Three essays on real estate risk and return

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THREE ESSAYS
ON
REAL ESTATE RISK AND RETURN

by

Peng Fei

_____________________________________________________________________

A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(PLANNING)

August 2009

Copyright 2009 Peng Fei

ii
Acknowledgments
The way through the PhD has been a rollercoaster of excitements and disappointments.
Now that I am close to the final station, I wish to thank the people who took me through
these years, pushing me uphill when the way was steep and enjoying the downhills with
me.
First and foremost, I am grateful to my dissertation committee members Dr.
Yongheng Deng, Dr. Cheng Hsiao, and Dr. Christian L. Redfearn for their support and
advice. Having them as supervisors has been extremely challenging and insightful. I
thank my advisor Dr. Yongheng Deng in particular for buying me the ticket for this ride,
and providing me enormous guidance, encouragement and support during my years at
USC. He has been very supportive not only on my studies and dissertation, but also on
my academic and career development. I am indebted to Dr. Hsiao for his excellent
teaching in econometrics and financial time series. In his class, the second essay in this
dissertation was developed. Dr. Redferan is a great mentor, and a nice friend. He always
provides critical thinking on my work and encouraged me all the way through my Ph.D.
study.
Special thanks are due to Dr. Xudong An, who treats me like a brother and gives
me lots of emotional and academic support. I’d also like to thank my qualifying
committee members Dr. Peter Gordon, Dr. Delores Conway and Dr. Selale Tuzel for
helping me develop my dissertation research questions. And I would like to pay my

iii
heartiest thanks to my colleagues and friends for their great help and invaluable
discussions throughout my graduate study. A far from complete list includes Letian Ding,
Xiaohui Gong, Mingye Zhang, Della Zheng, and Huanghai Li.
I am also grateful to the USC LUSK Center for Real Estate, the Real Estate
Research Institute and the Pension Real Estate Association for financial supports to my
PhD dissertation.
Last, but most deserved, I am deeply indebted to Kai Sun. This thesis would not
have been possible without her invaluable love, encouragement, and support. I thank my
parents for teaching me the value of commitment and overcoming every distance with
their constant and affectionate support. It is mainly to them I owe my best achievements
as a researcher, and above all as a person.

iv
Table of Contents

Acknowledgments......................................................................................................ii

List of Tables .............................................................................................................v

List of Figures..........................................................................................................vii

Abstract...................................................................................................................viii

Chapter 1: Idiosyncratic Risk and the Cross-section of Expected REITs Returns....1
Introduction........................................................................................................1
Related Research................................................................................................8
The Data...........................................................................................................13
Measuring Idiosyncratic Volatility in REITs...................................................17
The PCA Approach..................................................................................19
The Portfolio Approach ...........................................................................25
Links between Idiosyncratic Risk in REITs and Economic Variables............30
Time-Series Property of Idiosyncratic Risk in REITs .............................30
Economic Determinants of Idiosyncratic Risks.......................................38
Does Idiosyncratic Risk Really Matter? ..........................................................45
Robustness Checks...........................................................................................49
Conclusions......................................................................................................60

Chapter 2: Correlation and Volatility Dynamics in REIT Returns..........................64
Introduction......................................................................................................60
Literature Review.............................................................................................68
Methods............................................................................................................71
Data and Empirical Results..............................................................................75
The Data...................................................................................................75
Empirical Results.....................................................................................80
Conclusion .......................................................................................................75

Chapter 3: Persistence of U.S. Housing Returns: A Markov Chain Analysis .........95
Introduction......................................................................................................95
Related Research............................................................................................101
Methodology..................................................................................................103
Data and Empirical Results............................................................................101
The Data.................................................................................................101
The Results.............................................................................................112
Conclusion .....................................................................................................118

v
References..............................................................................................................119

vi

List of Tables

Table 1: Descriptive Statistics of Variables in the Sample......................................16

Table 2: Descriptive Statistics of Idiosyncratic Volatility Measures.......................33

Table 3: Do Monthly Idiosyncratic Volatilities Follow a Random .........................35
Walk Process? (Unit Root Test and Stationary Test for Idiosyncratic Volatility)

Table 4: Economic Determinants of Idiosyncratic Volatility in REITs ..................37

Table 5: Economic Determinants of Idiosyncratic Volatility in REITs ..................39

Table 6: Does Idiosyncratic Risk Really Matter?....................................................43

Table 7: Does Idiosyncratic Risk Really Matter?....................................................44
(Controlling for Lagged Portfolio Return)

Table 8: Does Idiosyncratic Risk Really Matter?....................................................49
(Controlling for Business Cycle Variables)

Table 9: Does Idiosyncratic Risk Really Matter?....................................................50
(Controlling for Business Cycle Variables and Lagged Portfolio Return)

Table 10: Summary Statistics for Portfolios Formed on the Conditional................53
Idiosyncratic Volatility

Table 11: Descriptive Statistics of the Respective Series 1987-2008......................73

Table 12: DCC GARCH models..............................................................................76

Table 13: Economic Determinants of Dynamic Correlations..................................81

Table 14: Relationship between Equity REIT Returns and.....................................82
Conditional Correlations (Monthly Data: Jan, 1987- May, 2008)

Table 15: Relationship between S&P 500 Returns and Conditional Correlations ..84

Table 16: Relationship between Mortgage REIT Returns.......................................85
and Conditional Correlations

vii
Table 17: Relationship between Hybrid REIT Returns and ....................................86
Conditional Correlations

Table 18: Transition Counts Matrix and Transition Probabilities Matrix ..............99

Table 19: S&P/Case-Shiller Home Price Indexes: MSA Coverage .....................104

viii
List of Figures

Figure 1: Time-varying 3 Largest Eigenvalues of the PCA Analysis .....................23

Figure 2: The Components of the Eigenvectors Corresponding Three ...................22
Largest Eigenvalues

Figure 3: Idiosyncratic Risk: Portfolio, PCA, and Cross-Section Measures ...........31

Figure 4: Ratio of Idiosyncratic Risk to Total Risk: Portfolio and..........................30
PCA Measures

Figure 5: Decomposition of Total Risk: PCA Approach.........................................34

Figure 6: Decomposition of Total Risk: Portfolio Approach ..................................34

Figure 7: Idiosyncratic Risk: Portfolio, PCA, and Cross-Section Measures ...........59

Figure 8: Conditional Volatility of REITs, Direct Real Estate and S&P 500..........82

Figure 9: Dynamic Correlation between the REITs and Direct Real Estate............83

Figure 10: Dynamic Correlation between the REITs and S&P 500 ........................84

Figure 11: U.S Nominal and Real Housing Price Index (HPI) Changes.................87

Figure 12: Metropolitan Regions in the S&P/Case-Shiller Home Price Indices...110

ix
Abstract

This dissertation consists of three essays addressing different aspects of real estate risk
and return. First two essays concern the securitized real estate, namely the Real Estate
Investment Trusts (REITs). The third one focuses on the private (unsecuritized) real
estate.
Essay 1, “Idiosyncratic Risk and the Cross-section of Expected REITs Returns”
investigates the inter-temporal relationship between idiosyncratic volatility and the cross-
section of expected REITs returns by introducing new measures of aggregate
idiosyncratic risk: the PCA approach which is based on random matrix theory and the
procedure of principal component analysis and the portfolio approach which is based on
mean-variance portfolio theory and the concept of gain from portfolio diversification.
The results show that conditional aggregate idiosyncratic volatility in REITs is a
significant factor in explaining the cross-sectional returns of REITs stocks.
Essay 2, “Correlation and Volatility Dynamics in REIT Returns” explicitly
examines correlation dynamics among REIT, direct real estate and stock asset classes
and investigate the presence of asymmetric responses in conditional variances and
correlations by utilizing the multivariate asymmetric dynamic conditional correlation
(AD-DCC) GARCH model. The findings in this essay actually resolve the debate by
academics and industry practitioners on the role of REITs in mixed asset portfolios,
questioning whether REITs actually provide exposure to the private real estate asset class
or simply represent additional exposure to common stocks. More importantly, this essay

x
documents a significant linkage between these correlations and REITs returns while the
patterns are distinguishable for different type of REITs.
Essay 3, “Persistence of U.S. Housing Returns: A Markov Chain Analysis”
analyzes the magnitude and stability of housing return persistence in the U.S. housing
market by adopting the Markov chain approach in both time series and cross-section
manners. The dynamics of housing returns are modeled by Markov chain processes that
estimate transitional probabilities from one state to another via a maximum likelihood
estimation method (MLE), and then the stationary of this transitional probability is tested
using the likelihood ratio test (LRT).

1
Chapter 1:
Idiosyncratic Risk and the Cross-section of Expected REITs Returns

Introduction
The traditional capital asset pricing model (CAPM) of Sharpe (1964), Lintner (1965), and
Black (1972) prescribes that only the non-diversifiable systematic risk should be
incorporated into asset prices and command a risk premium. Idiosyncratic risk, on the
other hand, should not matter because it can be completely excluded through
diversification according to modern portfolio theory. In reality, however, due to different
exogenous reasons, such as transaction costs, incomplete information, and institutional
constraints, and so on, it is infeasible for all investors to hold well diversified portfolios.
1

Moreover, Malkiel and Xu (2006) demonstrate that if one group of investors fails to hold
the market portfolio, the remaining investors will also be unable to hold the market
portfolio. The inability to hold the market portfolio will force investors to care about total
risks to some degree in addition to market risk (Merton, 1987). Therefore, idiosyncratic
volatility should be positively related to the cross-section of expected returns if investors

1
Transaction costs and taxes are the key reasons to restrict the portfolio holdings of investors. Employee
compensation plans or policies and private information are other motive for holding large positions of
individual stocks. Barber and Odean (2000) report the mean household’s portfolio contains only 4.3 stocks
(worth 47, 334 dollars), and the median household invests in 2.61 stocks (worth 16, 210 dollars).
Goetzmann and Kumar (2001) and Polkovnichenko (2001) also present the evidence on the lack of
diversification of the equity portfolios of household investors. Benartzi (2001) and Benartzi and Thaler
(2001) document that individuals hold a disproportionate amount of their pension plans in the stock of the
company they work for. Huberman (2001) surveys evidence that investors are prone to investing in familiar
stocks and ignore portfolio diversification.

2
demand compensation for imperfect diversification (see e.g., Levy, 1978; Merton, 1987;
Malkiel & Xu, 2002). In addition, idiosyncratic risk also has special implications to
derivatives investment because the pricing of derivatives is closely related to the total
volatility of underlying assets, which includes systematic risk as well as individual risk.
Meanwhile, idiosyncratic risk is particularly important to the arbitragers who try to
exploit the mispricing of the assets (e.g., Shleifer & Vishny, 1997; Ali et al., 2003).
However, the empirical results on the existence and direction of a tradeoff
between idiosyncratic risk and the cross-section of expected asset returns are still
inconclusive. Consistent with earlier research such as Lehmann (1990), Lintner (1965),
Tinic and West (1986), and Merton (1987), a number of recent studies (e.g., Malkiel &
Xu, 1997, 2006; Goyal & Santa-Clara, 2003; Fu, 2005; Jiang & Lee, 2004; Spiegel
&Wang, 2006) find a positive relation between idiosyncratic volatility and expected
returns at the firm or portfolio level. Other studies, however, do not support this positive
relation, where the cross-sectional relation has been found insignificant (e.g., Fama &
Macbeth, 1973; Longstaff, 1989; Bali et al., 2008) and sometimes even negative (Ang et
al., 2006; 2008). Hence, the debate on the role of idiosyncratic risk in the asset pricing is
far from over.
The purpose of this essay is to clarify the existence and significance of a relation
between idiosyncratic risk and the cross-section of expected stock returns with a
specification on Real Estate Investment Trusts (REIT) stocks. Whilst the relationship
between REIT returns and idiosyncratic volatility is not anticipated to be significantly

3
different from other common stocks, it is widely accepted that real estate assets and
property-related stocks have more restrictions to diversification due to high transaction
costs, low liquidity, significant asset heterogeneity and inherently localized and
segmented nature of real estate market.
2
Moreover, REITs also have a different
organizational structure
3
and enjoy a unique tax status. Therefore, all these distinct
features of real estate assets and the implied lack of diversification strongly suggest that
idiosyncratic risk may play a significant role in REIT returns. Actually some recent
empirical studies have already noted that the influence of idiosyncratic risk component in
REIT returns is growing over time (Clayton & MacKinnon, 2003; Tan, 2004; Anderson,
2005). Therefore REITs provide a unique and excellent opportunity to study the
relationship between idiosyncratic risk and the cross-section of expected stock returns
theoretically and empirically.
Furthermore, as for the investment strategy of REITs, although the benefits of
corporate focus, versus diversification are well documented in the REIT literature,
4
the
implications on REITs returns and risks are still not fully understood. For instance, Boer

2
The performance of REITs is inherently lined with the underlying illiquid real estate properties that are
prone to booms and busts, where the information about the direct real estate market fundamentals is
compounded in REITs (e.g., Chen et al., 1990; Giliberto, 1990; Gyourko & Keim, 1992). Taking a longer-
term view, Barkham and Geltner (1995) show a strong co-movement between REIT and lagged real estate
returns, the similar results are also found in Geltner and Rodriguez (1998).

3
Gyourko and Keim (1992) report that REITs differ from other common stocks in three aspects: (a) agency
problems are more severe in REITs, because of high dividend payouts, which lead to more intense scrutiny
of REIT performance; (b) There are corporate control differences when compared to other industries; (c)
REITs differ in how fast the information is disseminated and incorporated into their stock prices, which
may be because of the limited holdings of REITs stocks by institutional investors. Capozza and Seguin
(2003) observe that REITs with greater inside holdings tend to invest in assets with lower systematic risks.

4
See e.g., Capozz and Seguin (1999); Tan (2004); Boer et al. (2005), among others.

4
et al. (2005) found that there is a strong positive relation between real estate corporate
focus and firm-specific risk. That is to say, firm-specific risk increases with the degree of
corporate specification. If the idiosyncratic risk is not priced, REIT manager should shift
towards a more diversified real estate investment strategy instead of a more focused one.
A detailed study on the relation between idiosyncratic risk and REITs returns is,
therefore, timely warranted. Finally, given the fact that the performance of REITs is
intimately linked with the underlying illiquid real estate properties, the study on the inter-
temporal relationship between idiosyncratic risk and expected REIT returns will shed
new light on the temporal aspects of risk and return in real estate market. Actually there
is surprisingly little in real estate literature on this topic. Among few empirical papers
that aim to study the interactions between real estate returns and risk, Dolde and
Tirtiroglue (1997) use data sets for towns in Connecticut and near San Francisco from
1971 to 1994 and find evidence of time-varying volatility and positive relations between
conditional variance and returns. Very recently, using a disaggregate housing data, Miller
and Pandher (2008) find that idiosyncratic volatility plays a strong positive role on cross-
sectional housing returns and the relation is robust to the price level and socioeconomic
variation, while Ooi et al. (2007) provide the first analysis investigating the relationship
between conditional idiosyncratic volatility and expected REIT returns. These studies
provide important insights; however, several fundamental questions remain indistinct.
The questions include how to measure idiosyncratic risk correctly; why idiosyncratic risk

5
in REITs seems more volatile in some periods than in others; and whether and how
idiosyncratic volatility affects the cross-section of expected REIT returns.
To answer these questions, the predominant challenge is to measure and estimate
the idiosyncratic risks correctly. The previous studies compute the average idiosyncratic
volatility based on the variance decomposition or using the residuals from the one-factor
capital asset pricing model (CAPM) or three-factor model of Fama and French (1993)
assuming a certain parametric specification for the return generation. However, since
some critical assumptions of CAPM or three-factor model obviously contradict with the
market experience,
5
using the factor or CAPM based models in estimating idiosyncratic
volatility perhaps lead to inaccurate or inconsistent measures of diversifiable risk. To
obtain an accurate and unique measure of idiosyncratic risk, this essay firstly introduces
the model-independent measures of idiosyncratic risk, which do not require estimation of
market betas or correlations: (a) the Principal Component Analysis (PCA) approach
which extracts meaningful information from the correlation matrix of all REITs stocks;
and (b) the Portfolio Approach which is based on the concept of gain from portfolio
diversification.
Another difficulty encountered in empirical tests on asset pricing models are that
whilst the models are framed in expectations (ex-ante), the data are usually ex-post. In
order to explain the expected REIT returns, the theoretically correct variable should be
the expected idiosyncratic volatilities in the same period that the expected returns are

5
See Fama and French (2004) for a good summary about the shortcoming of CAPM and three factor
models.

6
measured. Since idiosyncratic volatilities are time-varying, some prior studies such as
Campbell et al. (2001), Fu (2005), and Ooi et al. (2007) have employed sophisticated
parametric ARCH or stochastic-volatility models to estimate the expected value of
idiosyncratic risk, while others use the lagged idiosyncratic risk as the best estimates of
their expected values. For this reason, this essay pays close attention to the time series
properties of the estimated idiosyncratic volatility in REITs, and finds that idiosyncratic
risk in REITs is volatile over time but actually highly auto-correlated. Dicker-Fuller tests
also show that the identified idiosyncratic volatility does follow a random walk process.
Hereby this essay uses the lagged idiosyncratic volatility as a good estimate of expected
idiosyncratic risk.
The third contribution of this study is to demonstrate that a significant fraction of
time-series variations in aggregate REIT idiosyncratic volatility can be explained by
fluctuations in macroeconomic variables such as the term spread, the credit spread,
inflation as well as the short interest rate. Although there is a long-term ongoing debate
in the finance literature about the pricing of idiosyncratic risk in the cross section of
stocks, to our knowledge, this is the first study that examines the links between
underlying economics conditions and time-varying idiosyncratic risks in REITs.
More importantly, this essay finds that, contrary to CAPM theory, a positive
relationship between idiosyncratic risk and the cross-section of expected REIT returns is
found statistically significant. This relation is still robust with a verity of robustness
checks.

7
This study relates to and builds upon several very important literatures. First, it
adds to the literature regarding theoretical asset pricing models and the role of
idiosyncratic risks in pricing assets returns (see, for example, Malkiel & Xu, 1997, 2006;
Ali et al., 2003; Goyal & Santa-Clara, 2003; Bali & Cakici, 2008; Fu, 2005, among
others) for this essay models idiosyncratic risk in a more appropriate way by introducing
model-independent measures, which is totally different to measuring idiosyncratic risks
directly by utilizing FF 3-factor model in previous literature. Moreover, this essay
provides new empirical evidences with a specification on REITs stocks that is expected
to be more exposed to idiosyncratic risks than other financial assets. This essay also
builds upon the literature dealing with REITs risk characteristics and return predictability
(see, Capozza & Seguin, 2003; Chaudry et al., 2004; Clayton & MacKinnon, 2003;
among others) by investigating idiosyncratic risk dynamics in REITs and their links to
prevailing economic conditions. It also expands the literature on the relations between
REITs market and the underlying direct real estate market (see, for example, Chaudry et
al., 2004; Anderson et al., 2005). Furthermore, this paper has implications for
diversification ability of REIT stocks in portfolio choices (see, e.g., Chun et al., 2004;
Dhar & Goetzmann, 2005) due to the close relation between idiosyncratic volatility and
correlation/variance with other financial assets. Finally, this essay has important
implications for REITs managers seeking to develop an optimal investment strategy
addressing the benefits of corporate focus, versus diversification (see, e.g., Capozza &
Seguin, 1999; Tan, 2004; Boer et al. 2005) because it has been well documented in the

8
finance literature that real estate firm-specific risk (idiosyncratic risk) is positively
related to the degree of corporate focus.
The remainder of this chapter is organized as follows. Section 2, Related
Research, reviews related studies to provide the relevant theoretical and empirical
background for the research design. Section 3, The Data, presents the descriptive analysis
of the data. Section 4, Measuring Idiosyncratic Volatility in REITs, introduces the
alternative measures methodologies of aggregate idiosyncratic risk. Section 5, Links
Between Idiosyncratic Risk in REITs and Economic Variables, presents the empirical
findings of the time series trends of idiosyncratic risks in REITs as well as the estimation
results for the relationship between idiosyncratic risk and economic variables. Section 6,
Does Idiosyncratic Risk Really Matter?, sets up the econometric models and presents the
results for interaction between idiosyncratic risk and the cross-section of expected REIT
returns. Section 7, Robustness Checks, presents the robustness checks. Section 7
concludes the chapter.

Related Research
Starting from the mean variance analysis, the traditional CAPM theory prescribes that
only systematic risk should be priced in equilibrium; any role for idiosyncratic risk is
completely excluded through diversification. In reality, however, CAPM theory has been
questioned by a growing number of both theoretical and empirical evidences for its
inadequacy to capture the complexity of rationality in action; therefore, several economic

9
theories in the literature have taken idiosyncratic risk into account. Levy (1978) derived a
modified CAPM model that relates the returns of stocks to their beta with the market as
well as their beta with respect to a market-wide measure of idiosyncratic risk. Merton
(1987) theorizes that in the presence of market frictions and incomplete information,
stocks with high idiosyncratic volatility have high expected returns because investors
cannot fully diversify away firm-specific risk and demand a premium for holding stocks
with high idiosyncratic risk
6
. Furthermore, Malkiel and Xu (2002) and Jones and Rhodes-
Kropf (2003) demonstrate that if one group of investors fails to hold the market portfolio
for exogenous reasons, the remaining investors will also be unable to hold the market
portfolio. Therefore, traditional CAPM model may not hold and idiosyncratic risk should
be priced to compensate rational investors for not being able to perfectly diversify
idiosyncratic risk. Some theoretical papers focus on the role of idiosyncratic risk in asset
pricing from non-traded assets (two prominent examples of non-traded assets are human
capital and private business) held by investors (see Jagannathan & Wang, 1996; Heaton
& Lucas, 1997, 2000; Storesletten et al., 2001). For example, Storesletten et al. find that
idiosyncratic risk in labor income helps explain equity returns. Some behavioral models,
like Barberis and Huang (2001), offer a different type of asset pricing model based on
prospect theory,
7
where investors are loss averse over the fluctuations of individual

6
In addition to incomplete information, there are a number of other factors that could also attribute to why
investors hold undiversified portfolios. They include market segmentation and institutional restrictions
including limitations on short sales, taxes, transaction cost, liquidity, imperfect divisibility of securities
(Merton, 1987; p. 488).
7
See Kahneman and Tversky (1979).

10
stocks that they own. They also predict that higher idiosyncratic volatility stocks should
earn higher expected returns.
On the empirical front, Lintner (1965) is perhaps the first study that considers the
role of idiosyncratic risk. Douglas (1969) and Linter find the variance of the residuals
from a market model is strongly significant in explaining the cross-section of average
stick returns, but Miller and Scholes (1972) suggest that several sources of bias may exist
in the analysis. A key study supporting the CAPM theory is Fama and Macbeth (1973)
who rejected the role of idiosyncratic risk in explaining the cross-sectional returns of
common stocks. Lehmann (1990), however, reaffirmed the results of Douglas in the
context of a careful econometric methodology. Similarly, Tinic and West (1986), Malkei
and Xu (1997, 2002) also present the evidence of the positive tradeoff between
idiosyncratic risk and the cross-section of expected stock returns. In a different context,
some indirect evidences regarding the role of idiosyncratic risk have also surfaced.
Bessembinder (1992) finds strong evidence that idiosyncratic risk was priced in foreign
currency and agriculture futures markets. Falkenstein (1996) found some evidence that
the equity holdings of mutual fund managers appeared to be related to idiosyncratic risk.
Using Swedish government lottery bonds, Green and Rydqvist (1997) find that these
bonds command a premium for the underlying risk that is idiosyncratic by construction.
Recent papers by Campbell et al. (2001), Goyal and Santa-Clara (2003) and Ang
et al. (2006) have rekindled the debate of whether idiosyncratic risk is priced in the equity
market. Campbell et al. note that idiosyncratic risk has increased in U.S stock market in

11
recent decades, and the similar trends in U.K market is also reported by Angelidis and
Tessaromatis (2004). Goyal and Santa-Clara claimed that the average stock variance,
which is demonstrated to be largely idiosyncratic risk, is significantly and positively
related to subsequent capitalization weighted market returns, while no relation exists
between market volatility and future market returns. However, Wei and Zhang (2005)
showed the positive relation between market return and lagged idiosyncratic volatility is
sample specific. When the sample of Goyal and Santa-Clara was extended by three years
(from 1999 to 2002), the positive relation between return and idiosyncratic risk
disappears. Adopting the sophisticated generalized autoregressive conditional
heteroskedasticity (GARCH) model to estimate the expected conditional idiosyncratic
volatilities, Fu (2005), Brockman and Schutte (2007), Spiegel and Wang (2006) and
Eliling (2006) all find a significantly positive relation between the estimated conditional
idiosyncratic volatilities and expected returns. Chua et al. (2007) model idiosyncratic
volatility as an auto regression AR(2) process and decompose it into an expected and an
unexpected components, they also find a significantly positive relationship between
expected return and expected idiosyncratic risk after controlling for the unexpected
idiosyncratic volatility.
However, a puzzling negative relation found recently by Ang et al. (2006) shows
that the US stocks with past high idiosyncratic volatility have low future risk-adjusted
returns. Furthermore, Ang et al. (2008) also demonstrate this effect is individually
significant in each G7 country and rule out the explanations based on trading frictions,

12
information dissemination, and higher moments. In other words, in contrast to the
existing literature, they document a negative intertemporal relation between the realized
idiosyncratic risk and future stock returns around the world. Huang et al. (2007) point out
that the results of An et al. (2008) are driven by monthly stock return reversals. After
controlling for the difference in the past-month returns, the negative relation between
average returns and the lagged idiosyncratic volatility disappears. Jiang et al. (2006)
argue that high idiosyncratic volatility and low future returns are both related to a lack of
information disclosure among firms with poor earnings prospects. Investors underreact to
earnings information in idiosyncratic volatility. Bali and Cakikci (2008) attribute the
contrasting results in previous studies to the differences of methodologies, particularly
the data frequency used to estimated idiosyncratic volatility, the weighting scheme used
to compute the average portfolio returns, breakpoints utilized to sort stocks into quintile
portfolios, and screening for size, price and liquidity. They conclude that there is no
robust, significant relation between idiosyncratic volatility and expected returns.
In real estate literature, there is surprisingly little on this topic. Ooi et al. (2007)
provide the first and the only analysis to investigate the relationship between conditional
idiosyncratic volatility and expected returns of REIT stocks and a significant positive
relation is found. Other authors, Clayton and Mackinnon (2003), and Anderson et al.
(2005) have observed a dramatic increase in the proportion of total volatility not
explained by the three common factors, namely stock, bond and direct real estate. This
suggests that the influence of idiosyncratic risk component on REIT returns is increasing

13
with time, which is consistent with the previous findings in stock market by Campbell et
al. (2001), Bekaert, et al. (2005). Chaudhry et al. (2004) examine the various
determinants of idiosyncratic risk and find that efficiency, liquidity and earnings
variability are the important determinants of idiosyncratic risk in REIT, whereas size and
capital do not. However, all these studies do not link idiosyncratic risk to cross-sectional
returns in REITs.
The existing evidence highlights the implications of idiosyncratic risk;
nevertheless, it is noteworthy that the estimation of aggregate idiosyncratic risk has so far
been model-dependent because in almost all previous studies the idiosyncratic risk is
computed by employing the residuals from CAPM model or three-factor model of Fama
and French (1993). Since the shortcomings of CAPM and FF models have been
documented in large number of literatures, it is reasonable to doubt obtaining an accurate
measure of idiosyncratic volatility through these models. To the best of knowledge, this
is the first study that examines the relationship between the conditional idiosyncratic risk
and the cross-section of expected REIT returns based on the model-independent measures
of idiosyncratic risks.

The Data
This study examines the publicly traded equity REITs in U.S markets during the period
from Oct 1, 1989 to June 30, 2008. The data of both the daily and monthly returns are
obtained from the Center for Research in Security Price (CRSP). At the end of 2007,

14
there were 118 equity REITs traded on the NYSE, the AMEX, and the NASDAQ with a
total market capitalization $288.7 bn. However, after omitting the equity REITs that have
not traded for more than five years, the number of equity REITs in the sample is not static
over the study period. Nineteen equity REITs are included in 1989, increasing to 109 at
the end of the sample period (as of May 2008). The study also includes the daily and
monthly data of National Association of Real Estate Investment Trusts (NAREIT) equity
REIT index which is widely used in real estate literature as the market capitalization
weighted benchmark for the equity REITs market.
8

To capture the economic determinants of conditional correlations among the
assets, this paper uses those macroeconomic variables widely used in finance literature
which include the term spread (the difference between the yields on 10-year and 1-year
Treasuries, TSPR), the credit spread (the difference between the yields on BAA-and
AAA-rated corporate bond, CSPR), Consumer Price Index (CPI) inflation rate (INF) and
three-month Treasure bill rate (TB3M) (e.g., Campbell & Shiller, 1988; Fama & French,
1989; Torous et al., 2005). The term spread is used to proxy for business cycle effects. As
noted by Fama and French, the term spread is closely related to short-term business cycle
as identified by the NBER. Given the importance of external debt financing to REITs, the
credit spread and the three-month Treasury bill is used as the proxy for availability of
credit with narrow credit spreads corresponding to an abundance of external credit
financing. CPI inflation rate captures the current economic activity. All these data, except

8
The FTSE NAREIT Equity REITs Index measures the performance of all publicly traded equity real
estate investment trusts traded on U.S. exchanges. See www.reit.com for a detailed description of index
construction.

15
the three-month Treasury bill rate, are taken from the FRED database. The three-month
Treasury bill rate is obtained from Ibboston Associates.
The panel A of Table 1 contains corresponding summary statistics of variables.
Both the average returns of equity REITs in my sample and the return of NAREIT equity
REITs index exhibit an average positive monthly return with left skewness and fat-tails.
However, the REITs in the sample has higher means and lower standard deviations than
the NAREIT equity REIT index return which is market capitalization weighted return and
includes all equity REITs publicly traded in the market. Considering my sample only
includes the REITs which have traded more than five years, the AR (1) model coefficient
is 0.316 for my sample portfolio which is much higher than 0.112 for the NAREIT equity
REITs index. This suggests that my portfolio may have some “survivorship bias”
9
which
derives from the tendency of failed REITs gradually being excluded from the market. In
previous literature, this potential problem has been obviously omitted; this study will do
the robustness check lately to address this problem. While the fluctuations in economic
variables are highly persistent in the sample, the correlations between the measures of
REITs returns and economic variables are presented in the Panel B of Table 1. The
unconditional correlation between REIT and the term spread is positive, which means the
REIT return are higher (lower) in periods when term spread is wider (narrower), while a
narrower (wider) credit spread is correlated with higher (lower) equity REITs returns.
This preliminary result has pointed to the importance of external debt financing and the

9
See Elton, et al. (1996) for a good summary about the “survivor bias.”

16
underlying credit channel of monetary policy transmission to the equity REITs
performance. That is to say, the widening of credit spread, consistent with tightening
conditions in credit markets and more expensive external debt financing, results to the
lower the REITs returns. Based on this observation, I conjecture that the idiosyncratic
risk in equity REITs will also be driven by such fluctuations in macroeconomic variables.
The intuition for this argument is based on the fact that the propagation and effects of
economic cycle fluctuations vary across different REITs companies because of
corresponding differences in capital structure, in availability of funds, in firm size and in
corporate underlying real estate assets compositions.

17
Table 1: Descriptive Statistics of Variables in the Sample
Period: Dec 1989 to Jun 2008
EREIT EREITi TSPR CSPR TB3M INF
Panel A: Descriptive Statistics
Mean 0.0176 0.0101 1.28% 0.85% 4.06% 0.74%
Median 0.0209 0.0117 0.97% 0.83% 4.50% 0.72%
Standard Deviation 0.0371 0.0403 1.08% 0.22% 1.72% 0.60%
Minimum -0.1342 -0.1458 -0.41% 0.55% 0.88% -1.18%
Maximum 0.1104 0.1094 3.29% 1.42% 7.90% 2.48%
Skewness -0.6163 -0.3984 0.2892 0.8805 -0.1725 0.2331
Kurtosis 4.3611 3.7933 1.7619 3.0849 2.4695 4.1453
AR(1) 0.316 0.113 0.996 0.998 0.992 0.902
Panel B: Correlation Matrix
EREIT 1 0.9232 0.1345 -0.0251 -0.0202 -0.1792
EREITi 1 0.0778 -0.0092 -0.0893 -0.1761
TSPR 1 0.2986 -0.6872 -0.0420
CSPR 1 -0.3335 0.1108
TB3M 1 0.1450
INF 1
Note: The table reports descriptive statistics of The table reports descriptive statistics of the
equally weighted equity REITs portfolio monthly return in our sample (EREIT), NAREIT
equity index monthly return (EREITi) and economic variables. The economic variables are
defined as follows: TSPR is the difference between the yield on 10-year and 1-year Treasuries,
CSPR is the difference between the yields on BAA- and AAA-rated corporate bonds, INF is
inflation computed as the growth of the CPI index and TB3M is the three-month Treasury bill
rate. All variables are measured on a month basis. Panel A reports the mean, the standard
deviation (denoted by Std), the AR(1) coefficient, the coefficients of skewness (Skew) and
kurtosis (Kurt), the minimum (Min) and maximum (Max) value in the sample. Panel B shows
the correlation matrix. The sample is 223 monthly observations from 1989:12 to 2008:6.

Measuring Idiosyncratic Volatility in REITs
Although the concept of idiosyncratic risk
10
is quite simple and widely accepted and
employed in the finance literature, how to measure idiosyncratic risk correctly is not a
simple thing and much more complex than what previous research assumed it would be.

10
Where there is no scope for confusion, we refer to standard deviation as “volatility” and “risk” in this
paper.

18
The idiosyncratic risk is unobservable, and is generally estimated using a return
generating process based on the variance decomposition (see, e.g Campbell et al., 2001;
Xu & Malkiel 2003) or using the residuals from the one-factor capital asset pricing model
(CAPM) or three-factor mode of Fama and French (1993). Early studies define the total
risk of an individual stock i as follows:
( ) ( )
2
ii
fM σ ζε =+ (1)
where () f M is the portion of total variance explained by the factor model M , and
()
i
ζ ε measures the idiosyncratic risk that is unique to the stock i and irrelevant to the
overall market. However, there are various models
12 3
,, M MM measuring the systematic
risk, which will end up with different measures of idiosyncratic risk for the same
individual stock i . Unless the model M is correct, the calculated idiosyncratic risk ( )
i
ζ ε
can not be accurate and unique. There has been plenty of literature addressing the
problems in CAPM model and three-factor model, so the model-dependent measures of
idiosyncratic risk maybe inconsistent and inaccurate. This study will introduce model-
independent measures of idiosyncratic risk to overcome above problems in previous
studies.
It is worth noting that instead of focusing on measuring idiosyncratic risk of
individual stock, this chapter pays special attention to the average idiosyncratic risk of the
equity REIT market since the key research question is whether idiosyncratic risk is priced
in expected returns in REITs market. Some studies like Bali et al. (2005), Brown and

19
Ferreria (2004), Angelidis and Tessaromatis (2007) have noted that the weighting scheme
used to compute the average aggregate idiosyncratic risk play a critical role in
determining the existence and significance of a relation between idiosyncratic risk and
the cross-section of expected returns. Goyal and Santa-Clara (2003) use the cross-
sectional average stock variance of all the stocks traded in the market as the indirect
measurement for the average idiosyncratic risk. This paper will measure the average
idiosyncratic risk directly by employing both the Principal Component Analysis (PCA)
and the Portfolio approaches different to previous studies.

The PCA Approach
The variance decomposition will also be the starting point for the PCA approach.
Considering a multi-factor model derivation of Equation (1), for each REIT stock i

%
1
m
i
iijj
j
RF β ε
=
= +
∑
(2)
This simple regression model decomposes stock returns into a systematic components
1
m
ij j
j
F β
=
∑
and an (uncorrelated) idiosyncratic component
%
i ε . The terms
j
F , 1,... j m =
represent returns of risk-factors associated with the market under consideration, which
can be thought of as the returns of “benchmark” portfolios representing systematic factors.
The traditional CAPM model uses “market portfolio” as one systematic factor F (e.g.,
the return on a capitalization-weighed index, such as the S&P 500) while Fama and

20
French (1993) suggest three systematic risk factors
j
F . This leads to the interesting
question of how to define the “factors” correctly in practice.
Not using explicit market variables in priori as proxies for systematic factors, the
PCA approach uses the Principal Component Analysis to extract underlying “systematic
risk factors” that affect all stocks in the market directly from data itself. This approach
uses the historical share-price data on a cross-section of N stocks going back, say,
M days for the estimation window and T days for the whole sample period. Denote the
returns of the N REIT stocks at time t by
1
{}
N
ii
R
=
, since different stocks have varying
levels of volatility, it is convenient to work with a normalized return
1
{}
N
ii
Y
=
,

i
ik
ik
i
R R
Y
σ
−
= , 1,..., kM = , 1,..., iN = (3)
where
1
1
M
i
ik
k
R R
M
=
=
∑

and
2
2
1
1
()
1
M
i i
ik
k
R R
M
σ
=
=−
−
∑
.
The NN × empirical correlation matrix C with element is defined by

1
1
1
M
ij ik jk
k
YY
M
ρ
=
=
−
∑

21
which is symmetric and non-negative definite. By solving the eigenvalues and
eigenvectors problem
iii
CX X λ = , 1,..., iN = the correlation matrix C can be diagonalized
into

'
CXDX = (4)
where D is the diagonal matrix with eigenvalues as the diagonal elements in a decreasing
order :

12
{ , ,..., }
N
D λλλ = ,
12
... 0
N
N λλλ ≥≥ ≥ ≥ ≥
and X is composed by the corresponding eigenvectors:
12 { , ,..., } N X xx x =
vvv

From the point of view of investment theories, each eigenvector x
v
can be considered as a
realization of an N -security portfolio
i
P (so-called eigenportfolio) with the weights equal
to the eigenvector components
() m
i
x , 1,..., mN = , 1,..., iN = .
There are some important properties of eigenportfolios. Firstly, for a non-
degenerate correlation matrix C , the eigenvectors are orthogonal to each other, that
means any pair of eigenportfolios
i
P and
j
P are independent which allows one to choose
such a portfolio, whose risk is independent of others. Secondly, the risk of these
egienportfolios can be easily related to the corresponding eigenvalues:

2
() ()
T
ii
ii i
RP P x Cx σ λ = ==
v v
(5)

22
Thus, the eigenvalue size is a risk measure and, as a consequence, the larger the
i
λ , the
larger the risk of the corresponding eigen-portfolio
i
P . In this way, if first m
eigenportfolios are used to proxy the market factors, in equation (2), the residual part
%
ε
will not be correlated with those “market” factors. Then its volatility can be regarded as a
measurement of the idiosyncratic risk. As noted by Meucci (2005), it is easy to prove that
the risk of market portfolio equals the sum of risks of these m eigenalues
11
()
mm
ii
ii
Var P λ
==
=
∑∑
and the aggregate idiosyncratic volatility
%
1
()
N
i
im
Var ε λ
=+
=
∑
.
Furthermore,
%
()
0 E ε = and
%
1
(,)0
m
i
i
Corr P ε
=
=
∑
. Finally and most importantly, applying the
Random Matrix Theory (RMT) to analyze the properties ofC , recent studies have found
that the largest eigenvalue and its corresponding eigenvector of C represent the influence
of the entire market on all stocks.
11
Therefore, the largest eigenvalue
1
λ corresponding
eigenportfolio 1 x
v
is a good proxy for the systematic factor F , which reflects the genuine
underlying “interactions” for all stocks.

11
The logic behind applying Radom Matrix Theory (RMT) is that if the properties of C conform to those of
a random correlation matrix, then it follows that the contents of the empirically measured C are random.
Conversely, deviations of the properties of C from those of a random correlation matrix convey information
about ‘‘genuine’’ correlations. Thus, by comparing the properties of C with those of a random correlation
matrix and separate the content of C into two groups: (a) the part of C that conforms to the properties of
random correlation matrices (‘‘noise’’) and (b) the part of C that deviates (‘‘information’’). Refer to Plerou
et al. (2002), Laloux et al. (2000) for details.

23
Figure 1: Time-varying 3 Largest Eigenvalues of the PCA Analysis
Dec89 Dec90 Dec91 Dec92 Dec93 Dec94 Dec95 Dec96 Dec97 Dec98 Dec99 Dec00 Dec01 Dec02 Dec03 Dec04 Dec05 Dec06 Dec07
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Explained Variance(%)
Year

Rank 1 eigenvalue
Rank 2 eigenvalue
Rank 3 eigenvalue

So this chapter adapts one factor model in equation (2) by taking the eigenvector
corresponding to the largest eigenvalue as the only systematic factor and the residual part
as the source of the idiosyncratic risk. The estimation window for calculating the
covariance matrix is 60 days. It is worth noting that sometimes it is possible that other
eigenvalues also contain the “market information,” under that condition, one-factor
model maybe underestimate the systematic risk and overestimate the idiosyncratic risk.
However, my estimation results suggest that generally the largest eigenvalue and its
corresponding eigenvector have included the market systematic information. Figure 1
exhibits the time-varying values of three largest eigenvalues of the correlation matrix.
One can find that the largest eigenvalue became to dominate the system after 1998 and is
well separated with other eigenvalues. This feature confirms the result of Plerou et al.

24
(2002) that the largest eigenvalue represents the market portfolio. Moreover, Figure 1
shows the components of the three largest eigenvalues’ corresponding eigenvectors at the
end of the sample period (on June 30, 2008), which indicates the weights on REITs in the
sample correspondingly. Since all components in the eigenvector corresponding to the
largest eigenvalue are positive and almost equal cross different REITs, it represents an
influence that is common to all stocks, while most components of the remaining
eigenvectors usually are near zero and only few are positive, which indicates their effects
only concentrate on several individual REITs.

Figure 2: The Components of the Eigenvectors Corresponding Three Largest
Eigenvalues
(Weights of 109 REITs at the end of the sample period)

0 20 40 60 80 100 120
-5
0
5
10
15
x 10
-3
First Eigenvector sorted by coefficient size
0 20 40 60 80 100 120
-0.5
0
0.5
1
1.5
2
Second Eigenvector sorted by coefficient size
0 20 40 60 80 100 120
-1
-0.5
0
0.5
Third Eigenvector sorted by coefficient size

25
The Portfolio Approach
The portfolio approach measuring average idiosyncratic risk in REITs is based on the
definition of idiosyncratic risk itself. Since idiosyncratic risk is firm specific and not
attributable to overall market volatility, which can be eliminated by complete
diversification. This means a fully diversified portfolio will not contain any idiosyncratic
risk, on the other hand, when the correlations among individual REIT stocks equal one,
there is no gain from diversification (nondiversified portfolio). Therefore, the portfolio
approach’s measure of average idiosyncratic volatility is defined as the difference
between the variance of the nondiversified portfolio and the variance of the fully
diversified portfolio.
In this way, based on the mean-variance portfolio theory and the concept of gain
from portfolio diversification introduced by Markowitz (1952, 1959), the portfolio
approach for average idiosyncratic risk is to calculate the difference of variances between
two constructed REITs portfolios. The return of the portfolio of n REIT stocks is defined
as:

,,,
1
n
pt it it
i
R wR
=
=
∑
(6)
The risk of the portfolio measured by the variance is given as:

222
,,, ,,,,,
11
2
nnn
pt it i t it j t ij t i t j t
iiji
www σ σρσσ
==>
=+
∑∑∑
(7)

26
where
2
, it
σ is the variance of excess return on stock i at time t ,
, ij t
ρ is the correlation of
excess returns on stock i and j , and
, it
w ,
, jt
w represent the weights of REIT stock i and
j .
It is well known that for a given weights, the lower the correlations
ij
ρ , the
smaller the portfolio variance, hence the larger the gain from diversification. For the non-
diversified portfolio, the REITs are perfectly correlated ( 1
ij
ρ = ), no diversification gains
are achieved. When 1
ij
ρ = in equation (7), the variance of nondiversified portfolio,
which contains both systematic risk and idiosyncratic risk, is written as:

2
2
,,,
1
n
pt i t it
i
w σσ
=
⎛⎞
=
⎜⎟
⎝⎠
∑
(8)
An equally-weighted portfolio of n REIT stocks can be viewed as a fully
diversified portfolio when n is large which does not contain any idiosyncratic risk. When
,
1
it
w
n
= , the variance of the portfolio return in equation (7) changes to:

22
,, ,,, 22
11
22
,,
11
2
,
12
11 1 1
1cov(,)
(1)/2
11
1
var cov
nnn
pt it ij t i t j t
iiji
nnn
pt i t i j
iiji
pt
nn
R R
nn n nn
average average
iance ariance nn
σσ ρσσ
σσ
σ
==>
==>
=+
⎛⎞
⎛⎞⎛⎞
=+−
⎜⎟ ⎜⎟ ⎜⎟
−
⎝⎠ ⎝⎠
⎝⎠
⎛⎞ ⎛ ⎞
⎛⎞
=+−
⎜⎟ ⎜ ⎟ ⎜⎟
⎝⎠
⎝⎠ ⎝ ⎠
∑∑∑
∑∑∑
(9)

27
when n is going to infinity, the first part in equation (9) will decrease to zero, and the
risk (variance) of portfolio return will be reduced to average of covariance of returns in
the portfolio. Since idiosyncratic is unique to a stock because it is related to the part of a
stock’s return that does not vary with returns on other stocks or market, the
1
n
portfolio
contains no idiosyncratic risk any more, and only the nondiversifiable (systematic) risk
contributes to the total risk of the portfolio. Hence, with a large number of different
stocks, the equally-weighted portfolio can be treated as a well-diversified portfolio, the
risk of which is due solely to the systematic risk of REITs stocks in the portfolio.
The concept of gain from portfolio diversification yields a model-independent
measure of aggregate idiosyncratic risk that does not require the estimation of market
beta or correlations:
2
2
,,, 1/,
1
2
22
,, ,
11 1
2
2
,,
1
()
111 1 1
1cov(,)
(1)/2
1
cov
n
titit nt
i
nn nn
tit it ij
ii iji
n
tit
i
wVarR
RR
nnn nnn
average
ariance n
ε
ε
ε
σσ
σσ σ
σσ
=
== =>
=
⎛⎞
=−
⎜⎟
⎝⎠
⎡ ⎤ ⎛⎞
⎛⎞ ⎛⎞⎛⎞
=− +−
⎢ ⎥ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟
−
⎝⎠ ⎝⎠ ⎝⎠ ⎢ ⎥
⎝⎠ ⎣ ⎦
⎛⎞ ⎛⎞
=−
⎜⎟ ⎜⎟
⎝⎠⎝⎠
∑
∑∑ ∑∑
∑
(10)
where
,
1
1
n
it
i
n
σ
=
∑
is the equally weighted average standard deviation of individual REIT
stocks. The difference between the variance of the non-diversified portfolio,
2
,
1
1
n
it
i
n
σ
=
⎛⎞
⎜⎟
⎝⎠
∑
,

28
and the variance of the completely diversified portfolio,
1/ ,
()
nt
Var R , yields the average
idiosyncratic variance
2
,t ε
σ in equation (10).
It is worth noting that despite the sophisticated theoretical optimization models
developed in the last 50 years and the advances in methods for reducing the variance of
portfolio, there are other three reasons choosing the
1
n
rule to build up the fully
diversified portfolio. First, it is easy to implement because it does not rely either on
estimation of the moments of REIT stock returns or on optimization. Second, such simple
allocation rule are widely used by investors continue to use for diversification.
12
Third,
recently DeMiguel et al. (2008) find that of the 14 models across seven empirical dataset,
none is consistently better than the naive portfolio diversification the
1
n
rule in terms of
Sharpe ratio, certainty-equivalent return, or turnover. In addition, this chapter also uses
the equity REITs market index (NAREIT index) and Minimum-variance portfolio as the
fully diversified portfolio to calculate the aggregate REIT idiosyncratic risk. There are no
significant differences to the following economic analysis.
For some day d in the montht , I firstly estimate the daily variance
2
, id
ε , using m -
days estimation time window (including the current day d ) in the equation (11). Then
following the Campbell et al. (2001), the daily variances within the month t are summed

12
For instance, Benartzi and Thaler (2001) document that investors allocate their wealth across using the
equally weighed portfolio rule. Huberman and Jiang (2006) find that participants tend to invest in only a
small number of funds offered to them, and that they tend to allocate their contributions evenly across the
funds that they use, with this tendency weakening with the number of funds used.

29
up to get the monthly volatility (variance)
2
, it
σ of REITi . So the monthly idiosyncratic
volatility is calculated based on the within-month daily data as:

22
,,
1
d
id i j
jd m
R
m
ε
=−
=
∑
(11)

22
,, ,
1
()
t
D
it it i d
d
Var R σ ε
=
==
∑
(12)
where m is the number of days in the estimation period.
,
()
it
Var R is the monthly
variance of REIT i in the month t ,
, ij
R is the return on REIT i at day j ,
2
, id
ε is the daily
variance of REIT i at day d . Note that this it not, strictly speaking, a variance measure
since I do not demean returns before taking the expectation. However, for short holding
periods, the impact of subtracting the means is minimal. Even for monthly excess return,
the expected square return overstates the variance by less than one percent of its level
(See Goyal & Santa-Clara, 2003)

.
13

Another key point to take not of it that in practice, the estimation window m for
estimating daily variance needs to be chosen carefully. I take 20 m = because Pesaran and
Pesaran (2007) suggest based on lots of empirical evidences that for daily returns a value
of lag value twenty tends to render the devolatized returns, nearly Gaussian, with

13
Based on a typical stock mean return of 1.08 percent and a standard deviation of 16.53, Goyal and Santa-
Clara (2003) show that the squared means term is irrelevant to calculate variances. Using daily data, French
et al. (1987) and Schwert (1989) also find the similar results.

30
approximately unit variances, for all asset classes foreign exchange, equities, bonds or
commodities.
14

Links Between Idiosyncratic Risk in REITs and Economic Variables
In this section, I firstly examine the time-series behaviors of idiosyncratic volatility
measured by model-independent approaches introduced in the section on Measuring
Idiosyncratic Volatility in REITs, and then I investigate the links between underlying
economic conditions and idiosyncratic risk in REITs.

Time-Series Property of Idiosyncratic Risk in REITs
Although many researchers have investigated the time-series behavior of REITs stock
market volatility, there is little empirical research on the time series property of
idiosyncratic volatility in REITs.
Figure 3 shows the historic idiosyncratic volatility pattern of REIT stocks publicly
traded in the US between 1989 and 2008 using three model-independent measures: the
PCA approach, the portfolio approach as well as the average stock variance approach.
The average stock variance approach is fundamentally an indirect measure of
idiosyncratic risk introduced by Goyal and Santa-Clara (2003), which is computed as the
arithmetic average of the monthly variance of each stock’s returns. Goyal and Santa-

14
For detailed empirical evidence in support of this claim see Section 6 of Pesaran and Pesaran (2007).

31
Clara find that over the period 1926-1999, the effect of idiosyncratic risk constitutes
almost 85 percent of the average stock variance measure. Since it is also model-
independent (but in an indirect way to measure the idiosyncratic risk), the results of the
average stock variance approach are also demonstrated in Figure 3 and Table 2 for a
comparison purpose.

Figure 3: Idiosyncratic Risk: Portfolio, PCA, and Cross-Section Measures
Dec89 Dec90 Dec91 Dec92 Dec93 Dec94 Dec95 Dec96 Dec97 Dec98 Dec99 Dec00 Dec01 Dec02 Dec03 Dec04 Dec05 Dec06 Dec07
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Idiosyncratic Risk
Year

IR from Portfolio
IR from PCA
IR from Cross-Section

Whilst idiosyncratic volatility of the average REIT fluctuates over time, some
discernable patterns can be observed clearly. Firstly, different to Campbell et al. (2001)
who find a noticeable increase in firm-level volatility over the period from 1962 to 1997,
my sample period shows that there is a decreasing trend for aggregate idiosyncratic
volatility in REITs but with a significant cyclical pattern. However, the historical pattern

32
estimated in model-independent approaches in this chapter is quite similar to Ooi et al.
(2007) who employ the Fama-French three factor model to measure the REIT
idiosyncratic volatility. Secondly, consistent with Ooi et al. (2007), the idiosyncratic risk
of REIT returns is especially low during the bullish market between 1995 and 1998 and
spikes quickly during the bad market conditions such as September 1998, April 2004 and
Jan 2008. Campbell et al. suggest that this characteristic of idiosyncratic volatility has
important implications for portfolio diversification at different stages of the business
cycle. Since idiosyncratic risk increases fast during the bad market time, diversification
will be more difficult and more individual stocks will need to be held in the market
downturn.
Following the Anderson et al. (2005), furthermore, I employ a variance
decomposition approach to examine the significance of the idiosyncratic component of
REIT return volatility. Specifically, its relative contribution to total REIT return volatility
is inferred by calculating the proportion of the variance of REIT returns due to the
idiosyncratic component, as follows:
22
,,
/
tpt ε
σ σ . The results over the sample period are
reported in Figure 4. Essentially the ratio of REIT idiosyncratic volatility to the total
volatility appears to decrease from above 0.9 in 1989 to around 0.5 in 2008, however, it is
obvious that the idiosyncratic volatility still accounts for most of the total volatility
exhibited by REIT stocks over the study period (average ratio is 81%). This finding is

33
consistent with Goyal and Stata-Clara (2003) and Ooi et al. (2007)
15
who also observe
that the total volatility of stock returns is largely idiosyncratic volatility. In addition,
Figure 5 and Figure 6 present the decomposition of total volatility into systematic risk
and idiosyncratic risk estimated by the PCA approach and the Portfolio approach,
respectively.

Figure 4: Ratio of Idiosyncratic Risk to Total Risk: Portfolio and PCA Measures
Dec89 Dec90 Dec91 Dec92 Dec93 Dec94 Dec95 Dec96 Dec97 Dec98 Dec99 Dec00 Dec01 Dec02 Dec03 Dec04 Dec05 Dec06 Dec07
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Idiosyncratic Variance Ratio
Year

Ratio from Portfolio
Ratio from PCA

15
Goyal and Santa-Clara (2003) find that over the period 1926-1999, idiosyncratic volatility is on average
80% of total volatility of common stocks. While, Anderson et al. (2005) observe that 62% of the monthly
return volatility of the NAREIT index is unrelated to any of the capital market factors in their asset pricing
model. Ooi et al. (2007) find that during 1990 to 2005, 78.3% of the monthly return volatility can not
explained by the three risk factors in Fama-French (1993) model.

34
Figure 5: Decomposition of Total Risk: PCA Approach
Dec89 Dec90 Dec91 Dec92 Dec93 Dec94 Dec95 Dec96 Dec97 Dec98 Dec99 Dec00 Dec01 Dec02 Dec03 Dec04 Dec05 Dec06 Dec07
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Risk from PCA analysis
Year

IR from PCA
SR from PCA
TR from PCA

Figure 6: Decomposition of Total Risk: Portfolio Approach
Dec89 Dec90 Dec91 Dec92 Dec93 Dec94 Dec95 Dec96 Dec97 Dec98 Dec99 Dec00 Dec01 Dec02 Dec03 Dec04 Dec05 Dec06 Dec07
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Risk from PCA analysis
Year

IR from Portfolio
SR from Portfolio
TR from Portfolio

35
Table 2 presents the time-series properties of REIT idiosyncratic volatility. The
time series mean, median and standard deviation of idiosyncratic risk measured by the
portfolio approach are the lowest among three approaches. This is reasonable because the
average stock variance approach actually measures the total risk instead of idiosyncratic
risk although idiosyncratic risk accounts for most of it, while for the PCA approach, this
study only uses the largest eigenvector as one market factor, and the other systematic
information possibly contained by the rest eigenvectors is neglected. With the above
consideration, both average approach and the PCA approach may overestimate the true
idiosyncratic volatility. Although the aggregate idiosyncratic volatilities vary
substantially over time, quite different to Fu (2005), who find that the mean
autocorrelation of idiosyncratic risk estimated by the FF-three factor model is only 0.33
at the first lag, my results show a highly persistent time series pattern in the model-
independent estimated idiosyncratic volatilities: the first lag auto-correlation coefficients
are all above 0.96 for three different approaches, besides, they have a slow decay rate.

36
Table 2: Descriptive Statistics of Idiosyncratic Volatility Measures
IR Mean Median
StdDe
v
Min Max Skew Kurt AR1
AR1:
12
Cross-
section
0.1464 0.1349 0.0502 0.0751 0.3153 0.8391 3.4048 0.9839
0.992
9
PCA 0.0799 0.0760 0.0269 0.0407 0.1592 0.5853 2.7572 0.9941
0.996
6
Portfoli
o
0.0674 0.0640 0.0191 0.0386 0.1307 0.7808 3.3291 0.9911
0.996
0
Note: This table presents descriptive statistics on returns and measures of idiosyncratic volatility.
The sample period is December 1989 to June 2008 (223 monthly observations). The variable “IR”
represents the idiosyncratic risks under different measure approaches. Cross-section approach, PCA
approach and portfolio approach, respectively. “Skew” is the skewness, “Kurt” is the kurtosis,
“AR1” is the first-order autocorrelation, and “AR1:12” is the sum of the first 12 autocorrelation
coefficients.

High autocorrelation coefficient and slow decay rate imply the possibility of
existence of unit root and non-stationarity of REIT idiosyncratic volatility. I carry out the
Augmented Dickey-Fuller (ADF) unit root test and the KPSS test to determine whether the
idiosyncratic risk of REITs is non-stationary and follows a random walk process or not. This
question is especially important for measuring the expected idiosyncratic risk. Whilst some
previous studies such as Fama and French (1993) and Ang et al. (2006) have employed the
lagged values of idiosyncratic risk as the best estimates of its expected value, Fu (2005)
argues that such approximation is only valid of the stock’s conditional volatility follows a
random walk process. In absence of evidences about the random walk process existing in
idiosyncratic risk, some researchers like Fu, Brockman and Schutte (2007), Spiegel and
Wang (2006) use EGARCH model to estimate the conditional idiosyncratic volatility. Table
3 reports the results of unit root tests. The optimal lag lengths for those tests are chosen
based on the Akaike’s Information Criterion (AIC). For each idiosyncratic volatility time

37
series, I have done the best-known ADF (see, e.g., Fuller, 1976; Dickey and Fuller, 1979)
and KPSS (see Kwiatkowski et al., 1992) econometric tests with and without time-trend
being included to test whether the estimated idiosyncratic risk follows a random walk
process. For all three idiosyncratic volatilities time series, according to the ADF critical
values of t-statistics, the null hypothesis of a random walk process can not be rejected; while
for the KPSS test, the null hypotheses that the idiosyncratic volatility is stationary are all
rejected. However, after taking first difference on these three time series, the time series
exhibit the stationary and non-existence of unit root. My results thus suggest that it is
appropriate to describe the aggregate REIT idiosyncratic volatility process as a random walk.
Put differently to Fu and others using this month’s REIT idiosyncratic volatility to
approximate the value in the next month should not introduce severe measurement errors.

38
Table 3: Do Monthly Idiosyncratic Volatilities Follow a Random Walk Process?
(Unit Root Test and Stationary Test for Idiosyncratic Volatility)
ADF Test t-statistics KPSS Test
Idiosyncratic
Risk
Levels Trends First Difference Levels Trends
First
Difference
Cross-section -2.1788(4) -3.2913(4) -11.4313
**
(3) 2.6634
**
(4) 0.1954
*
(4) 0.0396(3)
PCA -1.9011(18) -1.9807 (18) -3.5631
**
(17) 0.9008
**
(18) 0.0754(18) 0.0687(17)
Portfolio -1.6523(8) -1.8641(8) -8.3152
**
(7) 1.6036
**
(8) 0.1833
*
(8) 0.0438(7)
Note: **indicate the statistical test is significant at 1% level. * indicate the statistical test is significant at 5%
level. This table presents the results of unit root test for the idiosyncratic volatility measured by three different
approaches: cross-sectional variance approach (Goyal and Santa-Clara,2003), PCA approaches, and Portfolio
approach, respectively. The number in brackets means the optimal lag lengths for the ADF test and KPSS test
based on the Akaike’s Information Criterion (AIC).
(1) ADF Test is based on the model:
1
*
1
1
j
p
tt tj t
j
yy y u α
−
−−
=
Δ =Φ + Δ +
∑
, in the model the pair of hypotheses
0
:0 H Φ= versus
1
:0 H Φ< is tested based on the t -statistics of the coefficient Φ .
0
H is
rejected if the t -statistics is smaller than the relevant critical value*. If 0 Φ = (that is, under
0
H ) the
series has a unit root and is non-stationary.
(2) KPSS test the hypothesis that
0
:~ (0)
t
Hy I versus
1
:~ (1)
t
Hy I , the null hypothesis of stationary
is rejected for larger values of KPSS than critical values.

* Dicker-Fuller and KPSS Test Critical Value
1% 5% 10%
ADF Level Test -3.43 -2.86 -2.57
Time Trend Test -3.96 -3.41 -3.13
KPSS Level Test 0.739 0.463 0.347
Time Trend Test 0.216 0.146 0.119

Economic Determinants of Idiosyncratic Risks
Due to the intrinsic heterogeneity of direct real estate properties, the effects of
macroeconomic fluctuations and their speed of propagation differ across different real
estate assets. The differences depend on several factors, including the type of real estate,
the hedonic characteristics of real estate, as well as the composition of local economy.
Thus idiosyncratic volatility that is not attributable to overall market volatility, but

39
derived from the property’s specific volatility will play a significant role in determinants
of cross-sectional direct real estate returns. Recent research by Miller and Pandher (2008)
has already confirmed this claim with strong empirical evidences. On the other side, lots
of studies have suggested that there is a fundamental link between REITs and private real
estate returns, where the information about the direct real estate market fundamentals is
compounded into REITs returns (e.g.,, Chen, et al., 1990; Giliberto, 1990; Gyourko and
Keim, 1992). Taking a longer-term view, Barkham and Geltner (1995), Geltner and
Rodriguez (1998) show a strong co-movement between REIT and lagged real estate
returns. In this way, the idiosyncratic risk of REIT returns (securitized real estate) is
expected to be related to the economic conditions.
Another link between idiosyncratic risk and the economy operates through the
capital structure of individual REITs. Because of high dividend payouts, REITs have to
frequently obtain funds from the external capital markets. Different capital structure of
REIT will lead to different volatility pattern of returns even under the same changes of
external financing condition. Hence, the idiosyncratic risk will be related to the
conditions of credit markets and external debt financing, which are inherently linked to
the macro economy.
To investigate how changes in economic conditions affect the REITs aggregate
idiosyncratic volatility, I run the following regressions for each estimated idiosyncratic
volatility series, respectively.

40

1,
1
n
tt ktkit
k
IR IR X αγβ ε
−−
=
=++ +
∑
(13)
where
t
IR denotes the conditional aggregate REIT idiosyncratic volatility at time t , the
parameter γ is the auto-regression parameter which reflects how
1 t
IR
−
at timet -1 affect
the idiosyncratic volatility value
t
IR at time t ; the parameters
k
β are vector parameters
which measure the sensitivities of the REIT idiosyncratic volatility to the current and k -
lagged macroeconomic variable vector
tk
X
−
. Here I include an autoregressive term to
capture any time-series variation in
t
IR attribute to factors other than prevailing
macroeconomic conditions. These factors include the liquidity, the momentum effects,
and the corporate governance structure of REITs as well as idiosyncratic shocks.

41
Table 4: Economic Determinants of Idiosyncratic Volatility in REITs
Idiosyncratic PCA Portfolio Cross-sectional
Risk (1) (2) (1) (2) (1) (2)
Constant -0.014 -0.004 -0.010 -0.003 -0.025 -0.014
(0.008)
*
(0.004) (0.006)
*
(0.004) (0.016) (0.011)
TSPR 1.477 0.268 1.466 0.364 3.996 1.179
(0.104)
***
(0.107)
**
(0.111)
***
(0.103)
***
(0.322)
***
(0.292)
***
CSPR 0.681 0.237 1.709 0.433 2.225 0.925
(0.620) (0.301) (0.430)
***
(0.305) (1.241)
*
(0.869)
TB3M 2.252 0.211 1.075 0.288 2.510 0.826
(0.161)
***
(0.070)
***
(0.072)
***
(0.070)
***
（0.208)
***
(0.186)
***
CPI -0.123 0.069 0.088 0.096 -0.075 0.107
(0.216) (0.105) (0.150) (0.102) (0.432) (0.300)
LAG -- 0.873 -- 0.733 -- 0.704
(0.033)
***
(0.046)
***
(0.047)
***
2
adj
R 0.5199 0.8883 0.5364 0.7844 0.4512 0.7373
Note: This table reports the results from the regression by model-independent measures of idiosyncratic
volatility at time t on economic variables at time t with and without lagged idiosyncratic volatility at
time 1 t − for the PCA approach, the portfolio approach and average variance approach, respectively. The
economic variables are defined as in Section 3, and LAG is the time 1 t − idiosyncratic volatility. The standard
deviation values are in parentheses. Significance at 90%, 95% and 99% level is denoted with one, two and three
asterisks, respectively. The
2
adj
R goodness of fit measure. The sample is 223 monthly observations from
1989:12 to 2008:6.

Table 4 shows the results of regression estimations using the cotemporaneous
economic variables as repressors. The autoregressive term is excluded as well as included.
With including one-lagged idiosyncratic volatility, both the term spread (TSPR) and the
three-month Treasury bill rates (TB3M) have a positive effect on the idiosyncratic
volatility. Moreover, they are statistically significant at 1% level for all three different
model-independent measures of idiosyncratic volatility. Interestingly, all other economic
variables also enter the regression with a positive sign; however, the credit spread (CSPR)
and inflation (INF) are not statistically significant with 5% level. When lagged

42
idiosyncratic volatility is excluded, the regression estimates of Table 4 display the
marginal economic significance of all the economic variables. Again both the term spread
(TSPR) and the three-month Treasury bill rates (TB3M) enter each regression with a
positive sign at 1% significance level. The credit spread (CSPR) is found to be positively
significant at the 1% level for the portfolio measure as well as at the 10% level for cross-
sectional measure of idiosyncratic risk, but not significant for the PCA measure. In
addition, the high adjusted R-square in Table 4 also shows that the economic variables
capture most of variations in aggregate REIT idiosyncratic volatility measured by
different approaches.

43
Table 5: Economic Determinants of Idiosyncratic Volatility in REITs
Idiosyncratic Risk PCA Portfolio Cross-sectional
Constant -0.002 0.001 -0.003
(0.003) (0.003) (0.008)
TSPR(t) 0.209 0.220 0.722
(0.110)
*
(0.116)
*
(0.322)
***

TB3M(t) 0.174 0.189 0.531
(0.070)
**
(0.075)
**
(0.198)
***

LAG 1 1.111 0.693 0.805
(0.068)
***
(0.067)
***
(0.068)
***

LAG 2 -0.283 -0.003 -0.315
(0.102)
***
(0.080) (0.088)
***

LAG 3 -0.175 0.162 0.273
(0.102) (0.079)
**
(0.088)
***

LAG 4 0.171 -0.182 -0.114
(0.100) (0.080)
**
(0.087)
LAG 5 0.074 0.157 0.163
(0.068) (0.069)
**
(0.067)
**

2
adj
R 0.9047 0.7966 0.7595
Note: This table reports the results from the regression by model-independent
measures of idiosyncratic volatility at time t on economic variables at time t
with and without lagged idiosyncratic volatility at time 1 t − for the PCA
approach, the portfolio approach and average variance approach,
respectively. The economic variables are only TSPR and TB3M as defined in
The Dara Section, and LAG (k) is the time tk − idiosyncratic volatility. The
standard deviation values are in parentheses. Significance at 10%, 5% and
1% level is denoted with one, two and three asterisks, respectively. The
2
adj
R
goodness of fit measure. The sample is 223 monthly observations from
1989:12 to 2008:06.

Table 5 checks the significance of TSPR and TB3M with more auto-regression
terms of idiosyncratic risk being included. Surprisingly, both TSPR and TB3M are still
statistically significant at 10% level or better with a positive sign for all three different
measures of aggregate REIT idiosyncratic volatilities. These findings suggest that

44
economic variables, the term spread (TSPR) and short term interest rate (TB3M) do have
strong positive effect on the fluctuations of aggregate REIT idiosyncratic risk.
As previously conjectured, the observed effects of the economic variables on the
REIT idiosyncratic volatility are consistent with the differences in the propagations of
economic shocks across underlying real estate assets held by REITs as well as the
differences in the capital structure of REITs. The signs of the term spread coefficient in
these regressions imply that idiosyncratic volatility is high (low) in periods when the term
spread is wide (narrow). Because the term spread is closely related shorter-term business
cycle with being wider in a business downturn and narrower in a business cycle
expansion (e.g., Fama and French, 1989, Campbell et al., 1997), my findings indicate that
the aggregate idiosyncratic volatility of REIT returns are counter-cyclical that are large in
a recession and small in an expansion. This is also consistent with the observation in the
historical performance of idiosyncratic volatility in Figure 3. Furthermore, this claim is
also strongly supported by the evidence of the three month Treasury bill rate. The
idiosyncratic risk tends to be high (low) when the interest rate is high (low). Because of
the importance of external debt financing to the REIT market, the increasing of the
interest rates, consistent with the tightening conditions in credit markets and more
expensive external debt financing, results in not only the lower return of REIT returns
(downturn of the stock market too) but also higher the idiosyncratic volatility.

45
Does Idiosyncratic Risk Really Matter?
Now I explore the linkage between the cross-sectional expected REIT stock returns and
expected idiosyncratic volatility in REITs. From the theory perspective, the risk and
return tradeoff should be contemporaneous. Investors earn returns for bearing the risk in
the same period. A conventional practice is to use the realized return as the dependent
variable in cross-sectional regressions, where the realized returns are assumed to be an
unbiased estimate of expected returns.
16
The expected idiosyncratic volatility and other
control variables are put on the right side of the cross-sectional regressions as follows:

11
1
[] [ ]
n
ttt ktktt
k
REIV EX αγβ ε
−−
=
=+ + +
∑
, 1,2,..., tT = (14)
where the dependent variable
t
R is the realized monthly cross-section of REIT return at
time t ,
1
[]
t
E
−
⋅ stands for the function of expectation conditioned on the information set at
time 1 t − .
t
IV represents the idiosyncratic volatility of REIT portfolio returns.
kt
X
represents other explanatory variables of cross-sectional REIT returns. Here the cross-
section measure of REIT return is calculated by the equally-weighting scheme, and the
idiosyncratic volatility is measured by the portfolio approach.
17
The null hypothesis is
that 0 γ = , that is, the idiosyncratic risk is not priced. A priori, I expect that the γ

16
See, for example, Fama and French (1992), Chordia et al. (2001), Easley et al. (2002), among others.

17
We do not use the idiosyncratic volatility measured by the PCA approach as the dependent variable here
because the idiosyncratic volatility calculation process of the PCA approach depends on the estimation of
the first eigenvector of returns covariance matrix, which is based on the previous 60-day time window. This
may bring some time-delay effects on the estimation of current idiosyncratic volatility.

46
estimates to be positive if under-diversified investors demand higher return compensation
for bearing idiosyncratic risk.
It is crucial to have a quality estimate of the expected idiosyncratic
volatility
1
[]
tt
EIV
−
. Some previous studies use the lagged idiosyncratic value as the
expected value of idiosyncratic volatility (e.g., Goyal and Santa-Clara, 2003; Ang et al.
2006, 2008); however, other studies such as Fu (2005), Brockman and Schutte (2007),
and Eiling adopt the EGARCH model to estimate the conditional idiosyncratic volatility.
In this chapter, I use the lagged idiosyncratic volatility as a proxy for the expectation of
the current period’s idiosyncratic volatility, which has been justified by the high
persistence of the idiosyncratic volatility series measured by the portfolio approach in the
section on Time-Series Property of Idiosyncratic Risk in REITs as following a random
walk process.
18

Table 6 presents the results of regressions for the monthly cross-section of REITs
return. The first regression repeats the classic regression of market (portfolio) return on
the expected market (portfolio) volatility.
19
The extant literature that estimate the relation
between the market risk and returns in equity markets presents conflicting results on the
sign and significance of this coefficient. Campbell (1987) and Glosten et al. (1993) find a
significant negative relation, whereas French et al. (1987) and Ghysels et al. (2004) find a

18
The results are essentially the same if we replace idiosyncratic volatilities by the fitted values from an
ARMA model (see, Schwert, 1989, 1990).

19
The equally-weighted REIT portfolio can be also regarded as a measure of the REIT stock market.

47
significantly positive relation. However, most studies usually do not find a positive nor
statistically significant coefficient (e.g., Goyal and Santa-Clara, 2003; Turner et al. 1989;
Baillie and DeGennaro, 1990; Whitelaw, 1994). For my data and sample period, I find a
negative but insignificant coefficient.
20

Table 6: Does Idiosyncratic Risk Really Matter?
Dependent Variable: monthly equally-weighted portfolio return
Model Constant ()
P
t
EIV ()
P
t
ESV
2
adj
R (%)
1 0.016 -0.152 0.01
(0.005) (0.154)
2 -0.008 0.286 1.69
(0.009) (0.130)
**

3 -0.003 0.312 -0.205 2.03
(0.010) (0.131)
**
(0.154)
Note: This table presents the results of a one-month-ahead predictive regression
of excess equally weighted REIT portfolio return on explanatory variables. The
variable ()
t
EIV is the expected REIT idiosyncratic volatility measured by the
portfolio approach, ()
P
t
ESV is the REIT market volatility (systematic risk)
also measured by portfolio approach. The first row in each regression is the
coefficient, the second row is the standard deviation value, and significance at
10%, 5% and 1% level is denoted with one, two and three asterisks,
respectively.

The second regression shows that the expected idiosyncratic volatility is
positively significant in explaining the cross-section of expected REIT returns. The

20
Goyal and Santa-Clara demonstrate that the insignificant negative coefficient on the market volatility
should not be interpreted as a rejection of the ICAPM. The ICAPM postulates a positive partial relation
between the market’s return and its variance, after taking into account the covariance with all other state
variables. To the extent that the covariance of the market with all other state variable is not controlled in the
regression, following Goyal and Santa-Clara,our findings are not the evidence against the ICAPM.

48
coefficient γ is statistically significant at 5% level and the adjusted
2
R is about 1.7 %.
This significance actually increases when the third regression add the market volatility as
a second regressor. Including both volatility measures makes the coefficient on market
volatility insignificantly negative and the coefficient on the expected REIT aggregate
idiosyncratic volatility still positive at 5% significance level. The adjusted
2
R of this
regression is as high as two percent. Therefore, my findings strongly support that the
effect of idiosyncratic risk on the cross-section of expected REITs return is significant
both statistically and positively, and the idiosyncratic risk in REITs does matter!

Table 7: Does Idiosyncratic Risk Really Matter?
(Controlling for Lagged Portfolio Return)
Dependent Variable: monthly equally-weighted portfolio return
Model Constant ()
P
t
E IV ()
P
t
E SV
1
P
t
r
−

2
adj
R (%)
1 0.010 -- -- 0.150 1.68
(0.003)
***
(0.068)
2 -0.008 0.256 -- 0.134 2.92
(0.009) (0.130)
**
(0.068)
**

3 0.013 -- -0.097 0.142 1.40
(0.006)
**
(0.155) (0.069)
**

4 -0.004 0.278 -0.153 0.120 2.90
(0.010) (0.132)
**
(0.156) (0.069)
*

Note: This table presents the results of a one-month-ahead predictive regression of excess
equally weighted REIT portfolio returns on explanatory variables. The variable ()
P
t
EIV is
the expected REIT idiosyncratic volatility measured by the portfolio approach, ()
P
t
E SV is
the portfolio systematic risk volatility also measured by portfolio approach,
1
P
t
r
−
is the
lagged equally-weighted REIT portfolio return in excess of the 3-month T-bill rate. The
first row in each regression is the coefficient, the second row is the standard deviation
value, and significance at 10%, 5%, and 1% level is denoted with one, two and three
asterisks, respectively.

49
I test whether the importance of idiosyncratic risk derives from it being a proxy of
lagged REIT cross-sectional return. I control for this possibility by including the lagged
value in the regression and find that the inclusion of the lagged value does not alter the
significance and the sign of the coefficient on the idiosyncratic risk measure. Table 7
presents the results of regressions. The coefficients on expected idiosyncratic risk do not
change too much after including the lagged REIT market return as regressor. In the
second regression without the REIT market volatility, the coefficient γ on the expected
idiosyncratic volatility is 0.256 while the coefficient on the lagged market return is 0.134.
Both of them are consistently significant at 5% level. After including the market volatility
in the regression, the coefficient on the REIT market volatility keeps being insignificantly
negative in each regression. The coefficients on both the idiosyncratic risk and the lagged
market return are still positive with a 5% significance level. I, therefore, reject the
hypothesis that the idiosyncratic risk is a proxy for the lagged cross-sectional REIT return.
The results are robust with adding the lagged REIT cross-sectional return and the relation
between the idiosyncratic risk and the expected REIT returns remains positive and
statistically significant. Again the idiosyncratic risk in REITs is priced!

Robustness Checks
In this section, I confirm the positive relation between aggregate idiosyncratic volatility
and the cross-section of REIT expected return through a battery of robustness checks.
Firstly, I examine the pricing of aggregate idiosyncratic volatility in the cross-section of

50
REIT returns controlling for variables proxying for business cycle fluctuations; secondly,
adopting a portfolios sorting strategy,
21
I test the role of idiosyncratic volatility in
explaining the return differences of portfolios sorted on conditional idiosyncratic
volatility at the firm level; finally, I resample the data to exclude the possible “survivor’s
bias.”
A. Controlling for the Business Cycle
Alternatively, it maybe argued that abovementioned regression results are not due to the
pricing of idiosyncratic risk in the cross-section of expected REIT returns but rather
reflect time-varying expected returns for REITs. Chen et al. (1986), Campbell and Shiller
(1988b), Keim and Stambaugh (1986), Campbell (1991), Fama and French (1989) and
others show that expected returns of stocks vary with the state of economy and the
market return can be predicted by variables related to the economic cycle such as the
term spread, the Treasure bill rate and the credit spread. Since as previously documented,
aggregate REIT idiosyncratic risk also varies with the state of the economy, one potential
explanation for the regression results is that the aggregate idiosyncratic risk proxies for
time variation in the economy. Under this argument, directly adding the economic
variables that proxy for business cycle fluctuations to Regression (14) would render the
coefficient on
1
[]
tt
EIV
−
insignificant. Related arguments for including these additional
economic variables are provided by Scruggs (1998) and Plazzi et al. (2008) who
demonstrate that in the context of ICAPM model (Merton, 1973), it is important to

21
Chui et al. (2003) also adopted a similar approach to examine the payoffs of REIT portfolios constructed
based on the momentum and value effects.

51
include all variables correlated with the state variables that describe changes in the future
investment opportunity set
22
. Otherwise, regression such as equation (14) will suffer from
omitted variables bias.
Table 8 presents the regression results controlling for the economic variables.
The estimate of coefficient γ on the expected idiosyncratic volatility remains positive
and statistically different from zero at the 10% significance level or better for each
regression. The term spread (TSPR) is positively significant at 10% level to explain the
cross-section of expected REIT returns while the CPI is negatively significant at 1% in
the regression. In addition, adding the economic variables also enhance the adjusted
2
R of
regression obviously.
Furthermore, I run the regression with the economic variables as well as the
lagged cross-sectional REIT return values. Including the lagged REIT market return is to
control the possible return persistence. The regression results are shown in Table 8.
Again the expected aggregate idiosyncratic volatility in REITs still has positive and
statistical significant effects on the cross-section of expected REIT returns for each
regression. While the coefficient on the market risk remains insignificantly negative and
the lagged market rerun is positive but only significant at 10% significance level for the

22
The ICAPM (Merton, 1973) suggests that the demand for a risky asset depends on partly on the asset’s
ability o hedge against unfavorable shifts in the future investment opportunity set. In a multifactor world
where s state variables proxy for changes in the opportunity set, an asset’s return is partly attributable to the
covariance between the asset’s return and the return of the market portfolio and to the covariance between
the asset’s return and the return of the hedging portfolio.

52
forth regression that includes two risk measure, the credit spread (CSPR) and the lagged
market return as the regressors.
In summary, I find a positive and statistically significant relation between the
expected idiosyncratic risk in REITs and their cross-sectional returns. This relation is so
robust that does not appear to be merely proxying for fluctuations in the state of the
economy or the lagged cross-sectional REIT return value.

53
Table 8: Does Idiosyncratic Risk Really Matter?
(Controlling for Business Cycle Variables)

Dependent Variable: monthly equally-weighted portfolio return
Model Constant ()
P
t
E IV

()
P
t
ESV
1 t
TSPR
−

1
3
t
TB M
−

1 t
CPI
−

1 t
CSPR
−

2
adj
R
(%)

1 -0.005 0.256 -0.201 0.435 -- -- -- 3.11
(0.010) (0.133)
*
(0.152) (0.232)
*

2 0.003 0.396 -0.286 -- -0.241 -- -- 2.56
(0.011)
(0.142)
**
*

(0.162)
*
(0.161)
3 -0.001 0.351 -0.144 -- -- -0.942 -- 3.67
(0.010)
(0.131)
**
*

(0.155) (0.429)
***

4 -0.011 0.303 -0.279 -- -- -- 1.256 2.00
(0.013) (0.131)
**
(0.171) (1.281)
Note: This table presents the results of a one-month-ahead predictive regression of excess equally weighted
REIT portfolio returns on explanatory variables. The variable ()
P
t
EIV is the expected REIT idiosyncratic
volatility measured by the portfolio approach, ()
P
t
ESV is the portfolio systematic risk volatility also
measured by portfolio approach. The economic variables are defined as follows: TSPR is the difference between
the yield on 10-year and 1-year Treasuries, CSPR is the difference between the yields on BAA- and AAA-rated
corporate bonds, INF is inflation computed as the growth of the CPI index and TB3M is the 3-month Treasury
bill rate. The first row in each regression is the coefficient, the second row is the standard deviation value, and
significance at 10%, 5% and 1% level is denoted with one, two and three asterisks, respectively.

54
Table 9: Does Idiosyncratic Risk Really Matter?
(Controlling for Business Cycle Variables and Lagged Portfolio Return)
Dependent Variable: monthly equally-weighted portfolio return
Model Constant ()
P
t
EIV

()
P
t
E SV
1
P
t
r
−
1 t
TSPR
− 1
3
t
TB M
− 1 t
CPI
−

1 t
CSPR
−

2
adj
R
(%)

1 -0.006 0.232 -0.155 0.106 0.392 -- -- -- 3.69
(0.010) (0.134)
*
(0.155) (0.069)
(0.232
)

2 0.002 0.352 -0.226 0.106 -- -0.201 -- -- 3.12
(0.011) (0.144)
**
(0.166) (0.070) (0.162)
3 -0.002 0.319 -0.105 0.102 -- -- -0.857 -- 4.16
(0.010) (0.132)
**
(0.156) (0.069)
(0.431
)
**

4 -0.011 0.272 -0.217 0.115 -- -- -- 1.057 2.75
(0.012) (0.132)
**
(0.174) (0.069)
*
(1.278)
Note: This table presents the results of a one-month-ahead predictive regression of excess equally weighted
REIT portfolio returns on explanatory variables. The variable ()
P
t
EIV is the expected REIT idiosyncratic
volatility measured by the portfolio approach, ()
P
t
ESV is the portfolio systematic risk volatility also
measured by portfolio approach,
1
P
t
r
−
is the lagged equally-weighted REIT portfolio return in excess of the 3-
month T-bill rate. The economic variables are defined as follows: TSPR is the difference between the yield on
10-year and 1-year Treasuries, CSPR is the difference between the yields on BAA- and AAA-rated corporate
bonds, INF is inflation computed as the growth of the CPI index and TB3M is the 3-month Treasury bill rate. The
first row in each regression is the coefficient, the second row is the standard deviation value, and significance at
10%, 5% and 1% level is denoted with one, two and three asterisks, respectively.

55
B. Return Analysis of Portfolios Sorting on Conditional Idiosyncratic Risk
The cross-sectional results suggest a statistically significant and positive relation between
conditional idiosyncratic volatility and the cross-section of expected REIT returns.
However, so far the analysis is only focused on the REIT cross-sectional market level
because the idiosyncratic volatility measured by the portfolio approach actually is the
aggregate measure of REITs idiosyncratic risk. Now I focus on the REIT firm level by
examining the returns of portfolios formed on conditional idiosyncratic volatility. The
logic behind this is that since the aggregate idiosyncratic volatility can be priced by the
REIT market, at individual REITs firm level, idiosyncratic volatility should also play a
significant role in pricing of individual REIT stock. If individual REIT stocks with high
expected idiosyncratic risk have higher returns than REIT stocks with low expected
idiosyncratic risk, a zero-investment portfolio that is long in high idiosyncratic risk
REITs and short in low idiosyncratic risk REITs should earn a positive return.
More importantly, using the portfolio sorting method, Ang et al. (2006, 2008) find
a negative relationship between the lagged idiosyncratic risk and the cross-section of
expected stock returns. In their study, every month stocks are formed into five equal size
portfolios according to their corresponding idiosyncratic risk in the previous month; they
compared the risk-adjusted returns between the highest risk and lowest risk portfolios and
found the return difference is significantly negative. The negative relation is quite
puzzling because it suggests that stocks with lower idiosyncratic risk earned higher
average returns.

56
The key question here is whether the lagged value is a proper proxy of the
expected idiosyncratic volatility. Most of previous studies including both Fu (2005) and
Ang et al. (2006, 2008) use the three factor model of Fama and French (1993) to measure
the idiosyncratic volatility at firm level. The estimated idiosyncratic volatilities, unlike
some firm characteristics, are very volatile and not very persistent over time. A growing
literature studies since Fu have demonstrated that the time-series idiosyncratic volatility
at firm level by the three-factor model cannot be approximated by a random walk process.
Hence using the lagged idiosyncratic volatility to proxy the expected value will introduce
severe measurement errors. That’s why Fu (2005), Brockman and Schutte (2007), Spiegel
and Wang (2006) and Eliling (2006) all adopt the sophisticated GARCH models to
estimate the expected conditional idiosyncratic volatilities. However, as discussed in
Time-Series Property of Idiosyncratic Risk in REITs, the time-series idiosyncratic
volatility measured by the model-independent approaches instead of the three-factor
model exhibits a highly persistent pattern that does follow a random walk process.
Therefore, this study simply uses the lagged idiosyncratic value as an estimate of the
expected value and whether the lagged (expected) one will have a negative effect on the
expected returns is also examined below.

57
Table 10: Summary Statistics for Portfolios Formed on the Conditional
Idiosyncratic Volatility
Portfolios formed on Conditional Idiosyncratic Risk
Variables
Low 2 3 4 High
Port. Return 22.09% 19.09% 21.85% 20.49% 26.11%
SD 14.27% 14.41% 14.74% 15.82% 16.01%
Lagged IV 11.50% 15.45% 18.91% 23.96% 38.25%
Note: At the beginning of each month five portfolios are sorted based on the
conditional idiosyncratic risk estimated using the PCA model, which equals to the
idiosyncratic volatility for the last month. The first portfolio (Low) consists of the
20% of REITs stocks with the lowest idiosyncratic volatility and the last portfolio
(High) consists of the 20% of stocks with the highest idiosyncratic risk. Portfolios
are updated monthly. The “Port. Return” represents the time-series mean of the
equally weighted portfolio (annual) return. The “SD” shows the standard deviation
of the portfolio returns. The “Lagged IV” shows the time series mean of the equally
weighted portfolio idiosyncratic volatility. The sample is 223 monthly observations
from 1989:12 to 2008:06.

The procedure of the portfolio sorting methodology is similar to Ang et al. (2006,
2008). Based on the monthly idiosyncratic volatility at firm level estimated by the PCA
model, at the beginning of each month, I sort the expected idiosyncratic volatility
1,
[]
tit
EIV
−
to form five portfolios with equal number of REIT stocks. The first portfolio
contains the 20% of REITs that have the lowest idiosyncratic volatilities in the last month
and the last portfolio consists of the 20% of REIT stocks that have the highest
idiosyncratic volatilities. The portfolios are updated monthly. Table 10 presents the
descriptive statistics for these five portfolios. The mean lagged idiosyncratic volatility
(Lagged IV) increased from 11.5% for the first portfolio to 38.35% for the last portfolio.
The equally weighted mean return of portfolio with highest lagged idiosyncratic volatility

58
is 26.11%, while the portfolio return for the lowest lagged idiosyncratic volatility is
22.09%. So the highest-minus-lowest idiosyncratic risk portfolio which is zero-cost can
generate an average annual return 4%. This result is also consistent with Ooi et al. (2007),
who find the monthly return for the same zero-cost REIT portfolio strategy is around
0.41%, but Ooi et al. use the three factor model to measure the idiosyncratic risk and their
portfolios are also sorted on the expected values that are estimated by the EGARCH
model. Hence, my findings reaffirm that it is rewarding to follow a trading strategy of
constructing portfolios based on the idiosyncratic volatility of REIT stocks.

C. Controlling for Sample Biases
The number of REITs in our sample is not static over the study period; growing from 19
to 109 at the end of the sample period (as of June 2008). To examine whether the REITs
idiosyncratic risk measured by the portfolio approach are simply due to the increased
number of REITs in the sample or any survivor’s bias from including the new REITs and
excluding the failed REITs stocks over the study period, I also construct the idiosyncratic
volatility series using the fixed 40 original REITs that have been traded continuously
since September 1993. The resulting idiosyncratic volatility series are presented in Figure
7, which show the similar trends as observed earlier in Figure 1, suggesting that the
observed idiosyncratic volatility pattern for REITs is not driven by the addition of more
REITs over the study period. Furthermore, I also re-do the regression analysis based on

59
this fixed 40 REITs sample. The results and conclusions are still similar to the previous
analysis and do not have significant changes.
In addition, to ensure that the observed patterns in the volatility series are not
driven by outliers, I re-compute the idiosyncratic time series using both the PCA
approach and the portfolio approach by omitting the 5% observation at both ends of the
distribution. The time trend for the reconstructed series is similar to the observed in
Figure 1 and hence, is not reported for brevity.

Figure 7: Idiosyncratic Risk: Portfolio, PCA, and Cross-Section Measures
(With Fixed 40 REITs from Sep 93 –June 2008)
Sep93 Sep94 Sep95 Sep96 Sep97 Sep98 Sep99 Sep00 Sep01 Sep02 Sep03 Sep04 Sep05 Sep06 Sep07
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Idiosyncratic Risk
Year

IR from Portfolio
IR from PCA
IR from Cross-Section

60
Conclusions
This article investigates the intertemporal relation between idiosyncratic volatility and the
cross-section of expected REIT stock returns. First of all, this chapter introduces new
measures of aggregate idiosyncratic risk: (a) the PCA approach which is based on
random matrix theory and the procedure of principal component analysis; (b) the
portfolio approach which is based on mean-variance portfolio theory and the concept of
gain from portfolio diversification. The measure of idiosyncratic risk by the PCA
approach mainly relies on the estimation of the largest eigenvalue of the covariance
matrix for all REITs in the market. The measure of idiosyncratic risk by the portfolio
approach is defined as the difference between the variances of the non-diversified and
fully diversified REITs portfolios. The crucial difference between the new and the
existing idiosyncratic volatility measures is that the new measure does not depend on any
parametric specifications of the return generating process such as the CAPM model or
three-factor capital asset pricing model.
Second, relying on data over the 1989-2008 sample period, I analyze the time-
series properties of REITs idiosyncratic risk measured by both PCA and portfolio
approaches. The idiosyncratic volatility is found to be time varying and highly persistent
as following a random walk process, and more importantly, its fluctuations are affected
by the term spread and the three-month Treasury bill rate. Moreover, the REIT
idiosyncratic volatility is counter-cyclical, increasing in recessions and decreasing in
expansions. In particular, the idiosyncratic volatility is high (low) in periods when the

61
term spread is wide (narrow), and the idiosyncratic risk increases when interest rate
increases with tightening conditions in credit markets and more expensive external debt
financing.
Third, most importantly, my results show that idiosyncratic risk dominates the
total volatility of REIT returns and conditional idiosyncratic volatility is a significant
factor in explaining the cross-sectional returns of REIT stocks. The positive relationship
between expected idiosyncratic risk (also the lagged one in this study) and the cross-
section of expected REIT returns are statistically significant. This significant and positive
relationship is robust with a variety of robustness check. In particular, firstly, the positive
relation is confirmed by regression models with inclusion of the lagged cross-sectional
return and the macroeconomic variables as regressors, together and respectively.
Secondly, I also analyze the return differences of portfolios sorted by idiosyncratic risk
using portfolio sorting methods and find that the return of REITs portfolio with the
highest idiosyncratic risk is much higher than one with lowest idiosyncratic risk. Finally,
the positive relation is still robust with categorization of data over different sub-sample to
avoid the possible sample survivors’ bias. In addition, consistent with some previous
studies, I find that the lagged variance of the REITs market has no forecasting power for
the REITs market return.
My findings are based on a more proper and solid measure of idiosyncratic
volatility and consistent with Merton’s (1987) proposition that idiosyncratic volatility
should be positively related to the cross-section of expected returns if investors demand

62
compensation for imperfect diversification in presence of incomplete information. One
important implication is that the investment strategy of REITs should consider the
benefits of corporate focus versus diversification since REITs idiosyncratic risk has been
found be priced by the market and it is well documented that real estate corporate focus is
positively related to idiosyncratic risk in REITs. The results also have practical
applications for estimating the required return or cost of capital as well as the portfolio
formation and performance evaluation. Furthermore, the sample period in this article
show that there is a decreasing trend for aggregate idiosyncratic volatility in REITs
accompanying with a significant cyclical pattern. The characteristics of historical pattern
of idiosyncratic volatility in REIT returns have important implications for portfolio
diversification at different stages of the business cycle. Since idiosyncratic risk increases
fast during the bad market time, diversification will be more difficult and more individual
stocks will need to be held in the market downturn.
This article raises several additional issues. First, questions regarding the factors
that influence expected stock returns have long interested both academic and practitioner
audiences. While this research actually originates within the asset pricing literature to
answer whether idiosyncratic risk plays a role in expected returns, another tradition
derives from the market microstructure literature and looks at the relationship between
the liquidity and the expected returns. I do not consider the liquidity effect in this study.
An interesting direct for future research would be to design direct tests to empirically
disentangle the roles played by liquidity and idiosyncratic risk in REITs stock returns.

63
Another interesting question is whether a positive relation between idiosyncratic
risk and return is also present in the direct real estate market. One of the most important
benefits of REITs is that they allow investors to hold a more diversified real estate
portfolio. To the extent that our results are driven by the difficulty to diversify, the results
in private real estate market should be stronger.
Finally, the results presented in this article underscore the importance of short-
sale constraints in REITs stock market. With controlling for short-sale constraint, Miller
(1977) and Boehme et al. (2005) show that when short-sale constraint are absent,
idiosyncratic risk is positively correlated with future excess returns, a result consistent
with Merton (1987); However, when short-sale constraints are present the correlation
becomes negative: increased idiosyncratic volatility produce negative abnormal returns.
To date there has not been an attempt to empirically examine the relation between
idiosyncratic risk and REIT return with a special consideration on this short-sale
constraint effect. I hope the findings in this article will stimulate future work on this
important issue.

64

Chapter 2:
Correlation and Volatility Dynamics in REIT Returns

Introduction
Diversification plays an essential role in asset allocation. The core of modern portfolio
theory prescribes that investors should populate their portfolios with assets that are
composed by different types of assets with low correlations. Many studies have
documented that the returns to real estate have very low correlation with movements in
the stock market
23
. These studies infer that mean-variance optimal portfolios should
contain significant allocations to real estate. Nevertheless, due to some intrinsic
drawbacks such as illiquidity, high transaction costs and intensive management
involvement, direct investment in real estate can be time consuming and expensive.
Indirect real estate investment through securitization mainly the real estate investment
trusts (REITs) has offered investors a favorable alternative to invest in real estate assets.
The investors expect to achieve internal and external diversification through holding the
securitized real estate assets.
While there is a considerable body of literature addressing the return
characteristics of REITs, and the relationship between REITs and both the stock market
and the private direct real estate market, the literature regarding the dynamics in the

23
See, for example, Webb and Rubens (1987) and Firstenberg, et al. (1988).

65
correlations among REIT, direct real estate and stock returns remains very thin. There are
only few studies documenting that the diversification potential of REITs in the mixed-
asset portfolio appears to be time-dependent due to the volatile time-varying correlations
between REITs and other stocks (See, e.g., Giliberto, 1993; Mull and Soenen, 1997).
During some periods, for instance, from 1990 to 1992, the unconditional correlation rate
between REIT stocks and the general stock market is more than 70% which is almost as
triple as the correlation rate in 2003, although this number between equity REITs and the
S&P 500 index has averaged only about 35% since 1992.
An understanding of the correlations and volatility dynamics of REITs return in a
mixed-assets portfolio is extremely important to investors. A closely related underlying
economic motivation is whether or not the diversification ability of REITs in a multi-
asset portfolio will be seriously diluted owing primarily to the time-varying correlations
of REITs to real estate and stock returns. This could be attributed to the rising correlation
between REIT and stock market and decreasing one between REIT and private direct real
estate market. Some specific questions emerge as a consequence. First, what is the nature
and extent of the time-varying conditional correlations between REITs and stock returns
as well as the correlations between REITs and direct real estate returns? Is the REITs
market becoming more increasingly correlated with the real estate market right now?
This chapter tries to provide direct evidence evaluating the claim that REITs are now
“less like stocks and more like real estate” given the dramatic growth and maturation of
the REIT sector since 1992 (Ziering et al., 1997). The extant evidence in support of the

66
claim is primarily the decreased correlation between REIT and S&P 500 returns and the
inability of stock and bond factors to explain REIT return since early 1990s compared to
1970s and 1980s (Ghosh et al., 1996; Zierling et al., 1997; and McIntosh and Liang,
1998). Second, whether or not the conditional correlation between REITs and stock or
real estate returns is related to the future performance of REITs. Despite the important
implications of the nature of the correlations among REIT, real estate and stock returns,
relatively little research has been directed at the relationship between these conditional
correlations and the expected returns of REITs. Third, whether the dynamic correlations
are determined by the macroeconomic variables? Finally, the asymmetric volatility
phenomenon is found widely in the estimates of the second moment of equities, where
volatility increases more after a negative than after a positive shock of the same
magnitude. Surprisingly, asymmetries in conditional correlations have been neglected
obviously in the portfolio optimization involving REITs in literature. Is there any
asymmetry in the conditional correlation between REITs and regular stocks?
This chapter examines the dynamics in correlations and volatility of REITs, stock
and direct real estate returns using the monthly data from Jan 1987 to May 2008. While
there are extensive studies on the dynamics of stock markets conditional correlations and
portfolio diversification, far less attention has been devoted to such studies in the real
estate literature. This is mainly due to the lack of longer and high frequency time series
for real estate returns. Consequently, many real estate studies focus on the unconditional
real estate and REITs correlation with the broader market. Moreover, different to

67
previous studies, with considering the possible asymmetry in the conditional correlations,
this chapter investigate the time-varying conditional correlations and volatilities in REITs,
real estate and stock return series by adopting a new generalized autoregressive
conditionally heteroskedastic (GARCH) process, the asymmetric dynamic conditional
correlation (AG-DCC) GARCH technique. Then I test whether the estimated conditional
correlations are a good indicator to estimate the future returns of REITs and whether the
dynamics in correlations can be explained by the macroeconomic factors such as the term
and credit spreads, inflation and the short rate of interest.
This chapter makes two main contributions to the literature. First, it explicitly
examines correlation dynamics among REIT, direct real estate and stock asset classes and
investigate the presence of asymmetric responses in conditional variances and
correlations to negative returns. The findings in this paper actually resolve the long
debate by academics and industry practitioners on the role of REITs in mixed asset
portfolios, questioning whether REITs actually provide exposure to the private real estate
asset class or simply represent additional exposure to common stocks (see e.g., Clayton
and Mackinnon, 2003). Second, in addition to estimating the time-varying correlations
among these asset classes, I examine the economic determinants of correlations and
volatilities as well as the implications on future REIT returns. This allows quantifying the
time-varying diversification ability of REITs in a mixed assets portfolios and to better
understand the risk and return characteristics of REITs.

68
The results show that over the 1987-2008 time period, the correlations among
REIT, direct real estate and stock returns are time-dependent and volatile which can be
explained by macroeconomic variables such as the term and credit spreads, inflation and
the unemployment rate. However, different to the previous evidences of existing
significant asymmetry in correlations among financial assets (see e.g., Cappiello, et al.
2006), this study only find little asymmetry in the conditional correlations of REITs,
stock and direct real estate returns. I also find strong relationship between conditional
correlations and future REITs returns, while those patterns are distinguishable for
different types of REITs. In particular, when the correlation between REITs and S&P 500
index is the lowest, the future performance of REITs will be the best. For equity REITs,
there exists a robust relationship between correlations and future returns: the higher
(lower) correlation between equity REIT and direct real estate is, the higher (lower) the
future returns of equity REIT.

Literature Review
This chapter attempts to draw together two strands of real estate studies on the correlation
between REITs and other asset classes, which determines diversification benefits of
REITs in a mixed-asset portfolio.
The first strand involves examination of the behavior of property-backed security
vehicles in particular real estate investment trust shares in relation to the underlying
direct real estate investment market. Early studies of REIT performance generally found

69
that the relationship between REITs and unsecuritized real estate is quite limited (See,
e.g., Gyourko and Linneman, 1988; Scott, 1990). More recent studies documents that
REITs and private unsecuritized real estate returns have significantly different statistical
properties and the return of REITs has some unique characteristics of “hybrid securities”
(e.g., Liang, McIntosh and Webb, 1995; Ghosh, et al., 1996; Liang and McIntosh, 1998;
Seiler et al., 2001). However, on the other hand, some studies have suggested that there is
a fundamental link between REITs and private real estate returns, where the information
about the direct real estate market fundamentals is compounded in REITs (e.g., Chen, et
al., 1990; Giliberto, 1990; Gyourko and Keim, 1992). Taking a longer-term view,
Barkham and Geltner (1995) show a strong co-movement between REIT and lagged real
estate returns, the similar results are also found in Geltner and Rodriguez (1998).
The second stand seeks to analyze the nature and behavior of the correlations
between REITs and other financial assets. An extensive body of literature show that
macroeconomic variables that have been found to explain stock and bond returns and
risks have significant power in explaining REIT return and risks (Ling and Naranjo, 1997;
Peterson and Hseih, 1997, Karolyi and Sanfers, 1998; Calyton and Mackinnon, 2003).
However, the empirical results about the direction and extent of the relationship between
REITs and stock have been also mixed and sometimes contradictory depending on the
time periods or the methodology used in the studies. Brueggeman, et al. (1984) find a
strong negative correlation of REITs with bonds and stocks for the periods from 1972 to
1983. Different results are found in other studies during other time periods extended into

70
1990s, where positive correlations between REITs and other asset classes (stocks and
bonds) (Chen and Peiser, 1999; Hartzell, et al., 1999; and Clayton and MacKinnon, 2001
among others). Clayton and MacKinnon (2001) found that in the 1990s REIT returns
were closely linked to the real estate asset class suggesting they may be an acceptable
substitute for real estate in the portfolio.
The time-varying nature of correlations between REITs and other financial assets
has also been documented in the literature. Ghosh, et al. (1996) found that the
correlations between REIT and stock returns were not constant, and they declined over
time over time period 1985 to 1996. Liang and Whitaker (2000) also observed changing
correlations between of equity REITs and stocks with the Russell Real Estate Indexes
during the 1981 to 2000 period. Mull and Soenen (1997) concluded that the
diversification potential for REITs appears to be very time dependent. They found that
while REITs would have been a good investment vehicle in the 1990–1994 period, they
were not as attractive in the period 1985–1990. The time varying nature of the
correlations between REITs and other financial assets was also observed in the beta
measures on REIT returns. Goldstein and Nelling (1999) showed that the REIT betas
were time varying, with rising with the periods of high volatility and bad market
conditions.

71
Methods
Multivariate generalized autoregressive conditionally heteroskedastic (MGARCH)
models are deployed to explore the stochastic behavior of financial time series and, in
particular, to explain the behaviors of the return volatility and covariance over time
(Bollerslev, et al., 1992). Most of early MGARCH models parameterize time-varying
covariance in the sprit of Bollerslev (1990) where correlation coefficients are assumed to
be constant over the sample period. Although setting all conditional correlations to be
constant greatly simplifies estimation, the assumption is neither theoretically justified nor
robust to the empirical evidence. Tse and Tsui (2002) and Engle (2002) propose a new
class of models that both preserve the ease of estimation of the Bollerslev’s constant
correlation model but make the conditional correlation matrix time-dependent. However,
econometric specifications that explicitly model asymmetry in conditional covariances
and, specifically, conditional correlations are far less common
24
. Very recently, Cappiello,
et al. (2006) advance dynamic conditional correlation (DCC) models further by
considering the asymmetry effect in the correlations discussed above.
This chapter uses the asymmetric generalized dynamic conditional correlation
(AG-DCC) model of Cappiello et al. (2006) to model the dynamic conditional
correlations in stock, bond and foreign exchange markets. The AG-DCC process extends
previous specifications along two dimensions: it allows for series-specific news impact
and smoothing parameters and permits conditional asymmetries in correlation dynamics.

24
There exist studies that account for asymmetry in conditional covariances, see, for instance, Koutmos and
Booth (1995), Booth, et al. (1997), Scruggs (1998), and Christiansen (2000), however in their model
specifications, the correlation coefficients are assumed to be constant over the sample period.

72
The AG-DCC specification is well suited to examine correlation dynamics among
different asset classes and investigate the presence of asymmetric responses in
conditional variables and correlations to negative returns.
The AG-DCC model can be written as follows:

1
~(0, ),
tt t
rH
−
ℑ T t ,..., 1 = (1)
,
tttt
HD D = Γ (2)
where
t
r be a 1 n × vector of asset returns, which is assumed to be conditionally normal
with mean zero and covariance matrix
t
H ; and
1 t −
ℑ is the time 1 t − information set, and
t
D is the nn × diagonal matrix of time-varying standard deviations from univariate
GARCH models with
it
h on the i th diagonal, and
t
Γ is the time-varying correlation
matrix.
The AG-DCC model is designed to allow for three-stage estimation of the
conditional covariance matrix,
t
H . In the first stage, univariate volatility models are fit
for each of the assets and estimates of
it
h are obtained. As Engle and Sheppard (2001)
indicate, any univariate GARCH process that is covariance stationary and assumes
normally distributed errors can be used to model the variances. However, since the
conditional variance is an asymmetric function of past innovations, which increases
proportionately more after a negative than after a positive shock of the same magnitude,
the so-called asymmetric effects thus becomes another important issue in the applications

73
of the univariate GARCH models. Asymmetric GARCH models include Nelson’s (1991)
exponential GARCH model, Glosten et al. (1993)’s GJR-GARCH model and Zakorian’s
(1994) threshold-GARCH model. While in this chapter I tried and compared with all
these asymmetric models and the GJR-GARCH model specification is selected according
to the Bayesian information criterion (BIC).
The conditional variances follow a univariate GJR-GARCH (1, 1) specification:
22
,0 ,1 ,1 , 1 ,1 ,1
[0]
it i i it i i t i t i it
hI h ααε δ ε ε γ
− −− −
=+ + < + , n i ,..., 2 , 1 = (3)
where
,1 i
α measures the ARCH effect. The persistence of volatility (i.e. GARCH effect)
is measured by
i
γ . The unconditional variance is finite if 1
i
γ < ;
i
δ is the coefficient that
measures the asymmetric (leverage) effect;
,1
[0]1
it
I ε
−
< = if the innovation in last period
is negative,
,1
0
it
ε
−
< and otherwise
,1
[0]0
it
I ε
−
< = .
In the second stage, after the GJR-GARCH model are estimated, the standardized
residuals, /
it it it
rh η = , are used to estimate the parameters of the dynamic conditional
correlations, as follows:
''
11 1 1 1
(' ' ' ) ' ' '
tttttt
QAABBGNGA AGnnGBQB ηη
−− − − −
=Γ−Γ −Γ − + + + (4)
*1 *1
tt tt
QQQ
− −
Γ= (5)
where
'
[]
tt
E ηη Γ= , ] [
'
t t
n n E N = , A , B and G are n n × parameter matrices,
[0]
tt t
nI η η =< o ( [] I ⋅ is a 1 n × indicator function which takes on value 1 of the argument

74
is true and 0 otherwise, while “ o ” indicates the Hadamard product).
*
12
{ , ,... }
tttnt
Qdiag q q q = is a diagonal matrix with the square root of the i th
diagonal element of
t
Q on its i th diagonal position. Here as log as
t
Q is positive definite,
*
t
Q is a matrix which guarantees
*1 *1
tt tt
QQQ
− −
Γ= is a correlation matrix with ones on the
diagonal and every other element ≤ 1 in absolute value. Furthermore, a sufficient
condition for
t
Q to be positive definite for all possible realizations is that the intercept
parameters, (' ' ' ) AA BB GNG Γ− Γ− Γ− in Equation (4) is semi-definite and the initial
covariance matrix
0
Q is positive definite (see Ding and Engle, 2001 for further details).
Finally, the third stage conditions on the correlation intercept parameters
constraint to estimate the coefficients governing the dynamics of correlation. The
parameters are estimated by the Maximum Likelihood method assuming that the assets
returns are conditional Gaussian. However, because the AG-DCC generalization comes at
the cost of added parameters and complexity, which actually require
2
n parameters in
each correlation term, two simplified modifications of AG-DCC models are commonly
used as follows:
1) The asymmetric DCC model (A-DCC), where the matrices A , B and G are
replaced by scalars a ,b and g respectively. The Equation (5) will be reduced to
22 2 2 ' 2 ' 2
11 1 1 1
()
tttttt
Qa b gNa gnn bQ ηη
−−−− −
=Γ− Γ− Γ− + + + (6)

75
A necessary and sufficient condition for that to be
t
Q positive definite is

22 2
1 ab g δ + +< (7)
where δ = the maximum eigenvalue of
1/ 2 1/ 2
N
−−
ΓΓ .
2) The asymmetric diagonal DCC model (AD-DCC), where the matrices A ,
B and G are assumed to be diagonal. The AG-DCC representation reduces to
'' ' ' ' ' ' ' '
11 1 1 1
()
tttttt
Q ii aa bb N gg aa gg n n bb Q ηη
−−−− −
=Γ − − − + + + ooo o o (8)
where i is a vector of ones and a ,b and g are vectors containing the diagonal elements
of the matrices A , B andG , respectively. In this case, a sufficient condition for
t
Q to be
positive definite for all t is that the intercept,
'' ' '
() ii aa bb N gg Γ− − − oo is positive
semi-definite.

Data and Empirical Results
The Data
The empirical tests conducted in this chapter utilize the monthly return data on REITs,
common stocks and unsecuritized (private) real estate in the U.S market. The sample
ranges from Jan 1987 to May 2008, with 257 observations for each return time series.
The National Association of Real Estate Investment Trusts (NAREIT) indices including
the composite REIT index, the equity REIT index, the mortgage REIT index and the

76
hybrid REIT index are used in this study, allowing an analysis of not only the overall
REIT market but also the Equity, Mortgage, and Hybrid sub-markets. As the most widely
known stock index, the S&P 500 composite index is used as a proxy for the stock market.
The direct real estate price data is from the S&P/Case-Shiller® Home Price Indexes,
which utilize the weighted repeated sales methodology and only use transaction data on
single-family home re-sales to measure the real estate returns in the U.S. market. There
are other two major housing indexes commonly used in the literature to track the
performance of the U.S. housing market: the National Association of Realtors (NAR)
Indexes and the Office of Federal Housing Oversight (OFHEO) Indexes. However, the
NAR Indexes quote median values without recourse to a repeat sales methodology, which
creates a significant potential for bias, while the OFHEO indexes are confined to the
Fannie Mae and the Freddie Mac conforming mortgages, which are skewed to the lower
end of the housing market. This is a significant issue because, for instance, only
approximately one-sixth of housing in California is sold with a conforming mortgage. In
addition, the OFHEO indexes also utilize refinance appraisal data to supplement their
samples, which creates the possibility of bias that have an appraisal smoothing effect on
persistence of housing returns. Thus using monthly Case-Shiller Indexes should exorcise
the criticism for that appraisal smoothing effect. In this chapter, all the price index data,
t i
P
,
, at time t is transformed into the continuously compounded monthly returns as
follows:

77
) / log(
1 , , , −
=
t i t i t i
P P R , (9)
where
t i
P
,
and
1 , − t i
P are the closing prices of asset i at time t and 1 − t , respectively.
To capture the economic determinants of conditional correlations among the
assets, this paper uses those macroeconomic variables widely used in finance literature
which include the unemployment rate, the term spread (the difference between the yields
on 10-year and 1-year Treasuries), the credit spread (the difference between the yields on
BAA-and AAA-rated corporate bond), Consumer Price Index (CPI) inflation rate and 3
month Treasure bill rate (e.g., Campbell and Shiller, 1988; Fama and French, 1989;
Torous, Valkanov and Yan, 2005). All these data, except the 3-month Treasury bill rate,
are taken from the FRED database. The 3-month Treasury bill rate is obtained from
Ibboston Associates.

78
Table 11: Descriptive Statistics of the Respective Series 1987-2008
Period Jan 1987 to May 2008

Composite
REITs
Equity
REITs
Mortgage
REITs
Hybrid
REITs
S&P 500
Case&Shille
r HPI
Panel A: Descriptive
Statistics

Mean 0.0078 0.0092 0.0027 0.0032 0.0068 0.0041
Median 0.0097 0.0108 0.0098 0.0084 0.0111 0.0047
Standard
Deviation
0.0381 0.0391 0.061 0.0523 0.0428 0.008
Minimum -0.1656 -0.1653 -0.2758 -0.2249 -0.2454 -0.0285
Maximum 0.095 0.1038 0.1325 0.1928 0.1238 0.0227
Skewness -0.8019 -0.7432 -1.475 -0.9303 -1.1532 -0.6923
Kurtosis 5.349 5.2506 7.4193 6.9362 7.7786 4.6088
Sharpe Ratio 0.2047 0.2352 0.0443 0.0612 0.1589 0.5125
Panel B: Correlation Matrix
Composite
REITs
1 0.9849 0.6035 0.6873 0.4423 0.0617
Equity REITs 1 0.5120 0.6312 0.4500 0.0619
Mortgage
REITs
1 0.6519 0.3246 0.0604
Hybrid REITs 1 0.3497 0.1355
S&P 500 1 -0.0154
Case&Shiller
HPI
1
Note: Panel A in this table reports summary statistics for the 6 indexes returns used in this
paper. The standardized skewness and kurtosis are the skewness and kurtosis of the returns
standardized by their estimated standard deviation.
Panel B reports the unconditional correlations of indexes returns.

The panel A in Table 11 contains descriptive statistics for the data which exhibit
the expected properties of financial returns and real estate returns. All markets exhibit an
average positive monthly return with left skewness and fat-tails. As expected, the private
real estate has the highest Sharp ratio due to the least volatility/standard deviation of
returns. Without considering other factors such as transaction cost, direct real estate asset
do have a definite role in the formation of efficient portfolio. That’s why Webb, et al.

79
(1988) suggest that two-thirds of investment wealth should be allocated to real estate and
only one-third to financial assets. During the sample period, all REITs, except the hybrid
REIT sub-sector, are less volatile than the S&P 500 stock index; while the ex post
composite and equity REITs returns are higher than stocks. This reflects some kind of
superior risk and return characteristics of equity REITs since the high-tech bubble bursts.
The panel B in table 2.1 summarizes the unconditional correlations for REITs, stock and
real estate returns. It is found that although both are positive, the unconditional
correlations between REITs and stock are much higher than the correlations between
REITs and direct real estate returns during the sample period. This suggests that the
REITs and unsecuritized real estate returns have significantly different statistical
properties and that the degree of substitutability of REITs for private real estate in static
mixed-asset portfolios is quite limited. This is consistent with the previous studies that
little connection was found between REIT investment performance and unsecuritized
income-property (e.g., Corgel, et al., 1995; Seck, 1996 and Seiler, et al. 2001). Among
the three sub-sectors of REIT markets, the correlation between the equity REIT and the
S&P 500 returns is the highest, nearly to 0.45. Surprisingly, the hybrid REIT return has
the higher correlation coefficient to the unsecuritized real estate than the other two REIT
sectors. Finally, the unconditional correlation between housing index and the S&P 500 is
slightly negative. The similar result is also found by Goetzmann (1993). The negative
correlation indicates again that housing returns has strong diversification benefits to the
stock asset portfolio.

80
Empirical Results
Estimated Dynamic Correlations. As described above, two different parameterizations
are estimated for the dynamics of the correlations among different asset classes. The first
and simplest model is an asymmetric scalar DCC (see Equation (6)). Second, the full
diagonal version of AG-DCC model (see Equation (8), with diagonal matrices A ,
B andG ). According to the Akaike’s information criterion (AIC), the diagonal
asymmetric DCC specification outperforms the competing model. Table 12 reports the
estimation results for these two DCC specifications. Most parameters are significantly
different to zero, with exceptions noted in the table. Log-likelihood values also suggest
that the diagonal version significantly outperform the scalar specification. Interestingly,
most asymmetric terms
i
g are not statistically significant in the model at 95% confidence
level. This suggests that the correlations among the assets classes tested in the model may
not therefore be higher after a negative innovation than after a positive innovation of the
same magnitude. Although for an individual asset return series, there still exist the
significant asymmetric volatility phenomenon in fitting uni-variate GJR-GARCH model
in the first step. My finding that the absence of significant asymmetry in the conditional
correlations among REIT, direct real estate and stock returns is quite different to the
previous evidences found in the capital market, where both equities and bonds exhibit
significant asymmetries in conditional correlations (e.g., Cappiello, et al. 2006;
Christiansen, 2000). The possible reason for this difference maybe attributed to the
segmentation between real estate market and stock market and the unique “hybrid”

81
characteristics of return and risk in REITs. Actually in addition to possible explanations
for asymmetries in return volatility, little theoretical framework is available to justify the
empirical evidence of asymmetric correlations.

Table 12: DCC GARCH models

i
a
i
b
i
g
NAREIT Composite REIT 0.2114 0.9705 0.0210
*

NAREIT Equity REIT 0.2116 0.9686 0.0013
*

NAREIT Mortgage REIT 0.2298 0.9709 0.0486
NAREIT Hybrid REIT 0.2245 0.9704 0.0299
Case & Shiller HPI 0.1270 0.8886 0.0517
*

S&P 500 0.1383 0.9762 0.0071
*

Scalar model 0.2068 0.9719 0.0000
Note: This table reports parameter estimates for asymmetric DCC GARCH models (diagonal
and scalar models). * indicates insignificance at the 5% level.

Figure 8 contains the dynamic conditional volatility series for REITs, direct real
estate and stock returns. Except for the direct real estate returns, the conditional volatility
linkages among other assets were most evident during some certain tumultuous periods
such as black Tuesday in October 1987, the Iraqi invasion of Kuwait, and the Gulf war in
1990/1991. Interestingly, most recently the volatility of the mortgage REIT and the
hybrid REIT sectors exhibit a spike as the same as what happened in the conditional
volatility in private real estate returns. This seems the volatility linkages in these three
groups are strengthening increasingly.

82
Figure 8: Conditional Volatility of REITs, Direct Real Estate and S&P 500

Figure 9 plots the estimated correlations between REITs and private direct real
estate returns. The correlation is not high and fluctuates in a narrow range from -0.12 to
0.25. However, recently since 2007, the correlation is increasing. The pattern is generally
consistent with the previous findings that the correlation between REIT and direct real
estate is limited and recently the REIT can reflect more fundamentals in the direct real
estate market. The time-varying conditional correlations between REITs and stock
returns are exhibited in Figure 10. The correlations of the assets under consideration
show considerable variation. My findings actually resolve the mixed empirical evidences
about the correlations between REITs and stock returns. For instance, Ghosh, et al., (1996)
88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08
1
2
3
4
5
6
7
x 10
-3 FTSE NAREIT All REITs
Volatility
Year
88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08
1
2
3
4
5
6
x 10
-3 FTSE NAREIT Equity REITs
Volatility
Year
88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08
0
0.005
0.01
0.015
FTSE NAREIT Mortgage REITs
Volatility
Year
88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08
0
0.005
0.01
0.015
0.02
FTSE NAREIT Hybrid REITs
Volatility
Year
88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08
0
0.2
0.4
0.6
0.8
1
x 10
-3 Case&Shiller HPI
Volatility
Year
88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08
0
0.002
0.004
0.006
0.008
0.01
S&P 500
Volatility
Year

83
find that the correlations between REIT returns and S&P 500 declined over time
from1985 to 1996; while Mueller, et al. (1994) find that REIT returns are shown to
exhibit strong positive correlations with the S&P 500 from 1976 to 1993; Mull and
Soenen (1997) find that REITs is a good diversification vehicle in the 1990–1994 period,
they were not as attractive in the period 1985–1990. All of these empirical results are
consistent with the dynamic correlation patterns plotted in the Figure 10. In addition,
Figure 9 and Figure 10 show the obvious cyclical nature of the REIT returns sensitivities
to financial assets and unsecuritized real estate, which also reaffirms the findings by
Calyton and Mackinnon (2001).
Figure 9: Dynamic Correlation between the REITs and Direct Real Estate

88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08
-0.25
0
0.25
0.5
Correlation
Year

FTSE NAREIT All REITs-Case&Shiller HPI
FTSE NAREIT Equity REITs-Case&Shiller HPI
FTSE NAREIT Mortgage REITs-Case&Shiller HPI
FTSE NAREIT Hybrid REITs-Case&Shiller HPI

84
Figure 10: Dynamic Correlation between the REITs and S&P 500

Determinants of Dynamic Correlations. To test whether the dynamic conditional
correlations are determined by the economic fluctuations, I run the following regression
models for each return series separately:

,,10, 1,1 , it i i t i t i t t i t
Corr Corr X X d α ββ ε
−−
=+ + ++ (10)
where
, it
Corr denotes the conditional correlation between REIT i and other asset at time
t , the parameter
i
α is the auto-regression parameter which reflect how
,1 it
Corr
−
at time t -
1 affect the current correlation value
, it
Corr ; the parameters
0,i
β and
1,i
β are vector
parameters which measure the sensitivities of the conditional correlations to the current
88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08
0
0.25
0.5
0.75
Correlation
Year

FTSE NAREIT All REITs-S&P 500
FTSE NAREIT Equity REITs-S&P 500
FTSE NAREIT Mortgage REITs-S&P 500
FTSE NAREIT Hybrid REITs-S&P 500

85
and lagged macroeconomic variable vector
t
X . I also include a time dummy variable to
constraint the seasonal effect.
Since the macroeconomic variables always tend to correlate each other, I select
the best set of macroeconomic determinants into the regression by utilizing the Sequential
Elimination of Regressors approach to decide on possible constraints. This strategy
sequentially deletes those regressors which lead to the largest reduction of the selected
criterion until no further reduction is possible (see, Bruggemann and Lutkepohel, 2001
for more details).
Table 13 reports the results of estimation. Firstly, I find that the conditional
correlations between REITs and both unsecuritized real estate and stock returns are quite
persistent over the sample time period. The coefficient on lagged conditional correlation
keeps significance at a 1% level. Secondly, the results indicate the correlations are
significantly affected by the macroeconomic variables, although the best set of economic
regressors is a little different across different REIT sub-sectors. For example, the
correlation between equity REITs and direct real estate returns is mainly determined by
the lagged CPI inflation rate with the corresponding coefficient is statistically significant
at the 1% level. While as for the correlation between equity REITs and S&P 500 index,
the credit spread, the term spread and the unemployment rate enter the regression after
the SER subset selection. Those coefficients are also statistically significant at the 5%
level or better. Thirdly, I find the distinct effects of economic variables on the dynamics
in the correlations. The effects of the term spread on the correlation are always negative,

86
while the credit spread, CPI inflation rate and the unemployment rate always have
positive effects on the correlations. In addition, the magnitudes of those coefficients are
also different. Finally, the adjust R
2
coefficients of determination are around 0.9 for all
those regressions and indicates our models fit the data quite well.

Table 13: Economic Determinants of Dynamic Correlations

EREIT
and HPI
EREIT
and Stock
MREIT
and HPI
MREIT and
Stock
HREIT
and HPI
HREIT
and
Stock
Correlation Lag 1 0.938
***
0.918
***
0.969
***
0.931
***
0.955
***
0.957
***
Lag
0
--- 0.039
***
--- --- 0.028
**
--- Credit Spread
Lag
1
--- --- --- --- --- ---
Lag
0
--- --- --- --- 0.010
*
--- Inflation Rate
Lag
1
0.010
***
--- --- --- --- ---
Lag
0
--- -0.011
**
--- -0.009
**
--- --- Term Spread
Lag
1
--- --- --- --- --- ---
Lag
0
--- --- --- --- -0.021
*
--- Treasure Bill
Lag
1
--- --- --- --- 0.020
*
---
Lag
0
--- --- --- --- --- --- Unemployment
Rate
Lag
1
--- 0.013
**
--- 0.008
***
--- 0.003
**
2
adjust
R

0.8906 0.9384 0.9269 0.8989 0.9556 0.9318
Notes: 1. EREIT, MREIT and HREIT represent the equity REIT, mortgage REIT and hybrid REIT,
respectively; HPI denotes the Case & Shiller housing price index; Stock denotes the S&P 500 stock
index. 2. Significance at the 10%, 5% and 1% level is denoted with one, two and three asterisks,
respectively. And --- denote zero restriction is posed in this coefficient. The R
2
adj
goodness of fit
measure is also displayed.

Expected Returns and the Dynamic Correlations. Most financial decisions involve
a trade-off between risks and asset returns. The volatilities and correlations of assets

87
returns are often the major components of risk (Cappiello, et al., 2006). Since the
correlations between REITs and both private real estate and stocks evolve over time as
the economy changes, the time-varying conditional second moments may demand to be
compensated by returns. To understand the role of conditional correlations of REITs on
its future returns, I examine the historical impacts of the conditional correlations on REIT
sector performances via a non-parametric test. For each REIT sub-sector, the estimated
conditional correlations (between REIT and direct real estate returns, and between REIT
and stock returns, separately) are grouped into four regimes which are determined by
equally dividing the range of correlations during the sample time period, while the regime
ranking is also recorded for each month. Then using the re-sampled monthly correlations
data in every regime, I calculate the conditional probability of the loss and gain, as well
as the expected gain or loss for the next month.

88
Table 14: Relationship between Equity REIT Returns and Conditional Correlations
(Monthly Data: Jan, 1987- May, 2008)

* Risk free rate is assumed to be 3 % (annual)

Table 14 contains the results about the relationship between the equity REIT and
the conditional correlations. As for the correlation between equity REIT and direct real
estate represented by the Case & Shiller HPI, I find there is strong impact of the
correlation on the equity REIT returns. For example, as shown in the first part of Table
14, when the correlation is negative (the first regime -0.104 to -0.03) the odds are high
(about 53% monthly) that investors will only gains of 0.8% monthly which is much lower
than the average return of equity REIT (1%, see Table 11). However, as the correlations
increase from the first regime to the fourth regime, the expected return of equity REIT
also increase from 0.8% to 1.78% accordingly. These findings suggest that when the
Correlation between Equity REIT and Private Real Estate (Case & Shiller HPI)
Correlation % Chance % Chance If Up If Down Expected Volatiliy
Sharpe
Ratio
Regime
Range Up Month
Down
Month
Avg
Gain
Avg
Loss
Gain/
Loss
Next
Month
Next
Month
1
st

-0.104 - -
0.030
47.37% 52.63% 4.35% -2.39% 0.80% 4.33% 0.1072
2
nd
-0.030 - 0.043 61.39% 38.61% 3.11% -2.51% 0.94% 3.59% 0.1730
3
rd
0.043 - 0.116 64.71% 35.29% 3.21% -2.68% 1.13% 3.56% 0.2274
4
th
0.116 - 0.190 62.50% 37.50% 6.50% -6.08% 1.78% 7.53% 0.1592
Correlation between Equity REIT and Stock (S&P 500)
Correlation % Chance % Chance If Up
If
Down
Expected Volatility
Sharpe
Ratio
Regime
Range Up Month
Down
Month
Avg
Gain
Avg
Loss
Gain/Los
s
Next
Month
Next
Month
1
st
0.244 - 0.365 67.57% 32.43% 3.30% -2.46% 1.43% 3.51% 0.3186
2
nd
0.365 - 0.486 61.36% 38.64% 3.58% -3.05% 1.02% 4.11% 0.1657
3
rd
0.486 - 0.607 54.17% 45.83% 3.42% -2.52% 0.70% 4.08% 0.0876
4
th
0.607 - 0.728 55.26% 44.74% 2.71% -1.60% 0.78% 3.09% 0.1598

89
correlations between equity REITs and unsecuritized real estate is positively higher, the
performance of equity REIT is better. The second part in Table 14 indicates that the
correlation between equity REIT and S&P 500 has different impact on the equity REIT
returns. Contrary to the relationship between equity and direct real estate, here I find that
when the correlation between equity REITs and stock is low, the expected gain of REIT
is high, except in the fourth regime. However, Figure 10 shows that the fourth regime
(high correlation with stocks) actually took place only during 1988 -1989. In this way, I
find that the higher (lower) the relationship between equity REITs and unsecuritized real
estate (stock), the better the performance of REITs exhibits. To test whether or not equity
REITs is a superior diversifier to the stock market given the correlations between equity
REITs and the S&P 500 is time-varying; I calculate the S&P 500 return at the four
different regimes of the correlation between equity REIT and S&P 500. The results are
presented in Table 15. I find that both the return and sharp ratio of the equity REITs are
much better (the 1
st
and 3
rd
regimes) or comparable (the 2
nd
regime) to S&P 500.
Considering the fourth regime is really an exception, which is mainly due to the stock
market crash of 1987, my findings suggest that although the correlations between equity
and S&P 500 returns are time-varying, the equity REIT is still persistently a superior
diversifier to the S&P 500 portfolio.

90
Table 15: Relationship between S&P 500 Returns and Conditional Correlations
Correlation between Equity REIT and Stock (S&P500)
Correlation
%
Chance
% Chance If Up
If
Down
Expected Volatility
Sharpe
Ratio
Regime
Range
Up
Month
Down
Month
Avg
Gain
Avg
Loss
Gain/Loss
Next
Month
Next
Month
1
st

0.244 -
0.365
48.65% 51.35% 2.81% -4.71% -1.05% 4.59% -0.3039
2
nd

0.365 -
0.486
66.67% 33.33% 3.05% -2.78% 1.11% 3.56% 0.2209
3
rd

0.486 -
0.607
64.58% 35.42% 3.05% -4.21% 0.47% 4.95% 0.0186
4
th

0.607 -
0.728
63.16% 36.84% 3.87% -2.68% 1.46% 4.03% 0.2792
* Risk free rate is assumed to be 3% (annual)

However, the relationships between the correlations and the future returns in
mortgage REITs and hybrid REITs are mixed. Table 16 reports the results of correlations
impact on the mortgage REIT. When the correlations between mortgage REIT and
private real estate are in two extreme regimes: the first regime and the fourth regime, the
expected gains are much higher than the average mortgage REITs. Table 17 presents the
results of correlations impact on the hybrid REIT. However, one interesting finding is
that when the correlation between REIT and S&P 500 is in the first regime (lowest
correlation), whatever equity REIT, mortgage REITs or hybrid REIT, the return in next
month are always highest in all four regimes. This implies that the lowest conditional
correlations between REIT and stock markets are a good predicator for the best future
returns in REIT. At last, it needs to be mentioned that I found that all the above
observations are still robust when one moves to quarterly data.

91
Table 16: Relationship between Mortgage REIT Returns and Conditional Correlations
(Monthly Data: Jan, 1987- May, 2008)

Correlation between Mortgage REIT and Private Real Estate (Case & Shiller HPI)
Correlation
%
Chance
% Chance If Up
If
Down
Expected
Volatili
ty
Sharpe
Ratio
Regime
Range
Up
Month
Down
Month
Avg
Gain
Avg
Loss
Gain/Loss
Next
Month
Next
Month
1
st

-0.110 - -
0.024
67.57% 32.43% 4.53% 4.44% 1.62% 5.22% 0.2315
2
nd

-0.024 -
0.062
53.42% 46.58% 3.54% 3.99% 0.03% 5.23% -0.0662
3
rd

0.062 -
0.150
63.27% 36.73% 4.78% 6.98% 0.46% 7.36% -0.0102
4
th

0.150 -
0.230
65.22% 34.78% 4.88% 5.37% 1.31% 6.85% 0.1094
Correlation between Mortgage REIT and Stock (S&P 500)
Correlation
%
Chance
%
Chance
If Up
If
Down
Expected Volatility
Sharpe
Ratio
Regime
Range
Up
Month
Down
Month
Avg
Gain
Avg
Loss
Gain/Loss
Next
Month
Next
Month
1
st

0.148 -
0.273
64.29% 35.71% 5.87% 3.92% 2.37% 5.47% 0.3629
2
nd

0.273 -
0.398
60.44% 39.56% 4.05% 4.60% 0.63% 5.87% 0.0319
3
rd

0.398 -
0.523
61.11% 38.89% 3.73% 5.43% 0.16% 6.00% -0.0434
4
th

0.523 -
0.648
45.24% 54.76% 4.69% 3.48% 0.22% 5.62% -0.0332
* Risk free rate is assumed to be 3%(annual)

92
Table 17: Relationship between Hybrid REIT Returns and Conditional Correlations
(Monthly Data: Jan, 1987- May, 2008)

Correlation between Hybrid REIT and Private Real Estate (Case & Shiller HPI)
Correlatio
n
%
Chance
%
Chance
If Up If Down Expected Volatility
Sharpe
Ratio
Regime
Range
Up
Month
Down
Month
Avg
Gain
Avg
Loss
Gain/Loss
Next
Month
Next
Month
1
st

-0.120 - -
0.032
55.77% 44.23% 4.02% 5.35% -0.12% 6.13% -0.0901
2
nd

-0.032 -
0.054
61.67% 38.33% 2.76% 2.74% 0.65% 3.91% 0.0830
3
rd

0.054 -
0.141
56.36% 43.64% 3.83% 4.11% 0.37% 5.08% -0.0027
4
th

0.141 -
0.228
60.71% 39.29% 4.80% 5.24% 0.86% 7.39% 0.044
Correlation between Hybrid REIT and Stock (S&P 500)
Correlation % Chance % Chance If Up
If
Down
Expected Volatility
Sharpe
Ratio
Regime
Range Up Month Down Month
Avg
Gain
Avg
Loss
Gain/Loss
Next
Month
Next
Month
1
st

0.096 -
0.243
70.00% 30.00% 4.80%2.26% 2.69% 4.52% 0.5306
2
nd

0.243 -
0.390
60.27% 39.73% 3.62%5.71% -0.09% 5.99%
-
0.0846
3
rd

0.390 -
0.538
57.98% 42.02% 2.89%3.74% 0.10% 4.86%
-
0.0530
4
th

0.538 -
0.685
43.48% 56.52% 2.83%1.98% 0.11% 3.51%
-
0.0563

93
Conclusion
This chapter utilizes the AG-DCC GARCH model to explore correlation dynamics
between REITs and other two important assets: unsecuritized real estate and stock returns.
While a growing body of literature has shown both equities and bonds exhibit asymmetry
in conditional correlation, however, this study finds little asymmetry in the dynamic
conditional correlations among REIT, direct real estate and stock returns. Using the
monthly data of REITs, direct real estate and stock returns, this chapter finds the time-
varying conditional correlations in REITs can be explained by the macroeconomic
variables. Since the correlations are the main component of risk, this chapter also
investigates whether the future REIT return is related to the correlations between REITs
and stock returns/underlying direct returns. Although the patterns are distinguishable for
different type of REITs, generally there is strong relationship between conditional
correlations and future returns. Interestingly, I find that when the correlation between
REITs and S&P are lowest, the future performance of REITs is best. For equity REITs,
there exist strong relationships between correlations and future returns: the higher (lower)
correlation between equity REIT and direct real estate is, the higher (lower) the future
returns of equity REIT.
My results have important economic motivations and implications. First, this
chapter offers the direct evidence about the time-varying diversification power of REIT
stocks in a mixed-assets portfolio. Moreover, it concerns the potential integration of
REIT, direct real estate, and stock markets and the possibility of including information

94
about conditional correlations to design more optimal portfolio using REITs assets.
Second, my findings also build upon the literature dealing with REITs’ “hybrid” risk
characteristics and return predictability using conditional correlations with underlying
real estate and stocks return. Moreover, my findings give the first analysis of the
macroeconomic impacts on the dynamics in the conditional correlations and volatilities
in REITs. Finally, I find that although the correlations between equity REITs and S&P
500 returns are time-varying, the equity REIT is still persistently a superior diversifier to
the S&P 500 portfolio since 1990.

95

Chapter 3:

Persistence of U.S. Housing Returns: A Markov Chain Analysis

Introduction
The key stylized fact of the housing market is that intertemporal changes in house returns
exhibit strong persistence. Figure 11 plots real and nominal quarterly house returns for
the U.S., based on house price index data published by the Office of Federal Housing
Enterprise Oversight, which illustrates clearly that the positive housing returns tend to be
followed the positive ones, and vice versa for negative returns.

96
Figure 11: U.S Nominal and Real Housing Price Index (HPI) Changes
Source: Quarterly Data from BLS and OFHEO

The persistence of housing returns in the U.S. housing market deserves attentions
for several reasons. First, under the efficient market hypothesis, asset prices will adjust
immediately to reflect new information about fundamental value, not gradually over time
(see e.g., Meese and Wallace, 1994 for some important empirical work on this issue),
while persistence in housing returns apparently stands in stark contrast to the notion of an
efficient housing market. This persistence is usually explained in the sense that housing
market takes time to clear due to the frictions, such as high search and transaction costs.
Or it takes time for developers to bring new houses to market after an increase in demand
-3%
-2%
-1%
0%
1%
2%
3%
4%
1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006
Nominal HPI Change Real HPI Change

97
and to work off inventories when demand weakens (See, e.g., Englund and Ioannides,
1997; Lee and Ward, 2001). An alternative explanation for the persistence in house
returns advanced by Krainer (2002) who suggested that housing price dynamics depend
directly on macro economic variables, such as employment growth and changes in
personal income. It is well documented that returns of these macro economic variables
are persistent, so the housing returns also exhibit persistence. Others suggest that
persistence in housing returns results from the appraisal-based total return calculation.
Hence the persistence in housing return is a reflection of appraisal smoothing effect. In a
short, the economic explanations for the observed persistence in housing returns have not
reached a consensus among housing economists; so that underlying mechanism for
housing return persistence still needs to be studied further.
Moreover, the phenomenon of performance persistence in housing returns itself is
not understood thoroughly. The persistent housing return tends to offer a compelling
investment strategy for investors and homebuyers to purchase housing that has performed
well in the past and sell it when the housing market has performed poorly in prior periods.
The validity of such an investment strategy, however, depends on the magnitude and the
stability of the successive period performance relationship over time. However, most of
the existing literature is based on the ordinary linear parametric models (see e.g.,, Case
and Shiller, 1988, 1990; Meese and Wallace, 1994; Quigley, 1999), which may render
some spurious results (Young and Graff, 1996, 1997), that ignore the important dynamics
in the housing market observed over time.

98
Finally, owner-occupied houses compose a major part of private sector wealth in
the U.S. Almost two-thirds of the typical household’s portfolio in the United States is
represented by housing (Tracy, et al., 1999). According to Federal Reserve Board data
(Federal Reserve Statistical Release, June 5, 2008), by the end of first quarter, 2008,
United States has more than $22 trillion of real estate assets, almost $20 trillion of them
are residential in nature owned. Hence, housing return patterns do have a major impact on
the distribution of economic well-being and they should also be important in explaining
time-varying behaviors of household saving and consumption. There have been growing
amount of literatures addressing the effects of housing on household’s savings and
portfolio choices
25
.
The purpose of this chapter is to investigate the magnitude of housing return
persistence and its implication to the U.S. housing market. There has been a growing
volume of study analyzing the speculative bubbles in the U.S. housing market, which
suggests the possibility of nonlinear pattern exists in housing returns. The approach
adopted in this chapter is to model the dynamics of housing returns by a Markov chain
process that estimates transitional probabilities from one state to another via the
maximum likelihood estimation method (MLE)
26
. I also test the stationary of this
transitional probability using the likelihood ratio test (LRT). Given a panel data of

25
See Cocco (2005), Hu (2005), Bond et al. (2003), Campbell and Cocco (2003), Kullmann and Siegel
(2003), Davidoff (2002), Flavin and Yamashita (2002), Gu (2002), Capozza et al. (1997), Goetzmann
(1993), and Goetzmann and Ibbotson (1990) among others.

26
The modeling approach adopted in this paper is related to McQueen and Thorley (1991) and Lee and
Ward (2001).

99
housing price indexes, I define a Markov Chain { }
t
I by letting one state represent “high”
returns and the other represent “low” returns. Generally there are two ways to determine
whether housing return is “high” or “low”: one of them is based on time series approach
and the other is cross-sectional approach. Majority of the prior studies choose the former.
Based on the theoretical arguments made earlier and the empirical evidence from the data,
this chapter utilizes both time series and cross-sctional approaches to determine return’s
states in the Markov Chain process. In contrast to traditional time series based tests, the
advantage of the Markov Chain approach is that it addresses nonlinearity by allowing the
parameters (transition probabilities) to vary depending on a given sequence of prior states.
For example, this chapter estimates the probability of a below-average year given that
both prior years are above average. Moreover, the Markov chain methodology allows
relaxing the relatively stringent assumptions that are traditionally assumed. The Markov
chain analysis does not require housing returns to be normally distributed although it
does require the transition probability matrix to be stationary
27
, for example. Unlike the
previous tests such as Young and Graff (1996, 1997) and Graff et al. (1999), where only
annual data of housing returns is analyzed for persistency, this chapter utilizes the
housing data for different sample frequencies: annual returns, quarterly returns and
monthly returns, to test whether persistence behaviors of housing returns display
qualitatively and quantitatively distinct patterns across the test intervals since quite a few

27
Myer and Webb (1993, 1994), Young and Graff (1996) and Byrne and Lee (2001) have found that
housing returns are not normally distributed.

100
studies have shown that return persistence appears to be sensitive to the data frequency
employed.
28

This chapter provides additional evidence on serial correlation in house returns in
a nonlinear way. Using the MSA level data, this chapter finds the persistence behaviors
of housing returns exhibit significant spatial heterogeneity. Some MSAs have more
persistent and stable patterns of rerun dynamics than others. Furthermore the results show
the persistence patterns of housing return are significantly affected by sample frequencies.
The housing price data used in this study is the S&P/Case-Shiller® Home Price Index
which has been discussed recently as a tool for investment in the housing price index
derivatives (futures/options) market.
The remainder of the chapter is organized as follows: The following section,
Related Research, discusses the related research on the persistence of housing returns, the
section entitled Methodology discusses the Markov Chain model used in this study, and
Data and Empirical Results describes the data and reports the results of the transitional
probabilities and the likelihood ratio tests for annual, quarterly and monthly returns. The
conclusions are presented at the last section.

28
The empirical evidences are found both in capital market (see, e.g., Lo and MacKinlay, 1988; McQueeen
and Thorley, 1991; Fama and French, 1988; Kim et al., 1991) and in real estate market (see, e.g., Lee, 2002;
Graff and Young, 1997; Lizieri, and Ward, 2000).

101
Related Research
The issue of persistence in housing returns has been examined in a lot of studies. The
most widely quoted studies are those by Case and Shiller (1989, 1990). Case and Shiller
(1989) uses repeat transaction indices at the metropolitan area level to test whether
housing market are efficient, in the sense that excess housing returns are not forecastable.
Their results show that although individual house price changes are not very forecastable,
the year-to-year changes in housing price indices exhibit significant persistence. While
based on the quarterly housing panel data from 1970 to 1986, Case and Shiller (1990)
reports strong evidence of positive autocorrelation at short lags and weaker evidence of
negative autocorrelation at longer lags. Using the San Francisco area housing return data,
Meese and Wallace (1994) find consistently positive first-order autocorrelations ranging
from 0.15 to 0.64. Quigley (1999) finds most of the variation in log housing prices can be
predicted by previous price movements. A one- and two-period lag of housing price
explains more than 96 percent of the variation, however, the models have no power to
predict the “turning points” of housing prices, the specific periods in which price declines
reverse and when “bubbles” burst. The positive autoregression pattern in housing returns
found in the U.S market also holds in other countries which are reported by some studies
such as Hort (1995), Englund and Ioannides (1997), and Muellbauer and Murphy (1997).
Generally these studies are based on linear time series modeling of housing returns in the
sense that they relate all the observations in a series to the values of prior observations
using a linear function, which may render the spurious results and the spatial
heterogeneity in the persistence patterns of housing returns is also ignored.

102
More recently, instead of utilizing the linear parametric modeling, Young and
Graff (1996, 1997) and Graff, et al. (1999) have conducted non-parametric tests on the
persistence of housing returns. They calculated the returns of each house for each time
period, and the returns were grouped into quartiles while the quartile ranking was also
recorded for each period. As a result, the theoretical probability of repetitive quartile
rankings is 25% if the performance of housing return through time is serially independent.
Thus the statistically significant departure from 25% is deemed evidence of performance
persistence. Using annual returns from the NCRRIF over 1978 to 1994, Young and Graff
(1996, 1997) find that for the two extreme quartiles, the highest and the lowest ranks,
serial persistence was demonstrated with almost complete certainty from one year to the
next. In contrast, there is virtually little or no significant evidence to support persistence
in the second and third quartiles. Their results suggest that strong performance is
generally followed by continued strong performance, and weak performance is generally
followed by continued weak performance. The similar results are also found in Australia
real estate market (Graff et al., 1999) as well as securitized real estate market in the U.S.
(Graff and Young, 1997). These results motivate utilizing the more rigorous Markov
Chain approach to test the persistence of housing returns that condition on specific
sequence of “high” or “low” states systematically.

103
Methodology
In this section, a two-state Markov chain model is applied to analyze the persistence of
the annual, quarterly and monthly housing returns at MSA level respectively. The
Markov process was introduced around 1907, but did not come into general use till 1980s.
The process assumes that if the returns
it
R can be classified into two “states”, here
namely, “high” and “low” returns, the transitions of returns
it
R between these two states
over time can be regarded as a stochastic process. With a given set of states { }
it
S , it is
assumed possible to estimate the probability
it
P of returns
it
R moving from one state to
another. Let
it
R denote the housing return (monthly, quarterly, or annual) at time t in a
time series of returns for one MSAi , and let { }
it
S be given by

if 1
0
if
{
it i
it
it i
R
R
S
R
R
>
=
<
(1)
where i R is a measure to divide the return series into high and low return “states”. The
operation of a Markov chain model depends on the definition of a set of mutually
exclusive and comprehensive states. In most literatures the separation measure i R of
returns into two states is defined in a time series manner. For example, McQueen and
Thorley (1991) use the rolling average return of the prior 20 years as the benchmark to
measure high or low states for annual returns. The similar time series measures used to
determine the Markov chain states can also be found in Turner, et al. (1989), Engle and
Hamilton (1990), Gregory and Sampson (1987). However, the traditional time-series

104
measure in financial studies considers is conditional on its long-term means, recent
findings in real estate studies have indicated that the cross-sectional means of housing
returns maybe a more appropriate measure to predetermine the high and low states. For
housing market, due to the differences in the composition of local industry, the
distribution of household income, as well as other social-economic factors, such as
demographics, employments, etc, the effects of macroeconomic fluctuations and their
speeds of propagation will vary across metropolitan areas.
29
The Markov process states
series determined by the cross-sectional means of housing return across metropolitan
areas can capture fluctuations due to common aggregate factors as well as idiosyncratic
fluctuations arising from local shocks. In fact, the empirical studies conducted by Young
and Graff (1996, 1997) and Graff, et al. (1999) can also be regarded as some kind of the
cross-sectional way, which group housing returns into quartiles every time. The strong
persistence of housing return is found in the first and fourth quartiles. So from the
theoretical and empirical viewpoints, in this study, I define the i R not only in a
traditional time series way but also in a cross-sectional manner. Specifically for MSAi
the corresponding time series Markov Chain {}
it ST
is given by:

if 1
0
if
{
it i
it
it i
R
R
ST
R
R
>
=
<
(2)
where
i
R is the average housing return of MSA i over the sample time period. The cross
sectional Markov Chain {}
it SC
is determined as follows:

29
See, for example, Redfearn (2001); Carlino and DeFina (2003); Carlino and Sill (2001); Owyang, et al.
(2003); Fratantoni and Schuh, (2003), for more detailed discussion.

105

if 1
0
if
{
it t
it
it t
R
R
SC
R
R
>
=
<
(3)
where
t
R is the average housing return across all MSAs at time t .
The derived series { }
it ST
and { }
it SC
are two two-states Markov chains which
represent high returns as ones and low returns as zeros in both time series and cross-
sectional ways. To test the persistence of housing returns that conditions on specific
sequences of high or low return states, Markov chain models are estimated up to the third
order which estimates the probability of a below average (time series and cross-sectional)
given that three prior states are above average. For simplicity, here I take the second
order Markov chain model as an example. The methodology is similar to the third order
and the first order Markov chain models. Table 18 shows the forms of transition
probability matrix and transition counts matrix, where
00
N , …
11
M are the number of
observed occurrences of the associated transactions. For example,
N 00
indicates the
number of observations of the state if previous two states are 0 0, while
M 00
is the
number of observation of state 1 conditioning on previous states 0 0.The transition
probabilities for two-order Markov chain are defined as:
21
[0| , ]
ij t t t
Pi j
SS S S −−
== = = for } 1 , 0 { , ∈ j i

106
Table 18: Transition Counts Matrix and Transition Probabilities Matrix
Transition Count Matrix Transition Probability Matrix
Previous States Current State Previous States Current States

0 1 0 1
0 0
N 00

M 00
0 0
λ00

λ00
1 −
0 1
N 01

M 01
0 1
λ01

λ01
1 −
1 0
N 10

M 10
1 0
λ 10

λ 10
1 −
1 1
N 11

M 11
1 1
λ 11

λ 11
1 −
Note:
00
N , …
11
M are the number of observed occurrences of the associated transactions. For example,
N 00
indicates the number of observations of the state if previous two states are 0 0, while
M 00
is the
number of observation of state 1 conditioning on previous states 0 0.

The
λij
values can be estimated by the transition counts through maximum likelihood
estimation (MLE). Let ]' , , , [
11 10 01 00 λ λ λ λ
= Λ be the vector of transition probabilities.
Following Grant McQueen and Steven Thorley (1991), the likelihood function
) , ( π Λ L can be written as:
) 1 log( ) log( ) log( ) , (
11
00
λ λ
π π
ij ij
ij
ij ij M N
L − + + = Λ
∑
=
(4)
where π is the probability of the initial states and can be ignored if the size of time series
is large
30
. So ignoring the initial states, by setting the partial derivatives of the log

30
As noted by Nefti (1984), π can be ignored if the sample size T is large, so that the information
contained in π becomes negligible.

107
likelihood function equal to zero, that is to say, 0
) , (
=
∂
Λ ∂
λ
π
ij
L
, and solving for the four
parameters in terms of transition counts
ij
N and
ij
M , we get the maximum likelihood
estimate of the probability
λ ij
is
M N
N
ij ij
ij
+
.
If there is no persistence pattern in the housing returns, we can expect that the
probability of state 1 or state 0 will be the same for any prior two or three states sequence.
More specifically, the transition probabilities of a staying in a specific state for time
series Markov chain { }
it ST
or cross sectional Markov chain { }
it SC
would be no larger
than the chance of it changing to any other states, i.e. for the second order Markov chain
model, the null hypothesis is
λ λ λ λ 11 10 01 00
= = = . Moreover, to test the extent of
persistence the housing returns have, namely, whether a “low” return is more likely after
observing a sequence of three/two high returns or a sequence of three/two low returns, we
also test the null hypothesis that
000 111 λ λ
= and
λ λ 11 00
= .
The likelihood ratio test (LRT) is used to test the null hypothesis
λ λ 11 00
= , and
λ λ λ λ 11 10 01 00
= = = for the second order Markov chain model, as well as the null
hypothesis
000 111 λ λ
= for the third order Markov chain model:
χ
2
~ )]
~
( ) ( [ 2
n
L L LRT Λ − Λ = (5)
where ) ( Λ L and )
~
( Λ L are the log likelihood function using the unconstrained and
constrained MLE of the parameters, respectively. The likelihood ratio test is

108
asymptotically distributed with
2
n
χ with n degrees of freedom equal to the number of
restrictions.

Data and Empirical Results
The Data
The empirical analysis of this study is based on the S&P/Case-Shiller® Home Price
Index (CS HPI) from Jan 1987 to Jan 2008 instead of the OFHEO home price indices
traditionally used in the previous literatures (See, for example, Deng and Quigley, 2008;
Himmelberg, et al., 2005). Both OFHEO and CS housing price indices use the same
weighted repeated sales methodology. However, the OFHEO indexes do have some
intrinsic limits compared with the CS housing price indexes. First, the CS housing price
indexes use the real transaction home prices to establish the index while the OFHEO also
includes refinance appraisals which will result in the “appraisal smoothing bias” for
housing return measurement. This bias issue has been addressed in a substantial literature
(see, e.g., Giliberto, 1988; Geltner, 1989a, 1989b, 1991; Ross and Zisler, 1991; Edelstein
and Quan, 2005). Second, the OFHEO indexes are confined to Fannie Mae and Freddie
Mac conforming mortgages, which are concentrated towards the lower end of the housing
market. Third, the OFHEO index at MSAs level is quarterly issued while the CS index is
monthly based, which provide a unique opportunity to measure and model the housing
price changes in a shorter time interval. Finally, the CS indexes are currently used for

109
derivatives trading at the CME and the OTC market. Thus the study of CS home price
indexes will have more important implications for the emerging housing price index
derivatives market.
Recognized as one of the most authoritative and trustworthy home price change
measures, Case-Shiller Home Price Index tracks changes in the value of the residential
real estate market in 20 major metropolitan regions across the United States, and
consisted of these 20 indices, there are also two composite indices as aggregates of the 10
regions and 20 regions, respectively. Figure 12 maps these 20 metropolitan regions in the
S&P/Case-Shiller Home Price indices. The market value of housing stocks in these 20
major regions accounts for 42.5% of total value of the whole U.S housing market in
terms of 2000 census. Table 19 gives the details about the housing market values of the
covered metropolitans in the CS housing price indexes.

110
Figure 12: Metropolitan Regions in the S&P/Case-Shiller Home Price Indices

111

Table 19: S&P/Case-Shiller Home Price Indexes: MSA Coverage

• Aggregate Value for all Owner-Occupied Housing Units, 2000 Census

To correct the effect of inflation, all nominal monthly housing price indexes have
been deflated by the monthly CPI
31
. The analysis in this paper uses the real monthly S&P
Case/Shiller home price indices from Jan 1987 to Jan 2008. I do the analysis of the

31
See Figure 2 and Figure 3 for the difference between Nominal performance and real performance of the
S&P Case-Shiller Home Price Indices from 1987 to 2008.
Housing
Value
*

Composite
10
Composite
20 Census Division MSA
($mm)
% of
U.S
% of
Div
weight weight
New England Boston $220,448 2.3% 35.7% 7.4% 5.3%
Middle
Atlantic New York $826,909 8.5% 62.5% 27.2% 19.4%
Chicago $325,954 3.4% 21.3% 8.9% 6.3%
Detroit $189,917 2.0% 12.4% n/a 4.8%
East North
Central
Cleveland $80,009 0.8% 5.2% n/a 1.7%
West North
Central Minneapolis $126,237 1.3% 21.8% n/a 2.8%
Miami $154,650 1.6% 9.1% 5.0% 3.6%
Tampa $69,455 0.7% 4.1% n/a 1.5%
Atlanta $158,706 1.6% 9.4% n/a 3.9%
Charlotte $49,954 0.5% 3.0% n/a 1.3%
South Atlantic
Washington,
DC $237,472 2.5% 14.0% 7.8% 5.6%
West South
Central Dallas $143,355 1.5% 20.5% n/a 4.0%
Phoenix $117,756 1.2% 17.9% n/a 2.9%
Denver $108,884 1.1% 16.5% 3.7% 2.6% Mountain
Las Vegas $43,695 0.5% 6.6% 1.5% 1.1%
Los Angeles $549,808 5.7% 25.7% 21.2% 15.1%
San Diego $136,719 1.4% 6.4% 5.5% 3.9%
San Francisco $320,763 3.3% 15.0% 11.8% 8.4%
Portland $88,465 0.9% 4.1% n/a 1.9%
Pacific
Seattle $165,144 1.7% 7.7% n/a 3.9%
Composite 20
Index $4,114,301
42.50
%
Composite 10
Index $2,925,302
30.20
%

112
statistics of the monthly housing returns for 20 major metropolitans. The significant
positive first order and negative third order autocorrelation suggests the serial correlation
exists in monthly returns. The Ljung-Box (1987) portmanteau test Q (6) at lag 6 is used
to test the serial correlation in the monthly S&P Case-Shiller Home Price Indices. If the
monthly housing returns are serially uncorrelated, Q value will be asymptotically
2
n
χ ;
however, if a number of the sample autocorrelations are not close to zero, Q will be
inflated. The results indicate the serial correlations exist at the 1 percent significance
level across all MSAs in sample. The significant positive first order and negative second-
order autocorrelations and the Q statistics are both linear based tests of serial correlation
that portend return persistence found using the nonlinear Markov chain tests. On the other
side, the significant non-normality of return distribution was also found in most MSAs
except for Boston and New York by utilizing Jarque-Bera Test (Jarque and Bera, 1987).
The Results
The different sample frequency of returns: monthly, quarterly and annual returns are used
to establish the Markov chain processes which also display qualitatively distinct forms of
persistence patterns during the test intervals. However, there are no estimates for Atlanta,
Dallas, Detroit, Minneapolis, Phoenix, and Seattle because the time horizon covered by
the S&P Case/Shiller housing price index is too short to apply Maximum Likelihood
Estimation (MLE) methodology for these MSAs. I calculate the MLE of the transition
probabilities that result from dividing the monthly housing return series into high and low
returns. The states of high and low returns are defined in both time series and cross-

113
sectional manner. The unconstrained point estimates for
0
λ ,
00
λ ,
000
λ ,
1
λ ,
11
λ , and
111
λ have indicate that the presence of strong positive dependence in monthly housing
return. Actually most MSAs, except Chicago, Charlotte and Cleveland have the
probabilities
0
λ ,
00
λ ,
000
λ greater than 72% for time series states measure. That means the
probability of seeing a below-average monthly housing appreciation given two or three
prior months are below-average is generally 0.72 significantly above 0.5. Moreover, the
persistence of monthly housing returns behaves quite differently across MSAs. Some
MSAs show more significant positive serial persistence. Such as Los Angles, the
probability
000
λ of a low month given three prior months are also low
000
λ is 0.937, much
great than the probability
111
λ of a low month after a sequence of three high months which
is only 0.118. However, for Chicago, the persistence probability
000
λ is only 0.667 in
contrast to the
111
λ 0.434. The housing return persistence in Chicago is not as significant
as that in Los Angeles. In addition, the persistence of monthly housing return measured in
the cross-sectional way is negerally stronger than under the time series measure. The
result also indicates that for most MSAs, the conditional probability
0
λ ,
00
λ ,
000
λ is
generally higher under the cross-sectional states measure than that under the time series
states measure. This implies if the housing return in some MSA remains to be above the
average appreciation rate of the whole housing market for consecutive two or three
months, it still has a high possibility to beat the market in the following month. This
cross-sectional persistence in housing returns is consistent with the findings by Young

114
and Graff (1996, 1997). The stronger persistence of housing returns in the cross-sectional
manner has some particular implications for investors in real estate. It is compelling to
buy purchase the real estate that has performed well compared to the average return of
the market, and sell real estate that has performed, poorly in prior period.
The viability of such an investment strategy, however, depends on the stability of
the persistence of housing returns. I also did the analysis of the Likelihood Ratio Test
(LRT) for the monthly housing returns in both time series and cross-sectional manners.
We re-estimating the parameters with the constraint under hypothesis:
00 11
λ λ = ;
000 111
λ λ = and
00 11 10 01
λ λλ λ == = respectively, and substitute both the constrained and
unconstrained log likelihood into equation (5) to get the LRT value which is
asymptotically distributed with
2
n
χ with 1,1,3 n = degrees of freedom equal to the
number of restrictions, respectively. The hypothesis of
00 11 10 01
λ λλ λ = == is rejected at
the 1 percent marginal significance level both in time series and cross-sectional measure
of Markov chain states across different MSAs. Moreover, both hypothesis
00 11
λ λ = and
000 111
λ λ = are rejected with 99 percent confidence under cross-sectional manner for all
MSAs, which indicate the significant stability of persistence of housing returns in a cross-
sectional way. For time series measures of states, the hypothesis
00 11
λ λ = is also rejected
with 99% confidence in all MSAs, but condition on the specific sequences of prior three
successive low or high months; we can not reject the hypothesis
000 111
λ λ = for Charlotte
with 90% confidence. So in Charlotte, given a specific sequence of consecutive three low

115
states, the probability of a low state in the following month is not statistically significant.
The persistence pattern of housing returns will probably disappear for the fourth month.
For Chicago and Cleveland, we can reject the
000 111
λ λ = with 95% confidence and in
other MSAs the hypothesis
000 111
λ λ = can be rejected with 99% confidence.
The results show the MLE and the LRT for the quarterly housing returns
respectively which indicate that the persistence of quarterly return is not as statistically
significant as monthly return in most cities. The conditional probabilities
0
λ ,
00
λ ,
000
λ that
measure the persistence of housing returns in Charlotte, Cleveland and Chicago are
almost less than 0.5. That means there is no statistically significant persistence in
quarterly returns among these three MSAS. In addition, given the prior two or three
quarterly are “low” states in time series way, other two MSAs: Las Vegas and Portland
where the probability of being low in the current quarter is not statistically different to 0.5
and LRT test can not reject the hypothesis that
00 11
λ λ = and
000 111
λ λ = with 95%
confidence. So the persistence pattern in time series manner for Las Vegas and Portland
has not been notable or distinctive after observing two or three consecutive quarters of
the same state. Comparatively, quarterly housing returns’ Markov chain measured in a
cross-sectional way generally still show the significant persistence of housing returns in
most MSAs, no matter in second order or third order Markov chain model. Only Miami
and San Francisco do not show return persistence for third order Markov chain, where
000
λ are 0.563 and 0.417 respectively and the LRT test can not reject the hypothesis that

116
000 111
λ λ = with 95% confidence. The results for quarterly housing returns indicate that
for a given MSA, the persistence measured in a cross-sectional way is generally stronger
than that determined in a time series way. The persistence of housing return does
dependent on how to determine what is high or low states for housing returns. Moreover,
using quarterly date, the persistence becomes weaker for most MSAs compared with the
monthly housing return data. In some MSAs (Charlotte, Cleveland, Chicago, and
Portland), the persistence of quarterly housing returns is insignificant. However, Los
Angeles, San Diego and Washington continue to show the significant persistence in the
quarterly housing returns in a time series as well as in a cross-sectional manner.
My result shows that the significant serial persistence for annual returns in most
MSAs. Interestingly, the persistence pattern for annual returns is similar to the pattern for
the monthly returns. Except for Chicago and Cleveland, the conditional
probability
0
λ ,
00
λ ,
000
λ are consistently greater than 0.5, which means there is a
remarkable positive serial persistence in the annual returns. This finding is consistent
with the significant linear serial persistence behaviors in housing returns by some
previous literatures e.g., Hort (1995), Quigley and Redferan (1997) and Quigley (1999).
The LRT results for housing annual returns clearly indicate that return persistence
in annual housing returns exhibits the heterogeneity across MSAs. Charlotte, Chicago,
Cleveland have very low LRT values which can not reject both the hypothesis
00 11
λ λ = and the hypotheis
000 111
λ λ = , while Denver, New York and Tampa can easily
reject these hypothesis with 99% confidence level. We see this because the sample period

117
in S&P Case/Shiller Housing Price index is only 21 years (1987-2008) which results in
quite few observations of some specific sequences of high or low years in the sample,
especially for the third order Markov chain model. For example, there are consecutive
time series “low” years in Cleveland, so it can not use the sample data to test the
conditional probability
000
λ for Cleveland. Again, for a given MSA, the persistence
behavior measured in a cross-sectional way still have a different persistence behavior to
that measured in a time series manner. For example,
000
λ in a time series way for Chicago
is only 0.333 below 0.5 while
000
λ in a cross-sectional way is 0.857 much higher than 0.5.
In summary, the results indicate that first of all although housing returns
measured in different sample frequencies generally exhibit significant positive
persistence behaviors, the sample frequency has obvious effects on the serial persistence
in housing returns. The different sample frequencies of housing returns are shown to
exhibit various extents of positive persistence. In our sample, most MSAs in both
monthly and annual housing returns exhibit significant positive serial persistence up to
the third orders. While the persistence in quarterly housing return is not as strong as that
in the monthly housing return data. Actually five MSAs have not shown return
persistence in the third order Markov chain model. Second, the persistence of housing
returns exhibits significant spatial heterogeneity. The housing return in some MSAs is
more persistent than others, no matter in monthly, quarterly or annual housing returns.
Charlotte, Chicago, Cleveland have the least persistent housing returns, while Los
Angeles, Boston, San Francisco, San Diego, Tampa and Washinton have significant

118
persistence in housing returns. Third, the definition of the high or low return is a key
determination of the persistence of housing returns. For some MSAs, the Markov chain
defined in a time series way has a different persistence pattern to that defined in a cross-
sectional way.

Conclusion
This chapter introduces a two-state Markov chain model in both time series and cross-
sectional manners to analyze the persistence of the annual, quarterly and monthly housing
returns at MSA level respectively. The Markov chain model (up to the third order) used
in this study focuses on periods of the established trends and tests whether a low return is
more likely after observing a sequence of two/three high returns or a sequence of
two/three low returns. The findings show that the extent of return persistence is also
impacted by the sample frequency and the definition of the high/low states used in
Markov chain model. More importantly, this study finds the monthly, quarterly and
annual real housing returns exhibit significant persistence pattern in the sense that low
(high) returns tend to follow runs of low (high) returns. However, these persistent return
patterns have remarkable spatial heterogeneity, i.e., some MSAs have more persistent
returns in housing market than others. In Charlotte, Chicago, Cleveland housing returns
are not as persistent as in other MSAs, while Los Angeles, Boston, San Francisco, San
Diego, Tampa and Washington have more significant persistence in housing returns.

119
Though the underlying economic mechanism is not discussed in this paper, I hope the
findings in this chapter can stimulate future works on this important issue.

120
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Abstract (if available)

Abstract This dissertation consists of three essays addressing different aspects of real estate risk and return. First two essays concern the securitized real estate, namely the Real Estate Investment Trusts (REITs). The third one focuses on the private (unsecuritized) real estate.

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