On the Time-Series Properties of Real Estate Investment Trust Betas
Kevin C.H. Chiang*
College of Business Administration
Northern Arizona University
Flagstaff, AZ 86011-5066
Ming-Long Lee
Department of Finance
National Yulin University of Science and Technology
Touliu, Yulin
Taiwan 640
Craig H. Wisen
School of Management
University of Alaska Fairbanks
Fairbanks, AK 99775
USA
Real Estate Economics, Summer 2005, 33 (2)
On the Time-Series Properties of Real Estate Investment Trust Betas
Abstract
The relation between real estate investment trust (REIT) returns and stock market returns
is of significant importance to investors, practitioners, and academics. The temporal
properties of this relationship have a critical impact on the usefulness of REIT risk
estimates and portfolio allocations to this asset class. Recent studies have suggested a
decline in the market betas of equity real estate investment trusts (EREITs). This study
applies a rigorous statistical test of the hypothesis that the market betas of EREITs have
remained unchanged during the 1972 through 2002 time period. There is weak evidence
of a downward trend in EREIT betas using a single-factor model; however, the
hypothesis is not rejected when using a three-factor model.
2
On the Time-Series Properties of Equity Real Estate Investment Trust Betas
The stability of a risky security’s market beta is important to those who use the estimated
coefficient for performance evaluation, event studies, valuation, and asset allocation. A
number of recent studies have observed an apparent decline in the market betas of equity
real estate investment trusts (EREITs). If the decline is of statistical and economic
significance, then the implication is that estimates of EREIT betas that rely upon
historical returns are biased upwards. Although several explanations have been proposed
for the apparent decline in EREIT betas, no formal tests for a significant time trend have
been conducted. This paper rigorously tests the time-series properties of EREIT betas.
Related Literature
McIntosh, Liang, and Tompkins (1991) were the first to detect a decline in EREIT betas
during the 1974 through 1983 time period. Khoo, Hartzell, and Hoesli (1993) expanded
the McIntosh et al. sample period to 1970 to 1989, and provided additional evidence of a
temporal decline in EREIT betas. Khoo et al. applied a two-sample test for a regime shift
under the assumption of time independence. As will be shown below, however, beta
innovations are serially correlated. Khoo et al. also found that EREIT betas during the
1982 through 1989 period were significantly lower than the 1970 through 1981 period.
Although the current analysis does not contradict or support the findings of Khoo et al., it
does offer evidence that previous assertions of a temporal decline in REIT betas could be
erroneous.
3
This study is not the first to question the validity of previous evidence of a temporal
decline in EREIT betas. Liang, McIntosh, and Webb (1995) extended the focus of Khoo
et al. by examining intermediate-term variations in beta estimates, and found significant
shifts in return-generating regimes in the vicinity of 1983. Nevertheless, their results
(Figure 10) did not imply a declining trend in EREIT betas since bias in the study’s data
may have contributed to the absence of a declining trend.1
This study employs the Fama-French (1993) three-factor model and the Vogelsang (1998)
method to test the null hypothesis that EREIT betas have remained constant over time.
The Fama-French three-factor model is selected because Peterson and Hsieh (1997)
found that the Fama-French factors helped to explain EREIT pricing and performance.2
The Vogelsang (1998) test is applied primarily because of the method’s generality. This
method is useful when EREIT beta innovations are serially correlated, and when the
nature of the innovations is unknown. These features are desirable when testing for
deterministic time trends in EREIT betas because serial correlation is induced by the use
of rolling regressions to obtain time-series estimates of betas. The generality is also
beneficial since unit root tests often have very low power.
This study finds weak evidence for a decline in EREIT betas based upon a single-factor
model. However, when the three-factor model is used, the declining trend in EREIT
betas disappears. This study also uses the tests of Liang et al. to investigate whether
EREIT betas have shifted and, if so, when the changes occurred. Our results demonstrate
4
that detecting regime shifts in market betas is sensitive to both the nature of the data and
the asset pricing model that is used.
Statistical Methods
We employ rolling 60 month windows to obtain a series of EREIT beta estimates. The
asset pricing models include the one-factor model of Sharpe (1964) and the three-factor
model of Fama and French (1993). The one-factor regression is specified as:
Rp,t =  + b Rm,t + p,t
(1)
where Rp is the monthly EREIT excess return and Rm is the monthly market excess return.
Excess return is expressed as the difference between the monthly return on the market
portfolio and the monthly return of the 30-Day U.S. Treasury Bill. The three-factor
regression is as follows:
Rp,t =  + b Rm,t + s SMBt + h HMLt + p,t
(2)
where SMBt is the difference between the returns on portfolios composed of small and
big stocks, and HMLt is the difference between the returns on portfolios composed of
stocks with high and low BE/ME (book-to-market) ratios.
Next, we apply Vogelsang’s (1998) t-PST1 test to check for a deterministic trend. The tPST1 test is valid for errors that are integrated of order zero (I(0)), and for errors
integrated of order one (I(1)). Therefore, a priori knowledge about beta innovations, and
testing whether the innovations are I(0) or I(1), is not required. The t-PST1 test is based
on the following specification:
5

bt =  +  t + t
(3)


where  is the initial level of bt ,  is the average slope of time trend in bt , and t is a
serially correlated random process.
Testing for a time-trend in beta estimates is
essentially a test of whether the parameter  is different from zero. The t-PST1 test
statistic is specified as:
t-PST1 = T -1/2 tz exp(-c JT)
(4)
where T is the sample size, tz is the set of t-statistics for testing the null hypothesis that
the individual parameters in the partial-sums regression of equation (3) are zero, c is a
constant, and JT is a unit root statistic proposed by Park and Choi (1988) and Park (1990).
When the innovations in betas are known to be I(0), the specification of c = 0 is
appropriate and most powerful. In contrast, when it is unclear whether the innovations
are I(0) or I(1), c can be chosen such that the critical values of the PST1 test statistics are
same, whether t is I(0) or I(1). Therefore, different values for c are used for different
levels of statistical significance.
Because the asymptotic distribution of the t-PST1
statistic is nonnormal, statistical inferences are based upon on the critical values tabulated
in Vogelsang.
To investigate intermediate-term variations in EREIT betas, this study uses the cusum of
squares test applied by Brown, Durbin, and Evans (1975) and by Liang et al. (1995). The
method defines recursive residuals as:
6
wr =
Rp , r  xr ' Br  1
, r = k + 1, …, T
(1  xr ' ( Xr ' Xr ) 1 xr )1 / 2
(5)
where xr is the column vector of observations on k regressors, Br = (Xr’ Xr)-1Xr’ Xr, and
Xr’ = (x1, …, xr). Under the assumption that recursive residuals are stationary, 3 the test
statistic of cusum of squares is defined as:
r
w
Sr =
j
2
j  k 1
(6)
T
w
j
2
j  k 1
The lines of statistical significance are plotted and defined as  d2 + (r  k)/(T  k). If Sr
travels outside the lines of significance, the null hypothesis of a constant regression
relationship is rejected. According to Durbin (1969), d2 is 0.15483 and 0.12823 for the
1% and 5% level, respectively.4
In addition to the cusum of squares test, a standard likelihood ratio, Lr, can be used to
detect the point of change:
Lr = ½ r log(s12) + ½ (T  r) log(s22)  ½ T log(s2)
(7)
where s12, s22, and s2 are the ratios of the residual sums of squares to the number of
observations, when the regression is run on the first r observations, the remaining (T  r)
observations, and the T observations, respectively. The point of change occurs when Lr
reaches its minimum value.
7
Data
The monthly return from the Center for Research in Security Prices (CRSP) valueweighted index is used as the proxy for the return on the market portfolio. Monthly U.S.
Treasury Bill returns are retrieved from the CRSP database. Monthly SMB and HML
factor returns are provided by Kenneth French. Because SMB and HML are constructed
from equity returns, they are most appropriate for explaining the returns on equity
securities. As a result, the study focuses on the intertemporal changes in the riskiness of
EREITs.
The study uses monthly returns on the EREIT index of the National Association of Real
Estate Investment Trusts (NAREIT). The sample period is from January 1972 through
December 2002. There were 170 REITs in the index as of September 30, 2003. The
NAREIT index allows for greater comparability with prior REIT studies, albeit at the cost
of a higher level of survivorship bias.
The study also uses monthly returns on the Wilshire REIT index. The minimum market
capitalization within the NAREIT index was less than $5 million, whereas the minimum
market capitalization within the Wilshire REIT index was greater than $100 million.
Thus, one might expect a lower level of survivorship bias relative to the NAREIT index.
As of June 30, 2003, there were 88 REITs in the capitalization-weighted Wilshire REIT
index.5 The inception date of the Wilshire REIT index is September 1991. Since the
Wilshire REIT index was introduced in September 1991, returns prior to this date were
8
backfilled to January 1978.
The monthly returns on the Wilshire REIT index are
retrieved from the Datastream database. The sample period is from January 1978 through
December 2002.
Empirical Results
Times Series Estimates of Market Betas
The time-series regression results for the one-factor and three-factor models are reported
in Table 1. Panel A presents the one-factor regression results with the use of NAREIT
returns. Over the sample period beginning January 1972 and ending December 2002, the
beta of EREITs is 0.4734. The adjusted R-squared is 32%, suggesting that the one-factor
model provides a limited explanation of EREIT returns. To shed light on the evolution of
EREIT betas, the study splits the full sample period into three subsamples: January 1972
to February 1983, March 1983 to December 1991, and January 1992 to December 2002.6
March 1983 is used as the first cutoff to reflect the Tax Reform Act of 1981 (Liang et al.
(1995)). The second cutoff, January 1992, reflects the potential impact of the Tax
Reform Act of 1993 (Glascock, Lu, and So (2000)). Panel A of Table 1 reports the
regression results for the three subsamples. The betas for the three subsamples are
0.6531, 0.4903, and 0.2316. In addition, the regression results indicate that the low Rsquared value for the full sample is largely driven by the most recent subsample, in which
the adjusted R-squared is 8%.
9
Panel B of Table 1 reports the three-factor regression results using NAREIT index
returns. The beta and the adjusted R-squared are 0.5485 and 49%, respectively. The
betas for the three subsamples vary less when a three-factor model is used rather than a
one-factor model. Specifically, the market betas for the three subsamples are 0.5966,
0.5579, and 0.3980. The decrease in the market beta for the last subsample is largely
driven by 2002 returns. While not reported, the beta for the period beginning January
1992 and ending December 2001 is 0.5530  nearly identical to the market beta
estimated for the period beginning March 1983 and ending December 1991. The SMB
factor is more useful than the HML factor in describing EREIT returns in the first two
subsamples; however, the coefficient of the HML factor increases in the last two
subsamples. For the most recent subsample, the loading of 0.5228 on the HML factor is
even higher than that of 0.3980 on the market beta. The results are consistent with the
findings of Chiang and Lee (2002) and of Chiang, Lee, and Wisen (2004), who document
a value return component in EREIT returns.
This result is useful in resolving the
asymmetric REIT-beta puzzle of Goldstein and Nelling (1999) and Sagalyn (1990). 7
Moreover, the use of the three-factor model improves the R-squared for the three
subsamples, particularly for the most recent subsample.
Panels C and D of Table 1 report the one-factor and three-factor regression results using
the Wilshire REIT index. In general, the results are similar to those using the NAREIT
index. The notable difference is that during the earliest subsample, the use of Wilshire
REIT index produces high market beta estimates of 0.9718 and 0.9405 under the null of
the one-factor and the three-factor models, respectively. This is not surprising since the
10
Wilshire REIT returns are backfilled prior to September 1991. It is generally accepted
that backfilled data may contain a higher degree of data bias.
Long-Term Trend of Market Betas
Figure 1 depicts the evolution of EREIT betas over time using the NAREIT return index.
Betas of EREITs under the null of the one-factor model appear to exhibit a downward
trend over the sample period. Furthermore, the slope of the time series of betas is
particularly steep after 1995. The location of the switching point in 1983 (documented
by Liang et al. (1995)) is identified by Arrow 3.
In contrast to the single-factor model, use of the three-factor model suggests that EREIT
betas were stable over the past three decades, except in 2002. Before 2002, the maximum
market beta of EREITs is approximately 0.7, and the minimum market beta is
approximately 0.5.
Figure 2 plots the estimates of EREIT betas under the one-factor and three-factor models
using the Wilshire REIT index. Rolling beta estimates in Figures 1 and 2 have the same
pattern. That is, the beta estimates under the three-factor model are more stable over time
than those under the one-factor model. Figure 2 is also consistent with the notion that
Wilshire REIT returns yield particularly high market beta estimates during the period
when the index values were backfilled.
11
Before applying Vogelsang’s (1998) method to test for a deterministic time trend, one
should emphasize the difficulties in detecting such a trend. The main difficulty is that the
power of the test largely hinges on whether the innovations in market betas are I(0) or
I(1). If the process is stochastic, a trend that is visually obvious can be stochastic in
nature and yield no statistical significance when one attempts to test for a deterministic
trend. Therefore, a priori knowledge about the nature of the innovations in market betas
is useful to the extent that it can improve the power of the t-PST1 test.8
This study uses two versions of the t-PST1 test because of the difficulties in determining
whether the innovations in market betas are I(0) or I(1). The first version is aggressive
since it assumes that the innovations are I(0), and the second version is conservative since
it allows the nature of the innovations to remain unknown. The test results are reported
in Table 2. Panel A of Table 2 reports one-, two-, three-, four-, six-, and twelve-month
autocorrelation structures for beta estimates. It is evident that EREIT beta estimates
under both the one-factor and the three-factor models are highly serially correlated,
regardless of whether NAREIT or Wilshire REIT data is used. The result calls for an
econometric method that allows for serial correlation (e.g., Vogelsang (1998)). Panel B
presents the results of the Dicky-Fuller (1979) unit root test, the point-optimal test, and
the modified Dickey-Fuller test proposed by Elliot, Rothenberg, and Stock (1996). None
of these test statistics reject the null hypothesis of a unit root.
Panel C of Table 2 shows that when NAREIT data is used, the linear trend coefficient for
the one-factor betas is 10 times steeper than the coefficient for the three-factor market
12
betas. By specifying c = 0 when the innovations of betas are I(0), under the one-factor
model, the t-PST1 test statistic is -2.45, which is statistically significant at the 5% level.
This result indicates a downward linear trend in betas. In contrast, under the three-factor
model, the t-PST1 test statistic is -0.73, which is not statistically significant at the 1%
level or 5% level. As expected, the Wilshire REIT index yields steeper linear trend
coefficients because of the apparent inflation of beta estimates created in the early
backfill period. However, this does not have a material impact on statistical inferences
because the deterministic trend test is applied to the entire dataset. By specifying c = 0,
under the one-factor model the t-PST1 test statistic is -2.82, which is statistically
significant at the 1% level. In contrast, under the three-factor model the test statistic is 1.29, which is not statistically significant at the 1% level or 5% level.
When no a priori knowledge about the innovations of market betas is assumed, according
to Vogelsang (1998) the values of c should be specified as 0.494, 0.716, and 1.501 for the
10%, 5%, and 1% level of significance, respectively. With the use of NAREIT data, the
resulting t-PST1 test statistics are quite low, ranging from -0.00 to -0.25, and they are not
statistically significant at any level. The use of the Wilshire REIT index results in similar
t-PST1 test statistics, ranging from -0.00 to -0.06. Overall, these linear trend tests provide
weak evidence that there is a downward deterministic trend in market betas under the
one-factor model, and strong evidence that there is no deterministic trend in market betas
under the three-factor model.
13
Intermediate-Term Variations of Market Betas
Having documented the long-term evolution in market betas, the study turns to
intermediate-term variations. The sample period is from January 1972 to December 1989
and the cusum of squares test of Brown et al. (1975) is used to duplicate the methods and
approximate the time period of Liang et al. (1995). This duplication is necessary because
our previous results show that intermediate variations in betas are sensitive to whether the
data is backfilled, and to the degree of survivorship bias.
Consistent with Liang et al., the null hypothesis of a constant regression relationship is
rejected at the 1% level under the one-factor model and NAREIT index returns. Theses
results are displayed in Figure 3, Panel A. 9 While Liang et al. find a switching point in
the vicinity of 1983, this study uses the likelihood ratio test in equation (7) and identifies
three switching points: 1976, 1980, and 1986, which are labeled by Arrows 1, 2, and 4 in
Figure 1. The locations of the three switching points coincide with those of intermediateterm variations in market betas, a feature for which this test is designed to address. In
addition, the study finds that the null hypothesis is rejected at the 1% level under the
three-factor model. The testing results are depicted in Figure 3, Panel B. The study
identifies the same set of switching points using the three-factor model.
Applying the cusum of squares test to the more recent sample period, the study finds that
the null hypothesis of a constant regression relationship is rejected at the 1% and 5%
levels under the one-factor and the three-factor models, respectively.10 These results are
14
presented in Figure 3, Panels C and D. In addition, the likelihood ratio test identifies one
switching point with the one-factor model in 1998, and one switch point with the threefactor model in 2001. The two switching points, depicted as Arrow 5 and 6 in Figure 1,
may reflect market anticipation and the effectiveness of the REIT Modernization Act of
1999, respectively. Two observations are noteworthy. First, it seems reasonable that the
intermediate-term variations in market beta estimates under the three-factor model are
less volatile than those under the one-factor model, because the beta estimates under the
three-factor model exhibit a lower linear trend coefficient.
Second, the Revenue
Reconciliation Act of 1993 does not cause a regime shift. This is surprising because of
the impact the act had on the REIT industry.
The cusum of squares test using the Wilshire REIT index rejects the null hypothesis of a
constant regression relationship at the 1% level under both models during the January
1978 to December 1989 period. The testing results are depicted in Figure 4, Panels A
and B. 11 As expected, the likelihood ratio test again shows that testing results are
sensitive to the nature of data.
The test yields three switching points: 1980, 1983, and
1987 under both the one-factor and the three-factor model.
Finally, the cusum of squares test shows for the January 1990 to December 2002 period,
the null hypothesis of a constant regression relationship is rejected at the 5% and 1%
levels under the one-factor and the three-factor models, respectively. The testing results
are depicted in Figure 4, Panels C and D. The likelihood ratio test identifies two
15
switching points under the null of the one-factor model in the vicinities of 1998 and 2001,
and one point under the null of the three-factor model in the vicinity of 2001.
Conclusion
This study shows that EREIT betas exhibit long-term temporal stability during the 1972
through 2002 time period. This stability is premised upon the descriptive ability of the
Fama-French three-factor model. Consistent with the findings of Perterson and Hsieh
(1997), the three-factor model is more useful than the single-factor model in explaining
the variation in EREIT returns, and in providing stable estimates of market betas. One
caveat of these results is that a sharp decline in market beta occurs in 2002 under the
three-factor model. While this may be attributed to the random nature of sampling, it is
also possible that the decline is significant. The study also demonstrates that testing
regime shifts in intermediate-term variations of betas is sensitive to the nature of the data
and to the underlying asset pricing model. As a result, findings on intermediate-term
structural changes need to be considered within the context of data quality and model
validity.
These results have implications for decisions concerning asset allocation and portfolio
diversification. Under the null of the three-factor model, EREIT returns exhibit stable
exposure over time with respect to market risk. History may be a more useful guide for
the three-factor model than for the single-factor model in the context of diversification
and risk reduction exercises using EREITs.
16
Acknowledgements
The authors thank David Ling (the editor) and four anonymous referees for their helpful
suggestions. The authors also thank Kenneth French for providing size and book-tomarket factor series.
17
References
Brown, R.L., J. Durbin and J.M. Evans. 1975. Techniques for Testing the Constancy of
Regression Relationships over Time. Journal of the Royal Statistical Society 37: 149-163.
Chiang, K. and M. Lee. 2002. REITs in the Decentralized Investment Industry. Journal
of Property Investment & Finance 20: 496-512.
Chiang, K., M. Lee and C. Wisen. 2004. Another Look at the Asymmetric REIT-Beta
Puzzle. Journal of Real Estate Research 26: 25-42.
Durbin, J. 1969. Testing for Serial Correlation in Regression Analysis Based on the
Periodogram of Least-Squared Residuals. Biometrika 56: 1-15.
Dickey, D. and W. Fuller. 1979. Distribution of the Estimators for Autoregressive Time
Series with a Unit Root. Journal of the American Statistical Association 74: 427-431.
Elliot, G., T.J. Rothenberg and J.H. Stock. 1996. Efficient Tests for an Autoregressive
Unit Root, Econometrica 64: 813-836.
Enders, W. 1995. Applied Econometric Time Series. John Wiley & Sons, New York.
Fama, E.F. and K.R. French. 1993. Common Risk Factors in the Returns on Stocks and
Bonds. Journal of Financial Economics 33: 3-56.
Glascock, J.L., C. Lu and R.W. So. 2000. Further Evidence of the Integration of REIT,
Bond, and Stock Return. Journal of Real Estate Finance and Economics 20: 177-194.
Goldstein, A. and E.F. Nelling. 1999. REIT Return Behavior in Advancing and Declining
Stock Markets. Real Estate Finance 15: 68-77.
Khoo, T., D. Hartzell and M. Hoesli. 1993. An Investigation of the Change in Real Estate
Investment Trust Betas. Journal of American Real Estate and Urban Economics
Association 21: 107-130.
Liang, Y., W. McIntosh and J.R. Webb. 1995. Intertemporal Changes in the Riskiness of
EREITs. Journal of Real Estate Research 10: 427-443.
McIntosh, W., Y. Liang and D.L. Tompkins. 1991. An Examination of the Small Firm
Effect within the EREIT Industry. Journal of Real Estate Research 6: 9-18.
Park, J.Y. 1990. Testing for Unit Roots and Cointegration by Variable Addition. In
Advances in Econometrics: Cointegration, Spurious Regressions and Unit Roots, ed. by
T. Fomby and F. Rhodes, Jai Press, London.
18
Park, J.Y. and B. Choi. 1988. A New Approach to Testing for a Unit Root. Working
paper, Connell University.
Peterson, J. and C. Hsieh. 1997. Do Common Risk Factors in the Returns on Stocks and
Bonds Explain Returns on REITs? Real Estate Economics 25: 321-345.
Sagalyn, L. 1990. Real Estate Risk and the Business Cycle: Evidence from security
markets. Journal of Real Estate Research 5: 203-219.
Sharpe, W. 1964. Capital Asset Prices: A Theory of Market Equilibrium under
Conditions of Risk. Journal of Finance 19: 425-442.
Shumway, T. 1997. The Delisting Bias in CRSP. Journal of Finance 52: 327-340.
Vogelsang, T. 1998. Trend Function Hypothesis Testing in the Presence of Serial
Correlation. Econometrica 66: 123-148.
19
Table 1 ■ REIT time-series regression results for one-factor and three-factor models.
Jan. 1972 Dec. 2002
Panel A: One-Factor Model (NAREIT)
0.0084

(5.0)
B
0.4734
(13.2)
Adj. R2
0.32
Panel B: Three-Factor Model (NAREIT)
0.0054

(3.6)
B
0.5485
(15.9)
S
0.3734
(8.3)
H
0.4757
(9.4)
Adj. R2
0.49
Jan. 1978 –
Dec. 2002
Panel C: One-Factor Model (Wilshire)
0.0079

(3.9)
B
0.4982
(11.5)
Adj. R2
0.31
Panel D: Three-Factor Model (Wilshire)
0.0048

(2.7)
B
0.6131
(14.2)
S
0.4076
(7.2)
H
0.5304
(8.2)
Adj. R2
0.46
Jan. 1972 Feb. 1983
0.0099
(3.4)
0.6531
(11.0)
0.48
0.0062
(2.4)
0.5966
(9.5)
0.4585
(5.0)
0.2766
(2.8)
0.60
Jan. 1978 –
Feb. 1983
Mar. 1983 –
Dec. 1991
0.0074
(3.0)
0.4903
(9.5)
0.46
0.0074
(3.4)
0.5579
(10.6)
0.4965
(5.1)
0.4390
(4.2)
0.59
Mar. 1983 Dec. 1991
Jan. 1992Dec. 2002
0.0082
(2.8)
0.2316
(3.5)
0.08
0.0043
(1.7)
0.3980
(6.2)
0.3283
(4.9)
0.5228
(6.7)
0.32
Jan 1992 Dec. 2002
0.0155
(3.4)
0.9718
(10.6)
0.65
0.0041
(1.6)
0.5188
(10.0)
0.49
0.0075
(2.5)
0.1967
(2.9)
0.05
0.0087
(2.1)
0.9405
(8.9)
0.6520
(4.0)
0.4355
(2.6)
0.75
0.0041
(1.9)
0.5895
(11.6)
0.5489
(5.8)
0.4699
(4.6)
0.64
0.0040
(1.5)
0.3461
(5.1)
0.2568
(3.5)
0.4554
(5.4)
0.23
The dependent variable is either the NAREIT index or the Wilshire REIT index. The explanatory
factors (coefficients) consist of the market return (B), size (S), and book-to-price (H) series. The
CRSP value-weighted return is used to proxy the market. The size proxy, SMB, is the difference
in returns between portfolios with small capitalization and large capitalization equities. The
book-to-price proxy, HML, is the difference between the returns on high and low book-to-market
portfolios. t-Statistics are in parentheses.
20
Table 2 ■ Testing for the temporal stability of EREIT market betas.
NAREIT Index
One-Factor
Three-Factor
Wilshire REIT Index
One-Factor
Three-Factor
Panel A: Autocorrelation Structure
1
2
3
4
6
12
0.9899
0.9799
0.9676
0.9561
0.9310
0.8498
Panel B: Unit Root Test
Dicky-Fuller Statistic
Point-Optimal Statistic
Modified Dicky-Fuller Statistic
Panel C: Linear Trend Test
Linear Trend (10-4)
Aggressive Test (c = 0)
t-PST1
Conservative Test
10% Significance (c = 0.494)
t-PST1
5% Significance (c = 0.716)
t-PST1
1% Significance (c = 1.501)
t-PST1
0.9478
0.8992
0.8525
0.8000
0.7191
0.5416
0.9941
0.9876
0.9793
0.9710
0.9534
0.8921
0.9871
0.9726
0.9548
0.9345
0.8995
0.7800
-0.26
21.90
0.41
-1.48
3.85
-1.85
-1.23
96.91
1.29
-0.96
39.05
0.75
-12.82
-1.24
-23.23
-10.26
-2.45**
-0.73
-2.82***
-1.29
-0.25
(1.3)
-0.16
(1.3)
-0.06
(1.3)
-0.01
(1.3)
-0.09
(1.7)
-0.08
(1.7)
-0.01
(1.7)
-0.00
(1.7)
-0.00
(2.7)
-0.01
(2.7)
-0.00
(2.7)
-0.00
(2.7)
Two versions of the t-PST1 test are applied to the time series. The first version assumes that the
innovations in EREIT market betas are I(0) and is labeled as the “Aggressive Test”. The
alternative version allows the nature of the innovations to remain unknown and is labeled the
“Conservative Test.” Panel A reports the one-, two-, three-, four-, six-, and twelve-month
autocorrelation structure for market beta estimates. The statistics in Panel B account for the serial
correlation and present the testing results for the Dicky-Fuller (1979) unit root test, the pointoptimal test, and modified Dickey-Fuller test proposed by Elliot, Rothenberg, and Stock (1996).
The critical values for linear trend tests are contained in parentheses. NAREIT index results are
based upon the January 1972–December 2002 period. Wilshire REIT index results are based
upon the January 1978–December 2002 period. ** denotes statistically significance at the 5%
level. *** denotes statistically significance at the 1% level.
21
Figure 1 ■ Market beta estimates of EREIT using the NAREIT index.
Beta estimates use overlapping 60 month intervals and the NAREIT index. Based upon
the likelihood ratio test, the arrows labeled 1 through 6 identify the switching points in
EREIT market betas.
22
Figure 2 ■ Market beta estimates of EREIT using the Wilshire REIT index.
Beta estimates use overlapping 60 month intervals and the Wilshire REIT index. Based
upon the likelihood ratio test, the arrows labeled 1 through 4 identify the switching points
in EREIT market betas.
23
Figure 3 ■ Cusum of squares test of forward recursive residuals: NAREIT index
The cusum of squares test using NAREIT index returns is applied to the sample period
from January 1972 to December 1989. The solid line identifies the 1% significance
boundary and the dotted line identifies the 5% significance boundary. The null
hypothesis of a constant regression relationship is rejected at the 1% level under the onefactor model in Panel A. The null hypothesis is also rejected at the 1% level under the
three-factor model in Panel B. Panel C and Panel D apply the cusum of squares test to
the more recent sample period beginning January 1990 and ending December 2002. The
null hypothesis of a constant regression relationship is also rejected at the 1% and 5%
levels under the one-factor and the three-factor models, respectively.
24
Figure 4 ■ Cusum of squares test of forward recursive residuals: Wilshire REIT index
The cusum of squares test using Wilshire REIT index returns is applied to the sample
period from January 1978 to December 1989. The solid line identifies the 1%
significance boundary and the dotted line identifies the 5% significance boundary. The
null hypothesis of a constant regression relationship is rejected at the 1% level under the
one-factor (Panel A) and the three-factor models (Panel B). Panel C and Panel D apply
the cusum of squares test to the sample period from January 1990 to December 2002 for
the one-factor and three-factor models respectively. The null hypothesis of a constant
regression relationship is rejected at the 5% and 1% levels under the one-factor and the
three-factor models, respectively.
25
Endnotes
Liang et al.’s (1995) dataset include small, illiquid EREITs and their EREIT returns are retrieved from the
CRSP database. The delisting bias in the CRSP database of Shumway (1997) may be amplified by higher
chance of failures among small, illiquid EREITs.
1
2
Peterson and Hsieh (1997) demonstrate that REIT performance, in terms of the statistical significance of
the intercept term from an asset pricing regression, is sensitive to model specification. Because of this
sensitivity, one would expect that the time-series of market beta estimates could exhibit different time
trends under different model specifications.
3
Dicky-Fuller (1979) t-test shows that recursive residuals of EREITs are stationary.
4
Brown et al. (1975) also derive a first-moment test statistic, called the cusum test. Nevertheless, Liang et
al. (1995) show that the cusum of squares test is more powerful when statistical inferences are based on the
cusum of squares test. The current study focuses on the cusum of squares test in order to directly compare
the results with Liang et al.
The liquidity of the Wilshire REIT index’s constituent REITs is commensurate with that of other
institutionally held equity real estate securities. As of June 2003, the Wilshire REIT index listings are
largely equity properties with the following sector weights: office (20.96%), apartment (19.41%), regional
retail (14.44%), local retail (12.69%), diversified (12.23%), industrial (7.50%), hotels (4.44%), storage
(3.96%), manufactured homes (1.60%), factory outlets (1.38%), and cash (1.39%).
5
6
The study also experiments with other cutoff points in the vicinities of 1976, 1981, 1986, and 1988. The
results in Table 1 are not sensitive to the use of cutoff points.
7
The usefulness of the HML factor in describing REIT returns in the latest subsample may be due to the
wider participation of institutional investors who perceive REITs more like value stocks because real estate
rents frequently have upper limits in their annual increase (Chiang and Lee (2002)).
8
In general, one would expect the innovations in market betas to be I(0) because, as time goes to infinity,
market betas are unlikely to go to infinity. By definition, market beta is a ratio where the numerator is the
covariance in returns between EREITs and the market portfolio, and the denominator is the variance of the
market portfolio. Since the variance of any risky asset is positive, and the minimum possible return for any
risky asset is -100%, the only scenario in which market beta could approach infinity is when the maximum
possible return for EREITs, or the market portfolio, is positive infinity and asset return follows an I(1)
process. Because equity returns are skewed to the left, asset prices are an I(1) process, and asset returns are
an I(0) process. It is inconceivable that the return for EREITs or the market portfolio could be positive
infinity. Of course, one can argue that the answer to whether the innovations of market betas are I(0) or
I(1) should be determined by a unit root test. Unfortunately, it is generally accepted that the power of unit
root tests is frequently quite low. Therefore, failing to reject the null hypothesis of a unit root is not strong
evidence for the existence of a unit root. Enders (1995) provides a detailed discussion about the problems
in testing for unit roots.
9
The test results reported in Figure 3 are based on forward recursive residuals. The study also experiment
backward recursive residuals. Consistent with Liang et al. (1995), the testing results are not sensitive to
whether forward or backward recursive residuals are used.
10
Although we used the spilt date of December 1989 to facilitate the comparison between our results and
those established by Liang et al. (1995), this ad hoc choice does not have any material impact on statistical
inferences. Specifically, we also repeat the analysis without the use of a split date. The results remain the
same.
26
11
The test results reported in Figure 4 are based on forward recursive residuals. However, consistent with
Liang et al. (1995), we find that testing results are not sensitive to whether forward or backward recursive
residuals are used.
27