Discussion Paper No. 932

LOSING TRACK OF
THE ASSET MARKETS:
THE CASE OF HOUSING AND STOCK
Kuang-Liang Chang
Nan-Kuang ChenCharles Ka Yui Leung

April 2015

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Losing track of the asset markets: The case of housing
and stock∗
Kuang-Liang Chang

Nan-Kuang ChenCharles Ka Yui Leung†

April 17, 2015∗

Acknowledgement: An earlier version of the paper was circulated as “Would some models please giveme some hints?”. The current version is revised after receiving helping comments from (alphabeticalorder) Gianni Amisano, Wing Hong Chan, Marcelle Chauvet, Andrew Filrado, Xu Han, Ivan Jaccard,
Yuichiro Kawaguchi, Kyung-Hwan Kim, Fred Kwan, Rose Lai, Giovanni Lombardo, Thomas Lubik,
Joe Ng, Liang Peng, Daniel Preve, John Quigley, Tim Riddiough, Douglas Rolph, Anna Samarina, Jim
Shilling, Edward Tang, an anonymous referee, seminar participants at the Asian Econometric Society
meeting, AsRES-AREUEA meeting, Bank of International Settlements, City University of Hong Kong,
European Central Bank, European Real Estate Society meeting, NUS, and WEAI. Chen thanks the
National Science Council (Taiwan) for financial support. The work in this paper was partially supported
by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China[Project No. CityU 144709]. We are grateful for all their support. The usual disclaimer applies.
†
Correspondence: Chang: Department of Applied Economics, National Chiayi University, 580 Sinmin
Road, Chiayi City 60054, Taiwan. Tel.: 886-5-2732852, E-mail: klchang@mail.ncyu.edu.tw. Chen: Department of Economics, National Taiwan University, 21 Shuchow Road, Taipei 10021, Taiwan. Tel.: 8862-2351-9641 ext. 471, E-mail: nankuang@ntu.edu.tw. Leung: Department of Economics and Finance,
City University of Hong Kong, Kowloon Tong, Hong Kong. Tel.: 852-2788-9604, Fax: 852-2788-8806,
E-mail: kycleung@cityu.edu.hk.

1

Abstract
This paper revisits the relationships among macroeconomic variables and asset returns. Based on recent developments in econometrics, we categorize competing models
of asset returns into diﬀerent “Equivalence Predictive Power Classes” (EPPC). During
the pre-crisis period (1975-2005), some models that emphasize imperfect capital markets outperform an AR(1) for the forecast of housing returns. After 2006, a modelthat includes both an external finance premium (EFP) and the TED spread “learns andadjusts” faster than competing models. Models that encompass GDP experience a significant decay in predictive power. We also demonstrate that a simulation-based approachis complementary to the EPPC methodology.
Key words: monetary policy, financial market variables, Uni-variate Single-regime
Benchmark, Markov Regime Switching, forecasting
JEL classification: E50, G00, R00

2

“In view of the structural equality of explanation and prediction, it may be said that an
explanation ... is not complete unless it might as well have functioned as a prediction.”
Carl Hempel, 1942, The Function of General Laws in History.

1

Introduction

This paper contributes to the literature in a number of ways. First, it revisits the relationship between some macroeconomic variables and asset prices (housing and stock).
On the one hand, based on the eﬃcient market hypothesis (EMH), relevant information
has been summarized in the asset prices and hence introducing macroeconomic variables
should not improve our prediction for these asset prices.1 On the other hand, there is an
emerging strand in the literature suggesting that there are non-trivial interactions among
macroeconomic variables and asset prices, and therefore including macroeconomic variables would enhance our understanding of the asset prices.2 A simple way to distinguishthese two bodies of theory above would be to set the asset prices as the dependent variables and introduce the macroeconomic variables as independent variables, in addition to
including lagged asset prices. If the coeﬃcients of the macroeconomic variables are statistically insignificant, then the eﬃcient market hypothesis is confirmed. Conversely, if thecoeﬃcients of the macroeconomic variables are statistically significant, then the eﬃcient
market hypothesis is rejected and the macroeconomic variable-asset prices interactions
are indeed important.
While such an approach is intuitive and easy to improve, there are several shortcomings. First, if the asset price movements indeed generate a wealth eﬀect or collateraleﬀect,3 then there is a feedback eﬀect from asset prices to macroeconomic variables.
1

Clearly, there are diﬀerent views about the EMH and how we can test it. Among others, see Fama

(1970), Darrat and Glascock (1989), LeRoy (1989).
2
Among others, see Green (2002), Case, Quigley and Shiller (2005), Campbell and Cocco (2007) fora discussion of the wealth eﬀect that can be created by the asset price fluctuations; Leung (2004), Chen
and Leung (2008), Jin et al. (2012) and the reference therein for the interaction between housing prices
and macroeconomic variables through the collateral eﬀect.
3
Since the aggregate consumption constitutes almost 70% of the total GDP of the USA, and since
many countries target their export to the USA, the wealth eﬀect generated by asset price swings canhave important implications to the economies of both the U.S. and many of its trade partners. The

3

Thus, an endogeneity bias may occur. The minimum model that can embed the potential feedback among asset prices and macroeconomic variables would be the vector
autoregressive (VAR) models,4 but comparing VAR models is not always a trivial task.5
A common practice is to apply the test proposed by Diebold and Mariano (1995), whichallows for a bilateral model comparison. In the context of asset price movements, we may
want to compare more than two models as the rationale is quite clear. Diﬀerent theories
are “represented” by diﬀerent econometric models, in the sense of diﬀerent variables anddiﬀerent possible relationships among variables being highlighted.6 To compare which“theory” of asset price movements, it is important to employ an appropriate procedure
to select the “best performing model” among competing ones. Our second major contribution of the literature thus combines the work on multi-model performance comparisonby Mariano and Preve (2012) and the work on “model confidence set” by Hansen etal. (2011) in order to “eliminate” less competitive models sequentially. As a result,
we can categorize models into several “Equivalence Predictive Power Classes” (EPPC).
collateral eﬀect refers to the scenario that continuous declines in house prices can cause a quick decayof collateral quality and value, potentially leading to a credit crunch and subsequent rise in bankruptcy
and foreclosures.
4
Sims (1980b) makes a strong case why in a macroeconometric context, estimating a system of
equations, especially in the context of a dynamically interacting system, is econometrically more sensible
than single equation estimation.
5
It is well known that under some conditions, we can view a VAR model as a collection of uni-variate
regression. In fact, some standard computer packages deliver a separate 2 measure for each of theequation within a multi-variate VAR. Therefore, it is possible that one VAR model produces a higher“2 ” in the “stock price equation” and yet a lower “2 ” in the “house price equation” than the other
VAR. Yet both the “stock price equation” and “house price equation” are part of a dynamic system andhence it seems that reading the “individual equation 2 ” may not be suﬃcient.
6
Notice that the reduced form dynamics of many dynamic, stochastic general equilibrium models(DSGE) have a VAR representation (for instance, see Kan et al., 2004; Leung, 2014). While identification
can be an issue and we might not be able to recover the underlying DSGE model from estimated VAR,
comparing the empirical performance of a collection of VAR models might still provide an indirect test
for the models’ capacity to account for the data. In the current context, a VAR at least could provide
us some hints about whether regime-switching is important, which variables should be included in the
DSGE, etc. See Kapetanios et al. (2007), among others, for more discussion on this point. See also
Pagan and Robinson (2014) which show that some existing DSGE with imperfect capital market may
not explain the data very well.

4

Eﬀectively, we may be able to provide indirect evidence about which theories have thesame explanatory powers and which theories have a lower explanatory power than somerivals. Thus, the empirical results obtain herein might provide some reference for the
future development of theoretical modeling. The method is very simple, can be applied
to any finite number of models and very diﬀerent contexts, and hence it may have someindependent interests.7
This paper specifically focuses on U.S. aggregate data during the period 19752 −20121 and studies various versions of Vector Auto-Regressive (VAR) models, with andwithout regime-switching, which arguably represent diﬀerent views on the driving forceof the asset markets.8 While we have explained why VAR models should be used, ourchoice of regime switching model also seems non-controversial. First, the possibility of
regime switching in the macroeconomic and financial time series has long been studied
and verified.9 Among others, Chen and Leung (2008) show that in the presence ofcollateral constraints, bankruptcy possibility and asset price spillover, the relationshipbetween aggregate output and the real estate price can be very non-linear (piece-wise
continuous with diﬀerent slopes in diﬀerent segments) and hence may not be well captured
by widely used linear VAR model. In addition, Chang, Chen and Leung (2011) argue
that the regime switching model can be consistent with two stylized facts in the housing
market: (1) short-run predictability, which has been repeatedly documented since Caseand Shiller (1990), and (2) long-run non-profitability, which is a prerequisite of the longrun eﬃciency of the housing market.10 The inclusion of some monetary policy variable
would further justify the use of regime-switching model.11
7

For instance, see Kwan et al. (2015) for an application of this model comparison method on structuralmodels.
8
The National Bureau of Economic Research, among others, has also admitted that an economic
recession has started in the first quarter of 2008. When it will end, however, is still a topic for debate.
9
Again, the literature is too large to be reviewed here. Among others, see Hamilton (1994), Maheu
and McCurdy (2000).
10
Notice that if the true model is single-regime, if short-run predictability holds, then we cannot havethe long-run non-profitability at the same time. And there is a large literature on testing the housing
market eﬃciency based on this simple relationship. Among others, see Chang et al. (2012, 2013) for
more discussion.
11
For instance, Sims and Zha (2006) find that the changes of monetary policy “were of uncertain5

In this paper we focus on the return stock price index (SRET) and the house priceindex (HRET), as well as variables that may aﬀect the two asset returns. Our choices ofvariables are mainly guided by the previous literature on asset price dynamics and will be
explained in further details in the following section.12 For a more balance understanding,we conduct both in-sample forecasting (ISF) and out-of-sample forecasting (OSF) herein.
Our OSF takes two diﬀerent approaches: conditional expectations and simulation-based
methods.13 The merits of using the former approach have been widely discussed in theprevious literature. On the other hand, the confidence intervals may not be available
from the former approach. It is an important issue for the regime switching models,
because the system not only receive shocks within a regime, but also experiences thestochastic switching from one regime to another. Following Sargent, Williams and Zha(2006), we adopt the simulation-based approach to calculate the median path and theconfidence interval, with regular model updating.
There is clearly a large and recent literature on asset pricing.14 For instance, Maheu,
McCurdy and Song (2012) study the daily returns of equity indices for 125 years andfocus on the regime-dependence of the transition probability matrices. Grishchenko andtiming, not permanent, and not easily understood, even today” and that models which “treat policychanges as permanent, nonstochastic, transparent regime changes are not useful in understanding thishistory.” Casual observation also suggests that the conduct of monetary policy has changed over timealong changes in the chairman of the Fed and several episodes that dramatically aﬀect inflation andeconomic activity (such as oil price shocks). Thus it may be appropriate to explicitly allow for regimeswitching behavior in the study of asset returns and monetary policy.
12
For instance, there is a large class of dynamic equilibrium models suggesting that the house prices
and stock prices should be correlated. Among others, see Leung (2007), Kwan et al. (2015) and thereference therein.
13
14

We will provide more justifications on why this would be appropriate in later sections.
For instance, Welch and Goyal (2008), among others, argue that most if not all models fail topredict equity premium. Recently, Phillips (2013) proves that the confidence intervals in some of thepredictive regressions have zero converge probability and the corresponding statistics Q would indicate
predictability even when there is none.
The focus of this paper is diﬀerent. First, we emphasize on the model ability to account for both stock
price and housing price, rather than the equity premium. Second, we emphasize on how diﬀerent models“learn and adopt” since the 2006 housing price decline. Therefore, we will adopt a diﬀerent approach,as it will be explained in later sections.

6Rossi (2012) employ the monthly data from Consumer Expenditure Survey from 1984 to2012 to estimate the asset price model. Clark (2011) takes real-time data from 1985Q1
to 2009Q1 and shows that stochastic volatility improves the real time forecast of macroeconomic variables. There is also a strand of literature comparing the rate of returnsof real estate versus the stock market from an investment perspective.15 This paper
diﬀers from the literature in several dimensions. First, this paper takes a “dynamicalsystem approach”. Our estimation is multivariate and when regime-switching occurs in
our equation, it is actually the whole dynamical system switch from one regime to another. This clearly diﬀers from, and hence complements, some of the literature taking a
univariate approach. Second, we focus on aggregate data starting from 1975 and hence
we can cover a longer period. Since the oﬃcial, aggregate data of housing is in quarterly
frequency, we adjust the frequency of other variables to quarterly frequency, and hence
our work may complement earlier works that focus on the higher frequency movements ofasset returns. Third, we focus on the application of classical econometrics. Perhaps more
importantly, this paper compares not only how diﬀerent models predict for the in-sample
period (or, the “pre-crisis period”), but also how diﬀerent model performances evolve as
we recursively update the parameter estimates with new data in the out-of-sample period(or, the “post-crisis period”). To some extent, we examine the ability of diﬀerent models
to “learn and adopt” within changing asset markets. To our knowledge, these features
are not emphasized by the previous literature and our paper can supplement that.
The rest of the paper is organized as follows. Section 2 describes the econometricmodel and gives a statistical summary of the data. Section 3 presents the empiricalestimation results with the baseline model. Section 4 compare forecasting performancesacross models. Section 5 concludes.

15

Among others, see Ibbotson and Siegel (1984), Quan and Titman (1997, 1999).

7

2The Econometric Analysis

2.1

Data

The analysis of this paper is based on the U.S. data covering the period 19752−20121.
Limited by data availability, the dimensionality constraint of the econometric model, we
focus on the returns of the stock price index (SRET) and the house price index (HRET),
as stock and house are the most important assets for a typical household in the UnitedStates. Focus is placed on the asset return rather than the asset prices, because the latter
tend to be non-stationary while the former may be mean-reverting. Figure 1 provides
a visualization of these variables and clearly demonstrates two stylized facts: (1) the
negativity of the housing return in the recent years, and (2) the high volatility of stock
returns (relative to the housing returns).

(figure 1 about here)

The previous literature on asset price dynamics guides our choices of variables to be
included, on top of the asset returns. They include: the (3-month) federal funds rates(hereafter FFR) which is a measure of the US monetary policy; the term spreads (SPR)
which is a measure of the diﬀerence between long-term and short-term interest rates16 ;external finance premium (EFP), which is a measure of the degree of credit market
imperfect from non-financial firms’ perspective;17 the TED spread (TED), which is a
16

In this paper, SPR is defined as the discrepancy of the long term (10 years) interest rate and theshort term (3-month) counterpart.
17
In this paper, EFP is defined as the spread between high-rating corporate bond and the low-rating
one (Baa-Aaa).
There are a number of available series that have been used as the measure external finance premium.
Among these are the prime spread (prime loan rate - federal funds rate) and the corporate bond spread(Baa-Aaa), and the high-yield bond spread (Bbb-Aaa). De Graeve (2007) argues that the prime loan
spread provides a poor indication of financing conditions of firms which are typically considered vulnerable to credit market frictions, because it focuses on firms of the highest credit quality, to which financial
constraints pertain the least. Gertler and Lown (1999) show that the high-yield bond spread is strongly

8measure of the degree of credit market imperfect from banks’ perspective;18 and GDP
growth rates (GDP). The choices can be easily justified. For instance, our inclusionof monetary policy variable in the study of asset price dynamics is consistent with theliterature.19 On the one hand, the monetary policy is widely perceived to be influential
to both the stock and housing market. On the other hand, the Federal Reserve may
react to the stock market movement in some instances (e.g. Rigobon and Sack, 2003).
In this paper, we follow Sims (1980a) and choose FFR to represent the movement of
the monetary policy. Similarly, the GDP growth rate (GDP) seems to be a natural
choice for a proxy of “economic fundamental.”20 The term spread (SPR) is well-known
to contain information about future inflation, future real economic activities as well as
asset returns.21 Thus, it may be instructive to include the term structure as a (partly)“forward-looking variable” in the regression without taking any stand on the formation of
future inflation or interest rate expectation.22 The external finance premium (EFP) andassociated with both general financial conditions and the business cycle (as predicted by the financial
accelerator). However, the series started only in the early 1980s. Therefore, we choose corporate bondspread (Baa-Aaa) to be our measure of the external finance premium.
18
In this paper, the TED spread is defined as the diﬀerence between the 3-month Eurodollar deposit
rate and the 3-month T-bill rate.
The widely-used BBA LIBOR, compiled by the British Bankers’ Association, started only from January 1986. Therefore, we replace the 3-month LIBOR rate by 3-month Eurodollar deposit rate. These
two series are highly correlated. Both the corporate bond spread and the 3-month Eurodollar deposit
rate are from H.15 statistical release (“Selected Interest Rates”) issued by the Federal Reserve Board ofGovernors.
19
For instance, audiance in diﬀerent occasions suggest us to adopt the “financial stress index” (FSI).
However, FSI is available only from 1990 onwards, while other variables in our dataset are available since1975. Using FSI will force us to lose much information, including how the system react to dramatic events
such as the 1987 stock market crash. Furthermore, FSI may have higher predictive power with higher
frequency data, while this paper focuses on the quarterly frequency where GDP data is available.
20
In contrast, the aggregate consumption may be “too smooth” to account for the movement of stock
return very well. Thus, we use the GDP instead. The literature is too large to be reviewed here. Amongothers, see Mehra and Prescott (2003).
21
This statement has been confirmed by the data of the U.S. as well as other advanced countries. Fora review of the more recent literature, see Estrella (2005), Estrella and Turbin (2006), among others.
22
In the literature of term structure, a lot of eﬀorts have been devoted to verify the “expectation
hypothesis.” However, Collin-Dufresne (2004) shows that there are several versions of the expectation
hypothesis and they are not consistent with one another. Thus, the explicit formulation of the expecta-9the TED spread (TED) are included in the analysis because some recent literature whichhighlights the role of imperfect capital market find these variables important proxies forthe degree of capital market imperfection-ness.23 While EFP may capture the degree ofcapital market tightness faced by non-financial firms, TED spread, which is the diﬀerence
between the interbank rate and the riskfree rate, can be interpreted as a proxy for thecapital market tightness faced by banks. For instance, in a collateral-constrained economy
such as Kiyotaki and Moore (1997) present, the borrowing constraint, which holds at theequilibrium, would take the following form,µ¶future col lateral value
debt ≤ discount factor
Clearly, there are other variables that may be important in explaining the asset returns
during the sampling period. Unfortunately, not every potentially important variable isavailable since 1975, and some other variable may not be as useful as it may seem. For
instance, it has been suggested to us that the 30 year mortgage rate would improve theperformance of our models. In the appendix, we provide empirical evidence it may not
be the case, at least not for our sampling period. In addition, our “system approach”
also limit the total number of variables that could be included in the empirical analysis.
Thus, the current list tries to balance the economic validity and the data availability.
Our choice of sampling period is also constrained by data availability, as 1975 is
the earliest date that the U.S. quarterly data of housing price is available.24 Given our
selected group of VAR models, we examine the in-sample fitting for the period 19751 −tion may matter to the final empirical result.
23
For instance, Jin et al (2012) show that the movement of EFP can be related to the housing returns
in a DSGE model recently. In general, as surveyed by Bernanke and Gertler (1995), EFP is perceived asa measure of the “risk premium” and hence a reflection of the credit market conditions that faced by nonfinancial firms in the literature. For models which emphasize the role of imperfect capital market in the
propagation of shocks over the business cycles, see Bernanke, Gertler and Gilchrist (1999), Christiano,
Motto and Rostagno (2007), Davis (2010), among others. These papers did not explicitly model housing
though.
24
Another merit of choosing 1975 as the starting point is that it also avoids the first oil price crisis,
which may be a period of “indeterminacy,” especially in respect to the monetary policy, which will makethe empirical identification diﬃcult. Among others, see Lubik and Schorfheide (2004) for more analysis
on this.

10

20054 and the out-of-sample forecasts for the period beginning with 20061. We choose
20054 as the cut-oﬀ point, because the rise in the house price growth rate starting 1990s
peaked around the end of 2005.25 Furthermore, we allow the models to “learn” in later
sections in the sense that we would recursively re-estimate the model, for instance, from1975 to 2006, and use it to predict the asset prices in 2007, and then re-estimate again
by using the data from 1975 to 2007 in order to predict the asset prices in 2008, and soon. Thus, the in-sample we choose initially may not aﬀect our results as much as somerelated studies.
To be compatible with the quarterly house price index provided by the Oﬃce ofFederal Housing Enterprise Oversight (OFHEO), variables that are originally available
in monthly are transformed into quarterly. The S&P 500 stock price index is obtained
from the DataStream. We compute stock and housing returns by taking the growth ratesof the stock price index and housing price index respectively. Real GDP is taken fromthe Department of Commerce, Bureau of Economic Analysis. The federal funds rateis taken from H.15 statistical release (“Selected Interest Rates”) issued by the Federal
Reserve Board of Governors. As for the term spread, we follow Estrella and Trubin(2006) by choosing the diﬀerence between ten-year Treasury bond yield and three-monthT-bill rate, both of which are released by the Federal Reserve Board of Governors. Since
the constant maturity rates are available only after 1982 for 3-month T-bills, we use thesecondary market three-month T-bill rate expressed on a bond-equivalent basis.26
While these time series have all been studied in the literature, it may nevertheless be
instructive to present some “stylized facts” before any formal modeling. Table 1 gives
a statistical summary for the variables in the data. The stock returns have a higher
mean than housing returns (1.9 versus 1.1), and have an even larger volatility than the
housing returns (8.2 versus 1.3). The simple correlation coeﬃcients in Table 2 show that25

Clearly, there are substantial diversity in terms of when the asset price cycle ends. In this paper,

we would experiment diﬀerent end-dates in the robustness check section and the appendix also discusses
the related literature in more details.
26
The 3-month secondary market T-bill rate provided by the Federal Reserve System is on a discountbasis. We follow Estrella and Trubin (2006) by converting the three-month discount rate ( ) to a

365× 100bond-equivalent rate ():  = 360−91× 100 × 100. They argue that this spread provides an accurate

and robust measure in predicting U.S. real activity over long periods of time.

11

some variables are indeed highly correlated. For instance, the correlation between the
FFR and SPR is -0.6, that between the FFR and TED is 0.7, and that between EFP
and TED is 0.6. On the other hand, some other correlations are close to zero. Thus,
it is not clear which model will perform better a priori. To facilitate the comparison,models are constructed in some certain ways. As shown in Table 3a, for instance, Models
A to E would have FFR involved, which can highlight the potential role of monetary
policy in the asset return dynamics.27 Model F to H diﬀer from the previous ones as themonetary policy variable FFR is replaced by a financial market variable. Thus, model F
can be interpreted as model C with FFR replaced by EFP, model G as model E with
FFR replaced by EFP, and model H as model E with FFR replaced by SPR. Thus, acomparison of the Model A to E on the one hand, and the Model F to H on the other hand
would provide some information of the importance of monetary policy in the asset pricedynamics. Table 3b also shows how we choose the AR(1) as the Univariate, Single-regime
Benchmark (USB). Obviously, if, for instance, econometric models that include EFP asa variable outperform other econometric models, it would provide indirect support thatthe firm financing problem is important in explaining asset market returns. On the other
hand, if models that include TED spread outperform other models, it would suggestthat the interbank market is important in explaining the asset market returns. Thus,
by investigating the performance of diﬀerent models we would be informed about which
economic channels may be more important.

(Table 1, 2, 3a, 3b about here)

2.2

The Econometric Model

This paper takes a dynamical system approach in the sense that we estimate vector
autoregressive (VAR) models which include both asset returns and other macroeconomic
27

For the purpose of parsimony and model comparison, we set the lag period of all models to be one( = 1). It turns out that most models with one lag period have the lowest value of , comparedwith models having more than one lag periods. Details are available upon request.

12and financial variables and allow them to interact with one another. Some justifications
have been discussed and we simply re-organize them here. First, much of the literaturetakes a univariate approach and hence this paper, which takes an alternative approach, iscomplementary. Second, it is well-known that when the regressors are not distinguishable
from integrated processes, the conclusions about return predictability could be altered.28
Under the VAR approach, this issue would become less severe because it is less likely thatthe whole vector follows a unit root process than in the case of an individual variable. And
as we argued in the introduction, the reduced form dynamics of some dynamic, stochastic
general equilibrium models (DSGE) actually have a VAR representation. Hence, testing
the ability of a class of VAR models may provide an indirect test for a class of DSGE
models. Thus, the results in this paper could oﬀer some hints for the future developmentof DSGE models. In addition, while some of the existing literature tends to take oneof the returns as given and to use its movement to explain the other return, the VAR
approach naturally allows for dynamic interactions between the asset returns (housing
and stock) and other variables, as well as the feedback eﬀects among asset returns. In
other words, it avoids assigning one of the asset returns as an “exogenous variable”, which
could lead to potential endogeneity bias.29
Our econometric model is a regime-switching VAR, with lag length  for a (vector)
process  :
0 ( )  =  ( ) +X ( ) − +  ( ) 

(1)

=1

where we allow for all parameters, including intercept coeﬃcients, autoregressive coeﬃcients, and covariance matrix of stochastic terms to be contingent on the unobservablestate variable  ∈ . The regime-dependent coeﬃcients capture possible nonlinearities
or time variation in the lag structure of the model. The stochastic volatility allows forpossible heteroskedasticity of the stochastic terms.
The variables of interest  = (1  2    )0 is a  × 1 vector. The stochastic

intercept term  ( ) = (1 ( )  2 ( )    ( ))0 captures the diﬀerence in the intercept
28
29

Among others, see Torous et al (2004), Cochrane (2008).
Among others, see Sims (1980a, b) for more discussion on these issues and the potential biases that

could be eliminated by the VAR method.

13

under diﬀerent states. 0 ( ) is a  ×  state-dependent matrix that measures thecontemporaneous relationship between variables, and the econometric identification of
the model is obtained through restrictions on 0 ( ). In addition,  ( ) is a  × ()matrix with each element state-dependent  ( ),   = 1    = 1  . The
stochastic error term  will be explained below.
The corresponding reduced form of the above model can be obtained by pre-multiplying (1)by −10 ( ), which yields:
 =  ( ) +XΦ ( ) − +  ( ) 

(2)

=1

−1
−1
where  ( ) = −10 ( )  ( ), Φ ( ) = 0 ( )  ( ), and  ( ) = 0 ( )  ( ), = 1 2 . Φ ( ) is a  ×  matrix with each element which is state-dependent() ( ),   = 1    = 1  . We further define ( ) ≡  +  ( ) which will be explained below. The vector of stochastic error term  can be further
expressed as12
 ( )  = −10 ( )  = Λ ( ) where  is a  ×  diagonal matrix with diagonal elements 2 ,  = 1  , Λ ( ) is a
 ×  diagonal matrix with diagonal elements  ( ),  = 1  ,⎤⎡
1 ( )
0
···
0
⎥
⎢
⎥
⎢ 0

(
)
·
·
·
0
2

⎥
⎢
Λ ( ) = ⎢
⎥
.
.
.
.
..
..
..
..
⎥
⎢
⎦
⎣
0
0
· · ·  ( )which captures the diﬀerence in the intensity of volatility, and  ( ) is a vector of standard normal distribution,  ( ) ∼  (0 Σ ( )), where the covariance matrix is given
by

⎡

1

21 ( ) · · · 1 ( )⎤⎥⎢⎢ 12 ( )1· · · 2 ( ) ⎥⎥⎢Σ ( ) = ⎢
⎥
.
.
.
.
..
..
..
..
⎥
⎢
⎦
⎣
1
1 ( ) 2 ( ) · · ·14(3)

We also include an “atheoretical benchmark,” which is the AR(), i.e. an order- autoregressive process that is labelled as the Univariate, Single-regime Benchmark (USB).
This is clearly motivated by the “eﬃcient market hypothesis,” which conjectures thatall “relevant information” has been reflected in current (and potentially previous) period
price whereby additional variables, such as those provided by the VAR, are consideredinsignificant. This could be the case of the stock return. On the other hand, the housing
market is always being accused of not being as eﬃcient as the stock market, and hence thehousing market prediction can be improved with additional variables, including the stock
return. Hence, a comparison of model performance with USB provides not only a formof “eﬃcient market test,” but also an indirect test of the “cross-market informational
spillover.”30 For the linear VAR model, we include all 7 variables. For the regimeswitching models, we can only aﬀord to include 4 variables, which is a much shorter
list than the linear model, as we need to estimate parameters in each regime, plus thetransition probabilities. By design, it puts regime-switching models in a disadvantageous
position. If, however, the regime-switching models still outperform the widely used linear
model, then it suggests that regime changes may indeed be very important in the data.
In addition, if models with certain variable(s) consistently outperform alternatives, thensuch variable(s) may be important to account for in the asset return movements. Thus,
it may be important to include diﬀerent combinations of our listed variables in diﬀerent
models and test for the performance of those models.
For the linear VAR model, we include all 7 variables. For the regime switchingmodels, we can only aﬀord to include 4 variables, which is a much shorter list thanthe linear model, as we need to estimate parameters in each regime, plus the transition
probabilities. By design, this puts regime switching models in a disadvantageous position.
If, however, the regime switching models still outperform the widely used linear model,
then it suggests that regime changes may indeed be very important in the data. In
addition, if models with certain variable(s) consistently outperform alternatives, thensuch variable(s) may be important to account for in the asset return movements. Thus,
it may be important to include diﬀerent combinations of our listed variables in diﬀerent
30

Even for the case of aggregate output, it may still be a good idea to use an uni-variate AR(p) as thebenchmark. Among others, see Chauvet and Potter (2012) for more discussion.

15

models and test for the performance of those models.

2.3

Two-state Markov Process

Being severely constrained by the sample size, we assume that there are only two states,i.e.,  ∈  = {1 2}. The procedure for the identification of the regime of the economy
for a given period will be discussed below. The Markov switching process relates theprobability that regime  prevails in  to the prevailing regime  in  − 1,  ( =  |−1 = ) =  . The transition probabilities are assumed to be constant and the transition
matrix is given by:31

⎛

 =⎝

11

1 − 221 − 11

22

⎞

⎠

Given that the economy can be either in state 1 or state 2, the term  ( )   =1  , defined above, captures the diﬀerence in the intercept under diﬀerent states. For
convenience, we set  (1) = 0 for  = 1, thus  (2) measures the diﬀerence in theintercept between state 2 and state 1. Furthermore, we set the diagonal element of Λ ( )at state 1 to be unity, i.e.,  (1) = 1 so that if  (2)  1, then the intensity of volatility
in state 2 is larger than that in state 1, and vice versa. Since  ( ) is a vector of standard
normal distribution and  (1) is set to be one, the variance of    = 1  , at state
1 is 2 , and the variance is 2 (2) 2 .

2.4

Identification of Regimes

Since the state of the economy is unobservable, we identify the regime for given a time
period by Hamilton’s (1989, 1994) smoothed probability approach, in which the probability of being state  at time  is given by  ( | Ω ), where Ω = {1  2       }.
The idea is that we identify the state of the economy from an ex post point of view, andthus the full set of information is utilized. Notice that we only allow for two regimes
31

In principle, we could allow the transition probabilities to depend on the observed variables. However,

the time series we have assess to are relatively short and hence we make compromise in the modelling
choice.

16

in this paper, i.e.,  ∈  = {1 2}. Thus, if  ( =  | Ω )  05, then we identify the
economy most likely to be in state ,  = 1 2.32

2.5

Forecasting

After we estimate all the above models, we use the calculated probabilities of regime
switching for evaluating the forecasting performances of house and stock prices acrossvarious models, and then examine both in-sample and out-of-sample forecasting performances. We divide the sample into in-sample period 19752−20054 and out-of-sample
period 20061 and after. In any econometric model assessment, the calculation of confidence interval is important, because it provides a quantitative sense on whether the point
estimate of the coeﬃcient and predicted path of variables are “oﬀ the mark.” Clearly,
there is a large body of literature on “predictive regression” in asset pricing which haveapplied diﬀerent techniques to construct the confidence intervals. Phillips (2014) shows
through an analytical method and simulations that some commonly used methods in thatliterature may be misleading. In particular, Phillips presents that “the commonly used Q
test is biased towards accepting predictability and associated confidence intervals for the
regressor coeﬃcient asymptotically have zero converge probability in the stationary case.”
In light of these potential shortfalls, we conduct out-of-sample forecasting (OSF) in two
diﬀerent approaches. The first approach is the conventional conditional moment method.
Given the estimation window 19752−20054 and a forecasting horizon  = 1  4, theestimated parameters are used to forecast house and stock prices -step ahead outside
the estimation window, using the smoothed transition probabilities. The -step ahead
forecasted value of + based on model  and information at time , Ω , is given by (+ | Ω ) =

2
X
=1

 [+ | + =  Ω ] ×  (+ =  | Ω ) where  is the index for the state, and  ∈  . The estimation window is then updated
consecutively with one observation and the parameters are re-estimated. Again the step ahead forecasts of house and stock prices are computed outside the new estimation
32

In addition, we follow Francq and Zakoian (2001) to use spectrual radius to determine the stationarityof the regime-switching models. Due to the space limit, we report the results in the appendix. We findthat most models are stationary. Only Model B and C are marginally non-stationary.

17

window. The procedure is iterated until the final observation. The forecasts based on
this method is basically to compute the -step ahead conditional expectations of thevariable being predicted. Most existing (non-Bayesian) works follow this method.
The second approach is the simulation method. A merit of this approach is that
we eﬀectively simulate a “confidence interval” (CI) for ourselves and hence do not need
employ other approximation technique to construct the CI. In a sense, we avoid thecritique of Phillips (2013). The idea is simple. We simulate the path of the forecasted
values by repeated drawings. The procedure is as follows.
• (Step 1) We estimate the model using the estimation window 19752 − 20054 andobtain the parameters, transition probabilities, and variance-covariance matrix. Given
the estimation results we compute the smoothed probabilities to identify the regime at
the period 20054.
• (Step 2) Given the regime at the period 20054, we simulate the path of -stepahead regimes by random drawing,  = 1  4.33 Given this particular path of -stepahead regimes, we can obtain the path of predicted values of  ∈  from (2).
• (Step 3) We iterate step 1 and 2 for 50 001 times to obtain the median of the step ahead forecasted values during 20061−20064 and their corresponding confidence
intervals.

• We then update the sample with four more quarterly observations and repeat Step
1-3, including re-estimating the model, in order to simulate the path of predicted values
for the subsequent four quarters. This procedure is repeated till the end of our sample.
An advantage of the second approach over the first one is that this method takes fullaccount of the regime switching model by determining the path of future regimes using
random drawing, rather than simply taking expectations over transition probabilities.
Another advantage is that a confidence interval is naturally generated, which enhances
33

For example, suppose the regime identified at the time 20055 is state 1, we use the transition

probabilities 11 and 12 to generate the state at the period 20061. Specifically, we draw a value from a uniform distribution  [0 1]. The state at 20061 is state 1 if  ∈ (0 11 ), and is state 2 ifotherwise. Suppose we have identified the state at 20061 to be state 2, then we use the transition

probabilities 21 and 22 to generate the state at the period 20062. Therefore, we will be able tosimulate the path of -step ahead regimes.

18the evaluation of diﬀerent models’ forecasting performance. It should be noticed that theregime-switching nature of the model implies that the future forecast is path-dependentand hence the conventional way to construct confidence interval may not be valid.
To evaluate the performances of in-sample and out-of-sample forecasts, we follow theliterature to compute two widely-used loss function for  ∈  , which are the square loss
¯
´ P³
´2
´ P¯
³
³
¯  ¯



+| and the absolute loss function,  +| = ¯+| ¯,
function,  +| =where +| be the -step forecast error of model , +| ≡ + −   (+ | Ω ). For

future reference, when we employ the square loss function as criteria to select models,

we label it as “the square loss criterion” (SLC). Similarly, when we use the absolute loss
function as criteria to select models, we label it as “the absolute loss criterion” (ALC).
Clearly, the SLC tends to penalize “big mistakes” more than the ALC. As it will beclear later, our main conclusions do not depend on which criteria is used. We will then
combine these results with our newly proposed model comparison procedure, which will
be explained in more details in the following section.

2.6

Multi-lateral Model Comparison Procedures

On top of computing the diﬀerences in the loss functions for each model on the predictionof both assets (housing and stock), we need a statistical procedure to compare whether
those diﬀerences are significant. Before we explain our procedure, it would be helpful
to provide a quick review of the current practice. The existing literature of applied
works adopt the Diebold-Mariano test (henceforth DM test) and its variants to assessthe “relative performance” of two models in a bilateral manner.34 While Diebold andMariano (1995), Zivot (2004) provide the details, it is nevertheless instructive to outline
the test here, as our procedure is closely related to the DM test. In our notations, theDiebold-Mariano (henceforth DM) test is based on the “loss diﬀerential”  ,

34

¡
¢
¡
¢
 =  1+| −  2+| 

Diebold and Mariano test has been widely used in the literature. Among others, see Mariano andPreve (2011) for a review of the literature.

19

where () is some loss function. Clearly, if the two models have roughly the same
predictive power, the expectation of the loss diﬀerential will be zero,  [ ] = 0 If, instead,
Model 1 predicts better (worse) Model 2, the expected value of the loss diﬀerential will
be negative (positive).35 In practice,  is unlikely to be exactly zero. The question isthen how we can decide whether Model 1 is in fact “significantly better (or worse)” thanModel 2. One of the contributions of DM test is that it shows that some function of 
follows the standard normal distribution. Hence, a test statistics can be constructed sothat we can make judgement on models scientifically.
In application, researchers may need to choose among many alternatives. Typically,
researchers need execute the DM test repeatedly, and therefore the issue of ordering
naturally arises. For instance, consider the case of three competing models, A, B and
C. One may compare A against B first, and then compare the “winner” with C. One
may also compare A against C first and then compare the winner with B. Do these twoslightly diﬀerent ordering deliver the same final winner? Clearly, as the number of models
increase, the number of possible ordering increases significantly and the importance ofordering in model comparison might matter.36 Hansen et al (2011) develop a procedure
named as “model confidence set” (MCS). The idea is that we start with one modeland put that in the MCS. One then applies the DM test to compare that model with analternative. If both have the same predictive power in the sense that the “loss diﬀerential”
 is smaller than certain critical value, then both models will be kept in the MCS;
otherwise, only the one with high predictive power will stay in the MCS. We repeat theprocedure until we exhaust all the models we have. The models in the remaining MCS
35

The DM statistics will depend  which is an average value of   for diﬀerent period , and theco-variance of  and −   = 1 2 3  As shown by Zivot (2004), other things being equal, if model1 which consistently over-predict in some sub-period and then consistently under-predict in other subperiod, it is more likely to get not only a lower value of  in diﬀerent period t, but also a higher valueof co-variance  and −   = 1 2 3  As a result, model 1 is would be classified as under-perform thealternative model. See Zivot (2004) for more details.
36
In fact, the situation is analogous to the old Condorcet Paradox, in which the candidate who winsin every pair-wise situation may not be the winner when all candidates can be selected in one time. In
the present context, it means that the order of which model compares with which model would actuallyaﬀect the final outcome. For more discussion, see Austen-Smith and Banks (2000), among others.

20will therefore have the same predictive power, and by construction they are better thanthe models that are not selected into the MCS. Hence, while we still compare models ina bilateral manner, we can still compare any finite set of models.
Mariano and Preve (2012), on the other front, generalize the idea of the DM test
and compare several models simultaneously (MP test). The idea is simple. Consider thesituation with ( + 1) models. We then define the “j-th loss diﬀerential” ´³´³
 =  +| −  +1
+| ,  = 1 2 ,where () continues to denote some loss function. We collect these loss diﬀerential in avector,  = { },  = 1 2 . We then take average of it,
1 X
≡
 
 =1


¡
¢0 ³ ´−1 ¡¢bMariano and Preve (2012) prove that   −  Ω −  −→ 2 (in distribution),b is a consistent estimator of the population
where  is the mean of the distribution, Ωvariance-covariance matrix Ω and  is the degree of freedom. Thus, MP test enables us
to test whether all models of concern have the same predictive power.
In this paper we diﬀerentiate competing theories of asset prices, which are represented
by diﬀerent econometrics models. Therefore, it is natural to use the MP test rather thanthe DM test. Moreover, we need to define a procedure which enable us to categorize themodels into diﬀerent “equivalent classes,” each of them contains model with (statisticallyspeaking) the same predictive power. The procedure needs to be “robust” in the sensethat the final outcome (i.e. the ranking of diﬀerent models) would not be aﬀected by theordering of which models are being compared first. Our procedures are similar to Hansen
et al (2011) and here are the steps.
1. We consider  models that make predictions on the same economic variable. We
first rank models according to a criterion, such as SLC. Without loss of generality,
we assume that according to the chosen criteria, the predictive performance ofModel 1 is better than that of Model 2, which in turn is better than of Model 3,
and so on.
21

2. We conduct the MP test, in which the null hypothesis is that all models have thesame predictive power. If the hypothesis is not rejected, then by definition, all Nmodels have the same predictive power on a particular variable according to thechosen criteria.
3. If the null hypothesis is rejected, then we eliminate the model with the least predictive power. It can be easily identified as the models have been ranked according to
the predictive power in Step (1). We then repeat Step (2) until the null hypothesisof equal predictive power is accepted.
4. Assume that in Step (3), there are 1 models which are found to possess the equal
predictive power, 1  0. For future reference, they are referred to as Class 1
among the  models. By construction, there are ( − 1 ) models which do nothave the same predictive power as the models in Class 1. We now repeat Step (1)on these ( − 1 ) models until the null of equal predictive power is not rejected.
Assume that there are 2 models, 2  0, in the final list and they are identified
as Class 2.
5. If  = 1 + 2 , then the procedure stops. The set of models are divided into twoclasses, and models within each class have the same predictive power. Every modelin Class 1 has higher predictive power than any model in Class 2.
6. Now we consider the case that   1 + 2 . Again, by construction, there are( − 1 − 2 ) models which do not have the same predictive power as the models
in Class 1 or Class 2. We now repeat Step (1) on these ( − 1 − 2 ) models
until the null of equal predictive power is not rejected. Assume that there are 3
models, 3  0, in the final list, and they are identified as Class 3.
7. If  = 1 + 2 + 3 , the the procedure stops.
8. If not, we repeat Step (6) and construct Class 4.
9. We repeat Step (8) until all N models are categorized into diﬀerent classes. If there
are totally  class, then it must be that  = 1 + 2 +  +  22

10. We repeat the whole process with a change in the model evaluation criteria, e.g.
SLC being replaced by ALC.

Several remarks are in order. First, by definition,  ,  = 1 2 3. . . must be positive.
In other words, each “Equivalent Predictive Power Class” (EPPC) is non-empty. In
addition, since models are first ranked according to a given criteria, our elimination
procedure is easy to implement. Second, by construction, all models within the sameclass have the same predictive power, and any model in Class  will always have higher
predictive power than any model in Class , for any  , such that   . Third, it
is possible that for the same set of  models, the ranking will vary as the criteriachanges (say, from SLC to ALC), and hence the model can be categorized diﬀerently.
In other words, the classification of EPPC is criteria-dependent. In case the same setof  models have predictions on several economic variables (for instance, stock returnand housing return in this paper), it is also possible that the ranking of models also
varies with the variable we would like to predict. In other words, EPPC is also variabledependent. Notice also that this procedure still employs conventional criteria such as
SLC and ALC, and hence it facilitates a comparison with the literature. On the other
hand, our procedure allows us to compare a large number of competing models on a “fairground.” As computers become increasingly powerful and data availability improves overtime, we believe that comparison among a large number of alternative models may be
inevitable and the procedure we propose here can facilitate such a comparison. As we
present our empirical results, these features will become clear.

3

Estimation Results

Limited by data availability, we keep the model as parsimonious as possible. The detailsof the estimation results for the whole sampling period 1975Q2-2012Q1 are presented in
the appendix, and Table 4a provides a summary. In general, a model allowing for regime
switching attains a lower value of Akaike’s information criterion (AIC) and a higher loglikelihood value. Among all these models, the regime switching model G (EFP, TED,
23

SRET, HRET) has the best goodness of fit, i.e., a significantly lower value of AIC thanother models, suggesting that the credit market frictions and asset returns are indeed
significantly inter-related.
(Table 4a about here)
For the Markov switching model, recall that we set the volatility at regime 1  (1) = 1,
thus the element  (2) measures the relative volatility of regime 2 over regime 1. In the
appendix the figures show that the estimated values of relative volatility  (2) are all
significantly less than one for  = 1 and 2, which means that for both federal funds rate
and the spread the volatility in regime 2 is lower than in regime 1. On the other hand,almost all of the 3 (2) and 4 (2) are insignificant, suggesting that for the quarterly
stock and housing returns there is no significant diﬀerence in volatility across regimes.
Thus, we identify two regimes for this monetary policy tool: a high volatility regime(regime 1) and a low volatility regime (regime 2). Table 4b provides a summary of theestimated transition probabilities. It is clear that regimes are highly persistent, regardlessthe models. In particular, most models suggest 11 to be close to 080 and all models
suggest 22 to be higher than 093. They imply that the expected duration of regime 1 to
be around 1(1 − 08) = 50 quarters and that for regime 2 is not less than 1(1 − 093) =14 3 quarters.
(Table 4b about here)
Given the estimated parameters, transition probabilities, and variance-covariance matrices, we estimate the classification of regimes under diﬀerent models and report theresults in Table 5. Basically, these models show similar classifications of the regimes. For
those periods identified as regime 1, all models include the aftermath of the second oil
crises and P. Volcker being appointed as Chairman of the Federal Reserve.37 Interestingly, when the TED spread (TED) is included in Model E, G and H, the regime 1 also
includes the stock market crash in the 1987, suggesting that TED picks up the volatility
in the credit market after the stock market crash. It is also interesting that under Model
37

Among others, Goodfriend and King (2005), Goodfriend (2007) provides a summary of the historyof monetary policy during that period.

24

E, which is like Model D except that EFP is replaced by the TED spread, the changes in
regimes are much more frequent. In general, models that involve TED would experience
more regime switching, suggesting higher variability of the risk premium faced by financial intermediations. We also compute the smoothed probabilities for all from Model B
to Model H, as shown in Figure 2. It shows the probabilities of the economy being in
regime 1 (high volatility regime) at a given period. Since there are only 2 regimes, theprobabilities of being in regime 2 would be suppressed.
(Table 5, Figure 2 about here)

4

Forecasting

We now proceed to forecast stock and housing returns. As discussed above we first
conduct in-sample forecasting for the period 1975Q2-2005Q4 and then examine the outof-sample forecasts for the period 2006Q1-2011Q4, using respectively the expectationsbased and simulation-based methods.

4.1

In-Sample Forecasting

We compute the loss functions based on SLC and ALC of in-sample -step ahead forecasts,  = 1  4, for each variable across all models. Several findings are in order. First,
as shown in table 6a, the in-sample forecasts of asset returns are mixed. For the stock
returns, the model C (FFR, SPR, SRET, HRET) has the best performance. For housing
return, however, it is the Model E (FFR, TED, SRET, HRET) that out-perform allothers. Notice that both models contain the monetary policy variable FFR. It is true forwhether we use SLC or ALC. On the other hand, it seems that the performances in predicting stock returns across models are similar. We therefore implement our multi-lateralmodel comparison procedures and attempt to categorize models into diﬀerent “equivalentpredictive power class” (EPPC).38 Table 6b confirms our intuition. Whether we use SLC
38

As a robustness check, we also use the conventional Diebold-Mariano test bilaterally and obtain very

similar results. The details are in the appendix.

25

or ALC, we find that all models have the same predictive power in terms of explaining
the stock return during the in-sample period. In particular, no model has superior performance than an AR(1) process, which is the USB. It is consistent with the notion thatthe stock market is very eﬃcient in reflecting all the relevant information that adding
other variables in the statistical model does not provide any extra predictive power.
(Tables 6a, 6b about here)
For housing return, the situation is very diﬀerent. Several observations are in order.
First, whether we use SLC or ALC, model D, E, F, H have the same predictive power
and they are always in Group 1, meaning that they are at least as good as the othermodels. Second, whether we use SLC or ALC, model B (FFR, GDP, SRET, HRET)
and USB are always in the lowest group. This suggests that using the monetary policy(FFR) and economic growth rate alone are not suﬃcient in understanding the housing
market, at least for the in-sample period (1975Q2-2005Q4). Alternatively, we may saythat the information in monetary policy and economic growth has been reflected in the
housing return itself. The fact that AR(1) (which is also USB for the housing return)is always inferior to the four “good models” (Model D, E, F, H) suggests that there areimportant information in the financial market that enhances our ability to account for the
housing market. Notice however that such “cross-market informational spillover” is very
subtle. Table 6c also shows that Model G (EFP, TED, SRET, HRET) is always inferior
to the four good models (Model D, E, F, H). However, Table 3 shows that Model F issimply Model G with TED spread replaced by the term spread (SPR), and Model H issimply Model G with the external finance premium (EFP) replaced by the term spread(SPR). Does it mean that the term spread is so crucial in understanding the housing
return during that period? It does not seem to be the case, as both Model D and E,
which have the same predictive power as model F and H, do not contain SPR. Noticethat financial variables are correlated and hence some other variables may also containthe information that is relevant in predicting future housing returns. To summarize, ourestimations indicate that the fact that the “eﬃcient market hypothesis” does not apply
to the housing market during the in-sample period (1975 to 2005). The data are more26

consistent with models which emphasize on imperfect capital market, such as Christiano,
Motto and Rostagno (2007), Davis (2010), Jin et al. (2012), among others.

4.2

Out-of-Sample Forecasting via Conditional-Expectation Estimation

We now turn to the out-of-sample forecasting (OSF) of housing and stock returns beginning 20061, at the time when the growth of housing returns began to decline andthe sub-prime crisis started to unfold. Following the literature, we first conduct OSF
by using the conditional-expectations predictions. The appendix provides details of theout-of-sample -step ahead forecasts,  = 1  4, for each variable across all models.
Tables 6c summarize the results. In terms of forecasting stock returns, Model H (SPR,
TED, SRET, HRET) performs better than other models, both in terms of SLC and
ALC. In terms of forecasting housing returns, Model D (FFR, EFP, SRET, HRET) performs better than other models, also both in terms of SLC and ALC. Naturally we ask
whether the diﬀerence is statistically significant. Again, we adopt the same procedureand categorize models into diﬀerent “equivalent predictive power class” (EPPC). Table
6d reports the results, and several observations are in order. In terms of stock return
forecasting, we find that (1) Model B, E, H and USB are always in Group 1, which meansthat they are at least as good as other models, and (2) Model D and F are always in thelower group. This is true whether we use SLC or ALC. The ranking of Model A, C, andG will depend on which criteria is used, implying that there are models (especially D
an F) which under-perform the USB, which is the simple AR(1), in the out-of-sampling
forecasting of the stock return. Recall that for the in-sample forecasting, using the same
set of procedures and same set of models, we have found that all models have the same
predictive power. In other words, some models have actually deteriorated relative to the
USB (i.e. AR(1)) in terms of the ability of predicting the stock return. Notice thatboth model D and F involve EFP, it may suggest that the ability of EFP to track theaggregate stock return after the crisis is not as good as before.
(Tables 6c, 6d about here)
27

The case of OSF for the housing return is perhaps equally interesting. Table 6d clearlyshows that whether we use SLC or ALC, (1) Model C, D, F, G, H and USB are always
in Group 1, meaning that they have better predictive power, and (2), Model A, B, E are
always in Group 2, meaning that their predictive powers are not as good. Notice thatfor in-sample forecasting, whether we use SLC or ALC, Model B (FFR, GDP, SRET,
HRET) and USB are always in the lowest group. For the out-of-sample forecasting,Model B remains in the lowest group, yet the USB is “promoted” to the higher group. It
means that whether for the linear VAR with 7 variables (Model A), or regime-switching
VAR, no model in our list can out-perform the simple USB in terms of out-of-sample
forecasting of the housing return. Similar to the case of stock return forecasting, thissuggests that some models have deteriorated in terms of forecasting the housing return,at least relative to the simple AR(1) process. In particular, model A and B are the only
models that involve GDP and yet they are always in group 2, suggesting that GDP may
not be as useful in predicting the house price as before. Clearly, the model comparison
here are far from conclusive and future research can revisit the issue with more rigorous
tools.
Another interesting observation is that, perhaps the forecasting ability of a model has
become more asset-specific for the out-of-sample forecast.39 For instance, while model
D and F are always the inferior models for stock return out-of-sample forecasting, they
are always in the Group 1 for the housing return forecasting. Similarly, while Model B
and E are always the “better” models in terms of out-of-sample forecast for the stock
return, they are always the “not-as-good” models in terms of out-of-sample forecast
for the housing return. Future research may investigate further on this phenomenon ofasset-dependent forecasting performance.

39

Interestingly, this is also the case for structural models comparison. See Kwan et al. (2015), among

others, for more details.

28

4.3

Out-of-Sample Forecasting via Simulation

The results presented in the current section diﬀer in at least two important dimensions
from the results presented in the previous section. First, the previous sections provide
only information of the “relative performance” of diﬀerent models, as we use diﬀerent statistical tools and procedures to assess whether some models have more superior predictive
power than the others. In this section, we attempt to assess the “absolute performance”
of diﬀerent models by using simulation-based forecasting. Second, we aggregate the forecasting performance of each models during the whole out-of-sample period (2006 andafter) into some statistics and then compare across models in the previous section. In
the current section, we will compare the forecasting performance of diﬀerent periods ineach year, and then allow the model to be re-estimated with updated data, and thencompare again in the subsequent year. Thus, we allow models to “learn and improve”
and would like to see which model(s) are more successful in adjust the parameter withnew data and hence provide more accurate forecasting over time.
We more specifically consider a forecasting window of 4 quarters starting 20061,
with -quarter ahead forecasts,  = 1  4. After simulating the out-of-sample path20061 − 20064 based on observations up to 20054, the data are updated with four
observations and the parameters are re-estimated. The procedure is repeated until wehave updated the sample to include all observations from 1975 to 2010 to predict theasset returns in 2011. The purpose of this exercise is to see how the performances of themodels change when information is updated. The simulated paths together with their
80-percent confidence intervals can be visualized in Figure 3 for stock returns and Figure
4 for housing returns. Table 7a and 7b provide a summary of the performance of diﬀerent
models(Figure 3 and 4, Table 7a, 7b about here)
As shown in Figure 3a and 3b, the actual stock return and the predicted paths by
diﬀerent models are well within the boundaries of the 80% confidence intervals for all fivemodels. Thus, although the models do not predict what have actually happened in 2006
and 2007, the models’ predictions are not that “far oﬀ the mark.” Unfortunately, with the29collapse of the Lehman Brothers, virtually all models are disappointing in the predictionof 2008 returns (Figure 3c). Among them, Model G (EFP, TED, SRET, HRET) performs
worst in the sense that its 80-percent confidence interval does not even contain any of theactual quarterly return in 2008. As we include the data up to 20084 and re-estimate
the models, the prediction of 2009 by models significantly improve. As shown by Figure
3d, each model’s confidence interval contains at least one quarter of stock return within
the confidence interval. Among them, Model D (FFR, EFP, SRET, HRET), Model E(FFR, TED, SRET, HRET), Model F (EFP, SPR, SRET, HRET) and Model G containalmost the path of the whole year stock return. With the data of 2009 included andmodel updated, the prediction of 2010 is even better. As Figure 3e shows, each model’sconfidence interval contains at least one quarter of stock return. Interestingly, Model G,
which has the worst performance in 2008, becomes the best model of 2010 in the sensethat it is the only model whose confidence interval contains the whole path of quarterly
return of the year. The prediction of 2011 is similar. As Figure 3f shows, virtually allmodels contain most of the year return, and all models unfortunately “miss” the dropof stock return in 20113, as no model is able to generate a confidence interval whichcontains the stock return in 20113.
As shown in Table 7b and Figure 4, the prediction of housing returns is worse than
the stock counterpart. Figure 4a shows that every models generates a confidence intervalthat contains the path of the quarterly housing return of 2006yet as early as 2007, Figure
4b shows that the confidence intervals generated by our models fail to contain at least one
quarter of housing return. It should be notice that the same set of models successfully
contain the whole year path of stock return of the same year (2007). With 2007 data
included and model updated, the failure of 2008 is in a sense unexpected. Both Table
7b and Figure 4c show that the confidence intervals generated by more than half of ourmodels - namely, Model A (linear VAR with all 7 variables), Model B (FFR, GDP, SRET,
HRET), Model D (FFR, EFP, SRET, HRET), Model E (FFR, TED, SRET, HRET),
Model G (EFP, TED, SRET, HRET) - fail to contain any quarterly housing return in
2008. The remaining 4 models all miss at least one quarterly return of housing of theyear. Thus, while our models do not perform well in 2008 in predicting the stock return,30

the prediction of housing return in the same year is much worse. With the information of2008 included and model updated again, the prediction of 2009 is improved but perhaps
still disappointing. Recall that for the year 2009, the confidence interval generated by all
our models all contain some quarterly return of stock, suggesting that the enlargementof the sample with model updating might improve the stock return forecasting. In thecase of housing, Figure 4d shows that Model B continues to fail to contain any quarterly
return of housing in 2009. While Model A, D, E, G make some improvements, theconfidence interval generated by the Model F (EFP, SPR, SRET, HRET) fail to containany quarterly housing return. And if the year 2008 and 2009 are disappointing year of
housing return forecasting, the year 2010 is a year with positive surprises. As shownby Table 7b and Figure 4e show that all models successfully contain some quarterly
return of housing. Moreover, Model C (FFR, SPR, SRET, HRET), D, E and G areable to generate confidence intervals that contain the whole year housing return. For
the case of stock return, only Model G contains the whole year of stock return in theconfidence interval. In this sense, the prediction of housing returns in 2010 is a success.
Unfortunately, Figure 4f shows that all models fail to contain the drop in housing returnin 20111, although they contain at least some quarterly return in the later part of thesame year. It is comparable to the performance of stock return prediction of the same
year (2011).
In sum, it seems that the OSF of asset returns is particularly diﬃcult during thisperiod. In the case of stock return, Table 7a suggest that since 2008, most models will
miss at least one quarter of stock return, and while Model G “fails” in 2008, it becomesvery successful in 2010. This confirms the intuition that regularly incorporating new data
and re-estimating the model lead to better forecasting. In the case of the housing return,most of our models, namely, Model A, B, D, E, F, G all experience at least one “missingyear” on either 2008 or 2009, i.e. a year in which the confidence interval generated by
the model does not contain any quarterly return of the year. At the same time, when we
re-estimate the model with data up to the end of 2009, model C, D, E, G successfully
capture the year 2010. In this sense, model G (EFP, TED, SRET, HRET) seems to be
the “best learner” in the sense that while it made mistakes in the 2008 or 2009, when it is31

re-estimated with the data up to the end of 2009, it successfully captures the movements
in both stock return and housing return in 2010. Notice that according to table 7aand 7b, neither the USB for stock return nor the USB for the housing return enjoys a“perfect” year (i.e. the whole year asset return movement within the 80% confidence
interval) since 2006, suggesting that while the USB may be classified in the same EPPC
as other models during the out-of-sample period (2006 and after) as a whole, it may not“learn” as much and as fast as other models which incorporate other macroeconomic andfinancial variables.

4.4

Some Robustness Checks

Thus far, our analysis is based on using the data from 1975 to 2005 as in-sample, andthe periods after as out-of-sample, and then we progressively update the in-sample. Asa robustness check, we also re-estimate our models when we use the period from 1975
to 2006 as in-sample. Table 8 provides a summary and the details can be found in the
appendix. Notice that Table 8 is constructed analogous to Table 6 in order to facilitatecomparison. A few observations are in order. Table 8a shows that the best in-sample
forecasting performance comes from model C (FFR, SPR, SRET, HRET) for stock andmodel E (FFR, TED, SRET, HRET) for housing, which is the same as the results inTable 6a. While the details of the model classifications in Table 8b slightly diﬀer from
Table 6b, some principal findings sustain. First, most, if not all, models are equally goodin predicting stock return during the in-sample period. Second, in terms of predicting
housing return, model B (FFR, GDP, SRET, HRET) and the USB are often the worst.
Again, this supports the idea that GDP is not very helpful even for in-sample predictionof the housing return, and information about the future housing returns are reflectedin economic variables other than the current housing return. Table 8c shows that the
principal result of Table 6c also sustains, namely, model H (SPR, TED, SRET, HRET)
out-performs other models in terms of out-of-sample forecasting of stock returns, andmodel D (FFR, EFP, SRET, HRET) out-performs other models in terms of out-ofsample forecasting of housing returns. Table 8d, which provides the model classification32

in terms of out-of-sample forecasts is also similar to Table 6d. More specifically, themodel classification in terms of out-of-sample 4-quarter ahead forecasts of stock returnare identical. In terms of the counterpart of housing returns forecast, model A (linear
VAR with all 7 variables), model B which contains FFR and GDP and model E whichcontains FFR and TED are inferior to models C, D, F, G. A noticeable diﬀerence is thatnow model H which contains SPR and TED, and USB are also in the second class. Thus,
it seems that while the choice of choosing 2005 as the end of the in-sample might not be
the consensus among researchers, the results are not as sensitive as one may think. In
addition, the previous section has also presented our results based on a recursive approachof allowing models to “update and re-estimate” in terms of the out-of-sample forecasting.

(Tables 8a, 8b, 8c, 8d about here)

5

Concluding Remarks

Dramatic movements in asset prices often occupy media headlines and carry implications
in real economic activities, even political personnel changes. Yet in media and even academic circles are full of competing and sometimes even conflicting explanations. This
paper attempts to bring some explanations for these events to formal testing. Noticethat while an econometric model comparison is not equivalent to an explanation comparison, the exercise in this paper may provide us hints for future model development. It isespecially true if models contain certain variables consistently out-perform other models
which do not contain those variables. From a policy point of view, identifying the empirically more relevant model does not only satisfies intellectual curiosity, but may also
assist governments making more appropriate policy decisions. Thus, this paper attempts
to establish some stylized facts about the asset return dynamics in the aggregate level. In
particular, this paper separates the data into the pre-crisis period 1975-2005 (in-sample)
and the period after (2006-2011), and examines whether an estimated model based on
the pre-crisis period can have reasonable forecasting for the crisis period (out-of-sample
33

forecasting), with the eﬀect of other variables such as GDP growth and monetary policy
taken into consideration.
Our first contribution is to demonstrate how to categorize competing models into different “Equivalence Predictive Power Classes” (EPPC). We find that it can shed light on
the Eﬃcient Market Hypothesis (EMH) debate, as well as the structural-break discussion
on the asset markets. During the in-sample period, all models have the same predictive
power as the USB (Univariate, Single-regime Benchmark, i.e. AR()) on stock return.
It is consistent with the EMH that all information about the future stock returns has
been reflected in the current period stock return and hence additional variables do notenhance the forecasting. For the housing return, the situation is diﬀerent. Model D, E,
F, H consistently outperform the USB, suggest the existence of cross-market information
flow and the value of multi-variate modeling. For the out-of-sample period as a whole,
the picture changes. No model has superior predictive power than the USB on either
the stock return of housing return forecast. Whether we use the “square loss criterion”(SLC) or the “absolute loss criterion” (ALC), and Model A, B and E are shown to havelower predictive power than the USB in the housing return forecast. Thus, these results
seem to be consistent with the notion that there are some “structural changes” in thedeterminants of asset return dynamics since 2006. In particular, since both Model A
and B are the only models which contain GDP as a variable in the dynamical system,
the evidence then points to the possibility that GDP has lost its power in predicting the
housing return after the crisis period, perhaps due to the deep recession. Obviously, it
does not mean that “economic fundamentals” such as GDP is not important. One interpretation is that the information contained in the GDP has already been reflected in otherfinancial variables that we include in the econometric model, such as TED spread. Thisis consistent with the theoretical result in Telmer and Zin (2002) that with incomplete
markets, asset returns can predict (future) asset prices well.
Our second contribution shows that all models do not have the same capacity to “learn
and update.” Notice that the results based on EPPC clearly treat the model performance
during the out-of-sample period as a whole. However, it is possible that some models
make more mistake in the beginning of the financial crisis, and later improves significantly

34

with new data is supplied. To investigate such possibility, we conduct out-of-sampling
forecasting based on the simulation approach. We find that the USB of either the stock
return or the housing return never enjoys a perfect year (i.e. a year that the asset returnis totally inclusive within the confidence interval) since 2006. On the other hand, afterrevising with data by the end of 2009, Model G, which includes both the EFP and TED
spread, can include the actual asset returns of 2010 in the confidence intervals. In otherwords, while uni-variate models may be as good as any model in predicting asset returns
in the “long run,” they may learn “slower” than multi-variate ones in the “short-run”. It
also suggests that macroeconomic models, which emphasize the role of imperfect capital
market faced by non-financial firms as well as those faced by banks, may be promisingin enhancing the understanding of the stock and housing returns.40
In addition,we also methodologically demonstrate a few things in terms of multivariate modeling. First, the widely used linear VAR model (i.e. single regime) with 7
variables can actually under-perform than the regime-switching counterparts with only
4 variables most of the time. Perhaps more importantly, we demonstrate that we can
combine the multi-model comparison test of Mariano and Preve (2012) with the “Model
Confidence set” procedure developed by Hansen et al. (2011) to classify models into
diﬀerent EPPC. We also demonstrate that such EPPC approach can be complemented
by the simulation-based method used by Sargent, Williams and Zha (2006) and others tofurther distinguish the empirical performance of models within the same EPPC. Future
research may want to further explore along these lines.
This research can clearly be extended in several dimensions. First, it would be interesting to apply the current econometric framework to other economies, such as theEuropean Union countries. The current framework can also be applied to more disaggregate data. Also, it would be interesting to incorporate higher frequency variables andstudy the (potential) price discovery processes among diﬀerent asset markets. Needlessto say, one can re-examine the two asset markets with Bayesian econometrics. Finally,
one could develop a theoretical framework that mimics the stylized facts found in thispaper, which will further enhance our understanding of the interactions between the real
40

Among others, see Gertler and Kiyotaki (2010), Davis (2010), Andres and Arce (2012), Jin et al.

(2012).

35

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42

Tables
Table 1. Statistical Summary of Federal Funds Rate, Term Spread, Gross Domestic Production Growth Rate, External Finance Premium, Market
Liquidity, Stock Index Return and Housing Market Return (1975Q2-2012Q1)
FFR

SPR

GDP

EFP

TED

SRET

HRET

Mean
MedianMaximum
Minimum

5.808
5.417
17.780
0.073

1.634
1.789
3.611
-2.182

0.699
0.744
3.859
-2.328

1.125
0.997
3.023
0.560

0.867
0.605
3.333
0.097

1.910
2.436
18.952-26.431

1.123
1.206
4.533
-2.989Std. Dev.
Skewness
Kurtosis
Observations

3.806
0.785
3.838
148.000

1.343-0.7073.006
148.000

0.809-0.5366.270
148.000

0.473
1.596
5.914
148.000

0.750
1.643
5.216
148.000

8.204-0.7873.995
148.000

1.277-0.4904.735
148.000

Note: FFR denotes the federal funds rate, SPR denotes the term spread, GDP means the gross domestic production growth rate, EFP means theexternal finance premium, TED means the market liquidity, SRET means stock index return, and HRET means housing market return.

Table 2. Correlation Coefficients (1975Q2-2012Q1)
FFR
SPR
GDP
EFP
TED
SRET
HRET

FFR

SPR

GDP

EFP

TED

SRET

HRET

1.000

-0.608
1.000

0.031
0.050
1.000

0.290
0.097
-0.3601.000

0.722-0.410-0.250
0.638

0.016
0.011
0.148
-0.0500.266-0.2510.209-0.1921.000-0.1301.000-0.015-0.0071.000

Table 3a. List of Models
Model

Model Structure Variables

A

Linear

FFR, SPR, TED, EFP, GDP, SRET, HRET

B

Two-regime

FFR, GDP, SRET, HRETC

Two-regime

FFR, SPR, SRET, HRETD

Two-regime

FFR, EFP, SRET, HRETE

Two-regime

FFR, TED, SRET, HRETF

Two-regime

EFP, SPR, SRET, HRETG

Two-regime

EFP, TED, SRET, HRETH

Two-regime

SPR, TED, SRET, HRET

USB

Linear

Uni-variate, Single-regime, Benchmark (i.e. AR(1))

Key: (unless specified, all variables refer to quarterly data) FFR, Federal Fund Rate; SPR, term spread, which is equal to (10-year bond rate –FFR); TED, TED spread, which is equal to (3-month Eurodollar deposit rate - 3-month T-bill rate), a measure of market liquidity; EFP, External
Finance Premium, which is equal to corporate bond spread (Baa-Aaa), a measure of External Finance Premium; GDP, GDP growth rate; SRET,
Stock Market Return; HRET, Housing Market Return.

Table 3b. Choosing the Uni-variate, Single-regime Benchmark (USB) model
AIC

SBC

AR(1)
AR(2)
AR(1)
AR(2)
SRET
7.07747.0902
7.1381
7.1712HRET
2.75892.74642.81962.8274
Note: AIC refers to the Akaike information criterion. SBC refers to the Schwartz Bayesian information criterion.

(1) For SRET, in terms of both AIC and SBC, AR(1) model performs better than AR(2).
(2) For HRET, in terms of AIC, AR(2) is the marginally better (2.746 versus 2.759). However, in terms of SBC, AR(1) ismarginally better (2.820 versus 2.827). For parsimony purpose, we choose AR(1) specification.

Table 4a.

A Summary of Goodness of Fit for All Eight Models (1975Q2-2012Q1)
ModelsAIC

SBC

Model A
Model B
Model C
Model D

Single-regime model (FFR, SPR, TED, EFP, GDP, SRET, HRET)
Two-regime model (FFR, GDP, SRET, HRET)
Two-regime model (FFR, SPR, SRET, HRET)
Two-regime model (FFR, EFP, SRET, HRET)

12.1273
13.8681
12.7748
11.0063

13.8284
15.1237
14.0304
12.2619

Model E
Model F
Model G
Model H

Two-regime model (FFR, TED, SRET, HRET)
Two-regime model (EFP, SPR, SRET, HRET)
Two-regime model (EFP, TED, SRET, HRET)
Two-regime model (SPR, TED, SRET, HRET)

11.8574
10.6150
9.265611.7468

13.1130
11.8706
10.521213.0024

Model USB
Model USB

Uni-variate, Single-regime, Benchmark for (SRET)
Uni-variate, Single-regime, Benchmark for (HRET)

7.07742.75897.13812.8196

Note: AIC refers to the Akaike information criterion. SBC refers to the Schwartz Bayesian information criterion.

Table 4b. Estimated Persistence of Regimes among Models (1975Q2-2012Q1)

p11

p22

Model B

0.9427

0.9920Model C

0.95050.9914Model D

0.80600.9643Model E

0.78240.9386

Model F0.8226

0.9492Model G

0.79520.9370Model H

0.84780.9447

Table 5.

Identified periods of Regime 1

ModelsRegime 1

Model B
Model C

1978Q2-1982Q4
1979Q4-1986Q2

Model D
Model E

1975Q2-1975Q3
1975Q2-1976Q2
2007Q3-2008Q4
1975Q2Model F
Model G
Model H

1979Q4-1982Q4 1984Q1-1984Q4 2001Q1-2002Q1 2008Q4
1978Q2-1978Q4 1979Q3-1982Q4 1984Q2-1984Q4 1987Q2-1987Q4
2009Q2
1980Q2-1986Q1 2001Q1-2002Q1 2008Q4
2009Q2-2009Q3

2011Q3-2011Q4
1975Q2
1976Q2
1978Q4
2007Q3-2008Q4 2009Q2-2009Q3
1975Q2-1976Q2 1978Q2-1982Q3 1984Q2
2007Q3-2009Q2

1979Q3-1984Q4

1987Q2-1987Q4

1987Q2-1988Q3

2001Q1-2001Q4

Table 6a. A Summary of In-sample Forecasting Performances (4-Quarter Ahead Forecasts) (1975Q2-2005Q4)

Stock Returns

Housing Returns

SLC

ALC

SLC

ALC

Model A

Single-regime model (FFR, SPR, TED, EFP, GDP,
SRET, HRET)

57.58665.66860.75770.6488Model BTwo-regime model (FFR, GDP, SRET, HRET)

57.80425.64530.83040.7053Model C

Two-regime model (FFR, SPR, SRET, HRET)

57.1605

5.61880.7356

0.6691Model D

Two-regime model (FFR, EFP, SRET, HRET)

58.70395.66620.70130.6364Model E

Two-regime model (FFR, TED, SRET, HRET)

58.2775

5.65530.69800.6204Model F

Two-regime model (EFP, SPR, SRET, HRET)

60.6847

5.74250.73310.6502Model G

Two-regime model (EFP, TED, SRET, HRET)

59.05015.77070.82950.6727

Model H

Two-regime model (SPR, TED, SRET, HRET)

58.26235.63660.73080.6378USB

Uni-variate, Single-regime Benchmark

57.99665.68810.84000.6726

Key: SLC: Square Loss Criteria; ALC: Absolute Loss Criteria.

Table 6b. A Summary of EPPC (In-sample 4-Quarter Ahead Forecasts)

Stock Return
Housing Return

SLC
ALC
SLC
ALC

Class 1
All models
All models
D, E, F, H
A, C, D, E, F, H

Class 2//
A, C, G
B, G, USB

Class 3//
B, USB/

Note: SLC: Squared Loss Criterion; ALC: Absolute Loss Criterion. Our convention is that if i  j ,then any model in Class i has less predictive power than any model in Class j .

Table 6c. A Summary of Out-of-Sample Forecasting Performances (4-Quarter Ahead Forecasts)(2006Q4-2012Q1)
Stock Returns

Housing Returns

SLC

ALC

SLC

ALC

Model A

Single-regime model (FFR, SPR, TED, EFP, GDP,
SRET, HRET)

126.3311

9.46714.87301.8844Model BTwo-regime model (FFR, GDP, SRET, HRET)

119.1706

9.09404.86581.8975Model C

Two-regime model (FFR, SPR, SRET, HRET)

122.2287

9.33293.7581

1.6619Model D

Two-regime model (FFR, EFP, SRET, HRET)

127.3973

9.5067

3.69381.6136

Model E

Two-regime model (FFR, TED, SRET, HRET)

115.8617

8.80735.0151

1.9273Model F

Two-regime model (EFP, SPR, SRET, HRET)

133.7992

9.6332

3.83591.6712Model G

Two-regime model (EFP, TED, SRET, HRET)

127.3231

8.89304.05991.7108Model H

Two-regime model (SPR, TED, SRET, HRET)

109.6158

8.58354.2041

1.7652USB

Uni-variate, Single-regime Benchmark

113.6018

8.65294.5453

1.8221

Key: SLC: Square Loss Criteria; ALC: Absolute Loss Criteria.

Table 6d. A Summary of EPPC (Out-Of-Sample 4-Quarter Ahead Forecasts)

Stock Return
Housing Return

SLC
ALC
SLC
ALC

Class 1
A, B, C, E, H, USB
B, E, G, H, USB
C, D, F, G, H, USB
C, D, F, G, H, USB

Class 2
D, F, G
A, C, D, F
A, B, E
A, B, E

Note: SLC: Squared Loss Criterion; ALC: Absolute Loss Criterion. Our convention is that if i  j ,then any model in Class i has less predictive power than any model in Class j .

Table 7a. Is the forecasted Stock return within the 80% confidence interval?
Models

Predicting 2006

Predicting 2007

Predicting 2008

Predicting 2009

Predicting 2010

Predicting 2011

based on
1975-2005

based on 1975-2006

based on
1975-2007

based on
1975-2008

based on
1975-2009

based on
1975-2010

Model A

Yes

Yes

PartlyPartlyPartlyPartlyModel B

Yes

Yes

PartlyPartlyPartlyPartlyModel C

Yes

Yes

PartlyPartlyPartlyPartlyModel D

Yes

Yes

PartlyPartlyPartlyPartlyModel E

Yes

Yes

PartlyPartlyPartlyPartlyModel F

Yes

Yes

PartlyPartlyPartlyPartlyModel G

Yes

Yes

No

PartlyYes

PartlyModel H

Yes

Yes

PartlyPartlyPartlyPartly

USB

Yes

Yes

PartlyPartlyPartlyPartlyTable 7b. Is the forecasted Housing return within the 80% confidence interval?
Models

Predicting 2006

Predicting 2007

Predicting 2008

Predicting 2009

Predicting 2010

Predicting 2011

based on
1975-2005

based on 1975-2006

based on
1975-2007

based on
1975-2008

based on
1975-2009

based on
1975-2010

Model A

Yes

PartlyNo

PartlyPartlyPartlyModel B

Yes

PartlyNo

No

PartlyPartlyModel C

Yes

PartlyPartlyPartlyYes

PartlyModel D

Yes

PartlyNo

PartlyYes

PartlyModel E

Yes

PartlyNo

PartlyYes

PartlyModel F

Yes

PartlyPartlyNo

PartlyPartlyModel G

Yes

PartlyNo

PartlyYes

PartlyModel H

Yes

PartlyPartlyPartlyPartlyPartlyUSB

YesPartlyPartlyPartlyPartlyPartlyTable 8a. A Summary of In-sample Forecasting Performances (4-Quarter Ahead Forecasts) (1975Q2-2006Q4)
Stock Returns

Housing Returns

SLC

ALC

SLC

ALC

Model A

Single-regime model (FFR, SPR, TED, EFP, GDP,
SRET, HRET)

56.16615.58600.74630.6412Model BTwo-regime model (FFR, GDP, SRET, HRET)

56.3073

5.56550.81740.7001Model C

Two-regime model (FFR, SPR, SRET, HRET)

55.6801

5.5256

0.72320.6599Model D

Two-regime model (FFR, EFP, SRET, HRET)

56.68375.58490.72520.6521Model E

Two-regime model (FFR, TED, SRET, HRET)

56.7498

5.57310.69950.6216

Model F

Two-regime model (EFP, SPR, SRET, HRET)

58.9283

5.64070.71900.6447

Model G

Two-regime model (EFP, TED, SRET, HRET)

58.23905.68500.76220.6450Model H

Two-regime model (SPR, TED, SRET, HRET)

57.70335.73000.77270.6579USB

Uni-variate, Single-regime Benchmark

56.45315.60160.82180.6636

Key: SLC: Square Loss Criteria; ALC: Absolute Loss Criteria

Table 8b. A Summary of EPPC (In-sample 4-Quarter Ahead Forecasts)

Stock Return
Housing Return

SLC
ALC
SLC
ALC

Class 1
All models
A, B, C, D, E, F, G, USB
A, C, D, E, F, G, H
A, C, D, E, F, G, H, USB

Class 2/
H
B, USB
B

Note: SLC: Squared Loss Criterion; ALC: Absolute Loss Criterion. Our convention is that if i  j ,then any model in Class i has less predictive power than any model in Class j .

Table 8c. A Summary of Out-of-Sample Forecasting Performances (4-Quarter Ahead Forecasts)(2007Q4-2012Q1)
Stock Returns

Housing Returns

SLC

ALC

SLC

ALC

Model A

Single-regime model (FFR, SPR, TED, EFP, GDP,
SRET, HRET)

152.4294

11.01355.3034

1.9487

Model BTwo-regime model (FFR, GDP, SRET, HRET)

143.7629

10.54115.23281.9505Model C

Two-regime model (FFR, SPR, SRET, HRET)

147.2156

10.81143.95511.6780Model D

Two-regime model (FFR, EFP, SRET, HRET)

153.8418

11.0492

3.90001.6292Model E

Two-regime model (FFR, TED, SRET, HRET)

139.7478

10.20425.53642.0180Model F

Two-regime model (EFP, SPR, SRET, HRET)

161.8230

11.1754

4.11991.7119Model G

Two-regime model (EFP, TED, SRET, HRET)

153.6449

10.30144.39441.7674Model H

Two-regime model (SPR, TED, SRET, HRET)

132.4850

9.99654.64001.8598USB

Uni-variate, Single-regime Benchmark

137.1296

10.02365.0374

1.9263

Key: SLC: Square Loss Criteria; ALC: Absolute Loss Criteria.

Table 8d. A Summary of EPPC (Out-Of-Sample 4-Quarter Ahead Forecasts)

Stock Return
Housing Return

SLC
ALC
SLC
ALC

Class 1
A, B, C, E, H, USB
B, E, G, H, USB
C, D, F, G
C, D, F, G

Class 2
D, F, G
A, C, D, F
A, B, E, H, USB
A, B, E, H, USB

Note: SLC: Squared Loss Criterion; ALC: Absolute Loss Criterion. Our convention is that if i  j ,then any model in Class i has less predictive power than any model in Class j .

Figure 1aFederal Funds Rate (FFR), Term Spread (SPR), Percentage Changes in Gross Domestic Production (GDP), External FinancePremium (EFP)
FFR

SPR

20

4
3

16
2
12

1

8

0
-1

4
-2
0

-3
1980

1985

1990

1995

2000

2005

2010

1980

1985

GDP

1990

1995

2000

2005

2010

2000

2005

2010

EFP

4

3.5

3

3.0

2

2.5

1
2.0
0
1.5

-1

1.0

-2
-3

0.5
1980

1985

1990

1995

2000

2005

2010

1980

1985

1990

1995

Figure 1b Market Liquidity (TED), Stock Index Return (SRET), Housing Market Return (HRET)

TED

SRET

4

20
10

3

0
2
-10
1

-20

0

-30
1980 1985 1990 1995 2000 2005 20101980 1985 1990 1995 2000 2005 2010

HRET
6
4
2
0
-2
-4
1980 1985 1990 1995 2000 2005 2010

Figure 2a Smoothed Probabilities for Model B (FFR, GDP,SRET,HRET)(only the probability of Regime 1 will be shown)
Regime 1
1.0

0.8

0.6

0.4

0.2

0.0
1980

1985

1990

1995

2000

2005

2010

Figure 2b Smoothed Probabilities for Model C (FFR,SPR,SRET,HRET)
Regime 1
1.0

0.8

0.6

0.4

0.2

0.0
1980

1985

1990

1995

2000

2005

2010

Figure 2c Smoothed Probabilities for Model D (FFR,EFP,SRET,HRET)
Regime 1
1.0

0.8

0.6

0.4

0.2

0.0
1980

1985

1990

1995

2000

2005

2010

Figure 2d Smoothed Probabilities for Model E (FFR,TED,SRET,HRET)
Regime 1
1.0

0.8

0.6

0.4

0.2

0.0
1980

1985

1990

1995

2000

2005

2010

Figure 2e Smoothed Probabilities for Model F (EFP,SPR,SRET,HRET)
Regime 1
1.0

0.8

0.6

0.4

0.2

0.0
1980

Figure 2f

1985

1990

1995

2000

2005

2010

Smoothed Probabilities for Model G (EFP,TED,SRET,HRET)
Regime 1

1.0

0.8

0.6

0.4

0.2

0.0
1980

1985

1990

1995

2000

2005

2010

Figure 2g Smoothed Probabilities for Model H (SPR,TED,SRET,HRET)
Regime 1
1.0

0.8

0.6

0.4

0.2

0.0
1980

1985

1990

1995

2000

2005

2010

Figure 3a Simulation-Based Out-of-Sample Forecasts of Stock Returns with 80-Percent Confidence Interval (CI) from 2006Q1-2006Q4 Basedon Information Available at 2005Q4
Model A: Single-Regime (FFR,SPR,TED,EFP,GDP,SRET,HRET); Model B: Two-Regime (FFR,GDP,SRET,HRET); Model C: Two-Regime(FFR,SPR,SRET,HRET); Model D: Two-Regime (FFR,EFP,SRET,HRET); Model E: Two-Regime (FFR,TED,SRET,HRET); Model F:
Two-Regime (EFP,SPR,SRET,HRET); Model G: Two-Regime (EFP,TED,SRET,HRET); Model H: Two-Regime (SPR,TED,SRET,HRET);
Model SUB: Single-regime, Uni-variate AR(1) Benchmark
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Model C
CI for Model C

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Model SUB
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Figure 3b Simulation-Based Out-of-Sample Forecasts of Stock Returns with 80-Percent Confidence Interval (CI) from 2007Q1-2007Q4 Basedon Information Available at 2006Q4
Model A: Single-Regime (FFR,SPR,TED,EFP,GDP,SRET,HRET); Model B: Two-Regime (FFR,GDP,SRET,HRET); Model C: Two-Regime(FFR,SPR,SRET,HRET); Model D: Two-Regime (FFR,EFP,SRET,HRET); Model E: Two-Regime (FFR,TED,SRET,HRET); Model F:
Two-Regime (EFP,SPR,SRET,HRET); Model G: Two-Regime (EFP,TED,SRET,HRET); Model H: Two-Regime (SPR,TED,SRET,HRET);
Model SUB: Single-regime, Uni-variate AR(1) Benchmark
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Model C
CI for Model C

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CI for Model H
Model UB
CI for Model SUB

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CI for Model G
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Figure 3c Simulation-Based Out-of-Sample Forecasts of Stock Returns with 80-Percent Confidence Interval (CI) from 2008Q1-2008Q4 Basedon Information Available at 2007Q4
Model A: Single-Regime (FFR,SPR,TED,EFP,GDP,SRET,HRET); Model B: Two-Regime (FFR,GDP,SRET,HRET); Model C: Two-Regime(FFR,SPR,SRET,HRET); Model D: Two-Regime (FFR,EFP,SRET,HRET); Model E: Two-Regime (FFR,TED,SRET,HRET); Model F:
Two-Regime (EFP,SPR,SRET,HRET); Model G: Two-Regime (EFP,TED,SRET,HRET); Model H: Two-Regime (SPR,TED,SRET,HRET);
Model SUB: Single-regime, Uni-variate AR(1) Benchmark
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CI for Model C

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Model SUB
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Figure 3d Simulation-Based Out-of-Sample Forecasts of Stock Returns with 80-Percent Confidence Interval (CI) from 2009Q1-2009Q4 Basedon Information Available at 2008Q4
Model A: Single-Regime (FFR,SPR,TED,EFP,GDP,SRET,HRET); Model B: Two-Regime (FFR,GDP,SRET,HRET); Model C: Two-Regime(FFR,SPR,SRET,HRET); Model D: Two-Regime (FFR,EFP,SRET,HRET); Model E: Two-Regime (FFR,TED,SRET,HRET); Model F:
Two-Regime (EFP,SPR,SRET,HRET); Model G: Two-Regime (EFP,TED,SRET,HRET); Model H: Two-Regime (SPR,TED,SRET,HRET);
Model SUB: Single-regime, Uni-variate AR(1) Benchmark

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CI for Model C

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Model H
CI for Model H
Model SUB
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Figure 3e Simulation-Based Out-of-Sample Forecasts of Stock Returns with 80-Percent Confidence Interval (CI) from 2010Q1-2010Q4 Basedon Information Available at 2009Q4
Model A: Single-Regime (FFR,SPR,TED,EFP,GDP,SRET,HRET); Model B: Two-Regime (FFR,GDP,SRET,HRET); Model C: Two-Regime(FFR,SPR,SRET,HRET); Model D: Two-Regime (FFR,EFP,SRET,HRET); Model E: Two-Regime (FFR,TED,SRET,HRET); Model F:
Two-Regime (EFP,SPR,SRET,HRET); Model G: Two-Regime (EFP,TED,SRET,HRET); Model H: Two-Regime (SPR,TED,SRET,HRET);
Model SUB: Single-regime, Uni-variate AR(1) Benchmark
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Model ACI for Model A

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Model C
CI for Model C

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Figure 3f Simulation-Based Out-of-Sample Forecasts of Stock Returns with 80-Percent Confidence Interval (CI) from 2011Q1-2011Q4 Basedon Information Available at 2010Q4
Model A: Single-Regime (FFR,SPR,TED,EFP,GDP,SRET,HRET); Model B: Two-Regime (FFR,GDP,SRET,HRET); Model C: Two-Regime(FFR,SPR,SRET,HRET); Model D: Two-Regime (FFR,EFP,SRET,HRET); Model E: Two-Regime (FFR,TED,SRET,HRET); Model F:
Two-Regime (EFP,SPR,SRET,HRET); Model G: Two-Regime (EFP,TED,SRET,HRET); Model H: Two-Regime (SPR,TED,SRET,HRET);
Model SUB: Single-regime, Uni-variate AR(1) Benchmark
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Model C
CI for Model C

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Model H
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Figure 4a Simulation-Based Out-of-Sample Forecasts of Housing Returns with 80-Percent Confidence Interval (CI) from 2006Q1-2006Q4
Based on Information Available at 2005Q4
Model A: Single-Regime (FFR,SPR,TED,EFP,GDP,SRET,HRET); Model B: Two-Regime (FFR,GDP,SRET,HRET); Model C: Two-Regime(FFR,SPR,SRET,HRET); Model D: Two-Regime (FFR,EFP,SRET,HRET); Model E: Two-Regime (FFR,TED,SRET,HRET); Model F:
Two-Regime (EFP,SPR,SRET,HRET); Model G: Two-Regime (EFP,TED,SRET,HRET); Model H: Two-Regime (SPR,TED,SRET,HRET);
Model SUB: Single-regime, Uni-variate AR(1) Benchmark
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Figure 4b Simulation-Based Out-of-Sample Forecasts of Housing Returns with 80-Percent Confidence Interval (CI) from 2007Q1-2007Q4
Based on Information Available at 2006Q4
Model A: Single-Regime (FFR,SPR,TED,EFP,GDP,SRET,HRET); Model B: Two-Regime (FFR,GDP,SRET,HRET); Model C: Two-Regime(FFR,SPR,SRET,HRET); Model D: Two-Regime (FFR,EFP,SRET,HRET); Model E: Two-Regime (FFR,TED,SRET,HRET); Model F:
Two-Regime (EFP,SPR,SRET,HRET); Model G: Two-Regime (EFP,TED,SRET,HRET); Model H: Two-Regime (SPR,TED,SRET,HRET);
Model SUB: Single-regime, Uni-variate AR(1) Benchmark
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Figure 4c Simulation-Based Out-of-Sample Forecasts of Housing Returns with 80-Percent Confidence Interval (CI) from 2008Q1-2008Q4
Based on Information Available at 2007Q4
Model A: Single-Regime (FFR,SPR,TED,EFP,GDP,SRET,HRET); Model B: Two-Regime (FFR,GDP,SRET,HRET); Model C: Two-Regime(FFR,SPR,SRET,HRET); Model D: Two-Regime (FFR,EFP,SRET,HRET); Model E: Two-Regime (FFR,TED,SRET,HRET); Model F:
Two-Regime (EFP,SPR,SRET,HRET); Model G: Two-Regime (EFP,TED,SRET,HRET); Model H: Two-Regime (SPR,TED,SRET,HRET);
Model SUB: Single-regime, Uni-variate AR(1) Benchmark
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Figure 4d Simulation-Based Out-of-Sample Forecasts of Housing Returns with 80-Percent Confidence Interval (CI) from 2009Q1-2009Q4
Based on Information Available at 2008Q4
Model A: Single-Regime (FFR,SPR,TED,EFP,GDP,SRET,HRET); Model B: Two-Regime (FFR,GDP,SRET,HRET); Model C: Two-Regime(FFR,SPR,SRET,HRET); Model D: Two-Regime (FFR,EFP,SRET,HRET); Model E: Two-Regime (FFR,TED,SRET,HRET); Model F:
Two-Regime (EFP,SPR,SRET,HRET); Model G: Two-Regime (EFP,TED,SRET,HRET); Model H: Two-Regime (SPR,TED,SRET,HRET);
Model SUB: Single-regime, Uni-variate AR(1) Benchmark
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CI for Model C

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Figure 4e Simulation-Based Out-of-Sample Forecasts of Housing Returns with 80-Percent Confidence Interval (CI) from 2010Q1-2010Q4
Based on Information Available at 2009Q4
Model A: Single-Regime (FFR,SPR,TED,EFP,GDP,SRET,HRET); Model B: Two-Regime (FFR,GDP,SRET,HRET); Model C: Two-Regime(FFR,SPR,SRET,HRET); Model D: Two-Regime (FFR,EFP,SRET,HRET); Model E: Two-Regime (FFR,TED,SRET,HRET); Model F:
Two-Regime (EFP,SPR,SRET,HRET); Model G: Two-Regime (EFP,TED,SRET,HRET); Model H: Two-Regime (SPR,TED,SRET,HRET);
Model SUB: Single-regime, Uni-variate AR(1) Benchmark
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Figure 4f Simulation-Based Out-of-Sample Forecasts of Housing Returns with 80-Percent Confidence Interval (CI) from 2011Q1-2011Q4 Basedon Information Available at 2010Q4
Model A: Single-Regime (FFR,SPR,TED,EFP,GDP,SRET,HRET); Model B: Two-Regime (FFR,GDP,SRET,HRET); Model C: Two-Regime(FFR,SPR,SRET,HRET); Model D: Two-Regime (FFR,EFP,SRET,HRET); Model E: Two-Regime (FFR,TED,SRET,HRET); Model F:
Two-Regime (EFP,SPR,SRET,HRET); Model G: Two-Regime (EFP,TED,SRET,HRET); Model H: Two-Regime (SPR,TED,SRET,HRET);
Model SUB: Single-regime, Uni-variate AR(1) Benchmark
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Losing track of the asset markets: the case of housing and stock (April 2015)
APPENDIX

We are grateful to all suggestions and feedback. In response, we provide more details and some
supplementary results in this appendix, which has different parts.

Appendix A provides the details of the estimation results.
Appendix B provides the non-stationarity test for the regime-switching models.
Appendix C provides the results of the Diebold-Mariano tests.
Appendix D provides the LM test results of the residuals from different models.
Appendix E discusses the suggestion of using Case-Shiller index.
Appendix F shows that the consideration of mortgage rate may not be necessary once our variables,such as FFR, and others are included.
Appendix G shows all the graphs of smoothed probabilities for different regimes in different
models.
Appendix H provides the detailed results of the Mariano-Preve test.
Appendix I provides a simple review of the literature which chooses a similar boom-bust date as wedo.
Appendix J provides the detailed results when the in-sample is chosen to be 1975 to 2006.
Appendix K provides the detailed results when the in-sample is chosen to be 1975 to 2007.

Appendix A: Details of the Estimation Results
Table A-1 The Estimation Results for (FFR,SPR,TED,EFP,GDP,SRET,HRET)(1975Q2-2012Q1)
Parameterc1
1(11)
1(12)
1(13)
1(14)
1(15)
1(16)
1(17)
 12c2
1(21)
1(22)
1(23)
1(24)
1(25)
1(26)
1(27)
 22c3
1(31)
1(32)
1(33)
1(34)
1(35)
1(36)
1(37)
 32

Estimate-0.0632
0.9521**-0.01850.0326-0.04730.17830.01500.1820**0.7801***0.12910.03240.8321-0.2874*
0.3409-0.0457-0.0127-0.1051
0.3765***0.18940.0422-0.04360.6628***-0.03160.0018-0.0055-0.02640.1421***

S.E.
1.11670.4351
1.27750.26164.01801.46500.01870.07790.26940.88100.21680.62910.17042.25040.72910.01190.06920.10490.19540.10030.29490.13020.71400.52300.00780.07210.0459(continued next page)Table A-1 The Estimation Results for (FFR,SPR,TED,EFP,GDP,SRET,HRET) (Continued)
Parameterc4
1(41)
1(42)
1(43)
1(44)
1(45)
1(46)
1(47)
 42c5
1(51)
1(52)
1(53)
1(54)
1(55)
1(56)
1(57)
 52c6
1(61)
1(62)
1(63)
1(64)
1(65)
1(66)
1(67)
 62

Estimate
0.2517**0.00460.00940.1281*0.6960***-0.0224-0.0091**-0.02650.0384***0.03990.06330.1251-0.4813***0.13900.21880.0154*0.1457***0.4258***-1.91810.1758-0.1731-2.1358
3.31050.94680.02730.445164.6998***

S.E.
0.12120.01150.04520.07480.0871
0.05860.00380.03390.01170.51920.10280.23070.15950.77880.31000.00850.05700.07773.62370.95941.60974.0102
3.07790.89780.5652
1.65909.2548(continued next page)Table A-1 The Estimation Results for (FFR,SPR,TED,EFP,GDP,SRET,HRET) (Continued)
Parameterc7
1(71)
1(72)
1(73)
1(74)
1(75)
1(76)
1(77)
 72

 12
 13
 14
 15
 16
 17
 23
 24
 25
 26
 27
 34
 35
 36
 37
 45
 46
 47
 56
 57
 67

lnL
AIC

Estimate
S.E.
0.2744
0.46640.03830.06920.17430.12380.31680.2622-0.6332
0.43300.05810.3275-0.00290.03980.6504***0.07400.8264***0.1203-0.7719***0.0692
0.39690.3766-0.0579
0.40580.2686**0.1194-0.11870.2400-0.0001
1.0282-0.0852
0.34000.1092
0.3428-0.1549
0.11050.01440.3184-0.1646
0.81970.4720***0.13590.06890.1646-0.2184*
0.1145-0.1768
0.4194-0.1759*
0.1030-0.13240.1342-0.0302
0.16330.0885
0.0967-0.00370.5499-0.07640.3802-813.4167
12.1273

Notes:*, **, and *** represent the significance at
10%, 5%, and 1%, respectively.

Table A-2 The Estimation Results for (FFR,GDP,SRET,HRET) and (FFR,SPR,SRET, HRET)(1975Q2-2012Q1)

Parameerc1
1 (2)
1(11)
1(12)
1(13)
1(14) 12
1 (2)c2
 2 (2)
1(21)
1(22)
1(23)
1(24) 22
2 (2)c3
 3 (2)
1(31)
1(32)
1(33)
1(34) 32
3 (2)

Model B (FFR,GDP,SRET,HRET)
Regime 1
Regime 2
Estimate
S.E.
Estimate
S.E.
0.64595.4864 0.6459
5.4864-0.79675.49170.8869***0.3217 0.9470***0.01840.06530.3550 0.3165***
0.0716
0.09830.1246 0.0037
0.00510.37510.7392 0.0998***
0.03063.7607***
1.1723 3.7607***
1.17230.2331***0.04002.41184.6724 2.4118
4.6724-2.1901
4.7050-0.18140.2976 0.0110
0.01790.00620.1014 0.4137***0.08180.00000.1247 0.0164***0.00590.10860.4495 0.1271***
0.04861.3203***0.4182 1.3203***
0.4182
0.4697***0.079920.1490
16.2770 20.1490
16.2770-19.9915
16.2654-1.15450.9472 0.1053
0.25950.47390.9429 0.8703
1.3462-0.1306
0.2787 0.0357
0.1091-1.8127
2.2297 0.4312
0.875044.4884*** 10.6552 44.4884***
10.6552
1.23310.1727

Model C (FFR,SPR,SRET,HRET)
Regime 1
Regime 2
Estimate
S.E.
Estimate
S.E.
3.370910.2522 3.3709
10.2522-3.7369
10.16840.68050.6246 1.0043***0.0261-0.4163
0.7737 0.0814
0.05030.01690.1570 0.0130***
0.00500.44751.4950 0.1530***
0.03262.9799***0.9583 2.9799***
0.95830.2608***0.0448-0.1800
6.0660 -0.1800
6.06600.53185.9436
0.06800.3528 -0.0204
0.0306
0.9184*0.4715 0.8897***
0.0537-0.0649
0.0827 0.0006
0.0060-0.16931.0029 -0.0765***
0.02890.9795***0.2850 0.9795***
0.28500.4455***0.07533.484941.7203 3.4849
41.7203-1.028137.9679-0.2134
3.2006 -0.1091
1.02651.0294
3.1672 -0.5515
1.36590.09270.6556 0.0214
0.14970.54222.3035 0.5266
1.089546.0042*** 11.5218 46.0042***
11.5218
1.22040.1890(continued next page)Table A-2 The Estimation Results for (FFR,GDP,SRET,HRET) and (FFR,SPR,SRET, HRET)(Continued)

Parameterc4 4 (2)Model B (FFR,GDP,SRET,HRET)
Regime 1
Regime 2
Estimate
S.E.
Estimate
S.E.
4.2564*** 1.5149 4.2564*** 1.5149-4.1994***

1.5257

Model C (FFR,SPR,SRET,HRET)
Regime 1
Regime 2
Estimate
S.E.
Estimate
S.E.
3.5685*2.06653.5685*2.0665-3.9907*2.41101(42)-0.2212**0.09550.02440.0338-0.16150.12560.10560.0700-0.0384

0.18870.27060.1837-0.2963

0.19370.16530.13631(44)-0.02620.0493-0.0104

0.01040.02450.0223-0.01290.01340.18010.20600.6453***0.0819-0.15840.36450.6856***0.07574 (2)
 12
 13
 14
 23
 24
 34
P11
P22

0.9707***0.33140.9707***0.33140.3865***0.1206

0.3865***0.1206

0.89450.1664

1.4978*0.2676-0.6622***0.03330.05980.40140.06600.0144-0.2153-0.0878

0.14220.59340.14310.1497

(41)
1

(43)
1

2
4

0.3477**-0.5269**

0.13770.20730.1335-0.0201

0.09190.0918-0.4137**0.04440.2043-0.0587-0.2109-0.0523-0.2987

0.39350.27780.15230.2067-0.05650.1670-0.0783-0.11450.08200.1126
0.09880.0986-0.0705
0.2199-0.0476-0.19900.31260.27530.31550.2056

0.9427***(0.0372)
0.9920***(0.0081)-0.9124***0.9505***(0.1053)
0.9914***(0.0086)

lnL-964.2407-883.3381
AIC
13.8681
12.7748
Note:*, **, and *** represent the significance at 10%, 5%, and 1%, respectively. Values next to thee estimates oftransition probability Pii are standard deviations.

Table A-3 The Estimation Results for (FFR,EFP,SRET,HRET) and (FFR, TED,SRET, HRET)(1975Q2-2012Q1)

Parameterc1
1 (2)


1(12)(11)
1

Model D (FFR,EFP,SRET,HRET)
Regime 1
Regime 2
Estimate
S.E.
Estimate
S.E.
0.40421.7058 0.4042
1.7058-0.2192

1.7147

Model E (FFR, TED,SRET, HRET)
Regime 1
Regime 2
Estimate
S.E.
Estimate
S.E.
0.54110.6876 0.5411
0.6876-0.5605

0.69791.0196***0.10520.9702***0.01581.0290***0.08690.9624***0.0177-0.84141.3212-0.1803**

0.0836-0.7432

0.56700.11140.12721(14)

0.0932*0.04760.00220.00420.04190.03270.00510.00610.30350.49720.1415***0.03040.30690.21980.0883***1 (2)c2
 2 (2)3.2435***1.08803.2435***1.08802.3484***0.81602.3484***0.81600.2274***0.04050.2827***0.05420.8576***0.24340.8576***-0.6579***0.24340.24880.7896***

0.29430.7896***-0.7080**0.29430.29391(21)

0.00430.01800.00800.00500.06330.0386

0.0434***0.00581(23)
1(24) 22
2 (2)c3
 3 (2)
1(31)
1(32)
1(33)
1(34) 32
3 (2)

0.6681***0.18150.7648***0.04490.34560.23570.4285***0.0386-0.01140.0074-0.0087***

0.0019-0.0180

0.0130-0.00310.0021-0.2027***

0.0674-0.00540.0116-0.0403

0.08880.00330.01080.0889***0.02090.0889***0.02090.3276***0.08400.3276***0.08400.4015***0.06450.2549***0.0370-16.2521**

8.2731 -16.2521**
15.7362*8.27318.7014-4.11224.5255-4.11225.67204.52554.9842-0.1352

0.67700.18440.2990-0.5253

0.6682

0.33630.45938.38466.65491.6192

2.21103.48094.1813-0.3749

2.9023-0.1241

0.29100.04360.1225-0.03700.32580.03370.20823.7744*2.20080.20710.6802

3.0437**1.4118-0.55400.651484.2574***

21.5064 84.2574***

21.5064

87.2822***21.3439 87.2822***

(13)
1

2
1

(22)
1

0.82360.131621.34390.7869*0.1130(continued next page)Table A-3 The Estimation Results for (FFR,EFP,SRET,HRET) and (FFR, TED,SRET, HRET)(Continued)
Model D (FFR,EFP,SRET,HRET)
Model E (FFR, TED,SRET, HRET)
Regime 1
Regime 2
Regime 1
Regime 2
Paramete
Estimate
S.E.
Estimate
S.E.
Estimate
S.E.
Estimate
S.E.
c4
2.0783**0.8594 2.0783**
0.8594-0.0424
1.1684 -0.0424
1.1684 4 (2)-1.6651*
0.94980.25781.2087(41)
0.0517
0.0619 0.0509
0.03650.13000.0985 0.0130
0.04911(42)-0.9692
0.7966 -0.3145
0.3204-0.4374
0.8638 0.2737
0.2132
1(43)
0.03130.0199 -0.0097
0.0131
0.04650.0326 -0.0156
0.01171(44)
0.07190.2222 0.7005
0.07980.4447*0.2385 0.6679***
0.0837
1
2
0.5933***0.1555 0.5933***
0.15551.3805***0.3431 1.3805***
0.3431
4
4 (2)
1.18500.1884
0.6610***0.1158 120.17110.2826 -0.3211***0.08780.4746***0.1487 0.4847***0.0704
 13-0.2408*0.1248 -0.0517
0.1037-0.2376
0.1477 -0.0350
0.1200 14-0.0453
0.3667 0.0817
0.0796-0.05850.2470 0.0261
0.1079 23
0.18780.1870 -0.1889**
0.0951-0.4594***
0.1466 0.1616
0.1015
 24-0.5396***0.1480 -0.0083
0.0514-0.10120.2397 -0.0629
0.1126
 34
0.10590.1863 -0.0678
0.09850.03810.3192 -0.1594*
0.0946P11
0.8060***(0.0675)
0.7824***(0.0839)
P22
0.9643***(0.0160)
0.9386***(0.0245)
lnL-752.4659-815.4470
AIC
11.0063
11.8574
Note:*, **, and *** represent the significance at 10%, 5%, and 1%, respectively. Values next to thee estimates oftransition probability Pii are standard deviations.

Table A-4 The Estimation Results for (EFP,SPR,SRET,HRET) and (EFP,TED,SRET, HRET)(1975Q2-2012Q1)

Parameterc1
1 (2)
1(11)
1(12)
1(13)
1(14) 12
1 (2)c2
 2 (2)
1(21)
1(22)
1(23)
1(24) 22
2 (2)c3
 3 (2)
1(31)
1(32)
1(33)
1(34) 32
3 (2)

Model F (EFP,SPR,SRET,HRET)
Regime 1
Regime 2
Estimate
S.E.
Estimate
S.E.
0.7296***
0.2636 0.7296***0.2636-0.5681**0.2657
0.6616***0.1039 0.9043***0.0251-0.03210.0340 -0.0329***0.0077-0.0151**
0.0071 -0.0065***0.0012-0.0974
0.1045 -0.0098
0.00750.1204***0.0334 0.1204***
0.03340.2456***
0.03650.49760.8508 0.4976
0.8508-0.5735
0.88850.10540.4325 0.1975
0.16790.7736***0.1259 0.9450***
0.0444-0.0465*
0.0261 0.0095
0.0058-0.0669
0.1764 -0.0967***
0.02900.8517***0.2923 0.8517***0.29230.4228***0.0835-17.0212***5.1265 -17.0212***5.126523.0667***
5.99878.0003***2.2391 -3.5786
2.45612.0457**0.8949 -0.3452
0.6991-0.03380.3840 -0.0114
0.14583.9087**
1.7890 -0.0885
0.492968.1342***
13.9435 68.1342***
13.9435
0.89300.1252Model G (EFP,TED,SRET, HRET)
Regime 1
Regime 2
Estimate
S.E.
Estimate
S.E.
0.5626***0.1730 0.5626***0.1730-0.4569***
0.17290.5378***0.0834 0.8877***0.03050.1708***0.0600 0.0112
0.0208-0.0217***0.0070 -0.0046***0.0017-0.0506
0.0575 -0.0026
0.01010.0854***0.0269 0.0854***
0.02690.3436***
0.05971.2040***0.3525 1.2040***
0.3525-0.9280**0.3642-0.4471***0.1503 -0.1385
0.08910.7754***0.1248 0.6267***
0.0647-0.0204
0.0133 -0.0013
0.00310.04950.0822 0.0330**
0.01450.3048***0.0665 0.3048***0.06650.3290***0.0436-12.7965**
5.6932 -12.7965**
5.693217.6252***
6.36579.1034***2.0602 -2.9172
3.3965-1.7698
2.0466 1.0652
1.94370.15510.2154 -0.0219
0.12792.1608*1.2348 -0.2343
0.619468.4217*** 19.6088 68.4217*** 19.6088
0.90300.1479(continued next page)Table A-4 The Estimation Results for (EFP,SPR,SRET,HRET) and (EFP,TED,SRET,HRET)(Continued)
Model F (EFP,SPR,SRET,HRET)
Model G (EFP,TED,SRET, HRET)
Regime 1
Regime 2
Regime 1
Regime 2
Parameter
Estimate
S.E.
Estimate
S.E.
Estimate
S.E.
Estimate
S.E.
c4
2.4474***0.59522.4474***0.59520.20720.5430 0.2072
0.5430 4 (2)-2.1522***0.71660.5174
0.5835(41)-0.9926***0.3252 -0.0989
0.4520-0.5901*
0.3463 -0.5057*
0.27621(42)-0.1516*0.08280.0516
0.07640.4704*0.2613 0.4723**
0.18911(43)
0.0497**0.0217 -0.0174
0.01410.0782*** 0.0243 -0.0223**
0.0113
1(44)
0.29810.19570.7640***0.08330.5742*** 0.1394 0.6261*** 0.0742
1
2
0.6971***0.21530.6971***0.21531.0832*** 0.2912 1.0832*** 0.2912
4
4 (2)
1.05850.20260.7435**0.1173 12-0.0547
0.21670.10030.12280.4363**0.1707 0.1741*
0.0976
 13-0.0258
0.1456 -0.1884**
0.09580.01790.1834 -0.0249
0.1074 14
0.06070.2749 -0.1275
0.09680.25120.1932 0.0519
0.1162
 23
0.17800.13020.00460.1788-0.4192*** 0.1168
0.2075**0.0872 24-0.0860
0.2871 -0.2094**
0.0945-0.0595
0.2125 0.0019
0.0904 34-0.16310.17870.06320.0859-0.1905
0.1616 -0.1067
0.0908P110.8226***(0.0807)
0.7952***(0.0789)
P22
0.9492***(0.0225)
0.9370***(0.0300)
lnL-723.5095-623.6579
AIC
10.6150
9.2656
Note:*, **, and *** represent the significance at 10%, 5%, and 1%, respectively. Values next to thee estimates oftransition probability Pii are standard deviations.

Table A-5 The Estimation Results for (SPR,TED,SRET,HRET)
Model H (SPR,TED,SRET,HRET)
Regime 1

Regime 2

Parameter

Estimate

S.E.

EstimateS.E.

c1

0.8605**0.35070.8605**0.3507-0.6475*

0.3671

1 (2)


0.7534***0.10140.9285***0.03181(13)-0.1560

0.2116-0.2563***0.0867-0.02850.01750.0119**0.0058-0.1326*

0.0769-0.0532

0.03371 (2)

0.8689***0.26580.8689***0.26580.3963***0.0635c2

0.9804***0.9804***0.2697-0.7302***0.2790

(11)
1
(12)
1

(14)
1
2
1

0.2697

 2 (2)-0.1413**0.0666-0.0393**

0.0164



0.6087**0.13510.6601***0.0593



0.01580.0096-0.0092***

0.0027
 22
2 (2)-0.09470.08010.01020.01380.3418***0.0756

0.3418***0.07560.2434***0.0308c3-5.5126-5.5126

3.60417.4321*4.2638

(21)
1
(22)
1
(23)
1
(24)
1

3.6041 3 (2)


0.3514

1.2145-0.26850.64181(33)
1(34) 32
3 (2)

0.39681.7197

4.1531***1.0871-0.1298

0.1506-0.0537

0.12303.0544***0.8198-0.69500.629478.2822***

16.5663 78.2822***

16.5663

0.7828***0.1073

(31)
1
(32)
1

(continued next page)Table A-5 The Estimation Results for (SPR,TED,SRET,HRET)
Model H (SPR,TED,SRET,HRET)
Regime 1

Regime 2

Parameter

Estimate

S.E.

EstimateS.E.

c4

0.20280.58190.20280.5819-0.1368

0.6656

 4 (2)
-0.02210.22580.07730.06381(43)

0.06520.15700.3256*0.17780.00170.0346-0.0127

0.0135
 42
4 (2)

0.6925***0.19250.6871***

0.0912

1.5297***0.4401

1.5297***0.44010.6229***0.1214

(41)
1
(42)
1

(44)
1

 12
 13
 14
 23
 24
 34

-0.1347

0.2236-0.08010.12040.22230.1448-0.0338

0.1339-0.13560.2213-0.2543***0.0980-0.17230.17190.05550.1070-0.2133

0.14220.00610.1816-0.1038

0.1396-0.1427

0.0905P11

0.8478***(0.0638)

P22

0.9447***(0.0230)

ln L-807.2657

AIC
11.7468
Note:*, **, and *** represent the significance at10%, 5%, and
1%, respectively. Values next to the estimates of transition
probability Pii are standard deviations.

Table A-6 The Estimation Results for Model SUB
Model SUB
Stock Return

Housing Return

Parameter

Estimate

S.E.

EstimateS.E.

c1.8024**0.73250.3476***0.12030.05840.09160.6807***0.072366.6180***

9.38120.8874***0.1377

1


2

ln L

-520.7270-201.1615

AIC
7.07742.7589Note: **, and *** represent the significance at 5%, and 1%,
respectively.

Appendix B: Stationarity of different Regime-Switching models

Table B-1 Stationarity Results of Francq and Zakoian (2001)
Model

Model Structure

Variables

Spectral RadiusB

Two-regime

FFR, GDP, SRET, HRET 0.942

C

Two-regime

FFR, SPR, SRET, HRET

1.013

D

Two-regime

FFR, EFP, SRET, HRET

1.015

E

Two-regime

FFR, TED, SRET, HRET 0.992

F

Two-regime

EFP, SPR, SRET, HRET

0.834

G

Two-regime

EFP, TED, SRET, HRET

0.769

H

Two-regime

SPR, TED, SRET, HRET 0.852

Key: (unless specified, all variables refer to quarterly data) FFR, Federal Fund Rate; SPR, term
spread, which is equal to (10-year bond rate – FFR); TED, TED spread, which is equal to (3-month
Eurodollar deposit rate - 3-month T-bill rate), a measure of market liquidity; EFP, External FinancePremium, which is equal to corporate bond spread (Baa-Aaa), a measure of External FinancePremium; GDP, GDP growth rate; SRET, Stock Market Return; HRET, Housing Market Return.
It is clear that Model B and C are (marginally) non-stationary.

Appendix C: Results from Diebold Mariano tests
In the main text, we use the multi-lateral model comparison test of Mariano and Preve (2012). Since
the literature is full of examples of using the more traditional bilateral model comparison test of
Diebold and Mariano (1995), we also produce the corresponding results to facilitate thecomparison.
Table C-1 A Summary of Diebold and Mariano (1995) Statistics (In-sample 4-Quarter Ahead
Forecasts)
Stock Returns

Housing Returns

SLC

ALC

SLC

ALC

Model A

Single-regime model (FFR,
SPR, TED, EFP, GDP, SRET,
HRET)-0.3097-0.4418-0.9133-1.0650

Model BTwo-regime model (FFR, GDP,
SRET, HRET)-0.4282-0.2274-1.9474*-2.8435*

Model C

Two-regime model (FFR, SPR,
SRET, HRET)

---

---

-0.7190-2.5876*

Model D

Two-regime model (FFR, EFP,
SRET, HRET)-1.0046-0.4647-0.0825-0.8596

Model E

Two-regime model (FFR, TED,SRET, HRET)-0.9847-0.3806------

Model F

Two-regime model (EFP, SPR,
SRET, HRET)-1.7738*-1.0419-0.6186-1.1555

Model G

Two-regime model (EFP, TED,SRET, HRET)-1.3291-1.7457*-1.7517*-1.4553

Model H

Two-regime model (SPR, TED,SRET, HRET)-0.7629-0.1499-0.7542-0.6536

SUB

Single-regime, Uni-variate
AR(1) Benchmark-0.5010-0.5293-1.8833*-1.4664

Key: SLC: Square Loss Criteria; ALC: Absolute Loss Criteria.
Note: For stock returns, the DM test is used to compare the forecasting ability for model C andthe competing model. For housing returns, the DM test is used to compare the forecasting
ability for model E and the competing model. * Significant at 10% level of significance.
For stock returns, (1)For SLC, Model C is better than Model F(2)For ALC, Model C is better than Model G
For housing returns, (1)For SLC, Model E is better than Models B, G and SUB.
(2)For ALC, Model E is better than Models B and C.

Table C-2 A Summary of Diebold and Mariano (1995) Statistics (Out-of-sample 4-Quarter Ahead
Forecasts)
Stock Returns

Housing Returns

SLC

ALC

SLC

ALC

Model A

Single-regime model (FFR,
SPR, TED, EFP, GDP, SRET,
HRET)-1.5689-1.6794*-2.4232*-2.3882*

Model BTwo-regime model (FFR, GDP,
SRET, HRET)-1.9558*-2.8362*-3.2540*-4.4689*

Model C

Two-regime model (FFR, SPR,
SRET, HRET)-4.3008*-3.6151*-0.9058-2.1382*

Model D

Two-regime model (FFR, EFP,
SRET, HRET)-1.8491*-2.1131*------

Model E

Two-regime model (FFR, TED,SRET, HRET)-0.9668-1.1528-1.8916*-1.8734*

Model F

Two-regime model (EFP, SPR,
SRET, HRET)-2.0955*-2.3682*-0.5494-0.7063

Model G

Two-regime model (EFP, TED,SRET, HRET)-1.3451-0.7321-1.5640-1.3771

Model H

Two-regime model (SPR, TED,SRET, HRET)

---

---

-1.3386-1.2733

SUB

Single-regime, Uni-variate
AR(1) Benchmark-0.8101-0.3555-2.2557*-1.8146*

Key: SLC: Square Loss Criteria; ALC: Absolute Loss Criteria.
Note: For stock returns, the DM test is used to compare the forecasting ability for model H andthe competing model. For housing returns, the DM test is used to compare the forecasting
ability for model D and the competing model. * Significant at 10% level of significance.
For stock returns,(1)For SLC, Model H is better than Models B, C, D, F.
(2)For ALC, Model H is better than Models A, B, C, D, F.
For housing returns,(1)For SLC, Model D is better than Models A, B, E, SUB.
(2)For ALC, Model D is better than Models A, B, C, E, SUB

Appendix D: LM test for the residuals
In this appendix, we report the multivariate autocorrelation LM test (Johansen 1995: 22) for
standardized residuals. The null hypothesis is “no serial correlation for standardized residuals”.

Table ?

A Summary of Residual Tests for All Eight Models (1975Q2-2012Q1)
ModelsLM test(lag=10)Model A

Single-regime model (FFR, SPR, TED, EFP, GDP, SRET, HRET)

64.014*Model B
Model C
Model D
Model E

Two-regime model (FFR, GDP, SRET, HRET)
Two-regime model (FFR, SPR, SRET, HRET)
Two-regime model (FFR, EFP, SRET, HRET)
Two-regime model (FFR, TED, SRET, HRET)

16.25918.97821.07424.779*Model F
Model G
Model H

Two-regime model (EFP, SPR, SRET, HRET)
Two-regime model (EFP, TED, SRET, HRET)
Two-regime model (SPR, TED, SRET, HRET)

19.18424.992*19.550Note: The figure refers to the multivariate autocorrelation LM test of Johansen (1995) for
standardized residuals. *represents the significance at 10%.

Reference:

Johansen, S. 1995. Likelihood-based Inference in Cointegrated Vector Autoregressive Models. New
York: Oxford University Press.

Appendix E: the suggestion of using Case-Shiller Index
It has been suggested to us of employing the Case-Shiller Index (CSI) instead of the FHFA data
series. First, we would like to plot the two series for a comparison. Clearly, Case-Shiller index tends
to be more volatile, perhaps due to its focus on the top 30 cities, or even the top 10 cities. As
discussed in Glaeser and Gyourko (2008), among others, conclusions being drawn from those “topcities” may not be the same as the one from studies using all 102 metropolitan areas.
6
4
2
0
-2
-4
-6
-8
1980

1985

1990
HRET

1995

2000

2005

2010

CASH_SHILLER

In principle, we would like to employ the CSI at least as a robustness check. Unfortunately, the CSI
is available from 1987:Q1, which is much shorter than the FHFA series. And to employ much
shorter time series for multivariate study probably involves a switch in econometric technique andtherefore we leave that for future research.
We also find that the correlation coefficient between FHFA and S&P Case-Shiller housing returns is0.494 (1987:Q1-2012Q1), and thus the use of different indices may potentially generate different
results.

Reference:

Glaeser, E. and J. Gyourko, 2008, Rethinking Federal Housing Policy: how to make housing
plentiful and affordable, Washington, D.C.: American Enterprise Institute.

Appendix F: the inclusion of Mortgage Rate may not be necessary once our listed variables areincluded.
It has been suggested to us that introducing the 30 year Mortgage Rate may improve the results.
This appendix shows that it might not be the case. We first shows that the 30 Mortgage Rate, whichis available from online, http://research.stlouisfed.org/fred2/series/MORTG, is highly correlatedwith other varibles.
The solid line represents the 30-Year Conventional Mortgage Rate (MORTG30) (Not Seasonally
Adjusted) and the dotted line represents the FFR (Federal Fund Rate).

20

MORTG30
FFR

16

12

8

4

0
1980

MORTG30
FFR
EFP
SPR
TED

1985

MORTG30
1.000000
0.921546
0.441285-0.267886
0.686500

1990

1995

2000

2005

2010

The correlation coefficient matrix
FFR
EFP
SPR
0.921546
0.441285-0.267886
1.000000
0.289963-0.608354
0.289963
1.000000
0.096856-0.608354
0.096856
1.000000
0.722262
0.638450-0.410358

TED
0.686500
0.722262
0.638450-0.410358
1.000000

The correlation between FFR and MORTG30 is 0.921546.

Furthermore, we run a supplementary regression to show that Mortgage rate is mostly explained byour listed variables (FFR, EFP, SPR, TED). Hence, the introduction of it in the empirical analysis
will not generate additional information.
More specifically, we run the following regression:(Mortgage Rate) = a0+ a1*(SPR, Term Spread) + a2*(TED Spread) + a3*(EFP)+a4*(FFR)+(residual)
Table F1 Regression results
Dependent Variable: MORTG30
Method: Least Squares
Date: 08/19/13
Time: 16:39
Sample: 1975Q2 2012Q1
Included observations: 148
Variable

Coefficient

Std. Error

t-Statistic

Prob.

C
SPR
TED
EFP
FFR

1.186891
0.971697-0.269824
0.604615
0.965193

0.096209
0.027584
0.069512
0.085181
0.011652

12.33661
35.22738-3.881673
7.098020
82.83384

0.00000.00000.00020.00000.0000R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat0.989211
0.988909
0.321806
14.80891-39.65616
0.746341

Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)

8.826689
3.055711
0.603462
0.704719
3277.810
0.000000

Appendix G: The graphs of smoothed probabilities for different regimes in different models
Figure G-1 Smoothed Probabilities for Model B (FFR, GDP,SRET,HRET)
Regime 1

Regime 2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

1980

1985

1990

1995

2000

2005

2010

1980

1985

1990

1995

2000

2005

2010

Figure G-2 Smoothed Probabilities for Model C (FFR,SPR,SRET,HRET)
Regime 1

Regime 2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

1980

1985

1990

1995

2000

2005

2010

1980

1985

1990

1995

2000

2005

2010

Figure G-3 Smoothed Probabilities for Model D (FFR,EFP,SRET,HRET)
Regime 1

Regime 2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

1980

1985

1990

1995

2000

2005

2010

1980

1985

1990

1995

2000

2005

2010

Figure G-4 Smoothed Probabilities for Model E (FFR,TED,SRET,HRET)
Regime 1

Regime 2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

1980

1985

1990

1995

2000

2005

2010

1980

1985

1990

1995

2000

2005

2010

2005

2010

Figure G-5 Smoothed Probabilities for Model F (EFP,SPR,SRET,HRET)
Regime 1

Regime 2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

1980

1985

1990

1995

2000

2005

2010

1980

1985

1990

1995

2000

Figure G-6 Smoothed Probabilities for Model G (EFP,TED,SRET,HRET)
Regime 2

Regime 1
1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0
1980

1985

1990

1995

2000

2005

2010

1980

1985

1990

1995

2000

2005

2010

Figure G-7 Smoothed Probabilities for Model H (SPR,TED,SRET,HRET)

Regime 1

Regime 2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0
1980

1985

1990

1995

2000

2005

2010

1980

1985

1990

1995

2000

2005

2010

Appendix H:
This appendix provides more details of the results which have been summarized in the text.

Table H-1. A Summary of Mariano and Preve (2011) Statistics for Stock Returns(In-sample 4-Quarter Ahead Forecasts)
Selected models

S statistic

Sc statistic

Panel A: Squared loss criterion
A, B, C, D, E, F, G, H, USB
7.2842
7.2230Panel B: Absolute loss criterion
A, B, C, D, E, F, G, H, USB
7.97277.9057
Note: The test statistics S and Sc are used to compare the forecasting ability for various
competing models. * Significant at 10% level of significance.
Conclusion: All models have the same forecasting ability.

Table H-2. A Summary of Mariano and Preve (2011) Statistics for Housing Returns(In-sample 4-Quarter Ahead Forecasts)
Selected models
Panel A: Squared loss criterion
A, B, C, D, E, F, G, H, USB
A, B, C, D, E, F, G, H
A, C, D, E, F, G, H
A, C, D, E, F, H
C, D, E, F, H
D, E, F, H
A, B, C, G, USB
A, B, C, G
A, C, G
B, USB
Panel B: Absolute loss criterion
A, B, C, D, E, F, G, H, USB
A, C, D, E, F, G, H, USB
A, C, D, E, F, G, H
A, C, D, E, F, H
B, G, USBS statistic

Sc statistic

59.4120*42.3268*
14.7568*
12.6766*
12.6730*
5.754213.3869*
10.9644*2.46970.041258.9127*
41.9711*
14.6328*
12.5701*
12.5665*
5.705913.2744*
10.8722*2.44890.040842.6523*
17.0562*
12.1324*
6.6256
1.933442.2939*
16.9129*12.0305*
6.56981.9172

Note: The test statistics S and Sc are used to compare the forecasting ability for various
competing models. * Significant at 10% level of significance.
Conclusion: (1) For squared loss criterion:
Group 1: Models D, E, F, H have the same forecasting abilityGroup 2: Models A, C, G have the same forecasting abilityGroup 3: Models B, USB have the same forecasting ability
The order of forecasting performance: Group 1 > Group 2 > Group 3(2) For absolute loss criterion:
Group 1: Models A, C, D, E, F, H have the same forecasting abilityGroup 2: Models B, G, USB have the same forecasting ability
The order of forecasting performance: Group 1 > Group 2

Table H-3. A Summary of Mariano and Preve (2011) Statistics for Stock Returns (Out-of-sample
4-Quarter Ahead Forecasts)
Selected models
S statistic
Sc statistic
Panel A: Squared loss criterion
A, B, C, D, E, F, G, H, USB
50.5384*
48.2412*A, B, C, D, E, G, H, USB
33.7668*
32.2319*
A, B, C, E, G, H, USB
26.6326*
25.4220*
A, B, C, E, H, USB
8.01837.6538D, F, G
2.0174
1.9257
Panel B: Absolute loss criterion
A, B, C, D, E, F, G, H, USB
67.7885*
64.7072*
A, B, C, D, E, G, H, USB
40.7105*38.8600*
A, B, C, E, G, H, USB
23.9001*
22.8138*
B, C, E, G, H, USB
16.9517*
16.1812*B, E, G, H, USB
4.82074.6016
A, C, D, F
0.44620.4259Note: The test statistics S and Sc are used to compare the forecasting ability for various
competing models. * Significant at 10% level of significance.

Conclusion: (1) For squared loss criterion:
Group 1: Models A, B, C, E, H, USB have the same forecasting abilityGroup 2: Models D, F, G have the same forecasting ability
The order of forecasting performance: Group 1 > Group 2(2) For absolute loss criterion:
Group 1: Models B, E, G, H, USB have the same forecasting abilityGroup 2: Models A, C, D, F have the same forecasting ability
The order of forecasting performance: Group 1 > Group 2

Table H-4. A Summary of Mariano and Preve (2011) Statistics for Housing Returns (Out-of-sample
4-Quarter Ahead Forecasts)
Selected models
S statistic
Sc statistic
Panel A: Squared loss criterion
A, B, C, D, E, F, G, H, USB
18.0523*
17.2317*
A, B, C, D, F, G, H, USB
15.5727*
14.8649*
B, C, D, F, G, H, USB
15.4347*
14.7331*
C, D, F, G, H, USB
7.08356.7615A, B, E
0.28080.2680Panel B: Absolute loss criterion
A, B, C, D, E, F, G, H, USB
23.1800*
22.1263*
A, B, C, D, F, G, H, USB
21.5041*
20.5266*
A, C, D, F, G, H, USB
13.5882*
12.9705*
C, D, F, G, H, USB
6.30116.0146
A, B, E
0.29110.2779Note: The test statistics S and Sc are used to compare the forecasting ability for various
competing models. * Significant at 10% level of significance.
Conclusion: (1) For squared loss criterion:
Group 1: Models C, D, F, G, H, USB have the same forecasting abilityGroup 2: Models A, B, E have the same forecasting ability(2) For absolute loss criterion:
Group 1: Models C, D, F, G, H, USB have the same forecasting abilityGroup 2: Models A, B, E have the same forecasting ability
The order of forecasting performance: Group 1 > Group 2. Thus, whether we useThe thesquared loss criterion or absolute loss criterion, havethe same ranking is delivered.

Appendix I:
This appendix provides a simple review of the literature which chooses a similar boom-bust date as
we do.
AS it can be expected, the dating of bust regime varies across research papers, reflecting thedifference in modelling techniques as well as research focuses. Nevertheless, there seems to be acollection of studies which also select the beginning of 2006 as the start of bust phase of the cycle.
1. Huang (2014) extends Galvão’s (2006) SBTVAR framework and shows that a persistenthousing-bust phase in US begins in March 2006. This finding is in line with prior studies that
identify 2006 as the starting period of the recent housing bust in the U.S.
2. Ferreira, Gyourko, and Tracy (2010) point out that loan-to-value ratios of some local housing
markets are higher than 85% in 2006, and specify that the housing bust starts in 2006.
3. Huang and Tang (2012) argue that tight housing supply constraints, which result in supply
inelasticity, magnify the recent housing price bust since 2006.
4. In’t-Veld, Raciborski, Ratto, and Roeger (2011) propose that the decline in residential
investment since the early-2006 led to the economic recession, and identify the bust of the
housing bubble in 2006.
5. Khandani, Lo, and Merton (2013) estimate the potential losses of the US housing market due toa housing price bust from 2006.
6. Lutz, Molloy, and Shan (2011) address five channels through which the US housing market
influences state and local tax revenues. They provide estimates of a fall in sales tax revenue
from 2006 to 2009 and find that housing prices peak in the first quarter of 2006, followed by acollapse of housing price growths.
7. Nichols, Oliner, and Mulhall (2013) show that residential and commercial land price indexes
plunge after peaking in 2006.
8. Chen et al. (2014): According to FHFA house price index, the U.S. average house price hasgrown consecutively around 1.6% for the period 1995Q4 – 2005Q4, reaching a total of 89% in
capital gains. The returns on housing started to decline around 2006q1 and then precipitated in
the following quarters when the subprime mortgage problem aggravated. Using RW model, they
identify the boom-bust regime capturing the periods where housing returns exhibited highvolatility before mid-1980s and after 2005q4.
9. Kouwenberg and Zwinkels (2014): The real house price index reached a maximum in the firstquarter of 2006, when it was 48% higher than the fundamental value estimate, and theprolonged period of overvaluation ended in the first quarter of 2009, as the price eventually
dropped below the fundamental value.
10. Cetorelli and Goldberg (2012) show that internal capital markets of global banks also contributeto the international propagation of shocks. Using balance sheets data for chartered US banks
over the period 1980-2009, the sample is split over two samples, a pre-crisis period using
quarterly data on individual banks from “Call reports” from 1980Q1 to 2005Q4, and a crisis
period sample using weekly series on aggregate Assets and Liabilities of Commercial Banks in
the United States (Fed. H.8 Statistical Release).
11. Cetorelli and Goldberg (2010) confirm that global banks played a significant role in the crisis
transmission to emerging market economies. Using banking statistics data from the BIS and
IMF’s International Financial Statistics for 17 source countries from the OECD and 94
destination countries from emerging Europe, Asia, and Latin from 2006 to 2009 they examine
the relationships between adverse liquidity shocks on main developed-country banking systems
to emerging markets, isolating loan supply from loan demand effects. The sample is split in to apre-crisis period from 2006Q2 to 2007Q2 and a post-crisis period from 2008Q3 to 2009Q2,

leaving out the crisis event period between 2007Q3 and 2008Q2 when market functioning was
disrupted by the Lehman shock. Alternative datings were considered in robustness tests.
12. Bertay and Gong (2014) use an international panel to analyze the relationship between
securitization and macroeconomic performance. Our findings suggest that economic growth isnegatively related to country-level securitization activities, even before the 2007-2009 financial
crisis. The annual sample is split into 1995-2006 and 2006-2012.
13. Shiller (2007): Between the bottom in the 4th quarter of 1996 and the peak in the first quarter of
2006, real home prices rose 86% nationally in the United States.

Appendix J
This appendix provides the detailed results when 1975-2006 is chosen to be the in-sample period
and the years after are categorized as out-of-sample.

Table J-1

A Summary of Mariano and Preve (2011) Statistics for Stock
Returns (In-sample 4-Quarter Ahead Forecasts)

Selected models

S statistic

S c statistic

Panel A: Squared loss criterion
A, B, C, D, E, F, G, H, USB
12.0249
11.9271
Panel B: Absolute loss criterion
A, B, C, D, E, F, G, H, USB
14.0797*
13.9653
A, B, C, D, E, F, G, USB
7.14307.0850Note: The test statistics S and S c are used to compare the forecasting ability for various
competing models. * Significant at 10% level of significance.

Conclusion: (1) For squared loss criterions:
All models have the same forecasting ability.
(2) For absolute loss criterion:
Group 1: A, B, C, D, E, F, G, USB
Group 2: H
The order of forecasting performance: Group 1 > Group 2.

Table J-2
Selected models
Panel A: Squared loss criterion
A, B, C, D, E, F, G, H, USB
A, B, C, D, E, F, G, H
A, C, D, E, F, G, H
B, USB
Panel B: Absolute loss criterion
A, B, C, D, E, F, G, H, USB
A, C, D, E, F, G, H, USB

A Summary of Mariano and Preve (2011) Statistics for Housing Returns(In-sample 4-Quarter Ahead Forecasts)
S statistic

S c statistic

21.6824*
21.4527*
7.98310.009921.5062*
21.2782*
7.91820.009819.2663*9.801219.1096*
9.7216

Note: The test statistics S and S c are used to compare the forecasting ability for various competing models. * Significant at 10% level ofsignificance.
Conclusion: (1) For squared loss criterion:
Group 1: Models A, C, D, E, F, G, H have the same forecasting abilityGroup 2: Models B, USB have the same forecasting ability
The order of forecasting performance: Group 1 > Group 2(2) For absolute loss criterion:
Group 1: Models A, C, D, E, F, G, H, USB have the same forecasting abilityGroup 2: Model B
The order of forecasting performance: Group 1 > Group 2.

Table J-3
Selected models

A Summary of Mariano and Preve (2011) Statistics for Stock Returns(Out-of-sample 4-Quarter Ahead Forecasts)
S statistic

S c statistic

Panel A: Squared loss criterion
A, B, C, D, E, F, G, H, USB
81.8845*
77.3353*
A, B, C, D, E, G, H, USB
44.3031*41.8418*
A, B, C, E, G, H, USB
31.1086*29.3803*
A, B, C, E, H, USB
7.79457.3615
D, F, G
2.23462.1105
Panel B: Absolute loss criterion
A, B, C, D, E, F, G, H, USB
99.5143*
93.9857*
A, B, C, D, E, G, H, USB
50.8986*
48.0709*
A, B, C, E, G, H, USB
24.1955*
22.8513*
B, C, E, G, H, USB
15.5978*
14.7313*
B, E, G, H, USB
4.50194.2518
A, C, D, F
0.54390.5137
Note: The test statistics S and S c are used to compare the forecasting ability for various competing models. * Significant at 10% level ofsignificance.
Conclusion: (1) For squared loss criterion:
Group 1: Models A, B, C, E, H, USB have the same forecasting abilityGroup 2: Models D, F, G have the same forecasting ability
The order of forecasting performance: Group 1 > Group 2(2) For absolute loss criterion:
Group 1: Models B, E, G, H, USB have the same forecasting abilityGroup 2: Models A, C, D, F have the same forecasting ability
The order of forecasting performance: Group 1 > Group 2.

Table J-4
Selected models

A Summary of Mariano and Preve (2011) Statistics for Housing Returns(Out-of-sample 4-Quarter Ahead Forecasts)
S statistic

S c statistic

Panel A: Squared loss criterion
A, B, C, D, E, F, G, H, USB
25.0090*
23.6196*
A, B, C, D, F, G, H, USB
16.8688*
15.9316*
B, C, D, F, G, H, USB
16.8638*
15.9270*
C, D, F, G, H, USB
12.8406*
12.1272*
C, D, F, G, H
11.1509*10.5314*C, D, F, G
4.76034.4958A, B, E, H, USB
6.12485.7846
Panel B: Absolute loss criterion
A, B, C, D, E, F, G, H, USB
25.6580*
24.2326*
A, B, C, D, F, G, H, USB
25.4794*24.0639*
A, C, D, F, G, H, USB
17.8622*16.8699*
C, D, F, G, H, USB
15.8284*
14.9490*
C, D, F, G, H
11.9081*11.2465*
C, D, F, G
3.27593.0939A, B, E, H, USB
2.40582.2721
Note: The test statistics S and S c are used to compare the forecasting ability for various competing models. * Significant at 10% level ofsignificance.

Conclusion: (1) For squared loss criterion:
Group 1: Models C, D, F, G have the same forecasting abilityGroup 2: Models A, B, E, H, USB have the same forecasting ability(2) For absolute loss criterion:
Group 1: Models C, D, F, G have the same forecasting abilityGroup 2: Models A, B, E, H, USB have the same forecasting ability.

Appendix K
This appendix provides detailed results when the in-sample is chosen to be 1975 to 2007. We start with some summary tables, followed by thedetails.

Table K-1 A Summary of In-sample Forecasting Performances (4-Quarter Ahead Forecasts) (1975Q2-2007Q4)
Stock Returns
SLC
ALC
Model A

Single-regime model (FFR, SPR, TED, EFP, GDP,
SRET, HRET)
Model BTwo-regime model (FFR, GDP, SRET, HRET)
Model C
Two-regime model (FFR, SPR, SRET, HRET)
Model D
Two-regime model (FFR, EFP, SRET, HRET)
Model E
Two-regime model (FFR, TED, SRET, HRET)
Model F
Two-regime model (EFP, SPR, SRET, HRET)
Model G
Two-regime model (EFP, TED, SRET, HRET)
Model H
Two-regime model (SPR, TED, SRET, HRET)USB
Uni-variate, Single-regime Benchmark
Key: SLC: Square Loss Criteria; ALC: Absolute Loss Criteria

Housing Returns
SLC
ALC

54.84495.50730.80250.661554.9900
54.161455.364755.103157.304456.1486
56.106155.09035.48555.44085.50245.52935.59945.56625.61805.52220.86830.73890.77550.7726
0.73120.80900.79420.86760.7154
0.66290.66800.65100.65200.6634
0.66200.6835

Table K-2. A Summary of EPPC (In-sample 4-Quarter Ahead Forecasts)

Stock Return

SLC
ALC
SLC
ALC

Class 1
All models
All models

Class 2//

A, C, D, E, F, G, H
B, USB
A, C, D, E, F, G, H, USB
B
Note: SLC: Squared Loss Criterion; ALC: Absolute Loss Criterion. Our convention is that if i  j ,then any model in Class i has better predictive power than any model in Class j .

Housing ReturnTable K-3 A Summary of Out-of-Sample Forecasting Performances (4-Quarter Ahead Forecasts)(2008Q4-2012Q1)
Stock Returns
SLC
ALC
Model A

Single-regime model (FFR, SPR, TED, EFP, GDP,
SRET, HRET)
Model BTwo-regime model (FFR, GDP, SRET, HRET)
Model C
Two-regime model (FFR, SPR, SRET, HRET)
Model D
Two-regime model (FFR, EFP, SRET, HRET)
Model E
Two-regime model (FFR, TED, SRET, HRET)
Model F
Two-regime model (EFP, SPR, SRET, HRET)
Model G
Two-regime model (EFP, TED, SRET, HRET)
Model H
Two-regime model (SPR, TED, SRET, HRET)USB
Uni-variate, Single-regime Benchmark
Key: SLC: Square Loss Criteria; ALC: Absolute Loss Criteria.

Housing Returns
SLC
ALC

170.5441

11.6253

4.0734

1.6796157.0917
161.3897
169.8544
153.8335
172.0982
166.1691
142.7175
150.2610

10.8966
11.2335
11.5395
10.5620
11.2934
10.461210.212210.3168

3.73552.78912.65154.4722
3.17793.31973.63993.8935

1.6366
1.39271.3177
1.7894
1.46851.50391.63681.6767

Table K-4

Stock Return
Housing Return

SLC
ALC
SLC
ALC

A Summary of EPPC (Out-Of-Sample 4-Quarter Ahead Forecasts)
Class 1
B, C, E, H, USB
B, E, G, H, USB
C, D, F, G
C, D, F, G

Class 2
A, D, F, G
A, C, D, F
A, B, E, H, USB
A, B, E, H, USB

Note: SLC: Squared Loss Criterion; ALC: Absolute Loss Criterion. Our convention is that if i  j ,then any model in Class i has better predictive power than any model in Class j .

We will now present all the details which support the previous summary tables.

Table K-5 A Summary of In-sample Forecasting Performances (4-Quarter Ahead Forecasts)(1975Q2-2007Q4)

Model A
Model B
Model C
Model D
Model E
Model F
Model G
Model HUSB

Single-regime model (FFR, SPR, TED, EFP, GDP,
SRET, HRET)
Two-regime model (FFR, GDP, SRET, HRET)
Two-regime model (FFR, SPR, SRET, HRET)
Two-regime model (FFR, EFP, SRET, HRET)
Two-regime model (FFR, TED, SRET, HRET)
Two-regime model (EFP, SPR, SRET, HRET)
Two-regime model (EFP, TED, SRET, HRET)
Two-regime model (SPR, TED, SRET, HRET)
Uni-variate, Single-regime Benchmark

Stock Returns
MSE
MAE

Housing Returns
MSE
MAE

54.84495.50730.80250.661554.9900
54.161455.364755.103157.304456.1486
56.106155.09035.48555.44085.50245.52935.59945.56625.61805.52220.86830.73890.77550.7726
0.73120.80900.79420.86760.7154
0.66290.66800.65100.65200.6634
0.66200.6835

Table K-6 A Summary of Mariano and Preve (2011) Statistics for Stock Returns (In-sample 4-Quarter
Ahead Forecasts)
Selected models

S statistic

S c statistic

Panel A: Squared loss criterion
A, B, C, D, E, F, G, H, USB
10.8690
10.7834
Panel B: Absolute loss criterion
A, B, C, D, E, F, G, H, USB
7.72577.6648
Note: The test statistics S and S c are used to compare the forecasting ability for various competing models. * Significant at 10% level ofsignificance.

Conclusion: All models have the same forecasting ability.

Table K-7 A Summary of Mariano and Preve (2011) Statistics for Housing Returns (In-sample 4-Quarter
Ahead Forecasts)
Selected models
Panel A: Squared loss criterion
A, B, C, D, E, F, G, H, USB
A, C, D, E, F, G, H, USB
A, C, D, E, F, G, H
B, USB
Panel B: Absolute loss criterion
A, B, C, D, E, F, G, H, USB
A, C, D, E, F, G, H, USB

S statistic

S c statistic

18.2991*
15.0643*5.62170.000318.1550*
14.9457*5.57750.000313.7984*5.247113.6897*
5.2058

Note: The test statistics S and S c are used to compare the forecasting ability for various competing models. * Significant at 10% level ofsignificance.
Conclusion: (1) For squared loss criterion:
Group 1: Models A, C, D, E, F, G, H have the same forecasting abilityGroup 2: Models B, USB have the same forecasting ability
The order of forecasting performance: Group 1 > Group 2(2) For absolute loss criterion:
Group 1: Models A, C, D, E, F, G, H, USB have the same forecasting abilityGroup 2: Model B
The order of forecasting performance: Group 1 > Group 2.

Table K-8

A Summary of Out-of-Sample Forecasting Performances (4-Quarter Ahead Forecasts)(2008Q4-2012Q1)
Stock Returns
MSE
MAE

Model A
Model B
Model C
Model D
Model E
Model F
Model G
Model HUSB

Single-regime model (FFR, SPR, TED, EFP, GDP,
SRET, HRET)
Two-regime model (FFR, GDP, SRET, HRET)
Two-regime model (FFR, SPR, SRET, HRET)
Two-regime model (FFR, EFP, SRET, HRET)
Two-regime model (FFR, TED, SRET, HRET)
Two-regime model (EFP, SPR, SRET, HRET)
Two-regime model (EFP, TED, SRET, HRET)
Two-regime model (SPR, TED, SRET, HRET)
Uni-variate, Single-regime Benchmark

Housing Returns
MSE
MAE

170.5441

11.6253

4.0734

1.6796157.0917
161.3897
169.8544
153.8335
172.0982
166.1691
142.7175
150.2610

10.8966
11.2335
11.5395
10.5620
11.2934
10.461210.212210.3168

3.73552.78912.65154.4722
3.17793.31973.63993.8935

1.6366
1.39271.3177
1.7894
1.46851.50391.63681.6767

Table K-9 A Summary of Mariano and Preve (2011) Statistics for Stock Returns (Out-of-sample 4-Quarter
Ahead Forecasts)
Selected models

S statistic

S c statistic

Panel A: Squared loss criterion
A, B, C, D, E, F, G, H, USB
71.2009*
66.1151*
A, B, C, D, E, G, H, USB
55.4084*
51.4507*
B, C, D, E, G, H, USB
41.2602*38.3131*
B, C, E, G, H, USB
15.5858*
14.4735*
B, C, E, H, USB
7.24406.7266
A, D, F, G
0.38070.3535
Panel B: Absolute loss criterion
A, B, C, D, E, F, G, H, USB
68.1230*
63.2570*
B, C, D, E, F, G, H, USB
44.8673*
41.6625*
B, C, E, F, G, H, USB
27.5887*
25.6180*
B, C, E, G, H, USB
12.2081*
11.3361*
B, E, G, H, USB
2.90302.6957
A, C, D, F
5.99295.5649Note: The test statistics S and S c are used to compare the forecasting ability for various competing models. * Significant at 10% level ofsignificance.
Conclusion: (1) For squared loss criterion:
Group 1: Models B, C, E, H, USB have the same forecasting abilityGroup 2: Models A, D, F, G have the same forecasting ability
The order of forecasting performance: Group 1 > Group 2.
(2) For absolute loss criterion:
Group 1: Models B, E, G, H, USB have the same forecasting abilityGroup 2: Models A, C, D, F have the same forecasting ability
The order of forecasting performance: Group 1 > Group 2.

Table K-10 A Summary of Mariano and Preve (2011) Statistics for Housing Returns (Out-of-sample
4-Quarter Ahead Forecasts)
Selected models

S statistic

S c statistic

Panel A: Squared loss criterion
A, B, C, D, E, F, G, H, USB
25.6310*23.8002*
A, B, C, D, F, G, H, USB
19.2929*17.9149*
B, C, D, F, G, H, USB
19.0812*17.7183*
B, C, D, F, G, H
18.9057*
17.5553*
C, D, F, G, H
12.1488*
11.2810*C, D, F, G
4.16773.8700A, B, E, H, USB
5.65045.2468Panel B: Absolute loss criterion
A, B, C, D, E, F, G, H, USB
19.8506*
18.4327*
A, B, C, D, F, G, H, USB
18.9253*
17.5735*
B, C, D, F, G, H, USB
18.8464*17.5002*
B, C, D, F, G, H
18.1156*
16.8216*
B, C, D, F, G
11.2287*
10.4267*
C, D, F, G
4.05493.7653A, B, E, H, USB
5.19154.8207
Note: The test statistics S and S c are used to compare the forecasting ability for various competing models. * Significant at 10% level ofsignificance.

Conclusion: (1) For squared loss criterion:
Group 1: Models C, D, F, G have the same forecasting abilityGroup 2: Models A, B, E, H, USB have the same forecasting ability.
(2) For absolute loss criterion:
Group 1: Models C, D, F, G have the same forecasting abilityGroup 2: Models A, B, E, H, USB have the same forecasting ability.
The order of forecasting performance: Group 1 > Group 2.