A Common Real Estate Risk Premium in Time-Varying Returns of Real
Estate and Financial Assets
Leiting DENG
&
Tien Foo SING*
Department of Real Estate
National University of Singapore, Singapore
and
Hong WANG
School of Economics and Management
Tsinghua University, China
Abstract
The market integration hypothesis in the latent common market factors study by Mei
and Liu (1994) suggests that when the risk premiums of the investment can be
captured by common risk beta in overall stock, both and real estate markets, the
financial and real estate markets can be deemed to be integrated. This paper employed
the multi-factor latent variable model of Mei and Lee (1994) to test the predictability
of five asset markets and also to examine whether there are common risk premiums in
the five asset markets in Singapore. Our results were consistent with those found in
the study by Mei and Lee (1994), which showed that the three market factors,
inclusive of real estate market risk factor, contain sufficient market information to
explain the price variations in real estate and financial asset classes. By further
examining into the beta coefficient estimates of the commercial real estate and
property stock assets with respects to the three latent market factors. We observed
differences in the market behaviors in the five asset classes, even though the risk
premiums could be fully captured by the three market risk factors. It seems like
securitized real estate as represented by the property stock asset is more closely
integrated with the overall stock market, and less so with the direct real estate market
and bond market.
Keywords: Market integration, Generalized Method of Moments, Common Market
Factors, and Time Varying Risks.
* Corresponding Author: rststf@nus.edu.sg. Address: Department of Real Estate, National
University of Singapore, 4 Architecture Drive, Singapore 117566.We wish to thank the
participants at the European Real Estate Society Conference in Milan, Italy for comments
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A Common Real Estate Risk Premium in Time-Varying Returns of Real Estate
and Financial Assets
1.
Introduction
Market integration is an important research issue in the finance and real estate
literature. If there is no integration between different asset markets, investors can
efficiently diversify their portfolios to reduce systematic risks. In view of the
increasing trend in real estate securitization, it becomes more critical to investigate
whether different real estate markets are segmented from the capital market (Liu,
Hartzell, Greig and Grissom, 1990). Past empirical tests on market integration
predominantly focus on the asset and real estate markets in the US (see Liu & Mei,
1992, Mei & Lee, 1994, Ling and Naranjo, 1999). Other country studies used mainly
evidence of cointegration and Granger-causality relationships as an indirect way to
define integration of securitized and direct real estate markets with equity markets in
the UK (Lizieri and Satchell,1997), Australia (Wilson, Okunev and Ta, 1996) and
Hong Kong (Fu and Ng, 2001). Evidence of cross-country direct real estate markets
and capital markets integration was also observed in studies by Mei and Hu (2000),
Liu and Mei (1998), Liu and Mei (1999) and Bond, Karolyi and Sanders (2003).
The real estate investment trust market in Singapore is still at its infancy stage of
development. Investment in stocks of listed property companies is often regarded as a
close proxy for securitized claims in real estate asset. In an efficient market, the
performance of these securitized real estate assets is expected to reflect the market
value of the underlying real estate assets held in the portfolios of companies. This
hypothesis implies that investors can capture risk premiums embedded in the direct
real estate market by investing in property stocks. Empirical tests on the comovements between the property stock returns and the direct real estate returns in
Singapore have been conducted by Ong (1994 and 1995), Liow (1998 and 2001) and
Sing and Sng (2003). Liow (2004) provided evidence on the effects of common
macroeconomic risks in the commercial real estate markets. However, there are still
no studies thus far that empirically test the existence of common factors among
different asset markets in Singapore, and how the common risk factors affect the
predictability of the returns in the securitized and the direct real estate markets.
1
This paper aims to empirically investigate the common factors that affect the
movements in five different asset markets in Singapore: the overall stock market, the
property stock market, the bond market, the residential real estate market and the
commercial real estate market, using a multifactor latent-variable model. Consistent
with the findings in the US by Mei and Lee (1994), we also found three common
factors in overall stock market, bond market and direct real estate market that explain
the variations in excess returns of the five asset markets in Singapore. Our
predictability tests further showed that the residential real estate market is the most
predictable among the five markets. The variations in the responses of the excess
returns of five assets to the common market risk betas also suggest that there are
different degrees of integration across the asset classes. We also observe that
securitized real estate market is more closely integrated with overall-stock market,
and less so with the direct real estate and bond markets.
This paper is organized into six sections. Section 2 reviews the literature on the
market integration studies. Section 3 explains the asset pricing framework and the
empirical methodology. The descriptions of empirical data are provided in Section 4
followed by analyses of the empirical results in Section 5. Section 6 concludes the
findings of the paper.
2.
Literature Review
The notion of market integration across different asset markets has significant
implications for the well-accepted investment theories. If asset markets are not
integrated, the investors can seek to efficiently diversify their portfolios by holding
assets in these markets. Recent literature on market integration tests can be broadly
divided into three groups based on the empirical techniques applied in the respective
studies: (1) a multi-factor asset pricing approach that test the predictability and risk
premia associated with a-priori defined macro-economic drivers; (2) a latent multifactor approach that tests the common fundamental factors in the asset classes; and (3)
a cointegration approach that examines the long-term contemporaneous relationships
between different historical returns of different asset classes.
Applying Jorion and Schwartz (1986) technique in the standard Capital Asset Pricing
Model (CAPM) framework, Liu, Hartzell, Greig and Grisssom (1990) test the
2
integration between the equity market and the real estate markets that are represented
by equity REITs and non-farm commercial real estate in the US. Their empirical
results supported the integration between the equity REIT and the stock market, but
rejected the integration between the commercial real estate market and the stock
market.
For studies on Singapore market, Ong (1994) discovered a contemporaneous longterm relationship between property stock return, real estate return and 3-month
Treasury bills interest rate using a structural vector autoregressive (VAR) model. In
another study by Ong (1995) with data of securitized real estate and direct real estate
for the period from 1977 to 1992, the results contradicted those in his previous paper.
Liow (1998) confirmed the segmentation between the commercial real estate market
and the securitized real estate market using the same methodology. Using the
Generalized Autoregressive Conditional Heteroscedasticity in Mean (GARCH-M)
methodology, Sing and Sng (2003) found evidence of incremental information flowed
uni-directionally from the conditional volatility of unsecuritized market to the
securitized property market. They concluded that the two markets are partial
integrated.
The above literature tests the relationships of the returns between securitized real
estate markets and direct real estate markets without specifying the underlying factors.
Ling and Naranjo (1999) tested the predictabilities of the stock market and the real
estate market using pre-defined common macro-economic factors, which include the
growth rate of industrial production, the consumer consumption, the yield spread
between the long term and short term government bond, the unexpected inflation rate
and others.
Liu and Mei (1992), Mei and Lee (1994) applied the Generalized Method of Moments
(GMM) methodology to test common factors in the asset markets in the so-called
latent multi-factor framework, which assumes time-varying expected excess returns
for different asset classes. Liu and Mei (1992) confirmed that the real estate return
variations can be fully captured by the risk premiums in the stock and the bond
markets. Mei and Lee (1994) found an additional common real estate market risk
3
factor, and suggested that investors should not neglect real estate asset in their welldiversified portfolios.
Our empirical methodology follows the multi-factor latent variable model adopted in
Liu and Mei (1992) and Mei and Lee (1994). We correct the heteroskedascity and
serial correlation in the residual terms, and also allow for correlated residual terms in
our model using the Generalized Method of Moments (GMM) methodology. The
GMM methodology was also applied in the financial literature by Ferson (1990) to
test the time-varying multi-beta asset pricing model involving non-real estate asset
returns.
3.
Latent-Variable Asset Pricing Model Framework
The latent factor asset pricing framework used in this study is identical to that of Liu
and Mei (1992). On the assumption that capital markets are perfectly competitive and
frictionless, asset returns are generated by the following K-factor model:
K
r%i ,t +1 = Et [ r%i ,t +1 ] + ∑ β ik f%k ,t +1 + ε%i ,t +1
(1)
k =1
where r%i ,t +1 denotes the excess return of asset i held from time t to time t + 1 , which
is estimated as the difference between return on asset i and the risk-free interest rate.
Et [ r%i ,t +1 ] is the expected excess return on asset i which is allowed to vary through
time conditional on information known at the end of time t. β ik is the time-invariant
factor loadings with each of the K factors. The unexpected return on asset i equals the
sum of K-factor realizations, f%k ,t +1 times their betas β ik plus an idiosyncratic
error ε%i ,t +1 . We assume that E[ f%k ,t +1 ] = 0 and that E[ε%i ,t +1 ] = 0 .
If zero-beta return, Et [r%i ,t +1 ] , is not constant, we need to look at both the similarities in
betas and the co-movement of Et [r%i ,t +1 ] over time when analyzing the co-movement of
excess returns for two or more assets. In other words, it is possible for the excess
returns of two assets to move independently even though they share similar betas.
This problem will not occur, if the following linear pricing relationship holds:
K
Et [r%i ,t +1 ] = ∑ β ik λkt
(2)
k =1
4
where λkt is the “market price of risk” for the k-th factor at time t .
By assuming that the information set at time t consists of a vector of L forecasting
variables X nt , n =1… L (where X 1t is a constant), and that conditional expectations
are a linear function of these variables, we can write λkt as
L
λkt = ∑ θ kn X nt
(3)
n =1
and therefore equation (2) can be expressed as
K
L
L
k =1
n =1
n =1
Et [r%i ,t +1 ] = ∑ β ik ∑ θ kn X nt = ∑ α in X nt
(4)
where α in is the risk premium for the forecasting variable X nt . Equations (1) to (4)
collectively define the multifactor “latent-variable” model. The model implies that
expected excess returns are time-varying and can be predicted by the forecasting
variables in the information set. Equations (3) and (4) impose restrictions on the
α in coefficients, which is given as
K
α ij = ∑ βikθ kn
(5)
k =1
where β ik and θ kn are free parameters.
Based on the conditional excess returns Et [ r%i ,t +1 ] in Equation (4) with the pricing
restriction in Equation (5), we first obtain an unrestricted conditional excess return by
regressing excess returns on the forecasting variables:
L
Et [r%i ,t +1 ] = ∑ φin X nt
(6)
n =1
In the linear regression specified by Equation (6), the risk premium, φin , for X nt , does
not have to be the same as α in in Equation (4).
Next, we test for common systematic risk factors that affect asset returns in the
multifactor latent-variable model by testing the rank restriction, H0: α=ΘΒ. To test
the restrictions in equation (5), we first normalize the model by setting the factor
5
loadings of the first K assets as follows: β ij = 1 (if j = i ) and β ij = 0 (if j ≠ i ) for
1 ≤ i ≤ K . We partition the excess return matrix, [ R = ( R1 , R2 ) ], into R1 , which is a
T × K matrix of excess returns of the first K assets, and R2 , which is a T × ( N − K )
matrix of excess returns for the rest of the assets. Combining equation (4) and (5), we
can derive the following regression system:
R1 = X Θ + µ1
(7a)
R2 = X α + µ2
(7b)
where X is a T × L matrix of the forecasting variables, Θ is a matrix of θ ij , and α is
a matrix of α ij . If the linear pricing relationship in Equation (2) holds, the rank
restriction implies that the data should not be able to reject the null hypothesis H0:
α=ΘΒ, where B is a matrix of β ij elements. The objectives of this study are to use the
regression system in equations (7) to test the extent to which the forecasting
variables, X , can predict excess returns, and at the same time, to test the significance
of the rank restriction.
The unrestricted conditional excess returns of Equation (6) and the restricted
conditional excess returns of Equation (7) with the restriction in equation (5) were
estimated using Hansen’s (1982) Generalized Method of Moments (GMM)
methodology. The GMM technique allows for both conditional hetroskedasticity and
serial correlation in the error terms in the excess return regressions.
4.
Data
The data consist of quarterly excess returns for five asset markets in Singapore: the
overall stock market, the property stock market, the bond market, the residential real
estate market and the commercial real estate market. The sample period from 1988Q2
to 2003Q4 is dictated by the availability of the long-term government bond data.
The All-share Price Index (ALLSTOCK) published by the Singapore Exchange
(SGX) is used to represent the overall stock market performance. The SGX Property
Sub-sector Index (PPYSTOCK) represents the performance of securitized real estate
market. These two quarterly indices are obtained from the DataStream. The 5-year
6
government bond (BOND) representing the performance of the bond market is
obtained from the Monetary Authority of Singapore (MAS) database. The quarterly 5year bond yield ( rq ,t ) is a compounded rate, which is calculated from the annualized
yield ( ra ,t ), using the equation: [ rq ,t = (1 + ra ,t )1/ 4 − 1 ]. The quarterly residential (PPIR)
and office (PPIC) property price indices that proxy the performance of the residential
and commercial real estate markets are obtained from the Singapore Real Estate
Information System (REALIS) provided by the Urban Redevelopment Authority
(URA) of Singapore. REALIS contains property price indices estimated using
transaction prices of properties in the caveats lodged with the Singapore Land
Registry.
The quarterly returns of the five assets are calculated using the continuous return
formula by taking the difference of returns in natural logarithm terms between time t
and time t − 1 . The log-quarterly returns are subtracted by the risk-free rate, which is
represented by the 3-month Treasury Bill rate ( Raq ,t ) published by the MAS, to derive
at the excess quarterly returns for the respective asset classes. The quarterly T-Bill
yield ( Rqf ,t ) is a compounded rate, which is calculated from the annualized yield
( Raq ,t ) using the same equation for computing the quarterly yield of the 5-year bond.
The forecasting variables used to estimate the unrestricted and the restricted models
specified in Equations (6) and (7) are consistent with those widely used in previous
studies of conditional risk premiums on asset markets in the US literature. This set of
variables includes the real 3-month Treasury-bill (RTBill), the yield spread between
the 3-month commercial bill and the 3-month T-bill (DEFAULT), the growth rate in
industrial production output (GIP), the term-spread between the 5-year long term
government bond yield and the 2-year short term government bond yield (TERM),
and the quarterly unexpected inflation rate (UI). These variables are joint proxies for a
set of latent variables that determine the asset returns in Singapore markets.
Following Ling and Naranjo (1999), we define the real 3-month Treasury bill rate
(RTBILL) as the difference between the quarterly return on a 3-month T-bill rate and
the inflation rate, which is measured by the consumer price index (CPI). The yield
7
spreads between the 3-month commercial bill and the 3-month T-bill proxy the credit
risk factor, (DEFAULT), and the yield spreads between the 5-year long term
government bond and the 2-year short term government bond reflect the termstructure risk, (TERM). All the yield data are obtained from the MAS database. The
growth rate of industrial production, GIP, is defined as the percentage quarterly
change in industrial production. The GIP measures the one-quarter lagged change in
industrial production. Therefore, we move the GIP data forward by one quarter to
represent GIP changes in the same quarter (See Ling and Naranjo, 1999). The
quarterly index of industrial production is obtained from the “TREND” (Time Series
Retrieval and Dissemination) database by the Department of Statistics, Singapore.
The unexpected inflation (UI) is computed as the difference between realized inflation
during period t and expected inflation at the beginning of the same period t . The
expected inflation is calculated using the forecast errors of the first-order moving
average process, MA(1).
5.
Empirical Results
5.1.
Descriptive Statistics
Table 1 summarizes the descriptive statistics of the excess returns for the five asset
market and the forecasting variables. Figure 1 plots the excess returns for the five
asset markets. For each variable, we report the mean, standard deviation, and the first
order autocorrelation statistics. The residential real estate has the highest average
excess return of 1.0033%, and the risk of this asset measured by the standard
deviation is also relatively lower than other assets, with the exception of the
government bond. The property stock exhibits the lowest average excess return with a
negative value -0.6447% and the highest volatility 18.3353%. Property stock is the
worst performer among the five asset classes. The results are consistent with those
reported in Liow (2001). The negative average excess return on the commercial real
estate market was partly caused by the weak economic condition in Singapore in the
post 1997 financial crisis periods. Bond is the safest investment with the lowest risk
volatility of 0.1659%.
Table 1 also exhibits the correlations of excess returns between the five assets. A high
correlation between the overall stock market and the property stock market is
observed, which is consistent with the results in the previous studies (Liow, 1998 and
8
2001, Sing and Sng, 2003). There is also a strong correlation between the residential
real estate market and the commercial real estate market. However, the evidence of
pair-wise correlationship between the two real estate asset returns is not sufficed to
draw any conclusion on whether the two direct real estate sub- markets share the same
common risk factors. We will use the restricted model to test whether the residential
real estate market and the commercial real estate market are wholly integrated. The
correlation coefficients also seems to indicate that the property stock market is likely
to be closely driven by the risk factors common to the residential real estate market,
but not by those underlying the commercial real estate market. The bond market has
low correlations with the other four asset markets. The correlations between the
forecasting variables are also relatively weak.
[Insert Table 1 & Figure 1]
5.2.
GMM Regression Results
Table 2 reports the regression results of excess returns on the five forecasting
variables and a constant term. The excess returns of residential real estate exhibit the
most predictable behavior among the five asset classes with an adjusted R-square of
36.3%. The bond’s excess return is the least predictable with only 4.4% of the
variations in the excess returns explained by the regression.
[Insert Table 2]
The strong predictability of the residential real estate excess return is dependent on
the term structure and the unexpected inflation rate at a significance level of 10%. The
growth rate of industrial production is significant in explaining the variations in
excess returns of commercial real estate at a 5% level. An increase in industrial
production will trigger new demand for industrial, warehouse and office space, which
will in turn improve the rental and price performance of industrial and office
properties. It is interesting to note that different forecasting variables are incorporated
in the excess return generating processes for residential and commercial real estate
assets.
The predictability of the property stock and all-stock excess returns is dependent on a
same set of forecasting variables: the real 3-month Treasury bill, the growth rate of
industrial production, and the unexpected inflation, at a 5% significance level. The
9
negative signs for the real 3-month T-bill variable suggest that both the overall stock
and the property stock were poor hedges against inflation, and their returns react
inversely with the unexpected inflation changes. This results are consistent with those
found in Liu & Mei (1992) and Mei & Lee (1994). The property stock market shares
the same predictive variable in unexpected inflation albeit different sign with
residential real estate market. The GIP risk factor, on the other hands, explains the
variations in both the property stock and the commercial real estate markets.
For bonds, the excess return variations are explained by the zero-beta and the term
structure risk factors. A 1% increase in the spread between the government 5-year
bond yield and the government 2-year bond yield will cause the excess bond return to
increase by 0.512%. Mei & Lee (1994) also reported significant positive relationship
between term structure risk premium and the bond returns.
5.3.
Common Latent Risk Factors
Table 3 reports the regression results of the excess asset returns on the forecasting
variables based on the restricted model (4) with imposed restriction in Equation (5). In
the one-factor model, [k=1], the null hypothesis is that all asset returns are driven by
one systematic risk factor, which is assumed to be latent in the performance on the all
stock market. We normalize the beta for all-stock return to unity. If the null
hypothesis is not rejected, the all-stock return beta will be significantly captured in
other asset betas that reflect the ratio of time-varying excess returns of other assets to
that of overall stock. The statistic of the chi-square test on the restriction in Equation
(5) and the P-value rejects the null hypothesis at 5% significance level, which implies
that the stock market beta alone can not fully capture all common factors in the five
asset markets.
[Insert Table 3]
Next we test the two-factor latent variable model by assuming that the overall stock
and bond market betas can explain the variations in the time-varying excess returns of
the five assets. Two normalized betas are introduced for the overall stock market and
the bond market. For the overall stock market, the first beta is set to unity, and the
second beta is set to zero for the overall stock market. We also normalize the first beta
to zero and the second beta to unity for the bond market. The Panel B in Table 3
10
exhibits the regression results with the coefficient estimates that capture the responses
of other asset time varying excess returns towards the two common latent risk factors.
The two-factor latent model is again significantly rejected at the 10% level with a Pvalue 0.096. The rejection of the null hypothesis again suggests that there exist other
factors underlying the direct real estate market, which are not captured by the
common stock market and bond market factors.
We continue the test of three-factor model, [k=3], by adding a normalized beta that is
equal to unity for residential real estate asset, and the first two betas related to
common stock and bond market were set to zero for the residential real estate excess
return. Panel C represents the beta estimation of the three-factor model (K=3). The
chi-square statistic and the P-value of 0.217 indicate that the null hypothesis for the
three-factor model is not rejected at 10% significance level. The result, therefore,
implies that the time-varying excess returns for assets in the five markets are driven
by not more than the three common risk factors represented in the stock market, the
bond market and the direct real estate market. The result was consistent with the [k ≤
3] finding in Mei and Lee (1994), which confirms the significant of the real estate
factor premium in both Singapore and the US markets. The tests do not reject the null
hypothesis that the three-factor model contains all common market risk information
required to explain excess returns in the five asset markets. Investors can adequately
capture all market risk premiums by diversifying their investments in at least three
assets. According to Mei & Lee (1994), the direct real estate market and the financial
asset markets are deemed to be integrated.
Next, we further analyze the common risk premiums in the restricted and unrestricted
return generating processes for the commercial real estate and the property stocks. We
found similarities in beta coefficients of the property stocks with those of overall
stock market. The overall-equity market beta is the only risk factor, i.e. [βi1 = 1.505],
that is significant and positive in explaining the excess returns variations of property
stock. The bond market beta and the residential real estate market beta were not
significantly different from zero. The results suggest that there is overall stock market
information that will drives property stock price changes, and the price process of the
property stock market behaves more closely to the overall stock price changes. For the
11
direct commercial real estate, the bond market beta, [βi2 = -3.957], and the residential
real estate market beta, [βi3 = 0.668], were the two latent risk factors that are captured
in the excess returns of the commercial real estate asset. The bond market factor has a
negative relationship with the excess return of the commercial real estate. The
commercial real estate excess returns seem to be segmented with the overall stock
market since the overall stock beta coefficient was not significantly different from
zero, [βi1 = 0.039].
Figures 3, 4 and 5 plot the restricted and unrestricted excess returns of the property
stock and the commercial real estate for one, two, and three latent factors respectively.
Figure 5 shows that the restricted excess return moves more closely with the
unrestricted excess return in the property market when [k = 3] factors were included,
compared with the return trends in [k = 1] and [k = 2] models as shown in figures 3
and 4. The results suggest that the three-factor model is superior to the other two
models, which provide better predictability fit for the excess returns.
[Insert Figures 3, 4 and 5]
6.
Conclusion
In this paper, we employed the multi-factor latent variable model of Mei and Lee
(1994) to test the predictability of the five asset markets and also to examine whether
there are common risk premiums in the five asset markets in Singapore. Our results
were consistent with those found in the study by Mei and Lee (1994) on the US asset
markets. We found that three market factors, inclusive of real estate risk factor,
contain sufficient market information to explain the price variations of real estate and
financial asset classes.
In our study, five asset classes: all-stock, property stock, bond, commercial real estate
and residential real estate, were included. The residential real estate excess returns
were the most predictable with the highest adjusted R-square of 0.416. Two macrovariables: term spread and unexpected inflation were significant in the residential real
estate model. The predictability of all-stock and property stock excess returns was
both dependent on the same set of macro-economic variables: real T-bill rate (RTbill),
growth in industrial production (GIP) and unexpected inflation (UI).
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We also tested the models with different common risk factors. Like in Liu and Mei
(1992), the common equity and bond market remain significant in explaining the
variations in the five asset excess returns. When the residential real estate market
factor was included in the [k = 3] latent factor model, we could not reject the null
hypothesis that residential real estate market beta was the third factors in our fiveasset latent variable pricing model. The real estate market risk factor also appeared to
be significant in determining the risk premiums of real estate and other financial
assets. The same empirical observation was found in Mei and Lee (1994), which
implies consequently that the financial and real estate asset markets are integrated.
Investors can not capture all risk premiums without including direct real estate in their
portfolio.
By further examining into the beta coefficient estimates of the commercial real estate
and property stock assets for the three latent market factors, we found that property
stock market excess returns are mainly driven by overall-stock market risk. The bond
market and residential real estate market betas did not significantly explain the
variations in the property stock returns. On the other hands, commercial real estate
excess returns are more effectively predicted by the bond and the residential real
estate market risk factors, and the overall-stock market beta information was not so
relevant in this sub-sector of direct real estate market. Clearly, there are differences in
the market behaviors in the five asset classes, even though the risk premiums could be
fully captured by the three market risk factors. It seems like securitized real estate as
represented by the property stock asset is more integrated with the overall stock
market, and less so with the direct real estate market and bond market. Commercial
real estate, residential real estate and bond market are, however, more closely
integrated, based on the common risk factor behaviors.
There are limitations in this paper. The shortage of time-series bond market data in
different bond classes restrict the ways by which the term structure and credit risks
can be defined. In the future extension to the study, we could include other
macroeconomic variables such as the consumer consumption, the capitalization rates
of direct real estate, the dividend yield in the stock market and others to improve the
predictability of the excess returns for the asset markets.
13
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21(4): 366-382.
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15
Table 1: Summary Statistics
A) Historical Trends
Variable
Mean
S.D.
Ρ1
a) Dependent Variables
• Excess return on All Sing-Equities (ALLSTOCK)
• Excess return on All Property Equities (PPYSTOCK)
• Excess return on government bond (BOND)
• Excess return on residential real estate market (PPIR)
• Excess return on commercial real estate market (PPIC)
0.3976
-0.6447
0.4187
1.0033
-0.0545
13.0639
18.3353
0.1659
5.2376
6.3205
-0.038
-0.029
0.327
0.734
0.510
b) Independent Variables
• Real Yield on 3-month T-bill (RTBILL)
• Yield spread between long and short term bond (TERM)
• Yield spread between T-bill and commercial bill
(DEFAULT)
• Growth rate of Industrial Production (GIP)
• Unexpected Inflation (UI)
0.1021
0.2370
0.2136
0.0177
0.2877
0.4051
0.0952
0.2117
0.0370
0.8429
0.427
0.687
0.732
0.329
0.440
S.D. denotes standard deviation
B) Correlations between Excess Asset Returns
ALLSTOCK
PPYSTOCK
ALLSTOCK
1
0.8006
PPYSTOCK
1
BOND
PPIR
PPIC
P1 denotes first order autocorrelation statistics
BOND
0.1266
0.0980
1
C) Correlations between Forecasting Variables
RTBILL
TERM
DEFAULTR
RTBILL
1
-0.2833
-0.0149
TERM
1
-0.3921
DEFAULTR
1
GIP
UI
PPIR
0.4744
0.5005
0.1511
1
GIP
-0.0816
0.3211
-0.0439
1
PPIC
0.1921
0.1396
-0.0390
0.5583
1
UI
-0.8548
0.0699
0.1137
0.0631
1
16
Table 2: Results of Regressions that Predict the Unrestricted Excess Returns of Five Asset Classes
Model: ri ,t +1
= φ1Cons. + φ2 RTBILLt + φ3TERM t + φ4 DEFAULTt + φ5GIPt + φ6UI t + ε%i
Costant
RTbill
Term
Default
GIP
UI
R2
Adjusted R2
DW
5.128
-20.351*
-7.540
-1.143
113.241*
-8.636*
0.184
0.111
2.035
(0.901)
(-2.443)
(-0.412)
(-0.209)
(2.841)
(-2.140)
9.381
-32.753*
-13.567
-9.074
148.790*
-13.635*
0.216
0.146
2.111
(1.166)
(-3.203)
(-0.554)
(-1.148)
(2.179)
(-2.612)
Bond
0.273*
-0.018
0.512**
0.155
0.491
-0.043
0.122
0.044
1.851
(2.691)
(-0.146)
(1.756)
(0.835)
(1.261)
(-0.848)
PPIR
-4.146*
-1.791
18.560*
0.252
19.502
2.217**
0.416
0.363
0.921
(-2.410)
(-0.647)
(3.313)
(0.116)
(1.571)
(1.826)
-0.770
-1.146
-4.168
1.365
45.495*
2.744
0.258
0.191
1.116
(-0.232)
(-0.244)
(-0.401)
(0.385)
(2.775)
(1.403)
Asset Market
Allstock
Ppystock
PPIC
J-statistic = 2.69E-30
* Indicates significance level at 5%;
** indicates significance level at 10%.
The above results are based on the Simultaneously regressions of quarterly excess returns on each asset market at time t + 1 on independent variables, which
include the real 3-month T-bill yield, the yield spread between the long and the short term government bond, yield spread between the 3-month T-bill and the
3-month commercial bill, the growth rate of the industrial production and the unexpected inflation rate at time t using the equation (6). The sample period is
1988Q2-2003Q4. Regression coefficients are given by the first line of each row, while the t-statistics are given in parentheses in the second row.
17
Table 3: Estimation Results for Multi-factor Latent Variable Models
K
L
L
k =1
n =1
n =1
Et [r%i ,t +1 ] = ∑ β ik ∑ θ kn X nt = ∑ α in X nt
(4)
K
α ij = ∑ βikθ kn
k =1
(5)
A. The number of systematic factors in the economy equals one (K=1)
βi1
Estimated beta coefficient for the following asset:
Excess return on All Sing-Equities (ALLSTOCK)
Excess return on All Property Equities (PPYSTOCK)
Excess return on government bond (BOND)
Excess return on residential real estate market (PPIR)
Excess return on commercial real estate market (PPIC)
χ2-statistic of the restriction in equation (5):
Significant level:
S.D.
1.000*
0.429
0.729
0.071
0.147
0.576
1.361
0.210
0.211
33.599 (DF=20)
P=0.029
B. The number of systematic factors in the economy equals two (K=2)
βi1
S.D.
Estimated beta coefficient for the following asset:
1.000*
Excess return on All Sing-Equities (ALLSTOCK)
0.000*
Excess return on government bond (BOND)
7.756
3.524
Excess return on All Property Equities (PPYSTOCK)
18.207
5.603
Excess return on residential real estate market (PPIR)
6.045
-1.695
Excess return on commercial real estate market (PPIC)
χ2-statistic of the restriction in equation (5):
18.720 (DF=12)
Significant level:
P=0.096
βi2
S.D.
0.000*
1.000*
-8.615
-5.709
6.313
12.499
31.408
10.776
C. The number of systematic factors in the economy equals three (K=3)
βi1
S.D.
Estimated beta coefficient for the following asset:
1.000*
Excess return on ALLSTOCK
0.000*
Excess return on BOND
0.000*
Excess return on PPIR
0.230
1.505
Excess return on PPYSTOCK
0.112
0.039
Excess return on PPIC
χ2-statistic of the restriction in equation (5): 8.293 (DF=6)
Significant level:
P=0.217
βi2
S.D.
0.000*
1.000*
0.000*
-2.977
-3.957
3.091
1.461
βi3
S.D.
0.000*
0.000*
1.000*
-0.006 0.465
0.668 0.197
18
Figure 1: Excess returns for the five asset markets
Log-Excess Returns of Assets
80
60
40
20
0
-20
-40
19
90
2
19
91
Q
2
19
Q 92
2
19
93
Q
2
19
94
Q
2
19
Q 95
2
19
96
Q
2
19
97
Q
2
19
Q 98
2
19
99
Q
2
20
00
Q
2
20
Q 01
2
20
02
Q
2
20
03
2
19
89
Qtr-Year
Q
Q
Q
Q
2
2
19
88
-60
All-Stock
Property Stock
5-year Government Bond
Residential Real Estate
Commercial Real Estate
Figure 2: Restricted and Unrestricted expected excess returns on property stock
market and commercial real estate market (K=1)
Restricted and Unrestricted Excess Return
20
15
10
5
0
-5
-10
-15
-20
-25
19
Q 88
3
19
Q 89
3
19
Q 90
3
19
Q 91
3
19
Q 92
3
19
Q 93
3
19
Q 94
3
19
Q 95
3
19
Q 96
3
19
Q 97
3
19
Q 98
3
19
Q 99
3
20
Q 00
3
20
Q 01
3
20
Q 02
3
20
03
-30
Q
3
Qtr-Year
Unrestricted Property Stock Excess Return
Restricted Property Stock Excess Return
Unrestricted Commercial Real Estate Excess Return
Restricted Commercial Real Estate Excess Return
19
Figure 3: Restricted and Unrestricted expected excess returns on property stock
market and commercial real estate market (K=2)
20
15
Restricted and Unrestricted Excess Returns
10
5
0
-5
-10
-15
-20
-25
20
03
Q
3
20
02
20
02
4
Q
Q
1
2
20
01
20
00
Q
Q
3
19
99
19
99
4
19
98
1
Q
Q
Q
Q
2
19
97
19
96
3
4
1
Q
Q
Q
19
96
19
95
2
19
94
19
93
3
Q
4
Q
Q
1
19
93
19
92
19
91
2
Q
3
Q
Q
4
19
90
19
90
1
19
89
Q
Q
Q
2
3
19
88
-30
Unrestricted Property Stock Excess Return
Restricted Property Stock Excess Return
Unrestricted Commercial Real Estate Excess Return
Restricted Commercial Real Estate Excess Return
Figure 4: Restricted and Unrestricted expected excess returns on property stock
market and commercial real estate market (K=3)
30
Restricted and Unrestricted Excess Returns
20
10
0
-10
-20
-30
20
03
3
20
02
Qtr-Year
Q
20
02
4
Q
1
2
3
20
01
Q
20
00
Q
Q
4
1
19
99
19
99
Q
Q
Q
2
19
98
19
97
19
96
3
Q
4
1
19
95
19
96
Q
Q
Q
19
94
2
19
93
3
Q
1
4
19
93
Q
19
92
Q
19
91
2
Q
4
3
19
90
Q
19
90
Q
1
Q
2
Q
Q
3
19
88
19
89
-40
Unrestricted Property Stock Excess Return
Restricted Property Stock Excess Return
Unrestricted Commercial Real Estate Excess Return
Restricted Commercial Real Estate Excess Return
20