Time-varying expected momentum profits

Time-varying expected momentumprofitsDongcheol Kim a ,Tai-Yong Roh b ,Byoung-Kyu Min c ,⇑,Suk-Joon Byun baBusiness School,Korea University,Seoul,Republic of KoreabGraduate School of Finance,Korea Advanced Institute of Science and Technology (KAIST),Seoul,Republic of Korea cInstitute of Financial Analysis,University of Neuchatel,Neuchatel,Switzerlanda r t i c l e i n f o Article history:Received 15November 2013Accepted 11September 2014Available online 22October 2014JEL classifications:G12G14Keywords:MomentumTime-varying expected returnsMarkov switching regression model Business cycle Procyclicality Growth optionsa b s t r a c tThis paper examines the time variations of expected momentum profits using a two-state Markov switching model with time-varying transition probabilities to evaluate the empirical relevance of recent rational theories of momentum profits.We find that in the expansion state the expected returns of winner stocks are more affected by aggregate economic conditions than those of loser stocks,while in the recession state the expected returns of loser stocks are more affected than those of winner stocks.Consequently,expected momentum profits display strong procyclical variations.We argue that the observed momentum profits are the realization of such expected returns and can be interpreted as the procyclicality premium .We provide a plausible explanation for time-varying momentum profits through the differential effect of leverage and growth options across business cycles.Ó2014Elsevier B.V.All rights reserved.1.IntroductionThe cross-sectional difference in average stock returns across their recent past performance has become one of the most contro-versial issues in academia as well as industry since the pioneering work of Jegadeesh and Titman (1993).A simple momentum strat-egy buying recent winners and selling recent losers generates both statistically and economically significant profits.There are two explanations for the sources of these momentum profits in the literature:One is that momentum profits result from investors’irrational underreaction to firm-specific information (e.g.,Barberis et al.,1998;Daniel et al.,1998;Hong and Stein,1999;Jiang et al.,2005;Zhang,2006;Chui et al.,2010).Another is a rational risk-based explanation stating that momentum profits are realizations of risk premiums because winner stocks are riskier than loser stocks (e.g.,Conrad and Kaul,1998;Berk et al.,1999;Johnson,2002;Ahn et al.,2002;Bansal et al.,2005;Sagi and Seasholes,2007;Liu and Zhang,2008).In contrast to the extensive aforementioned literature on the cross-sectional aspects of momentum,the intertemporal aspects of momentum profits have received much less attention.Studies of the intertemporal aspects of momentum profits focus on procyc-lical time variations in momentum profits.Johnson (2002)and Sagi and Seasholes (2007)provide the theoretical insight that momen-tum profits are likely to be procyclical.According to Johnson (2002),winner stocks have higher exposure to growth rate risk than loser stocks.Since expected growth rates tend to be high in expansions and growth rate risk is accordingly high,expected returns on momentum portfolios should be higher in expansions than in recessions.In a similar vein,the model of Sagi and Seasholes (2007)suggests that winner stocks tend to have more valuable growth options in expansions than in recessions and such firms are riskier and associated with higher expected returns in expansions,since growth options are riskier than assets in place.There is also empirical evidence of the procyclicality of momen-tum profits.Chordia and Shivakumar (2002)show that profits of momentum strategies can be explained by a set of lagged macro-economic variables that are related to business cycles and payoffs to momentum strategies disappear after stock returns are adjusted for their predictability based on these macroeconomic variables.These authors also find that momentum trading delivers reliably positive profits only during expansionary periods but negative,/10.1016/j.jbankfin.2014.09.0040378-4266/Ó2014Elsevier B.V.All rights reserved.⇑Corresponding author.Tel.:+41327181574;fax:+41327181401.E-mail addresses:kimdc@korea.ac.kr (D.Kim),mooseroh@business.kaist.ac.kr (T.-Y.Roh),byoungkyu.min@unine.ch (B.-K.Min),sjbyun@business.kaist.ac.kr (S.-J.Byun).statistically insignificant profits during recessions.Theirfindings uncover procyclical time variations in momentum profits.Cooper et al.(2004)alsofind that momentum profits depend on the state of the market in a procyclical way.Average momentum profits are positive following periods of up markets but negative following periods of down markets.However,these authors interpret these results as consistent with the overreaction models of Daniel et al. (1998)and Hong and Stein(1999).1In our view,a possible reason behind the discrepancy in the above authors’different interpretations is that the above two stud-ies do not link time-series and cross-sectional properties of expected returns.For example,the empirical specification used by Chordia and Shivakumar(2002),regressing momentum payoffs on the lagged macroeconomic variables,does not impose a covari-ance between momentum portfolio returns and the pricing kernel. Thus,we cannot discriminate whether winners are riskier than los-ers or vice versa from their results.2Cooper et al.(2004)alsofind that asymmetries conditional on the state of the market complement the evidence of asymmetries in factor sensitivities,volatility,corre-lations,and expected returns and thus argue that asset pricing mod-els,both rational and behavioral,need to incorporate(or predict) such regime switches.Stivers and Sun(2010)show that time varia-tion in momentum profits can be tied to variation in the market’s cross-sectional return dispersion.They regard this return dispersion as a leading counter-cyclical state variable according to the theory of Gomes et al.(2003)and Zhang(2005).These authorsfind that the recent cross-sectional return dispersion is negatively related to the subsequent momentum profits and thus suggest that momentum profits are procyclical.3This paper aims to combine the time-series and cross-sectional implications of the profitability of momentum trading.As Fama (1991,p.1610)states,‘‘In the end,I think we can hope for a coher-ent story that relates the cross-section properties of expected returns to the variation of expected returns through time.’’This paper seeks to provide empirical evidence for such a coherent story for momentum.To do so,we adopt the two-state Markov switch-ing regression framework with time-varying transition probabili-ties by following Perez-Quiros and Timmermann(2000)and Gulen et al.(2011).Thisflexible econometric model allows us to combine the cross-sectional evidence on past stock returns with the time-series evidence on the evolution in conditional returns and to describe asymmetries in the response of momentum profits to aggregate economic conditions across the state of the economy by incorporating regime switches.We also examine a differential response in expected returns to shocks to aggregate economic conditions between winner and loser stocks across the state of economy and the procyclicality of momentum profits.By employing a similar approach,Perez-Quiros and Timmermann(2000)examine whether a differential response exists in expected returns to shocks to aggregate eco-nomic conditions between small and largefirms.Gulen et al. (2011)also examine a differential response in expected returns between value and growthfirms andfind strong counter-cyclicality of the value premium.Our paper is not thefirst to examine the pro-cyclicality of momentum profits.Chordia and Shivakumar(2002), Cooper et al.(2004)and Stivers and Sun(2010)already documented procyclical variations in momentum profits.Unlike the previous literature,however,this paper shows that the risks of winner and loser stocks are asymmetrical across business cycles and time-variation in riskiness is a driving force for time-variation in momentum profits.In particular,we provide a plausible explana-tion for why winners are riskier than losers in expansions but losers are riskier than winners in recessions.We document two mainfindings.First,in the recession state, loser stocks tend to have greater loadings on the conditioning mac-roeconomic variables than winner stocks,while in the expansion state winner stocks tend to have greater loadings on these vari-ables than loser stocks.In other words,in recessions loser(winner) stocks are most(least)strongly affected,while in expansions win-ner(loser)stocks are most(least)strongly affected.This indicates that returns on momentum portfolios react asymmetrically to aggregate economic conditions in recession and expansion states. Second,the asymmetries in winner and loser stocks’risk across the states of the economy lead to strong procyclical time-variations in the expected momentum profits.The expected momentum profit estimated from the Markov switching regression model tends to be positive and spike upward just before entering a reces-sion(i.e.,the peak of the business cycle),while it becomes negative during recessions,reaching a maximal negative value at the end of a recession(i.e.,the trough of the business cycle).The above two findings are robust to estimating exogenously the state transition probabilities and identifying the states,using alternative instru-mental variables in modeling state transition probabilities,and assuming the fat-tailed distribution of stock returns.We also examine the economic significance of out-of-sample predictability of the model by setting up a simple stylized trading rule based on the prediction.The results show that the economic significance of out-of-sample predictability is particularly significant when this trading rule is applied to loser stocks and during a recession state.Thefirst mainfinding above implies that the riskiness of winner and loser stocks is different across business cycles and,conse-quently,momentum profits are time-varying.We provide a plausi-ble explanation for time-varying momentum profits through the differential effect of leverage and growth options across business cycles.During expansions,growth options have a higher effect and leverage has a lower effect,and winner stocks tend to have greater growth options and lower leverage.As a result,winner stocks are riskier in expansions.On the contrary,during recessions, growth options have a lower effect and leverage has a higher effect, and loser stocks tend to have lower growth options and higher leverage.Thus,loser stocks are riskier in recessions.We argue that leverage and growth options are the underlying driving forces for the different riskiness of winner and loser stocks and for time-varying momentum profits.The remainder of this paper proceeds as follows.Section2dis-cusses the sources of time-varying momentum profits.Section3 presents a method to estimate the two-state Markov switching regression model with time-varying transition probabilities. Section4describes the data and the empirical results for the model fitted to momentum portfolios.Section5provides a plausible explanation for the observed time-varying momentum profits. Section6sets forth a summary and conclusions.2.Sources of time-varying expected momentum profitsIn his theoretical model,Johnson(2002)argues that stock prices are a convex function of expected growth,meaning that growth rate risk increases with growth rates and thus,stock price changes1Cooper et al.(2004)report that a multifactor macroeconomic model of returns,asused by Chordia and Shivakumar(2002),does not explain momentum profits aftercontrolling for market frictions.Additionally,these authors report that the macro-economic model cannot forecast the time-series of out-of-sample momentum profits,whereas the lagged return of the market can.Hence,they suggest that the laggedreturn of the market is the type of conditioning information that is relevant inpredicting the profitability of the momentum.2Chordia and Shivakumar(2002)admit to this weakness in their approach:‘‘We donot impose cross-sectional asset pricing constraints in this study.Proponents of thebehavioral theories may argue that,to be rational,the payoff to momentum strategiesmust covary with risk factors’’(p.988).3Stivers and Sun(2010)also show that the recent cross-sectional return dispersionis shown to be positively related to the value premium and thus suggest that thevalue premium is countercyclical.192 D.Kim et al./Journal of Banking&Finance49(2014)191–215(or stock returns)should be more sensitive to changes in expected growth when the expected growth is higher.If exposure to this risk carries a positive premium,expected returns rise with growth rates.Other things being equal,firms with large recent positive price moves(winners)are more likely to have had positive growth rate shocks thanfirms with large recent negative price moves(los-ers).Hence,a momentum sort will tend to sortfirms by recent growth rate changes and sorting by growth rate changes will also tend to sortfirms according to growth rate levels and hence by end-of-period expected returns.In other words,recent winners (losers)will tend to have both higher(lower)growth rate changes in the recent past and higher(lower)subsequent expected returns. Motivated by Johnson’s(2002)theoretical work,Liu and Zhang (2008)show that the risk exposure of winners on the growth rate of industrial production differs from those of losers.Assuming that the growth rate of industrial production is a common factor sum-marizingfirm-level changes of expected growth,these authors document that winners have temporarily higher average future growth rates than losers.More importantly,theyfind that the expected growth risk as defined by Johnson(2002)is priced and increases with expected growth.4In their theoretical model,Sagi and Seasholes(2007)show that afirm’s revenues,costs,and growth options combine to explain momentum profits and they exercise their theoretical insights to show that momentum strategies usingfirms with high revenue growth volatility and valuable growth options outperform tradi-tional momentum strategies.Their model suggests thatfirms with valuable growth options exhibit higher autocorrelation thanfirms without such growth options,becausefirms that performed well in the recent past are better poised to exploit their growth options. Since growth options are riskier than assets in place,suchfirms are riskier and are thus associated with higher expected returns. Winner stocks that have good recent performance are likely to have riskier growth options than loser stocks that have bad recent performance.Subsequently,winner stocks should earn higher expected returns than loser stocks.Importantly,the Sagi and Seasholes(2007)model implies that momentum profits should be procyclical:‘‘During up markets,firms tend to move closer to exercising their growth options,which tends to increase return autocorrelations.During down markets,firms tend to move closer tofinancial distress,which tends to decrease return autocorrela-tions’’(p.391).The above theoretical models suggest that momentum profits are procyclical.The expected growth rates mentioned by Johnson (2002)are high in expansions and growth rate risk is accordingly high.Since trading strategies based on momentum tend to have high exposure to this risk,their expected returns should be higher in expansions than in recessions.In a similar vein,procyclical stocks tend to have greater growth rate risk and more valuable growth options in expansions than in recessions and thus such firms are riskier and associated with higher expected returns in expansions.According to Johnson(2002)and Sagi and Seasholes (2007),recent winner stocks are likely to have greater growth rate risk and riskier growth options and should earn higher expected returns than recent loser stocks.Therefore,observed momentum profits(or returns on winner-minus-loser portfolios,hereafter WML)are realizations of such expected returns and can be interpreted as the procyclicality premium.3.An econometric model of time-varying expected returnsBased on Sagi and Seasholes’(2007)theoretical model,we argue that momentum profits are procyclical because of the extent of exercising growth options across business cycles.To empirically examine the procyclical behavior of momentum profits,the Markov switching regression framework is appropriate since it can accommodate the time-varying behavior of momentum profits across business cycles and business cycles can be regarded as states.In this regard,we employ the Perez-Quiros and Timmermann(2000)Markov switching regression framework with time-varying transition probabilities based on Hamilton(1989) and Gray(1996).Let r t be the return of a test asset in excess of the riskless return at time t and let X tÀ1be a vector of conditioning variables available up to time tÀ1used to predict r t.The Markov switching specification takes all parameters(the intercept term, slope coefficients,and volatility of excess returns)as a function of a single,latent state variable,S t.Specifically,r t¼b0;Stþb0S tX tÀ1þe t;e t$N0;r2S t;ð1Þwhere N0;r2Stdenotes a normal distribution with mean zero and variance r2St.In a two-state Markov switching specification,S t=1or S t=2,meaning that the parameters to be estimated are eitherh1¼b0;1;b01;r21ÀÁor h2¼b0;2;b02;r22ÀÁ.Since the above Markov switching model allows the risk and expected return to vary(or transit)across two states,it is neces-sary to specify how the underlying states evolve through time. We assume that the state transition probabilities follow afirst-order Markov chain as follows:pt¼ProbðS t¼1j S tÀ1¼1;y tÀ1Þ¼pðy tÀ1Þ;ð2Þ1Àpt¼ProbðS t¼2j S tÀ1¼1;y tÀ1Þ¼1Àpðy tÀ1Þ;ð3Þqt¼ProbðS t¼2j S tÀ1¼2;y tÀ1Þ¼qðy tÀ1Þ;ð4Þ1Àqt¼ProbðS t¼1j S tÀ1¼2;y tÀ1Þ¼1Àqðy tÀ1Þ;ð5Þwhere y tÀ1is a vector of variables publicly available at time tÀ1 and affects the state transition probabilities between times tÀ1 and t.Although the standard formulation of the Markov switching model assumes the state transition probabilities to be constant,it would be more reasonable to assume that the probability of staying in a state depends on prior conditioning information,y tÀ1,and thus is time-varying,since investors are likely to possess information about the state transition probabilities superior to that implied by the model with constant transition probabilities.The literature shows that the economic leading indicator(Filardo,1994;Perez-Quiros and Timmermann,2000),interest rates(Gray,1996;Gulen et al.,2011),or the duration of the time spent in a given state (Durland and McCurdy,1994;Mahue and McCurdy,2000)is used as prior condition information.We estimate the above two-state Markov switching model using maximum likelihood methods.5Let h=(h1,h2)denote the vector of parameters to be estimated in the likelihood function. The probability density function of the return,conditional on being state j,is Gaussian defined asfðr t j X tÀ1;S t¼j;hÞ¼1ffiffiffiffiffiffiffiffiffiffiffiffi2pr2jq expÀr tÀb0;jÀb0jX tÀ122r2j8><>:9>=>;ð6Þfor j=1,2.The information set X tÀ1contains X tÀ1,r tÀ1,y tÀ1,and lagged values of these variables.Then,the log-likelihood function is4Liu and Zhang(2008)alsofind that in many specifications this macroeconomic risk factor explains more than half of momentum profits and conclude that risk plays an important role in driving momentum profits.However,some papers report different results.For example,Grundy and Martin(2001)and Avramov and Chordia (2006)report that controlling for time-varying exposures to common risk factors doesnot affect momentum profits.Griffin et al.(2003)show that the model of Chen et al. (1986)does not provide any evidence that macroeconomic risk variables can explain momentum.5Another estimation approach is a Bayesian approach based on numerical Bayesian methods such as the Gibbs sampler and Markov Chain Monte Carlo methods(Kim and Nelson,1999).D.Kim et al./Journal of Banking&Finance49(2014)191–215193L ðr t j X t À1;h Þ¼X T t ¼1log ½/ðr t j X t À1;h Þ ;ð7Þwhere the density function /(r t j X t À1;h )is simply obtained by sum-ming the probability-weighted state probabilities across the twostates.It is defined as/ðr t j X t À1;h Þ¼X2j ¼1f ðr t j X t À1;S t ¼j ;h ÞProb ðS t ¼j j X t À1;h Þ;ð8Þwhere Prob (S t =j j X t À1;h )is the conditional probability of being in state j at time t given information at time t À1.The conditional state probabilities can be obtained from the standard probability theorem:Prob ðS t ¼i j X t À1;h Þ¼X2j ¼1Prob ðS t ¼i j S t À1¼j ;X t À1;h ÞProb ðS t À1¼j j X t À1;h Þ:ð9ÞBy Bayes’rule,the conditional state probabilities can be written asProb ðS t À1¼j j X t À1;h Þ¼f ðr t À1j S t À1¼j ;X t À1;y t À1;X t À2;h ÞProb ðS t À1¼j j X t À1;y t À1;X t À2;h ÞP j ¼1f ðr t À1j S t À1¼j ;X t À1;y t À1;X t À2;h ÞProb ðS t À1¼j j X t À1;y t À1;X t À2;h Þ:ð10ÞThe conditional state probabilities Prob (S t =i j X t À1;h )are driven by iterating recursively Eqs.(9)and (10)and the parameter estimates of the likelihood function are obtained (Gray,1996).Variations in the state probabilities are evidence that the conditional expected return is time-varying.4.Empirical results4.1.Data and model specificationWe use monthly excess returns (raw returns minus the one-month Treasury bill return)on the momentum decile portfolios as test assets.Momentum portfolios are constructed in accordance with Jegadeesh and Titman (1993)by sorting all stocks every month into one of 10decile portfolios based on the past six-month returns and holding the deciles for the subsequent six months.We skip one month between the end of the portfolio formation period and the beginning of the holding period to avoid potential micro-structure biases.All stocks in a given portfolio have equal weight.Portfolio 1is the past loser,while Portfolio 10is the past winner.Gulen et al.(2011)examine the time-varying behavior of the expected value premium and show that the expected value pre-mium displays strong countercyclical variations,while we show that the expected momentum profits display strong procyclical variations.To compare the opposite time-varying behaviors of these two stock return regularities,we match the beginning of the sample period with Gulen et al.(2011):Our sample period is from March 1960to December 2012.Table 1presents the mean,standard deviation,skewness,and kurtosis of monthly excess returns on the 10decile momentum portfolios.The mean excess returns monotonically increase from 0.369%per month for the past loser portfolio (Portfolio 1)to 1.127%per month for the past winner portfolio (Portfolio 10).The mean return on the WML is quite significant:0.758%per month.A distinct pattern is found in skewness,which almost monotonically decreases from 1.375for the loser portfolio to À0.661for the winner portfolio.Portfolios 1through 4are posi-tively-skewed,while Portfolios 5through 10are negatively-skewed.These results indicate that past (short-term)winners are preferred to past losers in the mean–variance framework,but this may not necessarily be true when considering the third moment,since positively-skewed portfolios should be preferred to nega-tively-skewed portfolios.This is consistent with the Arrow–Pratt notion of risk aversion.Loser portfolios tend to have greater kurto-sis than do winner portfolios.To show that momentum returns are asymmetrically affected by macroeconomic variables across states (or business cycles),we model the excess returns of each of the momentum portfolios as a function of an intercept term and lagged values of the relative three-month Treasury bill rate,the default spread,the growth in the monetary base,and the dividend yield.These variables are commonly used in the literature on the predictability of stock returns.As in Perez-Quiros and Timmermann (2000)and Gulen et al.(2011),we use the relative three-month Treasury bill rate (RREL)as a state variable proxying for investors’expectations of future economic activity.According to Fama (1981),an unobserved negative shock to real economic activity induces a higher Treasury bill rate through an increase in the current and expected future inflation rate.He argues that a negative correlation between stock returns and inflation is not a causal relation but is proxying for a positive relation between stock returns and real activity.Thus,the Treasury bill rate,which is an indicator of the short-term inter-est rate,tends to have a negative relation with stock returns (e.g.,Fama and Schwert,1977;Fama,1981;Campbell,1987;Glosten et al.,1993).Berk et al.(1999)present a theoretical modelTable 1Moments of monthly excess returns for ten decile momentum portfolios.Momentum portfolio Mean Standard deviation Skewness Excess Kurtosis q 1(r it )q 1r 2itÀÁLoser 0.3699.064 1.375 6.4910.201[0.000]0.213[0.000]20.549 6.7340.581 5.1930.220[0.000]0.138[0.001]30.722 6.0520.340 5.2510.224[0.000]0.143[0.000]40.751 5.5990.138 5.0710.213[0.000]0.104[0.009]50.799 5.294À0.108 4.4070.201[0.000]0.080[0.043]60.817 5.123À0.273 4.1740.185[0.000]0.055[0.163]70.863 5.112À0.419 4.3020.170[0.000]0.051[0.194]80.900 5.220À0.580 4.0250.157[0.000]0.048[0.224]90.979 5.532À0.709 3.7280.154[0.000]0.074[0.062]Winner 1.127 6.516À0.661 2.6640.162[0.000]0.114[0.004]WML0.7585.999À2.97117.0000.166[0.020]0.109[0.006]This table reports the mean,standard deviation,skewness,and excess kurtosis of excess returns (in percent)on the momentum portfolios which are constructed in accordance with Jegadeesh and Titman (1993).That is,all stocks are sorted every month into one of ten decile portfolios based on past six-month returns,and held for six months.Excess returns are calculated as the difference between monthly stock returns and the one-month Treasury bill rate.The data for the one-month Treasury bill rate are from Kenneth French’s Web site.‘WML’indicates ‘Winner’portfoliominus ‘Loser’portfolio.q 1(r it )and q 1r 2itÀÁare the first-order autocorrelations of the raw excess return and squared raw excess returns,respectively.Numbers in brackets indicate p -values of the first-order autocorrelations.The sample period is from March 1960to December 2012.194D.Kim et al./Journal of Banking &Finance 49(2014)191–215predicting that changes in interest rates will affect expected stock returns differently acrossfirms and providing a direct link between cross-sectional dispersions of expected stock returns and interest rates.Interest rates should be a true cause of ex post stock returns, because an increase(decrease)in the real interest rate induces a reduction(increase)in stock values.The default spread(DEF)is defined as the difference between yields on Baa-rated corporate bonds and10-year Treasury bonds from the Federal Reserve Economic Data at the Federal Reserve Bank of St.Louis and is included to capture the effect of default pre-miums.Fama and French(1989)show that the major movements in DEF seem to be related to long-term business cycle conditions and the default spread forecasts high returns when business condi-tions are persistently weak and low returns when conditions are strong.Indeed,the default spread is one of the most frequently used conditioning variables in predicting stock returns(e.g.,Keim and Stambaugh,1986;Fama and French,1988;Kandel and Stambaugh,1990;Jagannathan and Wang,1996;Chordia and Shivakumar,2002).The growth in the money base(MB)is defined as the12-month log-difference in the monetary base reported by the St.Louis Fed-eral Reserve.This variable is included in the conditional mean equation,since this variable affects stock returns through changes in macro liquidity(or moneyflow liquidity)and eventually micro liquidity(or transaction liquidity)in stock markets.6This variable also affects stock returns through shocks in monetary policies that can affect aggregate economic conditions.In particular,Fama (1981)argues that it is important to control for money supply when establishing the inflation-future real economic activity proxy story.The dividend yield(DIV)is defined as the sum of dividend pay-ments accruing to the Center for Research in Security Prices(CRSP) value-weighted market portfolio over the previous12months divided by the contemporaneous level of the index at the end of the month.The standard valuation model indicates that stock prices are low relative to dividends when discount rates and expected returns are high and vice versa.Thus,the dividend yield (usually measured by the ratio of dividends to price)varies with expected returns.Thus,the dividend yield proxies for time-varia-tion in the unobservable risk premium.There is ample empirical evidence that the dividend yield predicts future stock returns (e.g.,Keim and Stambaugh,1986;Campbell and Shiller,1988; Fama and French,1988;Kandel and Stambaugh,1990).7 To capture the movements of momentum portfolio returns,we specify Eq.(1)by including the above-mentioned return predict-able variables in the following conditional mean equation:r it¼b i0;St þb i1;StRREL tÀ1þb i2;StDEF tÀ1þb i3;StMB tÀ2þb i4;St DIV tÀ1þe it;ð11Þwhere r it is the monthly excess return for the i th decile momentumportfolio at time t,e it is the normally distributed random error termwith mean zero and variance r2i;St,and S t={1,2}.The regressors arelagged by one month.The conditional variance of excess returns,r2i;S t,is allowed to depend only on the state of economy:ln r2i;St¼k i;St:ð12ÞWe do not include autoregressive conditional heteroskedasticity(ARCH)effects in the conditional volatility equation.Table1showsthefirst-order autocorrelations of the raw excess returns and thesquared raw excess returns in each of the10decile momentumportfolios.All ten portfolios exhibit a significant positivefirst-orderautocorrelation at the one percent level.Only six squared rawexcess returns out of the ten portfolios have significantfirst-orderautocorrelation coefficient estimates at thefive percent level.Theseresults indicate that ARCH effects are less important in the condi-tional volatility in our framework.Following Gray(1996)and Gulen et al.(2011),we model thetime-varying state transition probabilities to be dependent onthe level of short interest rates,Treasury bill rates,as follows:p it¼Prob S it¼1j S itÀ1¼1;y tÀ1¼U p i0þp i1RREL tÀ1ÀÁ;ð13Þ1Àp it¼Prob S it¼2j S itÀ1¼1;y tÀ1;ð14Þq it¼Prob S it¼2j S itÀ1¼2;y tÀ1¼U p i0þp i2RREL tÀ1ÀÁ;ð15Þ1Àq it¼Prob S it¼1j S itÀ1¼2;y tÀ1;ð16Þwhere RREL tÀ1is the relative three-month Treasury bill rate calcu-lated as the difference between the current Treasury bill rate andits12-month backward moving average,and U(Á)is the cumulativeprobability density function of a standard normal variable.8Forrobustness checks,we also use two alternative instrumentalvariables in modeling state transition probability equations insteadof the relative three-month Treasury bill rate:the Composite LeadingIndicator and the industrial production growth rate.However,theresults are qualitatively similar,as reported in Section4.5.4.2.Estimation results4.2.1.Identifying the statesTable2reports the estimation results of the parameters in Eqs.(11)–(16)for portfolios P1(loser),P2,P4,P6,P8,and P10(winner).9The constant parameter estimates in the conditional mean equationin state1(^b i0;1Þare much lower than those in state2(^b i0;2Þin allmomentum portfolios.The constant term in state1monotonicallyincreases across the portfolios from the loser to the winner portfoliosand is more precisely estimated.Eight out of10constant terms instate1are significantly estimated at the1%level and all10constantterm estimates are negative.In contrast,there is no particular pat-tern in the constant term in state2and any of the10constant termsare not significantly estimated.The conditional standard deviationestimate in state1ð^r i;1Þis greater than that in state2ð^r i;2Þin allportfolios.All conditional volatilities are highly significantlyestimated.Schwert(1990)and Hamilton and Lin(1996)find thatreturn volatilities are higher in recession periods than in expansionperiods.Theirfindings are verified with historical National Bureauof Economic Research(NBER)business cycle dates.These resultsmay indicate that state1is the recession state and state2is the6Instead of the growth in monetary base,we also include the inflation rate. However,the results are qualitatively similar.The reason that the inflation rate can be included is that since both economic theory and traditional idea imply that stock returns and inflation should be positively correlated,since equities are‘‘hedges’’against inflation because they represent claims to real assets.However,the United States and other industrialized countries exhibit a significant negative correlation between inflation and real stock returns in the post-war periods(e.g.,Fama and Schwert,1977;Fama,1981;Geske and Roll,1983;Danthine and Donaldson,1986; Stulz,1986;Kaul,1987,1990;Marshall,1992;Boudoukh et al.,1994;Bakshi and Chen,1996).This negative correlation between inflation and real stock returns is often termed the stock return–inflation puzzle.Many authors have tried to resolve this puzzle(e.g.,Fama,1981;Marshall,1992;Geske and Roll,1983;Kaul,1987).In contrast to existing evidence of a negative relation at short horizons,Boudoukh andRichardson(1993)find evidence to suggest that long-horizon nominal stock returns are positively related to both ex ante and ex post long-term inflation.7Ang and Bekaert(2007)report that the dividend yield does not univariately predict excess returns,but the predictive ability of the dividend yield is considerably enhanced,at short horizons,in a bivariate regression with the short rate.8Instead of the relative Treasury bill rate,we also use the one-month Treasury bill rate in the state transition probability equation.However,the results are quite similar.The results are available upon request.9The estimation results for portfolios P3,P5,P7,and P9are not reported because of space constraint.The results are available upon request.D.Kim et al./Journal of Banking&Finance49(2014)191–215195。