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Housing consumption-based asset pricing and residential mortgage default risk
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Housing consumption-based asset pricing and residential mortgage default risk

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Content



HOUSING CONSUMPTION-BASED ASSET PRICING AND RESIDENTIAL
MORTGAGE DEFAULT RISK




by


Minye Zhang




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)




December 2009





Copyright 2009                                                                                                Minye Zhang







ii
ACKNOWLEDGMENTS

I would like to express my deepest gratitude to my supervisor, Professor
Yongheng Deng for his guidance, support and patience in guiding me through the
completion of my dissertation. He is a knowledgeable, warmhearted and tough-minded
person with profound insights in real estate economics and finance studies. I am grateful
for being his Research Assistant as well as his student in the past 5 years. It has been a
great pleasure and honor to learn from him and work with him closely. He has cultivated
my interest in mortgage credit risk research and his continual support was the driving
force behind me. He has also offered me different opportunities to explore myself. In my
hard times, I always received his kind encouragement and help. These years of study
have been the most important experience of my life. I wish to thank him for his valuable
suggestions for moving this study forward with a goal towards perfection.  
I wish to thank my dissertation committee members Professor Peter Gordon and
Professor Cheng Hsiao from Department of Economics for their careful reading of my
dissertation, consideration in choosing the defense date and their helpful feedback and
suggestions. They provided much more help than what I expected. My heartfelt thanks to
Professor Peter Gordon for sparing particular time to talk with me regarding my
dissertation research after Professor Yongheng Deng’s leave from USC. He even helped
me prepare my job application materials. Special thanks go to Professor Cheng Hsiao for
always being willing to help me in both my dissertation study and job hunting.  


iii
I’d also like to thank my qualifying committee members Dr. Chris Redfearn, Dr.
and Raphael Bostic for helping me develop my dissertation research questions. My deep
thanks go to Dr. Delores Conway for her help in my teaching assistant work with her, her
warmhearted support in my job searching, and her enlightening conversations. I learn a
lot of teaching skills and communication skills from her. Particular thanks to Dr. Gary
Painter for his kind support in my attendance of various academic activities in the U.S.
and overseas. My gratitude goes out to Xudong An for giving me helpful suggestions and
comments in both my Ph.D. studies and life in the U.S. Special thanks to Professor Stuart
Gabriel for being an amazing supervisor while I was a first and second Ph.D. student at
USC. Thanks for his invaluable suggestions and giving me a deeper understanding in the
field of real estate finance. Thanks to all the faculties and staffs I have met at School of
Planning, Policy and Development at USC, for sharing their experiences and taking care
of me, especially Ms. Christine Wilson, June Muranaka, Ms. Nina Tibayan, and Ms.
Juliet Musso. Thanks to the rest of the wonderful members in School of Planning, Policy
and Development and Marshall School of Business at USC for creating a stimulating and
entertaining environment.
My special thanks also to Professor Leslie Young and Professor Suzanne Young,
for his useful suggestions on my long-term career objective to be a good researcher and
their encouragement during the whole Ph.D. study period in the U.S.  
Thanks to Xiaoxin Zhang, Pengyu Zhu, Della Zheng, Lin Zhang, Lingqian Hu,
and Mingxia Sheng for their friendships and sharing my happiness and sadness. Thanks
them for sharing the knowledge of their research interest.  

iv
I reserve my deepest heartfelt feelings to my family. Thanks to my dear wife
Luyao Zheng for her enduring love and care over the years. Special thanks to my parents,
for their endless love and encouragement throughout the good and bad times. Thanks to
my elder brother Jinye Zhang and his wife Jie Cai for their unending support from
Canada.  

v
TABLE OF CONTENTS

ACKNOWLEDGMENTS .................................................................................................. ii

LIST OF TABLES............................................................................................................ vii

LIST OF FIGURES ......................................................................................................... viii

ABSTRACT....................................................................................................................... ix

CHAPTER 1: INTRODUCTION........................................................................................1
1.1 Research Motivations..........................................................................................1
1.2 Research Questions and Approaches.................................................................. 5
1.3 Summary of Main Findings and Results............................................................. 6
1.4 Expected Contributions.....................................................................................10
1.5 Dissertation Outline..........................................................................................11

CHAPTER 2: LITERATURE REVIEW...........................................................................12
2.1 Asset Pricing Literature .................................................................................... 12
2.2 Related Mortgage Credit Risk Literature.......................................................... 15

CHAPTER 3: THEORETICAL MODEL .........................................................................18
3.1 Set Up................................................................................................................18
3.1.1 Consumers.................................................................................................18
3.1.2 Technology...............................................................................................19
3.2 Pricing Kernel...................................................................................................20
3.3 Analytical Solutions..........................................................................................24
3.3.1 Dynamics of the system............................................................................ 25
3.3.2 Equilibrium solutions of
,1 Ct
r
+
and its excess returns ............................. 28
3.3.3 Conditional mean and volatility of
1 t
m
+
and risk free rate of return
, f t
r 32
3.3.4 Housing return
,1 ht
r
+
................................................................................. 38
3.3.5 Special case: consumption growth with a fixed component..................... 44


vi
CHAPTER 4: DATA .........................................................................................................48
4.1 Data on Non-housing Consumption.................................................................. 48
4.2 Data on Housing Consumption......................................................................... 49
4.2.1 The problem of NIPA housing series........................................................ 49
4.2.2 Construction of comprehensive housing quality index............................. 50
4.2.3 Housing expenditure.................................................................................53
4.2.4 Housing price-rent ratio
, ht
q and housing return..................................... 54
4.2.5 Observed properties of the economy ........................................................ 55

CHAPTER 5: EMPIRICAL RESULTS ............................................................................61
5.1 Parameter Estimations......................................................................................61
5.1.1 Non-housing consumption growth and expenditure share........................ 61
5.1.2 Parameters of the pricing kernel ............................................................... 63
5.2 GMM Estimations Based on Standard CCAPM............................................... 69

CHAPTER 6: PREDICTING CREDIT RISKS.................................................................72
6.1 Estimation and Prediction of the State Variables ............................................. 72
6.2 Forecasting of Housing Price and Interest Rate................................................ 75
6.3 The Cox Proportional Hazard Model................................................................ 77
6.3.1 Data and method ....................................................................................... 77
6.3.2 Results.......................................................................................................81

CHAPTER 7: CONCLUSIONS ........................................................................................86

BIBLIOGRAPHY..............................................................................................................93

APPENDICES ...................................................................................................................99
A1. Pricing Kernel...................................................................................................99
A2. Log Real Return on a Claim to Aggregate Consumption............................... 101
A3. Excess Returns and Risk Free Rate of Return ................................................ 105
A4. Approximation of Real Growth of Housing Consumption............................. 106
A5. Log Real Return on a Claim to Housing Consumption .................................. 107
A6. Estimations Using Standard CCAPM............................................................. 112
A6.1 Methodology...............................................................................................112
A6.2 Data sample.................................................................................................113
A6.3 Results of unrestricted GMM and MLE estimations .................................. 114
A6.4 Results of restricted GMM and MLE estimations ...................................... 118


vii
LIST OF TABLES  

Table 4.1: Summary Statistics of Quarterly Data (1975 Q1 – 2008 Q4).......................... 60

Table 5.1: Parameter Estimations of log Non-housing Consumption Growth
1 t
g
+
........ 62

Table 5.2: Parameter Estimations of log Expenditure Share ( )
1
log
t
α
+
.......................... 63

Table 5.3: Parameter Estimations of intratemporal elasticity of substitution ε ............. 64

Table 5.4: Single Asset GMM Estimations for Housing-Consumption Based CAPM .... 67

Table 5.5: Multiple Asset GMM Estimations for Housing-Consumption Based CAPM. 68

Table 5.6: Single Asset GMM Estimations for Standard CCAPM (1952 Q1-- 2008 Q4) 70

Table 5.7: Multi-Asset GMM Estimations for Standard CCAPM (1952 Q1-- 2008 Q4). 71

Table 6.1: Statistics Summary of FHA Loan Data............................................................ 78

Table 6.2: Maximum likelihood estimations for Deng et al (2000) Hazard Model.......... 79

Table A1: Simple Statistics (1952 Q1 to 2008 Q4)   …………………….114

Table A2: Single Asset GMM Estimations (1952 Q1 -- 2008 Q4)……….115

Table A3: Multiple Assets GMM Estimations (1952 Q1 -- 2008 Q4)…....117

Table A4: Single Asset Maximum Likelihood Estimations (1952 Q1 – 2008 Q4) …....119

Table A5: Multiple Assets Maximum Likelihood Estimations (1952 Q1 -- 2008 Q4) ..120

viii

LIST OF FIGURES

Figure 1.1: Interest Rate and Real Housing Price Growth (1994 Q1- 2008 Q4)................ 2

Figure 3.1: Log Expenditure Share and Time Trend......................................................... 26

Figure 4.1: Comparison among Constant Quality Housing Price Index, Non-quality
Adjusted Housing Price Index, and Quality Index ......................................... 52

Figure 4.2: Relative Price, Relative Consumption, and Expenditure ratio....................... 56

Figure 4.3: Housing Sale Price Index, Housing Rent Price Index, and Housing Price-
Dividend Ratio................................................................................................ 57

Figure 6.1: 1-Quarter to 4-Quarter Ahead Forecast for State Variables............................ 74

Figure 6.2: Housing Rental Index (1970-2008)................................................................ 76

Figure 6.3: Actual Prepayment-Default CPR vs. One-Quarter-Ahead Forecasted CPR .. 83



ix
ABSTRACT

Housing is a macro asset category and has significant impact on the whole
economy. In recent years, some consumption-based asset pricing (CCAPM) literature
states that the optimal consumption-saving/investment decision depends not only on
aggregated consumption but also on composition between housing and non-housing. This
study adopts an Epstein-Zin recursive utility specification to set up a housing
consumption-based capital asset pricing model (HCCAPM) which models the housing
both as an asset and as consumption good, to study the impact of housing consumption
and long-run consumption risks on asset pricing. The study introduces an equilibrium
asset pricing model with housing and presents long-run risks and cointegration between
expenditure share uncertainty and economic growth as other two factors that drive asset
prices. The model reveals that the household not only concerns with uncertainty on
overall consumption and consumption composition between non-housing and housing
service, but also concerns with the long-run uncertainty of the consumption streams. This
study analytically solves the pricing kernel in terms of state variables and derives
equilibrium solutions for two systematic determinants of mortgage credit risk-- the risk
free rate of return and the real return of housing asset.  
In empirical part, the study utilizes the Unobserved Components Method (UCM)
to decompose the unobserved exogenous state variables and GMM approach to estimate
the preference parameters in the pricing kernel. Based on the estimation results, the study
analyzes the mutual interactions between the interest rate and housing return in terms of

x
state variables. This dissertation finds that households are not willing to substitute
consumptions over time. It also provides potential explanations about why housing is on
average more uncertain than other risky assets. Compared with the homeownership ratio,
I argue that the housing quality index is an important alternative indicator of well-being
the shield provides. The fact that the housing consumption per agent had declined since
1991 to 2008 raises questions about the sustainability of the U.S. homeownership policy.  
In the application part, I incorporate these two mutual interacted factors into an
option-based proportional hazard model to forecast risks of FHA-insured mortgage
default and prepayment. I find that this model could deliver quite accurate predictions of
the impacts of macroeconomic dynamics on credit risks than the hazard model that
assumes the risk free rate of return and housing prices are independent processes.  
Key Words:  
Housing consumption, asset pricing, interest rate, housing return, long-run risks

1
CHAPTER 1:  INTRODUCTION
1.1 Research Motivations
Housing plays a double-fold role in the economy -- can be viewed simultaneously
as an asset and a consumption good – and it is critical in asset pricing. On one hand,
housing is a major asset category. Household’s residence is usually the largest single
asset it owns. It accounts for a very large fraction of national wealth. The estimated
market value of the housing stock in the United States is $24.1 trillion at the end of 2005
(Davis and Heathcote, 2007), two times of the 2005 GDP. On the other hand, housing is
the most important single consumption good. Housing is a durable consumption good
which could provide housing services. According to the U.S. Department of Commerce,
housing expenditure is $5.84 trillion in 2007, accounting for 17.7% of aggregated
expenditure.  
Distinctive from other consumption goods, housing is durable, heterogeneous,
expensive and spatially fixed, and very costly to trade in the market. Besides, the stock of
housing changes slowly and the supply is highly inelastic in the short-run, because it
takes roughly a two-year period to develop a new housing project. Hence, housing is
prone to oversupply in prosperity and demonstrates a cyclical nature. These features
make housing sensitive in both magnitude and persistence to the overall economic shocks
and particularly, to the long-run uncertainty. Housing has important macroeconomic
impacts on the pricing of universal assets, not only on the pricing of housing related
assets. Besides, the effect of housing sector on the economy has been further amplified if

2
the housing assets are over-leveraged through mortgage lending and securitization, just as
one can find in this subprime and financial crisis.

Figure 1.1: Interest Rate and Real Housing Price Growth (1994 Q1- 2008 Q4)
-2
0
2
4
6
8
10
1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008
Percentage
Interest Rate Real Housing Price Growth

Figure Description: Interest rate is obtained from Fed. Reserve Bank at St. Louis; Housing price data are
OFHEO (Office of Federal Housing Enterprise Oversight)’s seasonal adjusted repeat-sales housing price
index (HPI).

The booming U.S. mortgage market and the fast development of a mortgage-
backed-securities (MBS) market in the past decades plays a very critical role in helping
households achieve homeownerships. However, since the middle of 1990s, particularly
during the period of 2001-2003, triggered by the low interest rate, the housing price
increased very fast. The expectation that housing prices would continue to appreciate
coupled with the homeownership expending policy in both the Clinton and Bush
administrations had further encouraged many borrowers to obtain mortgages that they

3
cannot afford, therefore exaggerated the housing bubble and resulted in a rapid expansion
of the subprime industry. Subprime loans made up 12.75% of the $10.2 trillion mortgage
market in 2006 and subprime originations accounted for nearly 23 percent of total
residential originations. Since most of the new mortgage products including subprime
have not been enough time to go through a full housing business cycle. As the housing
price cools down and the interest rate goes up, the sharp rise in subprime and prime
foreclosures placed the whole economy in jeopardy. Overbuilding caused home prices to
decline beginning in the summer of 2006. By May 2008, housing price had fallen 18.4%.
Moody's Economy.com estimates that 8.8 million homeowners, or about 10.3% of homes,
will have zero or negative equity by the end of the February 2008. This resulted in an
escalating number of subprime house foreclosures. The collapse of the subprime industry
has triggered a global financial crisis since the middle 2007. The U.S. Treasury Secretary,
Henry Paulson called the bursting housing bubble “the most significant risk to our
economy.”
The subprime crisis raises red flags to the mortgage originators, investors and
regulators. It’s not only time to rethink the free market ideology, homeownership policy,
and the high-risk mortgage lending and securitization practices, but it’s time to rethink
the credit risk pricing methodology as well. It reveals that current mortgage pricing
research is insufficient to provide accurate risk valuation in a dynamic process with many
unanticipated interactions among systematic factors (such as housing price and interest
rate) in the course of market adjustments. Therefore, a pricing model from
macroeconomic perspective may effectively resolve the problem caused by this puzzle.  

4
Despite the fact that housing is recognized as an import industry, there is a shortage of
theoretical research on the relationship between housing and the whole economy, the
relationship between mortgage credit risks and macroeconomic conditions or systematic
risk factors, and the relationship among the systematic risk factors of credit risks. This
paper attempts to fill some of these gaps.
Although, recent mortgage pricing studies recognize that besides the interest rate,
housing prices are another important determinant of mortgage termination, but they only
consider houses as underlying assets of mortgage, not a macroeconomic factor. They also
assume housing price and interest rate as independent stochastic processes. But the
subprime crisis reveals that the relationship between these two systematic factors is
complicated. On one hand, as a fundamental economic risk factor, the interest rate could
not only affects property value and real estate cycles, but also affect unemployment and
income and thus affects the residential demand. On the other hand, property as a major
consumption good, could also affect the interest rate. It is questionable that current two-
factor mortgage valuation models (structural-form models such as, Kau et al. (1992), Kau
et al. (1995), Kau and Keenan (1995), and An (2007); reduced form models such as Deng,
Quigley, and Van Order (2000))) can deliver an adequate prediction in the absence of
above interactions. Hence, a deep understanding of the relation between residential
mortgage default risk and systematic factors, such as housing price and interest rate
becomes critical.  



5
1.2 Research Questions and Approaches
This research hopes to provide insights to the relationship between macro
economy and housing industry, and the relationship two systematic determinants of the
residential mortgage default risk – housing price (housing return) and interest rate
through a theoretical model.  This research also aims at examining how these
relationships evolve over time and, particularly, how the housing price and interest rate
dynamics jointly drives the evolvement of residential mortgage default risk.  
So far, this dissertation has four basic research questions:  
First and most important research question is to introduce a theoretical asset
pricing framework – the housing-consumption based asset pricing model (HCCAPM) –
based on the observed properties implied by the historical data of the whole economy and
housing market. Based on some assumptions on the economy in which the housing and
non-housing aggregated consumption shocks are the only sources of uncertainty, I can
derive the stochastic discount factor (pricing kernel).  
Second, using this theoretical model to get the analytical solutions of the interest
rate and housing return (or housing price) as the functions of the time-variant exogenous
state variables and examine the observed properties of the housing and macroeconomic
data, such as consumptions, consumption composition, risky and riskless asset prices,
price-dividend ratio (price-rent ratio for housing asset), etc.
Third, the study utilizes the Unobserved Components Method (UCM) to estimate
the parameters of the underlying economic forces and utilizes the Generalized Method of
Moments (GMM) approach to estimate the preference parameters in the pricing kernel.

6
Based on the estimation results, the study will analyze the mutual interactions between
the interest rate and housing return in terms of those state variables.  
Furthermore, the study employs the model to predict interest rate and housing
prices by means of the Kalman filter. Based on these predicted interest rates and housing
prices and the observed default events of individual loans (FHA dataset), the study will
utilize a reduced form option-based proportional hazard model proposed by Deng,
Quigley, and Van Order (2000) to predict competing risks of FHA-insured mortgage
default and prepayment and examine the predictive performance of my model.  

1.3 Summary of Main Findings and Results
Numerous general economics studies investigate the dynamics of some
macroeconomic variables, such as interest rate and bond default yield spread, and the
business cycle of some specific industries, such as tourism and banking industry.
Nevertheless, few studies emphasize on the nature of housing price dynamics while the
real estate is increasingly regarded as a major asset class (such as He (2002))
1
. Although,
most previous mortgage credit risk studies assume that the interest rate and property
prices are two independent individual processes, I believe that they are mutually
interacted as stated by some studies such as Cochrane (1991a/b and 1996), Kullmann,
(2002), Piazzesi et al (2006 and 2007) and Lustig and Van Nieuwerburgh (2006). The
Piazzesi et al (2007) is an important work in recent asset pricing literature which attempt
                                               
1
He (2002) argues that the real estate factor could be regarded as a macro factor in the APT framework because real
estate is a major asset class, and almost all firms have operating expenses under this category. A significant portion of
corporate assets, on average as high as 25%, is real estate related.



7
resolve the risk-free rate puzzle, equity premium puzzle, and volatility puzzle by
incorporating the housing service as a unique consumption good and therefore the
stochastic housing consumption as the second risk factor into the pricing kernel. However,
Piazzesi et al (2007)’s work is far from conclusive because no empirical evidence is
presented to support their arguments. Fillat (2008) extends the Piazzesi et al (2007)’s
study by introducing the Epstein and Zin (1989) recursive utility to deal with the long-run
risk issue but he cannot give the analytical solution for the pricing kernel and only
consider the intertemporal elasticity substitution 1 ψ = special case, therefore restricts its
explanatory power of using housing as a measure of long-run risk in asset pricing. Their
job also has mismeasurement problem on the housing consumption. Most importantly it
ignores the long-run consumptions risks in the model setup.  
My study uses a new housing consumption-based capital asset model (HCCAPM)
to study the relationship between the housing price and interest rate dynamics – the key
two determinants of mortgage credit risks. Similar with Piazzesi et al (2007) and Fillat
(2008)’s work, it models the housing both as an asset and a consumption good and
includes the housing consumption into the pricing kernel. Unlike their work, my paper
develops a housing quality index as the measurement of housing consumption instead of
using the NIPA housing quantity index to avoid the mismeasurement issue. I find that as
the homeownership ratio kept increase, the housing consumption per household have
declined 14% since year of 1991 which is contrary to the original intention of the U.S.
homeownership policy. Besides the homeownership ratio, I argue that the housing quality
index is an important indicator to measure the housing well-beings.

8
I assume that the utility function of a representative agent takes the Epstein and
Zin (1989) recursive utility specification in order to capture the long-risks implied by the
observed properties of the data. Using the non-housing consumption as numeraire, a
general specification of the pricing kernel is derived. Based on the assumptions on the
dynamics of economy and the stochastic generating process of price-dividend ratio of the
claim to aggregate consumption, my paper numerically solve the pricing kernel in terms
of four time-variant exogenous state variables. Equilibrium analytical solutions of real
log housing return and risk free rate of return in terms of state variables are also obtained.
This model reveals that the household not only concerns with the consumption growth
and consumption composition between the non-housing and housing service, but also
concerns with the long-run risks introduced by the Epstein and Zin recursive utility. I also
find that the model is helpful to resolve the risk-free puzzle in the standard asset pricing
model.  
In the empirical section, I adopt the Unobserved Components Method (UCM)
technique to estimate the parameters of the economic forcing dynamics and utilize the
Generalized Method of Moments (GMM) approach to estimate the preference parameters
in the pricing kernel. The data applied are quarterly data ranging from the first quarter of
1975 to the fourth quarter of 2008. Several key results are found. First, as expected, the
value of risk aversion parameter γ is well above 1 and the values of inverse of the
intertemporal elasticity of substitution (IES) 1/ ψ is negative but larger than 1 −
( 11/ 0 ψ −< < ). This means that households are not willing to substitute consumption
over time. This finding proves that the IES doesn’t have to be nonnegative, and it is

9
consistent with many previous empirical works in asset pricing. Second, the coefficient
on the conditional variance
2
t
σ is significantly positive for housing asset and is negative
for riskless asset. It means expected higher economic volatility will make the agents buy
more riskless asst and buy less risky housing asset therefore pushing the housing return
up and the interest rate down. Third, as suggested by the model, a negative 1/ ψ also
means that the coefficient on log expenditure share is negative. This relationship is
strongly supported by the data. Furthermore, my model could well explain the disparities
of the coefficients on the expected consumption growth and constant term between the
risky asset and riskless asset, which could not be well resolved in the standard CCAPM,
single-good Epstein-Zin Utility CCAPM, and the Piazzesi et al (2007)’s housing CCAPM
frameworks. I denote these disparities as the long-run risk that introduced by the joint
impacts of the Epstein-Zin utility and cointegration between the expenditure share and
expected consumption growth. The fear of the long-horizon uncertainty of the
consumption streams leads to a risk premium in the constant term and results in a higher
sensitivity on expected consumption growth for returns of risky assets such as housing.  
In the application section, I incorporate these two mutual interacted factors – risk free
rate of return and housing return (housing price) into an option-based proportional hazard
model to forecast risks of FHA-insured mortgage default and prepayment. I find that this
model could deliver quite accurate predictions of the impacts of macroeconomic
dynamics on credit risks than the hazard model that assumes the risk free rate of return
and housing prices are independent processes.


10
1.4 Expected Contributions
In theoretical study aspect, the study is expected to contribute theoretical research
of the asset pricing and mortgage default risk by providing a new asset pricing model
from a time series perspective which also pays special attention on the dynamics and
patterns of two systematic factors and their mutual interactions into account. My thinking
is mostly inspired by recent discussions of dynamic credit risk modeling in general
finance studies. But no previous study examines the dynamics and interactions among the
systematic factors of default risk in the residential mortgage area.  
One of the major contributions of this study is the development of a housing
quality index as the measurement of the housing consumption. While previous literature
use the NIPA housing quantity index as housing consumption, I argue that the NIPA
quantity index is actually an index of the total number of tenant- and owner-occupied
housing units. It includes the location related amenities, unit area, and housing structure
attributes and hence it is a better measurement of housing service.  
In the empirical side, on the basis of my theoretical framework, the model is
expected to deliver a more adequate prediction of long term dynamics of residential
credit risk than previous literature. It could also provide hedging implications for
portfolio risk management. More and more scholars believe that the subprime crisis is an
extremely expensive cost to learn a new financial innovation. My study tries to provide
some business and policy-related implications for the subprime crisis because my
interpretations on the default risk arise from the underlying driving forces and from a
long-horizon perspective rather than from a static and cross-sectional perspective.  

11
1.5 Dissertation Outline
The reminder of the dissertation is organized as follows. Chapter 2 is the related
literature review in the asset pricing and mortgage credit risk studies. Chapter 3 presents
the theoretical model including the model setup and analytical solutions of the pricing
kernel, risk free rate of return and housing return. Chapter 4 describes the data sources
and method of how to construct a comprehensive housing quality index. Chapter 4 also
discusses the observed properties of the economy. Chapter 5 reports the empirical results
of the housing consumption-based asset pricing model. Chapter 6 is an application of the
theoretical model on the credit risk prediction based on a reduced form hazard pricing
model. The last Chapter gives results, contributions, and questions for future research.

12
CHAPTER 2:  LITERATURE REVIEW
Basically, my research questions are motivated by two broad strands of literature -
- the capital asset pricing models (CAPM) and mortgage credit risk models. Section 2.1
reviews the related multiple-factor CAPM models and Consumption-based CAPM
models. Section 2.2 summaries default and prepayment risk literature.  

2.1 Asset Pricing Literature
From the paper of Merton (1973), more and more researchers accepted the view
that housing has significant effect on the general asset pricing.  Although the results have
been well accepted, the mechanism of this effect is widely debated. To explain this,
various asset pricing theories have been presented. These theories can be classified into
two general categories: multiple-factor CAPM models and consumption-based CAPM
models.
(1) Multiple-factor CAPM models
Merton (1973) first introduces owner-occupied housing into the classical
intertemporal capital asset pricing model (CAPM) of Sharp (1964) and Lintner (1965).
This paper provides more insights in asset pricing. Merton (1973) treats the owner-
occupied housing as both a consumption goods and an asset. Since housing consumption
reflects household's expectation about future asset returns, it is natural to assume that the
housing variable contains information about financial asset returns. Based on this
presumption, Merton derives the analytical solution of a multi-factor pricing model.  

13
Grossman and Laroque (1990) proposed an influential work based on the theory
of optimal consumption and portfolio selection in the presence of illiquid housing. They
argue that the standard, one-factor, market portfolio based CAPM does not hold in this
environment and consumption based capital asset pricing model (CCAPM) also fails hold.
But Grossman-Laroque model relies on the assumption of constant house price, and
absence of the nondurable consumption good. Other papers, such as Beaulieu (1993),
Eberly (1994), and Dunn (1998), Flavin and Yamashita (2002), Cocco (2005), Yao and
Zhang (2005) also present similar models which studies particular implications based on
Grossman and Laroque (1990)’s setup.  Recently, Flavin and Nakagawa (2008)
generalizes the Grossman and Laroque (1990) model by relaxing the above assumptions.
House price is a stochastic process and both the housing and nondurable consumption are
included in the utility function. By introducing an endogenously determined but
infrequently adjusted state variable, Flavin and Nakagawa (2008)’s housing model
generates many of the implications of the Grossman-Laroque model, such as smooth
nondurable consumption, state-dependent risk aversion, and a small elasticity of
intertemporal substitution.  
In empirical studies side, He (2002) argues that the real estate factor could be
regarded as a macro factor in the arbitrage pricing theory framework of Ross (1976)
because real estate is a major asset class. His study indicates that along with the stock and
bond factors, the unsecuritized real estate market factor plays an important role in
explaining excess returns on industrial stocks. Cochrane (1991a/b, 1996) investigates real
estate investment as a pricing factor in a production-based asset pricing approach and

14
find real-estate investment growth predicts stock returns. Kullmann (2002) confirms
Cochrane’s result. Moreover, he finds the important component in real-estate investment
is residential real estate, not commercial real estate.  
(2) Consumption Based CAPM models
Consumption based CAPM (CCAPM) model is an expansion of the CAPM. The
CCAPM uses consumption measures, in terms of a consumption beta, in the calculation
of an asset’s expected return.  
Davis and Martin (2005) first introduces housing service into a neoclassical
consumption based CAPM to explain the equity puzzle. They find that households are not
willing to substitute housing service for consumptions. But their job doesn’t get an
analytical solution of the pricing kernel. Piazzesi et al (2006 and 2007) are important
work in recent asset pricing literature which attempt to resolve the risk-free rate puzzle,
equity premium puzzle, and volatility puzzle by incorporating the housing service as a
unique consumption good and therefore the stochastic housing consumption as the
second risk factor into the pricing kernel. However, Piazzesi et al’s work is far from
conclusive because no empirical evidence is presented to support their arguments. Fillat
(2008) extends the Piazzesi et al (2007)’s study by introducing the Epstein and Zin (1989)
recursive utility to deal with the long-run risk issue but his paper does not give the
analytical solution for the pricing kernel and only consider the intertemporal elasticity
substitution (IES) 1 ψ =  special case. Like Piazzesi et al (2007), Fillat (2008) have
mismeasurement problem of the housing expenditure and housing consumption.  


15
2.2 Related Mortgage Credit Risk Literature
Recent empirical mortgage pricing studies have increased in realism and
sophistication in the past decade. One important progress is that some studies incorporate
both the default and prepayment risks into one structural framework.  
In the mortgage risk measurement side, as summarized by Deng and Quigley
(2002), recent research on the economic behavior of mortgage holders yields three well-
known insights. First, the contingent claims model provides a coherent and useful
framework for analyzing borrower behavior. Second, the joint of the prepayment and
default options is important in explaining behavior. Third, competing risks models
provide a convenient analytical tool for analyzing borrower behavior.
Schwartz and Torous (1989) attempt to estimate the parameters of a proportional
hazard model for prepayment and default decisions. Their results indicate significant
regional differences in prepayment and default behavior. But they cannot formulate the
competing relationship between the call and put option. Kau et al (1992) and Kau et al.
(1995) first outline the theoretical relationships among the options. They also recognize
the necessity of incorporating both the interest rate and the house price as state factors to
capture the interaction of default and prepayment. However, their estimation methods for
proportional hazard model still have some difficulties in handling the competing risks.
Plus, they perform no empirical estimation or testing of their structural models.
Deng, Quigley and Van Order’s (2000) reduced-form model introduces maximum
likelihood estimation of competing risks hazard model, which provide a solution to
empirical modeling of competing risks of prepayment and default with proportional

16
hazard model. Their model estimates the heterogeneity simultaneously with baseline
hazards associated with prepayment and default functions. Their results find the
significant heterogeneity among borrowers.  
In the mortgage valuation side, a recent working paper of Kau et al. (2004)
develop a new reduced-form approach to estimate default and prepayment processes
jointly, and thereby, to value mortgages. However, compared with the flourish in
mortgage risk measurement studies, the risk valuation side is understudied.  
Recent mortgage credit risk studies recognize that besides the interest rate, house
prices are another important systematic determinant of mortgage termination, but they
only consider their direct impact on default behavior (structural-form models such as,
Kau et al. (1992), Kau et al. (1995), Kau and Keenan (1995), and An (2007); reduced
form models such as Deng, Quigley, and Van Order (2000)). When house prices fall, an
increase in the likelihood of default is expected. Structural form models also assume
these two systematic factors as independent stochastic processes. But the general finance
and economics literature and recent practical experience including the subprime crisis
reveal that the relationship between these two systematic factors is complicated. On one
hand, as a fundamental economic risk factor, the interest rate could not only affects
property value and real estate cycles, but also affect unemployment and income and thus
affects the housing demand. On the other hand, property as a major consumption good,
could also affect the interest rate according to some asset pricing studies. Moreover,
evidence also indicates the mutual interaction between default and the state of macro-
economy (Koopman and Lucas, 2005, 2007). It might be questionable that current two-

17
factor mortgage valuation models can deliver an adequate theoretical framework in
absence of the above interactions. Hence, can the existing mortgage credit risk model be
revised in this regard leaves challenges for future research.

18
CHAPTER 3:  THEORETICAL MODEL
3.1 Set Up
3.1.1 Consumers
Consider an economy in which there are many identical infinitely-lived agents and
they aim to maximize their expected value of the time-additive intertemporal discounted
utility function:
0
t
t
t
E u δ
∞
=
⎡⎤
⎢⎥
⎣⎦
∑ .  
Their preferences over the aggregate consumption
t
C take the Epstein and Zin
(1989) and Weil (1989) recursive utility specification
()
() ()( ) ( )
()( )
1
1
1
1
1
1
1
1
1
1
1111 1
,
11
ttt
tttt
CEu
uuCE u
γ
ψ
ψ
ψ
γ
δδ δ γ
δγ
−
−
−
−
−
+
+
⎧⎫
⎪⎪
− ++ − − − ⎡⎤
⎨⎬
⎣⎦
⎪⎪
⎩⎭
== ⎡⎤
⎣⎦
−−
      (3.1)
The parameter ψ is the intertemporal elasticity of substitution (IES). It implies that
consumers will smooth their consumption given the expected time profile of real interest
rates and lifetime wealth. Thus, consumers respond to an increase in current income by
raising both current and future consumption (Mao, 1989). A higher value of ψ means the
agents are more willing to substitute aggregate consumption over time.  
The coefficient of relative risk aversion (RRA) γ determines the curvature of the
value function. (0,1) δ ∈ represents the subjective discount factor of uncertainty. The
Epstein-Zin preference provides a convenient method to disentangle the intertemporal
elasticity of substitution ψ from the coefficient of relative risk aversion γ . The

19
expected value operator conditional on information set at time t is denoted as ( ) .
t
E . The
expectation term in the Epstein-Zin preference implies that agents care about the
continuation utility value relative to its risk adjustment (Hansen, Heaton, and Li, 2008).  
This Epstein-Zin preference has been used in many asset pricing literatures (such
as, Attanasio and Weber 1989; Epstein and Zin 1991; Normandin and St-Amour 1998;
Epaulard and Pommeret 2001; Bansal and Yaron, 2004).  

3.1.2 Technology
Following Locus (1978), I assume there are two perishable consumption goods
(fruits) which are produced by two assets (trees) with positive net supply: housing and
non-housing asset. At time t, The house pays a stream of housing services (housing
consumption)
t
s with price
s
t
p and the non-housing asset pays a stream of housing
services
t
c with price
c
t
p . The aggregated consumption
t
C is denoted as an index that is
composed of two goods -- housing service h
t
and non-housing consumption c
t
:  
()
11
1
( , )             0
ttt t t
Cfcs c ws
ε
εε
ε
εε
ε
−−
−
⎛⎞
== + >
⎜⎟
⎝⎠
                                                (3.2)
The parameter ε is the intratemporal elasticity of substitution between housing and
non-housing consumption. A higher value of ε means the agents are more willing to
substitute consumption across housing and non-housing within time t .
The intra-temporal first order condition (FOC) results in the relationship between
the marginal rate of substitution and relative price,  

20
1
(, ) 1
(, )
c
tctt t
s
tstt t
pfcs c
pfcs s
ε
ω
−
⎛⎞
==
⎜⎟
⎝⎠
                                                                                   (3.3a)
This FOC implies that the relative price move in opposite directions with relative
consumption because of 0 ε > .  
Based on equation (3.3a) it is obvious to derive the expenditure ratio  
1
1
1
c
tt t
t s
tt t
pc c
z
ps s
ε
ω
−
⎛⎞
==
⎜⎟
⎝⎠
                                                                                          (3.3b)
It is noteworthy that if ε is greater than 1, the expenditure ratio
t
z is positively
correlated to relative consumption
t
t
c
s
but negatively correlated to relative price
c
t
s
t
p
p
. This
feature is consistent with the empirical observations.  
Further, applying the FOC, the price index
C
t
P associated with the aggregate
consumption
t
C could be derived as
() ()
() ( )
11
1
1
11
1
11
1


cs
tt t t
C
Ccs tt
t tt
t
tt
pc p s
PC
Ppp
C
cs
ε
εε
ε
εε
εε
ε
ε
ε
εε
ε
εε
ω
ω
ω
−−
−
−−
−
−−
−
⎡⎤
+
⎢⎥
⎣⎦
⎡ ⎤
== = +
⎢ ⎥
⎣ ⎦
⎛⎞
+
⎜⎟
⎝⎠
               (3.4)

3.2 Pricing Kernel
To derive the pricing kernel which uses the non-housing consumption as the
numeraire, we begin with the pricing kernel that uses the aggregate consumption as

21
numeraire. Epstein and Zin (1989), Weil (1989), and Bansel and Yaron (2004) show that
the asset stochastic discount factor (pricing kernel) with recursive preferences is:
1 1
1,1
t
tCt
t
C
MR
C
θ
ψ
θθ
δ
−
− +
++
=                                                                                           (3.5)
where
,1 Ct
R
+
is the real gross return on the asset that deliver aggregate consumption
t
C as
dividends in period t . Parameter (0,1) δ ∈ is the time discount factor and the parameter
1
11/
γ
θ
ψ
−
≡
−
. The sign of θ is determined by the value of γ and ψ . Many previous (such
as Bansel and Yaron (2004), Mehra (2003), Epstein and Zin (1989), and Mehra and
Prescott (1985)) asset pricing studies argue that the risk aversion parameter γ is well in
excess of 1. The opinions on IES ψ are widely divergent, many studies, such as Hansen
and Singleton (1982, 1983), Eichenbaum, Hansen, and Singleton (1986), and Guvenen
(2000) argue that IES is larger than 1, while, Campbell (1999) and Hall (1988) estimate it
to be smaller than 1. The study by Hansen and Singleton (1988) even produces a negative
elasticity estimate depending on the data set used. However, as Bansel and Yaron (2004)
argue, an IES greater than 1 (or 1/ 1 ψ < ) is critical for capturing the observed negative
correlation between consumption volatility and price-dividend ratios of assets.  
It is worth noting that in special case when the inverse of the IES
1
ψ
equals to RRA
γ ,  1 θ = and the pricing kernel will collapse to the standard form with power utility.  
However, we are more interested to the pricing kernel which uses the non-housing
consumption as the numeraire. Applying the formula for the pricing kernel

22
() ()
( ) ( )
() ( )
11
11
tt
ttt
tt
uC f c
MuEu
uC f c
++
++
′′
′=×
′′
                                                               (3.6)
I can solve the different derivatives from (3.1) and (3.2), and substitute them into (3.6).
After canceling and rearranging I obtain the expression:  
()
1
1
1
1
1
1
1
1 1 1
1 1
1
1
1
t
t tt t
t
tt
t
t
s
cEV c
M
cV
s
c
ε
ψ
ε
ε
ε
γ
ψ
ψ γ
ε
ε
ω
δ
ω
−
−
−
−
+
−
−
+ + +
+ −
+
⎛⎞
⎛⎞
⎜⎟
+
⎜⎟
⎜⎟
⎛⎞ ⎛ ⎞
⎝⎠
=
⎜⎟
⎜⎟ ⎜ ⎟
⎝⎠ ⎝ ⎠ ⎜⎟
⎛⎞
+⎜⎟
⎜⎟
⎜⎟
⎝⎠ ⎝⎠
                                        (3.7)
where
1 t
V
+
denotes the value function. Epstein and Zin (1989) and Weil (1989) show that  
()
()
1
1
11
11
1
11
1
1 1
,1
1
tt t
Ct
tt
EV C
R
VC
γ
γγ
ψ
ψ ψ
ψ
δ
−
−−
−−
−
+ +
+
+
⎛⎞
=
⎜⎟
⎝⎠
                                                                     (3.8)
Also, using the FOC (3.3a) one finds that  
1
1
1
1 11
1
1
=
1
t
t tt
tt
t
t
s
c Cc
Cc
s
c
ε
ε
ε
ε
ε
ε
ω
ω
−
−
+
+ ++
−
⎛⎞
⎛⎞
⎜⎟
+
⎜⎟
⎜⎟
⎝⎠
⎜⎟
⎜⎟
⎛⎞
+⎜⎟
⎜⎟
⎜⎟
⎝⎠ ⎝⎠
                                                                               (3.9)
Substituting (3.8) and (3.9) into (3.7), I find that
()
11 1
1
1
1
11
1 1
1
11
1
1 1
1
1 1
1
1 1
1 ,1 1
1
1
t
t t
t Ct
t
t
t
s
c c
MR
c
s
c
εεγ
ψψ ψ
ε
ε
ε
ψ
γ γ
γ
ψ
ψ ψ
ψ
ε
ε
ω
δ
ω
−− +
−
⎛⎞
−−
⎜⎟ ⎛⎞
⎝⎠
⎜⎟ − −
+
−
−
⎜⎟ −
− − ⎜⎟
−
⎜⎟ + +
⎝⎠
+ + −
⎛⎞
⎛⎞
⎜⎟
+
⎜⎟
⎜⎟
⎛⎞
⎝⎠
=
⎜⎟
⎜⎟
⎝⎠⎜⎟
⎛⎞
+⎜⎟
⎜⎟
⎜⎟
⎝⎠ ⎝⎠
                          (3.10)


23
Further, define the expenditure share on non-housing consumption  
c
tt
t cs
tt t t
pc
pcps
α =
+
                                                                                                   (3.11)
and along with the FOC, it is easy to derive that
1 t
t
α
α
+
=
1
1
1
1
1
1
1
t
t
t
t
s
c
s
c
ε
ε
ε
ε
ω
ω
−
−
+
+
−
⎛⎞
⎛⎞
⎜⎟
+
⎜⎟
⎜⎟
⎝⎠
⎜⎟
⎜⎟
⎛⎞
+⎜⎟
⎜⎟
⎜⎟
⎝⎠ ⎝⎠
                                                                                     (3.12)
Therefore, the pricing kernel (3.10) could be rewritten as
1 1
11 ,1
t
tt Ct
t
MG R
τ
θ
θ θ ψ
α
δ
α
−
− +
++ +
⎛⎞
=
⎜⎟
⎝⎠
                                                             (3.13)
where
1
1
t
t
t
c
G
c
+
+
= is the gross growth rate of non-housing consumption. Parameter
()
11 1
1
1
11
ε εγ
ψψψ
τ
ε
ψ
−+ + −
≡
⎛⎞
−−
⎜⎟
⎝⎠
and
1
11/
γ
θ
ψ
−
≡
−
.
This model reveals that the household not only concerns with the non-housing
consumption growth but also concerns the composition among the non-housing and
housing consumptions. The consumption risk is captured by the term
/
1 t
G
θ ψ −
+
and the
composition risk is captured by the third term
1 t
t
τ
α
α
+
⎛⎞
⎜⎟
⎝⎠
. The last term
1
,1 Ct
R
θ −
+
means that
the agents are also concerned with the real return of aggregated goods which is
introduced by the recursive preference. Since the return of the aggregate consumption

24
contains the information about the expectation of future dividend streams, this term
reflects the long-run risks. The composition risk factor emerged from the change in the
expenditure share and long-run risks emerged from the uncertainty on the utility
continuation, generate higher equity premium than under the standard model. Thus my
model can help explain equity premium puzzle.  
Piazzesi et al (2006, 2007)’s asset pricing model could be interpreted as a special
case of the general model when 1 θ = (1/ ψ γ = ). They believe that the pricing kernel is
higher in states that non-housing consumption growth is lower and the expenditure share
on non-housing consumption tomorrow is higher than today.  

3.3 Analytical Solutions
Although I derive the model numerically, noted that the real return on a claim to
aggregate consumption
,1 Ct
R
+
is unobservable and the mechanisms that generate the
dynamics of non-housing consumption growth
1 t
G
+
and expenditure shares
1 t
α
+
are not
clear. In this subsection, I will utilize an approximate analytical framework to model the
forcing process of
1 t
G
+
and
1 t
α
+
along with
,1 Ct
R
+
. Then I derive the analytical solutions
and estimate the preference parameters.  
As a convention, throughout this dissertation, I use capital letters to denote gross
returns or growth (such as
,1 Ct
R
+
and
1 t
G
+
) and use lowercase letters to denote log
(continuously compounded) returns or growth (such as
()
,1 ,1
log
Ct C t
rR
++
= and
()
11
log
tt
gG
++
= ).  

25
Therefore, the logarithm form of the pricing kernel is
() ( )
11 1 ,1
log log 1
tt t Ct
mg r
θ
θδ τ α θ
ψ
++ + +
=− +Δ +−                                            (3.14)
where τ and θ are defined as before.  

3.3.1 Dynamics of the system
I assume the only uncertainties in economy are the shocks to housing and non-
housing consumptions. I model the growth of non-housing consumption
1 t
g
+
containing a
unobservable time trend component
t
μ , a unobservable persistent stationary
autoregressive component
t
x which determines the shocks of the economy, and a time-
varying volatility component
t
σ which represents the fluctuating economic uncertainty.  
11 tt t tt
gx μ ση
++
=+ +
11 tt t μ
μ μσ ζ
++
=+
11
                                      ( 1 1)
tt xt
xx e ρσρ
++
=+ −< <
( )
22 2 2
11
                 (0 1)
tt wt
w σσ νσ σ σ ν
++
=+ − + < <
11 1
, , ,  ~ . . . (0,1)
tt t t
ew iidN η ζ
++ +
                                                                     (3.15)
For the tractability of the model, I assume the shocks
1 t
η
+
,
1 t
ζ
+
, and
1 t
e
+
are
independent and normally distributed random variables with mean zero and variance one.
Therefore the gross aggregated consumption growth is lognormally distributed.  
I impose several additional parameters
μ
σ ,
x
σ , ρ ,
w
σ ,
2
σ and ν to calibrate the
dynamics of the non-housing consumption growth. The trend term
t
μ is a unit root

26
process with conditional variance
2
μ
σ . The parameter ρ determines the persistence of
auto-regression component
t
x and
2
x
σ is its conditional variance. The consumption
conditional variance
2
t
σ represents the time-variant economic uncertainty and it is
assumed to be a mean-reverting process with long-term mean
2
σ , mean reversion rate ν ,
and conditional variance
2
w
σ . Since the economy always adjusted gradually, I expect the
mean reversion rate 01 ν >> .  

Figure 3.1: Log Expenditure Share and Time Trend
-0.18
-0.17
-0.16
-0.15
-0.14
-0.13
-0.12
-0.11
-0.1
Mar-75
Mar-77
Mar-79
Mar-81
Mar-83
Mar-85
Mar-87
Mar-89
Mar-91
Mar-93
Mar-95
Mar-97
Mar-99
Mar-01
Mar-03
Mar-05
Mar-07
Log Expenditure Share
log Expenditure Share


The log expenditure share
1 t
α
+
is assumed to follow a mean-reverting process with
mean reversion rate β , unconditional mean ( ) at
α
μ + , and conditional variance
2
,t α
σ .  

27
()
1,1
log log              (0 1)
tt tt
at at u
ααα
αμ β α μ σ β
++
=+ + − − + < < (3.16a)
Historical data indicates that the
1
log
t
α
+
series demonstrate a time trend pattern
(particularly from 1983) and empirical regression results also shows that the conditional
variance of
1
log
t
α
+
changes with the move of consumption growth. To capture this
properties, I assume the conditional mean is trending and the conditional variance
2
,t α
σ to
be linear in the expected growth of the log non-housing consumption
tt
x μ + . And I
expect
1
0 a > .
()
2
,0 1 tt t t
E aa x
α
σμ ⎡⎤=+ +
⎣⎦
                                                                            (3.16b)
In order to derive the analytical solutions of model (3.14), following Bansel and
Yaron (2004)’s method, I use the approximation of
,1 Ct
r
+
proposed by Campbell and
Shiller (1988).  
,1 0 1 ,1 , 1
log
C t Ct Ct t
rqq C κκ
++ +
+ −+Δ                                                                  (3.17a)
2
,0 1 2 3 4
log
Ct t t t t
qA Ax A A A μ σα ++ + +                                                             (3.18)
where
, Ct
q denotes the log price-consumption ratio
2
of the claim to aggregate
consumption
t
C . The parameter
0
κ and
1
κ are constants that can be estimated from the
mean of
, Ct
q
3
.
                                               
2
Or price-dividend ratio of the asset that deliver aggregated consumption as its dividends each period
,,
/
Ct t c Ct
qCPw =
,
, Ct
w represents the wealth of asset.  
3
According to Campbell and Shiller (1988),
0
κ and
1
κ could be approximated by
( ) ( )
1, ,
exp 1 exp
Ct Ct
qq κ ⎡⎤ =+
⎣⎦

and
() ( )
01 1 1 1
1 log 1 log κκ κ κ κ =− − −
. Bansal and Yaron (2004) estimate
1
0.997 κ =
for monthly frequency data.
Therefore, I can derive that
1
0.991 κ =
and
0
0.0514 κ =
for quarterly frequency data.  



28
Using formula (3.9) and (3.12), I rewrite (3.17a) as:  
()
,1 0 1 , 1 , 1 1
log
1
Ct C t Ct t t
rqqg
ε
κκ α
ε
++ + +
+− + + Δ
−
                                           (3.17b)
Since
1 t
g
+
and
1
log
t
α
+
are the only sources of uncertainty in the economy and
1 t
g
+

could be further decomposed into state variables
t
μ ,
t
x , and
t
σ , the solutions to
1 t
g
+
and
1
log
t
α
+
can provide a complete solution to return
,1 Ct
r
+
by using the equation (3.18) and
(3.17b).  

3.3.2 Equilibrium solutions of
,1 Ct
r
+
and its excess returns
Since the asset pricing condition
( )
1,1
exp 1
tt it
Em r
++
⎡ ⎤ + =
⎣ ⎦
must hold for any asset i
()
,1 ,1
log
it i t
rR
++
= including the risk-free asset
,1 f t
r
+
and aggregate consumption claim asset
,1 Ct
r
+
, we have the Euler equation
()
11 ,1
exp log log 1
tt tCt
Eg r
θ
θδ τ α θ
ψ
++ +
⎡⎤ ⎛⎞
−+Δ + =
⎢⎥ ⎜⎟
⎝⎠ ⎣⎦
                                      (3.19)
Substituting dynamics of consumption growth and expenditure shares (3.15) and
(3.16) along with the approximation (3.18) and (3.17b) into Euler equation (3.19) and
yields the equation in terms of the state variables
t
x ,
t
μ ,
t
σ , and log
t
α .  
Since the Euler equation must be satisfied for all values of the four state variables,
utilizing the fact that  
1,1 1 ,1
1
var 0
2
tt it tt it
Em r m r
++ ++
++ + = ⎡⎤ ⎡⎤
⎣⎦ ⎣⎦
                                                              (3.20)

29
and after some straightforward algebra (see Appendix A1 and A2 for details), one can
solve the coefficients on the price-consumption ratio,
0
A ,
1
A ,
2
A ,
3
A , and
4
A  
11
1
1
1
1+
1
a
A
π
ψ θ
κρ
−
=
−

11
2
1
1
1+
1
a
A
π
ψ θ
κ
−
=
−

()
()
() ( )
()
2
3
11
11 11 1
=
21 21
A
θ ψψ γ
κν κν
−− −
=
−−

()
()
4
1
1-
-1
A
φ β
θκ β
=
() ( )( )
()( )()
2
01 3 4
0
22 2 10
1
log 1 + 1
1
=
1
1
2
xw
AA at
A
a
at
α
αμ
δκ κ ν σ β μ
φθ π
κ
βμ λ λ λ
θθ
⎧⎫
⎡ ⎤ ++ − − + +
⎣ ⎦
⎪⎪
⎨⎬
−
−+ + + + +
⎪⎪
⎩⎭
                      (3.21)
where
1
1
εγε
φ
ε
−−
=
−
,
1
11/
γ
θ
ψ
−
=
−
,
( )
()
2
2
1
1 2
1
1
2-1
φκ
π
κβ
−
= ,
11 x x
A λ κσ = ,
12
A
μμ
λ κσ = , and
13 ww
A λ κσ = .
It’s worthy noticing that, the sign of
1
A ,
2
A , and
3
A is determined by the
magnitude of  IES, ψ . Bansel and Yaron (2004) argues that an IES that is larger than 1 is
critical to capturing the negative relationship between the economic volatility
2
t
σ and
price-consumption ratio
, Ct
q . This argument is a sufficient condition but not a necessary
one for
1
1
ψ
< . One might immediately find that an negative IES could also capture the
negative relationship between
2
t
σ and
, Ct
q .

30
Since we already know that the risk aversion parameter γ is well above 1, and ε is
greater than 1, hence φ should be greater than 0. If the IES satisfies
1
1
ψ
< ,
4
A is positive
and
3
A is negative. In this case, if the economics uncertainty
2
t
σ is smaller, or/and the
expenditure share of non-housing consumption is larger (this reflects that the economy is
in prosperity, because the housing price might take more rapid growth than that of non-
housing counterparts, hence mitigate the housing service expenditure), agents will try to
buy risky assets, therefore pushing up the price-dividend ratio
, Ct
q of the aggregate
consumption claim asset.  
The signs of
1
A and
2
A are indeterminate because of the term
11
a π
θ
. Since 0 θ <
(given
1
1
ψ
< ) and
1
a is expected to be positive, it follows that
11
0
a π
θ
< . If
1
A and
2
A are
positive, the intertemporal substitution dominates wealth effect. In this case, the agents
will try to purchase more assets as the expected non-housing consumption growth (
t
x
and
t
μ ) becomes higher. If  
1
A and
2
A are negative, the wealth effect dominates the
substitution effect, the agents will try to engage in more consumption activities compared
with investment in risky assets as the expected non-housing consumption growth (
t
x and
t
μ ) becomes higher.  
As show in Table 4.1, the empirical historic data of the U.S. from 1975 to 2008
documents a negative correlation between the conditional volatility
2
t
σ of the non-
housing consumption growth and price-dividend ratio of main risky asset categories such

31
as housing and stocks, and a positive correlation between the expenditure share log
t
α
and asset price-dividend ratios. All these evidences strongly imply that
1
1
ψ
< and it is
consistent with lots of pervious literature, such as Hansen and Singleton (1982), Guvenen
(2000), and Bansel and Yaron (2004). I will discuss it in details in the following sections.
The data also shows a negative correlation between the expected growth of non-housing
consumption
1 t
g
+
and asset price-dividend ratio.  
Using the equilibrium solutions (3.21), and the equation (3.15), (3.18) and (3.17b), I
can derive a complete solution to return
,1 Ct
r
+
(see Appendix A2. for details). Therefore, it
immediately follows that the conditional mean and variance of
,1 Ct
r
+
are
() ( )
()
()
()( ) ()
22 2 10
,1
2 11
1
log
2
11 1 1 1
       log
2
tCt x w
tt t t
at a
Er
a
x
α
μ
τβ μ θ π
δλλλ
θ θ
ψγ τ β π
μ σα
ψθ θ
+
−+
=− − − + + − ⎡⎤
⎣⎦
−− − ⎛⎞
+− + − +
⎜⎟
⎝⎠
                (3.22a)
() ()
2
22 2 2
,1 1 4 0 1
var
1
tCt x w t t t
rAaax
μ
ε
λ λλ κ μ σ
ε
+
⎛⎞
=+ + + + + + + ⎡⎤
⎜⎟
⎣⎦
−
⎝⎠
              (3.22b)
The excess return of any asset i over the real risk free rate of return
, f t
r
4
is
determined by the conditional covariance between the return
,1 it
r
+
and pricing kernel
1 t
m
+

and conditional variance of
,1 it
r
+
:
,1 , 1 ,1 ,1
1
cov , var
2
tCt ft t t Ct t Ct
Er r m r r
+++ +
−=− − ⎡⎤⎡ ⎤ ⎡⎤
⎣⎦⎣ ⎦ ⎣⎦
.

                                               
4
Note that the risk free rate at time
1 t +
is known ahead of time. Therefore,
, f t
r represent the one-period risk free rate
of return in time 1 t + .  



32
Utilizing the results of equation (1.22), it is easy to compute that (see Appendix A3)
()
() ()
22 2 2
,1 ,
2
14 14 0 1
11
22
1
                       
21 1
tCt ft x w t
tt
Er r
AA aax
μ
θλ λ λ γ σ
εε
θκ τ κ μ
εε
+
⎛⎞ ⎛ ⎞
−= − + + + − + ⎡⎤
⎜⎟ ⎜ ⎟
⎣⎦
⎝⎠ ⎝ ⎠
⎡⎤
⎛⎞⎛⎞⎛⎞
−+ − + + +
⎢⎥
⎜⎟⎜⎟⎜⎟
−−
⎝⎠⎝⎠⎝⎠
⎢⎥
⎣⎦
     (3.23)

3.3.3 Conditional mean and volatility of
1 t
m
+
and risk free rate of
return
, f t
r
To rewrite the pricing kernel (3.14) in terms of state variables
t
x ,
t
μ ,
t
σ , and
log
t
α , I substitute the equation (3.15), (3.16), (3.17a) and (3.18) into equation (3.14) and
obtain  
()()( ) ()
()
()()
() ()
() ( )
( )
()
11 011
01 1 2 1
22 2
01 3 1
4011
2
01 2 3 4
log 1 log
        1
log
log
ttttt t ttt
txt t t
twt
tttt
tt t t
mx at aaxu
AA x e A
Aw
Aat aa xu
AAx A A A
α
μ
αα
θ
θδ μ ση χ β μ α μ
ψ
ρσ μ σζ
κκ σ νσ σ σ
θ
μβ αμ μ
μσ α
++ +
++
+
+
⎡ ⎤
=− ++ + − +− + + +
⎣ ⎦
⎛⎞
++ + + +
⎜⎟
⎜⎟
++ −+ + −
⎜⎟
+−
⎜⎟
⎜++ − + + + ⎟
⎝⎠
++ + + +()
1 tt tt
x μση
+
⎡⎤
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥ ++
⎣⎦
(3.24)
where
1
1
ε γ
χ
ε
−
=
−
. It is easy to compute the innovation to the pricing kernel and
conditional variance as followings

33
[] ()( )( )
( ) ( )
()( )
() () ()( ) ( )
2
01 0 13
1
14
2
41
11
log 1 1
1
11 1
               1 1 1 1 log
2
tt
tt t t
AA
Em at
Aat
xA
α
α
κκ κ σ ν
θδ χ β μ θ
κμ β
μ γγ σ χ β θ κβ α
ψψ
+
⎡⎤ +− + − +
=+ − ++−
⎢⎥
+−
⎢⎥
⎣⎦
⎛⎞
−+ + − − +− − + − − ⎡⎤
⎜⎟⎣⎦
⎝⎠
      (3.25)
[ ] ( ) ( ) ( )
() ( )
( )
11 11 1 12 1 13 1
14 0 1 1 1
11 1
                         1
tt t xt t wt
tt t tt
mEm A e A A w
Aaa xu
μ
θκ σ θ κ σζ θ κ σ
θκ χ μ γση
++ + + +
++
−=− +− +−
+− + + + − ⎡⎤
⎣⎦
        (3.26)
[] () ( ) () ( ) ( )
2 2
22 2 22
11401
var 1 + 1
tt x w t t t
mAaax
μ
θ λλ λ θ κ χ μ γσ
+
=− + + − + + + + ⎡⎤
⎣⎦
  (3.27)
Both the conditional mean and variance of log pricing kernel is time-variant and are
determined by the conditional mean ( )
tt
x μ + and conditional variance of the growth of
non-housing consumption
2
t
σ .  
Using the fact that [ ]
,1
1
ft t t
REM
+
= , It’s easy to follows that
[] []
[] () ( ) []
,1 1
11 ,1 1
1
var
2
1
   log log 1 var
2
ft tt tt
tt t t t Ct t t
rEm m
Eg E E r m
θ
θδ τ α θ
ψ
++
++ + +
=− −
=− + − Δ + − − ⎡⎤ ⎡⎤
⎣⎦ ⎣⎦
   (3.28a)
This equation could also be transferred in this form:  
[] () []
,1 1 ,1, 1
11 1
log log var
2
ft t t t t t Ct f t t t
rEgE Err m
τ θ
δα
ψθ θ θ
++ + +
−
⎡⎤ =− + − Δ + − − ⎡⎤
⎣⎦ ⎣⎦
   (3.28b)
Substituting the equilibrium solutions of
,1 Ct
r
+
into (3.28b) and I obtain the formula
of log real risk free rate of return in terms of four state variables.  

34
( ) ( )
()
( )
()
()
()
2
2
22 2 0 0
, 14
2
2
2 1
114
11 1
log
2212
1
111
1 11
     log
21 2 2
ft x w
tt t t
at a a
rA
a
aA x
α
μ
τβ μ θ θετ
δλλλ κ
θ εθ
γ
τβ ψ θετ
κμ σ α
ψεθ θ
−+ − −
⎛⎞
=− − + + + + + −
⎜⎟
−
⎝⎠
⎛⎞
+−−
⎜⎟
⎡⎤
− −
⎛⎞
⎝⎠
++ + − + + +
⎢⎥
⎜⎟
−
⎝⎠
⎢⎥
⎣⎦
 (3.29)
Here I obtain the general form of log real risk free rate of return.  
The effect of consumption risk of the risk free rate of return is represented in the
term ()
1
tt
x μ
ψ
+ and
()
2
1
111
2
t
γ
ψ
σ
⎛⎞
+− −
⎜⎟
⎝⎠
. Since it is already known that 1 γ >
5
and the
inverse of IES
1
1
ψ
< , if it also satisfies
1
1
ψ
>− , the sign of the term
()
1
11 1
2
γ
ψ
⎛⎞
+− −
⎜⎟
⎝⎠

should be negative. That means when the volatility of non-housing consumption growth
(fluctuating economic uncertainty
2
t
σ ) becomes higher, it encourages savings and hence
lowers the interest rate. This property is consistent with the evidence in the historical data.  
However, the effect of consumption risk on the expected consumption growth
tt
x μ + depends on the sign of
1
ψ
. In the case that
1
01
ψ
< < , when non-housing
consumption
tt
x μ + is expected to grow, agents try to borrow and consequently the risk
free rate of return rises. While in the case that
1
0
ψ
< , if non-housing consumption
tt
x μ + is expected to grow, agents would like to buy more riskless asset (engage in more
                                               
5
Previous literature documents that risk aversion parameter γ is well in excess of 1. Such as Mehra and Prescott
(1985) and Mehra (2003) argue that a reasonable value of risk aversion is around 10. Bansel and Yaron (2004) think
7.5-10 is a reasonable range.  



35
savings) hence decreasing the risk free rate of return. But the problem is: different from
the standard CCAPM and Piazzesi et al (2007)’s model, in my framework,
1
ψ
is not the
sole term on
tt
x μ + . Given
1
1
ψ
< , the sign of
1
11/
γ
θ
ψ
−
=
−
should be negative. Also, we
already know that
1
0 a > . So it is easy to find out that the joint effects of all three terms
on
tt
x μ + depends on the magnitude of the parameters
1
ψ
, γ , and
1
a .  
The presence of expenditure share log
t
α in the pricing kernel introduces
composition risks effects to the risk free rate of return. The first effect of composition risk
is represented by the term
( ) ( ) 1 at
α
τβ μ
θ
−+
− and
( ) 1
log
t
τβ
α
θ
−
. The sign of both
terms depends on the sign of
1
ψ
too
6
. Based on the negative correlation between log
t
α
and  
,1 f t
r
+
as shown in Table 4.1, one might expect that
1
ψ
to be negative. Noted that
since () at
α
μ + is the conditional mean of log
t
α , I expect their joint effect to be zero.
The uncertainty of the expenditure share introduces another composition risk represented
                                               
6
According to the expression
()
()
11 1
1
1
11
εεγ
ψ ψψ
τ
ε
ψ
−+ + −
≡
−−
, it easy to derive that if
1
0
ψ
< then 0 τ > , also because I
already know that  0 θ < , then I have
( ) 1
0
τβ
θ
−
<
. If  
1
10
ψ
>> , then
( ) 1
0
τβ
θ
−
>
.  



36
by term
2
0
2
a τ
θ
− and ()
2
1
2
tt
a
x
τ
μ
θ
−+ . The joint effect is expected to be negative because
( ) ()
2
,0 1 tt t t
Eaa x
α
σμ =+ + .  
Noted that Epstein-Zin recursive utility introduces three extra terms:
()
2
0
14
1
21
a
A
θ ε
κ
ε
−
⎛⎞
+
⎜⎟
−
⎝⎠
, ()
2
114
1
21
tt
aA x
θε
κμ
ε
−
⎛⎞
++
⎜⎟
−
⎝⎠
, and
()
22 2
1
2
xw μ
θ
λ λλ
−
++ .
Since 0 θ < along with
( ) ( )
2
,0 1 tt t t
E aa x
α
σμ =+ + , it implies that the joint effect of first
two terms should be negative and the last one is also negative. Therefore, the presence of
expenditure share factor and recursive utility in the pricing kernel lowers the risk-free
rate and can helps resolve the interest rate puzzle in the standard consumption-based
CAPM.  
When we use the standard power utility form instead of the Epstein-Zin preference,
taking the inverse of the IES
1
ψ
equal to RRA γ ( 1 θ = ), in this case, the formula (3.29)
will collapse to Piazzesi et al (2007)’s specification:  
() ()
2
22
,,,, 2
11
lo g 1 1 lo g
22
ftttztt t
r
αα
τ
δτβμ σ μ σ τβα
ψψ
=− − − − + − + −
where
,t
at
αα
μ μ =+ , ( )
2
,0 1 ttt
aa x
α
σμ =+ + and ( )
, zt t t
x μμ =+ are conditional mean
and variance of the expenditure share and conditional mean of the non-housing
consumption growth, respectively.
In addition, if only considering the consumption growth as the only source of risk
and excluding the expenditure share from the pricing kernel, that is, setting 1 τ = , the

37
Piazzesi et al (2007)’s model could be further simplified to the standard consumption-
based asset pricing specification (Lucas, 1978; Cochrane, 2002):  
()
2
, 2
11
log
2
fttt t
rx δμσ
ψψ
=− + + −
My model is an extension of Piazzesi et al (2007)’s model in at least two aspects.
First, I use an Epstein-Zin preference in the model setup; therefore introduce the long-run
risk originated by this recursive utility function. It can be interpreted as the fear of the
innovations or disparities between realized utility and expected utility of a stream of
future consumption (Fillat, 2008). The long-run risk is important in asset pricing because
the agent’s horizon on the investment-consumption tradeoff might be longer than the
previous asset pricing model had assumed. Housing is critical in asset pricing not only
because housing is a major asset class but also because it plays a twofold role in the
economy -- as both a consumption good and as a financial asset simultaneously. On one
hand, housing is durable, expensive and spatially fixed, and very costly to trade in the
market, which makes it particularly sensitive to the overall economic shocks and long-run
uncertainty. On the other hand, the sale price
h
t
p of the housing asset has a direct impact
on the consumption of housing service
t
s ; its impact on housing rental or imputed rental
value of owner-occupied houses
s
t
p is marginal and only via its impact on housing
service
t
s . This mutual mechanism also adds more long-run risk in the pricing kernel.  
As mentioned above, the presence of Epstein-Zin utility helps resolve the interest
rate puzzle by introducing three terms. Note that the negative term

38
()
2
114
1
21
tt
aA x
θε
κμ
ε
−
⎛⎞
++
⎜⎟
−
⎝⎠
introduced by Epstein-Zin utility may help explain why the
risk free rate of return is less sensitive to the consumption growth than other risky assets.  
Second, based on the observed properties of the data, I assume the shocks to log
expenditure share
2
,t α
σ are heteroskedastic and correlated with the expected consumption
growth
tt
x μ + . This restriction imposes a cointegration relationship between
1
log
t
α
+
and
1 t
g
+
-- the two forcing processes of the economy. This setup introduces a positive term
2
1
2
a τ
θ
− in the coefficient on the expected consumption growth ( )
tt
x μ + . The empirical
estimation result shows that my model could provide better goodness of fit than the naïve
consumption based model and Piazzesi et al (2007)’s model do.  

3.3.4 Housing return
,1 ht
r
+

Using the standard approximation proposed by Campbell and Shiller (1988), I have
,1 2 3 ,1 , 1
log
ht ht ht t
rqq s κκ
++ +
+− +Δ                                                                       (3.30)
2
,0 1 2 3 4
log
ht t t t t
qB Bx B B B μ σα ++ + +                                                              (3.31)
where
, ht
q denotes the log price-consumption ratio (sale price-rent ratio) of the claim to
housing consumption (housing service)
t
s . The parameter
2
κ and
3
κ are constants which
could be estimated from the mean of
, ht
q . Based on the 1975-2008 quarterly data, I
estimate
2
κ to be around 0.0776 and
3
κ to be 0.985. Applying linear approximation for

39
equation (3.30) leads to a convenient formula in terms of
1 t
g
+
and
1
log
t
α
+
(see
Appendices A3).  
1
,1 2 3 ,1 , 1
log
11
t
ht h t ht t
t
rqqg
ε α
κκ
ε α
+
++ +
Δ
+− + +
−−
                                                  (3.32)
where ( ) exp
t
at
α
αμ =+ is the conditional mean of expenditure share.  
Substituting equation (3.15) and (3.31) into (3.32), one can rewrite the housing
consumption claim asset return
,1 ht
r
+
in terms of exogenous state variables. Since the
equation
1,1 1 ,1
1
var 0
2
tt st tt st
Em r m r
++ ++
++ + = ⎡⎤ ⎡⎤
⎣⎦ ⎣⎦
must be satisfied for all values of the
four state variables, the solution coefficients for
t
x ,
t
μ ,
2
t
σ , and log
t
α on the price-
consumption ratio are:  
()
()( )
()
()( )
() ( )( )
()()
()( )
() () () ()
() ()
()()
()
22 2 10
2
23 3 4
0
3
22
22
31 1 1 3 2 1 2
2
2
33 1 3 3 4 1 4
log 1
log 1 1
2
1
11 1
=
11
1
11
11
22
11
11
2211
xw
t
x
w
t
at
at
a
B Bat at
B
BA B A
BA B A
α
α
μ
αα
μ
φ
δβμ
θ
θδ θ χ β μ
θπ
λλ λ
θ
ε
κ κ ν σ βμ βμ
εα
κ
κθ κ σ κ θ κ σ
ε
κθ κ σ κ θ κ
εα
⎡⎤
+− +
⎢⎥
−− + − +
⎢⎥
⎢⎥
++ + +
⎢⎥
⎣⎦
⎡⎤ ++ − + − + + − +
⎣⎦
−−
−
++− + +− +
+− + + +−
−−
2
0
a χ
⎧⎫
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨⎬
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎛⎞
⎪⎪
+
⎜⎟
⎪⎪
⎝⎠ ⎩⎭
 
1
112
1
3
1
1+
1
a
B
π
π π
ψθ
κρ
⎛⎞
−−+
⎜⎟
⎝⎠
=
−

1
112
2
3
1
1+
1
a
B
π
π π
ψθ
κ
⎛⎞
−−+
⎜⎟
⎝⎠
=
−


40
()
()
2
3
3
11
21
B
θ ψ
κν
−
=
−

()
()
( )
()()( )
4
33
1- 1-
-1 1 1 -1
t
t
B
φβ αε β
θκ β ε α κ β
=+
−−
                                                           (3.33)
where I denote
2
π :  
()
()()
2
214 34
1
1
211
t
AB
ε
πθ κ χκ
εα
⎡⎤
=− ++ +
⎢⎥
−−
⎣⎦
                                             (3.34)
Since I already know that the autoregressive coefficient 11 ρ − << , risk-aversion
parameter 1 γ > , IES 0 ψ > , mean reversion rate coefficient 11 ν − << and
3
κ is very
close to but less than unity, it is obvious that the sign of
3
B is determined by the
magnitude of  IES, ψ . If the inverse of IES
1
1
ψ
< ,
3
B is negative. In this case, the
economy becomes less uncertain, the agents will try to buy more housing asset, pushing
up the price-dividend ratio
, ht
q of the housing asset.  If  
1
1
ψ
> , the decrease in economy
uncertainty
2
t
σ leads to more consumptions compared with investment of risky assets,
therefore decreasing price-dividend ratio
, ht
q of the housing consumption claim asset.
The signs of
1
B ,
2
B , and
4
B are indeterminate from the equation (3.33).  
Since the historic data reported in Table 4.1 indicates a negative correlation
between
2
t
σ and
, ht
q , it also implies that
1
1
ψ
< . The empirical data also indicates a strong
negative correlation between
t
x and
, ht
q , and a strong positive correlation between

41
expenditure share
t
α and
, ht
q . These evidences suggest that
1
B and
2
B are negative is
positive. It also implies that the wealth effect dominates the substitution effect. The
increase in expected aggregate consumption growth leads to more consumption,
decreasing price-dividend ratio
, ht
q of the housing consumption claim asset.  
Based on the solutions (3.32) the conditional mean and conditional variance of  
,1 ht
r
+
can be written as
()
()
( ) ( )
()
() () () ()
() ()
()()
()
22 2 10
,1
22
22
31 1 1 3 2 1 2
2
2
2
33 1 3 3 4 1 4 0
1
112
111
log
2
11
         1 1
22
11
        1 1
2211
1
        +
tht x w
x
w
t
a
Er at
BA B A
B AB Aa
a
μα
μ
θθ θ π τ β
δλλλ μ
θθ
κθ κ σ κ θ κ σ
ε
κθ κ σ κ θ κ χ
εα
π
ππ
ψθ
+
−− −
=− + + + + − + ⎡⎤
⎣⎦
−+− − +−
⎛⎞
−+− − + +− +
⎜⎟
−−
⎝⎠
⎛
⎛⎞
+− + −
⎜⎟ ⎜
⎝⎠
⎝
()
()
()
2
1
11
1
log
2
       
tt t t
x
γ
τβ ψ
μσ α
θ
⎛⎞
−−
⎜⎟
− ⎞
⎝⎠
++ +
⎟
⎠
  (3.35)
()()
()
,1 , 1 3 1 1 3 2 1 3 3 1 1
34 0 1 1
                         
11
ht t h t x t t w t t t
tt t
t
rEr Be B B w
B aa xu
μ
κσ κ σζ κ σ ση
ε
κμ
εα
++ + + + +
+
−= + + + ⎡⎤
⎣⎦
⎛⎞
++ + +
⎜⎟
−−
⎝⎠
                     (3.35b)
()
()()
() ()
22 2 2 2 2 2 2
,1 3 1 2 3
2
34 0 1
var
                 
11
       
tht x w t
tt
t
rB B B
B aa x
μ
κσ σ σ σ
ε
κμ
εα
+
++ + ⎡⎤
⎣⎦
⎛⎞
++ + +
⎜⎟
−−
⎝⎠
                                  (3.36)
The sign of the coefficient on economic volatility depends on the magnitude of
1/ ψ . If the 1/ ψ is less than 1,
( ) ( ) 11 1
0
2
ψγ −−
> . In this case, when the economy

42
becomes less uncertain, the agents expect the housing asset to be less risky and they will
try to buy more assets, therefore lowering the log real return
, ht
r ; when the economy
becomes more volatile, the housing asset is expected to be more risky, therefore raising
the housing return. The empirical evidence from this aspect also suggests that 1/ 1 ψ < .
Compared with the coefficients on expected consumption growth ()
tt
x μ + between
housing asset and the aggregate claim asset, an extra term ()
11 2
a π π − is found for
housing asset. Since
1
a is expected to be positive, and it’s not difficult to prove that
12
π π > , it is easy to follow that ( )
11 2
a π π − are positive. Hence, the model helps explain
why the housing asset is more sensitive to the consumption growth than other risky assets
as a whole. The intuition is that, unlike other assets, housing is also a main consumption
good, hence any shocks on the housing sale price could also affect the relative
consumption
t
t
c
s
and non-housing consumption growth which will in-turn affect the
housing sale price. This feedback effect amplifies the volatility of housing price and
increase the Beta of housing asset.  
The excess return of the
,1 ht
r
+
could be expressed as below
[] ()()
, 1 , 1 ,1 ,1
1 1 ,1 ,1 ,1
1
cov , var
2
1
                     var
2
tht ft t t ht t ht
tt t t ht t ht t ht
Er r m r r
Em Em r E r r
+++ +
++ + + +
−=− − ⎡⎤ ⎡ ⎤ ⎡⎤
⎣⎦ ⎣ ⎦ ⎣⎦
⎡⎤
=− − − − ⎡⎤⎡⎤
⎣⎦⎣⎦
⎣⎦
          (3.37)
Substituting the (3.26), (3.35), (3.35b) and (3.36) into (3.37), yields the excess
return of the
,1 ht
r
+
 

43
() ()
()
()()
()()
22 2 2 2 2
,1 , 1 3 1 1 3 1 1 3 2 2 3 2
22 2 2
13 3 3 3 3
14 3 4
3
11
11
22
11
                     1
22
1
11
                     
1
2
tht ft x
wt
t
Er r AB B AB B
AB B
AB
B
μ
θ κκ κ σ θ κ κ κ σ
θκκ κ σ γ σ
ε
θκ χ κ
εα
κ
+
⎡⎤⎡ ⎤
−= − − + − − ⎡⎤
⎣⎦
⎢⎥⎢ ⎥
⎣⎦⎣ ⎦
⎡⎤⎛⎞
+− − + −
⎜⎟
⎢⎥
⎣⎦⎝⎠
⎛⎞
−− +
⎜⎟
−−
⎝⎠
+
−
()()
() ()
01 2
4
11
tt
t
aa x μ
ε
εα
⎡⎤
⎢⎥
⎢⎥
++
⎢⎥
⎛⎞
⎢⎥
+
⎜⎟
⎢⎥
−−
⎝⎠ ⎣⎦
  (3.38)
It could be simplified as
() ()
()
22 2 2 2 2
, 1 , 1 311 3 1 1 3 3 3 3 3
22 2
13 3 3 3 3
2
2
1
12 14 0
11
11
22
1
                     1
2
1
                      +
21 2
tht ft x w
w
Er r AB B AB B
AB B
kA a a
θκκ κ σ θ κκ κ σ
θκκ κ σ
θεπτ
ππ
εθ θ
+
⎡⎤⎡ ⎤
−= − − + − − ⎡⎤
⎣⎦
⎢⎥⎢ ⎥
⎣⎦⎣ ⎦
⎡⎤
+− −
⎢⎥
⎣⎦
⎡⎤
−
⎛⎞
−− + − + +
⎢⎥
⎜⎟
−
⎝⎠
⎢⎥
⎣⎦
() ()
2
1
1
2
tt t
x μ γσ
⎛⎞
++ −
⎜⎟
⎝⎠
(3.38b)
I know that risk-aversion parameter γ is well above 1, parameter
1
κ and
3
κ are
very close to but less than 1. As discuss before, the signs of
1
A ,
2
A ,
3
A ,
1
B ,
2
B , and
3
B
are determined by whether the magnitude of IES, ψ greater or less than 1, and by
definitions,
11
AB ≈ ,
22
AB ≈ , and
33
AB ≈ . If 1 ψ > , then
1
11/
γ
θ
ψ
−
=
−
is negative. Hence,
the first three terms are positive and coefficient on conditional volatility --
1
2
γ
⎛⎞
−
⎜⎟
⎝⎠
is
positive. It implies that when the economy becomes more uncertain, the excess return of
the housing asset will become larger to compensate the risk caused. Details of all
derivations are reported in the Appendix A5.  


44
3.3.5 Special case: consumption growth with a fixed component
For simplicity of the general case, one can specify the dynamics for consumption
growth rate with a fixed component μ , instead of a time-variant trend component
t
μ .
The forcing processes of the economy therefore are defined  
11 tttt
gx μ ση
++
=+ +
11
                                                       ( 1 1)
tt xt
xx e ρσρ
++
=+ −< <
( ) ( )
22 2 2
11
                               0 1
tt wt
w σσ νσ σ σ ν
++
=+ − + < <
() ( )
1,1
log log           0 1
tt tt
at at u
ααα
αμ β α μ σ β
++
=+ + − − + < <
2
,0 1 tt t
Eaax
α
σ ⎡⎤=+
⎣⎦

11
, , ~ ... (0,1)
tt t
eu iidN η
++
                                                                                    (3.39)
The log pricing kernel and its conditional mean and variance are rewritten in terms
of three state variables
t
x ,
t
σ , and log
t
α as following:  
()()( )
()
() () ()
()
()
()()
11 011
22 2
01 1 3 1
01
4011
2
01 3 4 1
log 1 log
        1 log
log
tttt t tt
txt t wt
ttt
tt t t tt
mx at aaxu
AA x e A w
Aataaxu
AAx A A x
α
αα
θ
θδ μ ση χ β μ α
ψ
ρσ σ νσ σ σ
κκ
θ μβ α μ
σαμ ση
++ +
++
+
+
⎡⎤
=− ++ + − +− + +
⎣⎦
⎡⎤ ⎛⎞
++ + + − + +
⎢⎥ ⎜⎟
+−
⎢⎥ ⎜⎟
+− +−−++
⎝⎠ ⎢⎥
⎢⎥
++ + + + +
⎢⎥
⎣⎦
  (3.40)
[] ( ) ( )
() ( ) ( ) ( )( ) ()
() ()( ) ( )
1
2
01 0 13 14
2
41
log 1
              1 1 1 1
11 1
              1 1 1 1 log
2
tt
tt t
Em at
AA A at
xA
α
α
θδ χ β μ γμ
θκ κ κ σ ν κ μ β
γγσ χ β θ κβ α
ψψ
+
=+ − + −
+− + − + − + + −
⎛⎞
−+ − − +− − + − − ⎡⎤
⎜⎟⎣⎦
⎝⎠
  (3.41)

45
[] ()() () ( )
2 2
22 22
11401
var 1 + 1
tt x w t t
mAaax θ λλ θ κ χ γσ
+
=− + − + + + ⎡⎤
⎣⎦
                  (3.42)
where
1
11/
γ
θ
ψ
−
=
−
,
1
1
ε γ
χ
ε
−
=
−
,
11 x x
A λ κσ = ,
13 ww
A λ κσ = .
1
A ,
3
A and
4
A are defined in
equation (3.21). The expression of
0
A is:  
() ( )( )
()( )()
2
01 3 4
0
22 11 1 0
1
log 1 + 1
1
=
1
1
11
2
xw
AA at
A
aa
at
α
α
δκ κ ν σ β μ
πφ θ π
κ
μβμ λλ
ψ θθ θ
⎧⎫
⎡⎤ ++ − − + +
⎣⎦
⎪⎪
⎨⎬
⎛⎞
−
−+ + − + + + +
⎪⎪
⎜⎟
⎝⎠⎩⎭
        (3.43)
where
1
1
εγε
φ
ε
−−
=
−
and
( )
()
2
2
1
1 2
1
1
2-1
φκ
π
κβ
−
= .
The aggregated consumption claim asset return
,1 Ct
r
+
could be rewritten as  
() ( ) ( )
()
()
()()
()( )
22 2
01 1 3 1
,1 0 1
4011
2
01 3 4 1
01 1
log
        log
        1 log
1
txt t wt
Ct
ttt
tt t t tt
ttt
AA x e A w
r
Aataaxu
AAxA A x
at a a x u
αα
α
ρσ σ νσ σ σ
κκ
μβ α μ
σα μ ση
ε
βμ α
ε
++
+
+
+
+
⎡⎤
++ + + − +
⎢⎥
+
⎢⎥
++ − − + +
⎣⎦
−+ + + + + +
⎡⎤
+− +− + +
⎣⎦
−
             (3.44)
Substituting the solutions of
1
A ,
3
A and
4
A as defined in equation (3.21) and
0
A of
equation (3.43) into it yields the final expressions of conditional mean and variance  
() ( )
()
()
()( ) ()
22 10
,1
2 11
1
log
2
11 1 1 1
       log
2
tCt x w
tt t
at a
Er
a
x
α
τβ μ θπ
δλλ
θθ
ψγ τ β π
μ σα
ψθ θ
+
−+
=− − − + − ⎡⎤
⎣⎦
−− − ⎛⎞
+− + − +
⎜⎟
⎝⎠
                 (3.45b)
()
2
22 2
,1 1 4 0 1
var
1
tCt x w t t
rA aax
ε
λ λκ σ
ε
+
⎛⎞
=+ + + + + ⎡⎤
⎜⎟
⎣⎦
−
⎝⎠
                                  (3.46b)


46
The expression of log real risk free rate of return
, f t
r
()( )
()
( )
()
()
2
2
22 0 0
, 14
2
2
2 1
114
11 1
log
22 12
1
111
1 11
        log
21 2 2
ft x w
tt t
at a a
rA
a
aA x
α
τβ μ θ θ ετ μ
δλλ κ
θ εθψ
γ
τβ ψ θετ
κσα
ψεθ θ
−+ − −
⎛⎞
=− − + + + + − +
⎜⎟
−
⎝⎠
⎛⎞
+−−
⎜⎟
⎛⎞
− −
⎛⎞
⎝⎠
++ + − + +
⎜⎟
⎜⎟
⎜⎟
−
⎝⎠
⎝⎠
      (3.47)
where
()
11 1
1
1
11
ε εγ
ψψψ
τ
ε
ψ
−+ + −
≡
⎛⎞
−−
⎜⎟
⎝⎠
and
1
11/
γ
θ
ψ
−
≡
−
.
Thus, the excess return of a claim to aggregate consumption
,1 , tCt ft
Er r
+
− ⎡⎤
⎣⎦
is
()
()
22 2
,1 ,
2
14 1 4 0 1
11
22
1
                       
21 1
tCt ft x w t
t
Er r
AA aax
θλ λ γ σ
εε
θκ τ κ
εε
+
⎛⎞ ⎛ ⎞
−=− + + − + ⎡⎤
⎜⎟ ⎜ ⎟
⎣⎦
⎝⎠ ⎝ ⎠
⎡⎤
⎛⎞⎛ ⎞ ⎛ ⎞
−+ − + +
⎢⎥
⎜⎟⎜ ⎟ ⎜ ⎟
−−
⎝⎠⎝ ⎠ ⎝ ⎠
⎢⎥
⎣⎦
            (3.48)
The expression of housing consumption claim asset return
,1 ht
r
+
in terms of
exogenous state variables and its conditional mean and variance can be rewritten as
() ( ) ( )
()
()
()()
()()
()( )
22 2
01 1 3 1
,1 2 3
4011
2
01 3 4 1
01 1
log
        log
        1 log
11
txt t wt
ht
ttt
tt t t tt
ttt
t
BB x e B w
r
Bataaxu
BBx B B x
at a a x u
αα
α
ρσ σ νσ σ σ
κκ
μβ α μ
σα μ ση
ε
βμ α
εα
++
+
+
+
+
⎡⎤
++ + + − +
⎢⎥
+
⎢⎥
++ − − + +
⎣⎦
−+ + + + + +
⎡⎤
+−+−++
⎣⎦
−−
            (3.49)

47
()
()
( ) ( )
()
() () () ()
() ()
()()
()
22 2 10
,1
22
22
31 1 1 3 2 1 2
2
2
2
33 1 3 3 4 1 4 0
1
112
111
log
2
11
         1 1
22
11
        1 1
2211
1
        +
tht x w
x
w
t
a
Er at
BA B A
B AB Aa
a
μα
μ
θθ θ π τ β
δλλλ μ
θθ
κθ κ σ κ θ κ σ
ε
κθ κ σ κ θ κ χ
εα
π
μππ
ψθ
+
−− −
=− + + + + − + ⎡⎤
⎣⎦
−+− − +−
⎛⎞
−+− − + +− +
⎜⎟
−−
⎝⎠
⎛
⎛⎞
+− −+
⎜⎟
⎝⎠
⎝
()
2
2
1 1
1log
2
       
tt t
x
τβ θ
σα
ψθ
− ⎞⎛ ⎞
−− +
⎜⎟⎜⎟
⎠⎝ ⎠
  (3.50)
()()
()
2
22 2 2 2 2 2
,1 3 1 3 3 3 4 0 1
var
11
tht x w t t
t
rB B B aax
ε
κσ κ σ σ κ
εα
+
⎡⎤
+++ + + ⎡⎤
⎢⎥
⎣⎦
−−
⎣⎦
       (3.51)
where
1
B ,
3
B and  
4
B are defined in equation (3.33) . The
0
B expression is:  
() ( ) ( )( )
( ) () ( ) ()( )
()()
()( ) () ()
() ()
2
01 0 3 4 0
2
21 2 3 3 4
2
2
0
31 1 1
3
2
2
33 1 3 2 0
log 1 1 1
11 1 1
1
1 =
11
1
11 2
1
1
2
x
t
w
AA A at A
aBBat
B
at B A
BA a
α
αα
α
θδ θ κ κ νσ β μ
γπ μ χ β μ κ κ ν σ β μ
ε
βμ κ θ κ σ
κ
εα
κθ κ σ π
⎧⎫
⎡⎤ ⎡⎤ +− + + − + − + −
⎣⎦ ⎣⎦
⎪⎪
⎪⎪
⎡ ⎤ +− + + − + + − + − +
⎣ ⎦
⎪⎪
⎪⎪
⎨⎬
+−+++−
−
⎪⎪
−−
⎪⎪
⎪⎪
++− +
⎪⎪
⎩⎭
  (3.52)
The excess return of the log real housing return
,1 ht
r
+
is  
() ()
()()
()()
()()
()
22 2 2 2 2
, 1 , 1 311 3 1 1 3 3 3 3 3
14 3 4
2
01 2
34
11
11
22
1
11
1
                     
2
1
21 1
             
tht ft x w
t
tt
t
Er r AB B AB B
AB
aax
B
θκκ κ σ θ κκ κ σ
ε
θκ χ κ
εα
γ σ
ε
κ
εα
+
⎡⎤⎡ ⎤
−= − − + − − ⎡⎤
⎣⎦
⎢⎥⎢ ⎥
⎣⎦⎣ ⎦
⎡⎤ ⎛⎞
−− +
⎢⎥ ⎜⎟
−−
⎢⎥ ⎝⎠
⎛⎞
+++−
⎜⎟ ⎢⎥
⎝⎠
⎛⎞
⎢⎥
−+
⎜⎟
⎢⎥
−−
⎝⎠ ⎣⎦
() ()
()
22 2 2 2 2
1 311 3 1 1 3 3 3 3 3
2
2
2 1
12 14 0 1
11
      1 1
22
11
                   
21 2 2
xw
tt
AB B A B B
kA a a x
θκκ κ σ θ κκ κ σ
θεπτ
ππ γ σ
εθ θ
⎡⎤⎡ ⎤
=− − + − −
⎢⎥⎢ ⎥
⎣⎦⎣ ⎦
⎡⎤
−
⎛⎞ ⎛⎞
+− − + − + + + −
⎢⎥
⎜⎟ ⎜⎟
−
⎝⎠ ⎝⎠
⎢⎥
⎣⎦
 (3.53)

48
CHAPTER 4:  DATA
4.1 Data on Non-housing Consumption
To measure non-housing consumption quantitative and price, as a convention, I
use the National Income and Product Accounts (NIPA) from the Bureau of Economic
Analysis, the U.S. Department of Commerce. The NIPA tables report quarterly nominal
personal dollar consumption expenditures starting from year 1947 on three main
categories -- durable goods, nondurable goods, and services, and some sub-categories in
each main category, such as food, housing, transportation, clothing and shoes, etc. The
NIPA tables also report a quarterly price index and a quarterly quantity index for each
main and sub category. Although the NIPA price and quantity index are criticized heavily
for their measurement bias by many researchers, such as Hobijn (2003) and Piazzesi et al
(2007), the NIPA is the only source I could obtain for the non-housing expenditure, price
and quantity index, and housing service price index. In next subsection, I will propose a
more accurate measure of the housing service consumption and housing expenditure.  
I use the aggregate consumption of durable goods, nondurable goods, and services
and exclude the housing service to measure the non-housing expenditure. All the original
expenditure data are from NIPA Table 4.1.5.5. “Gross Domestic Product, Expanded
Detail”. I adjusted all expenditure data by the U.S. population.
s
t
p is the NIPA housing
price index from line 13 of the NIPA Table 4.1.5.4. I define it as the housing service price
index (index number for base year 2000 is 100). Non-housing consumption price index
c
t
p could be approximated by the formula:  

49
Housing Expenditure
Aggregate consumption price index
Aggregate Expenditure
Housing Expenditure
1
Aggregate Expenditure
s
t
c
t
NIPA
p
NIPA
p
NIPA
NIPA
−×
=
−
          (4.1)
After computing the
c
t
p , I can derive the
t
c via dividing the population adjusted
NIPA non-housing expenditure by
c
t
p .  

4.2 Data on Housing Consumption  
4.2.1 The problem of NIPA housing series  
Following the Bureau of Economic Analysis (1990) paper regarding the
methodology of the NIPA, I know that the housing service sub category includes the
service provided by both the rental and owner-occupied housing. “Separate estimates are
prepared for owner-occupied permanent site dwellings, for owner-occupied mobile
homes, for tenant-occupied permanent site dwellings, and for tenant-occupied mobile
homes. For each type of dwelling, “the estimate of PCE
7
is essentially the product of the
number of occupied units and an appropriate rent per unit”. Benchmark estimates are
based primarily on data from the decennial census of housing (COH) and survey of
residential finance (SRF).”  
Rent per unit includes both monetary rents paid by tenants and an imputed rental
value for owner-occupied dwellings. The estimates use rent or homeowners’ equivalent
rent adjusted judgmentally for changes in the quality of the housing stock. Therefore, the
NIPA housing price index is actually a constant quality housing consumption per unit
                                               
7
PCE means personal consumption expenditures.  



50
price index. Meanwhile the quantity index is the index of the total number of tenant- and
owner-occupied housing units, thus the quantity index cannot be treated as the housing
consumption for a typical agent. For this reason, I think that Piazzesi et al (2007) and
Fillat (2008) have mismeasurement problem of the housing expenditure and housing
consumption for identical agents. For more details, please see the Bureau of Economic
Analysis (1990).  
I argue that the NIPA housing service price index
s
t
p (CPI rent) is not subject to
the mismeasurement problem. The problem with NIPA expenditure and quantity series
are their treatment of housing quality and per unit space, which are the most two
important housing consumption factors but are not reflected appropriately in NIPA
housing quantity index and housing service dollar expenditure. Since every household
need a unit to live either by renting or by owning, a comprehensive housing quality index
which includes the geographic location related amenities, housing structure attributes,
and the physical size (area per housing unit) is a better measurement of housing service
t
s per agent.  

4.2.2 Construction of comprehensive housing quality index  
The comprehensive housing quality index is derived by dividing the average
housing price index by the constant quality housing index.  
t
Average housing price index
t h
t
s
p
=                                                                       (4.2)


51
Constant quality housing price index
h
t
p is a weighted average of (1) the Census
constant quality price index of new houses (Laspeyres Index), and (2) Office of Federal
Housing Enterprise Oversight (OFHEO)’s seasonal adjusted repeat-sales housing price
index. The OFHEO HPI index starts in 1975 and it tracks quarterly house price indexes
for single-family detached properties using data on conventional conforming mortgage
transactions obtained from Freddie Mac and the Fannie Mae (Calhoun, 1996). Starting
from 1964, the Census Laspeyres Index is a supplement of the HPI and it focus on the
quarterly prices of new single-family houses sold including value of lot. The data used
for computing the Laspeyres index are obtained from the U.S. Census Bureau's Survey of
Construction. Both the HPI and Laspeyres index are calculated in current dollars and
have controlled the housing quality -- changes in size, amenities, and location.  
In order to summate these two indexes, I use the 1975 actual new single family sale
price and existing single family sale price to impute a “levels” series for both prices.  
( )
1
   
h
tt t t t t
p HPI stock Laspeyres Index stock stock
−
=× + × −                                    (4.3)
where
t
stock is the quarterly estimates of the total housing inventory for the United
States and it is obtained from the U.S. Census Bureau. The results show that the
Laspeyres Index has very slight impact on the overall constant quality price index due to
the small weight of
1 tt
stock stock
−
− .
Non-quality adjusted housing index is a weighted average of (1) National
Association of REALTORS (NAR) average sales price of existing single family homes,
and (2) the U.S. census average sales prices of new homes.  

52
The NAR average sales price of existing single family homes reflects quarterly re-
sale prices of existing homes and does not include new construction. The data covers the
years 1968 to present and is not adjusted for inflation. The data of average sales prices of
new homes are obtained from the U.S. Census Bureau. The original data are in monthly
frequency ranging from 1975 to present.  
The base year using for the constant quality price index
h
t
p , Non-quality adjusted
price index, and quality index is 1991 1
st
quarter; indexes value are set to 100.  

Figure 4.1: Comparison among Constant Quality Housing Price Index, Non-quality
Adjusted Housing Price Index, and Quality Index
0
50
100
150
200
250
Dec-74
Dec-76
Dec-78
Dec-80
Dec-82
Dec-84
Dec-86
Dec-88
Dec-90
Dec-92
Dec-94
Dec-96
Dec-98
Dec-00
Dec-02
Dec-04
Dec-06
Dec-08
Housing Price indexes
(1991 Q1 base year)
Constant Quality index Non-quality adjusted index Quality Index


Figure 4.1 indicates that the quality index
t
s crept up steadily from 76.3 in year
1975 to 100.0 in 1991. From 1991 to 1998, the quality index keeps stable at around 95.
From 1998, contradicting to the soaring of constant quality housing price index
h
t
p , the

53
housing quality index declined. This is probably because that the sharp growth of the
housing price leaded to a less affordability of the housing service.  
It’s noteworthy that, in contrast to the intention of the U.S. government
sustainable homeownership policy, the data show that couple with the soaring of the
housing price index since 1991
8
, the average housing service a household consumed
declined about 14% between 1991 and 2007. The burst of the housing bubble implies that
the pursuit of high homeownership ratio may be misleading or even could be harmful to
both the housing industry and the whole economy as it may encourage oversupply of
housing and the raise the housing price in unsustainable ways. Therefore, besides the
homeownership ratio, I argue that the housing quality index is an important indicator to
measure the housing well-beings.

4.2.3 Housing expenditure  
In order to obtain the expenditure ratio of housing consumption to the non-
housing consumption, I need to compute the level of the housing service expenditure.  
As discussed before, I don’t believe that the NIPA housing expenditure series could be an
accurate estimate due to its ignoring of the housing quality change (including not only the
physical size and amenities of the house, but also its geographic location). So I compute
the housing service expenditure index by the following equation:
   
s
tt t
Houing Service Expenditure p s = ×                                                           (4.4)
                                               
8
Data shows that the constant quality housing price index doubled from value of 100 the first quarter 1991 to 223 in
the second quarter of 2007. In the same period, the quality index dropped from 100 to 86.2, an amazing 14% decline.  




54
However, one of the difficulties facing researchers is that both housing rents and
sale prices have been available only as indexes and not as levels leading to severe
identification problems (Davis and Martin, 2005). What’s more, the quality index which I
utilize as the measurement of housing service is also available as index.  To solve this
problem, I employ the data from the 2007 Consumer Expenditure Survey (CEX)
9
,
therefore I can derive the housing expenditure to total consumption ratio in 2007
approximately 13.6%. Since I already have the NIPA aggregate consumption
expenditures, it is easy to estimate the level series for housing service expenditure
starting in 1975.  

4.2.4 Housing price-rent ratio
, ht
q and housing return  
The rent-to-price ratio has long been used as a measure of residential house price
valuation; roughly speaking, this ratio is the equivalent to the earnings-price ratio for
stocks and the capitalization rate for commercial properties (Davis and Martin, 2005).
The rent-to-price ratio (or its inverse price-rent ratio) is a critical indicator in my housing
consumption-based asset pricing model.  
By using the 2000 Decennial Census of Housing (DCH)
10
rents and prices data
obtained from the U.S. Bureau of Census and using the growth rates of housing rents and
prices calculated from the housing consumption price index
s
t
p and non-housing
                                               
9
For details please see CEX Table 7 --“Housing tenure and type of area: Average annual expenditures and
characteristics, Consumer Expenditure Survey, 2007.” The housing expenditure only counts in the expenses on the
shelter of owned dwellings. Expenses on the utilities, fuels, and public services, housekeeping supplies, and household
furnishings and equipment are not included. Data are downloadable thought the hyperlink:
http://www.bls.gov/cex/#tables
10
Data of Decennial Census of Housing are downloadable thought the hyperlink:
http://www.census.gov/hhes/www/housing/census/histcensushsg.html



55
consumption price index
c
t
p , I am able to estimate a “level” series for both housing
prices and rents from 1975 to 2008, then the housing price-dividend ratio
, ht
q (sale price-
rent ratio). The real housing return series are derived by the following equation.
()
1
11
,1 11
,1
11
1
= =
h
h
t
tt
hs c c
ht tt t t
ht hc h c
tt t t
p
pI
q
p pp p
R
p pp p
+
++
+ ++
+
++
+−
+
××                                               (4.5)
where
1
h
t
p
+
is the housing price index at time 1 t + and
1 t
I
+
is the housing improvements
and repairs to housing expenditure ratio. It is calculated by dividing the quarterly
residential improvements and repairs by the NIPA housing service expenditure quarterly
series. The residential improvements and repairs data are available in Table --“Historic
Expenditures for All Residential Properties - Seasonally Adjusted Annual Rate: Quarterly
1966-2007”
11
in the U.S. Census Bureau website. With the non-housing consumption as
numeraire price, the appropriate deflator for nominal housing return is
1
c
t
c
t
p
p
+
.  

4.2.5 Observed properties of the economy
As discussed before, the expenditure ratio
t
z (or expenditure ratio
t
α ) moves in
the same direction with the relative consumption
t
t
c
s
but opposite to relative price
c
t
s
t
p
p
, if
and only if 1 ε > . This feature is consistent with the evidence shown in Figure 4.2.  
                                               
11
The residential improvements and repairs include the (1) maintenance and repairs which are incidental costs that
keep a property in ordinary working condition; and (2) improvements which consist of general improvements, additions
and alterations, and major replacements. The Census Bureau has discontinued the Survey of Residential Alterations and
Repairs in fourth quarter 2007; therefore, this table is no longer being updated. Data are downloadable thought the
hyperlink: http://www.census.gov/const/www/c50index.html



56
Figure 4.2 plots the relative price (black line), relative consumption (red line), and
expenditure ratio (gray line) quarterly series from first quarter 1975 to fourth quarter
2008. This figure indicates that the housing services become relative expensive overtime,
and meanwhile relatively less housing services have been consumed just as FOC (3.3a)
implies. The expenditure ratio moves closely with the relative consumption. It strongly
indicates that the ε is greater than 1.  

Figure 4.2: Relative Price, Relative Consumption, and Expenditure ratio
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Dec-74
Dec-76
Dec-78
Dec-80
Dec-82
Dec-84
Dec-86
Dec-88
Dec-90
Dec-92
Dec-94
Dec-96
Dec-98
Dec-00
Dec-02
Dec-04
Dec-06
Dec-08
Relative Consumptions and Prices
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
Expenditure Ratio
Relative Price Relative Consumption Expenditure Ratio


Next figure depicts the quarterly series of housing sale price index
h
t
p , housing
rental price index
s
t
p , and price-rent ratio
, ht
q constructed in section (4.25). The housing
sale price became to take off in the beginning of 1999. In the second quarter of 2006, the
price-rent ratio reached its historical high --103.99. Since the summer of 2006, the

57
housing bubble burst and the housing price coupled with the price-rent ratio declined
steeply. The big up and down fluctuations in the ratio have led to concerns over the
stationarity of housing market.  

Figure 4.3: Housing Sale Price Index, Housing Rent Price Index, and Housing Price-
Dividend Ratio
0
50
100
150
200
250
Dec-74
Dec-76
Dec-78
Dec-80
Dec-82
Dec-84
Dec-86
Dec-88
Dec-90
Dec-92
Dec-94
Dec-96
Dec-98
Dec-00
Dec-02
Dec-04
Dec-06
Dec-08
Indexes
0
10
20
30
40
50
60
70
80
90
100
Price-rent Ratio
Housing Rental Price index Housing Sale Price Index Price-rent Ratio


To lay the groundwork, Table 4.1 reports the mean, variance, and correlation
matrix of the main data series from sample period 1975 Q1 - 2008 Q4 for housing
consumption-based asset pricing model. This is because the data on housing consumption,
housing return, and expenditure share only available starting from 1975 Q1. All data
employed are in quarterly frequency. The first and second column are log relative
consumption log
t
t
c
s
and log relative price
log
c
t
s
t
p
p
and their correlation is negative (-95.9%)

58
just as model predicts. The third to fifth column reports the statistics of three time-variant
state variables of the model – expenditure share
t
α , autoregressive component of non-
housing consumption growth
t
x , and conditional volatility of non-housing consumption
growth
2
t
σ . These variables are estimated from equation system (3.39) through an
unobserved component method (UCM). Real log risk-free rate of return
, f t
r and real log
stock return
, mt
r are calculated from risk-free rate of return and stock return which are
obtained in the Kenneth R. French’s website. More precisely, risk-free rate of return is
one-month Treasury bill rate (from Ibbotson Associates) and stock return is the value-
weight return on all NYSE, AMEX, and NASDAQ stocks (from CRSP 2009 January
file). Stock PD ratio
, mt
q is downloaded from the Kenneth R. French’s website – table
named “Portfolios Formed on Dividend Yield”; Housing PD ratio (price-rent ratio) is
calculated following the procedure in section 4.2.4.
The correlation coefficient between expenditure share and relative consumption is
as high as 95.3%. This evidence is consistent with what I find in Figure 4.2 and supports
that ε is greater than 1. Another important empirical property of the economy is
1
1
ψ
< .
Just as equation (3.29) predicts, the conditional volatility
2
t
σ is negatively correlated to
the log real risk free rate of return
, f t
r . Noted that the conditional volatility is positively
correlated to log real housing return
, ht
r and log stock market return
, mt
r . Based on
equation (3.22a) and (3.35), it suggests that the inverse of IES is smaller than 1.  

59
One can find that my model introduces a term
( ) 1
log
t
τβ
α
θ
−
into the formula of
risk free rate of return
, f t
r (equation (3.47)), and the expected return of housing asset
,1 tht
Er
+
⎡⎤
⎣⎦
and other risky asset
,1 tCt
Er
+
⎡ ⎤
⎣ ⎦
as a whole (equation (3.45b) and equation
(3.50)). As discussed before, the sign of
( ) 1 τ β
θ
−
depends on the sign of
1
ψ
. Based on
the negative correlation between log
t
α and  
, f t
r , (also
,1 ht
r
+
and
,1 Ct
r
+
) as shown in Table
4.1, one might expect that
1
ψ
to be negative, precisely,
1
10
ψ
−<< .
Table 4.1 also indicates that autoregressive component of non-housing
consumption growth
t
x commoves with the risk free rate of return, housing return, and
stock market return. Also, I find that the actual housing return is more sensitive to the
consumption growth than the returns of the riskless asset and the stock market. What’s
more, the correlation between consumption growth and housing price-dividend ratio
, ht
q
is negative. All these properties could be well explained by the model.  


60
        Table 4.1: Summary Statistics of Quarterly Data (1975 Q1 – 2008 Q4)
log
t
t
c
s

log
c
t
s
t
p
p
t
α
t
x
2
t
σ
, f t
r
, mt
r
, ht
r
, mt
q
, ht
q
Variables
log relative
consumption
log
relative
price
Exp.
Share
Auto Reg
Component
Conditional
Volatility
Risk-free
return
Real
Stock
Return
Real
Housing
Return
Stock PD
ratio
Housing
PD ratio
Mean 0.139 -0.004 0.864 0.0001 6.29 E-05 0.005 0.019 0.017 -0.881 79.946
Variance 0.040 0.011 0.000 2.76 E-06 5.15 E-10 4.14 E-05 0.007 0.000 7.682 E05 70.949
Correlation Matrix
log relative consumption 1          
log relative price -95.9% 1        
Expenditure Share 95.3% -83.2% 1        
Auto Reg Component -7.8% 2.1% -12.6% 1      
Conditional Volatility -24.3% 23.3% -21.5% -14.3% 1      
Real Risk-free return -28.4% 16.0% -38.4% 15.9% -5.1% 1    
Real Stock Return -11.3% 7.3% -14.4% 15.6% 7.2% 9.3% 1    
Real Housing Return 3.9% -9.7% -0.6% 50.9% 3.3% 6.7% 15.3% 1  
Stock PD ratio -8.1% 5.8% 9.8% -1.5% -3.2% 1.2% 0.9% -4.7% 1  
Housing PD ratio 62.8% -50.0% 69.3% -41.3% -4.9% -34.9% -12.6% -21.4% 1.5% 1
Note: (1) This table reports the summary statistics over sample period 1975 Q1 – 2008 Q4.
(2) Log relative consumption is defined as the logarithm of the non-housing consumption index
t
c divided by housing quality index
t
s , where
t
s is constructed in equation (4.3).
t
c is calculated following the procedure in section 4.1. Log relative price is defined as the logarithm of the
non-housing price index
c
t
p divided by housing rent price index
s
t
p .
c
t
p is calculated by equation (4.1). Expenditure share
t
α is defined in
equation (3.11). Autoregressive component of non-housing consumption growth
t
x and conditional volatility of non-housing consumption
growth
2
t
σ are estimated from equations system (3.39). Real log risk-free rate of return
, f t
r , real log stock return
, mt
r are calculated from risk-
free rate of return and stock return in the Kenneth R. French’s website. Stock PD ratio
, mt
q is downloaded from the Kenneth R. French’s
website. Housing PD ratio (price-rent ratio) is calculated following the procedure in section 4.2.4.  


61
CHAPTER 5:  EMPIRICAL RESULTS  

5.1 Parameter Estimations  

5.1.1 Non-housing consumption growth and expenditure share
The decomposition of economic time series into underlying components which
have a direct interpretation (such as trend, seasonal, cyclical and irregular components)
has a long tradition in time series analysis literature (such as Harrison and Stevens 1976;
Harvey, 1989; and Harvey, 2001). These models are called “Unobserved Components
Models” (UCM) or also called “Structural Models in the time series literature in contrast
to the ARIMA "Reduced-form" representations (Diebold, 1992). In the traditional UCMs,
optimal estimation of the unobservable components can be obtained by means of some
kind of filtering procedure (such as Kalman filter in Harvey (1989), Hodrick–Prescott
filter in Hodrick and Prescott (1997)). However, Diebold (1992) argues that there are no
obvious evidences that the structural time series models are better than ARIMA models
and very little is known about the comparative predictive performance of the structural
model.  
My time series model for the non-housing consumptions (equation (3.15)) shares
the characteristics of both the structural model and ARIMA-ARCH model. It contains a
trend component, an autoregressive component, and an irregular term (error term) with
conditional heteroscedasticity. Since the NIPA consumption is already seasonality
adjusted, it’s not necessary to include the seasonal component. Also, for the simplicity


62
of deriving the analytical solutions of the real log risk-free return
, f t
r and housing return
, ht
r , cyclical component is also not included. The estimation results show that the
conditional variance
2
μ
σ of the trend component
t
μ is not different from zero (P value is
0.997). Hence I set
2
0
μ
σ = and utilize the special case of the model (equation (3.39)).
The parameter estimations and P-values for the special case are reported in Table 5.1 The
estimation uses a two-stage procedure. In the first stage, I use UCM procedure to estimate
the level component μ ,  and parameters ρ and
x
σ for the auto-regressive component
and output the irregular terms (error terms). The second stage use a AR(1) model with a
constant term to regress the square irregular terms to estimate parameter
2
σ , ν , and
w
σ .  

Table 5.1: Parameter Estimations of log Non-housing Consumption Growth
1 t
g
+


Auto-Reg. Component
t
x
Conditional Variance of  
Irregular Component
t
σ

Level
Component
μ

Damping
factor
ρ
Conditiona
l variance
x
σ
Long term
mean
2
σ
Mean-reverting
rate
ν
Conditional
variance
w
σ
Coefficient 0.0053  0.652 7.83E-06 7.14E-05 0.250 2.63E-09
P-value <.0001 0.001 0.1684 <.0001 0.0002 <.0001
Note: (1) This table reports the parameter estimation results over sample period 1945 Q1 – 2008 Q4 for
unobserved-component model (part of equation (3.39)).  
()
11
11
22 2 2
11
                 
tttt
tt xt
tt wt
gx
xx e
w
μση
ρσ
σσ νσ σ σ
++
++
++
=+ + ⎧
⎪
=+
⎨
⎪
=+ − +
⎩

(2) Log non-housing consumption growth is defined as
( )
1
log /
tt
cc
+
, where
t
c is calculated
following the procedure in section 4.1.



63
My time series model for the log expenditure share (equation (3.16)) is a AR(1)
model with a constant term, a time trend, and a heteroskedastic error term. Parameter
estimations and corresponding P-values are reported in Table 5.2.  

Table 5.2: Parameter Estimations of log Expenditure Share ()
1
log
t
α
+


Conditional Variance
,t α
σ

Initial  
mean
α
μ
Time  
Trend
a
Damping
factor
β

Constant Term
0
a
1
a
Coefficient -0.1642 0.0003 0.864  -0.004 1.55E+02
P-value 0.0014 0.0037 <.0001 <.0001 0.0096
R-Square 0.9248      
Durbin-Watson 2.3535
# #
     
Note: (1) This table reports the parameter estimation results over sample period 1975 Q1 – 2008 Q4 for
autoregressive model (part of equation (3.39)).  
()
()
1,1
2
,0 1
log log
                                       
tt tt
tt t
at at u
Eaax
ααα
α
αμ β α μ σ
σ
++
=+ + − − + ⎧
⎪
⎨
=+
⎪
⎩

(2) Expenditure share
t
α is defined in equation (3.11). Autoregressive component of non-housing
consumption growth
t
x and conditional volatility of non-housing consumption growth
2
t
σ are
estimated from equations system (3.39) through an unobserved component method (UCM).
(3) Durbin Watson statistics superscripted by
# #
means that the test doesn’t reject the null hypothesis
that first order autocorrelation coefficient is equal zero at 5% significance level;
#
means that the test is
inconclusive; otherwise, reject the null at 5% significance level.

5.1.2 Parameters of the pricing kernel  
The estimation of intratemporal elasticity of substitution ε is based on log form of the
first order condition (FOC). The result is reported in the following table.  
log constant log
c
tt
s
tt
cp
sp
ε
⎛⎞ ⎛ ⎞
=−
⎜⎟ ⎜ ⎟
⎝⎠ ⎝ ⎠
                                                                   (5.1)


64
Table 5.3: Parameter Estimations of intratemporal elasticity of substitution ε
Coefficient P-value
ε 1.976** <.0001
Adjusted R-Sq 0.9179  

Since I already obtained the analytical solutions of the log real risk free rate of
return
, f t
r and conditional mean of the real housing return
,1 ht
r
+
in terms of state variables
t
x ,
2
t
σ , and log
t
α in equation (3.47) and (3.50). In these two equations, coefficients on
these state variables are non-linear functions of preference parameter γ and ψ . Therefore,
I can uses the generalized method of moments (GMM) approach of Hansen and Singleton
(1982, 1983) to estimate the risk-aversion parameter γ and intertemporal elasticity of
substitution ψ . The instrument variables are the up to 6
th
order lagged values of state
variables and log real return of riskless asset and housing asset.  
Parameter estimates and the results of overidentifying restrictions tests of single-
asset GMM estimations are reported in the Table 5.4
12
. Except the 2-order lag results, the
estimations are quite consistent within the same asset. Several key results are found. First,
the model generates quite good goodness of fit for housing asset due to the higher
adjusted R-square (ranging from 20% to 28%). Unexpectedly, for housing asset, the
values of inverse of IES 1/ ψ are positive although it is still less than 1. But none of the
estimations of 1/ ψ are significantly different from zero. The positive 1/ ψ is caused by
                                               
12
Since the results for the 2-order lag results are quite inconsistent with higher-lag and 0-lag (OLS) results, I discuss
the findings mainly based on other lag results. This might be caused by the autocorrelations between the independent
variables and their lower-lag instruments.  




65
the positive coefficient on log expenditure share. However, this coefficient is also not
significant from zero. As expected, the value of risk aversion parameter γ is well above
1 for housing asset. The coefficient on the conditional variance
2
t
σ is positive and
statistically significant for housing asset. It means expected higher economic volatility
will make the agents buy less risky housing asset therefore pushing the housing return up.  
Second, the GMM estimations in Table 5.4 indicates that the model produce an
acceptable prediction for riskless asset (risk free rate of return). The adjusted R-square is
around 15%, smaller than that of housing asset. As expected, the 1/ ψ is estimated to be
significantly negative but larger than 1 − ( 11/ 0 ψ −<< ) which is suggested by the
negative estimated coefficient on log expenditure share. This finding is consistent with
many previous empirical works using the standard CCAPM, such as Hansen and
Singleton (1982, 1983, and 1988) and Davis and Martin (2005). It means that households
are not willing to substitute consumption over time. It challenges the claims of Piazzesi et
al (2006, 2007) and Fallet (2008). As expected, the estimations of the coefficients on
conditional variance of non-housing consumption growth
2
t
σ is negative. This is due to
the agents try to engage in more savings if consumption growth become more uncertain,
hence damping the interest rate.  
Third, as implied by the equation (3.49) and (3.50), a negative 1/ ψ also means
that the coefficients
() 1 τ β
θ
−
on log expenditure share is negative. This relationship is
strongly supported by the results of the riskless asset in Table 5.4 and the multi-asset
GMM estimations in Table 5.5. My explanation of this negative relationship is that a


66
higher expenditure share expectation is a signal of higher realized asset prices (see the
lowest two cells in the third column of Table 4.1; therefore a lower future expected asset
returns. In contrast to the model, I find an insignificantly positive coefficient on log
t
α
for housing asset. This could be partially explained by the agent’s worry about the
housing price bubble because a higher expected expenditure share is often accompanied
with a higher price-rent ratio, a sign of overinvestment of the housing market. Indeed, the
correlation between housing price-rent ratio and expenditure ratio suggested by the data
from 1975 Q1 to 2008 Q4 is as high as 69.3% (see Table 4.1).  
The last but not the least, my model could well explain the disparities of the
coefficients on the expected consumption growth
t
x
and constant term between the risky
asset and riskless asset, which could not be well resolved in the standard CCAPM, single-
good Epstein-Zin Utility CCAPM, and the Piazzesi et al (2007)’s housing CCAPM
framework. From equation (3.53), the coefficients for the excess housing return is
2
2
1
11 2 14
1
21 2
akA
πθε τ
ππ
θ εθ
⎡⎤
−
⎛⎞
−− − + +
⎢⎥
⎜⎟
−
⎝⎠
⎢⎥
⎣⎦
. At a glance of the model one could find that
this term is introduced by the joint impacts of the Epstein-Zin utility and cointegration
between the expenditure share and expected consumption growth. Actual consumption-
savings decisions depend not only on the uncertain overall size of future consumption
bundles and their uncertain composition, but also on the uncertain long-run consumption
streams. The long-run risks as introduced by the recursive utility function as the fear of
the long-horizon uncertainty of the consumption, jointly pushing up the magnitude of the
constant term and Beta on expected consumption growth in risky asset returns.  


67
       Table 5.4: Single Asset GMM Estimations for Housing-Consumption Based CAPM
Coefficients on  
Preference
Parameters
 Model Statistics
Constant
Auto Reg
Component
Conditional
Variance
Log Exp.
Share
1/IES RRA  Overidentifying Test
Asset
Type
Order
of Lags

t
x

2
t
σ log
t
α

1/ ψ
γ

Adj.
R_Sq
Chi-Square DF P-Value
-0.02** 0.26 -38.06** -0.19**  -0.70** 252.87  
0
(0.00) (0.43) (0.03) (<.0001) (<.0001) (0.22)
0.151 NA NA NA
-0.02** 0.60 42.33 -0.17**  -0.60** -213.27
2
(0.00) (0.12) (0.68) (0.00)  (0.00) (0.66)  
0.092 0.76  4 0.944  
-0.02** 0.58 -61.48 -0.16**  -0.82 7.25  
4
(<.0001) (0.14) (0.47) (<.0001) (0.94) (0.99)
0.155 6.80  10 0.744  
-0.02** 0.56* -74.14* -0.20**  -0.72** 520.46  
Riskless
Asset

6
(0.00) (0.06) (0.10) (<.0001) (<.0001) (0.27)
0.159

14.00  16 0.599  
0.02* 3.53** 63.64** 0.08  0.29 179.03  
0
(0.06) (<.0001) (0.01) (0.35) (0.35) (0.11)
0.263 NA NA NA
0.03* 5.52** 310.62 0.21  1.01 -6780.60
2
(0.08) (<.0001) (0.21) (0.12) (0.05) (0.98)
0.031 5.09  4 0.278  
0.02* 4.48** 138.83 0.09  0.33 412.53  
4
(0.07) (<.0001) (0.22) (0.33) (0.33) (0.39)
0.277 11.23  10 0.340  
0.02* 4.91** 195.11** 0.10  0.37 616.99  
Housing
Asset

6
(0.09) (<.0001) (0.03) (0.21) (0.21) (0.17)
0.205

17.47  16 0.356  
Note: (1) This table reports the single asset GMM parameter estimation results and over period 1975 Q1 – 2008 Q4 for equation (3.47) and (3.50).
(2) Expenditure share
t
α is defined in equation (3.11). Autoregressive component of non-housing consumption growth
t
x and conditional
volatility of non-housing consumption growth
2
t
σ are estimated from equations system (3.39) through an unobserved component method .
(3) P-values shown in parentheses below the parameter estimations measure the two-tails significant level of null hypothesis that the parameter
is not different from zero. P-values for the overidentifying restrictions test measure the significant level of null hypothesis that the moment
condition restrictions could be satisfied.
(4) Estimated coefficients or Chi-square statistics superscripted by ** are significant at 5% confidence level and * are significant at 10% level.


68
Table 5.5: Multi-Asset GMM Estimations for Housing-Consumption Based CAPM
Coefficients on
Preference
Parameters
Constant
Auto Reg
Component
Conditional
Variance
log Exp.
Share
1/IES RRA
Equation

t
x

2
t
σ log
t
α 1/ ψ
γ
Adjust
R_sq
 
Riskless
Asset
-0.01**
(0.03)
0.66**
(0.04)
-16.61** -0.12**

0.128

-0.44**
(0.00)
58.98*
(0.10)
Housing
Asset
0.00
(0.45)
3.76**
(<.0001)
41.87** -0.12
 
0.205
Note: (1) This table reports the multiple asset GMM parameter estimation results and overidentifying
restrictions test over period 1975 Q1 – 2008 Q4 for equation (3.47) and (3.50).
(2) Expenditure share
t
α is defined in equation (3.11). Autoregressive component of non-housing
consumption growth
t
x and conditional volatility of non-housing consumption growth
2
t
σ are
estimated from equations system (3.39) through an unobserved component method (UCM).
(3) P-values shown in parentheses below the parameter estimations measure the two-tails
significant level of null hypothesis that the parameter is not different from zero. P-values for
the overidentifying restrictions test measure the significant level of null hypothesis that the
moment condition restrictions could be satisfied.
(4) Estimated coefficients or Chi-square statistics superscripted by ** are significant at 5%
confidence level and * are significant at 10% level.

Table 5.5 reports the estimation results of multiple-asset GMM using both the
riskless asset and housing asset together. Based on the empirical results, I could get some
main findings. First, the goodness of model fit doesn’t change too much compared with
single-asset GMM estimations: 12.8% for riskless asset and 20.5% for risky housing asset.
Second, the results are quite consistent with all my expectations -- 1/ ψ is significant
negative and risk aversion parameter γ is significantly positive and well above 1. Third,
the loadings on the log expenditure share is negative (this is implied by the negative
1/ ψ ). Fourth, the loading on conditional variance is negative for riskless asset but
positive for housing asset. Based on equation (3.47) and (3.50), this could be satisfied if
and only if 11/ 1 ψ −< < and γ is well above 1. The estimation results based on my


69
model strongly support these relationships implied by the data. Fifth, the estimated results
exhibit that the loading of the expected consumption growth for housing asset is much
higher than that for riskless asset (3.76 vs. 0.66). This property could be well captured by my
model.  

5.2 GMM Estimations Based on Standard CCAPM
To provide additional evidence on the source of dynamics of the riskless asset and
risky asset return and provide more information to robust my model, I also utilize GMM
and MLE to estimate the preference parameters based on the standard CCAPM and eight
kinds of assets: riskless asset, housing asset and other six stock portfolios -- small cap
growth portfolio (SGRAT), small cap blend portfolio (SBRAT); small cap value portfolio
(SVRAT); large cap growth portfolio (BGRAT), large cap blend portfolio (BBRAT),
large cap value portfolio (BVRAT). The return data on six stock portfolios are obtained
from Kenneth R. French Website
13
.  
The results of single asset and multiple GMM estimations are reported in the
following tables. Based on the single asset GMM estimation results (see Table 5.6), the
statistics of the overidentifying restriction test show that the standard CCAPM cannot
generate satisfying results for housing and riskless asset. The results from the multiple
asset GMM estimation show no consistent estimations for 1/ ψ . Details of this robust
check are discussed in the Appendix A6.  

                                               
13
The return data on six stock portfolios are from Kenneth R. French Website. The dataset contains these information is
titled “Fama/French Benchmark Factors -- Quarterly”. The data are downloadable thought the hyperlink:
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ftp/F-F_Benchmark_Factors_Quarterly.zip


70
Table 5.6: Single Asset GMM Estimations for Standard CCAPM (1952 Q1--2008 Q4)
Overidentifying Restrictions Test
Asset
Number
of Lags
δ
1
ψ

Chi-Square DF P-Value
, f t
R 6
0.990 **
(<.0001)
0.114
(0.635)
74.68** 11 0.000  
, ht
R 6
0.979 **
(<.0001)
0.745**
(0.012)
25.83** 11 0.007
SGRAT 6
0.939 **
(<.0001)
-6.127*
(0.093)
14.39  11 0.212  
SBRAT 6
0.948 **
(<.0001)
-2.029
(0.462)
19.27* 11 0.056  
SVRAT 6
0.954 **
(<.0001)
-0.482
(0.862)
18.81* 11 0.065  
BGRAT 6
0.955 **
(<.0001)
-2.719
(0.257)
11.59 11 0.395  
BBRAT 6
0.962 **
<.0001
-1.392
(0.499)
9.78  11 0.551  
BVRAT 6
0.965 **
(<.0001)
-0.062
(0.979)
15.32  11 0.168  
Note: (1) This table reports parameter estimate results and overidentifying restrictions test for standard
CCAPM using single asset GMM estimations.
(2) P values shown in parentheses below the parameter estimations measure the two-tails
significant level of null hypothesis that the parameter is not different from zero.
(3) P-values for the overidentifying restrictions test measure the significant level of null
hypothesis that the moment condition restrictions could be satisfied.
(4) Estimated coefficients or Chi-square statistics superscripted by ** are significant at 5%
confidence level and * are significant at 10% level.











71
Table 5.7: Multi-Asset GMM Estimations for Standard CCAPM (1952 Q1-- 2008 Q4)
Overidentifying Restrictions Test No. of
Lags
δ
1
ψ

Chi-Square DF P-Value
1
0.969**
(<.0001)
2.751**
(<.0001)
103.4** 61 0.001  
2
0.974**
(<.0001)
-1.107**
(<.0001)
117.8  117 0.463  
4
0.971**
(<.0001)
0.468**
(<.0001)
115.8  229 1.000  
6
0.977**
(<.0001)
0.086
(0.156)
113.8  341 1.000  
Note: (1) This table reports parameter estimate results and overidentifying restrictions test for CCAPM
using multiple-asset GMM estimations.
(2) P values shown in parentheses below the parameter estimations measure the two-tails
significant level of null hypothesis that the parameter is not different from zero.
(3) P-values for the overidentifying restrictions test measure the significant level of null hypothesis
that the moment condition restrictions could be satisfied.
(4) Estimated coefficients or Chi-square statistics superscripted by ** are significant at 5%
confidence level and * are significant at 10% level.


72
CHAPTER 6:  PREDICTING CREDIT RISKS  
One of the objectives of this study is to investigate and mutual interactions
between two systematic risk factors – interest rate and housing price – of the mortgage
credit risks. In this chapter, I will try to apply my model to study the impact of
macroeconomic factors on loan performance.  
First, I use Kalman filter to estimate the unobservable state variables based on
data sample from 0 to t quarter and forecast out the one- to four-quarter ahead; Second,
based on the housing consumption based CAPM model I already have, I predict the
housing prices and interest rate; Third, however, in order to examine the systematic credit
risks based on the loan-level data, I need to use a reduced form model to control the
covariates of individual loans and to estimate the model implied hazard rate (conditional
prepayment and/or default rate) for individual loans caused by the systematic risk
components only. Lastly, I evaluate the prediction performance of my method.  

6.1 Estimation and Prediction of the State Variables
Optimal estimates of the trend and autoregressive components can be obtained by
means of the Kalman filter. The estimate of the unobservable state can be updated by
means of a Kalman filtering procedure as new observations become available. Predictions
are made by extrapolating these estimated components into the future. A UCM can
therefore not only provide forecasts, but can also, through estimates of the components,
present a set of stylised facts of the economy (Harvey and Shephard, 1993).


73
Recall section 3.3, I assume the only uncertainties in economy are the shocks to housing
and non-housing consumptions. I model the growth of non-housing consumption
1 t
g
+

containing a unobservable time trend component
t
μ , a unobservable persistent stationary
autoregressive component
t
x , and a time-varying volatility component
t
σ which
represents the fluctuating economic uncertainty. The log expenditure share
1
log
t
α
+
is
assumed to follow a mean-reverting process with mean reversion rate β , unconditional
mean ( ) at
α
μ + , and conditional variance
2
,t α
σ . The
2
,t α
σ is assume cointegrated with
t
x .
Also assume
11
, , ~ ... (0,1)
tt t
eu iidN η
++

11 tttt
gx μ ση
++
=+ +
11
                                                       ( 1 1)
tt xt
xx e ρσρ
++
=+ −< <
( ) ( )
22 2 2
11
                               0 1
tt wt
w σσ νσ σ σ ν
++
=+ − + < < (6.1a)
() ( )
1,1
log log           0 1
tt tt
at at u
ααα
αμ β α μ σ β
++
=+ + − − + < <
()
2
,0 1
exp
tt
aax
α
σ =+                                                                                           (6.1b)
The study employs quarterly data in both estimation and prediction. The UCM
(equation (6.1a)) regresses on the non-housing consumption growth
t
g data from first
quarter of 1975 (set as quarter 0) to quarter t (starting from fourth quarter of 1991) and
obtains a model-based decomposition of the series (
i
x ,
2
i
σ , and constant μ, 1,2,..., it = ),
at the same time forecasting the values of the response series of the component series 1-
quarter ahead (
1
1
q
t
x
+
and
12
1
q
t
σ
+
) to 4-quarter ahead (
4
4
q
t
x
+
and
42
4
q
t
σ
+
). Next, I use the
equation (6.1b) to estimate parameter a , β , and
α
μ , and then forecasts
1
1
log
q
t
α
+
to


74
-0.18
-0.16
-0.14
-0.12
-0.1
Mar-75
Mar-77
Mar-79
Mar-81
Mar-83
Mar-85
Mar-87
Mar-89
Mar-91
Mar-93
Mar-95
Mar-97
Mar-99
Mar-01
Mar-03
Mar-05
Mar-07
Mar-09
log Expenditure Share 1
log Exp. share 1-Quarter 2-Quarter 3-Quarter 4-Quarter
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
Mar-47
Mar-51
Mar-55
Mar-59
Mar-63
Mar-67
Mar-71
Mar-75
Mar-79
Mar-83
Mar-87
Mar-91
Mar-95
Mar-99
Mar-03
Mar-07
Auto Reg Com ponent 1
AutoReg 1-Quarter 2-Quarter 3-Quarter 4-Quarter
0
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
Mar-47
Mar-51
Mar-55
Mar-59
Mar-63
Mar-67
Mar-71
Mar-75
Mar-79
Mar-83
Mar-87
Mar-91
Mar-95
Mar-99
Mar-03
Mar-07
Conditional Variance  1
Conditional Var. 1-Quarter 2-Quarter 3-Quarter 4-Quarter
4
4
log
q
t
α
+
by extrapolating the series. Hence, I get the forecast of all state variables based
on Q1 1975 to Q4 1991 data.  
Following the same procedure and move forward the t to the first quarter of 1992,
I could get the forecast of state variables at the first quarter of 1992. Repeat this
procedure and I can get the forecasting of state variables series ranging between first
quarter of 1992 and fourth quarter of 2008 (for 1-quarter ahead forecast). Figure 6.1
exhibits forecasting results of state variables (1-Quarter to 4-Quarter ahead).

Figure 6.1: 1-Quarter to 4-Quarter Ahead Forecast for State Variables







75
6.2 Forecasting of Housing Price and Interest Rate
Note that, for simplicity, this section only discusses the 1-quarter-ahead forecast
case. One can easily deduct to 2-quarter-ahead, 3-quarter-ahead, and 4-quarter-ahead
forecast cases.
In order to capture the information of the base year, I employ difference equations
to calculate the forecasted interest rate and housing return. Based on equation (3.47) and
(3.50), I have:  
()
()
()
2
2
2 1
,114
1
111
1 11

21 2 2
1
         log
f t tt
t
a a
raA x
γ
τβ ψ θετ
κ σ
θψ ε θ
τβ
α
θ
⎛⎞
+−−
⎜⎟
⎛⎞
− −
⎛⎞
⎝⎠
Δ=− + + + − Δ + Δ
⎜⎟
⎜⎟
⎜⎟
−
⎝⎠
⎝⎠
−
+Δ
         (6.3)
()
()
2
2 1
,1 1 1 2
111
+1
2
1
                  log
tht t t
t
a
Er a x
τβ πθ
ππσ
θψ θ ψ
τβ
α
θ
+
−⎛⎞⎛⎞
⎛⎞
Δ ≈− − − + Δ−−Δ ⎡⎤
⎜⎟ ⎜⎟⎜⎟ ⎣⎦
⎝⎠
⎝⎠⎝⎠
−
+Δ
                    (6.4)
where
()
11 1
1
1
11
ε εγ
ψψψ
τ
ε
ψ
−+ + −
≡
⎛⎞
−−
⎜⎟
⎝⎠
,
1
11/
γ
θ
ψ
−
≡
−
.
( )
()
2
2
1
1 2
1
1
2-1
φκ
π
κβ
−
= and
2
π is denoted in
equation (3.34). Therefore, the estimated forecast of real log interest rate and real log
housing return at time 1 t + are
,, ,1 ftft ft
rr r
−
=Δ +                                                                                                           (6.5)
,1 , 1 , ht t ht h t
rEr r
++
≈Δ + ⎡⎤
⎣⎦
                                                                                               (6.6)


76
Further, the estimated forecast of nominal interest rate
, f t
NR
and housing price
index
1
h
t
p
+
at time 1 t + can be expressed as
()
1
,,
exp 1
c
t
ft f t c
t
p
NR r
p
+
=×−
                                                           (6.5)
() ()
1
1,1 1
exp 1
c
hhs
t
tht tt c
t
p
p rppI
p
+
++ +
=××− −                                                   (6.6)
where exogenous variable
1
c
t
c
t
p
p
+
is the inverse of deflator which is measured by the price
ratio of the numeraire (non-housing consumption index);
h
t
p is the housing sale price at
time t ,
1
s
t
p
+
is the estimated forecast of housing consumption index (rental price index).
I is an estimated variable that represents the housing improvements and repairs to
housing expenditure ratio.
Figure 6.2:  Housing Rental Index (1970-2008)
20
40
60
80
100
120
140
160
180
Dec-70
Dec-72
Dec-74
Dec-76
Dec-78
Dec-80
Dec-82
Dec-84
Dec-86
Dec-88
Dec-90
Dec-92
Dec-94
Dec-96
Dec-98
Dec-00
Dec-02
Dec-04
Dec-06
Dec-08
Index
Housing Rental Price index

Figure Description: Housing rental price index
s
t
p is the housing service price index (base year 1991 Q1).
Data are come from National Income and Product Accounts (NIPA), Bureau of Economic Analysis, the
U.S. Department of Commerce.  


77
Compared with the high volatility of the housing sale price, the housing rental
price index is very smooth and predicable (see Figure 6.2). I use a unit root process with
draft
11
0.9015
ss
tt
pp
++
=+  to forecast housing rental price index. The P-value for constant
is less than 0.0001, and the total R-square is as high as 99.9%.  

6.3 The Cox Proportional Hazard Model  
Starting from Schwartz and Torous (1989), proportional hazard assumption has
been applied widely to modeling loan termination behavior. As summarized by Deng and
Quigley (2002), the jointness of the prepayment and default options is important in
explaining behavior and competing risks models provide a convenient analytical tool for
analyzing borrower behavior.  
In this paper, I follow the option-based proportional hazard model developed by
Deng-Quigley-Van Order (2000) to isolate the systematic factors from the heterogeneities.
The model introduces a convenient but powerful solution to modeling the competing
risks of prepayment and default with a proportional hazard assumption. The estimate
method of Deng-Quigley-Van Order (2000) is Maximum Likelihood estimation.  
6.3.1 Data and method
The mortgage data is a sample of FHA-insured home purchase loans originated
during the 1992-1996 period. The sample consists of 120,342 loans. All mortgage loans
are fully amortizing, most with as long as thirty-year terms. The individual loan records
contain information of loan, borrower, property-related characteristics, such as location
(MSA), and termination date of each loan and reason for termination.  


78
Table 6.1: Statistics Summary of FHA Loan Data
Variable Name Label MEAN STD
score1 Credit Scores < 620 (Dummy) 0.1852 0.3884
score2 Credit Scores 620~680 (Dummy) 0.3150 0.4645
score3 Credit Scores 680~740 (Dummy) 0.3206 0.4667
black Black (Dummy) 0.1245 0.3302
asian Asian (Dummy) 0.0175 0.1309
hispanic Hispanic (Dummy) 0.1159 0.3201
other Others (Dummy) 0.0128 0.1123
LTV Loan-to-Value Ratio 0.9389 0.0607
hei20_38 Housing Exp. to Income Ratio 20~38% (Dummy) 0.6388 0.4803
hei38_ Housing Exp. to Income Ratio > 38% (Dummy) 0.0112 0.1050
dti20_41 Debt to Income Ratio 20~41% (Dummy) 0.8090 0.3931
dti41_53 Debt to Income Ratio 41~53% (Dummy) 0.1564 0.3633
dti53_ Debt to Income Ratio >53% (Dummy) 0.0081 0.0896
IBD Buydown (Dummy) 0.0237 0.1520
hval_l Log of Property Appraisal Value 11.2135 0.3709
shrtmor Mortgage Term < 30 Years (Dummy) 0.0365 0.1876
central Central City Location (Dummy) 0.4339 0.4956
RURAL Rural (Dummy) 0.0690 0.2535
firstbuy First Time Home Buyer (Dummy) 0.6741 0.4687
new New House (Dummy) 0.0747 0.2630
cbunmard Unmarried Co-borrower (Dummy) 0.1087 0.3113
singlem Single Male (Dummy) 0.1938 0.3952
singlef Single Female (Dummy) 0.1972 0.3979
DEPENDNT Number of Dependents 0.7744 1.1093
lqass_l Log Value of Liquid Asset 8.4660 1.5807
age_25 Borrower Age < 25 (Dummy) 0.1141 0.3179
age25_35 Borrower Age 25~35 (Dummy) 0.5454 0.4979
age35_45 Borrower Age 35~45 (Dummy) 0.2722 0.4451
income_l Log Value of Household Income 8.0452 0.3985
Total Observations 115,406  
Note: (1) This table reports the statistics summary of FHA-insured home purchase loans originated during
the 1992-1996 period.  
(2) All loans are grouped according to the rounded mortgage rate and origination month. Groups
with observation frequency less than 50 are excluded. Therefore, I only base on 115,406 loans in this
study.  



79
Table 6.2: Maximum likelihood estimations for Deng et al (2000) Hazard Model
Prepayment Default
Variable
Estimations T-stat Estimations T-stat
Call Option for Credit Scores < 620 (Dummy) 4.54 (10.67) 2.18 (1.99)
Call Option for Credit Scores 620~680 (Dummy) 4.22 (13.36) 1.53 (1.57)
Call Option for Credit Scores 680~740 (Dummy) 5.38 (18.19) 1.67 (1.38)
Call Option for Credit Scores >740 (Dummy) 5.83 (15.94) 2.71 (1.42)
Put Option for Credit Scores < 620 (Dummy) -4.05 (6.71) 2.89 (3.95)
Put Option for Credit Scores 620~680 (Dummy) -2.65 (5.60) 1.85 (3.22)
Put Option for Credit Scores 680~740 (Dummy) -1.96 (5.24) 2.66 (5.15)
Put Option for Credit Scores >740 (Dummy) -1.53 (4.28) 3.07 (4.20)
Credit Scores < 620 (Dummy) -0.04 (0.65) 1.76 (5.78)
Credit Scores 620~680 (Dummy) 0.01 (0.10) 1.41 (4.83)
Credit Scores 680~740 (Dummy) -0.02 (0.40) 0.76 (2.53)
Black (Dummy) -0.50 (10.19) 0.50 (4.05)
Asian (Dummy) 0.01 (0.11) -0.25 (0.52)
Hispanic (Dummy) -0.28 (6.07) 0.19 (1.43)
Others (Dummy) -0.34 (3.36) -0.02 (0.07)
SMSA Unemployment Rate -0.12 (15.00) 0.05 (2.30)
Loan-to-Value Ratio -1.00 (5.26) 0.21 (0.24)
Housing Exp. to Income Ratio 20~38% (Dummy) 0.11 (2.67) 0.21 (1.24)
Housing Exp. to Income Ratio > 38% (Dummy) 0.08 (0.56) 0.19 (0.34)
Debt to Income Ratio 20~41% (Dummy) 0.01 (0.15) -0.43 (1.25)
Debt to Income Ratio 41~53% (Dummy) 0.10 (1.17) -0.27 (0.75)
Debt to Income Ratio >53% (Dummy) 0.04 (0.24) -0.75 (1.05)
Buydown (Dummy) 0.14 (1.85) 0.10 (0.32)
Log of Property Appraisal Value 0.16 (2.64) 0.18 (0.80)
Mortgage Term < 30 Years (Dummy) -0.03 (0.49) -0.98 (2.12)
Central City Location (Dummy) 0.04 (1.51) -0.15 (1.53)
Rural (Dummy) 0.05 (0.90) -0.29 (1.28)
First Time Home Buyer (Dummy) -0.19 (6.70) 0.07 (0.64)
New House (Dummy) -0.03 (0.60) -0.04 (0.18)
Unmarried Co-borrower (Dummy) -0.02 (0.36) -0.08 (0.43)
Single Male (Dummy) -0.01 (0.18) 0.27 (2.09)
Single Female (Dummy) -0.11 (2.72) -0.23 (1.51)
Number of Dependents -0.08 (5.78) 0.07 (1.60)
Log Value of Liquid Asset 0.01 (1.41) -0.07 (2.04)
Borrower Age < 25 (Dummy) 0.44 (7.94) 0.15 (0.80)
Borrower Age 25~35 (Dummy) 0.28 (6.48) -0.21 (1.37)
Borrower Age 35~45 (Dummy) 0.07 (1.46) -0.15 (0.94)
Log Value of Household Income 0.22 (3.27) -0.53 (2.02)
Note: (1) This table reports the MLE estimation results of Deng- Quigley-Van Order (2000) competing
risk hazard model for FHA-insured home purchase loans data originated during the 1992-1996
period  



80
Starting from the origination month, for each individual loan I can use equation
(6.5) and (6.6) to forecast nominal interest rate
, f t
NR
and housing price
1
h
t
p
+
. And then to
calculate the intrinsic value of exercising the call option (prepayment) and put option
(default). Following Deng-Quigley-Van Order (2000)’s method. From their definition of
the options, I find that the forecasted value of the put and call option are conditional on
interest rate and housing prices.  
The forecasted call option for each individual FHA loan is calculated through the
formula:  
,,
,
,
 
ii i i
ii
mr
ik ik
ik m
ik
VV
Forecasted Call Option
V
ττ
τ
+ +
+
−
=
                                                         (6.7)
While the put option is forecasted by
( )
,
,
log log
 
ii ii
m
ik k
ik
VH
Forecasted Put Option
ττ
ω
++
⎛⎞ −
=Φ⎜⎟
⎜⎟
⎝⎠

                                   (6.8)
where
()
,
1
,
1
ii
ii
ii
TM k
m
i
ik s
s
fk
P
V
NR
τ
τ
−
+
=
+
=
+
∑
 is forecasted market value of mortgage at time
ii
k τ + ,
()
,
1 1
ii
ii
TM k
r i
ik s
s
i
P
V
r
τ
−
+
=
=
+
∑
is the forecasted par value of mortgage at time
ii
k τ + ; and
i
r is
mortgage rate for house i ;
i
TM is the mortgage term;
i
k
is the seasoning period of the
mortgage after origination at time
i
τ ;
,
ii
f k
NR
τ +
is the forecasted interest rate; and P
i
is the
monthly mortgage payment; ( ) Φ ⋅ is CDF of standard normal distribution; ω is an
estimated variance following the method in Deng-Quigley-Van Order (2000) page 305.
Housing estimated price
ii
k
H
τ +
at time is forecasted with the formula:  


81
ii
ii
i
h
k
ki h
p
HC
p
τ
τ
τ
+
+
=
                                                                                            (6.9)
where
i
C is the purchase price of house i at time
i
τ ;
ii
h
k
p
τ +
is the forecasted price index at
time
ii
k τ + .
i
h
p
τ
is the actual price index at origination date
i
τ .  
The mortgage rates of each loan are rounded to the nearest 50 basic points. All
loans are grouped according to the rounded mortgage rate and origination month. I
exclude the groups which contain less than 50 loans. It leaves us 7 coupon rate groups:
6.5%, 7%, 7.5%, 8%, 8.5%, 9% and 9.5%, and 45 months of origination, totally 235 loan
groups, 115,406 loans. Therefore, I can calculate the one-quarter-ahead forecast of the
default and prepayment hazard rate for each individual loan. The results of the group
hazard rates are the weighted average results of each loan. The forecasted results of
selected groups are reported and discussed in next section.  

6.3.2 Results
Figure 6.3 exhibits comparison between the actual prepay-default CPR and one-
quarter-ahead forecasted prepay-default CPR. The two-quarter-ahead, three-quarter-
ahead, and four-quarter-ahead forecasted CPRs are similar with the one--quarter-ahead
results.  
From the figure I can find out that the model implied forecasted results track the
trend of the actual CPR but the model series are smoother than the actual series. It
indicates that my housing consumption based asset pricing model can help predict the
trend of the credit cycles. But, I find that the model results are not sensitive to the


82
short-term shock to the credit risks, such as the shock in the end of year 1994 for the
group originated in 1992 May with coupon rate 9.0%. This is partially because the
forecasted state variables and housing prices and interest rate are in quarter frequency. I
use the linear interpolation to generate the monthly forecasted estimate results.  


83
Figure 6.3: Actual Prepayment-Default CPR vs. One-Quarter-Ahead Forecasted
CPR  
FHA: 1992-May-9.0%
(High Coupon Rate & High Prepayment)
Observations: 1169
0
0.1
0.2
0.3
0.4
0.5
0.6
6/1992
10/1992
2/1993
6/1993
10/1993
2/1994
6/1994
10/1994
2/1995
6/1995
10/1995
2/1996
6/1996
10/1996
2/1997
6/1997
10/1997
2/1998
6/1998
10/1998
2/1999
6/1999
Actual CPR Q-1 Ahead Forecast
FHA: 1992-Sept-7.5%
(Low Coupon Rate & Low Prepayment)
Abservations: 446
0
0.1
0.2
0.3
0.4
0.5
0.6
10/1992
2/1993
6/1993
10/1993
2/1994
6/1994
10/1994
2/1995
6/1995
10/1995
2/1996
6/1996
10/1996
2/1997
6/1997
10/1997
2/1998
6/1998
10/1998
2/1999
6/1999
Actual CPR Q-1 Ahead Forecast
FHA: 1993-Jan-7.5%
(Low Coupon Rate & Low Prepayment)
Observations: 303
0
0.1
0.2
0.3
0.4
0.5
0.6
2/1993
6/1993
10/1993
2/1994
6/1994
10/1994
2/1995
6/1995
10/1995
2/1996
6/1996
10/1996
2/1997
6/1997
10/1997
2/1998
6/1998
10/1998
2/1999
6/1999
Actual CPR Q-1 Ahead Forecast
FHA: 1993-Feb-8.5%
(High Coupon Rate & High Prepayment)
Observations: 1077
0
0.1
0.2
0.3
0.4
0.5
0.6
3/1993
7/1993
11/1993
3/1994
7/1994
11/1994
3/1995
7/1995
11/1995
3/1996
7/1996
11/1996
3/1997
7/1997
11/1997
3/1998
7/1998
11/1998
3/1999
Actual CPR Q-1 Ahead Forecast
FHA: 1994-May-8.0%
(Low Coupon Rate & Low Prepayment)
Observations: 1235
0
0.1
0.2
0.3
0.4
0.5
0.6
6/1994
10/1994
2/1995
6/1995
10/1995
2/1996
6/1996
10/1996
2/1997
6/1997
10/1997
2/1998
6/1998
10/1998
2/1999
6/1999
Actual CPR Q-1 Ahead Forecast
FHA: 1994-Jul-9.0%
(High Coupon Rate & High Prepayment)
Observations: 946
0
0.1
0.2
0.3
0.4
0.5
0.6
7/1994
11/1994
3/1995
7/1995
11/1995
3/1996
7/1996
11/1996
3/1997
7/1997
11/1997
3/1998
7/1998
11/1998
3/1999
Actual CPR Q-1 Ahead Forecast


84
Figure 6.3, Continued: Actual Prepayment-Default CPR vs. One-Quarter-Ahead
Forecasted CPR
FHA: 1995-Jan-7.5%
(Low Coupon Rate & Low Prepayment)
Observations: 337
0
0.1
0.2
0.3
0.4
0.5
0.6
2/1995 6/1995 10/1995 2/1996 6/1996 10/1996 2/1997 6/1997 10/1997 2/1998 6/1998 10/1998 2/1999 6/1999
Actual CPR Q-1 Ahead Forecast
FHA: 1995-Jan-9.5%
(High Coupon Rate & High Prepayment)
Observations: 815
0
0.1
0.2
0.3
0.4
0.5
0.6
2/1995 6/1995 10/1995 2/1996 6/1996 10/1996 2/1997 6/1997 10/1997 2/1998 6/1998 10/1998 2/1999 6/1999
Actual CPR Q-1 Ahead Forecast
FHA: 1996-May-7.5%
(Low Coupon Rate & Low Prepayment)
Observations: 1082
0
0.1
0.2
0.3
0.4
0.5
0.6
6/1996 10/1996 2/1997 6/1997 10/1997 2/1998 6/1998 10/1998 2/1999 6/1999
Actual CPR Q-1 Ahead Forecast
FHA: 1996-Aug-9.0%
(High Coupon Rate & High Prepayment)
Observations: 992
0
0.1
0.2
0.3
0.4
0.5
0.6
9/1996 1/1997 5/1997 9/1997 1/1998 5/1998 9/1998 1/1999 5/1999
Actual CPR Q-1 Ahead Forecast

It’s worthy of note that my model only focuses on the effects of the nationwide
level economic factors (consumption and consumption-composition) on housing price
and interest rate, and the mortgage credit risks. The MSA-level volatility which is
important to credit risk pricing is not taken into consideration both in the asset pricing
model and the forecasting using Deng et al (2000) hazard model. Housing market
business cycles are quite localized. Loans within a MSA tend to be highly correlated --


85
they operate in the same regulatory environment, are similarly sensitive to
macroeconomic shocks, and are exposed to similar market supply and demand
fluctuations. Thus, the microstructure-bias in mortgage credit risk behavior within one
MSA is smaller than those across different MSAs. This is because house can be
decomposed into two components: structure and associated land. Though they are traded
as a bundle, structures and land are quite different. From 1975 to 2006, the value of land
has increased three times as fast as the value of house structures, and thus land price has
become an increasingly important component of wealth. By the second quarter of 2006,
residential land in the U.S. was valued at $11.6 trillion, accounting for 46% of the value
of the housing stock and 88% of GDP (Davis and Heathcote, 2007). Meanwhile, the land
prices are volatile: from 1975 to 2006, the real land prices are 3.3 times as volatile as real
house structure prices. In regions where land prices have surged, such as Los Angeles,
San Francisco, and Boston, the land’s share of house value are around 80% or higher in
2004
14
(Davis and Palumbo, 2007). Therefore, have much higher foreclosure rates than
the national average, and these areas deserve particular attention from the federal, state,
and local policymakers.  
                                               
14
The land’s shares of house value for Los Angeles, San Francisco, and Boston in 1998 are 0.649, 0.811, and 0.600
respectively, and those in 2004 are 0.787, 0.885, and 0.757.  




86
CHAPTER 7:  CONCLUSIONS  
My study introduces an equilibrium housing consumption-based capital asset
model (HCCAPM) to examine the relationship between the housing price and interest
rate dynamics. Similar with Piazzesi et al (2007)’s work, it models the housing both as an
asset and as consumption good. Unlike their work, instead of using the NIPA housing
quantity index as the measurement of housing consumption, my dissertation develops a
housing quality index as housing consumption. Since every household need a unit to live
either by renting or by owning, the housing quality which includes the location related
amenities, unit area, and housing structure attributes is a better measurement of housing
service. One interesting finding through investigating the housing quality index is that
contradicting to the intention of the U.S. government homeownership policy, the average
housing service a household consumed declined over time with the growing up of the
housing price index since 1991. And the burst of the housing bubble indicates that this
policy may not be sustainable. The pursuit of the higher homeownership ratio may be
misleading or even could be harmful to both the housing industry and the whole economy.  
I assume that the utility function of a representative agent takes the Epstein and
Zin (1989) recursive utility specification rather than the standard power utility function in
order to capture the long-risks implied by the observed properties of the data. The long-
run risk is important in asset pricing because the agent’s horizon on the investment-
consumption tradeoff might be longer than the previous asset pricing model had assumed.
What’s more, as a durable and expensive asset and consumption good, housing asset
were purchased on high-margin and the general mortgage terms are from 15 to 30 years.


87
Hence, housing is particularly subject to the overall economic shocks and long-run
uncertainty. This effect will be amplified if the housing assets are over-leveraged through
relaxed-standard mortgage lending and aggressive securitization, just as one can find in
this subprime and financial crisis – 20 percent drops in housing price index were enough
to place the whole economy in jeopardy.  
Using the non-housing consumption as numeraire, a general specification of the
pricing kernel is derived (equation (3.10) and equation (3.14)). Based on the assumptions
on the dynamics of economy and the stochastic generating process of price-dividend ratio
of the claim to aggregate consumption in section 3.31 , my dissertation numerically solve
the pricing kernel of the HCCAPM model (in equation (3.24)) in terms of four time-
variant exogenous state variables --
t
x (autoregressive component of non-housing
consumption growth),
t
μ (trend component of non-housing consumption growth),
2
t
σ
(conditional variance of non-housing consumption growth), and log
t
α (log expenditure
share of non-housing expenditure). Equilibrium analytical solutions of real log housing
return and risk free rate of return in terms of state variables are also obtained in equation
(3.29) and (3.35) respectively. A special case which assumes a fixed trend component μ
based on the evidence from my data sample is also discussed. The accordingly analytical
solutions of pricing kernel, real log housing return and risk free rate of return in terms of
state variables are also presented in the dissertation. Besides the consumption risks and
composition risks, the model presents long-run risks and cointegration between
expenditure share uncertainty and expected economic growth as two extra factors that
drive asset prices. It reveals that the household not only concerns with the consumption


88
growth and consumption composition between the non-housing and housing service, but
also concerns with the long-run consumption risks introduced by the Epstein-Zin
recursive utility and the uncertain of expenditure share positively cointegrated with
expected economic growth. I find that the model is helpful to resolve the interest rate
puzzle and equity puzzle.  
In the empirical section, I focus on the special case and adopt the Unobserved
Components Method (UCM) technique to estimate the parameters of the economic
forcing dynamics (in equation (3.39)) and utilize the Generalized Method of Moments
(GMM) approach to estimate the preference parameters in the pricing kernel (in equation
(3.14)). The data applied are quarterly data ranging from the first quarter of 1975 to the
fourth quarter of 2008. Several key results are found. First, the model generates quite
good goodness of fit (for housing asset adjusted R-squares are from 20% to 28%, and for
riskless asset adjusted R-squares are around 15%). As expected, the value of risk aversion
parameter γ is well above 1 and the values of inverse of the intertemporal elasticity of
substitution (IES) 1/ ψ is negative but larger than 1 − (1 1/ 0 ψ −<< ). This finding is
consistent with many previous empirical works using the standard CCAPM, such as
Hansen and Singleton (1982, 1983, and 1988). It means that households are not willing to
substitute consumption goods over time. It challenges the claims of Piazzesi et al (2006,
2007). Second, the coefficient on the conditional variance
2
t
σ is significantly positive for
housing asset and is negative for riskless asset.  It means expected higher economic
volatility will make the agents buy more riskless asst and buy less risky housing asset
therefore pushing the housing return up and the interest rate down. Third, as suggested by


89
the equation (3.49) and equation (3.50), a negative 1/ ψ also means that the coefficients
on log expenditure share is negative. This relationship is strongly supported by the data.
Furthermore, my model could well explain the disparities of the coefficients on the
expected consumption growth
t
x
and constant term between the risky asset and riskless
asset, which could not be well resolved in the standard CCAPM, one-consumption-good
Epstein-Zin Utility CCAPM, and the Piazzesi et al (2007)’s housing CCAPM framework.
I denote these disparities as the long-run risk that introduced by the joint impacts of the
Epstein-Zin utility and cointegration between the expenditure share and expected
consumption growth. The fear of the long-horizon uncertainty of the consumption
streams leads to a risk premium in the constant term and causes a higher sensitivity on
expected consumption growth for returns of risky assets such as housing.  
In the application section, I incorporate these two mutual interacted factors – risk
free rate of return and housing return (housing price) into an option-based proportional
hazard model to forecast risks of FHA-insured mortgage default and prepayment. I find
that this model could deliver quite useful predictions of the shocks of macroeconomic
dynamics on credit risks movement.  
The study contributes to the asset pricing by providing a new general theatrical
framework to study the long-run consumptions on the pricing of the general assets. The
study also contributes to the asset pricing study with new empirical evidence and
perspectives. My study finds that the UCM method is very useful in identifying the latent
driving forces behind the aggregate consumption and composition dynamics.
Furthermore, the housing consumption based model is also helpful to explain the risk-


90
free rate puzzle, equity puzzle, and why housing asset is more risky than the overall non-
housing risky asset.  
One of the major contributions of my paper is the development of a housing
quality index as the measurement of the housing consumption. It includes the location
related amenities, unit area, and housing structure attributes and therefore it is a better
measurement of housing service. Compared with the homeownership ratio, I argue that
the housing quality index is an important alternative indicator of well-being the shield
provides.  
The study also contributes to the mortgage credit pricing study by providing more
understanding of the impacts of macro economy on loan performance and providing a
potential way to predict the credit risks caused by the macro and long run risk factors.
While the previous studies examine the termination behaviors of loans in cross-sectional
perspective, I argue that the long-run macro economic risks, mortgage finance, and
human behavior (overreaction to past information that pushes the asset prices
above/below their fundamental values and the error correction thereafter) are critical to
understand current subprime and financial crisis. Both the residential real estate and
finance are heavily capitalized industries. They tends to oversupply in prosperity or when
there is other positive information, and traps into heavy losses during the subsequent
economic recession. Therefore, a special note of caution is necessary when the market is
oversupply. A public policy that is capable to help reduce oversupply is a wise way to
pursue. It could create favorable market conditions for both the housing and finance
industry, and therefore help improve their performance in the long-run.  


91
It could also provide hedging implications for portfolio risk management. More
and more scholars believe that the subprime crisis is an extremely expensive cost to learn
a new financial innovation and they don’t tend to think the subprime itself as a devil. My
study tends to support such a argument and tries to provide some business and policy-
related implications for the subprime crisis from a long-horizon perspective.  
Admittedly, the empirical research is in nonexperimental settings and thus it is
limited by data availability. Most the macro economic data, such as consumption,
inflation and housing rent information are only available in quarterly frequency basis,
which makes it hard to accurately estimate the conditional variance of them. What’s the
more, the sample period for housing service consumption and expenditure is short, so the
explanatory power of some empirical evidence could be subject to the these two
problems. On the other hand, it is still difficult to justify the reliability of the preference
parameters estimated from different assets (such as housing asset and stocks) and make a
horse race for the comparative predictive performance of different asset pricing models.
In this vein, the explanations of this study are only suggestive and an open area of
research. However, I believe this study could contribute new evidence and perspectives to
the justification of effect of long run consumption risks on asset prices.  
My results also raise at least three interesting questions for future research. First,
although my asset pricing model well explained the disparities of coefficients on the
expected consumption growth and constant term between the risky asset and riskless
asset, it cannot explain in theory why housing is positively sensitive to expenditure share.
Also, the estimated magnitude of parameters are heavily depends on the data period span
and the estimation methods applied.  I know of no explanation for these puzzles.  


92
Second, I use a well developed reduced form hazard model to estimate the
mortgage termination rates. It’s hard to construct a structural model to incorporate the
systematic factors and study their mutual interactions on credit risks or mortgage rates.  
Finally, as discussed before the housing market business cycles are quite localized,
and the MSA level risk is equally as important as the macro economic risk factors. Hence,
how to take these factors into consideration is an interesting topic for future research.  


93
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99
APPENDICES  
A1. Pricing Kernel  
The forcing processes of the economy are defined in equation (3.15) and (3.16) are
11 tt t tt
gx μ ση
++
=+ +
11 tt t μ
μ μσζ
++
=+
( )
11
                                                       1 1
tt xt
xx e ρσ ρ
++
=+ −< <
( ) ( )
22 2 2
11
                               0 1
tt wt
w σσ νσ σ σ ν
++
=+ − + < <
()
1,1
log log           (0 1)
tt tt
at at u
ααα
αμ β α μ σ β
++
=+ + − − + < <
()
2
,0 1 tttat
aa x
α
σ μσς =+ + +
11 1
, , , , ~ ... (0,1)
tt t t a
eu iidN ηζσ
++ +
                                                                        (A1)
The pricing kernel of this economy is given by  
() ( )
11 1 ,1
log log 1
tt t Ct
mg r
θ
θδ τ α θ
ψ
++ + +
=− +Δ +−                                                 (A2)
where
()
11 1
1
1
11
ε εγ
ψψψ
τ
ε
ψ
−+ + −
≡
⎛⎞
−−
⎜⎟
⎝⎠
and
1
11/
γ
θ
ψ
−
≡
−
. Since the asset pricing condition
1, 1
1
tt it
EM R
++
= ⎡⎤
⎣⎦
or
( )
1,1
exp 1
tt it
Em r
++
⎡⎤ + =
⎣⎦
must hold for any asset i
()
,1 ,1
log
it i t
rR
++
= including the risk-free asset
,1 f t
r
+
and aggregate consumption claim asset
,1 Ct
r
+
, we have the Euler equation


100
() ( )
11 ,1,1
exp log log 1 1
tt t Ctit
Eg rr
θ
θδ τ α θ
ψ
++ ++
⎡⎤ ⎛⎞
−+Δ +− + =
⎢⎥ ⎜⎟
⎝⎠ ⎣⎦
                       (A3)
The approximation of log real return on the aggregated consumption claim asset
,1 Ct
r
+
is
()
,1 0 1 , 1 , 1 1
log
1
Ct C t Ct t t
rqqg
ε
κκ α
ε
++ + +
+− + + Δ
−
                                                (A4)
The price-consumption ratio is assumed to be a linear combination of all four
exogenous state variables
2
,0 1 2 3 4
log
Ct t t t t
qA Ax A A A μ σα ++ + +                                                                (A5)
To rewrite the pricing kernel (A2) in terms of state variables
t
x ,
t
μ ,
t
σ , and log
t
α ,
I substitute the equation (A1), (A4) and (A5) into equation (A2) and obtain  
()
( ) ( )
()
()
()()
() ()
() ()
( )
11
01 1
01 1 2 1
22 2
01 3 1
4011
2
01 2 3 4
1log
log
        1
log
lo
t
ttttt
tt t
txt t t
twt
tttt
tt t
at
mx
aa xu
AA x e A
Aw
Aataaxu
AAxA A A
α
μ
αα
βμ α
θ
θδ μ ση χ
ψ
μ
ρσ μ σζ
κκ σ νσ σ σ
θ
μβ α μ μ
μσ
++
+
++
+
+
−+− ⎡⎤
=− ++ +⎢⎥
++ + ⎢⎥
⎣⎦
⎛⎞
++ + + +
⎜⎟
⎜⎟
++ −+ + −
⎜⎟
+−
⎜⎟
⎜+ − − + + + ⎟
⎝⎠
++ + +
()()
1
g
tt t tt
x αμ ση
+
⎡⎤
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥ ++ +
⎣⎦
 (A6)
where
1
1
ε γ
χ
ε
−
=
−
. It is easy to compute the conditional mean and variance of log pricing
kernel in the followings


101
[] ()( )( )
( ) ( )
()( )
()( ) ()( )
() ( ) ( )() ( )
2
01 0 13
1
14
12 2 1 1 1
2
31 4 1
11
log 1 1
1
              1 1 1 1
              1 1 1 1 1 log
tt
tt
tt
AA
Em at
Aat
AAAAx
AA
α
α
κκκσ ν
θδ χ β μ θ
κμ β
θθ
θκ μ θ κ ρ
ψψ
θκν σ χ β θ κβ α
+
⎛⎞ +− + −
=+ − + +−⎜⎟
⎜⎟
++ −
⎝⎠
⎡⎤⎡ ⎤
+− + − − + +− + − − +
⎢⎥⎢ ⎥
⎣⎦⎣ ⎦
+− − +− − + − − ⎡⎤
⎣⎦
(A7)
[ ] ( ) ( ) ( )
() ( )
( )
11 11 1 12 1 13 1
14 0 1 1 1
11 1
                   1
tt t xt t wt
tt t tt
mEm A e A A w
Aaa xu
μ
θκ σ θ κ σζ θ κ σ
θκ χ μ γση
++ + + +
++
−=− +− +−
+− + + + − ⎡⎤
⎣⎦
          (A8)
Further it follows that
[] () ( ) () ( )
2 2
22 2 22
11401
var 1 + 1
tt x w t t t
mAaax
μ
θ λλ λ θ κ χ μ γσ
+
=− + + − + + + + ⎡⎤⎡ ⎤
⎣⎦⎣ ⎦
       (A9)
where I denote
11 x x
A λ κσ = ,
12
A
μ μ
λ κσ = , and
13 ww
A λ κσ = .  
The conditional variance of log pricing kernel is determined by the conditional
mean ()
tt
x μ + and conditional variance of the growth of non-housing consumption
2
t
σ .

A2. Log Real Return on a Claim to Aggregate Consumption  
Euler equation (A3) implies that  
()
11,1
exp log log 1
tt tCt
Eg r
θ
θδ τ α θ
ψ
++ +
⎡⎤ ⎛⎞
−+Δ + =
⎢⎥ ⎜⎟
⎝⎠ ⎣⎦
                                      (A10)
I substitute the approximation (A4) and (A5) along with (A1) into Euler equation
(A10) and yield the equation in terms of the state variables
t
x ,
t
μ ,
t
σ , and log
t
α in the
following:


102
()
()()
() ()
() ()
( )
()()
()( )
1
01 1 2 1
22 2
01 3 1
4011
2
01 2 3 4 1
log
exp
log
log
1log
tt tt
txt t t
twt
t
tttt
tt t t t t tt
t
x
AA x e A
Aw
E
Aataaxu
AAx A A A x
at
μ
αα
α
θ
θδ μ ση
ψ
ρσ μ σζ
κκ σ νσ σ σ
θ
μβ α μ μ
μσ αμ ση
φβμ α
+
++
+
+
+
−++ +
⎛⎞ ⎛⎞
++ + + +
⎜⎟ ⎜⎟
⎜⎟ ⎜⎟
++ −+ + −
⎜⎟ ⎜⎟
+
⎜⎟ ⎜⎟
⎜+ − − + + + ⎟
⎜⎟
⎝⎠
⎜⎟
⎜⎟
++ + + + + +
⎝⎠
−− − ()
( ) 01 1
1
tt t
aa xu μ
+
⎡⎤ ⎛⎞
⎢⎥ ⎜⎟
⎢⎥ ⎜⎟
⎢⎥ ⎜⎟
⎢⎥ ⎜⎟
⎢⎥ ⎜⎟
⎢⎥ ⎜⎟
=
⎢⎥ ⎜⎟
⎢⎥ ⎜⎟
⎢⎥ ⎜⎟
⎢⎥ ⎜⎟
⎢⎥ ⎜⎟
⎢⎥ ⎜⎟
++ +
⎢⎥ ⎜⎟
⎝⎠ ⎣⎦
(A11)
where
1
1
εγε
φ
ε
−−
=
−
.  
Since the Euler equation (A11) must be satisfied for all values of the four state
variables, utilizing the fact that a normal distributed stochastic variable z means
[] []
1
var
2
tt
Ez z
z
t
Ee e
+
⎡⎤ =
⎣⎦

Setting
1,1 tit
zm r
++
=+ I can obtain the equation
1,1 1 ,1
1
var 0
2
tt it tt it
Em r m r
++ ++
++ + = ⎡⎤ ⎡⎤
⎣⎦ ⎣⎦
                                                              (A12)
It simply implies that  
()()
2
11 1 1 4 1
1
0
2
tttt t
xAxAxx A ax
θ
θκ ρ θκ φ
ψ
−+ − + + + =
()()
2
12 2 1 4 1
1
0
2
tt tt t
AA A a
θ
μθ κ μ μ μ θκ φ μ
ψ
−+ −+ + =
()
2
22 2
13 3
1
0
2
tt t
AA
θ
θκν σ σ θ σ
ψ
⎛⎞
−+ − =
⎜⎟
⎝⎠

( )
14 4
log log 1 log 0
tt t
AA θκ β α θ α φ β α −−− =                                                   (A13)


103
Solving (A13), I obtain
11
1
1
1
1+
1
a
A
π
ψ θ
κρ
−
=
−

11
2
1
1
1+
1
a
A
π
ψ θ
κ
−
=
−

()
()
() ( )
()
2
3
11
11 11 1
=
21 21
A
θ ψψ γ
κν κν
−− −
=
−−

()
()
4
1
1-
-1
A
φ β
θκ β
=                                                                                                     (A14)
where
()
()
2
2
1
1 2
1
1
2-1
φκ
π
κβ
−
= . Obviously,
1
π is larger than zero.  
Substituting the solutions we derived in (A14) into (A11), I find that
() ( )( ) ( )
()( )()
2
01 0 3 4 0
2
22 2
10
log 1 1
10
2
xw
AA A at A
at a
α
αμ
θδ θκ κ νσ β μ
θ
φβ μ λ λ λ π
⎡⎤
++ + − + − + −
⎣⎦
+− ++ + + + =
                    (A15)
Rearranging this equation then I obtain
() ( )( )
()( )()
2
01 3 4
0
22 2 10
1
log 1 + 1
1
=
1
1
2
xw
AA at
A
a
at
α
αμ
δκ κ ν σ β μ
φθ π
κ
βμ λ λ λ
θθ
⎧⎫
⎡ ⎤ ++ − − + +
⎣ ⎦
⎪⎪
⎨⎬
−
−+ + + + +
⎪⎪
⎩⎭
                      (A16)
Using equation (A1), (A4) and (A5), one can rewrite the approximation of the
aggregated consumption claim asset return
,1 Ct
r
+
as following


104
()()
() ()
() ()
( )
()()
()( ) ( )
01 1 2 1
22 2
,1 0 1 3 1
4011
2
01 2 3 4 1
01 1
log
        log
        1 log
1
txt t t
Ct t w t
tttt
tt t t t t tt
tttt
AA x e A
rA w
Aataaxu
AAxA A A x
at a a x u
μ
αα
α
ρσ μ σζ
κκ σ νσ σ σ
μβ α μ μ
μσ α μ ση
ε
βμ α μ
ε
++
++
+
+
+
⎡⎤
++ + + +
⎢⎥
⎢⎥
++ −+ +
⎢⎥
⎢⎥
+−−+ + +
⎢⎥
⎣⎦
−+ + + + + + +
⎡
+− +− + + +
⎣
−
⎤
⎦
                (A17)
Therefore, I have
( )
() ( ) ( )
()
()( ) ()
2
,1 0 0 1 0 3 4
2
14 4
1
11
1
11 1 1 1
       log
21
tCt
tt t t
Er A at A A A
xAA
αα
εβ
κμκσνμβ
ε
ψγ ε β
μσκβ α
ψε
+
−
⎡ ⎤ =−+ + + + − + − ⎡⎤
⎣⎦ ⎣ ⎦
−
−− − ⎡⎤
++ − + − −
⎢⎥
−
⎣⎦
(A18a)
Further, I have
()
,1 ,1 1 1 1 1 2 1 1 3 1
14 0 1 1 1
                           
1
Ct t C t x t t w t
tt t tt
rEr Ae A A w
Aaa xu
μ
κσ κ σζ κ σ
ε
κμ ση
ε
++ + + +
+ +
−= + + ⎡⎤
⎣⎦
⎛⎞
++ + + +
⎜⎟
−
⎝⎠
                        (A19)
Therefore, it immediately follows that the conditional variance of
,1 Ct
r
+
is
()()() ()
2
2
22
2
, 1 1 1 12 1 3 14 0 1
var
1
tCt x w t t
rA A A A aax
μ
ε
κσ κ σ κ σ κ σ
ε
+
⎛⎞
=+ + + + + + ⎡⎤
⎜⎟
⎣⎦
−
⎝⎠
    (A20a)
where
0
A ,
1
A ,
2
A ,
3
A , and
4
A is defined in equation (A14) and (A16). After some
algebra, it’s easy to obtain
() ( )
()
()
()( ) ()
22 2 10
,1
2 11
1
log
2
11 1 1 1
       log
2
tCt x w
tt t t
at a
Er
a
x
α
μ
τβ μ θ π
δλλλ
θ θ
ψγ τ β π
μ σα
ψθ θ
+
−+
=− − − + + − ⎡⎤
⎣⎦
−− − ⎛⎞
+− + − +
⎜⎟
⎝⎠
               (A18b)


105
() ()
2
22 2 2
,1 1 4 0 1
var
1
tCt x w t t t
rAaax
μ
ε
λ λλ κ μ σ
ε
+
⎛⎞
=+ + + + + + + ⎡⎤
⎜⎟
⎣⎦
−
⎝⎠
              (A20b)

A3. Excess Returns and Risk Free Rate of Return  
According to the asset pricing theory, the excess return of any asset i over the real
risk free rate of return
, f t
r
15
is determined by the conditional covariance between the
return
,1 it
r
+
and pricing kernel
1 t
m
+
and conditional variance of
,1 it
r
+
. Therefore, the excess
return of a claim to aggregate consumption
,1 , tCt ft
Err
+
− ⎡⎤
⎣⎦
is
[] ()()
,1 , 1 ,1 ,1
1 1 ,1 ,1 , 1
1
cov , var
2
1
                     var
2
tCt ft t t Ct t Ct
t t t t Ct t C t t Ct
Er r m r r
Em Em r E r r
+++ +
++ + + +
−=− − ⎡⎤ ⎡ ⎤ ⎡⎤
⎣⎦ ⎣ ⎦ ⎣⎦
⎡⎤
=− − − − ⎡⎤⎡⎤
⎣⎦⎣⎦
⎣⎦
       (A21)
Utilizing the results given in equation (A8), (A19), and (A20) into it and it follows
that  
[] ()()
()
,1 , 1 1 , 1 , 1 , 1
22 2 2
2
14 14
1
var
2
11
                     
22
1
                       
21 1
tCt ft t tt t CttCt tCt
xw t
Er r E m E m r Er r
AA
μ
θλ λ λ γ σ
εε
θκ τ κ
εε
++++ + +
⎡⎤
−=− − − − ⎡⎤ ⎡⎤ ⎡⎤
⎣⎦ ⎣⎦ ⎣⎦
⎣⎦
⎛⎞ ⎛ ⎞
=− + + + − +
⎜⎟ ⎜ ⎟
⎝⎠ ⎝ ⎠
⎡
⎛⎞⎛⎞⎛⎞
−+ − +
⎜⎟⎜⎟⎜⎟
−−
⎝⎠⎝⎠⎝⎠
() ()
01 tt
aa x μ
⎤
++
⎢⎥
⎢⎥
⎣⎦
   (A22)
where
()
11 1
1
1
11
ε εγ
ψψψ
τ
ε
ψ
−+ + −
≡
⎛⎞
−−
⎜⎟
⎝⎠
and
1
11/
γ
θ
ψ
−
≡
−
.
                                               
15
Note that the risk free rate at time 1 t + is known ahead of time




106
According to the fact that [ ]
,1
1
ft t t
REM
+
= , It’s easy to follows that
[] []
[] () ( ) []
,1 1
11 ,1 1
1
var
2
1
   log log 1 var
2
ft tt tt
tt t t t Ct t t
rEm m
Eg E E r m
θ
θδ τ α θ
ψ
++
++ + +
=− −
=− + − Δ + − − ⎡⎤ ⎡⎤
⎣⎦ ⎣⎦
  (A23)
Equation (A23) could also be rewritten in the following
[] () []
,1 1 ,1, 1
11 1
log log var
2
ft t t t t t Ct f t t t
rEgE Err m
τ θ
δα
ψθ θ θ
++ + +
−
⎡⎤ =− + − Δ + − − ⎡⎤
⎣⎦ ⎣⎦
  (A24)
Substituting (A9) and (A22) into it and I obtain the expression of log real risk free
rate of return
()( )
()
( )
()
()
()
2
2
22 2 0 0
, 14
2
2
2 1
114
11 1
lo g
2212
1
111
1 11
     lo g
21 2 2
ft x w
tt t t
at a a
rA
a
aA x
α
μ
τβμ θ θετ
δλλλ κ
θ εθ
γ
τβ ψ θετ
κμ σ α
ψεθ θ
−+ − −
⎛⎞
=− − + + + + + −
⎜⎟
−
⎝⎠
⎛⎞
+−−
⎜⎟
⎡⎤
− −
⎛⎞
⎝⎠
++ + − + + +
⎢⎥
⎜⎟
−
⎝⎠
⎢⎥
⎣⎦
          (A25)

A4. Approximation of Real Growth of Housing Consumption  
Since this dissertation set the non-housing consumption as numeraire in this paper,
I need to rewrite the real growth of housing consumption
1
log
t
s
+
Δ in terms of growth of
non-housing consumption
1 t
g
+
and log expenditure shares
1
log
t
α
+
.  
I first define the non-housing consumption expenditure ratio
cs
ttt tt
zpc ps = . It is easy to
derive the relationship between
t
z and
t
α
1
t
t
t
z
α
α
=
−
                                                                                                           (A26)


107
Linearizing log
t
z Δ around the point of
tt
α α =
16
, I can derive the approximation
1
1
log
log
1
t
t
t
z
α
α
+
+
Δ
Δ=
−
                                                                                           (A27)
Utilizing the FOC (3.3a) and definition of
t
α in equation (3.11), I can obtain:
1
11 1 tt t
tt t
sc z
sc z
ε
ε −
++ +
⎛⎞
=
⎜⎟
⎝⎠
                                                                                              (A28)
Rewrite it in logarithm form  
1
11 1 1
log
log log
111
t
tt tt
t
sg z g
ε εα
ε εα
+
++ ++
Δ
Δ= + Δ = +
−−−
                                      (A29)

A5. Log Real Return on a Claim to Housing Consumption  
Using the standard approximation of
,1 ht
r
+
proposed by Campbell and Shiller (1988).  
,1 2 3 ,1 , 1
log
ht ht ht t
rqq s κκ
++ +
+− +Δ                                                                     (A30)
where
, ht
q denotes the log price-consumption ratio (sale price-rent ratio) of the claim to
housing consumption (housing service)
t
s . The parameter
2
κ and
3
κ are constants which
could be estimated from the mean of
, ht
q . Applying (A29) leads to a formula in terms of
1 t
g
+
and
1
log
t
α
+
.  
1
,1 2 3 ,1 , 1
log
11
t
ht h t ht t
t
rqqg
ε α
κκ
ε α
+
++ +
Δ
+− + +
−−
                                                  (A31)

                                               
16
Note that
() exp
t
at
α
αμ =+





108
The variable
, ht
q is assumed to be a linear combination of state variables
2
,0 1 2 3 4
log
ht t t t t
qB Bx B B B μ σα ++ + +                                                             (A32)
Substituting equation (A1) and (A32) into (A31), one can rewrite the housing
consumption claim asset return
,1 ht
r
+
in terms of exogenous state variables
()()
() ()
() () ()
( )
()()
()()
()( ) ( )
01 1 2 1
22 2
,1 2 3 3 1
4011
2
01 2 3 4 1
01 1
log
      log
      1 log
11
txt t t
ht t w t
tttt
tt t t t t tt
tttt
t
BB x e B
rB w
Bataaxu
BBx B B B x
at a a x u
μ
αα
α
ρσ μ σζ
κκ σ νσ σ σ
μβ α μ μ
μσ α μ ση
ε
βμ α μ
εα
++
++
+
+
+
⎡⎤
++ + + +
⎢⎥
⎢⎥
++ −+ +
⎢⎥
⎢⎥
+−+ + + +
⎢⎥
⎣⎦
−+ + + + + + +
⎡
+−+−+++
⎣
−−
⎤
⎦
           (A33)
Combining (A6), (A33) and substituting in (A12) gives the following equations
()( )
11 1 3 1 1 2 1
10
tttt tttt
xAxAxxBxBxxax
θ
θκ ρ κ ρ π
ψ
−+ − − + + − + + =
()( )
12 2 3 2 2 2 1
10
tttt tttt
AA B B a
θ
μ θ κ μ μμ κ μ μμ π μ
ψ
−+ − − + + − + + =
()()
2
22 2 2 2
13 3 3 3 3
1
10
2
tt t t t
AA B B
θ
θκνσ σ θ σ κνσ σ
ψ
⎛⎞
−− +− + − =
⎜⎟
⎝⎠

() ()
()
()()
()
14 4 3 4 4
1 log 1 log log log
                         1 log 1 log 0
11
tt t t
tt
t
AA B B θκβ α θ α κβ α α
ε
χβ α β α
εα
−−− + −
−− − − =
−−
                (A34)
where I denote
2
π
()
()()
2
214 34
1
1
211
t
AB
ε
πθ κ χκ
εα
⎡⎤
=− ++ +
⎢⎥
−−
⎣⎦
                                             (A35)


109
and
() ( ) ( )( ) ( )
()( ) ( ) ()( )
()()
()( )
() () () ()
() ()
2
01 0 3 4 0
2
23 0 3 4
0
22
22
31 1 1 3 2 1 2
2
2
33 1 3 2 0
log 1 1 1
111
1
11
11
1
0
2
1
t
x
w
AAA atA
at B B B at
Bat
BA B A
BA a
α
α α
α
μ
θδ θ κ κ νσ β μ
χβ μ κ κ νσ β μ
ε
βμ
εα
κθ κ σ κ θ κ σ
κθ κ σ π
⎡⎤
+− + + − + − + −
⎣⎦
⎡ ⎤ +− + + + + − + − +
⎣ ⎦
−+ − +
−−
⎡⎤
+− + + − +
⎢⎥
+=
⎢⎥
+− +
⎣⎦
              (A36)
Solving (A34), I obtain
1
112
1
3
1
1+
1
a
B
π
π π
ψθ
κρ
⎛⎞
−−+
⎜⎟
⎝⎠
=
−

1
112
2
3
1
1+
1
a
B
π
π π
ψθ
κ
⎛⎞
−−+
⎜⎟
⎝⎠
=
−

()
()
2
3
3
11
21
B
θ ψ
κν
−
=
−

()
()
( )
()()( )
4
33
1- 1-
-1 1 1 -1
t
t
B
φβ αε β
θκ β ε α κ β
=+
−−
                                                             (A37)
Substituting the solutions (A37) into (A36), I obtain that
()
()
()( )
()( )
() ( )( )
()()
()( )
() () () ()
() ()
()()
()
2
03
01 0
4
2
23 3 4
0
22
3
22
31 1 1 3 2 1 2
2
2
33 1 3 3 4 1 4
1
log 1 1
1
11 1
1 11
=
1
11
11
22
11
11
2211
t
x
w
t
AA
Aat
Aat
BB at at
B
BA B A
BA B A
α
α
αα
μ
νσ
θδ θ κ κ χ β μ
βμ
ε
κ κ ν σ βμ βμ
εα
κ
κθ κ σ κ θ κ σ
ε
κθ κ σ κ θ κ χ
εα
⎡⎤ ⎛⎞ +−
+− + − + − + ⎢⎥ ⎜⎟
⎜⎟
+− +
⎢⎥
⎝⎠ ⎣⎦
⎡⎤ ++ − + − + + − +
⎣⎦
−−
−
++− + +− +
+− + + +− +
−−
2
0
a
⎧⎫
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨⎬
⎪⎪
⎪⎪
⎪⎪
⎛⎞ ⎪⎪
⎜⎟ ⎪⎪
⎝⎠ ⎩⎭
    (A38)


110
Further it is simplified to  
()
()( )
()
()( )
() ( )( )
()()
()( )
() () () ()
() ()
()()
()
22 2 10
2
23 3 4
0
3
22
22
31 1 1 3 2 1 2
2
2
33 1 3 3 4 1 4
log 1
log 1 1
2
1
11 1
=
11
1
11
11
22
11
11
2211
xw
t
x
w
t
at
at
a
B Bat at
B
BA B A
BA B A
α
α
μ
αα
μ
φ
δβμ
θ
θδ θ χ β μ
θπ
λλ λ
θ
ε
κ κ ν σ βμ βμ
εα
κ
κθ κ σ κ θ κ σ
ε
κθ κ σ κ θ κ
εα
⎡⎤
+− +
⎢⎥
−− + − +
⎢⎥
⎢⎥
++ + +
⎢⎥
⎣⎦
⎡⎤ ++ − + − + + − +
⎣⎦
−−
−
++− + +− +
+− + + +−
−−
2
0
a χ
⎧⎫
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨⎬
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎛⎞
⎪⎪
+
⎜⎟
⎪⎪
⎝⎠ ⎩⎭
  (A38b)
Substituting the (A37) into (A33), yields  
( ) ( ) ( ) ( )
()( )
()()
()( )
()
()
()()
() ( ) ()
2
,1 2 3 0 3 4 0
31 1 3 2 2
2
33 3 3 4 4
2
23 0 3 4
11
1
                + 1 1
11
1
               log
11
             1 1
tht
tt
t
tt
t
Er B B B at B
at
BB x B B
BB B B
BB B
α
α
α
κκ νσ β μ
εβ μ
κρ κ μ
εα
εβ
κν σ κ β α
εα
κκ νσ β μ
+
=+ + − + − + − + ⎡⎤
⎣⎦
−+
−+ + − +
−−
⎛⎞ −
+− + −−
⎜⎟
−−
⎝⎠
=+ + − + − −
()( )
()()
()
()
0
2
2 1
112
1
11
1 11
                + 1 log
2
t
tt t t
at
B
ax
α
εβ μ
εα
τβ πθ
ππμ σ α
ψθ ψ θ
−+
+
−−
− ⎛⎞ ⎛⎞
⎛⎞
−−+ + − − +
⎜⎟ ⎜⎟ ⎜⎟
⎝⎠
⎝⎠ ⎝⎠
     (A39)
Then substituting equation (A38b) into it, I get the conditional mean of  
,1 ht
r
+
 
()
()
( ) ( )
()
() () () ()
() ()
()()
()
22 2 10
,1
22
22
31 1 1 3 2 1 2
2
2
2
33 1 3 34 1 4 0
1
112
111
log
2
11
         1 1
22
11
        1 1
2211
1
        +
tht x w
x
w
t
a
Er at
BA B A
BA B A a
a
μα
μ
θθ θ π τ β
δλλλ μ
θθ
κθ κ σ κ θ κ σ
ε
κθ κ σ κ θ κ χ
εα
π
ππ
ψθ
+
−− −
=− + + + + − + ⎡⎤
⎣⎦
−+− − +−
⎛⎞
−+− − + +− +
⎜⎟
−−
⎝⎠
⎛⎞
⎛⎞
−−+
⎜⎟ ⎜
⎝⎠
⎝
()
()
2
2
1 1
1log
2
       
tt t t
x
τβ θ
μσ α
ψθ
− ⎛⎞
+− − +
⎟⎜⎟
⎠⎝⎠
  (A39b)


111
Hence, it’s easy to obtain the conditional variance
()()
()
,1 , 1 3 1 1 3 2 1 3 3 1 1
34 0 1 1
                         
11
ht t h t x t t w t t t
tt t
t
rEr Be B B w
B aa xu
μ
κσ κ σζ κ σ ση
ε
κμ
εα
++ + + + +
+
−= + + + ⎡⎤
⎣⎦
⎛⎞
++ + +
⎜⎟
−−
⎝⎠
                       (A40)
()
()()
() ()
22 2 2 2 2 2 2
,1 3 1 2 3
2
34 0 1
var
                 
11
       
tht x w t
tt
t
rB B B
B aa x
μ
κσ σ σ σ
ε
κμ
εα
+
++ + ⎡⎤
⎣⎦
⎛⎞
++ + +
⎜⎟
−−
⎝⎠
                                  (A41)
We already know that the excess return of the
,1 ht
r
+
could be expressed as below
[] ()()
, 1 , 1 ,1 ,1
1 1 ,1 ,1 ,1
1
cov , var
2
1
                     var
2
tht ft t t ht t ht
tt t t ht t ht t ht
Er r m r r
Em Em r E r r
+++ +
++ + + +
−=− − ⎡⎤ ⎡ ⎤ ⎡⎤
⎣⎦ ⎣ ⎦ ⎣⎦
⎡⎤
=− − − − ⎡⎤⎡⎤
⎣⎦⎣⎦
⎣⎦
          (A42)
Substituting the (A8), (A40), and (A41) into (A42), yields the excess return of the
,1 ht
r
+

() ()
()
()()
()()
22 2 2 2 2
,1 , 1 3 1 1 3 1 1 3 2 2 3 2
22 2 2
13 3 3 3 3
14 3 4
3
11
11
22
11
                     1
22
1
11
                     
1
2
tht ft x
wt
t
Er r AB B AB B
AB B
AB
B
μ
θ κκ κ σ θ κ κ κ σ
θκκ κ σ γ σ
ε
θκ χ κ
εα
κ
+
⎡⎤⎡ ⎤
−= − − + − − ⎡⎤
⎣⎦
⎢⎥⎢ ⎥
⎣⎦⎣ ⎦
⎡⎤⎛⎞
+− − + −
⎜⎟
⎢⎥
⎣⎦⎝⎠
⎛⎞
−− +
⎜⎟
−−
⎝⎠
+
−
()()
() ()
01
2
4
11
tt
t
aa x μ
ε
εα
⎡⎤
⎢⎥
⎢⎥
++
⎢⎥
⎛⎞
⎢⎥
+
⎜⎟
⎢⎥
−−
⎝⎠ ⎣⎦
(A43a)





112
It could be simplified as
() ()
()
22 2 2 2 2
, 1 , 1 311 3 1 1 3 3 3 3 3
22 2
13 3 3 3 3
2
2
1
12 14 0
11
11
22
1
                     1
2
1
                      +
21 2
tht ft x w
w
Er r AB B AB B
AB B
kA a a
θκκ κ σ θ κκ κ σ
θκκ κ σ
θεπτ
ππ
εθ θ
+
⎡⎤⎡ ⎤
−= − − + − − ⎡⎤
⎣⎦
⎢⎥⎢ ⎥
⎣⎦⎣ ⎦
⎡⎤
+− −
⎢⎥
⎣⎦
⎡⎤
−
⎛⎞
−− + − + +
⎢⎥
⎜⎟
−
⎝⎠
⎢⎥
⎣⎦
() ()
2
1
1
2
tt t
x μ γσ
⎛⎞
++ −
⎜⎟
⎝⎠
(A43b)

A6. Estimations Using Standard CCAPM
A6.1 Methodology
Asset pricing is a fundamental problem in finance research. Standard Consumption-
based asset pricing Model (CCAPM) developed by Lucas (1978), asserts that individuals
hold assets in order to optimize their intertemporal consumption. That is, individuals seek
to maximize a time-additive intertemporal discounted utility function that depends upon
the stochastic consumption: ()
0
t
t
t
UC δ
∞
=
∑

Following Hansen and Singleton (1982), by assuming a standard power utility
function,  
()
()
1
1
1
1
t
t
C
UC
ψ
ψ
−
=
−
                                                                                              (A44)
where ψ is the intertemporal elasticity of substitution IES and
1
ψ
is the relative risk
aversion parameter. Agent maximizes the above utility function subject to the sequence


113
of budget constraints and yields the following Euler equations in the estimation form (or
moment condition):
()
,1
10
tjt
EU
+
−=                                                                                              (A45)
where
1
,1 1 , 1 jttjt
UGR
ψ
δ
−
++ +
=
For 1,...,8 j = in this study, where
1 t
G
+
is the gross consumption growth (growth
ratio)
1
/
tt
CC
+
and
,1 j t
R
+
is the gross real return on asset j.  
The study uses the generalized method of moments (GMM) approach of Hansen and
Singleton (1982) and maximum likelihood (ML) estimation method (Amemiya, 1977 and
Hansen and Singleton, 1983) to estimate the risk-aversion parameter
1
ψ
and time-
preference parameter δ (discount factor) and to test moment condition (A45).  

A6.2 Data sample
Data are quarterly data and the sample period is from the first quarter of 1952 to the
fourth quarter of 2008 except for housing return data. Due to the limit of the availability
of the OFHEO housing price index (HPI), housing return is only starting from the first
quarter of 1975. The gross return data on six stock portfolios are obtained from Kenneth
R. French Website. Consumption data are from the National Income and Product
Accounts (NIPA) tables released by Bureau of Economic Analysis. Simple statistics and
correlation matrix among variables are reported in the following table.  



114
Table A1: Simple Statistics (1952 Q1 to 2008 Q4)
Panel A: Simple Statistics

t
G SGRAT SBRAT SVRAT BGRAT BBRAT BVRAT
, f t
R
, s t
R
MEAN 1.005 1.031 1.041 1.047 1.029 1.032 1.037 1.012 1.026
STD 0.005 0.132 0.104 0.107 0.088 0.074 0.085 0.007 0.010
Panel B: Correlation Matrix among Variables

t
G SGRAT SBRAT SVRAT BGRAT BBRAT BVRAT
, f t
R
, s t
R
t
G 1        
SGRAT 0.114 1        
SBRAT 0.133 0.942 1      
SVRAT 0.137 0.880 0.970 1      
BGRAT 0.107 0.851 0.806 0.728 1    
BBRAT 0.089 0.797 0.863 0.828 0.861 1    
BVRAT 0.141 0.771 0.866 0.879 0.784 0.900 1  
, f t
R
-0.168 -0.076 -0.062 -0.068 -0.072 -0.070 -0.063 1  
, s t
R
0.190 0.149 0.131 0.149 -0.029 0.014 0.026 -0.097 1
Note: (1) This table reports the Pearson correlation coefficients.  
(2) Data are quarterly gross return.  
(3) The label
1 t
G
+
represents consumption growth ratio; SGRAT represents real gross return on small
cap growth portfolio; SBRAT represents real gross return on small cap blend portfolio; SVRAT
represents real gross return on small cap value portfolio; BGRAT represents real gross return on
large cap growth portfolio; BBRAT represents real gross return on large cap blend portfolio;
BVRAT represents real gross return on large cap value portfolio;
, f t
R is the real gross return on
riskless asset; and
, s t
R .represents the real gross return on housing asset.  


A6.3 Results of unrestricted GMM and MLE estimations  
GMM estimation follows Hansen and Singleton (1982)’s method. The instrument
variables are the up to 1, 2, 4, or 6 lagged values of consumption ratio
t
G and gross real
return of j asset
, j t
R .  



115
Table A2: Single Asset GMM Estimations (1952 Q1 -- 2008 Q4)
Overidentifying Restrictions Test
Asset
Number
of Lags
δ
1
ψ

Chi-Square DF P-Value
1
0.976 *
(<.0001)
-2.256 *
(<.0001)
41.16 * 1 0.000  
2
0.992 *
(<.0001)
0.340
(0.309)
49.79 * 3 0.000  
4
0.992 *
(<.0001)
0.529
(0.101)
58.41 * 7 0.000  
, f t
R
6
0.990 *
(<.0001)
0.114
(0.635)
74.68 * 11 0.000  
1
0.980 *
(<.0001)
1.049
(0.183)
13.24* 1 0.000
2
0.978 *
(<.0001)
0.696
(0.327)
20.71* 3 0.000
4
0.977 *
(<.0001)
0.288
(0.413)
25.06* 7 0.001
, ht
R
6
0.979*
(<.0001)
0.745*
(0.012)
25.83 11 0.007
1
0.918 *
(<.0001)
-10.347
(0.064)
0.12  1 0.734  
2
0.919 *
(<.0001)
-9.922*
(0.042)
0.26  3 0.967  
4
0.937 *
(<.0001)
-7.008
(0.099)
9.61  7 0.212  
SGRAT
6
0.939 *
(<.0001)
-6.127
(0.093)
14.39  11 0.212  
1
0.933 *
(<.0001)
-5.361
(0.175)
0.32  1 0.571  
2
0.936 *
(<.0001)
-4.801
(0.183)
0.53  3 0.913  
4
0.950 *
(<.0001)
-2.343
(0.461)
10.93  7 0.142  
SBRAT
6
0.948 *
(<.0001)
-2.029
(0.462)
19.27 11 0.056  
1
0.934 *
(<.0001)
-4.153
(0.281)
0.21  1 0.647  
2
0.939 *
(<.0001)
-3.597
(0.315)
1.14  3 0.769  
4
0.957 *
(<.0001)
-0.273
(0.930)
10.07 7 0.185  
SVRAT
6
0.954 *
(<.0001)
-0.482
(0.862)
18.81  11 0.065  
1
0.949 *
(<.0001)
-4.263
(0.170)
2.19 1 0.139  
2
0.951 *
(<.0001)
-3.918
(0.185)
2.64  3 0.451  
4
0.958 *
(<.0001)
-2.487
(0.341)
10.44  7 0.165  
BGRAT
6
0.955 *
(<.0001)
-2.719
(0.257)
11.59 11 0.395  
1
0.958 *
(<.0001)
-2.198
(0.452)
1.64  1 0.201  
2
0.959 *
(<.0001)
-2.020
(0.470)
1.88  3 0.597  BBRAT
4
0.964 *
(<.0001)
-0.986
(0.689)
6.91  7 0.439  


116
Table A2, Continued: Single Asset GMM Estimations (1952 Q1 -- 2008 Q4)
6
0.962 *
<.0001
-1.392
(0.499)
9.78  11 0.551  
1
0.964 *
(<.0001)
0.192
(0.951)
2.21  1 0.137  
2
0.964 *
(<.0001)
0.010
(0.997)
2.43  3 0.487  
4
0.968 *
(<.0001)
0.316
(0.904)
8.56  7 0.286  
BVRAT
6
0.965 *
(<.0001)
-0.062
(0.979)
15.32  11 0.168  
Note: (1) This table reports parameter estimate results and overidentifying restrictions test for CCAPM
using single asset GMM estimations.  
(2) P values shown in parentheses below the parameter estimations measure the two-tails significant
level of null hypothesis that the parameter is not different from zero.
(3) P-values for the overidentifying restrictions test measure the significant level of null hypothesis
that the moment condition restrictions could be satisfied.
(4) Estimated coefficients or Chi-square statistics superscripted by * are significant at 5%
confidence level.

Parameter estimates and the results of overidentifying restrictions tests of single-
asset GMM estimations are reported in the Table A2. As expected, all of the estimates of
δ are significant and exceed 0.9 but are less than unity for all eight assets. This result is
the same with those of Hansen and Singleton (1982). Overidentifying restrictions tests
are also reported in this table. The estimates of 1/ ψ are quite diversified, however, I find
that the higher number of lag orders the less the absolute values are. Overidentifying
restrictions tests provide evidence against the model when riskless asset and housing
asset is included as the return for all number of lags since P-value of the overidentifying
restrictions tests are less than the 0.05 critical-level.  
Table A3 reports the GMM multiple-asset estimation results. Similar with the results
in Table A2, the estimates of δ are significant and exceed 0.9 but are less than unity for
multiple-asset model in all number of lags. In lag=1 and 2, overidentifying restrictions


117
tests reject the null that the moment condition restrictions
( )
,1
10
tjt
EU
+
−= could be
satisfied when taking all 8 assets into consideration together. I guess it maybe due to the
riskless asset and housing asset are included in the multiple-asset model.  

Table A3: Multiple Assets GMM Estimations (1952 Q1 -- 2008 Q4)
Overidentifying Restrictions Test
No. of
Lags
δ
1
ψ

Chi-Square DF P-Value
1
0.969**
(<.0001)
2.751**
(<.0001)
103.4** 61 0.001  
2
0.974**
(<.0001)
-1.107**
(<.0001)
117.8  117 0.463  
4
0.971**
(<.0001)
0.468**
(<.0001)
115.8  229 1.000  
6
0.977**
(<.0001)
0.086
(0.156)
113.8  341 1.000  
Note: (1) This table reports parameter estimate results and overidentifying restrictions test for CCAPM
using multiple-asset GMM estimations.  
(2) P values shown in parentheses below the parameter estimations measure the two-tails
significant level of null hypothesis that the parameter is not different from zero.
(3) P-values for the overidentifying restrictions test measure the significant level of null hypothesis
that the moment condition restrictions could be satisfied.  
(4) Estimated coefficients or Chi-square statistics superscripted by ** are significant at 5%
confidence level and * are significant at 10% level.

To comparison, I present the results in Table A4 and Table A5 from the estimation
of 1/ ψ and δ using the method of maximum likelihood (MLE) under the assumption
that asset return is lognormally distributed which I label as “Restricted MLE”. Except
estimations of riskless asset, all other 6 assets have quite similar parameter values.  




118
A6.4 Results of restricted GMM and MLE estimations
Maximum likelihood estimation follows Hansen and Singleton (1982 and 1983)’s
method. The results of restricted model are obtained using the procedure described in
these two papers which assumes that all gross asset return are lognormally distributed and
,1
log
j t
R
+
have sixth-order vector autoregressive (VAR) representation. The unrestricted
model is an unrestricted sixth-order vector autoregression.  
The parameter estimates using restricted model and the likelihood ratio tests of
single-asset GMM estimations are reported in the Table A4. The estimations of 1/ ψ and
δ  are similar with those estimated by GMM procedure. I also find that the estimated
standard errors of parameters from MLE are larger than the corresponding standard errors
from GMM. This finding is consistent with Hansen and Singleton (1982) paper. P-value
of riskless asset and small value portfolio (SVRAT) is less than 5%. It implies that the
riskless asset, housing asset, and SVRAT returns cannot satisfy the moment condition
restrictions. The MLE provide more evidence against the moment restrictions for riskless
asset.  


119
Table A4: Single Asset Maximum Likelihood Estimations (1952 Q1 – 2008 Q4)
Restricted MLE
Unrestricted
MLE
Likelihood Ratio Test
Asset
Order
of
VAR
δ
1
ψ

Log
Likelihood
Log Likelihood
Chi -
Square
DF P-Value
SGRAT 6
0.919*
(<.0001)
-10.196
(0.088)
979.11 985.00 11.764 11 0.382
SBRAT 6
0.930*
(<.0001)
-5.971
(0.176)
1031.00 1039.00 16.000 11 0.141
SVRAT 6
0.931*
(<.0001)
-4.782
(0.258)
1029.00 1040.00 22.000* 11 0.024
BGRAT 6
0.927*
(<.0001)
-3.564
(0.303)
1061.00 1067.00 12.000 11 0.364
BBRAT 6
0.961*
(<.0001)
-1.441
(0.602)
1097.00 1102.00 10.000 11 0.530
BVRAT 6
0.964*
(<.0001)
-0.013
(0.996)
1074.00 1081.00 14.000 11 0.233
, f t
R 6
0.951*
(<.0001)
-7.232*
(0.004)
1830.00 1856.00 52.000* 11 0.000
, ht
R 6
0.981*
(<.0001)
0.726*
(0.008)
1070.00 1064.00 19.000* 11 0.000
Note: (1) This table reports parameter estimate results using single asset MLE and the LR test of the
restricted model for the CCAPM.  
(2) Restricted model assumes that consumption ratio and asset gross returns are lognormal
distributed and they have 6th-order VAR representation, while unrestricted model is an
unrestricted sixth-order vector autoregression.
(3) P values shown in parentheses below the parameter estimations measure the two-tails
significant level of null hypothesis that the parameter is not different from zero.
(4) P-values for the likelihood ratio test measure the significant level of null hypothesis that the
asset returns are lognormally distributed and moment condition restrictions could be satisfied.  
(5) Estimated coefficients or Chi-square statistics superscripted by * are significant at 5%
confidence level.

Table A5 reports the MLE multiple-asset estimation results. Noticed that the δ is larger
than unity and P-value of LR test is less than 5%. This result rejects the null that the
moment condition restrictions could be satisfied while taking all 8 assets into
consideration together. This may be caused by the including of riskless asset and housing
asset.  


120
Table A5: Multiple Assets Maximum Likelihood Estimations (1952 Q1 -- 2008 Q4)  
Restricted MLE
Unrestricted
MLE
Likelihood Ratio Test
δ
1
ψ

Log
Likelihood
Log Likelihood Chi-Square DF P-Value
1.002 *
(<.0001)
2.924 *
(<.0001)
4045.00  4421.00  752.000 * 341 0.000  
Note: (1) This table reports parameter estimate results using multiple-asset MLE and the LR test of the
restricted model for the CCAPM.  
(2) Restricted model assumes that consumption ratio and asset gross returns are lognormal
distributed and they have 6th-order VAR representation, while unrestricted model is an
unrestricted sixth-order vector autoregression.
(3) P values shown in parentheses below the parameter estimations measure the two-tails
significant level of null hypothesis that the parameter is not different from zero.
(4) P-values for the likelihood ratio test measure the significant level of null hypothesis that the
asset returns are lognormally distributed and moment condition restrictions could be satisfied.
(5) Estimated coefficients or Chi-square statistics superscripted by * are significant at 5%
confidence level. 
Abstract (if available)
Abstract Housing is a macro asset category and has significant impact on the whole economy. In recent years, some consumption-based asset pricing (CCAPM) literature states that the optimal consumption-saving/investment decision depends not only on aggregated consumption but also on composition between housing and non-housing. This study adopts an Epstein-Zin recursive utility specification to set up a housing consumption-based capital asset pricing model (HCCAPM) which models the housing both as an asset and as consumption good, to study the impact of housing consumption and long-run consumption risks on asset pricing. The study introduces an equilibrium asset pricing model with housing and presents long-run risks and cointegration between expenditure share uncertainty and economic growth as other two factors that drive asset prices. The model reveals that the household not only concerns with uncertainty on overall consumption and consumption composition between non-housing and housing service, but also concerns with the long-run uncertainty of the consumption streams. This study analytically solves the pricing kernel in terms of state variables and derives equilibrium solutions for two systematic determinants of mortgage credit risk-- the risk free rate of return and the real return of housing asset. 
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Asset Metadata
Creator Zhang, Minye (author) 
Core Title Housing consumption-based asset pricing and residential mortgage default risk 
School School of Policy, Planning, and Development 
Degree Doctor of Philosophy 
Degree Program Planning 
Publication Date 09/27/2009 
Defense Date 04/16/2009 
Publisher University of Southern California (original), University of Southern California. Libraries (digital) 
Tag housing consumption,OAI-PMH Harvest 
Place Name USA (countries) 
Language English
Contributor Electronically uploaded by the author (provenance) 
Advisor Deng, Yongheng (committee chair), Gordon, Peter (committee member), Hsiao, Cheng (committee member) 
Creator Email minyezha@usc.edu,zhangminye@gmail.com 
Permanent Link (DOI) https://doi.org/10.25549/usctheses-m2619
Unique identifier UC1235787 
Identifier etd-Zhang-3264 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-256404 (legacy record id),usctheses-m2619 (legacy record id) 
Legacy Identifier etd-Zhang-3264.pdf 
Dmrecord 256404 
Document Type Dissertation 
Rights Zhang, Minye 
Type texts
Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection) 
Repository Name Libraries, University of Southern California
Repository Location Los Angeles, California
Repository Email cisadmin@lib.usc.edu
Tags
housing consumption