IV M l (-2) -0.38457 0.094469 -4.07 GDP(-l) -0.00991 0.017254 -0.57 Bond Rate -0.00368 0.001609 -2.28 1992.1 Intercept 4.861598 3.520021 1.38 - M l(-l) 1.588392 0.151125 10.51 2000.IV M l (-2) -0.59165 0.141427 -4.18 GDP(-l) -0.30954 0.249802 -1.24 Bond Rate -0.01120 0.003227 -3.47 M2 1980.IV Intercept 0.963669 0.367020 2.63 - M 2(-l) 1.511705 0.087687 17.24 2000.IV M2(-2) -0.45983 0.100682 -4.57 GDP(-l) -0.11382 0.055842 -2.04 Bond Rate -0.00146 0.000612 -2.39 1992.1 Intercept -0.27114 1.131228 -0.24 - M 2(-l) 1.246219 0.174734 7.13 2000.IV M2(-2) -0.24918 0.173719 -1.43 GDP(-l) 0.021131 0.094927 0.22 Bond Rate -0.00159 0.001623 -0.98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 2.5.3 Aggregation Over Heterogeneous Units Estimating macro relations always involves aggregation over micro units. Aggregation is valid if the disaggregate relation is linear and if all micro relations have identical parameter values or if the distribution of the micro variables remains constant over time (e.g., Stoker (1993)), Theil (1954)). Fomi and Lippi (1997), Fomi and Lippi (1999), Granger (1980), Lewbel (1992), Lewbel (1994), Trivedi (1985), Zaffaroni (2001), have shown that the dynamics o f aggregate variables can be very different from the dynamics o f disaggregate variables with the presence of parameter heterogeneity. Under the assumption that mjt, y it are 1(1) processes, it is also possible to have stable micro relations (cointegrating relations among micro units) but unstable macro relations (no cointegrating relation among macro relation). To see this consider a cointegrating relation among micro units o f the form, ~ 7— y it ~ 7 ^ —r/ = vu > (2-5.1) W / 1~ r i where vit denotes a zero mean 1(0) process. Let mt = mit , y t = y it and wu = yu I yt • LEM M A 1: Suppose (2.5.1) holds for all i. The aggregate relation mt - (iyt - crt - a is 1(0) if and only if either (or both) of the following conditions holds: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (ii). If bi ^ bj 1 ~ / i 1 ~ Y i , then wt ' must lie on the null space of d t , for all t where wt'={wlt,...,wNt) fo ralU d('= ---- P , - —^ -----P U -r i i- r N and | y i |< 1 for all PROOF: From (2.5.1) we have E n b{ v-N c i L v # a i , yt A-ii=1 Zj/'=1 v it w l- r t) 1~ri f> where ut =Y^= \Vit is 1(0). Let c = and « = 'T!i=\~L- > then i - r , l - T i mt - p y t -c r t - a = m t i Z i r - ™ « - p ! - / / yu +ut L Since y t is 1(1), ut is 1(0) if and only if X,=i— — wi( - P=XiL i 1-7. Yi 1 — P k , =o l ~Y i for all t, where £,=i wh = 1 by construction. This condition can hold for all t when either (i) or (ii) or both hold. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 COLLARY: A sufficient condition for LEMMA l(i) to hold is that all b. = b , y. = y and |^ |< 1. A sufficient condition for LEMMA l(ii) to hold is wit = xvj for all t. Lemma 1 states that under parameter heterogeneity there could be cases that even though there exist stable relations among all micro units, the aggregate relation can b, bj still exhibit unit root phenomenon. For instance, if xvu * wis and — - — 1 - f t 1 - y t then m t ~ P y t ~ crt ~ a = y t +ut (2.5.2) Even though all \yi \<\, because ^ ._ j-—— wit * (3 and because y t is 1(1) mt - f}yt - crt - a will behave like an 1(1) process.5 5 Of course, as long as there is one yt near 1 or equal to 1 for (2.3.1), the spectral density of (2.5.2) can become unbounded T -* o o even when wit = xvt for all t. Zaffaroni (2001) shows that even all yt are constrained between [0,1), as long as the tail density of yt takes the form cb (l - y)b with 0 <cb < oo, b < -1 / 2, as y t approaches 1 from the left, the spectral density of (l / C 1 _ TiLY^uit becomes unbounded where L denotes the lag operator. However, this condition actually requires the distribution of yt heavily clustered near 1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 Figures 4 and 5 present the distribution of yt for MF1 and MF2. The largest yt for MF1 is 0.78 for Nigata prefecture (about 2.1% of the total MF1 demand in 1992)and Figure 4 The Distribution of Lagged MF1 Coefficients Figure 5 The Distribution of lagged MF2 Coefficients Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 for MF2 is 0.93 for the Mie prefecture (slightly above 2.4% o f the total MF2 demand in 1992), hence appears to reject the hypothesis that instability observed at the aggregate level is due to one of the yi is one or near one. Figure 6 presents the distribution o f wit. Figure 6 Prefecture Income Weight Distributions Over 1992 - 1997 Q Q 35 Q O S QQ 2 Q015 0 0 1 1 1 .5 2 2 5 3 35 4 45 5 55 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 The prefecture income weights do not stay constant over time either. This information appears to indicate that micro heterogeneity could be the source of discrepancy between the aggregate and the disaggregate analysis for MF1 and MF2. 2.6 A Simulation Study of Aggregate and Disaggregate Behavior Although in section 2.5 we show that ignoring parameter heterogeneity across prefecture could be the reason o f inducing an unstable aggregate money demand function, in our micro analysis we use a log-linear specification. This implies that aggregation is in effect nonlinear because the aggregate variables are not defined as the sum o f variables in logarithms, but are defined as the logarithm of the sum of micro variables. The nonlinear aggregation makes it much more difficult to drive mathematical relationships between aggregate and disaggregate relations. In this section we resort to simulation methods to show that it is indeed possible to generate unit root phenomenon with insignificant income coefficient when there exists a stable relation among micro units if the parameters are different across prefectures and the aggregation is nonlinear. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 2.6.1 The Simulation Design We artificially generate time series data for each prefecture based on observed stylized facts. Therefore, we perform the simulation in the following steps: (i). Generate the logarithm of prefecture income as yu=Mi + yi,t-i+nit, C 2 -6-1) where /z( - is the drift parameter, r/it is the i.i.d. normally distributed error term with mean 0 and variances derived from the actual prefecture data. The initial value of prefecture income is set at each prefecture’s 1992 level. (ii). Generate bond rate as a random walk without drift, rt = ri - i + * t > (2-6-2) where the error term et is normally distributed with mean 0 and variance equal to variance of 5 year bond rate over the year 1992 - 1997. The initial value of simulated bond rate is set at the actual 1992 level. (iii). MF1 and MF2 are simulated according to mu = + bty it + + ai + s it, (2.6.3) where we use the estimated coefficients from the random coefficients models, {yt, 6,., ct, ai) to generate the disaggregate money demand data, and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 s it is the corresponding error term with mean 0 and variance equal to the prefecture level variances based on the actual data. (iv). Transform the simulated prefecture level logarithm data into level data. Sum the disaggregate data to get aggregate data. 2.6.2 Results Based on Simulated Data The estimation results based on simulated data are very similar to what we have obtained in section 2.3. The lagged dependent coefficients are near 1 and the income coefficients are either insignificant o r have the wrong sign. T able 17 provides the least squares estimates of the simulated M l and M2 equation. They are of similar magnitude to those obtained from the actual data (see Table 15). To see whether the above result is a pure chance event, we repeat the above simulation 500 times and plot the histogram for the estimated lagged dependent variable coefficients in Figures 7 and 8. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 Table 17 Least Squares Estimation for the Simulated M l and M2 Dependent Variable Variable Parameter Estimate t value M l M l(-l) 0.987828 63.027525 Real GDP 0.149237 0.855213 Bond Rate -0.003981 -1.269199 Constant -0.944954 -0.787662 M2 M 2(-l) 0.987361 1375.615056 Real GDP -0.970071 -5.734554 Bond Rate -0.061830 -12.879337 Constant 13.093577 11.257642 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 7 Histogram for Simulated M l(-l) Coefficients 300 1 ----------- 1 ------------1 ------------1 ------------1 ----------- 1 ------------r Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 Figure 8 Histogram for Simulated M 2(-l) Coefficients 1.015 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 We also repeat the simulation by randomly assigning disaggregate coefficients to different prefectures income data. The results are plotted in Figures 9 and 10. Figure 9 Histogram of Simulated Ml(-1) Coefficients When Disaggregate Coefficients are Randomly Assigned Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 Figure 10 Histogram for Simulated M 2(-l) Coefficients When Disaggregate Coefficients Are Randomly Assigned 1801 - 160 - 140 - 120 - 100 - 80 - 6 0 - 40 - 2 0 - 0 - 0.85 0.9 0.95 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.05 54 As one can see from these figures, the possibility o f obtaining near unit root in aggregate data is quite high. They appear to support the contention that the diametrically opposite results between the aggregate and the disaggregate analysis are largely due to ignoring heterogeneity in the micro units in the aggregate analysis. 2.7 Choice between Aggregate Model or Disaggregate Models Section 2.5 demonstrates analytically that it is possible to have stable micro relations but unstable macro relations when heterogeneity in the micro units are present. Section 2.6 uses simulation to show that the diametrically opposite inferences between aggregate and disaggregate analysis are possible with the features o f money demand data observed in Japan. Granted that there is indeed a stable relation between money demand, income and interest rate at the prefecture level in Japan, the questions naturally arise are that shall we favor using aggregate data to predict the aggregate outcomes o r study the impact o f policy changes or shall we first obtain disaggregate predictions or outcomes o f policy changes, then sum the disaggregate outcomes to obtain the aggregate outcomes. In this section we first compare the prediction performance then compare the implication of policy changes using aggregate and disaggregate models. Before we compare the prediction performance between the aggregate and disaggregate models, we note that when parameter heterogeneity is present in the disaggregate log linear AR(1) model, Lewbel (1992) has shown that if the same log- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 linear form is used, the aggregate model is AR( oo). Since higher order lag dependent variables are not statistically significant, we try to see if some nonlinear models would perform better than the log-linear model. We consider three alternatives: translog, Box-Cox transform and the inverse interest rate specifications as suggested by Zellner (1996). Table 18 presents the estimates from translog transformation. Table 18 Translog Estimation for the Aggregate Data"’ * Dependent Variable Sample Period Variable Parameter Estimate Standard Error t-value M l 1980.IV Intercept -250.56100 224.94780 -1.11 - Real GDP 33.10157 28.83521 1.15 2000.IV Bond Rate 5.90375 6.66412 0.89 GDP*GDP -1.03097 0.92418 -1.12 Rate*Rate 0.18397 0.04541 4.05 GDP*Rate -0.42975 0.42740 -1.01 M2 1980.IV Intercept -290.60145 100.07668 -2.90 - Real GDP 39.15582 12.82845 3.05 2000.IV Bond Rate -4.61941 2.96479 -1.56 GDP*GDP -1.25104 0.41116 -3.04 Rate*Rate 0.13351 0.02020 6.61 GDP*Rate 0.27601 0.19015 1.45 Translog estimation is performed for the static money demand function only. The sub period 1992.1 - 2000.IV aggregate money demand functions cannot be estimated by the static translog functions due to multicollinearity. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 The implied long-run income elasticity o f translog model is 0.99 and interest rate elasticity is -0.1 for M l and the implied long run income elasticity is 1.33 for M2 and the implied long run interest rate elasticity is 0.058 when the logarithm of real GDP and bond rate are at the mean value of 15.28 and 1.39 respectively. Table 19 Box-Cox Transformation for the Aggregate Data Dependent Variable Explanatory Transformation Coefficient Chi P Value Variable Parameter Estimates Square Pr > Chi square Case 1: Transform left hand side only M l Real GDP Bond Rate Constant -2.17 1.72e-20 143.481 0.000 -1.93e-15 53.748 0.000 .459831 M2 Real GDP Bond Rate Constant 0.45 .0003714 220.094 0.000 -4.751773 1.801 0.180 676.9516 Case 2: Transform left hand side and right hand side with different parameters M l Real GDP Bond Rate Constant -0.74 -1.55 .0000321 146.847 0.000 -1.55 -1.49e-10 70.601 0.000 0.6437668 M2 Real GDP Bond Rate Constant -1.45 -0.27 9.23e+07 284.779 0.000 -0.27 -.0030717 24.428 0.000 -6.35e+07 Case 3: Transform left hand side and right hand side with different parameters, but do not transform bond rate M l Real GDP Bond Rate Constant 2.87 -1.67 6.94e-30 140.403 0.000 -2.29e-12 42.544 0.000 .5979897 M2 Real GDP Bond Rate Constant -1.4 -0.64 170979.4 264.486 0.000 -2.57e-07 2.507 0.113 . | -122015.2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. However, the long run interest rate elasticity has the wrong sign for M2. Table 19 presents the estimates from the Box-Cox transformation. The Box-Cox transformation model using the procedure of Poirier and Melino (1978) yield practically zero income and interest elasticities. Table 20 presents the estimates using the inverse bond rate as explanatory variables. Table 20 Estimating Money Demand Using the Inverse of Bond Rate Dependen t Variable Sample Period Variable Parameter Estimate Standard Error t-value M l 1980.IV Intercept 0.17062 0.16659 1.02 - R M l(-l) 0.92301 0.02362 39.08 2000.IV Real GDP 0.05828 0.02167 2.69 Inv(BondRate) 0.10325 0.02238 4.61 1992.1 Intercept -2.32939 2.10248 -1.11 - R M l(-l) 0.93073 0.02551 36.49 2000.IV Real GDP 0.21403 0.14709 1.46 Inv(BondRate) 0.07074 0.02701 2.62 M2 1980.IV Intercept 1.05001 0.28429 3.69 - RM2(-1) 1.06804 0.03015 35.42 2000.IV Real GDP -0.13612 0.04779 -2.85 Inv(BondRate) 0.00705 0.00653 1.08 1992.1 Intercept -1.17412 0.94521 -1.24 - RM2(-1) 1.00222 0.02851 35.16 2000.IV Real GDP 0.07420 0.06569 1.13 Inv(BondRate) 0.00306 0.01326 0.23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 For the full sample period, the long run income elasticities is 0.75 for M l and 3.06 for M2, and the long run interest rate elasticities are -1.3 for M l and -1.01 for M2. For the sub sample period of 1992.1 - 2000.IV, the inverse of bond rate coefficients are not significant and logarithm of real GDP has the wrong signs. Moreover, the lagged dependent coefficients both exceed 1. Therefore we cannot draw consistent conclusion about the long-run relations based on the inverse o f bond rate for both full and sub-sample periods. In short the nonlinear models not only imply implausible income and interest rate elasticities, they also have statistically insignificant parameter estimates for the higher order terms. Moreover, neither do the nonlinear models predict better than ARIMA or log-linear model (see Table 8), therefore, for the aggregate models we shall focus our comparison using ARIMA or model (2.3.1). In choosing between whether to predict aggregate variables using aggregate or disaggregate equations (H aor Hd), Grunfeld and Griliches (1960) suggests using the criterion of Choosing Hd if e'd ed < e'a ea, otherwise choosing H a, where ed and ea are the estimates of the errors in predicting aggregate outcomes under H d and H a, respectively. Table 21 presents the within sample fit comparisons in the first row and the post-sample prediction comparison in the second row. Both Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 criteria unambiguously favor predicting aggregate outcomes by summing the outcomes from the disaggregations6. Table 21 Error Sum of Square (ESS) and Predicted Error Sum of Squares (PES) for Disaggregate and Aggregate Data M l M2 Aggregate Data Disaggregate Data Aggregate Data Disaggregate Data ESS 3.78 xlO 9 1.35xl06 3 .5 9 x l0 4 3 7.45 xlO 4 2 PES 2.51x10'° 5 .7 5 x l0 7 9 .5 5 x l0 4 5 2.04 xlO 4 3 We next perform simulation studies to evaluate the impact of changing interest rates on the aggregate money demand using either the macro or the micro equations. We shock the economy by lowering the interest rate for 50% and observe the evolution of the money demand at the prefecture level and then get aggregate money demand from the prefecture money demand. The prefecture money demand is modeled by the random coefficient model specified in (2.4.1). To use the macro equations to evaluate the interest rate shock impact, we assume that we have available only the 6 This simulation study may also be viewed as numerical generalization of Pesaran’s (1999) results that if the static disaggregate model is correctly specified, the mean-squared error of optimal aggregate forecast is larger than the corresponding mean-squared error of forcasting the aggregate based on the disaggregate model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 aggregate data that have been constructed from the prefecture level and do not have information on prefecture level data. Three aggregate series need to be constructed for this simulation study. The first one is the “true aggregates” before and after the interest rate shock. We simulate the disaggregate data using the stylized facts from the actual Japan’s money demand data, and we shock the disaggregate system by setting period 101 interest rate as half of that in period 100. The second series is the “simulated aggregates”, assuming that there is no information about the disaggregate data generating processes but only have information about the simulated 100 periods of “true aggregates”. These observation points are used to perform OLS estimation and the resulted coefficients are used to evaluate the response o f this aggregate system before and after the interest rate shock. In this way the “simulated aggregates” are constructed. Finally, the “simulated disaggregates” are simulated in the following steps. The first step is to use a random coefficients model to estimate coefficients based on the disaggregate data of the first 100 periods. In the second step we use the estimated coefficients to construct prefecture level money demand after the interest rate shock, and aggregating them to obtain the estimated aggregate response. The error sum of squares for predicting true system’s responses after interest shock is much smaller for the disaggregate system as compared to the aggregate system. For a typical simulation, the error sum of square for M l is 5.75xlO 7 for the disaggregate Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 system, and 2.51 xlO 1 0 for the aggregate system; the error sum o f squares for M2 is 2.04xlO 4 3 for the disaggregate system, and it is 9.55 xlO 45. As the absolute numbers of the error sum of squares are very large, we further look at the responses paths from the disaggregate and aggregate system. Figure 11 contains the changes of both the “true aggregate” and “simulated disaggregates”, with the dashed line shows the “simulated disaggregates” and the solid line shows the “true aggregates”. The top two figures are for M l and the bottom two are for M2, and the left half show the short run responses while the right half show the long run responses. This figure shows that the series o f the “true aggregates” and the simulated disaggregate responses are very similar. Both lines show the increase o f money demand in the short run and then convergence to certain equilibrium level in the long run. Figure 12 plots the responses of both the macro equations and the disaggregate equations together with the true aggregates. Again, the top two figures are for M l and the bottom two are for M2, and the left half show the short run responses while the right h alf s how t he 1 ong r un r esponses. T he t wo v ery c lose 1 ines d enote “ true aggregates (solid line)” and “simulated disaggregates (dashed line)” respectively, and the third line denotes “simulated aggregates”. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 Figure 11 Simulated Responses of the “True Aggregates” and “Simulated Aggregates” from Disaggregate Equations to Interest Rate Shock x 10 x 10 3 2 1 0 100 150 0 50 3 2 1 0 300 100 200 0 x 10 21 0.5 0 50 100 150 x 10 21 15 10 5 0 0 100 200 300 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 Figure 12 “True Aggregates” and The Simulated Responses from the Aggregate and Disaggregate Equations 5 4 3 2 1 0 100 200 0 300 3 2 1 0 0 50 150 100 3 2 1 0 0 100 200 300 0.5 0 50 100 150 These figures show that the macro equation responses are very different from the true ones. Because o f the near unit root phenomenon, even though the short-run interest elasticity is small, it would almost imply the presence o f "liquidity trap" in the long-run. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 Both the prediction comparison and policy evaluation unambiguously p oint out that when micro heterogeneity is present, it is better to generate the aggregate outcomes by summing the disaggregate outcomes from the disaggregate equations than using the aggregate model to generate aggregate outcomes. 2.8 Conclusion In this section we study the issues of whether Japan has a stable money demand function under the low interest rate policy using both aggregate and disaggregate data. There are two notable results. First, the only agreeable information is that the interest rate coefficients although are statistically significant, they are not o f the magnitude to im ply the presence of liquidity trap in the short run. The panel data estimated short-run semi interest rate elasticity based on random coefficient models is -0.0546 for MF1 and -0.0095 for MF2. The long-run interest elasticities are -0.145 for MF1 and -0.0185 for MF2. Here we have mostly reported findings in terms of semi-log form for the interest rate. We have tried alternative functional forms for the interest rate such as logarithmic transformation or the inverse transformation. Both specifications could imply the presence of liquidity trap if the estimated coefficients have the right magnitude. However, the alternative functional forms do not fit the data or yield post-sample prediction as well as the semi-log form reported here. Second, the estimated relationships between money demand and income are diametrically opposite to each other between the aggregate and the disaggregate data analysis. The estimated relationships using aggregate data are sensitive to the time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 period covered. Depending on the sample period used, they are either of wrong signs or statistically insignificant or yield unbelievable magnitudes. For example, the estimated long-run income elasticities is 75.23 for M l and 11.04 for M2 in the 90's (see Table 4). On the other hand, the disaggregate data analysis yields a stable relationship between money demand and income and are consistent with economic theory. The estimated short-run income elasticity for MF1 and MF2 are 0.88 and 0.47, respectively. The long-run income elasticities are 2.56 for MF1 and 1.01 for MF2 based on random coefficients models. These results appear to be consistent with the broad average relationship observed between M2 and income in Japan. The average growth rate for M2 in the 80's is about 9.34%, the inflation rate is 1.98%, with a real M2 growth rate o f 7.36%. The real growth rate o f GDP during this period is 4.13%. Had interest rate stayed constant, the ratio of these two numbers, 1.78, gives a measure of long-run income elasticity of the demand for real M2. In the 90's, the average growth rate o f M2 is 2.69%, the inflation rate is 0.14%. The real growth rate of M2 is about 2.55%. The real GDP growth rate during this period is about 1.38%. The ratio of real growth rate of M2 and real GDP is about 1.85. Taking account of the fact that 1.78 or 1.85 is the average measure assuming interest rate stayed constant, while in fact the five-year bond rate had fallen from 9.332 at 1980.1 to 5.676 at 1989.IV and further to 1.289 at 2000.IV and the estimated impact of interest rate changes on money demand, these figures are indeed very close to the estimated long-run income elasticities based on disaggregate data analysis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 6 However, if there indeed exists a stable real money demand equation, then the elementary argument that “The monetary authorities can issue as much money as they like. Hence, if the price level were truly independent o f money issuance, then the monetary authorities could use the money they create to acquire indefinite quantities of goods and assets. This is manifestly impossible in equilibrium. Therefore, money issuance must ultimately raise the price level, even if nominal interest rate are bounded at zero.”(Bemanke (2001)) presumably should hold. Then why did monetary authorities failed to stimulate aggregate demand and prices in the 90's? If the estimate is of any guide, it is not because o f the ineffectiveness of the low interest rate policy, but perhaps because that money supply did not increase as much as desired by the monetary authority. In the 80's, the average growth rate of M2 is about 9.34%, yet inflation rate (GDP deflator) is only 1.98% (and real GDP growth rate o f 4.13%). In the 90's, the average growth rate of M2 is only 2.69%, with an inflation rate of 0.14% (and real GDP growth rate o f 1.38%). This significant drop in the growth rate o f money supply is mainly due to the reluctance o f commercial banks to make loans to small and medium-sized enterprises because o f the erosion of their capital due to the accumulation of non-performing assets after the bubble burst in 1990. In fact, the growth rate of high-powered money is about 5.67% in the 90's (relative to 8.08% in the 80's). It is the ineffectiveness o f the transmission of the growth o f high-powered money to the growth o f M2 that led to the slowdown of the growth o f money supply. It appears that the challenge to the monetary authority to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 find a way to increase the supply o f money cannot be resolved through monetary means alone and complementary fiscal policies have to be implemented. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 3 Controlling Individual Heterogeneity in Evaluating the Effectiveness of Repeated Job Search Services 3.1 Introduction In Chapter 2 we have shown the importance of controlling individual heterogeneity in dynamic linear models. In this chapter we investigate the role o f individual heterogeneity when the dependent variable is discrete. The aim of this chapter is to provide a measure of the effects o f repeated job search services (JSS) on the employment rate o f female welfare recipients who participated in the WorkFirst program of the state of Washington. Most of the existing program evaluation studies consider the identification and estimation of one-time treatment effect of active labor market policies (see, for example, the survey by Heckman, LaLonde and Smith (1999)). However, welfare recipients often repeatedly take different training services from the same training program. A common practice to evaluate the effect o f a particular program is to lump client's participations in different services together, as if she/he had taken only one treatment. As noted by Heckman, LaLonde and Smith (1999), such practice is subject to aggregation bias Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 since different services may often reflect different economic models and different estimation strategies. The WorkFirst program is the implementation of the Federal Temporary Assistance for Needy Family (TANF) program in the state of Washington. Initiated in August 1997, its main goal is to help financially struggling families to find jobs, keep themselves off unemployment, and get better jobs. Emphasizing that getting a low- paying job now is better than waiting for a high-paying job in the future, the WorkFirst program has a process that emphasizes its main activities, job search services (JSS). Nevertheless, TANF recipients have been returning to welfare in greater numbers, with over 70 percent of the entrants to welfare being former TANF recipients during the sample period (1998.11 - 2000.IV). To efficiently allocate the limited resources, policy makers are particularly interested in finding out whether it is efficient to provide Job Search Services (JSS) to the same clients repeatedly. The population o f interest is female TANF recipients between age 25 - 35 in this chapter. However, our data set is not a balanced panel data as some o f the early studies using panel data (e.g., Bassi (1984), Ashenfelter and Card (1985), Heckman and Hotz (1989)). A significant feature of our data is that clients entered and left the program at different time periods. Table 22 shows the number o f new clients entering the program in each quarter over the sample period o f 1998.11 - 2000.IV. Only about 3.4 percent clients have the complete treatment history from quarter Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 1998.13 to 2000.IV. There is also the issue of right censoring because the data period ends at 2000.IV. If we restrict our attention to the sub-sample of clients that entered and left a training program in the sample time periods, it would greatly reduce available observations. Moreover, misleading inference may occur if the clients are not randomly selected from the whole population. If we start with a later period to keep as many observations as possible, participation histories would be incomplete for those who entered the program later, hence makes the estimation of effects of repeated job search services difficult. Table 22 Frequency Distribution of All Observations Over Time Time Frequency Percent Cumulative Frequency Cumulative Percent 1998.11 1628 3.40 1628 3.40 1998.III 2961 6.18 4589 9.59 1998.IV 3912 8.17 8501 17.76 1999.1 4070 8.50 12571 26.26 1999.11 4538 9.48 17109 35.74 1999.III 5188 10.84 22297 46.57 1999.IV 5045 10.54 27342 57.11 2000.1 5457 11.40 32799 68.51 2000.11 5417 11.31 38216 79.82 2000.Ill 4976 10.39 43192 90.22 2000.IV 4684 9.78 47876 100.00 In this chapter we propose a transition probability model o f being employed or staying on employment as a means to take account of issues arising from s ample attrition, sample refreshment and duration dependence. Being state dependent, a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 transition probability model also allows us to accommodate dynamics in a simple format. We first estimate such a model by assuming that (1) the participation of JSS is not endogenous so that there is no bias stemming from selection on unobservables; and (2) there are no unobserved individual specific effects. To check the validity o f these assumptions, we also suggest three estimators that relax one or both o f the above two assumptions: a nonlinear two-stage least square estimator that deals with the endogeneity of participation decisions, a generalized conditional maximum likelihood estimator that allows for state-dependent fixed effects and slope coefficients, and a conditional nonlinear Two-Stage Least Square estimator that allows for both unobserved individual heterogeneity and endogeneity o f the participation decision. The Hausman (1978) specification tests indicate that the above two assumptions are not contradicted by our sample information, therefore we shall discuss our empirical findings from the model treating participation decisions as exogenous and without the presence o f unobserved individual specific effects. Our findings show that the repeated job search services do have positive and significant effects on the employment rate o f initially unemployed clients, but their impacts are not significant for those who are already employed. Furthermore, the probability of employment is also influenced by the duration in employment or unemployment, family factors, education level, geographic and local labor market conditions as well as other welfare services. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 Section 3.2 introduces the model and section 3.3 presents the estimation results. Diagnostic checking is discussed in section 3.4. In this section, we propose a nonlinear two-stage least square estimator to control for the endogeneity of the JSS participation decision. We also generalize Honore and Kyriazidou (2000)'s conditional maximum likelihood estimator to allow for state-dependent individual specific effects and slope coefficients, and provide a conditional nonlinear two-stage least square estimator that allows for both the unobserved individual heterogeneity and the endogeneity o f participation decision. Hausman test statistics are calculated to check the validity of the corresponding assumptions. Conclusions are in Section 3.5. Detailed descriptions of our data are in Appendices. 3.2 The Model 3.2.1 A Transitional Probability Model for the Outcomes We are interested in evaluating how JSS influences the probability o f employment for WorkFirst clients. In this section we propose a transitional probability framework to take account issues of sample attrition, sample refreshment and duration dependence. Let y it be the independent binary indicator that that takes the value 1 if the ith client is employed and the value 0 if otherwise. It is constructed based on the quarterly earnings data documented by Employ Security Department of the State of Washington. A Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 client is employed if she is reported to have nonzero quarterly earnings and unemployed if otherwise. We assume that where tt and Tt denote the first period and last period that client i is observed and function of previous JSS and strictly exogenous socio-demographic variables, x it. A client is considered to have taken JSS if the records show that she has taken at least one JSS within one quarter. As over 95 percent participants took no more than 3 JSS over the period of 1998.11 - 2000.IV, we will focus on the evaluation o f the effects of at most three JSS treatments. We consider the effects of previous period JSS treatment on current y it since one JSS can last up to 12 continuous weeks. A client taking at most 3 JSS before period t is in one of the four possible potential states: (1) she has not taken any JSS; (2) she has taken one JSS; (3) she has taken two JSS; and (4) she has taken three JSS. Let d™ be mutually exclusive dummies such that 1 , if^ * >0 0, if otherwise, (3.2.1) yf* d enotes t he p otential s tate g iven y l t-\ =s, s = 0,1, w hich i s a ssumed to be a 1, if exactly m JSS has (have) been taken before period t, 0, if otherwise, m = 0,1,2,3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 We assume that JSS participations influence the probability o f employment through its impact on potential outcomes. Depending on the realization of dit , y it can take one of the four possible forms, y s™ *, m= 0,1,2,3, where yftm* denotes the corresponding potential outcome when d™= 1, w=0,l,2,3. At time t, yf* and y-tm* has the following relation, y S = 3 i y f + (l - S I ' p b f + (l - S I f e f + (l - S i ) y f )) (3.2.2) =y f + d i { y f - y f )+ d l { y f - y f )+ d l { y f - y f ) (3.2.3) = y f + d l r ! ? + d Z r f (3.2.4) where ? S ' = y f - y f , r f = y f - y f , and f f - y f - y f m e w m the cumulative effect of m JSS over no treatment, m — 1,2,3, for the zth individual at rth time period, when the no-treatment state is treated as the base state, i.e., dff = 1 and d l = d Z = d > = o .7 There is also a one-to-one relation between the cumulative effect and marginal effect. Let 7 Equation (3.2.2) and (3.2.3) are equivalent since d ™ , m=0,1,2,3, are mutually exclusive. In other words, the interactive terms of d™ are always zero. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Dit = [0,0,0], when d£ = 1, Dit = [1,0,0], when dft = 1, Dit = [1,1,0], when d£ = 1, Dit = [1,1,1], when dft =1. In this notation, equation (3.2.3) can also be written as yi - y f * 4 ( y f - y f )+ 4 ( v f - y f )+ 4 U 3 * - y f ) - y f + D „ r ‘ u , (3.2.5) where D„ = [ 4 , f ]. and y'u = [ y f y f ,y-f ]'. The value of /"m easu res the marginal impact of the mth job search service on the ith individual at time t conditional on her last period employment status 5 . The average treatment effect for the mth job search service is However, yftm* is not observable. We shall therefore define the treatment effect of the mth JSS on client / in terms o f the changes in the probability o f employment Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 conditional on last period's employment status s. For simplicity, we assume that y-tm* can be decomposed as the sum of the effect of observables x i(, g sm(xit), and the effects o f unobservables, u - tm, y f = g ’m ( x » h < , "I = 04,2,3. (3.2.6) Since Pr( v,f ~ 11 vVj-i, ) = Pr(v,f > 0; , xtI), the effect of the mth JSS on the probability o f employment for client i at period t conditional on last period’s employment status s is defined as A f =Pr(y,r* > 0 |y f,M ,^ ,)= Pr()'.? < ''''1 ) ' >01 (3 2 J ) s = 0,1, m = 1,2,3. The average treatment effect (ATE) of the mth JSS conditional on last period’s employment status s is defined as A7re = 4 c )= £[pr(y,f* > 0 1y ^ , x „ ) - P r l y f ^ ’ > 0 1 y „ . K ,x u )J(3.2.8) The treatment o f the treated (TT) of the mth JSS is defined as A ^ = 4 A 2 " K ? = l). (3-2.9) ATE is the mean impact of the mth JSS if clients are randomly assigned to the JSS. It is of interest if one is interested in estimating the impact of the mth JSS on a randomly selected clients. TT is the mean impact of the mth JSS on those clients who actually have taken the mth JSS compared to what they would have been had they Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 not taken any. It is of interest if the same selection rule for treatment applies in the future. Because it is impossible to simultaneously observe y™* and y ^ m~^ , m= 1,2,3, we cannot directly estimate AS ™ TE and A ^ . In essence, to evaluate treatment effect is to deal with a missing data problem. If we approximate the no-treatment outcomes of the treated group by observed outcomes from the control group and calculate the treatment effect by ~ Z y j t —J — I y j t , m = 1,2,3 (3.2.10) N m je<t>m N m - 1 1 where N m and N m_\ are the number of clients who have taken the mth and the (m- l)th treatment, respectively, and O m and are the sets that include the corresponding clients respectively. Equation (3.2.10) converges to E^ t = 11 =s,d" = l j - e(v, = 11 = s,d" = o) = 4 > r( y r ‘ >0|_>V_, = »,<*,” = l)-P r(y?" -1)- > 0 | = s,d" = l) +Pr(y;'” -')- > 0 | = ,,rf» = l)-Pr(y; ^ > 0 | ^ . , = *.</," =o)} — A j y + rst , where e r = ^Prl)-,?” - 1 *' > o I = s , d r = l j - P r ^ " - 0* > 0 1 = M ," = o ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 is the bias for estimating the effect of the mth treatment conditional on employment status s resulted from using control group to approximate the no treatment state o f the treatment group. If Bfm= 0, then (3.2.10) provides a consistent estimate of A fy. When «*" ± d % \ y ut_x, x it (3.2.11) where x it'= [xit' t h e treatment assignment is free from the influence of unobserved factors affecting yftm* conditional on > x it). Condition (3.2.11) is called Conditional Independence Assumption (Cl) by Rosenbaum and Rubin (1983) or Ignorable Treatment Assignment assumption by Heckman and Robb (1985) and Holland (1986). Then > 0 |A ,_, >0|y,v_, =0,**) In other words, under Cl, conditional on x , ATE( x )=TT( x ). For the unconditional bias Bfm to equal to zero we also need ** -L rfa lJV t (3.2.12) When condition (3.2.12) does not hold, we say that the selection bias is due to observables (Hsiao (2003b)), or selection on observables. In other words, for B fn= 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 unconditionally, we need that there is neither selection on observables nor unobservables. Usually, the x it that affects the potential outcome may also affect the participation decision. It is hard to assume that { x it ± d™ | y it_j ). Therefore, we shall analyze the treatment effects by simultaneously controlling the impacts o f x it and JSS. We c * R assume a linear structure for y it = x itfis +Dity*t +u°t . (3.2.13) We shall derive our model specification and inference under the Cl assumption in this section. We then consider methods o f testing Cl assumption and methods of estimating ATE or TT conditioning on x it when Cl is violated in section 3.5. 8 Under this assumption there is no decaying treatment effect. While it is possible to generalize our model specification (3.2.13) or (3.2.14) to take account the decaying effect, it substantially complicates our model specification without much gain in insight because our sample only covers a relatively short period from 1998 to 2000. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 Let Pisk = P r[yit = k \ y it_x = s ,x u l s,k = 0,1, be the transition probability that the rth individual is in state j in period M and k in period t, then Pisi = PrU / = 11 yi,t-1 = x it) (3.2.14) = P r[xitp s +Ditys i t + us it >o) - F ( x itfis +Dityft )= FiU 5 = 0,1, and F denotes a certain cumulative distribution function. In this study, we assume F takes the logit form, F s exp(xjtfis +Du7a) U 1 + exp[xitp s +Ditys it) (3.2.15) s = 0,1, t = tj +1,...,7). Since the lagged values of the initial employment status are not observed, the initial state y it is approximated by an unconditional specification, y * u , = Q(*i)+ s iti. i = 1 ^ (3.2.16) Pn^VrW , = 1 l^iJ - 1 T- where x t = -------------Y lx i t , and Q denotes a monotonic function. The means of the Tt -tf+l explanatory variables are used instead of x t '= (a:^. ',...,x iT. 'j as it usually yields better finite sample results when the variation of x it over time is limited (Hsiao (2003a). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 The likelihood function for individual i is Li =Vr{yiti,...,yiTi) ■= E l P r(yfl|^_i)Pr(yff/) t=tj+i = f t h l l , ''flo(1 ‘w )h ' _1^0l” '^oo(1 "W)f l' ,’ ','_ 1 V o,'" '(l-P o)('"®')- (3-2-17) t= t:+ l Let y t = [yit. ,...,yiT. )• Under the assumption that u)t and uft are independent across individuals, the likelihood function for all N individuals takes the form i- 1 t=tj + 1 Equation (3.2.18) is similar to the likelihood functions of the binary qualitative response models. Because the log likelihood function is the sum o f the log likelihoods o f the job-holder group (yi t_\ =1), o f the job-seeker group = 0) and o f the initial states, (/?' ,y]) and (fl° ,y° ) can be estimated separately using the data points from the job-holder group and those o f the job-seeker group respectively. Let 0s = be a mx 1 vector of unknown parameters, and 0s e Q s. Let wit = (xit', Dit')'. Assume that A l. The parameter space ®s is an open bounded subset of the Euclidean m- space. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 A2. { w } is uniformly bounded in i and t and l i m ^ ^ —X ,Z zH ’ifH V is a n finite nonsingular matrix, where n denotes the total number o f observations over i and t. Furthermore, the empirical distribution o f { wit } converges to a distribution function. Under these assumptions, the MLE of the model (3.2.15) is consistent and asymptotically normally distributed.9 3.2.2 Evaluation of the Conditional and Unconditional Impacts We estimate the likelihood function (3.2.18) to provide information on two questions. First, how do repeated JSS and personal characteristics affect the employment rate of the job seekers and the job holders, respectively? Second, what are the overall impacts of JSS and other characteristics on the employment rate regardless o f an individual's previous period employment status? The conditional impact of the mth JSS for job seekers and job holders can be calculated from 9 The proof follows straightforwardly from that of Amemiya (1985, Chapter 9.2) since under (3.2.15), A1 and A2 imply that the first and the second derivative of exist and are bounded. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 = i I y t-1= s’d ( = i> xt)— fi{ y t = i I y t-i ~s , < ^ t = xt) 5 = 0,1, tn = 1,2,3. (3.2.19) If x t is randomly drawn, E^yt = 1 1 y t_x = )can be approximated by the sample average o f the predicted probabilities o f those who have d™ — d, d - 0,1. When the explanatory variables are continuous, under the assumption that the transition probabilities are logistically distributed conditional on x it, we calculate the impact of one-unit change of Xy on Pis X using d p isi _ P ] t x A x i j P j ) (3 .2 .2 0 ) dXy 1 + exp [ x ij p ff dP The population impact of a one-unit change of x • is J— l — dF(xi). Assuming that dxu Xjj is randomly distributed given an individual is in state s, this impact can be approximated by 1 f)P < 3-2-21> N i dX :: To investigate the unconditional impacts of JSS and other socio-demographic variables regardless o f an individual's employment status, the equilibrium or Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 marginal probability, n it, need to be calculated. Since the unconditional probabilities at period t and period t-1 follow the relation 1 0 1 __ 1 > f l l r o o r 1 T 1 1 z P o o 0 1 «r° (3.2.22) the equilibrium probabilities satisfy it 1 I T » 1 z P 1 ----- © o r r ■ ■ i i o 1 i z P © o 0 1 i o i (3.2.23) Solving (3.2.23) yields the unconditional probability o f being employed as \-F } +Ft° (3.2.24) and the unconditional probability o f being unemployed is 1 - F 1 - F l + F ? ’ (3.2.25) where F® and F- are defined in (3.2.14). In this study n\ and n® are evaluated at x • = — — 1 N tl ' 1 0 We have assumed that PJk conditional on X; is time invariant. 1 1 The transitional probability framework also allows one to trace out an individual's dynamic path of from its initial state or towards the future by recursively substituting (3.2.22). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 Using (3.2.24) and (3.2.25), we can calculate the impact of a one-unit change of Xy on the unconditional probability o f being employed by calculating first-order derivative o f n\ to Xy: dn) d*y (l - P n j + Pm )' \P m P niPm P \j + pm pi\Qpm P oj)- (3.2.26) dft' We can then replace (3.2.19) and (3.2.21) by n\ and — - to evaluate the marginal dxij (or equilibrium) impact of JSS. 3.3 Findings 3.3.1 Impacts on the Job-seeker Group The explanatory variables are grouped into following categories: (i) WorkFirst participations. This category contains information on client's participations in JSS and other WorkFirst activities. In addition to JSS, the WorkFirst program also provide supportive activities, including alternative services (AS) for clients who cannot participate in JSS directly due to problems like drug abuse and family violence, and post-employment services (PS) for those who have got at least part time job. Appendix 1 provides the detailed information about these activities and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 how clients are introduced into each o f these activities.12 ( ii) Employment history. We have included the number of quarters in a given state to control for the impacts of duration dependence, (iii). Welfare history, (v) Family information, (vi) Race and ethnicity, (vii) Language and education, (viii) Geographic information, (ix) Local economy, (x) Time. Appendix 2 provides the definitions o f the key variables used in this study. Table 23 and Table 24 provide descriptive statistics for initially unemployed clients and initially employed clients respectively. The impacts o f JSS as well as these explanatory variables on the probability o f being employed for job seekers are presented at the third and forth columns of Table 25. 1 2 Another category o f supportive activities is education and training (ET). W e have excluded clients w ho have taken ET in our study to avoid confounding effects since there exists on-going debates about whether work-first strategy or education-first is better. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 Table 23 Descriptive Statistics For Initially Unemployed Individuals* Variable Category Variable Name Mean Std Minimum Maximum WorkFirst LJSS1 0.38 0.48 0.00 1.00 Participation LJSS2 0.10 0.30 0.00 1.00 LJSS3 0.05 0.21 0.00 1.00 Ltotal AS 0.26 0.73 0.00 12.00 Ltotal PS 0.19 0.58 0.00 8.00 Employment lunemploycount 0.28 0.50 0.00 3.00 History lemploycount 1.85 1.37 0.00 11.00 Welfare history lafdcnow 23.94 13.97 1.00 55.00 Family num adit 1.13 0.37 0.00 4.00 num chid 2.39 1.26 0.00 12.00 Ageyoungest 5.49 3.62 -1.00 18.00 Married 0.14 0.35 0.00 1.00 Race Whites 0.62 0.48 0.00 1.00 Blacks 0.18 0.38 0.00 1.00 Hispanics 0.14 0.34 0.00 1.00 Language and English 0.95 0.23 0.00 1.00 Education grade 12 0.15 0.36 0.00 1.00 Geographic region 1 0.16 0.37 0.00 1.00 Information region2 0.16 0.37 0.00 1.00 region3 0.08 0.28 0.00 1.00 Local economy Unemployrate 5.74 2.51 2.57 15.87 Time year98 0.16 0.36 0.00 1.00 year99 0.41 0.49 0.00 1.00 quarter 1 0.21 0.41 0.00 1.00 quarter2 0.23 0.42 0.00 1.00 quarter3 0.27 0.44 0.00 1.00 * Number of observations: 18,647 for all variables. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 Table 24 Descriptive Statistics for Initially Employed Individuals * Variable Category Variable Name Mean Std Minimum Maximum WorkFirst LJSS1 0.42 0.49 0.00 1.00 Participation LJSS2 0.12 0.32 0.00 1.00 LJSS3 0.05 0.22 0.00 1.00 Ltotal AS 0.69 1.18 0.00 18.00 Ltotal PS 0.04 0.24 0.00 6.00 Employment lunemploycount 2.21 1.66 0.00 11.00 History lemploycount 0.25 0.49 0.00 4.00 Welfare history lafdcnow 23.21 14.18 1.00 55.00 Family num_adlt 1.21 0.44 0.00 5.00 num chid 2.41 1.32 0.00 12.00 Age_youngest* * 4.81 3.83 -2.00 18.00 Married 0.21 0.40 0.00 1.00 Race Whites 0.70 0.46 0.00 1.00 Blacks 0.12 0.33 0.00 1.00 Hispanics 0.10 0.30 0.00 1.00 Language and English 0.89 0.31 0.00 1.00 Education grade12 0.14 0.34 0.00 1.00 Geographic regionl 0.13 0.33 0.00 1.00 Information region2 0.13 0.33 0.00 1.00 region3 0.07 0.26 0.00 1.00 Local economy Unemployrate 5.51 2.30 2.57 15.87 Time year98 0.19 0.39 0.00 1.00 year99 0.38 0.49 0.00 1.00 quarter 1 0.19 0.39 0.00 1.00 quarter2 0.25 0.43 0.00 1.00 quarter3 0.28 0.45 0.00 1.00 Number of observations is 29,229 for all variables. Age o f the youngest child is calculated based on the child’s age at the WorkFirst’s beginning quarter, 1997.IV. Therefore, if a child is bom after 1997.IV, his/her age is shown as a negative number. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 Table 25 MLE and NL2S Estimations for Initially Unemployed Clients O .y -1 = °) Category Variable MLE Estimation NL2S Estimation Coefficient Standard Error Coefficient Standard Error Intercept -1.288755 0.0857847 -1.288699 0.3209854 WorkFirst Participation LJSS1 LJSS2 LJSS3 LtotalAS LtotalPS 0.3208265 0.0974651 0.127637 -0.063875 0.1732402 0.0336228 0.0460288 0.0623509 0.0136483 0.054055 0.3210588 0.0971165 0.1276976 -0.064106 0.1733488 0.8014488 2.6166248 6.2028023 0.0418666 0.0783051 Employment History Lunemploycount Lemploycount -0.121042 0.1768895 0.0100561 0.0291352 -0.121247 0.1769557 0.0244949 0.076339 Family N um adlt Married -0.184392 -0.135805 0.0407723 0.0450123 -0.184336 -0.135767 0.0531489 0.0479468 Race Whites Blacks Hispanics -0.139019 0.0886408 0.1162661 0.0401963 0.0526743 0.0488666 -0.138914 0.0886086 0.1163025 0.0682148 0.0757805 0.0514206 Education English Grade 12 0.5122075 0.2229153 0.0576963 0.0390711 0.5122995 0.2229834 0.1662142 0.0423111 Time Year98 0.2476005 0.0370489 0.2476764 0.1510935 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 We note from c olumns 3 and 4 o f Table 2 5 that for j ob-seekers: First, job search services have significant impacts on the probability of employment ( at 5 % level). The first JSS has the biggest impact with an estimated coefficient o f 0.32, and the second JSS and the third JSS have less but similar magnitudes of impact (0.10 versus 0.12). Second, the longer an individual stays unemployed, the less likely she will find a job (the estimated coefficient of duration o f unemployment is -0.12). These two results make a strong case for the state to provide job search services to unemployed individuals quickly to get them stay out of unemployment. Third, family factors, race, language, education level and locations play important roles in determining the employment rate o f female welfare recipients. If a client has less adults in the household, is single, is black or Hispanic, she has a higher chance to be employed. If a client is better educated, her chance of being employed is also higher. Finally, variables on geographic information, local unemployment rate and welfare history are excluded from the final specification because they are statistically insignificant for job-seekers' employment status, after we control for employment history and other characteristics. The impacts of one-unit change in explanatory variables to the probability of being employed (job-seeker group) are reported in column 3 of Table 27. This table shows the first JSS increases the probability o f being employed by 8.45 percent on the treated group, the second JSS increases it by a further 2.15 percent, and the third JSS increases it by an additional 0.8 percent respectively. The impacts o f other Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 explanatory variables are also presented in Table 27 but are not described in detail here. Table 26 MLE and NL2S Estimations for Initially Employed Clients (yi,t-1 =1) Category Variable MLE Estimation NL2S Estimation Coefficient Standard Error Coefficient Standard Error Intercept 1.3670399 0.107082 1.3669012 0.3064295 WorkFirst Participation LJSS1 LJSS2 LJSS3 LtotalAS LtotalPS 0.0397998 0.0124084 0.0677785 -0.107349 0.034416 0.0383007 0.0592132 0.080673 0.0197488 0.0271343 0.0407879 0.0126616 0.0679399 -0.107377 0.034242 1.515584 4.0119876 4.9785368 0.0206929 0.0342981 Employment History Lunemploycount Lemploycount -0.058854 0.1060398 0.0316035 0.0126613 -0.058877 0.1062231 0.0406632 0.0169491 Family N um adlt Married -0.107615 0.051051 0.0472479 0.0503581 -0.107726 0.0510632 0.0478171 0.0557685 Race Whites Blacks Hispanics -0.079711 0.0056359 0.0175155 0.0483444 0.0590434 0.0542377 -0.079871 0.0056498 0.0173954 0.057891 0.089708 0.0719317 Education English Grade 12 -0.429121 0.1902404 0.0833468 0.0457895 -0.429282 0.1902577 0.1318459 0.0644864 Time Year98 0.3847283 0.0496395 0.3847448 0.0972272 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 3.3.2 Impacts on the Job-holder Group We present the impacts of all the explanatory variables on the probability of staying employed at columns 3 and 4 of Table 26. It shows that quite a few variables that are significant for the job seekers turn out to be insignificant for the job-holders. All the impacts of the job search services are insignificant. The estimated coefficients for the first, the second and the third JSS are 0.04, -0.01 and 0.06, respectively, and the corresponding standard errors are 0.04, 0.06 and 0.08, respectively. Total number of previous participations in post employment services (PS) also seems to have no significant impact on employment rate. Neither does marital status nor the race dummy for black or ethnicity dummy for Hispanic appear to be significant. In addition, unemployment duration is a negative but insignificant factor in determining the employment rate o f the job-holders. On the other hand, the longer an individual is employed, the higher the probability that she will stay employed next period. This seems to suggest that once a client is employed, what matters is not unemployment history, but her employment history. Column 4 of Table 27 reports the impacts o f one-unit change in explanatory variables to the probability of staying employed (job-holder group). As the estimated coefficients for all o f the JS S are insignificant, it appears that j ob search services have negligible impacts on the job-holders. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 Table 27 Mean Group Impacts and Equilibrium Impacts* Variable Category Parameter Group Impacts Equilibrium Impacts Job-Seeker Group Job-holder Group WorkFirst LJSS1 0.0845 0 0.037 Participation LJSS2 0.0215 0 0.003 LJSS3 0.008 0 0.007 Ltotal AS -0.012 -0.0175 -0.027 Ltotal PS 0.026 0 0.024 Employment lunemploycount -0.020 0 -0.0019 History Lemploycount 0.029 0.019 0.044 Family num adit -0.028 -0.0267 -0.051 Married -0.069 0 -0.021 Race Whites -0.0289 -0.0306 -0.039 Blacks 0.0465 0 -0.025 Hispanics 0.0329 0 0.016 Language and English 0.1055 -0.0815 0.005 Education grade 12 0.0484 0.0292 0.054 Local economy Unemployrate 0 -0.003 -0.003 Time year98 0.0831 0.0616 0.0899 Insignificant variables are considered as having coefficient zero in calculating the unconditional equilibrium effects, and its long run impacts on job-seekers (job-holders) are considered as zero. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 3.3.3 The Equilibrium Impacts Not only the job search services’ impacts on job-seekers and job-holders are o f our concern, the evaluation of the repeated JSS on the employment rate o f low income individuals regardless of her initial employment status is also o f interest. Column 5 of Table 27 presents the equilibrium impacts of the repeated JSS participations as well as other characteristics. Column 5 shows that in the long run, the first job search service increases the probability of being employed by 3.7 percent, the second job search increases it by a further 0.3 percent, and the third job search service increases it by an additional 0.7 percent. Therefore, overall, job search services have had positive effects for all kinds of clients regardless o f their employment histories. Overall, columns 3, 4 and 5 o f Table 27 show that the longer one stays unemployed, the less chance one has for employment. On the other hand, the longer o ne s tays employed, the higher the chance that she stays employed in the future. This information may shed light on the debate between the education-first or the employment-first strategy. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 3.4 Diagnostic Checking The inference reported on section 3.3 requires the validity of the conditional independence assumption. It is also assumed that there are no unobserved individual specific effects that can affect the potential outcome y s it . If the above two assumptions do not hold, then our maximum likelihood estimates are biased. In this section we propose methods to relax the above assumptions and then check whether these assumptions are valid. We start with the possibility that there are unobserved characteristics that can affect the selection decision, that is, we allow for the possibility that selection is due to unobservables. Then w e generalize Honore and Kyriazidou’s (2000) conditional maximum likelihood estimator to allow for the presence o f unobserved individual specific effects in the main equation assuming no selection on unobservables. Finally, a conditional nonlinear two-stage least square estimator is proposed to allow for both selection due to unobservables and unobserved individual heterogeneity. Hausman (1978) statistics are constructed to test for the validities o f the Cl assumption and the nonexistence of unobserved individual specific effects. Our sequential (or conditional) procedures allow the use o f more sample observations conditional on other assumptions being valid or relaxing the need for good instruments because consistent estimation methods that simultaneously relax both assumptions impose severe restrictions on the data that can lead to significant Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 loss of sample information. Moreover, the conditional testing procedures are more powerful to detect the alternative if the conditional event is true. Pedagogically, it is also much simpler to show the validity of proposed procedures before presenting a simultaneous test of Cl and the presence o f unobserved individual specific effects. 3.4.1 Controlling for the Selection on Unobservables In this subsection w e consider the situation that the Cl assumption does not hold. Instead of specifying a complete model of the interactions between potential or actual outcomes with sequential participation decisions that may depend on interim outcomes in a dynamic optimization framework, as long as the employment status is given by (3.2.13), we can use a limited information framework to take account of the issue that after controlling for the observed characteristics the treatment decision can still be correlated with the unobserved personal characteristics in the potential or actual s tate e quation. W e s eparate y it i nto t wo g roups d epending o n t he v alue of .Uv-i equals 0 or 1. Let y s it be those y it where y i t_x =s, 5=0,1. Then y u = F u + tlu (3.4.1) We note that with probability F# , yft = 1 and with probability (1- F°t ), yft = 0, it follows that )= Fts t (l - Fft )+ (l - F£ =0. Therefore, if we can find instruments zlt such that EzitT J it = 0 and Ez-Uwit has full column rank, where Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 w'it = {xit ', D it'), t hen w e c an a pply A memiya's (1974) nonlinear two-stage least squares estimator to obtain a consistent estimator of 0 s in a limited information framework. Z'={z1' Z i = \ ^ it.,...,ZiTt)- The nonlinear two-stage-least-square (NL2SLS) estimator of 0s is defined as the solution that minimizes Ss = ( Y - F s)'Z(Z'Zy1Z ( Y - F s). (3.4.2) We assume that A3. If 0 s * 0*s , P r [P r( / = 11 w,0s) * P r ( / = 11 tv,0*s)]>O. A4. lim — y t Zit Zit ' exists and is nonsingular. n 1 dFs A5. lim—XrZ/Z/f converges in probability uniformly in 0 * . n d0s' 1 dFs A6. phrn-'E t'ZtZ it— - L « is full rank. n Q0S' 0 1 d2F s A7. lim—'Zi'ZtZit------------- converges in probability to a finite matrix uniformly n 80/ 8 0 s ' in 0s e Q s. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 Following the proofs of Amemiya (1974), we can show that Theorem 1 Under A1 - A7, the value of 0 s that minimizes (3.4.2) is consistent with asymptotic covariance matrix Asy C ov0s NL2SLS = [ g 'Z{Z'Z)~1Z 'g Y x \g 'Z ( ? Z)~x Z 'V Z (? Z)~l Z 'g \ x \?Z (Z 'Z )~ 1Z 'g [ X , QFS where G is the stacked matrix o f , and V is a diagonal matrix with diagonal Q0S' elements equal to F(s t (l - Fft ). We use socio-demographic variables that are excluded from x it as instruments. Among them are regional dummies and number of children in the household. Using the MLE estimates o f 0 s as initial values, we apply quadratic hill climbing procedure (Quandt (1983)) to the method of score to iterate until convergence. The convergence criteria is 0.001. Column 5 and column 6 o f Table 25 present the NL2SLS estimates and the corresponding standard errors for initially unemployed clients, and columns 5 and 6 of Table 26 presents the NL2SLS estimates and the corresponding standard error for those who already have jobs when they took JSS. These two Tables show that the MLE estimates and the NL2SLS estimates are quite similar. Because the NL2SLS Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 estimates are consistent regardless of the validity of the Cl assumption while the MLE estimates is only consistent when the Cl assumption holds, and also because the MLE estimates are efficient under the Cl assumption, a Hausman test statistic can be constructed to test the validity of the Cl assumption. The calculated Hausman statistic is 2.66 for those who are initially unemployed and 0.54 for those who already have jobs. They are not significant at 5% significance level for a Chi square distribution with 16 degrees o f freedom. Therefore, the Cl assumption is not contradicted by the information of the data in the present study. 3.4.2 Controlling for Individual specific Effects As the WorkFirst dataset is a panel dataset, it is possible to control for unobserved individual characteristics using conditional maximum likelihood type estimator. In this subsection we propose to generalize Honore and Kyriazidou's (2000) conditional maximum likelihood estimator to allow for state-dependent individual specific effects as well as slope coefficients. Suppose that the error term can be decomposed into two parts, u s u = a f + 4 (3.4.3) where a? denotes the unobserved individual specific effects given y t M =s, and eft is the error term that has zero mean and variance cr^ . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 Given y it_x =s, t = t( + 1,..., 7}, yf* can be written as yit = a \ + x itfil + Dity x + 4 . when 5 = 1 (3.4.4) yf* = a? + x uP° + Dity Q + 4 , when 5 = 0 (3.4.5) Equations (3.4.4) and (3.4.5) can be combined into one equation: y it = a ? (l + S iy i4_x)+ x it {p° + byi> t_x)+ Dit (y 0 + gyi> t_x)+ s it, (3.4.6) where s it = s\ty it_x + 4 (l- y ^ - 1), 4 = 4 ( 1 + ^)> = /? ° + 6 ,and y l = y ° + g . 3.4.2.1 The Conditional Maximum Likelihood Estimator If a) and a? are treated as randomly distributed, it is fairly straightforward to write out the likelihood functions provided their conditional distributions given x ( - can be specified. However, the consistency of the estimated parameters depends on whether the conditional distributions of or- and a f are correctly specified. Moreover, even if the distribution assumptions o f a \ and orf are correctly specified, the estimation can be quite involved due to multiple integrations of a] and a? over the sample period. On the other hand, if a] and a f are treated as fixed, there is no need to specify their distributions conditional on x { a priori. Therefore, we focus on fixed effect models. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 We extend Honore and Kyriazidou's (2000) conditional maximum likelihood method to allow for state-dependent individual specific effects as well as slope coefficients. Conditional on x it, w e assume that s\t and sjj follow a standard type I extreme value distribution. Rewrite equation (3.4.6) as y*t = a ?(l + S ty it_i)+ wit (oQ + c y ^ )+ s it , where wit'= (x it',D it'), 0o'= (p ° ',y ° j, and c'= (b',g'). Then Pr = l\w it,a ? ,S i,y it.,...,yi> t_1) exp[orf (l + Siyi>t_ i) + wit {O q + cyit_i ) j 1 + exp[or? (l + Sty iJt_x)+ wit (dQ + cyijt_x)] (3.4.7) The initial observed employment status is approximated with all the observed covariates and the unobserved state-dependent individual specific effects: yiti = a ? ,<?,-)+ sit., i = 1 where wt '= [wit. w iT. 'j. The probability o f employment in the initial employed status is a function o f wt , a f , and St : PiH = P r ( y ^ - = 11 a i » s i )• (3.4.8) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 Equation (3.4.7) allows both the individual specific effects and the slope coefficients to be state dependent. When St = 0 for all i, and c = (y,0...,0), equation (3.4.7) becomes J 1 , o * ) e x p ^ f + ^ A + ^ v - i ] = 1 1 w i t ’a i >s i . y u t >•••>y tjt -1 j = -— o ---------------------- : ’ ( 3 -4 -9 ) 1 + exp[a, + wu0q + /y/jM ] which is of the same form as equation (6 ) in Honore and Kyriazidou (2000). Therefore, the model considered in Honore and Kyriazidou (2000) can be treated as a special case for this model. For e ase o f exposition, we first consider the case that 7} - t t = 3 and consider two events: A = {y u t > y i(ti+ 1 ) = °>T 4+ 2) = U yi(ti+ 3)}, B = {yiH ^ 4 + i ) = 1 ,^ 4 + 2) = o , ^ 4 +3) } . To simplify notations, we denote = T™, * 4 +m) = x im, m = 0,1,2,3 in the remaining section. The probability o f event A conditional on wit, a ® , and St is Pr(a | wu, a ? ,St)= p i0y‘° ( 1 - p i0 f *-yio x 1 + exp {a} (1 + Sty iQ ) + wu (0Q + cyiQ )} exp {orf + wi2e Q} exp {yBa ^ (1 + < ? ,■ ) + y i3 wi3 (< 9 0 + c)} 1 + exp {orf + wf 2 0 o } 1 + exp {a? (1 + 81) + wi3 (0O + c)} Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 and the probability o f event B conditional on wit, a f , and St is Pr(*, > < 1 + exp {at (1 + Sty i0) + wu (0O + cyi0)} i x exp(y i3yvi30o + y i3a f ) 1 + exp { o r,0 (1 + St) + wi2 (0O + c)} 1 + exp(cz; ° + rvi30o) Therefore, even under the Honore and Kyriazidou (2000) assumption that wi2 = wi3, V r (A \A u B ,w it,a®, St) still depends on individual specific effects a® and 8 t since V x{A \A vjB ,w it,a f ,8 1 )= 1 + exp j(w(7 - wi2 )0O + (wu y iQ - wi3y l3 )c + a? St (yi0 - y i3 )} ( ) Nevertheless, a f and St will disappear if y i0 = y i3 is further imposed. In this case, Pr(^ I BjWftjOCj ,S ^ = — — y, r - \~ r • (3.4.11) 1 + e x p |K 7 - wi2)0O + [wu y i0 - wi3y i3)c\ Similarly, the conditional probability Pr(f? \A y jB , wit ,a f , St ) does not depend on cci and Si when wi2 = wi3 and y i0 = y i3 hold. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 We have used x it to denote all the relevant continuous variables and Dit to be the only remaining binary variables o f interest in wit , 13 We first select observations that satisfy D i2 = Di3, then follow Honore and Kyriazidou (2000) by using a kernel function with weights that depend inversely to x i2 - x i3 to replace the indicator function l( w i2 = w i3). We propose to estimate 0O and c and by maximizing the objective function N Y M y a + y a =i)-i(yfo-J'(3 = «)• 1(0 ,2 - A s = o )-k i=1 r \ X j2 -X i3 V J In e x p ( [ w / i + ( w i l y iQ - W j 3 y g ) e } m 1 + exp {(wu - wi2 )0Q + (w u y i0 - wi3y B )c } (3.4.12) with respect to 0 and c over the parameter space, where K f \ x i2 ~ x i3 V a n j denotes a kernel density function that gives more weight to those observations whose x i2 is closer to x i3, and a n is a bandwidth that shrinks toward 0 as n increases. Compared with the objective function in Honore and Kyriazidou (2000) (their equation (6 )), equation (3.4.12) contains two additional indicator functions: one for 13 Since time invariant binary variables (like sex, race) will disappear after first differencing, and since the first-differences of time dummies do not have support at zero, the job service indicators appear to be the only relevant binary variables in the present study. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 the observed employment status, l(y i0 - y i3 = 0 ), and the other for the requirements on the treatment variables, \{Di2 - D i3 = 0). Equation (3.4.10) shows that the individual specific term disappears if (i) 8t = 0 for all i and/or (ii) y i0 - y i3 = 0. When (i) holds, individual specific effect is not only time independent but also state independent as in Honore and Kyriazidou (2000). When (i) fails, (ii) needs to be valid to allow for state dependent individual specific effects as well as slope coefficients. These conditions may be satisfied when the data set is large. 3.4.2.2 Asymptotic Properties We establish the consistency of our estimator in the following theorem. Theorem 2 Let qt = [wu - wi2 wu y i0 - wi3y i3\ (p = {00’,c ’\ h i ( ? ) = ifoi + y n = 0 • ifoo - T /3 = 0) • l(A-2 - As = 0) xln ^exp(gj(pyi X ^ 1 + exp (q w ) (3.4.13) Let the following assumptions hold: C l. i= l,-,N } is a random sample o f N observations from a distribution that satisfies (3.4.7). C2. The true value o f the parameters o f interest, < p §, is in the parameter space T which is a compact subset of the Euclidean K-space ( R K ), where K=k+q+l. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 C3. (i) The random vector x i2 - x i3 conditional on Dl2 - Di2 l = 0 is absolutely continuously distributed with density function /(•). /(•) is bounded from above, strictly positive and has support in the neighborhood of zero, (ii) P r(l)/2 = D t3)> 0. C4. e |h » <7 - wi2 ||2 | yl| and E|H»( 7 y i0 - wi3y i3 ||2 | A^ are bounded on their supports, where A = [(jc;2 = x i3),Di2 = Di3] for assumptions (C4) - (C6 ). C5. The function E(h(<p)\ A) is continuous in a neighborhood of zero for all (p e ¥ . C6 . The functions E((h> /7 - wi2 ){w u - wi2)\ A) and E((” u y i0 -W u y n )'(>W ;o -W i2 yi3) \ A ) have ful1 column rank in the neighborhood o f zero. C7. K : R — » • R is a function o f bounded variation that satisfies:(i) supveij | AT(v) |< oo, (ii) J| AT(v) (r/v < oo, and (iii) = 1 . C8 . crn is a sequence o f positive numbers that satisfies: a n — » 0 as n -» oo. Let (p be the solution to the problem MoxI l K tpe'V ;'=1 N ( x i2 - x i3' \ G n J hi{<p), (3.4.14) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 Assumptions (C2) to (C5) are the regularity conditions required for the objective function to converge to a nonstochastic limit, which is uniquely maximized at (p0 by a law o f large numbers. Assumption (C6 ) is required for the identification of 0O and c . Assumptions (C7) and (C8) are standard for kernel density estimation. The above assumptions are quite similar to those imposed in Honore and Kyriazidou (2 0 0 0 ), except that we separate the continuous explanatory variables and discrete variables i n o ur m oment c onditions and i n k emels. S imilar t o t hat o f H onore and Kyriazidou (2000), the convergence rate is much slower than root-«. It is at rate ( V/2 (n (T n ) • The assumptions for asymptotic normality are similar to those imposed in Honore and Kyriazidou (2000) except that in addition to conditioning on x i2 = x i3, we also need to condition on Di2 = D i3. The proof o f consistency and asymptotic normality follows straightforwardly to those in Honore and Kyriazidou (2000). When Ti - t i > 3 , we consider the events A = = 0 , y ^ +1),...yiV _ ,,y i7 = , — { T i c ’ T - • •»T / ,j--l >y i s ~ l > ^ i ( s + l ) >—y i j t - \ > yit = ^ > 3 ;»(/l-+ l)> 3 ;/,(7J—/,■)}> for any 0 < s < t < Tt - . When n > i s+ 1 = wi t+1 are satisfied, we have Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 P r (A \A u B ,fv it,a j> ,S i )= 1 + exp {wis - wit )0O + ( h ^ ^ , - wijt+1y ijt+ x )c + a? S t [yis_x - y lJ+x)] 1 ______________________ l ( / - s > l ) + e x p[(M ';,s+i>’i,s+i ~ w < t y i , t - \ ) c + a i 5 i G 'i^+i " y u t - 1)] When y i> s -\ = y if+ 1 and y is+x = y i> t_x hold, the individual specific effects are cancelled out so that we get Pr(a | A u B, wit, a f , S( )=---------- 17--------------r 7^ -------------------------------- r-i 1 + exp[(wfs - wit )0O + \p b y its_x - wU t+ 1y u+x jcj ^ 1________________ \(t - s > l) + exp[(wi,s+1y i> s + l - w > uy ijt_x )c] ’ where l(f - s > l) is an indicator function that takes value 1 if t>s. The advantage of the fixed effect approach is that we don't have to specify the distributions o f the unobserved individual specific effects, nor do we need to know their relations with the explanatory variables (Hsiao(2003b)). The disadvantage o f this method is that the key assumption that x is+1 '~xit+1' has support around 0 excludes time dummies. Moreover, as the individual specific effects cannot be estimated, we cannot calculate the conditional or unconditional impacts of JSS and other explanatory variables as we did in section 3.3. The consistency conditions also further limit the number of available observations used in estimation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 3.4.2.3 Control or Not to Control Individual Specific Effects Whether individual specific effects need to be controlled is critical for the adequacy of the model presented in section 3.4.2. The conditional MLE remains consistent when individual specific effects are not present. Significant information loss, however, incurs as the conditional MLE greatly restricts data points: only about 10 percent o f the observations used in MLE are qualified for the conditional MLE. Furthermore, as few clients have taken the third JSS, the conditional MLE fails to converge when the third JSS dummy is included. Table 28 presents the estimated coefficients for the first two JSS from MLE (model 1) and conditional MLE (model 2). The estimated coefficients and standard errors are in columns 3 and 4 for model 1, and in columns 5 and 6 for model 2. For initially unemployed clients, the estimated coefficients o f the first JSS are both significant at 5% level; further, they are very close (0.32 versus 0.346). The second JSS has significant impact in model 1 but not so in model 2 , probably due to the reduction of available observations that satisfy the consistency conditions o f the conditional MLE. For those who are already employed, the estimated impacts o f the first two JSS are insignificant both in model 1 and in model 2. The Hausman statistic for misspecification is merely 0.18 for the job-seeker group, which is not significant at a chi-square distribution with two degrees o f freedom (5.99 at 5% level). These results appear to suggest that there is no evidence of the presence o f significant unobserved individual heterogeneity conditioning on observed clients characteristics. They Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 appear to further confirm that using model 1 is adequate to evaluate the effectiveness of repeated job search services. Table 28 With or Without Individual Heterogeneity* Without Individual Heterogeneity With Individual Heterogeneity Probability of being Parameter Chi-Square Parameter Standard Error employed LJSS1_0 0.3228 91.40 0.35 0.176 L JSS20 0.1053 5.19 0.20 0.24 Probability LJSS1 1 0.044 1.128 of staying employed LJSS21 -0.016 0.066 0.14 0.25 LJSS1 1- L JSS10 LJSS2 1- L JSS20 -0.09 0.29 L JSS10 and L JSS20 are the impacts o f the first and the second JSS on the probability of being employed respectively. L JS S ll and LJSS21 are the impacts o f the first and the second JSS on the probability of staying employed respectively. 3.4.3 Controlling for both Selection Bias and Individual Specific Effects We have presented the nonlinear two-stage least square estimator that allows for selection due to unobservables but no unobserved individual heterogeneity, and the conditional maximum likelihood estimator that controls for unobserved individual specific effects but no selection due to unobservables. If both selection due to unobservables and unobserved individual heterogeneity a re present, the above two estimators remain biased. In this subsection we propose a conditional nonlinear two- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill stage least square estimator that is constructed to allow for unobserved individual heterogeneity as well as selection on unobservables: U inS = I " K ,L ,(y, - F,)z, 'f c * K ^ z , f z , (y, - F, ) where Lt = 1 (ya + y i2 = l) • l(yi0 - y i 3 > = 0) • 1 (Di2 - D i3 = 0), K t = K r ^ x i 2 ~ x i3 \ J Ft = _ exP(^f^) as jn equation (3.4.13), z t '= and z t is vector of l + e x p (^ j instruments for individual i. Under A1 - A7 and C l - C8 , the conditional nonlinear two-stage least squares estimator is consistent. Just like the conditional maximum likelihood estimator, the conditional nonlinear two-stage least square estimator greatly reduces the number of available observation points. Moreover, it needs good instruments. Hence, it is unlikely to yield accurate estimates. The MLE estimates are efficient but not consistent when both the no selection on unobservables assumption and the no inidividual heterogeneity assumption are violated. Again we use Hausman test to check whether the null hypothesis of no selection on unobservables assumption and no inidividual heterogeneity holds. The resulted Hausman test statistics is merely of magnitude o f 0.1, hence does not reject Cl and no individual effects assumptions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 3.5 Conclusion In this study we have evaluated the effectiveness of repeated job search services and individual characteristics on the employment rate of the prime-age female TANF recipients in Washington State. We have suggested a transition probability framework to deal with the complicated issues of sample attrition, sample refreshment and duration dependence. We estimated conditional and unconditional impacts o f the sequential job search services for job seekers and those who are already employed. We also proposed a nonlinear two-stage least square method to allow for selection due to unobservables, a generlized conditional maximum likelihood estimator to allow for state-dependent fixed effects and slope coefficients, and conditional nonlinear Two-Stage Least Square estimator to allow for both unobserved individual heterogeneity and selection due to unobservables. The specification tests indicated that the Cl assumption and the no individual specific effects assumption conditional on included socio-demographic variables were not contradicted by the information of the data. Our findings show that Job Search Services do have positive and significant effects on the employment rate of those who are initially unemployed, w ith the first JSS increases the probability of being employed by 8.45 percent, the second JSS increases it by a further 2.15 percent, and the third JSS increases it by an additional 0.8 percent for the treated group. But the impacts of JSS are insignificant for those who are already employed. Combining these findings with the finding that for each Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 113 additional quarter that an individual stays unemployed, her chance of being employed is decreased by 0.19 percent, it makes a strong case for the state to introduce job search services quickly to those who are unemployed so as to mitigate the self-enhancing effect of unemployment. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 Concluding Remarks 114 In this dissertation we use the actual and simulated data to show that ignoring heterogeneity among micro units can yield grossly misleading inference and predictions. The availability o f panel data provides the possibility to check on the validity of “representative agent” assumption so prevalent in economic analysis, both theoretically and empirically. In the case of linear models, we have derived conditions for the existence of stable aggregate relations with the presence of micro heterogeneity in chapter 2. However, these conditions are fairly restrictive and it is rare that they will be satisfied in empirical analysis. Fortunately, econometric techniques are fairly well developed to analyze linear panel data models (Hsiao (2003b)). In chapter 2 we use a Bayesian Hierarchical method to estimate a dynamic random coefficient model for the Japanese money demand function. We show that there was a stable money demand function and there was no evidence of liquidity trap. The panel data estimated short- run semi interest rate elasticity is -0.0546 for MF1 and -0.0095 for MF2. The long- run interest elasticities are -0.145 for MF1 and -0.0185 for MF2. The estimated short-run income elasticity for MF1 and MF2 are 0.88 and 0.47, respectively. The long-run income elasticities are 2.56 for MF1 and 1.01 for MF2 based on random coefficients models. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 In the case o f nonlinear models, the issue of controlling heterogeneity among micro units become much more complicated. There was no general solution. The problem has to be dealt with on a case-by-case base. In chapter 3 we propose a transitional probability model to analyze the effectiveness o f repeated job search services using unbalanced panel data. We propose a conditional MLE to control the impact of unobserved heterogeneity and a conditional NL2SLS estimator to simultaneously control the impact of simultaneity and unobserved heterogeneity. However, they impose severe restrictions on the data. With a total observation o f 47,876 in the Washington State WorkFirst program, only about 10% observations would satisfy the requirement. Moreover, to control the simultaneity it also requires the existence o f good instruments, which are difficult to find. On the other hand, if conditional on the individual socio-demographic variables, there is no evidence of unobserved heterogeneity, all sample observations can be used to estimate the model. The Hausman specification tests indicate that the no unobserved heterogeneity assumption and the no simultaneity assumption are not violated and we can make inferences based on the transition probability model. Our findings show that Job Search Services do have positive and significant impacts on the employment rate of those who are initially unemployed, but the impacts of JSS are insignificant for those who already have jobs. We also find evidence of self-enhancing effect of unemployment, which m akes i t a s trong c ase f or t he s tate t o i ntroduce j ob s earch services quickly to those who are unemployed. The two empirical studies in this dissertation show that it is important to test for the conditional homogeneity Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 assumption in empirical analysis to make the best use o f available sample information. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 117 References [1] Akaike, H. (1973), “Information Theory and an Extension of the Maximum Likelihood Principle,” Proceedings o f Second International Symposium on Information Theory, B. M. Petrov and F. Csaki (eds.), Akademia Kiado, Budapest, 267-281. [2] Amemiya, T. (1974), “The Nonlinear Two-Stage Least-Squares Estimator,” Journal of Econometrics, Vol. 2,105 -110. [3] Amemiya, T. (1985), Advanced Econometrics, Harvard University Press, Cambridge, Massachusetts. [4] Anderson, T. W. (2001), “A Study of Asymptotic Distributions of Canonical Correlations and V ectors i n H igher-Order C ointegrated M odels,” Technical Report No. 2001 - 3. [5] Ashenfelter, O.C. (1978), “Estimating the Effect o f Training Programs on Earnings,” The Review o f Economics and Statistics, Vol. 60(1), 648 - 660. [6 ] Ashenfelter, O.C. and C. Card (1985), “Using the Longitudinal Structure on Earnings to Estimate the Effect of Training Programs,” The Review o f Economics and Statistics, Vol. 67(4), 648 - 60. [7] Bassi, L. J. (1984), “Estimating The Effect of Training Programs with Non- Random Selection,” The Review o f Economic and Statistics, Vol. 66(1): 36 - 43. [8 ] Bemanke, B. S. (2001), “Japanese Monetary Policy: A Case of Self Induced Paralysis?” Japan's Financial Crisis and its Parallels to U.S. Experience, R. Mikitaui and P. A. Posen (eds.), Institute for International Economics, Bank of Japan. [9] Bohn, H. (1995), “The Substitutability o f Budget Deficits in a Stochastic Economy,” Journal o f Money, Credit, and Banking, Vol. 27, 257 - 271. [10] Box, G. E. P. and G. M. Jenkins (1970), Time Series Analysis, Forecasting and Control, San Francisco, Holden Day. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 [11] Doi, T. (2000), “ Wagakuni n i o kera Kokurai n o Ji zokukanousei t o Zaiser Unei,” Keizai Bunseki Seisaku Kenkyu no Shiten Series, Vol. 16, 9 -15. [12] Friedman, B. M. and A. J. Schawarz (1991), “Alternative Approaches to Analyzing Economic Data”, American Economic Review, Vol. 81, 39 - 49. [13] Fujiki, H., C. Hsiao, Y. Shen (2001), “Is There a Stable Money Demand Function Under the Low Interest Rate Policy? — A Panel Data Analysis,” Monetary and Economic Studies, Vol. 19. [14] Fujiki, H., K. Okina, et al. (2001), “Monetary Policy Under Zero Interest Rate: Viewpoints of Central Bank Economists,” Monetary and Economic Studies, Vol. 19 (s-1), 89-130. [15] Fomi, M. and M. Lippi (1997), Aggregation and the Micro Foundations o f Dynamic Macroeconomics, Oxford, Oxford University Press. [16] Fomi, M. and M. Lippi (1999), “Aggregation of Linear Dynamic Microeconomic Models,” Journal o f Mathematical Economics, 31(1), 131- 158. [17] Goldfeld, S. M. and D. E. Sichel (1990), “The Demand For Money,” Handbook o f Monetary E conomics, B . M. Friedman and F. H. Hahn (eds), Amsterdam, North Holland, 300 - 356. [18] Granger, C. W. J. (1980), “Long Memory Relations And the Aggregation of Dynamic Models,” Journal o f Econometrics, Vol. 14, 227 - 238. [19] Grunfeld, Y. and Z. Griliches (1960), “Is Aggregation Necessarily Bad?” Review o f Economic and Statistics, Vol. 42,1 -13. [20] Hausman, J. (1978), “Specification Tests in Econometrics,” Econometrica, Vol. 46(6), 1251-1271. [21] Heckman, J.J., and V. J. Hotz (1989). “Choosing Among Alternative Nonexperimental Methods for Estimating the Impact of Social Programs: The Case o f Manpower Training,” Journal o f American Statistical Association, Vol. 84(408), 862 - 874. [22] Heckman, J.J., R. J. LaLonde, and J.A. Smith (1999), “The Economics and Econometrics of Active Labor Market Programs,” Handbook o f Labor Economics, O. Ashenfelter and D. Card (eds.), Vol. IV, 1865-2073. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 [23] Heckman, J., and Robb R. (1985). “Alternative methods for Evaluating the Impact of Intervention,” Longitudinal Analysis o f Labor Market Data, Heckman J. and Singer, B. (eds.). [24] Heckman J. and Robb R. (1986), “Alternative Methods for Solving the Problem of Selection Bias in Evaluating the Impact of Treatments on Outcomes,” Drawing inferences from self-selected samples, Wainer H. (eds.). [25] Holland, P. (1986), “Statistics and causal inference,” Journal o f the American Statistical Association, Vol. 81, 945-960. [26] Honore, B. E., and E. Kyriazidou, (2000), “Panel Data Discrete Choice Models with Lagged Dependent Variables,” Econometrica, Vol. 68(4), 839 - 874. [27] Hsiao, C. (1997a), “Cointegration and Dynamic Simultaneous Equations Models,” Econometrica, Vol.65, 647 - 670. [28] Hsiao, C. (1997b), “Statistical Properties of the Two Stage Least Squares Estimator Under Cointegration.” Review o f Economic Studies, Vol.64, 385 - 298. [29] Hsiao, C. (2000), “Panel Data Methods,” A companion to Theoretical Econometrics, Badi Baltagi (eds.), Blackwell Publishers, 349 - 365. [30] Hsiao, C. (2001a), “Identification and Dichotomization o f Long- and Short- Run Relations o f Cointegrated Vector Autoregressive Models,” Econometric Theory, Vol. 17, 889-912. [31] Hsiao, C. (2001b), “Economic Panel Data Methodology,” International Encyclopedia o f the Social and Behavioral Sciences, N. J. Snelser and P. B. Bates (eds.), Elsevier. [32] Hsiao, C. (2003a), Analysis o f Panel Data, Cambridge University Press. [33] Hsiao, C. (2003b), “Program Evaluation,” memio. [34] Hsiao, C. and A. K. Tahmiscioglu (1997). “A Panel Analysis o f Liquidity Constraints and Firm Investment,” Journal o f American Statistical Association, Vol. 92, 455 - 465. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 [35] Hsiao, C., M. H. Pesaran, and A.K. Tahmiscioglu (1999), “Bayes Estimation o f short-run coefficients in dynamic panel data models,” Analysis o f Panels and Limited Dependent Variable Models, K. L. C. Hsiao, L. F. Lee and M. H. Pesaran (eds.), Cambridge, Cambridge University Press, 268-296. [36] Hsiao, C. and S . Y . W ang (2003), “ Estimation o f Nonstationary S tructural VAR”, mimeo. [37] Johansen, S. (1988) “Statistical Analysis of Cointegration Vectors,” Journal o f Economic Dynamics and Control, Vol.12, 231-254. Reprinted in Long-run Economic Relationships, Readings in Cointegration, R.F. Engle and C.W.J. Granger (eds.), Oxford University Press (1991). [38] Johansen, S. (1991), “Estimation and Hypothesis Testing o f Cointegration Vectors in Gaussian Vector Autoregressive Models”, Econometrica, Vol. 59(6), 1551 - 1580. [39] Klein, L. (1953), A Textbook o f Econometrics, Prentice Hall. [40] Laidler, D. E. W. (1969), The Demand fo r Money: Theories and Evidence, Scranton, International Textbook Company. [41] Lewbel, A. (1992), “Aggregation with Log-Linear Models,” Review o f Economic Studies, Vol. 59, 635 - 642. [42] Lewbel, A. (1994), “Aggregation and Simple Dynamics,” The American Economic Review, Vol. 84(4), 905 - 918. [43] Miyao, R. (1996), “Does a Cointegrating M2 Demand Relation Really Exist in Japan?" Journal o f the Japanese and International Economies, Vol. 10 (2), 169-180. [44] Nakashima, K. and M. Saito (2000), “Strong Money Demand and Nominal Rigidity: Evidence from the Japanese Money Market Under the Low Interest Rate Policy,” mimeo. [45] Nerlove, M. (1958), The Dynamics o f Supply: Estimation o f Farmers' Response to Price, Baltimore, the Johns Hopkins Press. [46] Pesaran, M. H. and R. Smith (1995), “Estimating Long Run Relationships for Dynamic Heterogeneous Panels, ” Journal o f Econometrics Vol.6 8 , 79-113. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 121 [47] Pesaran, M. H., R. Smith, and K. S. hn, (1996), “Dynamic Linear Models for Heterogeneous Panels,” The Econometrics o f Panel Data, L. Matyas and P. Sevestre (eds.), Kluwer Academic Publishers. [48] Quandt, R. E., (1983) “Computational Problems and Methods,” Handbook o f Econometrics, Vol. I, Z. Griliches and M.D. Intriligator (eds.), North-Holland Publishing Company. [49] Poirier, D. J. and Melino, A., 1978, “A Note on the Interpretation of Regression Coefficients within a Class of Truncated Distributions,” Econometrica, Vol. 46 (5), 1207-09. [50] Rosenbaum, P.R. and D. B. Rubin (1983), “The central role of propensity score in observational studies for causal effects,” Biometrika, Vol. 70,41 -55 [51] Schawarz, G. (1978), “Estimation of the Dimension of a Model,” Annals o f Statistics, Vol.6 , 461-464. [52] Stoker, T. M. (1993), “Empirical Approaches to the Problem o f Aggregation Over Individuals,” Journal o f Economic Literature, Vol. XXXI, 1827 - 1874. [53] Theil, H. (1954), Linear Aggregation o f Economic Relations, Amsterdam, North Holland. [54] Trivedi, P. K. (1985), “Distributed Lags, Aggregation and Compounding: Some Econometric Implications,” Review o f Economic Studies, Vol. 52, 19 - 35. [55] Zaffaroni, P. (2001), “Contemporaneous Aggregation o f Linear Dynamic Models in Large Economies,” mimeo. [56] Zellner, A. (1969) “On the Aggregation Problem: A New Approach to an Old Problem,” Economic Models, Estimation, and Risk Programming: Essays in Honor o f Gerhard Tintner, K. Fox et al. (eds.), Berlin, Springer Verlag, 365- 374. [57] Zellner, A. (1996), “Models, Prior Information and Bayesian Analysis,” Journal o f Econometrics, Vol.75, 51 - 6 8 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 122 Appendices Appendix 1 Descriptions o f the WorkFirst Activities This appendix provides some detailed information about the WorkFirst program. It describes client eligibility, categories of assistance, and procedures of WorkFirst activities assignment. 1 Eligibility. The WorkFirst program serves three groups: (i). Parents and children aged sixteen or older who receive cash assistance under TANF, general assistance for pregnant women (GA-S) or state family assistance (SFA) programs, (ii). Parents who no longer receive cash assistance and need some continuing support to remain self- sufficient; and (iii) Low income parents who support their family without applying for or relying on cash assistance. (Source: WAC 388 - 310 - 0100) 2 Categories o f Services. Three categories o f assistances are provided to eligible clients, a Agency provided services. The main activities are: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 123 i. Alternative Services: Services provided to clients who are unable to work or look for a job because of problems with substance abuse, domestic violence, temporary disabilities, or dependent care. ii. Job Search Services: Eligible participants are referred to Employment Security Department for Job Search, where they must actively seek employment by making a minimum number o f contacts as specified by the local WorkFirst office. Services may include a job search workshop, where clients are taught job search skills in a classroom setting. Job Search clients also have access to a resource room with personal computers and printers; hands-on help with job applications, letters, and resumes etc. Any individual who receives federal TANF benefits, and is capable of participating in the WorkFirst program, is required to do so. Since eligibility of JSS is m ainly d etermined b y i ncome r ather than b y employment status, both initially employed and initially unemployed clients can participate in JSS. Participants are required to work, look for work or prepare for work in the WorkFirst program, therefore, JSS is one of WorkFirst's main activities. Exceptions from the work requirement may be made for parents with chronic and severe disabilities, older adults caring for related children, people caring for disabled family members, parents of children younger than 1 2 months, and teen parents enrolled in a high school degree or equivalency program. Pregnant women and parents o f infants may choose to work and/or receive services such as parenting, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 124 nutrition and work preparation classes (source: Washington WorkFirst homepage (http://www.workfirst.wa.gov/about/faq.htm )). As one JSS may last up to 12 weeks, we consider clients took one JSS if she took one or more o f the above forms of service within one quarter. iii. Post-employment services: These services are provided to those who have already been employed. Through mentors, job-specific education, career planning and other services, these services intend to help clients stay employed and find higher-paying jobs. iv. Education and training services are particularly reserved to clients who work 20 hours or more per week, or those who are still unable to work after a period of time. In the present study, we focus on clients who took job search services and/or other services, but excluding those who took education and training services. v. Other activities. Clients unable to find unsubsidized employment may be directed to unpaid work experience, on-the-job training, subsidized employment, job-specific vocational education, job skills training, or other services designed to improve employability. There activities will be provided to clients only after they have first participated in the Job Search services. b Support services. These services help a client become employable, find a job, and stay employed. Support services include childcare, transportation, wage Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 subsidy, purchase of clothing and other personal goods or services, education and other services, c Financial assistance. Eligible clients will receive welfare benefits paid in cash, food stamps, and medical assistance. Once employed, the cash assistance will reduce 50 cents per dollar earned. In this study we focus on evaluating the impacts of Job search services in the category o f agency provided services. 3 Procedures of activities assignment Each client will start her participations in the WorkFirst program at the Department of Social and Health Services (DSHS). DSHS determines her eligibility mainly based on her income level. If she is eligible, DSHS will give her orientation to the program and assign her to a case manager. The case manager will work together with the client over the process. First, the case manager works on evaluating her employability, mainly to identify potential barriers to employment, like domestic violence, mental illness, and substance abuse. The employability evaluation allows the case manager to determine if the client needs additional services or activities to be successful at job search. An Individual Responsibility Plan (IRP) will then be developed for the client based on her employability evaluation, which states her responsibilities and what support the state will provide to help her succeed. Whether a client should participate in JSS, ET, AS or PS and which agency should they go for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 126 help will be specified in the IRP. For example, if she is assigned to JSS activities, she will be referred to the Employment Security Department (ESD) of the Washington State. With the help of ESD, she is supposed to take the assigned activities to work, look for work or prepare for work. In her next appointment with the case manager, they will again work together to evaluate her employability and revise her IPR. If she still fails to find a job, she may be referred to take JSS again. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 127 Appendix 2 Variable Definitions Variable Category Variable Name Definitions WorkFirst Participation LJSS1 LJSS2 LJSS3 LtotalA S Ltotal_PS Indicator for whether the first Job Search Services (JSS) had been taken before period t. Indicator for whether the second JSS had been taken before period t. Indicator for whether the third JSS had been taken before period t. Total number of Alternative Services (AS) before period t. Total number of Post employment Services (PS) before period t. Employment History lunemploycoun t lemploycount Total unemployed quarters before period t. Total employed quarters before period t. Welfare history lafdcnow Total quarters in AFDC and/or TANF before period t. (AFDC is the predecessor of TANF). Earnings history Preeam Earnings at the quarter that client entered the WorkFirst. Family numadlt numchld Age_youngest Married Number of Adults in the Assistance Unit. Number of Children in the Assistance Unit. Age o f the youngest child in the Assistance Unit. Calculated based on the first quarter that WorkFirst began, 1997.IV. Marital status. 1 indicates married. Race and ethinicity Whites Blacks Hispanics 1 indicates client is white. 1 indicates client is black. 1 indicates client is Hispanics. Language and Education English grade 12 Language indicator. 1 indicates client can speak English. Education indicator. 1 indicates client’s highest grader higher than 12. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 128 Appendix 2 Variable Definitions (continued) Variable Category Variable Name Definition Geographic Information region 1 region2 region3 Location indicator. 1 indicates client is from Region 1. Location indicator. 1 indicates client is from Region 2. Location indicator. 1 indicates client is from Region 3. Local economy Unemployrate The unemployment rate of the county that client is in. Time year98 year99 quarterl quart er2 quarter3 Year indicator. 1 indicates the record is in year 1998. Year indicator. 1 indicates the record is in year 1999. Quarter indicator. 1 indicates the record is in quarter 1. Quarter indicator. 1 indicates the record is in quarter 2. Quarter indicator. 1 indicates the record is in quarter 3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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Shen, Yan
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Individual heterogeneity and program evaluation
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