ISSN (Print) 0473-453X

Discussion Paper No. 1124

ISSN (Online) 2435-0982

HOW DIDTHE ASSET MARKETS CHANGE
AFTER THE GLOBAL FINANCIAL CRISIS?

Kuang-Liang ChangCharles Ka Yui Leung

March 2021

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

How did the asset markets change after the Global
Financial Crisis?
Kuang-Liang Chang and Charles Ka Yui Leung
This version: March 2021
Abstract
The Global Financial Crisis (GFC) changes the relative economic riskiness and risk-adjustedperformance of different asset markets. While the empirical distribution for stock return shifted to
the right and became more concentrated around the mean after the GFC, the real estate market
counterparts moved to the left and became more spread out. The economic risk of the OFHEO and
Case-Shiller housing indices was smaller than the counterpart of the equity REIT (EREITs) market
before the financial crisis, it substantially increased. Also, the economic performance of the
OFHEO and Case-Shiller housing indices decreased after the financial crisis. They are below theperformance indices of the stock and EREITs markets. The ex-post real estate premium vanishes.
If we presume the "best model" to be the same before and after the GFC, we could severely
misestimate the risk after the GFC.

Key Words: economic index of riskiness, risk-adjusted-performance index, real estate markets,
stock markets
JEL Classifications: C50, G32, R30

Acknowledgements
The authors gratefully acknowledge the comments and valuable suggestions from Man Cho, Yuichiro Kawaguchi,
Jin Man Lee, Jay Sa-aadu, the seminar participants at the annual conference of Chinese Society of Housing Studies,
Asian Real Estate Society, and international conference of Taiwan Finance Association. Chang thanks the grantfrom Ministry of Science and Technology of Taiwan (MOST 104-2410-H-415-004) and Leung thanks the City
University of Hong Kong, The Joint Usage/Research Center at ISER, Osaka University and Grant-in-aid for
Research Activity, Japan Society for the Promotion of Science (15H05728, 20H05631), and the International Joint
Research Promotion Program of Osaka University for financial support. Part of the research is conducted when
Leung visits the Hoover Institution, whose hospitality is gratefully acknowledged. Joe Ng provides capable research
assistance.

Correspondence
Chang, Department of Political Economy, National Sun Yat-sen University, Kaohsiung City,
Taiwan (R.O.C.), kuangliangc@mail.nsysu.edu.tw ; Leung, Department of Economics andFinance, City University of Hong Kong, Kowloon Tong, Hong Kong, kycleung@cityu.edu.hk

1. Introduction

This paper focuses on two crucial aspects of the recent Global Financial Crisis(GFC). First, it is the underestimation and mispricing of risk. 1 Second, investors
significantly change their portfolios after the GFC. 2 Also, central banks andgovernments responded with significant policy changes, such as adopting
unconventional monetary policies by the major central banks and different "fiscal
stimulus packages." With all these changes, the pre-crisis statistical models for theempirical asset return dynamics may or may not provide a good approximation after
the GFC. 3
Therefore, this paper first re-assesses the risk and risk-adjusted return (or"performance") of major assets, such as stock, equity real estate investment trusts(EREITs), and housing markets by utilizing different measures of economic risk andperformance. 4 Second, this paper provides a simple method to assess whether statisticalmodels built before the GFC continue to be useful in describing the return dynamics
after the GFC. Implicitly, we adopt the statistics literature view that "all models arewrong, but some models are more useful than the others." 5 Hence, a model can lose
the usefulness after some dramatic events such as the GFC. We devote a separate section

1

For instance, shortly after the collapse of Lehman Brothers, J. C. Trichet (January 2009), then Presidentof the European Central Bank (ECB), stated that "… the appropriate identification, assessment andhandling of risks in the financial sector are the key issue to be considered most carefully amid the current
global financial turmoil." (italics added).
2
For instance, the Wall Street Journal (2012) reported, "The landscape has changed for the asset
management industry in the wake of the global financial crisis. Investors are skittish after seeing theirportfolios shrink in value…. admits Emad Mansour, chief executive officer of the Qatar First
Investment Bank (QFIB). 'There is a general lack of interest by investors—both individuals andinstitutional—to plow money into markets that are directionless.'" (italics added).
3
For instance, Leung and Ng (2019) find that in business cycle frequency, many relationships among
the housing market variables and macroeconomic variables have changed significantly after the GFC.
4
Heaton and Lucas (2000) examine several waves of data from the Survey of Consumer Finance. The
portfolio weights vary across years and levels of net worth. Stock accounts for 14~33% of theportfolio. Real estate accounts for 30~60% of the portfolio, cash accounts for 5~16% of the portfolio,bonds account for 3~11% of the portfolio, and pension accounts for less than 10% of the portfolio.
5
See Lv and Liu (2014), among others, for more discussion on this point.

for more discussion on this issue.
Risk and performance measurement are not trivial tasks. 6 Since we cannot
directly observe risk, there are many different risk measures, and correspondingly,
various estimates of risk-adjusted-performance -- such diversity in measurement
matters. For instance, Homm and Pigorsch (2012) investigate performance rankings
across 25 hedge funds and find that hedge fund rankings change dramatically when adistinct performance index is used. We choose an economic performance indicator thatremains valid for non-normal distribution, as asset returns are typically non-normal. 7
Economic reasons also drive our concerns for useful statistical models of asset
returns dynamics. For both academic researchers and practitioners, statistical models
summarize the history, guide future forecasts, and quantify our evaluation of risk andperformance. For instance, if the asset return is independent and identically distributed(i.i.d.), as some previous studies assume, then the current period return would not help
us predict future yield. However, if asset return is positively and serially correlated, ahigh performance observed in the current period would suggest a high return in the next
period. It would thus alter our forecast for the near future. 8 Many "early warning
systems" adopted by policymakers also assume some form of intertemporal dependence.

6

The Sharpe ratio is a common and widely accepted measure for comparing the relative performance ofasset markets. Aumann and Serrano (2008) consider those specific characteristics are necessary for theright risk measurement. They include continuity, positive homogeneity, subadditivity, and monotonicity.
They demonstrate that widely used risk measures such as standard deviation and value-at-risk are
inadequate for measuring price risk due to monotonicity failure concerning first-order dominance. Hommand Pigorsch (2012) recommend an economic performance index for estimating a risky asset's
performance. That index has vital properties, such as scale invariance, monotonicity concerning firstorder and second-order dominance, and generalized continuity.
7
For instance, see Homm and Pigorsch (2012) and Schulze (2014) for analytically solvable examples.
Note that the analytical solutions of the economic risk and performance indices are not always available.
Chen et al. (2014), among others, utilize a numerical method to solve the economic index of riskinessfor the student-t distribution, left-skewed log-normal distribution, and right-skewed log-normal
distribution. They show that the economic index of riskiness is related to the mean, skewness, andkurtosis. Furthermore, Homm and Pigorsch (2012) observe no significant difference in the economic
index of riskiness and the economic performance index for parametric specification and non-parametricspecification.
8
See Anand et al. (2016) for more discussion on this point.

Therefore, choosing a model with the "right" type of dependency is essential. This paper
would estimate different models, each of which implicitly assumes another form of"intertemporal correlation."
Since housing plays a vital role in the GFC, we include its return dynamics andcompare that with other financial assets. 9 Consequently, the highest frequency we canuse is monthly data. Thus, this study would complement previous research, which uses
higher frequency data and focuses only on financial assets (Mondria and QuintanaDomeque, 2013; Mollah et al. 2016; Lehkonen, 2015).
Constrained by the sample size, we would proceed with relatively simple models
that we can estimate more accurately. They include four commonly used specifications,
including the auto-regressive (AR) model, AR-GARCH model, Markov-switching AR
model, and Markov-switching AR-ARCH model. While these models may be relatively
simple, various authors have adopted these specifications to study asset returns'
dynamic behaviors: stock, REIT, and real estate. 10 We, therefore, adopt these models
in this study to facilitate the comparison. We then conduct a formal model-comparison
to select the "best" model from our set of candidate models for the asset returns in eachperiod (pre-crisis and post-crisis). Our empirical strategy allows for the possibility that,for some asset, the "best" model that describes the return dynamics before the GFC may
no longer be so in the post-crisis era. As we have briefly reviewed, many changes havetaken place in the asset market, and we cannot rule out such a possibility. We will
provide more discussion on this point later.
This paper builds on several existing lines of literature. Our focus on the risk and9Many authors have discussed this point. Among others, see Hendershott et al. (2010), Malpezzi(2017) for a review of the literature.
10
For example, Dueker (1997), Maheu and McCurdy (2000), Engle and Patton (2001), Ang and Bekaert(2002), Crawford and Fratantoni (2003), Cotter and Stevenson (2006), Bredin et al. (2007), Miles (2008a,
2008b), Chen et al. (2009), Liow et al. (2009, 2011), Ashley and Patterson (2010), Chang (2010), Tsai etal. (2010), Zhou and Kang (2011), Chang et al. (2012) and Karoglou et al. (2013).

performance of different asset classes is related to the research on the “asset return
premium,” including the “equity premium puzzle” highlighted by Mehra and Prescott’s(1985), “real estate premium puzzle” by Shilling (2003). 11 In this paper, we computedifferent riskiness measures and the risk-adjusted performance of various asset classes
before and after the GFC. Therefore we contribute to the discussion on those puzzles.
This paper also builds on emerging literature insights, which compares the
dynamic properties of different assets. 12 This paper departs from these studies by
including the possibility that different asset returns could be proxied by various
statistical models during different times. This paper also relates to the "structuralchange" literature, which assumes the same statistical model applies, but a change inparameter values has occurred. 13 With some abuse of notations, we may label suchparameter change as “structural change in the intensive margin” (SCIM). At the sametime, this paper allows for an additional possibility that even the statistical model haschanged, which we may label as “structural change in the extensive margin” (SCEM).
In this paper, our model selection procedure chooses the "optimal model," and hence
differentiate the SCIM from the SCEM.
There is also an ongoing debate on the impacts on the financial market and the real
economy of the unconventional monetary policy (UMP) and large scale asset purchase(LSAP) that the major central banks have conducted. 14 Our focus is more on the(potential) changes in the asset return distribution and the statistical models that we canuse to approximate the asset return dynamics after the GFC, which would include not

11
For the equity premium puzzle, see Mehra (2006) for a survey. For the real estate premium puzzle,see, for instance, Seiler et al. (1999), Cheng et al. (2010b), and Lin and Liu (2008), and their references
to related discussions.
12
It includes Quan and Titman (1999), Glascock et al. (2000), Liow and Yang (2005), Chang et al.
(2011), etc.
13
See Bai and Perron (1998), Chong (2001), Hansen (2001), Perron (2006), among others.
14
Among others, see Bernanke (2018), D’Amico and King (2013), Eksi and Tas (2017), Eser and
Schwaab (2016), Hamilton (2018), Steeley (2015), Weale and Wieladek (2016), and the reference
therein.

only the effects of UMP but also other factors. Hence, this paper is related to thatliterature but with a different research focus.
The rest of this paper is organized as follows. Section 2 introduces the indices tomeasure economic risk and economic performance, the models we use to characterize
the return processes and the Monte Carlo simulation method. Section 3 reports thedifferent models' forecasting performance and explains their classification into different"equivalent predictive power classes" (EPPC). We also discuss empirical findings. The
last section concludes the paper.

2. The economic performance measure and empirical specification
This paper utilizes the economic performance index developed by Aumann andSerrano (2008), Homm and Pigorsch (2012) to reinvestigate financial assets' risk andperformance. For each asset during each sampling period, we allow its return dynamics
follows one of the four widely used models, which essentially assumes the returns
correlate over time in different ways. They are the AR, AR-GARCH, Markov-switchingAR, and Markov-switching AR-ARCH models. 15 We adopt the Monte Carlosimulation method (MCSM) to compute asset returns' economic performance, as noclosed-form solutions are available given these choices. Since alternative empiricalmodels and alternative performance indices would be developed in the future, and they
are unlikely to have closed-form solutions, this paper also demonstrates how aperformance comparison can be conducted using the MCSM.

2.1 Economic index of riskiness and economic performance index
Although the Sharpe ratio is a widely accepted method used to compare financial

15

Among others, see Bollerslev (2009), Chauvet and Piger (2008), and Duprey and Klaus (2017) for areview of the literature.

assets' relative performance and is easily implemented, it lacks monotonic stochastic
dominance and axiomatic justifications. Recently, Homm and Pigorsch (2012) propose
an alternative economic performance index (EPI), which is defined as follows:
EPI =𝐸𝐸(𝑟𝑟𝑡𝑡 −𝑟𝑟𝑓𝑓,𝑡𝑡 )(1)

𝐴𝐴𝐴𝐴(𝑟𝑟𝑡𝑡 −𝑟𝑟𝑓𝑓,𝑡𝑡 )where 𝑟𝑟𝑡𝑡 is the nominal return of a risky asset; 𝑟𝑟𝑓𝑓,𝑡𝑡 is the return of a risk-free asset;

𝑟𝑟𝑡𝑡 − 𝑟𝑟𝑓𝑓,𝑡𝑡 is the excess return; E denotes the expectation operator; and 𝐴𝐴𝐴𝐴(𝑟𝑟𝑡𝑡 − 𝑟𝑟𝑓𝑓,𝑡𝑡 )refers to the economic risk index of Aumann and Serrano (2008). Like the Sharpe ratio,

the EPI measure is also a risk-adjusted index. Aumann and Serrano (2008, TheoremA) prove that for any given gamble 𝑔𝑔, there is a unique positive number 𝑅𝑅(𝑔𝑔) thatsatisfies the equation
𝐸𝐸�𝑒𝑒 −𝑔𝑔/𝑅𝑅(𝑔𝑔) � = 1.

(2)

Aumann and Serrano (2008) prove that their index has several desirable properties.
However, we refer readers to the original papers to discuss the EPI and AS indices due
to space limitations. We provide some highlights of these discussions in the appendix.

2.2 The time-series dynamics of excess return
This paper employs four widely-adopted models of the time-series dynamics ofexcess returns. Our first model is the simple autoregressive (AR) model. It is often
regarded as an "atheoretical benchmark." Formally, it is given by𝑟𝑟𝑡𝑡 − 𝑟𝑟𝑓𝑓,𝑡𝑡 = 𝜃𝜃0 + 𝜃𝜃1 (𝑟𝑟𝑡𝑡−1 − 𝑟𝑟𝑓𝑓,𝑡𝑡−1 ) + 𝜀𝜀𝑡𝑡

𝜀𝜀𝑡𝑡 = √𝜔𝜔𝑧𝑧𝑡𝑡 ,𝑧𝑧𝑡𝑡 ~𝑁𝑁(0,1)

(3a)
(3b)

Where 𝑟𝑟𝑡𝑡 is the return of the asset, 𝑟𝑟𝑓𝑓,𝑡𝑡 is the risk-free return, 𝜃𝜃0 and 𝜃𝜃1 areparameters that describe the autoregressive process; 𝜔𝜔 is the variance; 𝑧𝑧𝑡𝑡 is a

Gaussian random error with a mean of zero and a variance of one.

The second model is the AR-GARCH model, which Bollerslev (1986) developed

to capture the fact that the innovation's variance may be time-varying. It is formulated
as follows:
𝑟𝑟𝑡𝑡 − 𝑟𝑟𝑓𝑓,𝑡𝑡 = 𝜃𝜃0 + 𝜃𝜃1 (𝑟𝑟𝑡𝑡−1 − 𝑟𝑟𝑓𝑓,𝑡𝑡−1 ) + 𝜀𝜀𝑡𝑡
𝜀𝜀𝑡𝑡 = �ℎ𝑡𝑡 𝑧𝑧𝑡𝑡 ,𝑧𝑧𝑡𝑡 ~𝑁𝑁(0,1)

(4a)
(4b)

2
ℎ𝑡𝑡 = 𝜔𝜔 + 𝛼𝛼𝜀𝜀𝑡𝑡−1+ 𝛽𝛽ℎ𝑡𝑡−1(4c)where 𝜔𝜔, α, and β are parameters that describe the conditional variance process and2
conditional variance ℎ𝑡𝑡 is a linear function of 𝜀𝜀𝑡𝑡−1
and ℎ𝑡𝑡−1 . The persistence ofconditional variance is captured by the sum of parameters α and β.

The third model captures the idea that the relationship between the current andprevious period returns could change across different regimes. More specifically, theMarkov Switching Autoregressive (MS-AR) model is an AR specification with two
different dynamical processes, and can be specified as follows:
𝑟𝑟𝑡𝑡 − 𝑟𝑟𝑓𝑓,𝑡𝑡 = 𝜃𝜃0,𝑠𝑠𝑡𝑡 + 𝜃𝜃1,𝑠𝑠𝑡𝑡 (𝑟𝑟𝑡𝑡−1 − 𝑟𝑟𝑓𝑓,𝑡𝑡−1 ) + 𝜀𝜀𝑡𝑡,𝑠𝑠𝑡𝑡

𝜀𝜀𝑡𝑡,𝑠𝑠𝑡𝑡 = �𝜔𝜔𝑠𝑠𝑡𝑡 𝑧𝑧𝑡𝑡 ,𝑧𝑧𝑡𝑡 ~𝑁𝑁(0,1)

(5a)
(5b)

where 𝑠𝑠𝑡𝑡 is a state variable that controls the pattern of regime switches. Constrainedby data availability, we assume in this paper that the state variable has two states: 1 and2. The evolution specification of 𝑠𝑠𝑡𝑡 is assumed to depend on the last period’s statevariable. A logistic function is used to ensure that the transition probabilities of the statevariable are positive values. The four transition probabilities are given by:
P(𝑠𝑠𝑡𝑡 = 1|𝑠𝑠𝑡𝑡−1 = 1) = 𝑃𝑃11 =P(𝑠𝑠𝑡𝑡 = 2|𝑠𝑠𝑡𝑡−1 = 1) = 𝑃𝑃12 =P(𝑠𝑠𝑡𝑡 = 1|𝑠𝑠𝑡𝑡−1 = 2) = 𝑃𝑃21 =P(𝑠𝑠𝑡𝑡 = 2|𝑠𝑠𝑡𝑡−1 = 2) = 𝑃𝑃22 =exp(𝑝𝑝)

(5c)

1

(5d)

1

(5e)

exp(𝑞𝑞)(5f)

1+exp(𝑝𝑝)1+exp(𝑝𝑝)1+exp(𝑞𝑞)1+exp(𝑞𝑞)

Our fourth model, MS-AR-ARCH, is an extension of the AR-ARCH specification

and has the following dynamics: 16
𝑟𝑟𝑡𝑡 − 𝑟𝑟𝑓𝑓,𝑡𝑡 = 𝜃𝜃0,𝑠𝑠𝑡𝑡 + 𝜃𝜃1,𝑠𝑠𝑡𝑡 (𝑟𝑟𝑡𝑡−1 − 𝑟𝑟𝑓𝑓,𝑡𝑡−1 ) + 𝜀𝜀𝑡𝑡,𝑠𝑠𝑡𝑡

𝜀𝜀𝑡𝑡,𝑠𝑠𝑡𝑡 = �ℎ𝑡𝑡,𝑠𝑠𝑡𝑡 𝑧𝑧𝑡𝑡 ,𝑧𝑧𝑡𝑡 ~𝑁𝑁(0,1)

2ℎ𝑡𝑡,𝑠𝑠𝑡𝑡 = 𝜔𝜔𝑠𝑠𝑡𝑡 + 𝛼𝛼𝑠𝑠𝑡𝑡 𝜀𝜀̅𝑡𝑡−1

(6a)
(6b)
(6c)

where 𝑠𝑠𝑡𝑡 is a state variable, and 𝜀𝜀̅𝑡𝑡−1 is the conditional mean of 𝜀𝜀𝑡𝑡−1 , given theinformation set Ω𝑡𝑡−2 . The transition probabilities of the state variable are the same as
those described in the MS-AR model.

2.3 The Monte Carlo simulation method
Although the four models of asset return dynamics considered in this paper are
somehow standard, closed-form solutions for the EPI and economic risk index are
unavailable. Therefore, we follow the approach of Chen et al. (2014) to compute the
EPI and economic risk index. Formally, we consider the following unconstrained
minimization problem:
min1

𝑟𝑟𝑡𝑡 −𝑟𝑟𝑓𝑓,𝑡𝑡

�𝑁𝑁 ∑𝑁𝑁
𝑡𝑡=1 𝑒𝑒𝑒𝑒𝑒𝑒 �− 𝐴𝐴𝐴𝐴(𝑟𝑟 −𝑟𝑟
𝑡𝑡

𝑓𝑓,𝑡𝑡

� − 1�
)2(7)

The indices can be obtained through the following steps:
Step 1: Estimate the parameters of the time-series model using the quasi-maximum
likelihood estimation method.
Step 2: Simulate 100,000 observations of excess returns through a time-series
process where the model parameters are estimated from Step 1.
Step 3: Calculate the economic index of riskiness, and the sample mean using thelast 50,000 observations generated from Step 2. 17 Following Chen et al. (2014), thenumerical optimization procedure is implemented using the OPTMUM procedure of

16

See also Haas et al. (2004), Kim and Hwang (2018), Liu (2006), among others.
The first 50,000 observations are omitted to remove the possible influence of starting values on
simulated data.

17

the Gauss program.
Step 4: Calculate the economic performance index through the sample mean andthe economic index of riskiness obtained from Step 3.
A merit of this method is that it is simulation-based. Hence, the small sample issue
is not a concern in our calculation.

3. Data and empirical findings
3.1 Data
This paper investigates the economic index of riskiness and several asset market
indices' economic performance at a monthly frequency. 18 The indices include theStandard and Poor’s 500 Stock Index, Dow Jones Industrial Average Index, Nasdaq
Composite Index, FTSE NAREIT All Equity REITs Price Index, Office of Federal
Housing Enterprise Oversight (OFHEO) purchase-only index, and S&P/Case-Shiller
U.S. National Home Price Index. As the GFC might have changed the market
participants' expectations and investment strategies, we allow for the possibility that thestatistical model that works well with a particular asset before the crisis may no longer
work well after. In this paper, the pre-crisis period covers 2000:m1-2006:m6. The postcrisis period covers 2009:m1-2019:m12, and the “in-crisis” period is avoided. 19 We
also avoid the Russian financial crisis's potential effect in 1998 by starting our sample
in 2000.
18

We do not have access to data that more frequent than monthly observations. Studies have examined
the joint dynamics of different assets at a quarterly rate (e.g., Chang et al., 2011, 2012, 2013). However,the sample size is relatively small in that case, and the estimation of pre-crisis versus post-crisis subsample could be difficult. Therefore this study, which is based on monthly data, naturally complements
the previous studies.
19
There are different classifications of the pre-crisis versus post-crisis periods. See Dungey et al.
(2015) for a detailed analysis. Since we employ monthly data, we cannot divide the sample into morethan two sub-periods.
Several studies suggest that lending and trading behaviors during the GFC are very different from that
before the crisis. Hence, it may be essential to examine the pre-crisis and post-crisis periods separately.
Among others, see Afonso et al. (2011), Brunnermeier (2009), Gorton (2010), Ivashina and Scharfstein(2010) for more details.

Our data come from the usual sources. Standard and Poor's 500 Stock Index and
the Dow Jones Industrial Average Index are obtained from Datastream. The OFHEO
purchase-only Index and the Equity RETIS Index are obtained from the Federal
Housing Finance Agency and the REIT.com website. The Nasdaq Composite Index andthree-month Treasury Bill interest rate come from the Federal Reserve Bank of St.
Louis. The S&P/Case-Shiller Index comes from S&P Dow Jones Indices LLC. The
annualized excess return is derived from by subtracting the three-month Treasury Billrate from the annualized return of a given financial asset.
According to the transaction process, transaction time and transaction cost, thethree stock indices and the Equity REITs index are labeled as "high-liquidity assets" inthis paper. In comparison, the two national housing indices are considered as "lowliquidity assets." Figures 1 and 2 provide the time-series plots of annualized excess
returns for the pre- and post-crisis periods. The time-series patterns for the high- andlow-liquidity assets are entirely different. In Figure 1, the annualized excess returns forhigh-liquidity assets increase significantly around 2003. However, the counterpart ofthe low-liquidity assets first increase and then decline sharply before the GFC. Figure
2 shows that the high-liquidity assets' returns decrease slightly over time, while the lowliquidity assets' counterparts significantly increase around 2011-2013.
(Figures 1 and 2 about here)

3.2 Empirical findings
In the main text, we use the root mean square error (RMSE) to compare the insample forecasting performance across four empirical models. 20 Both the return
forecasts and variance forecasts are examined. Table 1 reports the return forecasts and20

We also adopt the MAE (mean absolute error) and report the corresponding results in the appendix.
Compared to MAE, RMSE tends to “punish more” for significant forecast error and hence might be moresuitable for the study of risk in this paper.

variance forecasts for the pre-crisis period. 21 For the post-crisis period, the forecasting
performance for return forecasts and variance forecasts is reported in Table 2. The MSAR model has a smaller RMSE than the other three models in most cases for the precrisis period. 22 The MS-AR model seems to be more accurate for return forecasts
during the post-crisis period, but it is not always superior for variance forecasts.

(Tables 1-2 about here)

While the root-mean-square errors across models are not the same, the difference
might not be statistically significant in differentiating their forecast performance.
Combining the results in Hansen et al. (2011), Mariano and Preve (2012), Kwan et al.
(2015), Chang et al. (2016) propose a procedure to classify competing models into
different equivalent predictive power classes (EPPCs). Models in the same class havethe same predictive power, while models in Class 1 have higher predictive power thanthe models in Class 2. The square loss criterion (SLC) is used here to implement the
EPPC procedure. 23 Tables 3 and 4 summarize the relative model performance for the
pre- and post-crisis period, respectively. The predictive power classes in the pre- andpost-crisis periods are slightly different. Panel A of Table 3 shows that the four models
have the same forecasting ability in return forecasts. Panel B of Table 3 shows that thefour models can be divided into two classes for three different stock returns. For the
SP500 stock return, the MS-AR and MS-ARCH models belong to Class 1, and they
produce more accurate forecasts than the AR and AR-GARCH models, which areclassified as Class 2. However, for the EREITs, OFHEO housing index, and Case21

See Gray (1996) for more details about the forecasting error of conditional variances.
The AR-ARCH model is selected for two housing returns. For the S&P/Case-Shiller return, the MSARCH model does not converge.
23
We also repeat the analysis with the absolute loss criterion (ALC). Again, SLC tends to impose a more
massive penalty on "large error" than ALC. The results are similar and are reported in the appendix.
22

Shiller housing index, the four specifications' variance forecasts are not statisticallydifferent. In other words, for stock returns, models differ not in terms of forecasting theperformance, but in terms of predicting the risk in the pre-crisis period. For real estaterelated assets, there is no difference in terms of return or variance.
In the post-crisis period, Panel A of Table 4 shows that the four models do not havea statistically significant difference in return forecasts for SP500 returns, Down Jonesstock returns, and two housing returns (OFHEO and Case-Shiller). Moreover, thevariance forecasts of the four models for three stock returns are not significantly
different. Hence, models display no difference in forecasting stock returns in the postcrisis era. For the Case-Shiller return, the MS-AR, and MS-ARCH models produce
more accurate forecasts in return (Class 1) than the AR, AR-ARCH models (Class 2).
However, for the OFHEO return, the AR, AR-ARCH, and MS-ARCH models predict
better than the MS-AR, and the latter is therefore considered Class 2.

(Tables 3-4 about here)
We define the "best model" as the model which delivers the smallest RMSE forClass 1 models. We then use the estimated parameters to simulate each of the asset
returns. 24 We then use the Epanechnikov kernel density to find the empirical
distribution based on the simulated data, using Silverman's bandwidth (1992). We
simulate it for 100,000 periods and then drop the first half (sometimes regarded as the“training period”), using only the second half to “plot” the empirical distribution andother calculations (such as performance measures). Because we use 50,000 periods,“small sample bias” is not a concern.
The empirical distributions of asset returns are plotted in Figures 3. The solid
24

If the best model is non-stationary, then a second-best model is chosen. If the second best model is
non-stationary, a third best model is used to simulate the observations. This paper uses the third-bestmodel for the S&P/Case-Shiller housing return during the pre-crisis period.

(dotted) line shows the simulated distribution for the pre-crisis (post-crisis) period. The
empirical distributions across different asset markets are very different. Besides, its
empirical distributions before and after the crisis also differ for any given asset market.
For instance, in the pre-crisis period, the SP500 stock return and OFHEO housing return
have a bimodal distribution, and the remaining asset markets have a unimodal
distribution. In the post-crisis period, the OFHEO housing returns have an asymmetricbimodal distribution, while the high-liquidity markets have an asymmetric unimodal
distribution. 25(Figures 3 and 4 about here)

Compared with the empirical distribution in the pre-crisis period return, the postcrisis distributions for the three stock market indices shifts to the right. The value rangesof the three stock market indices in the post-crisis period are narrower. It means that
there is a decrease in extreme negative returns and a reduction in return variation for
financial asset markets. In contrast, the empirical distributions of the OFHEO and CaseShiller's housing returns shift to the left in the post-crisis period. And the ranges of the
distributions are more extensive in the post-crisis period than in the pre-crisis period,
suggesting an increase in extreme negative return and growth in return variation. For
the EREITs returns, the empirical distribution is more or less symmetric around themean in both the pre- and post-crisis periods. The observed distribution in the postcrisis period has "a little shorter left tail and a longer right tail fatter left tail" than the
pre-crisis period, which also suggests a decrease in extreme negative return and anincrease in extra-ordinary positive returns. 26
25

Bimodal distribution or “twin peaks” have been studied in the economics literature. For instance, see
Quah (1996).
26
It should be clear that while our sample may be relatively small, the “small sample issue” (SSI) should
apply to both pre-crisis and post-crisis sub-samples. SSI by itself would not make “model switching”
more or less likely to happen. For instance, MS-AR(1) is the best model to explain the housing returnNotice that Figure 3 is based on the ex-post data. When agents need to makeeconomic decisions in real-time, they may not be sure of the proper distribution andwould need to take the uncertainty of variances into considerations. Figure 4 provides
such an estimate. The empirical distribution of the (implied) variance for the post-crisis
period is more concentrated around the mean than that for the pre-crisis period for all
asset returns, which means that the uncertainty of variances becomes less severe postcrisis period. For instance, the "best model" for Down Jones returns in the post-crisis
period is an AR specification with constant variance. Hence, Figure 4 shows that its
distribution of the variance of return collapses to a vertical line during the post-crisis
period. One possible explanation is the active participation of the U.S. government in
the asset markets through TARP and QE policies are useful in shortening an otherwise
more prolonged recession and reducing the uncertainty of variance (Brookings, 2018;
Liang et al., 2018).
Table 5 summarizes different measures of risk for the best performing models.
Panels A and B show the variances and the 1% value-at-risk estimates, respectively. 27
Several observations are in order. The values of variance and value-at-risk are more
substantial for the more liquid assets than the less liquid ones, regardless of the period.
Moreover, according to the variance and value-at-risk, the risk measure for three stock
returns dramatically decreases after the financial crisis. This observation is consistent
with Figures 3(a)-3(c) that the range of stock returns' empirical distributions falls. The
observed distributions also move to the right after the GFC, thus decreasing the risk
measures.
On the other hand, the variances and value-at-risk measures for two housing(S&P/Case-Shiller) for both the pre-crisis and post-crisis periods. SSI would not make models that
implied “more risk” to be selected. The results of our model selection procedures suggest a decrease inrisk in financial asset returns and an increase in housing returns in the post-crisis period at the same time.
27
The value-at-risk measure at α% is defined as the αth percentile of the distribution of excess
returns multiplied by -1. The larger the value-at-risk, the larger the negative return.

markets increase after the GFC. This observation is consistent with Figures 3(e)-3(f)that the range of the empirical distributions of housing returns increases. The observed
distributions of housing returns move to the left after the GFC, thus increasing the risk
measures. For the EREITs returns, the variance increases, but the value-at-risk measuredecreases after the GFC. The value-at-risk substantially decreases for the three stock
returns and EREITs returns after the financial crisis. Specifically, there is a consistentfinding that the variance and value-at-risk reduce after the financial crisis for stock
assets and housing assets. There is no consistent finding for EREITs returns. 28
Furthermore, the view that real estate assets have smaller risks than high-liquidity assets
is confirmed.
(Table 5 about here)

Panels C and D of Table 5 report the 𝐴𝐴𝐴𝐴𝑁𝑁 and AS values, respectively. 29

Because the simulated sample mean is negative for the three stock markets, the 𝐴𝐴𝐴𝐴𝑁𝑁

and AS are not estimated here. 30 As Figure 3 shows, the empirical distribution of thesimulated data is distinct from the normal distribution. Thus, the AS may provide a

better measure of the risk. On the other hand, 𝐴𝐴𝐴𝐴𝑁𝑁 is widely used, and hence we alsooffer the 𝐴𝐴𝐴𝐴𝑁𝑁 value for reference.

For real estate-related assets, the 𝐴𝐴𝐴𝐴𝑁𝑁 value is smaller than the AS value for the

EREITs and S&P/Case-Shiller housing returns during the pre-crisis period, suggesting28

Our results are different from Zhou and Anderson (2012) due to several reasons. First, they use daily
data while we use monthly data. Second, their sampling period is from 1993 to 2009, while we explicitly
compare the pre-crisis (before the Global Financial Crisis) period from the post-crisis period. Third, they
cover nine REIT market across the nations while we compare different asset markets within the UnitedStates.
29
When the excess return is normally distributed, the AS index coincides with the 𝐴𝐴𝐴𝐴𝑁𝑁 index. See the
Appendix for more details.
30
The economic index of riskiness proposed by Aumann and Serrano (2008) operates under theassumption that the ex-ante excess return is non-negative. When the mean of asset returns is negative,the economic index is undefined. The intuition is that if the ex-ante excess return is negative, then theinvestor does not need to hold this risky asset as a risk-free asset is a better choice. In this paper, we focus
on the excess ex-post return, which is possible to be negative.

an underestimation of risk. Simultaneously, the 𝐴𝐴𝑆𝑆𝑁𝑁 value is larger than the AS value

for the OFHEO purchase-only housing returns, which indicates an overestimation ofrisk. Furthermore, the two housing markets' risk measure is smaller than that for the
EREITs market, irrespective of the adopted economic index.
After the GFC, the risk measure for two housing returns (OFHEO purchase-only
and S&P/Case-Shiller) dramatically increases, regardless of the type of economic index
of riskiness adopted. More specifically, for the OFHEO housing return, the AS
increases from 0.625 to 2.192, and the 𝐴𝐴𝐴𝐴𝑁𝑁 , increases from 0.791 to 2.150. The AS
index for the S&P/Case-Shiller housing returns increases from 1.451 to 9.603. For the

AS measure, the economic risk measure for the S&P/Case-Shiller is higher than thatfor the more liquid assets during the post-crisis period. The S&P/Case-Shiller housing
returns show the highest risk, followed by the Nasdaq stock returns, ERIETs, Dow
Jones stock returns, SP500 stock returns, and OFHEO purchase-only housing returns.
The housing markets have a more considerable risk than the stock markets and RERITs
market. Finally, we also confirm the literature finding that the risk ranking across assets
is sensitive to the choice of risk measures.
(Table 6 about here)

Table 6 summarizes the performance measures for the best specification. The
economic performance measures include the Sharpe ratio, 𝐸𝐸𝐸𝐸𝐸𝐸𝑁𝑁 , and 𝐸𝐸𝐸𝐸𝐸𝐸. As Panel A
of Table 6 shows, the Sharpe ratios for the three stock markets are negative before thefinancial crisis but are positive after the financial crisis. The Sharpe ratios of the three
real-estate-related assets (EREITs, OFHEO purchase-only, and S&P/Case-Shiller)
decrease after the financial crisis. Panel A of Table 6 also shows that in terms of the
Sharpe ratio, the housing returns are higher than the more liquid assets (stock and REIT)
only in the pre-crisis period. It suggests that the asset markets do not compensate forthe illiquidity of housing in the post-crisis period. Panels B and C of Table 6 further
confirm that, during the pre-crisis period, the two housing market indices outperformthe EREITs index. However, after the financial crisis, the illiquid housing does not
perform as well as the more liquid assets, irrespective of the economic performancemeasures. In other words, the illiquidity premium for housing vanishes in ex-post terms.
Furthermore, the high- and low-liquidity assets' performance rankings do not change
the Sharpe ratio and EPIN. The EPIN and EPI measures show a slight difference inperformance ranking. 31
One may wonder how it can be possible that the less liquid assets deliver lower
risk-adjusted returns than other investments in equilibrium. We conjecture that several
explanations are possible. Unlike financial assets, housing is traded in a lumpy fashion:
we either sell the whole housing unit or not. Macroprudential measures also limit the
number of mortgage loans and participation in the housing market (IMF, 2014). At thesame time, the Obama government has imposed many urban and housing policies,
which may change the incentives of the housing market participants (Agarwal andVarshneya, 2020; DeFilippis, 2016). This paper focuses on estimating and comparing
the risk and returns across different asset markets before and after the GFC, leaving
lower returns in the housing market issue for future research.

4. How important is the model selection procedure?
We receive a common question from different conferences and seminars: how
important is the model selection procedure in the current context? How would it affectour results? To address this question, we conduct the following experiment. Consider
an asset 𝑖𝑖 , 𝑖𝑖 = 1,2, … Assume that the best model to explain the return of asset 𝑖𝑖
31

Furthermore, the performance rankings of the high- and low-liquidity assets do not change the Sharpe
ratio and 𝐸𝐸𝐸𝐸𝐸𝐸𝑁𝑁 . The 𝐸𝐸𝐸𝐸𝐸𝐸𝑁𝑁 and EPI measures show a slight difference in performance ranking.

during the pre-crisis period is 𝑗𝑗(𝑖𝑖). Notice that if the best model in the pre-crisis period
maintains the best model in the post-crisis period, there is no need for model selection.

We simple re-estimate the parameters with the post-crisis period data of asset 𝑖𝑖. For
future reference, we label this re-estimated model as 𝑗𝑗′(𝑖𝑖). Thus, 𝑗𝑗(𝑖𝑖) and 𝑗𝑗′(𝑖𝑖) differ
only in parameter values, but not functional forms.

On the other hand, we have already selected the best model for asset 𝑖𝑖 during thepost-crisis period in the previous section, based on Kwan et al. (2015), Chang et al.
(2016). For future reference, we label that model as 𝑘𝑘(𝑖𝑖). If model 𝑘𝑘(𝑖𝑖) and 𝑗𝑗′(𝑖𝑖) are
the same, our model selection procedure simply confirms that the same model maintains
its supremacy over the other models. For instance, MS-AR(1) is the best model toexplain the SP500, Nasdaq, and EREITs returns for both the pre-crisis and post-crisis

periods. On the other hand, in the previous section, we have seen that 𝑘𝑘(𝑖𝑖) and 𝑗𝑗′(𝑖𝑖)can be different models. For instance, while the best model for OFHEO purchase-only

return in the pre-crisis period is MS-AR(1), it is replaced by MS-ARCH in the postcrisis period. Notice that we select the “best model” by minimizing the sum of some(conditional) forecast error. In Figure 5, we plot the distributions of return of the data,
and the counterpart implied by 𝑘𝑘(𝑖𝑖) and 𝑗𝑗′(𝑖𝑖). Here are more detailed remarks.
(Figure 5 and 6 about here)(1) SP500, Nasdaq, and EREITs have the same best model before and after the Global
Financial Crisis.
(2) The difference between the best pre-crisis model applies to the post-crisis period,
and the best post-crisis model is insignificant for Dow Jones.
(3) For the S&P/Case-Shiller index, the empirical distribution of actual data is a
Bimodal distribution. Yet the observed distribution evaluated from the pre-crisis bestmodel apply to the post-crisis period is a single-peak distribution. Hence, if we insist
on the best model in the pre-crisis period continues to be the best model in the postcrisis period, the bias could be enormous. Now, through the model selection procedure,
our best model in the post-crisis period also generates a bimodal distribution, whichmimics the distribution of the data.
In Figure 6, we apply the same logic to model-implied variance. The
distributions generated by 𝑗𝑗′(𝑖𝑖) (i.e., when we assume the same model maintains tobe the best before and after the crisis) are, in general, very different from thatgenerated by 𝑘𝑘(𝑖𝑖), the model selected by our model comparison procedure. And inmany cases, the distribution implied by 𝑘𝑘(𝑖𝑖) appears to be “closer” to that generated
by 𝑗𝑗′(𝑖𝑖). Here are more detailed remarks. 32(1) Notice that the "true variance" is unknown. Only the empirical distributions forvariances of the best pre-crisis model apply to the post-crisis period, and the best post-

crisis model is plotted.
(2) The best model for Nasdaq does not change before and after the GFC.
(3) For EREITs, OFHEO purchase-only, and S&P/Case-Shiller, the pre-crisis bestmodel applies to the post-crisis period has a more considerable variance than the postcrisis best model refers to the post-crisis period. If we ignore the structural change anduse the same model found in the pre-crisis period, the "uncertainty about variance" will
be overestimated.
(4) For SP500, the risk is more spread out in the post-crisis best model than the precrisis counterpart. Thus, the pre-crisis best model applies to the post-crisis period will
32

Notice that as the bandwidth suggested by Silverman (1992) is used, h =0.9(𝑁𝑁)1/5

𝑄𝑄 −𝑄𝑄1

min(�𝑉𝑉(𝑋𝑋), 3

1.349), where 𝑄𝑄3 is the third quartile and 𝑄𝑄1 is the first quartile. For the MS-AR

model, there are two different variances. When the 𝑄𝑄3 − 𝑄𝑄1 = 0, the maximum variance replaces the
𝑄𝑄3 , and the minimum variance replaces 𝑄𝑄1 .

under-estimate the “uncertainty about variance.”(5) In Figure 6(b) (Dow Jones), the shape of the empirical distribution of variancegenerated by the best model in the pre-crisis period and the counterpart in the postcrisis period is very similar. But, the post-crisis best model's empirical distribution
applies to the post-crisis period is slightly in the right of that of the pre-crisis best model
apply to the post-crisis period. If we ignore the structural change and use the samemodel found in the pre-crisis period, the "possibility of occurring higher variance" will
be underestimated.

In sum, our paper demonstrates that even for the same asset, there is a need tochoose the "best model" in different periods. If the best model selection mechanism isignored, the return and risk can be significantly mismeasured.

5. Conclusions
This paper began with an insightful observation by Mr. J. C. Trichet that aninappropriate measure of risk is crucial. An inadequate economic risk measure can leadto suboptimal investment strategies for individuals and governments' misdirected policy.
Several authors have made outstanding contributions in this area. For instance, Aumann
and Serrano (2008) develop an economic index of riskiness to evaluate an asset's pricerisk. Homm and Pigorsch (2012) develop a performance index and show that their index
is superior to the usually adopted Sharpe ratio. Recently, in a review of the GFC,
Malpezzi (2017) also observes that "…Many risks are 'fats in the tails,' and anyway
rare events do occur…" (italics added). This paper compares relative risk and relative
performance across housing, REITs, and stock markets in light of these contributions.
Our models allow for different forms of "fat tails in risk." This paper selects the "best"model for each asset return before and after the GFC based on the recent econometrics

literature. We find that the "best" model does change in some cases.
Furthermore, the risk measures and risk-adjusted-performance ranks alsosignificantly change after the GFC. In particular, the empirical distributions of the
OFHEO purchase-only housing returns and S&P/Case-Shiller returns shiftconsiderably to the left and become more spread out after the GFC, resulting in an
increase in variance and growth in value-at-risk measure. The two housing markets
have the same structures of economic riskiness. The economic risk increases for
OFHEO purchase-only and S&P/Case-Shiller housing returns after the GFC. The expost real estate premium virtually disappears after the financial crisis.
This paper can be extended in different directions. We have established that theriskiness and risk-adjusted-performance of the major asset markets change significantly
after the Global Financial Crisis. However, we have not investigated the causes andmechanisms. Future research should fill the research gap. We should also expand theanalysis to asset markets outside the U.S. One may also explore whether the comovement among different asset returns changes after the GFC. 33 Some of these
directions are being pursued and will hopefully enrich our understanding of the asset
markets and financial crises.

33

Among others, see Leung and Ng (2019) and the reference therein.

References
Afonso, G., Kovner, A., Schoar, A., 2011. Stressed, not frozen: The Federal Funds
market in the financial crisis, Journal of Finance, 66, 1109-1139.
Agarwal, S., Varshneya, S., 2020. Financial crisis and the US mortgage markets – A
review, mimeo.
Anand, A., Li, T., Kurosaki, T., Kim, Y.S., 2016. Foster-Hart risk and the too-big-to-failbanks: An empirical investigation, Journal of Baniking & Finance 68, 117-130.
Ang, A., Bekaert, G., 2002. International asset allocation with regime shifts. Review of
Financial Studies, 15, 1137-1187.
Ashley, R. A., Patterson, D. M., 2010. A test of the GARCH(1,1) specification for dailystock returns, Macroeconomic Dynamics, 14, 137-144.
Aumann, R. J., Serrano, R., 2008. An economic index of riskiness, Journal of Political
Economy, 116, 810-836.
Bai, J., Perron, P., 1998. Estimating and testing linear models with multiple structuralchanges, Econometrica, 66, 47-78.
Bernanke, B., 2018. The real effects of the financial crisis, Brookings Papers on
Economic Activity, 49(2), 251-342.
Bollerslev, T., 1986. Generalized autoregressive conditional heteroscedasticity, Journalof Econometrics, 31, 307-327.
Bollerslev, T., 2009. Glossary to ARCH (GARCH), Duke University, mimeo.
Bredin, D., O’Reilly, G., Stevenson, S., 2007. Monetary shocks and REIT returns,
Journal of Real Estate Finance and Economics, 35, 315-331.
Brookings Institution, 2018, Charting the Financial Crisis: U.S. Strategy andOutcomes, mimeo.
Brunnermeir, M., 2009. Deciphering the liquidity and credit crunch 2007-2008, Journalof Economic Perspectives, 23, 77-100.
Chang, K. L., 2010. House price dynamics, conditional higher-order moments, anddensity forecasts. Economic Modelling, 27, 1029-1039.
Chang, K. L., N. K. Chen, N. K., Leung, C. K. Y., 2011. Monetary policy, term
structure and real estate return: Comparing REIT, housing and stock, Journal ofReal Estate Finance and Economics, 43, 221-257.
Chang, K. L., N. K. Chen, N. K., Leung, C. K. Y., 2012. The dynamics of housing
returns in Singapore: How important are the international transmission
mechanisms? Regional Science and Urban Economics, 42, 516-530.
Chang, K. L., N. K. Chen, N. K., Leung, C. K. Y., 2013, In the shadow of the UnitedStates: The international transmission effect of asset returns, Pacific Economic
Review, 18(1), 1-40.

Chang, K. L., N. K. Chen, N. K., Leung, C. K. Y., 2016. Losing track of the asset
markets: The case of housing and stock, International Real Estate Review, 19, 435492.
Chauvet, M., Piger, J., 2008. A comparison of the real-time performance of businesscycle dating methods, Journal of Business and Economic Statistics, 26, 42-49.
Chen, W. S., So, K. P., Lin, M. H., 2009. Volatility forecasting with double Markov
switching GARCH models, Journal of Forecasting, 28, 681-697.
Chen, Y. T., Ho, K. Y., Tzeng, L. Y., 2014. Riskiness-minimizing spot-futures hedge
ratio, Journal of Banking and Finance, 40, 154-164.
Cheng, P., Lin, Z., Liu, Y., Zhang, Y., 2010a. Illiquidity and portfolio risk of thinly
traded assets, Journal of Portfolio Management, 36, 126-138.
Cheng, P., Lin, Z., Liu, Y., Zhang, Y., 2010b. Real estate allocation puzzle in the mixedasset portfolio: Fact or fiction? mimeo.
Chong, T. T. L., 2001. Structural Change In AR(1) Models, Econometric Theory, 17(01),87-155.
Cotter, J., Stevenson, S., 2006. Multivariate modeling of daily REIT volatility, Journalof Real Estate Finance and Economics, 32, 305-325.
Crawford, G. W., Fratantoni, M. C., 2003. Assessing the forecasting performance ofregime-switching, ARIMA and GARCH models of house prices. Journal of Real
Estate Finance and Economics, 31, 223-243.
D’Amico, S., King, T., 2013. Flow and stock effects of large-scale treasury purchases:
Evidence on the importance of local supply, Journal of Financial Economics, 108,425-448.
DeFilippis, J. ed., 2016. Urban Policy in the Time of Obama, Minneapolis: Universityof Minnesota Press.
Dueker, M. J., 1997. Markov switching in GARCH processes and mean-reverting
stock-market volatility, Journal of Business and Economic Statistics, 15, 26-34.
Dungey, M., Milunovich, G., Thorp, S., Yang, M., 2015. Endogenous crisis dating andcontagion using smooth transition structural GARCH, Journal of Banking &Finance, 58, 71-79.
Duprey, T., Klaus, B., 2017. How to predict financial stress? An assessment of Markov
switching models, ECB Working Paper No. 2057.
Eiling, E., Gerard, B., 2015. Emerging equity market comovements: Trends
and macroeconomic fundamentals, Review of Finance, 19, 1543–1585.
Eksi, O., Tas, B., 2017. Unconventional monetary policy and the stock market’s
reaction to Federal Reserve policy actions, North American Journal of Economics
and Finance, 40, 136-147.
Engle, R. F., Patton, A. J., 2001. What good is a volatility model? Quantitative Finance,

1, 237-245.
Eser, F., Schwaab, B., 2016. Evaluating the impact of unconventional monetary policy
measures: Empirical evidence from the ECB’s Securities Markets Programme,
Journal of Financial Economics, 119, 147-167.
Foster, D. P., Hart, S., 2009. An operational measure of riskiness, Journal of Political
Economy, 117, 785-814.
Glascock, J. L., Lu, C., So, R. W., 2000. Further evidence on the integration of REIT,
bond, and stock returns, Journal of Real Estate Finance and Economics, 20, 177194.
Gorton, G., 2010. Information, liquidity, and the (ongoing) panic of 2007. American
Economic Review, Papers and Proceedings, 99, 567-572.
Gray, S. F., 1996. Modeling the conditional distribution of interest rates as a regimeswitching process, Journal of Financial Economics, 42, 27-62.
Haas, M., Mittnik, S., Paolella, M., 2004. A New Approach to Markov-Switching
GARCH Models, Journal of Financial Econometrics, 2(4), 493–530.
Hamilton, J., 2018. The efficacy of large-scale asset purchases when the short-term
interest rate is at its effective lower bound, Brookings Papers on Economic
Activity, 49 (2), 543-554.
Hansen, B., 2001. The new econometrics of structural change: Dating breaks in U.S.
labour productivity, Journal of Economic Perspectives, 15(4), 117-128.
Hansen, P., Lunde, A., Nason, J., 2011. The model confidence set, Econometrica,
79(2), 453-497.
Heaton, J., and Lucas, D., 2000, Portfolio choice and asset prices: The importance ofentrepreneurial risk, Journal of Finance 55, 1163–1198.
Hendershott, P., Hendershott, R., Shilling, J., 2010. The mortgage finance bubble:
Causes and corrections, Journal of Housing Research, 19, 1-16.
Homm, U., Pigorsch, C., 2012. Beyond the Sharpe ratio: An application of the AumannSerrano index to performance measurement, Journal of Banking and Finance, 36,
2274-2284.
International Monetary Fund IMF, 2014, Staff guidance note on macroprudential
policy—Detailed guidance on instruments, mimeo.
Ivashina, V., Scharfstein, D., 2010. Bank lending during the financial crisis of 2008.
Journal of Financial Economics, 97, 319-338.
Karoglou, M., Morley, B., Thomas, D., 2013. Risk and structural instability in US homeprices, Journal of Real Estate Finance and Economics, 46, 424-436.
Kim, Y., Hwang, E., 2018. A dynamic Markov regime-switching GARCH model andits cumulative impulse response function, Statistics & Probability Letters,
139, 20-30.

Kwan, Y. K., Leung, C. K. Y. and Dong, J., 2015. Comparing consumption-based asset
pricing models: The case of an Asian city, Journal of Housing Economics, 28, 1841.
Lehkonen, H., 2015. Stock market integration and the global financial crisis, Review
of Finance, 19, 2039–2094.
Leung, C. K. Y., Ng, C. Y. J., 2019, Macroeconomic aspects of housing. In Hamilton,
J., Dixit, A., Edwards, S., Judd, K. (eds), Oxford Research Encyclopedia of
Economics and Finance, Oxford University Press.
Liang, N., McConnell, M. M., Swagel, P., 2018, Evidence on outcomes, Brookings
institution, mimeo.
Lin, Z., Liu, Y., 2008. Real estate returns and risk with heterogeneous investors, Real
Estate Economics, 36, 753-776.
Liow, K. H., Chen, Z., Liu, J., 2011. Multiple regimes and volatility transmission insecuritized real estate markets, Journal of Real Estate Finance and Economics, 42,295-328.
Liow, K. H., Ho, K. H. D., Ibrahim, M. F., Chen, Z., 2009. Correlation and volatility
dynamics in international real estate securities markets, Journal of Real Estate
Finance and Economics, 39, 202-223.
Liow, K. H., Yang, H., 2005. Long-term co-memories and short-run adjustment:
securitized real estate and stock markets, Journal of Real Estate Finance andEconomics, 31, 283-300.
Liu, J. C., 2006. Stationarity of a Markov-switching GARCH model, Journal of
Financial Econometrics, 4(4), 573–593.
Lv, J., Liu, J., 2014. Model selection principles in misspecified models, Journal of theRoyal Statistical Society, Series B, 76, 141-167.
Maheu, J. M., McCurdy, T. H., 2000. Identifying bull and bear markets in stock returns,
Journal of Business and Economic Statistics, 18, 100-112.
Malpezzi, S., 2017. Residential real estate in the U.S. financial crisis, the GreatRecession, and their aftermath, Taiwan Economic Review, 45, 5-56.
Mariano, R., Preve, D., 2012, Statistical tests for multiple forecast comparison,
Journal of Econometrics, 169, 123-130.
Mehra, R., 2006. The equity premium puzzle: A review, Foundations and Trends in
Finance, 2, 1-81.
Mehra, R., Prescott, E. C., 1985. The equity premium: a puzzle, Journal of Monetary
Economics 15, 145-161.
Miles, W., 2008a. Boom-Bust cycles and the forecasting performance of linear and nonlinear models of house prices, Journal of Real Estate Finance and Economics, 36,
249-264.

Miles, W., 2008b. Volatility clustering in U.S. home prices. Journal of Real Estate
Research, 30, 73-90.
Mollah, S., Shahiduzzaman Quoreshi, A.M.M., Zafirov, G., 2016. Equity market
contagion during global financial and Eurozonecrises: Evidence from a dynamic
correlation analysis, Journal of International Financial Markets, Institutions &Money, 41, 151-167.
Mondria, J., Quintana‐Domeque, C., 2013. Financial Contagion and Attention
Allocation, Economic Journal, 123, 429-454.
Perron, P., 2006. Dealing with structural breaks. In Patterson, K., Mills, T. C. (Eds.),
Palgrave Handbook of Econometrics, Vol. 1:Econometric Theory, Palgrave
MacMillan, Basingstoke, Hampshire, UK, 278-352.
Quah, D. T., 1996. Twin peaks: Growth and convergence in models of distribution
dynamics, Economic Journal, 106, 1045-1055.
Quan, D., Titman, S., 1999. Do real estate prices and stock prices move together? Aninternational analysis, Real Estate Economics, 27, 183-207.
Riddiough, T., 2020. Pension funds and private equity real estate: History,
performance, pathologies, risks, mimeo.
Rosenblatt, M., 1952. Remarks on multivariate transformation, Annals of Mathematical
Statistics, 23, 470-472.
Schulze, K., 2014. Existence and computation of the Aumann-Serrano index of
riskiness and its extension, Journal of Mathematical Economics, 50, 219-224.
Seiler, M. J., Webb, J. R., Myer, N. F. C., 1999. Diversification issues in real estate
investment. Journal of Real Estate Literature, 7, 163-179.
Shilling, J., 2003. Is there a risk premium puzzle in real estate? Real Estate Economics,
31, 501-525.
Silverman, B. W., 1992. Density Estimation for Statistics and Data Analysis. London:
Chapman & Hall.
Steeley, J., 2015.The side effects of quantitative easing: Evidence from the UK bondmarket, Journal of International Money and Finance, 51, 303-336.
Taylor, J. B., 2009. Getting Off Track: How Government Actions and Interventions, Caused,
Prolonged and Worsened the Financial Crisis. Stanford, CA: Hoover Institution Press.
Trichet, J. C., 2009. (Under-)Pricing of risks in the financial sector, speech, ECB.
Tsai, I. C., Chen, M. C., Ma, T., 2010. Modelling house price volatility states in the UK
by switching ARCH models, Applied Economics, 42, 1145-1153.
Wall Street Journal, 2016. Global Financial Crisis spurs evolution in the asset
management industry, 4th August.
Weale, M., Wieladek, T., 2016. What are the macroeconomic effects of asset purchases?

Journal of Monetary Economics, 79, 81-93.
Zhou, J., Anderson, R. I., 2012. Extreme risk measures for international REIT market,
Journal of Real Estate Finance and Economics, 45, 152-170.
Zhou, J., Kang, Z., 2011. A comparison of alternative forecast models of REIT volatility,
Journal of Real Estate Finance and Economics, 42, 275-294.

Table 1 RMSE for Pre-crisis period
AR(1)AR(1)-GRACH(1,1)

MS-AR(1)MS-ARCH(1)Panel A: return forecasts
SP500
Dow Jones
Nasdaq6.106
5.676
12.810

6.106
5.676
12.963

5.788
5.688
12.2495.800
5.834
12.628EREITsOFHEO Purchase-only
S&P/Case-Shiller

5.894
0.636
0.627

5.906
0.656#
0.656#

5.589
0.564
0.543

5.701
0.636
---

Panel B:variance forecasts
SP500
Dow Jones

55.70845.76355.69145.78341.98241.96842.30947.940NasdaqEREITsOFHEO Purchase-only
S&P/Case-Shiller

317.318
57.3482.333
1.775

293.258
58.5932.450#
1.940#

241.14147.8681.934
1.487

269.89442.6962.398---

Note: # means that the AR(1)-ARCH(1) model is employed. --- means that the MS-ARCH(1)model cannot be estimated. The lowest RMSE is in Italic.
Table 2 RMSE for Post-crisis period
AR(1)AR(1)-GRACH(1,1)

MS-AR(1)MS-ARCH(1)Panel A: return forecasts
SP500

5.257

5.261

5.229

5.543

Dow Jones
NasdaqEREITsOFHEO Purchase-only

4.989
6.239
6.790
0.749

5.002
6.244
6.859
0.773#

4.987
6.227
6.347
0.751

5.004
6.229
6.803
0.700

S&P/Case-Shiller

0.694

0.694#

0.670

0.658

48.17441.49669.090111.770

47.71341.74469.012112.04748.35941.55168.98279.63053.12842.51069.013109.080

3.393
3.164

3.451#
3.203#

3.414
1.986

1.648
3.067

Panel B:variance forecasts
SP500
Dow Jones
NasdaqEREITsOFHEO Purchase-only
S&P/Case-Shiller

Note: # means that the AR(1)-ARCH(1) model is employed.

Table 3 Summary of EPPC for Pre-crisis period in terms of SLC
Class 1

Class 2SP500
Dow Jones
NasdaqEREITsAll models
All models
All models
All models

\
\
\
\

OFHEO Purchase-only
S&P/Case-Shiller

All models#All models#\\Panel B: variance forecasts
SP500
Dow Jones
NasdaqMS-AR, MS-ARCH
AR,AR-GARCH,MS-AR
MS-AR,MS-ARCH

AR, AR-GARCH
MS-ARCHAR,AR-GARCH

Panel A: return forecasts

EREITsAll models\
OFHEO Purchase-only
All models#\
S&P/Case-Shiller
All models#\Notes: SLC refers to square loss criterion. The predictive power of Class 1 is better
than that of Class 2. # means that the AR(1)-ARCH(1) model is employed.
Table 4 Summary of EPPC for Post-crisis period in terms of SLC
Class 1

Class 2

Panel A: return forecasts
SP500

AR,AR-GARCH,MS-AR

MS-ARCHDow Jones
NasdaqEREITsOFHEO Purchase-only

AR,AR-GARCH,MS-AR
All models
All models
AR,AR-ARCH,MS-

MS-ARCH\

S&P/Case-Shiller

ARCH
MS-AR,MS-ARCH

AR,AR-ARCH

Panel B: variance forecasts
SP500
Dow Jones
Nasdaq

All models
All models
All models

\
\
\

\
MS-AR

EREITsAll models\
OFHEO Purchase-only
All models#\
S&P/Case-Shiller
AR,MS-AR,MS-ARCH
AR-ARCH
Notes: SLC refers to square loss criterion. The predictive power of Class 1 is better
than that of Class 2. # means that the AR(1)-ARCH(1) model is employed.

Table 5 Summary of risk measures for the best specification
Pre-crisis

Post-crisis

Panel A: Variance
SP500
Dow Jones
NasdaqEREITs291.507149.7441285.749
129.655

77.96371.226111.895
678.5106.648
1.587#

12.8998.505

Panel B: Value-at-Risk (probability=1%)
SP500
37.729Dow Jones
29.881
Nasdaq101.410

7.953
8.731
10.325EREITsOFHEO Purchase-only
S&P/Case-Shiller

20.9081.704
6.219#

8.928
3.981
6.751

Panel C: ‫ܵܣ‬ேSP500
Dow Jones

-----

3.408
3.340

NasdaqEREITsOFHEO Purchase-only
S&P/Case-Shiller--3.9200.791
0.083#

3.699
16.183
2.150
8.530

Panel D: AS
SP500

---

3.029

Dow Jones
NasdaqEREITsOFHEO Purchase-only----9.686
0.625

3.200
3.993
3.308
2.192

S&P/Case-Shiller

1.451#

9.603

OFHEO Purchase-only
S&P/Case-Shiller

Notes: --- means that the risk measure cannot be calculated. # means that the third bestmodel is employed due to stationarity.

Table 6 Summary of performance measures for the best specification
Pre-crisis

Post-crisis

Panel A: Sharpe ratio
SP500
Dow Jones-0.306-0.120

1.295
1.263

NasdaqEREITsOFHEO Purchase-only
S&P/Case-Shiller-0.4891.452
1.629
7.575#

1.430
0.805
0.835
0.171

Panel B: ‫ܫܲܧ‬ே
SP500

---

3.356

Dow Jones
NasdaqEREITsOFHEO Purchase-only----4.218
5.310

3.192
4.090
1.295
1.395

S&P/Case-Shiller

114.749#

0.058

Panel C: EPI
SP500
Dow Jones
NasdaqEREITs------1.707

3.776
3.332
3.788
6.337

OFHEO Purchase-only
S&P/Case-Shiller

6.722
6.576#

1.368
0.052

Notes: --- means that the performance measure cannot be calculated. # means that thethird best model is employed due to stationarity.

Figures
Figure 1. Annualized excess returns for Pre-Crisis period(2000m1-2006m6)(a)SP500(b)Dow Jones

40

40

0

0

-40

-40

-80

-80

2001

2002

2003

2004

2005

2001

2002

(c)Nasdaq
40

0

0

-40

-40

-80

-80

2002

2003

2004

2004

2005

(d)EREITs

40

2001

2003

2005

2001

(e)OFHEO Prchase-only

2003

2004

2005

(f)S&P/Case-Shiller

12

12

10

10

8

8

6

6

4

4

2

2

0

2002

0
2001

2002

2003

2004

2005

2001

2002

2003

2004

2005

Figure 2 Annualized excess returns for Post-Crisis period(2009m1-2019m12)(a)SP500(b)Dow Jones

80

80

60

60

40

40

20

20

0

0

-20

-20
10

11

12

13

14

15

16

17

18

19

10

11

12

13

(c)Nasdaq

14

15

16

17

18

19

17

18

19

18

19

(d)EREITs

80

80

60

60

40

40

20

20

0

0

-20

-20
10

11

12

13

14

15

16

17

18

19

10

11

(e)OFHEO Purchase-only

12

13

14

15

16

(f)S&P/Case-Shiller

12

12

8

8

4

4

0

0

-4

-4

-8

-8
10

11

12

13

14

15

16

17

18

19

10

11

12

13

14

15

16

17

Figure 3. Empirical distributions for returns in terms of minimum RMSE
Note: The third best model is employed for the S&P/Case-Shiller return in the Pre-crisis
period because the best and second best models are non-stationary.

(continued)

Figure 4. Empirical distributions for variances in terms of minimum RMSE
Note: The third best model is employed for the S&P/Case-Shiller return in the Pre-crisis
period because the best and second best models are non-stationary.

(continued)

Figure 5 Empirical distributions for returns of the data after GFC, the best performing
models in the Post-crisis period, and Post-crisis period.

(continued)

Figure 6 Empirical distributions for (implied) variances of the best performing modelin the Post-crisis period, and the best Pre-crisis model applies to Post-crisis period

Appendix

This appendix consists of several parts.
Appendix A provides additional results, including when MAE (instead of RMSE) isused, the classification of EPPC when ALC (instead of SLC) is used, return forecast,and variance forecast.
Appendix B provides more discussion on the economic performance index and theeconomic risk index.
Appendix C briefly explains the classifications of models into different “equivalentpredictive power classes” (EPPC).
Appendix D provides more discussion on the “counter-factual” experiments presented
in the text.

Table A-1 MAE for Pre-crisis period
AR(1)AR(1)-GRACH(1,1)

MS-AR(1)MS-ARCH(1)Panel A: return forecasts
SP500
Dow Jones
NasdaqEREITsOFHEO Purchase-only
S&P/Case-Shiller

4.668
4.594
9.392
4.560
0.317
0.356

4.667
4.590
9.479
4.561
0.302#
0.338#

4.601
4.557
9.024
4.303
0.253
0.270

4.584
4.612
9.152
4.423
0.262
---

Panel B:variance forecasts
SP500
Dow Jones
NasdaqEREITsOFHEO Purchase-only
S&P/Case-Shiller

39.47032.090179.39735.3500.612
0.553

39.41631.929172.69434.7000.388#
0.317#

31.35328.551
153.22731.6880.497
0.388

31.46929.957165.64229.7900.508---

Note: # means that the AR(1)-ARCH(1) model is employed. --- means that the MS-ARCH(1)model cannot be estimated.
Table A-2 MAE for Post-crisis period
AR(1)AR(1)-GRACH(1,1)

MS-AR(1)Panel A: return forecasts
SP500
Dow Jones
NasdaqEREITsOFHEO Purchase-only
S&P/Case-Shiller

3.929
3.840
4.748
4.964
0.416
0.360

3.951
3.859
4.770
4.968
0.439#
0.359#

3.893
3.840
4.732
4.777
0.420
0.378

4.268
3.848
4.736
4.862
0.430
0.314

Panel B:variance forecasts
SP500
Dow Jones
NasdaqEREITsOFHEO Purchase-only
S&P/Case-Shiller

29.96926.12842.07651.6870.781
0.687

30.66326.22045.22158.2810.658#
0.416#

30.00526.18742.17440.6810.735
0.684

39.1935
26.93542.33549.9850.575
0.477

Note: # means that the AR(1)-ARCH(1) model is employed.

MS-ARCH(1)Table A-3

Smmary of EPPC for Pre-crisis period in terms of ALC
Class 1

Class 2

Panel A: return forecasts
SP500
Dow Jones
NasdaqEREITsOFHEO Purchase-only
S&P/Case-Shiller

All models
All models
All models
All models
AR-ARCH,MS-AR,MS-ARCH
MS-AR

\
\

Panel B: variance forecasts
SP500
Dow Jones
NasdaqEREITsOFHEO Purchase-only
S&P/Case-Shiller

MS-AR, MS-ARCH
AR-GARCH,MS-AR,MS-ARCH
All models
All models
All models#
AR-ARCH,MS-AR

AR, AR-GARCH
AR
\
\
\

\
\
AR
AR, AR-ARCH

AR

Notes: ALC refers to absolute loss criterion. The predictive power of Class 1 is better
than that of Class 2. # means that the AR(1)-ARCH(1) model is employed.
Table A-4 Summary of EPPC for Post-crisis period in terms of ALC
Panel A: return forecasts
SP500
Dow Jones
NasdaqEREITsOFHEO Purchase-only
S&P/Case-Shiller

Class 1

Class 2

AR,AR-GARCH,MS-AR
All models
All models
All models
AR
MS-ARCHMS-ARCH\\\AR-ARCH,MS-AR,MS-ARCH
AR,AR-ARCH,MS-AR

Panel B: variance forecasts
SP500
All models
Dow Jones
All models
Nasdaq
All models
EREITs
AR,MS-AR,MS-ARCH
OFHEO Purchase-only
All models#
S&P/Case-Shiller
All models#\\\
AR-GARCH\

Notes: ALC refers to absolute loss criterion. The predictive power of Class 1 is better
than that of Class 2. # means that the AR(1)-ARCH(1) model is employed.

Table A-5 Estimation results in terms of RMSE of return forecasts
Sp500
Pre-crisis
𝜃𝜃0,1

Dow Jones

Post-crisis

Pre-crisis-10.604**(5.197)2.404**(1.201)
0.531**(0.213)0.758***(0.081)40.959***(15.587)
18.615***(5.063)4.154**(1.808)2.353***(0.907)-0.309(0.424)0.810***(0.057)4.746(5.038)26.467***(3.103)p

3.971***(1.327)4.172(2.823)1.413(1.378)4.346***(1.312)

1.643(23.444)7.072(31.457)

𝑃𝑃11
𝑃𝑃22

0.982
0.985

0.804
0.987

0.838
0.999

-205.813-367.118

𝜃𝜃0.2
𝜃𝜃1,1
𝜃𝜃1,2
𝜔𝜔1

𝜔𝜔2

-0.154(0.733)Post-crisis

0.889***(0.059)32.214***(5.633)-2.477(23.905)
2.141**(0.957)0.034(6.315)0.802***(0.059)9.122(103.287)
24.848***(2.972)

𝛼𝛼1

𝛼𝛼2
𝛽𝛽1

𝛽𝛽2

𝑞𝑞

Log(L)-208.239-363.134

Notes: One asterisks (*), two asterisks (**), and three asterisks (***) refer to significance at 10%, 5%,
and 1% levels, respectively. Standard errors are in parentheses. Log(L) refers to the logarithm of thelikelihood function. Values in parentheses represent standard errors.

Table A-6 Estimation results in terms of RMSE of return forecasts
NasdaqPre-crisis
𝜃𝜃0,1

EREITsPost-crisis

Pre-crisis

Post-crisis-17.855(11.825)
1.065(1.347)0.669***(0.201)0.909***(0.058)360.409**(143.162)
58.424***(15.900)

9.861***(3.747)2.644**(1.188)0.358(0.229)0.823***(0.067)3.676(2.807)42.186***(5.242)-1.730***(0.649)6.119***(2.033)
0.976***(0.102)0.683***(0.085)1.931(1.652)37.184***(8.462)34.150(71.940)
2.389***(0.849)0.632(0.867)0.782***(0.050)35.656(1712.747)
30.723***(4.539)p3.774*(2.093)4.274(1.565)1.316(1.106)3.830**(1.550)2.036*(1.185)3.851**(1.518)2.820**(1.399)4.879**(2.411)

𝑃𝑃11
𝑃𝑃22

0.978
0.986

0.788
0.979

0.884
0.979

0.944
0.992

-248.483-388.404-202.689-381.954

𝜃𝜃0.2
𝜃𝜃1,1
𝜃𝜃1,2
𝜔𝜔1

𝜔𝜔2
𝛼𝛼1

𝛼𝛼2
𝛽𝛽1

𝛽𝛽2

𝑞𝑞

Log(L)

Notes: One asterisks (*), two asterisks (**), and three asterisks (***) refer to significance at 10%, 5%,
and 1% levels, respectively. Standard errors are in parentheses. Log(L) refers to the logarithm of thelikelihood function. Values in parentheses represent standard errors.

Table A-7

Estimation results in terms of RMSE of return forecasts
OFHEO Purchase-only

Pre-crisis𝜃𝜃0,1

Post-crisis

S&P/Case-Shiller
Pre-crisis

Post-crisis

0.393(1.535)0.432***(0.106)0.734**(0.370)0.940***(0.017)1.402(1.209)0.037***(0.006)-2.669***(0.957)0.269***(0.050)0.020(0.330)0.950***(0.010)0.533(0.478)0.107***(0.026)0.027(0.406)
0.487**(0.192)-2.230(2.152)0.275***(0.105)
1.142***(0.395)0.979***(0.013)0.870(0.856)
0.046***(0.009)-0.322***(0.049)0.207***(0.013)0.963***(0.012)0.955***(0.003)0.058**(0.027)0.005***(0.002)0.123(0.266)99.988 × 10−2 ***(2.881 × 10−4 )p3.054*(1.839)3.502**(1.782)

4.211**(1.649)5.390**(2.423)2.969*(1.546)3.508*(1.871)2.127***(0.396)2.842***(0.405)

𝑃𝑃11
𝑃𝑃22

0.955
0.971

0.985
0.995

0.951
0.971

0.893
0.945

-12.148-79.239-14.494-8.305

𝜃𝜃0.2
𝜃𝜃1,1
𝜃𝜃1,2
𝜔𝜔1

𝜔𝜔2
𝛼𝛼1

𝛼𝛼2
𝛽𝛽1

𝛽𝛽2

𝑞𝑞

Log(L)

Notes: One asterisks (*), two asterisks (**), and three asterisks (***) refer to significance at 10%, 5%,
and 1% levels, respectively. Standard errors are in parentheses. Log(L) refers to the logarithm of thelikelihood function. Values in parentheses represent standard errors.

Table A-8 Estimation results in terms of RMSE of variance forecasts
Sp500
Pre-crisis
𝜃𝜃0,1
𝜃𝜃0.2
𝜃𝜃1,1
𝜃𝜃1,2
𝜔𝜔1

𝜔𝜔2

-10.604**(5.197)2.404**(1.201)
0.531**(0.213)0.758***(0.081)40.959***(15.587)
18.615***(5.063)Dow Jones

Post-crisis
1.927**(0.978)0.810***(0.074)9.318(10.250)

𝛼𝛼1

0.160
(0.116)𝛽𝛽1

0.531(0.401)

𝛼𝛼2

𝛽𝛽2
p

3.971***(1.327)4.172(2.823)

𝑃𝑃11
𝑃𝑃22

0.982
0.985

𝑞𝑞

Log(L)-205.813

Pre-crisis-4.775(4.352)
1.257(1.402)0.625***(0.235)0.832***(0.111)37.648**(18.876)
21.794***(5.958)Post-crisis
2.098***(0.816)0.804***(0.055)24.888***(2.803)3.846*(2.298)4.094*(2.458)-367.856-206.293-363.136

Notes: One asterisks (*), two asterisks (**), and three asterisks (***) refer to significance at 10%, 5%,
and 1% levels, respectively. Standard errors are in parentheses. Log(L) refers to the logarithm of thelikelihood function. Values in parentheses represent standard errors.

Table A-9 Estimation results in terms of RMSE of variance forecasts
NasdaqPre-crisis
𝜃𝜃0,1

EREITsPost-crisis

Pre-crisis

Post-crisis-17.855(11.825)
1.065(1.347)0.669***(0.201)0.909***(0.058)360.409**(143.162)
58.424***(15.900)

9.861***(3.747)2.644**(1.188)0.358(0.229)0.823***(0.067)3.676(2.807)42.186***(5.242)-0.865(0.909)9.038**(4.007)
0.946***(0.085)0.595***(0.154)4.371*(2.561)45.293*(26.734)
0.776(0.618)0.000(0.368)34.150(71.940)
2.389***(0.849)0.632(0.867)0.782***(0.050)35.656(1712.747)
30.723***(4.539)p3.774*(2.093)4.274(1.565)1.316(1.106)3.830**(1.550)4.688***(1.190)
4.180(3.655)2.820**(1.399)4.879**(2.411)

𝑃𝑃11
𝑃𝑃22

0.978
0.986

0.788
0.979

0.991
0.985

0.944
0.992

-248.483-388.404-202.717-381.954

𝜃𝜃0.2
𝜃𝜃1,1
𝜃𝜃1,2
𝜔𝜔1

𝜔𝜔2
𝛼𝛼1

𝛼𝛼2
𝛽𝛽1

𝛽𝛽2

𝑞𝑞

Log(L)

Notes: One asterisks (*), two asterisks (**), and three asterisks (***) refer to significance at 10%, 5%,
and 1% levels, respectively. Standard errors are in parentheses. Log(L) refers to the logarithm of thelikelihood function. Values in parentheses represent standard errors.

Table A-10

Estimation results in terms of RMSE of variance forecasts
OFHEO Purchase-only

Pre-crisis𝜃𝜃0,1

Post-crisis

S&P/Case-Shiller
Pre-crisis

Post-crisis

0.393(1.535)0.432***(0.106)0.734**(0.370)0.940***(0.017)1.402(1.209)0.037***(0.006)-2.669***(0.957)0.269***(0.050)0.019(0.330)0.950***(0.010)0.533(0.478)0.107***(0.026)0.027(0.406)
0.487**(0.192)-2.230(2.152)0.275***(0.105)
1.142***(0.395)0.979***(0.013)0.870(0.856)
0.046***(0.009)-2.339***(0.881)0.087(0.057)-0.057(0.383)0.985***(0.010)0.709(0.771)0.174***(0.022)p

3.054*(1.839)3.502**(1.782)4.212**(1.649)5.390**(2.423)2.969*(1.546)3.508*(1.871)

3.277**(1.583)4.877**(2.109)

𝑃𝑃11
𝑃𝑃22

0.955
0.971

0.986
0.995

0.951
0.971

0.964
0.992

-12.148-79.239-14.494-74.225

𝜃𝜃0.2
𝜃𝜃1,1
𝜃𝜃1,2
𝜔𝜔1

𝜔𝜔2
𝛼𝛼1

𝛼𝛼2
𝛽𝛽1

𝛽𝛽2

𝑞𝑞

Log(L)

Notes: One asterisks (*), two asterisks (**), and three asterisks (***) refer to significance at 10%, 5%,
and 1% levels, respectively. Standard errors are in parentheses. Log(L) refers to the logarithm of thelikelihood function. Values in parentheses represent standard errors.

Appendix B: More discussion on the economic performance index and economic risk
index
Although the Sharpe ratio is a widely accepted measure for comparing relative
performance among financial assets due to its easy implementation, the lack ofmonotonic stochastic dominance is its greatest weakness. Instead, Homm and Pigorsch(2012) propose a different economic performance index (EPI), which satisfies bothduality and homogeneity axioms, and is defined as follows:
EPI =𝐸𝐸(𝑟𝑟𝑡𝑡 −𝑟𝑟𝑓𝑓,𝑡𝑡 )(B1)

𝐴𝐴𝐴𝐴(𝑟𝑟𝑡𝑡 −𝑟𝑟𝑓𝑓,𝑡𝑡 )where 𝑟𝑟𝑡𝑡 is the nominal return of a risky asset; 𝑟𝑟𝑓𝑓,𝑡𝑡 is the return of a risk-free asset;
𝑟𝑟𝑡𝑡 − 𝑟𝑟𝑓𝑓,𝑡𝑡 is the excess return; E denotes the expectation operator; and 𝐴𝐴𝐴𝐴(𝑟𝑟𝑡𝑡 − 𝑟𝑟𝑓𝑓,𝑡𝑡 )refers to the economic risk index of Aumann and Serrano (2008). Similar to the Sharpe
ratio, the EPI measure is also a risk-adjusted index. The main difference between the
EPI and the Sharpe ratio is that the riskiness of a risky asset is proxied by 𝐴𝐴𝐴𝐴(𝑟𝑟𝑡𝑡 −𝑟𝑟𝑓𝑓,𝑡𝑡 ) for the EPI and by the standard deviation of excess returns for the Sharpe ratio.

Aumann and Serrano (2008) derive the economic index of riskiness in the case of
the CARA (constant absolute risk aversion) utility function. According to Theorem B
of Aumann and Serrano (2008), the riskiness of an asset can be derived when the
expected utility of an investor who holds a risky asset is identical to the expected utilityof an investor who does not hold a risky asset. The indifference condition for an investor
holding or not holding a risky asset can be expressed as follows:E �−exp �−𝛼𝛼�𝑊𝑊𝑡𝑡 + 𝑟𝑟𝑡𝑡 − 𝑟𝑟𝑓𝑓,𝑡𝑡 ��� = − exp(−α𝑊𝑊𝑡𝑡 )(B2)

where 𝛼𝛼 is the coefficient of absolute risk aversion and 𝑊𝑊𝑡𝑡 is the initial wealth.
After defining α =E �exp �−1, Equation (E2) can be rewritten as follows:

𝐴𝐴𝐴𝐴(𝑟𝑟𝑡𝑡 −𝑟𝑟𝑓𝑓,𝑡𝑡 )𝑟𝑟𝑡𝑡 −𝑟𝑟𝑓𝑓,𝑡𝑡

𝐴𝐴𝐴𝐴(𝑟𝑟𝑡𝑡 −𝑟𝑟𝑓𝑓,𝑡𝑡 )�� = 1(B3)

where 𝐴𝐴𝐴𝐴(𝑟𝑟𝑡𝑡 − 𝑟𝑟𝑓𝑓,𝑡𝑡 ) is the economic index of riskiness. This index is negatively

related to the coefficient of absolute risk aversion, but does not depend on the wealthlevel.
When the excess return is normally distributed, Homm and Pigorsch (2012) prove
the nonlinear relationship between the economic performance index and the Sharpe
ratio. Their relationship has the following form:
𝐸𝐸𝐸𝐸𝐸𝐸𝑁𝑁 =𝐸𝐸(𝑟𝑟𝑡𝑡 −𝑟𝑟𝑓𝑓,𝑡𝑡 )𝐴𝐴𝐴𝐴𝑁𝑁 �𝑟𝑟𝑡𝑡 −𝑟𝑟𝑓𝑓,𝑡𝑡 �=2×�

𝐸𝐸(𝑟𝑟𝑡𝑡 −𝑟𝑟𝑓𝑓,𝑡𝑡 )�𝑉𝑉(𝑟𝑟𝑡𝑡 −𝑟𝑟𝑓𝑓,𝑡𝑡2

� = 2 × 𝑆𝑆𝑆𝑆2)

where 𝑉𝑉(𝑟𝑟𝑡𝑡 − 𝑟𝑟𝑓𝑓,𝑡𝑡 ) is the variance of excess return, and SR is the Sharpe ratio.

(B4)

It is clear from Equation (B4) that 𝐸𝐸𝐸𝐸𝐸𝐸𝑁𝑁 and SR have a positive relationship.
Even though the values for 𝐸𝐸𝐸𝐸𝐸𝐸𝑁𝑁 and SR differ, the performance ranking of financial
assets remains unchanged no matter which performance criterion is used. Evidently, if
the assumption of normality is imposed on financial asset returns, the 𝐸𝐸𝐸𝐸𝐸𝐸𝑁𝑁 can beeasily calculated using only the first two moments, rather than the higher moments.

Appendix C: Classification of models into different “equivalent predictive powerclasses” (EPPC).
This appendix draws heavily on Kwan et al. (2015), Chang et al. (2016).
Consider the situation with ( K + 1) models. We then define the “ j -th loss
differential” d j ,t=d j ,t L ( etj+ h|t ) − L ( etj++h1|t ) , j = 1,..., K
where L (.) continues to denote some loss function. We collect these loss differential
in a vector, dt = {d j ,t } , j = 1,..., K . We then take average of it,d≡

1 P∑ dt .
P t =1()( ) ()

 −1 d − µ → χ 2Mariano and Preve (2012) (MP hereafter) prove that P d − µ ' Ωk is a consistent(in distribution), where µ is the mean of the distribution, Ωestimator of the population variance-covariance matrix Ω , and k is the degree of
freedom. Thus, MP test enables us to test whether all models of concern have thesame predictive power.
Then we define a procedure which enable us to categorize the models into
different “equivalent classes,” each of them contains model with (statisticallyspeaking) the same predictive power. Our procedures are similar to Hansen et al.
(2011) and here are the steps.
1. We consider N 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 of Model 1 isbetter than that of Model 2, which in turn is better than of Model 3, and so on.
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 thechosen criteria.
3. If the null hypothesis is rejected, then we eliminate the model with the leastpredictive 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 hypothesis ofequal predictive power is accepted.
4. Assume that in Step (3), there are N1 models which are found to possess the equal
predictive power, N1 > 0 . For future reference, they are referred to as Class 1 among
the N models. By construction, there are ( N − N1 ) models which do not have thesame predictive power as the models in Class 1. We now repeat Step (1) on these( N − N1 ) models until the null of equal predictive power is not rejected. Assume that
there are N 2 models, N 2 > 0 , in the final list and they are identified as Class 2.

= N1 + N 2 , then the procedure stops. The set of models are divided into two
5. If N
classes, and models within each class have the same predictive power. Every model inClass 1 has higher predictive power than any model in Class 2.
6. If, instead, N > N1 + N 2 . Again, by construction, there are ( N − N1 − N 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 ( N − N1 − N 2 ) models until the null of equal predictive
power is not rejected. Assume that there are N₃ models, N 3 > 0 , in the final list, andthey are identified as Class 3.
7. If N = N1 + N 2 + N 3 , 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 g classes, then it must be that N = N1 + N 2 + ... + N g .

Appendix D: More discussion on the “counter-factual” experiments presented in themain text.

Recall that in the main text, we conduct a simple experiment. We consider the
scenario in which the best model selected in the pre-crisis period to be the best modelin the post-crisis period, with parameters re-estimated. We then compare the
distribution generated by this “pseudo-best model” with the “truly best model,” which
is selected by the formal model selection procedure. In the main text, we presented thegraphs. Here we provide more detailed comments, which are divided into two parts.
The first part is related to the return forecast and the second is related to the varianceforecast. We begin with the return forecast.
(1) SP500 and EREITs have the same best model before and after Global Financial
Crisis.
(2) The difference between the pre-crisis best model apply to post-crisis period andthe post-crisis best model apply to post-crisis period is insignificant for Dow Jones,
Nasdaq and OFHEO index.
(3) For the S&P/Case-Shiller index, the empirical distribution of true data is a
Bimodal distribution. The empirical distribution evaluated from the pre-crisis bestmodel apply to post-crisis period is a single-peak distribution. The bias is larger.
The empirical distribution evaluated from the post-crisis best model apply to postcrisis period is a two-peak distribution. Its empirical distribution is similar with that oftrue data. Specifically, if we do not consider the possibility of changing in best model,
the “risk of return” and the probability that occurring lower return will be
underestimated.
We now present the remarks related to the variance forecast.
(1) Notice that the “true variance” is unknown. Only the empirical distributions forvariances of the pre-crisis best model apply to post-crisis period and the post-crisis bestmodel apply to post-crisis period are plotted.
(2) The best model for S&P/Case-Shiller index does not change before and after Global
Financial Crisis.
(3) The bandwidth suggested by Silverman (1992) is used.
0.9

𝑄𝑄 −𝑄𝑄1

h = (𝑁𝑁)1/5 min(�𝑉𝑉(𝑋𝑋), 3

1.349

)

where 𝑄𝑄3 is the third quartile and 𝑄𝑄1 is the first quartile.
For the MS-AR model, there are two different variances. When the 𝑄𝑄3 − 𝑄𝑄1 = 0, the𝑄𝑄3 is replaced by the maximum of variance and 𝑄𝑄1 is replaced by the minimum of

variance.
(4) For SP500, Dow Jones and EREITs, the pre-crisis best model apply to post-crisis
period has larger variance than the post-crisis best model apply to post-crisis period.
That is, if we ignore the structural change and use the same model found in the precrisis period, the “uncertainty about variance” will be overestimated.
(5) For OFHEO, the risk is more spread out in the post-crisis best model apply to postcrisis period than in pre-crisis best model apply to post-crisis period. The pre-crisis bestmodel applies to post-crisis period will under-estimate the variation of risk.
(6) In Figure 6(c) (Nasdaq), the shape of empirical distribution of variance is very
similar for the pre-crisis best model apply to post-crisis period and the post-crisis bestmodel apply to post-crisis period. But, the empirical distribution of the post-crisis bestmodel apply to post-crisis period is slightly in the left of that of pre-crisis best model
apply to post-crisis period. That is, if we ignore the structural change and use the samemodel found in the pre-crisis period, the “occurrence of lower variance” will be
underestimated.