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Journal of Banking & Finance 49 (2014) 191–215 Contents lists available at ScienceDirect Journal of Banking & Finance journal homepage: www.elsevier.com/locate/jbf Time-varying expected momentum profits Dongcheol Kim a, Tai-Yong Roh b, Byoung-Kyu Min c,⇑, Suk-Joon Byun b a Business School, Korea University, Seoul, Republic of Korea Graduate School of Finance, Korea Advanced Institute of Science and Technology (KAIST), Seoul, Republic of Korea c Institute of Financial Analysis, University of Neuchatel, Neuchatel, Switzerland b a r t i c l e i n f o Article history: Received 15 November 2013 Accepted 11 September 2014 JEL classifications: G12 G14 Keywords: Momentum Time-varying expected returns Markov switching regression model Business cycle Procyclicality Growth options a b s t r a c t This 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. Ó 2014 Elsevier B.V. All rights reserved. 1. Introduction The cross-sectional difference in average stock returns across their recent past performance has become one of the most controversial issues in academia as well as industry since the pioneering work of Jegadeesh and Titman (1993). A simple momentum strategy 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). ⇑ Corresponding author. Tel.: +41 32 718 15 74; fax: +41 32 718 14 01. E-mail addresses: [email protected] (D. Kim), [email protected] (T.-Y. Roh), [email protected] (B.-K. Min), [email protected] (S.-J. Byun). http://dx.doi.org/10.1016/j.jbankfin.2014.09.004 0378-4266/Ó 2014 Elsevier B.V. All rights reserved. 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 procyclical time variations in momentum profits. Johnson (2002) and Sagi and Seasholes (2007) provide the theoretical insight that momentum 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 momentum profits. Chordia and Shivakumar (2002) show that profits of momentum strategies can be explained by a set of lagged macroeconomic 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, 192 D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 statistically insignificant profits during recessions. Their findings uncover procyclical time variations in momentum profits. Cooper et al. (2004) also find 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).1 In our view, a possible reason behind the discrepancy in the above authors’ different interpretations is that the above two studies 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 covariance between momentum portfolio returns and the pricing kernel. Thus, we cannot discriminate whether winners are riskier than losers or vice versa from their results.2 Cooper et al. (2004) also find that asymmetries conditional on the state of the market complement the evidence of asymmetries in factor sensitivities, volatility, correlations, and expected returns and thus argue that asset pricing models, both rational and behavioral, need to incorporate (or predict) such regime switches. Stivers and Sun (2010) show that time variation 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 authors find that the recent cross-sectional return dispersion is negatively related to the subsequent momentum profits and thus suggest that momentum profits are procyclical.3 This 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 coherent 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 switching regression framework with time-varying transition probabilities by following Perez-Quiros and Timmermann (2000) and Gulen et al. (2011). This flexible 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, PerezQuiros and Timmermann (2000) examine whether a differential response exists in expected returns to shocks to aggregate economic conditions between small and large firms. Gulen et al. (2011) also examine a differential response in expected returns between value and growth firms and find strong counter-cyclicality 1 Cooper et al. (2004) report that a multifactor macroeconomic model of returns, as used by Chordia and Shivakumar (2002), does not explain momentum profits after controlling for market frictions. Additionally, these authors report that the macroeconomic 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 lagged return of the market is the type of conditioning information that is relevant in predicting the profitability of the momentum. 2 Chordia and Shivakumar (2002) admit to this weakness in their approach: ‘‘We do not impose cross-sectional asset pricing constraints in this study. Proponents of the behavioral theories may argue that, to be rational, the payoff to momentum strategies must covary with risk factors’’ (p. 988). 3 Stivers and Sun (2010) also show that the recent cross-sectional return dispersion is shown to be positively related to the value premium and thus suggest that the value premium is countercyclical. of the value premium. Our paper is not the first to examine the procyclicality 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 explanation for why winners are riskier than losers in expansions but losers are riskier than winners in recessions. We document two main findings. First, in the recession state, loser stocks tend to have greater loadings on the conditioning macroeconomic variables than winner stocks, while in the expansion state winner stocks tend to have greater loadings on these variables than loser stocks. In other words, in recessions loser (winner) stocks are most (least) strongly affected, while in expansions winner (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 recession (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 instrumental 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. The first main finding above implies that the riskiness of winner and loser stocks is different across business cycles and, consequently, momentum profits are time-varying. We provide a plausible 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 timevarying momentum profits. The remainder of this paper proceeds as follows. Section 2 discusses the sources of time-varying momentum profits. Section 3 presents a method to estimate the two-state Markov switching regression model with time-varying transition probabilities. Section 4 describes the data and the empirical results for the model fitted to momentum portfolios. Section 5 provides a plausible explanation for the observed time-varying momentum profits. Section 6 sets forth a summary and conclusions. 2. Sources of time-varying expected momentum profits In 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 changes 193 D. Kim et al. / Journal of Banking & Finance 49 (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 than firms with large recent negative price moves (losers). Hence, a momentum sort will tend to sort firms by recent growth rate changes and sorting by growth rate changes will also tend to sort firms 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 summarizing firm-level changes of expected growth, these authors document that winners have temporarily higher average future growth rates than losers. More importantly, they find that the expected growth risk as defined by Johnson (2002) is priced and increases with expected growth.4 In their theoretical model, Sagi and Seasholes (2007) show that a firm’s revenues, costs, and growth options combine to explain momentum profits and they exercise their theoretical insights to show that momentum strategies using firms with high revenue growth volatility and valuable growth options outperform traditional momentum strategies. Their model suggests that firms with valuable growth options exhibit higher autocorrelation than firms without such growth options, because firms that performed well in the recent past are better poised to exploit their growth options. Since growth options are riskier than assets in place, such firms 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 to financial distress, which tends to decrease return autocorrelations’’ (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. 4 Liu and Zhang (2008) also find 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 does not 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. 3. An econometric model of time-varying expected returns Based 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 rt be the return of a test asset in excess of the riskless return at time t and let Xt1 be a vector of conditioning variables available up to time t 1 used to predict rt. 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, St. Specifically, rt ¼ b0;St þ b0St X t1 þ et ; et N 0; r2St ; ð1Þ where N 0; r2St denotes a normal distribution with mean zero and variance r2St . In a two-state Markov switching specification, St = 1 or St = 2, meaning that the parameters to be estimated are either h1 ¼ b0;1 ; b01 ; r21 or h2 ¼ b0;2 ; b02 ; r22 . Since the above Markov switching model allows the risk and expected return to vary (or transit) across two states, it is necessary to specify how the underlying states evolve through time. We assume that the state transition probabilities follow a firstorder Markov chain as follows: pt ¼ ProbðSt ¼ 1jSt1 ¼ 1; yt1 Þ ¼ pðyt1 Þ; ð2Þ 1 pt ¼ ProbðSt ¼ 2jSt1 ¼ 1; yt1 Þ ¼ 1 pðyt1 Þ; ð3Þ qt ¼ ProbðSt ¼ 2jSt1 ¼ 2; yt1 Þ ¼ qðyt1 Þ; ð4Þ 1 qt ¼ ProbðSt ¼ 1jSt1 ¼ 2; yt1 Þ ¼ 1 qðyt1 Þ; ð5Þ where yt1 is a vector of variables publicly available at time t 1 and affects the state transition probabilities between times t 1 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, yt1, 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; PerezQuiros 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.5 Let 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 as 8 2 9 > < = rt b0;j b0j X t1 > 1 f ðr t jXt1 ; St ¼ j; hÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffi exp > > 2r2j : ; 2pr2j ð6Þ for j = 1,2. The information set Xt1 contains Xt1,rt1,yt1, and lagged values of these variables. Then, the log-likelihood function is 5 Another 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). 194 D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 Lðr t jXt1 ; hÞ ¼ T X log½/ðr t jXt1 ; hÞ; ð7Þ t¼1 where the density function /(rtjXt1;h) is simply obtained by summing the probability-weighted state probabilities across the two states. It is defined as /ðrt jXt1 ; hÞ ¼ 2 X f ðrt jXt1 ; St ¼ j; hÞProbðSt ¼ jjXt1 ; hÞ; ð8Þ j¼1 where Prob(St = jjXt1;h) is the conditional probability of being in state j at time t given information at time t 1. The conditional state probabilities can be obtained from the standard probability theorem: ProbðSt ¼ ijXt1 ; hÞ ¼ 2 X ProbðSt ¼ ijSt1 ¼ j; Xt1 ; hÞ j¼1 ProbðSt1 ¼ jjXt1 ; hÞ: ð9Þ By Bayes’ rule, the conditional state probabilities can be written as ProbðSt1 ¼ jjXt1 ;hÞ f ðr t1 jSt1 ¼ j;X t1 ; yt1 ; Xt2 ;hÞProbðSt1 ¼ jjX t1 ;yt1 ; Xt2 ;hÞ ¼ P2 : j¼1 f ðr t1 jSt1 ¼ j;X t1 ; y t1 ; Xt2 ;hÞProbðSt1 ¼ jjX t1 ;y t1 ; Xt2 ;hÞ ð10Þ The conditional state probabilities Prob(St = ijXt1;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 results 4.1. Data and model specification We use monthly excess returns (raw returns minus the onemonth 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 10 decile 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 microstructure biases. All stocks in a given portfolio have equal weight. Portfolio 1 is the past loser, while Portfolio 10 is the past winner. Gulen et al. (2011) examine the time-varying behavior of the expected value premium and show that the expected value premium 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 1960 to December 2012. Table 1 presents the mean, standard deviation, skewness, and kurtosis of monthly excess returns on the 10 decile 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 1.375 for the loser portfolio to 0.661 for the winner portfolio. Portfolios 1 through 4 are positively-skewed, while Portfolios 5 through 10 are negativelyskewed. 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 negatively-skewed portfolios. This is consistent with the Arrow–Pratt notion of risk aversion. Loser portfolios tend to have greater kurtosis 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 interest 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 model Table 1 Moments of monthly excess returns for ten decile momentum portfolios. Momentum portfolio Mean Standard deviation Skewness Excess Kurtosis q1(rit) Loser 2 3 4 5 6 7 8 9 Winner WML 0.369 0.549 0.722 0.751 0.799 0.817 0.863 0.900 0.979 1.127 0.758 9.064 6.734 6.052 5.599 5.294 5.123 5.112 5.220 5.532 6.516 5.999 1.375 0.581 0.340 0.138 0.108 0.273 0.419 0.580 0.709 0.661 2.971 6.491 5.193 5.251 5.071 4.407 4.174 4.302 4.025 3.728 2.664 17.000 0.201 0.220 0.224 0.213 0.201 0.185 0.170 0.157 0.154 0.162 0.166 q1 r2it [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.020] 0.213 0.138 0.143 0.104 0.080 0.055 0.051 0.048 0.074 0.114 0.109 [0.000] [0.001] [0.000] [0.009] [0.043] [0.163] [0.194] [0.224] [0.062] [0.004] [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 onemonth Treasury bill rate. The data for the one-month Treasury bill rate are from Kenneth French’s Web site. ‘WML’ indicates ‘Winner’ portfolio minus ‘Loser’ portfolio.q1(rit) and q1 r 2it are 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 1960 to December 2012. D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 predicting that changes in interest rates will affect expected stock returns differently across firms 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 and 10-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 premiums. 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 conditions 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 the 12-month log-difference in the monetary base reported by the St. Louis Federal Reserve. This variable is included in the conditional mean equation, since this variable affects stock returns through changes in macro liquidity (or money flow liquidity) and eventually micro liquidity (or transaction liquidity) in stock markets.6 This 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 payments accruing to the Center for Research in Security Prices (CRSP) value-weighted market portfolio over the previous 12 months 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-variation 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 predictable variables in the following conditional mean equation: r it ¼ bi0;St þ bi1;St RRELt1 þ bi2;St DEF t1 þ bi3;St MBt2 þ bi4;St DIV t1 þ eit ; ð11Þ 6 Instead 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 and Richardson (1993) find evidence to suggest that long-horizon nominal stock returns are positively related to both ex ante and ex post long-term inflation. 7 Ang 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. 195 where rit is the monthly excess return for the ith decile momentum portfolio at time t,eit is the normally distributed random error term with mean zero and variance r2i;St , and St = {1,2}. The regressors are lagged by one month. The conditional variance of excess returns, r2i;St , is allowed to depend only on the state of economy: ln r2i;St ¼ ki;St : ð12Þ We do not include autoregressive conditional heteroskedasticity (ARCH) effects in the conditional volatility equation. Table 1 shows the first-order autocorrelations of the raw excess returns and the squared raw excess returns in each of the 10 decile momentum portfolios. All ten portfolios exhibit a significant positive first-order autocorrelation at the one percent level. Only six squared raw excess returns out of the ten portfolios have significant first-order autocorrelation coefficient estimates at the five percent level. These results indicate that ARCH effects are less important in the conditional volatility in our framework. Following Gray (1996) and Gulen et al. (2011), we model the time-varying state transition probabilities to be dependent on the level of short interest rates, Treasury bill rates, as follows: pit ¼ Prob Sit ¼ 1jSit1 ¼ 1; yt1 ¼ U pi0 þ pi1 RRELt1 ; 1 pit ¼ Prob Sit ¼ 2jSit1 ¼ 1; yt1 ; qit ¼ Prob Sit ¼ 2jSit1 ¼ 2; yt1 ¼ U pi0 þ pi2 RRELt1 ; 1 qit ¼ Prob Sit ¼ 1jSit1 ¼ 2; yt1 ; ð13Þ ð14Þ ð15Þ ð16Þ where RRELt1 is the relative three-month Treasury bill rate calculated as the difference between the current Treasury bill rate and its 12-month backward moving average, and U() is the cumulative probability density function of a standard normal variable.8 For robustness checks, we also use two alternative instrumental variables in modeling state transition probability equations instead of the relative three-month Treasury bill rate: the Composite Leading Indicator and the industrial production growth rate. However, the results are qualitatively similar, as reported in Section 4.5. 4.2. Estimation results 4.2.1. Identifying the states Table 2 reports the estimation results of the parameters in Eqs. (11)–(16) for portfolios P1 (loser), P2, P4, P6, P8, and P10 (winner).9 The constant parameter estimates in the conditional mean equation ^i0;1 Þ are much lower than those in state 2 (b ^i0;2 Þ in all in state 1 (b momentum portfolios. The constant term in state 1 monotonically increases across the portfolios from the loser to the winner portfolios and is more precisely estimated. Eight out of 10 constant terms in state 1 are significantly estimated at the 1% level and all 10 constant term estimates are negative. In contrast, there is no particular pattern in the constant term in state 2 and any of the 10 constant terms are not significantly estimated. The conditional standard deviation ^ i;1 Þ is greater than that in state 2 ðr ^ i;2 Þ in all estimate in state 1 ðr portfolios. All conditional volatilities are highly significantly estimated. Schwert (1990) and Hamilton and Lin (1996) find that return volatilities are higher in recession periods than in expansion periods. Their findings are verified with historical National Bureau of Economic Research (NBER) business cycle dates. These results may indicate that state 1 is the recession state and state 2 is the 8 Instead 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. 9 The estimation results for portfolios P3, P5, P7, and P9 are not reported because of space constraint. The results are available upon request. 196 Table 2 Parameter estimates for the univariate Markov switching model of excess returns on ten decile momentum portfolios. Parameters In State 1 Parameters P2 P4 P6 P8 Winner WML HML Mean equation: ^i0;1 Constant, b 0.179 0.116 0.108 0.083 0.063 0.003 0.176 0.064 ^i1;1 RREL, b (3.33) 1.300 (3.17) 0.917 (3.40) 0.789 (2.97) 0.783 (2.20) 0.576 (0.13) 0.074 (2.21) 1.226 (1.26) 0.124 ^i2;1 DEF, b (1.17) 3.779 (1.18) 1.791 (1.11) 1.146 (1.23) 0.329 (0.84) 0.347 (0.12) 1.107 (0.71) 4.886 ^i3;1 MB, b (2.17) 0.150 (1.52) 0.076 (1.05) 0.050 (0.34) 0.046 (0.34) 0.031 (1.30) 0.044 ^i4;1 DIV, b (1.66) 3.047 (1.15) 2.285 (0.73) 2.392 (0.65) 2.150 (0.32) 1.902 (2.93) (3.18) (3.38) (3.41) 1.576 (11.4) 83.64 Transition probability parameters: 1.485 1.460 ^ i0 Constant, p (11.1) (10.4) 102.3 88.25 ^ i1 RREL,p In State 2 Loser P2 P4 P6 P8 Winner WML HML ^i0;2 b 0.019 0.014 0.010 0.009 0.007 0.001 0.020 0.042 ^i1;2 b (1.47) 0.153 (1.38) 0.276 (1.07) 0.399 (1.05) 0.475 (0.74) 0.582 (0.06) 1.065 (0.81) 0.912 (2.48) 0.056 (0.11) 0.782 ^i2;2 b (0.50) 0.248 (1.14) 0.053 (1.88) 0.125 (2.30) 0.135 (2.64) 0.142 (3.49) 0.580 (1.57) 0.828 (0.14) 0.654 (1.78) 0.106 (0.46) 0.006 ^i3;2 b (0.49) 0.113 (0.13) 0.000 (0.36) 0.035 (0.45) 0.033 (0.44) 0.034 (1.23) 0.004 (0.86) 0.109 (1.39) 0.044 (0.60) 0.516 (0.66) 2.531 (0.06) 1.456 ^i4;2 b (1.43) 0.261 (0.00) 0.247 (0.99) 0.122 (1.40) 0.036 (1.33) 0.090 (0.11) 0.466 (1.00) 0.727 (0.54) 1.416 (3.02) (0.97) (1.61) (1.40) (0.78) (1.00) (0.59) (0.18) (0.43) (1.43) (1.24) (3.14) 1.521 1.498 1.278 0.207 0.041 (10.1) 115.3 (10.3) 41.40 (8.14) 11.10 60.90 108.9 44.86 44.87 34.14 37.41 32.90 13.28 31.57 62.32 (2.78) (2.40) (1.74) (1.79) (1.57) (0.84) 0.051 (19.1) 0.038 (19.7) 0.034 (19.6) 0.032 (18.6) 0.034 (20.2) 0.039 (13.6) 0.013 0.014 (3.76) (3.32) (3.03) (2.92) (3.03) (2.41) Error term volatility: ^ i;1 0.133 Std dev, r (13.4) 0.097 (15.8) 0.080 (14.8) 0.071 (15.1) 0.072 (15.1) 0.081 (18.5) Log-likelihood value: 732.0 906.8 1014.5 1057.2 1038.6 879.8 0.052 0.025 p^ i2 r^ i;2 The following univariate two-state Markov switching model is estimated for excess returns on each momentum decile portfolio i: r it ¼ bi0;St þ bi1;St RRELt1 þ bi2;St DEF t1 þ bi3;St MBt2 þ bi4;St DIV t1 þ eit eit N 0; r2i;St ; St ¼ f1; 2g pit ¼ P Sit ¼ 1jSit1 ¼ 1 ¼ U pi0 þ pi1 RRELt1 ; 1 pit ¼ P Sit ¼ 2jSit1 ¼ 1 qit ¼ P Sit ¼ 2jSit1 ¼ 2 ¼ U pi0 þ pi2 RRELt1 ; 1 qit ¼ P Sit ¼ 1jSit1 ¼ 2 ; where rit is the monthly excess return for a given decile portfolio and Sit is the regime indicator. RREL is the relative three-month Treasury-bill rate calculated as the difference between the current T-bill rate and its 12-month backward moving average, DEF is the spread between Moody’s seasoned Baa-rated corporate bond yield and 10-year treasury bond yield, MB is defined as the 12-month log-difference in the monetary base reported by the St. Louis Federal Reserve, and DIV is the sum of dividend payments accruing to the CRSP value-weighted market portfolio over the previous 12 months divided by the contemporaneous level of the index. Numbers in parentheses are t-statistics (parameter estimates divided by standard errors). ‘WML’ indicates ‘Winner’ portfolio minus ‘Loser’ portfolio, and ‘HML’ is the highest book-to-market decile portfolio minus the lowest book-to-market decile portfolio. The sample period is from March 1960 to December 2012. D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 Loser D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 expansion state. A further analysis of the identification of the states will be presented in Section 4.3. 4.2.2. Estimation of the conditional mean equations The estimation results of the conditional mean Eq. (11) for portfolios P1 (loser), P2, P4, P6, P8, and P10 (winner) are presented in Table 2. The difference in the coefficient estimates between the winner (P10) and loser (P1) portfolios is reported in the column under WML. We also estimate the same Markov switching regression model for each of 10 decile book-to-market portfolios to compare with the results of WML.10 We report only the difference in the coefficient estimates between the value (highest book-to-market) portfolio and the growth (lowest book-to-market) portfolio in the column under HML in Table 2 to compare the results of WML with those of HML. The coefficient estimates on the relative three-month Treasury bill rate (RREL) are all negative for the 10 momentum portfolios ^i1;1 and b ^i1;2 Þ, which means that when in both states 1 and 2 (b the short-term interest rate increases (relative to the average of the prior 12 months) in the previous month, the returns of all momentum portfolios decrease in the current month. There is a systematic pattern in these coefficients across the portfolios. In recession state, moving from the loser to the winner portfolio, the coefficient on RREL increases monotonically from 1.300 (with t-statistic of 1.17) to 0.074 (with t-statistic of 0.12). The difference in the coefficients between the winner and loser portfolios (WML) is 1.226 (with t-statistic of 0.71). In the expansion state, however, the coefficient decreases monotonically from 0.153 (with t-statistic of 0.50) to 1.065 (with t-statistic of 3.49). The difference in the coefficients between the winner and loser portfolios (WML) is 0.912 (with t-statistic of 1.57). This evidence indicates that in the recession state, interest rate changes have a greater negative impact on loser stocks than winner stocks, while in the expansion state, interest rate changes have a greater negative impact on winner stocks than loser stocks. The coefficient estimates on the default spread (DEF) in both ^i2;1 and b ^i2;2 Þ exhibit a systematic variation across the states (b portfolios in both states. In the recession state, moving from the loser to winner portfolios, the coefficient estimate on DEF ^i2;1 Þ decreases largely monotonically from 3.779 (with t-statis(b tic of 2.17) to 1.107 (with t-statistic of 1.30). The difference in the coefficient estimates between the winner and loser portfolios is marginally statistically significant: 4.886 (with t-statistic of 1.78). In the expansion state, however, the coefficient ^i2;2 Þ increases monotonically from 0.248 estimate on DEF (b (with t-statistic of 0.49) to 0.580 (with t-statistic of 1.23). The difference in the coefficient estimates between the winner and loser portfolios is 0.828 (with t-statistic of 0.86). The negative value of WML in the recession state and the positive value of WML in the expansion state indicate that loser stocks are more affected by the credit condition of the market in the recession state than are winner stocks, but the reverse occurs in the expansion state. This implies that momentum profits are procyclical. These above findings are exactly the opposite to those of Gulen et al. (2011), who find the counter-cyclicality of the value premium (measured by HML). We find that moving from the growth portfolio (with low book-to-market) to the value portfolio (with high book-to-market), the coefficient on DEF increases monotonically in the recession state, while it decreases monotonically in the expansion state. This pattern is exactly the opposite of that in 10 Returns on the 10 decile book-to-market portfolios were obtained from Kenneth French’s website at Dartmouth College (http://mba.tuck.dartmouth.edu/pages/faculty/ken.french). 197 the momentum portfolios. Table 2 reports that the differences in the coefficient estimates on DEF between the value portfolio and the growth portfolio (HML) are 0.782 and 0.654 in the recession and expansion states, respectively. These results indicate that value stocks are more affected by the credit condition in the market than growth stocks in the recession state, but the reverse occurs in the expansion state; that is, the value premium is counter-cyclical, which is consistent with Gulen et al. (2011). The coefficient estimates on the growth in monetary base (MB) ^i3;1 and b ^i3;2 Þ do not show any particular pattern in both states (b across the portfolios and are all statistically insignificant. One noteworthy thing, however, is that the coefficient estimates on MB are all positive in recession state but are mixed in sign in expansion state, which means that an increase in money supply produces higher expected returns of the testing portfolios during recession periods but does not during expansion periods. The coefficient estimates on the dividend yield (DIV) in both ^i4;1 and b ^i4;2 Þ also exhibit a systematic variation across states (b the portfolios in both states. In the recession state, moving from the loser to winner portfolios, the coefficient estimate on DIV ^i4;1 Þ decreases monotonically from 3.047 (with t-statistic of (b 2.93) to 0.516 (with t-statistic of 0.97). The coefficient estimates are all positive and mostly significant at the five percent level. The difference in the coefficient estimates between the winner and loser portfolios is marginally statistically significant: 2.531 (with t-statistic of 1.61). In the expansion state, however, the ^i4;2 Þ increases monotonically from 0.261 coefficient estimate (b (with t-statistic of 0.78) to 0.466 (with t-statistic of 1.43). The difference in the coefficient estimates between the winner and loser portfolios is 0.727 (with t-statistic of 1.24). These results indicates that dividend yields have a positively greater impact on stock returns in recession periods than in expansion periods and that loser (winner) stocks are more greatly affected in the recession (expansion) state by cash flow shocks from dividends. This evidence also means that momentum profits are procyclical. Note that the differences in the coefficient estimates on DIV between the value and growth portfolio (HML) are 1.456 (with t-statistic of 1.40) and 1.416 (with t-statistic of 3.14) in the recession and expansion states, respectively. These are also exactly opposite in sign to those of WML and are used as evidence of the counter-cyclicality of the value premium by Gulen et al. (2011). In sum, in the recession state, the loser portfolio has a greater sensitivity to all four conditioning macroeconomic variables than does the winner portfolio. Specifically, the coefficient estimates of the loser versus winner portfolios on the variables, RREL, DEF, MB, and DIV, are 1.300 (t-statistic of 1.17) vs. 0.074 (t-statistic of 0.12), 3.779 (t-statistic of 2.17) vs. 1.107 (t-statistic of 1.30), 0.150 (t-statistic of 1.66) vs. 0.044 (t-statistic of 0.60), and 3.047 (tstatistic of 2.93) vs. 0.516 (t-statistic of 0.97), respectively. The coefficient estimates of the loser portfolio are mostly statistically significant, while those of the winner portfolio are insignificant. In the expansion state, however, we observe an opposite pattern in the coefficient estimates to the case of the recession state. That is, the winner portfolio tends to have a greater sensitivity to these variables (except for MB) than does the loser portfolio. Specifically, the coefficient estimates of the loser versus winner portfolios on RREL, DEF, and DIV are 0.153 (t-statistic of 0.50) vs. 1.065 (t-statistic of 3.49), 0.248 (t-statistic of 0.49) vs. 0.580 (t-statistic of 1.23), and 0.261 (t-statistic of 0.78) vs. 0.466 (t-statistic of 1.43), respectively. Contrary to the case in the recession state, the coefficient estimates of the winner portfolio are marginally statistically significant in the expansion state, while those of the loser portfolio are insignificant. These above results indicate that loser stocks are riskier in recession periods than winner stocks, while winner stocks are riskier in expansion periods than loser stocks. 198 D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 Table 3 Tests for equality of the slope coefficients across states in the Markov switching model. Unrestricted log likelihood value Restricted log likelihood with bik;St ¼1 ¼ bik;St ¼2 for k ¼ 1; 2; 3; 4 [p-value] Unrestricted log likelihood value Restricted log likelihood with bik;St ¼1 ¼ bik;St ¼2 for k ¼ 1; 2; 3; 4 [p-value] Loser Decile 2 Decile 3 Decile 4 Decile 5 732 723 [0.00] 907 900 [0.01] 973 965 [0.00] 1014 1008 [0.01] 1042 1037 [0.05] Decile 6 Decile 7 Decile 8 Decile 9 Winner 1057 1052 [0.04] 1055 1051 [0.09] 1039 1034 [0.07] 999 994 [0.08] 880 876 [0.08] The following univariate Markov switching model is estimated for excess returns on each momentum decile portfolio i: rit ¼ bi0;St þ bi1;St RRELt1 þ bi2;St DEF t1 þ bi3;St MBt2 þ bi4;St DIV t1 þ eit eit N 0; r2i;St ; St ¼ f1; 2g pit ¼ P Sit ¼ 1jSit1 ¼ 1 ¼ U pi0 þ pi1 RRELt1 ; 1 pit ¼ P Sit ¼ 2jSit1 ¼ 1 qit ¼ P Sit ¼ 2jSit1 ¼ 2 ¼ U pi0 þ pi2 RRELt1 ; 1 qit ¼ P Sit ¼ 1jSit1 ¼ 2 ; where rit is the monthly excess return for a given decile portfolio and Sit is the regime indicator. RREL is the relative 3-month T-bill rate calculated as the difference between the current T-bill rate and its 12-month backward moving average, DEF is the spread between Moody’s seasoned Baa-rated corporate bond yield and 10-year treasury bond yield from the FRED, MB is defined as the 12-month log-difference in the monetary base reported by the St. Louis Federal Reserve, and DIV is the sum of dividend payments accruing to the CRSP value-weighted market portfolio over the previous 12 months divided by the contemporaneous level of the index. Numbers in parentheses are t-statistics (parameter estimates divided by standard errors). The likelihood ratio tests are conducted on the null hypothesis that the coefficient are equal across states, i.e., bik;St ¼1 ¼ bik;St ¼2 for k = 1,2,3,4,f or momentum portfolio i. The sample period is from March 1960 to December 2012. This is consistent with the notion that momentum profits are procyclical. 4.2.3. Tests for equality of the slope coefficients across states The previous results show that returns on the momentum portfolios react asymmetrically to the macroeconomic conditioning variables across states. To confirm the differential responses of momentum returns to aggregate economic conditions in the recession and expansion states, it is necessary to test whether the coefficients on the four conditioning variables (the relative Treasury bill rate, the default spread, the growth in monetary base, and the dividend yield) are equal across states. We employ a likelihood ratio test for the null hypotheses: bik;St ¼1 ¼ bik;St ¼2 for k = 1,2,3,4 for each of the 10 momentum portfolios. Table 3 reports the unrestricted and restricted log-likelihood values and p-values from a standard v2 distribution for the 10 momentum portfolios. Six out of the 10 portfolios have a p-value less than 5% and all portfolios have p-values less than 10%, indicating that the null hypothesis is strongly rejected. It is statistically confirmed, therefore, that the conditional mean equation is state-dependent and the responses of momentum profits to the conditioning variables are asymmetric across states. Perez-Quiros and Timmermann (2000) and Gulen et al. (2011), the bivariate Markov switching regression model is as follows. 0 Let r t ¼ r Lt ; rW be a (2 1) vector consisting of excess returns t on the loser and winner portfolios, rLt and rW t , respectively. Then, the joint conditional mean equation is specified as follows: rt ¼ b0;St þ b1;St RRELt1 þ b2;St DEF t1 þ b3;St MBt2 þ b4;St DIV t1 þ et ; where bk;St for k = 1,2,3,4, and et is a (2 1) vector of normal residuals with P P mean zero and covariance matrix St ; St ¼ f1; 2g. Here St is a positive semidefinite (2 2) covariance matrix of the residuals from the loser and winner portfolios’ excess returns in state St. For estimation convenience, we assume the form of the conditional covariance matrix as follows: ! 8 X > > > ¼ ki;St ln > > < ii;St !1=2 !1=2 > X X X > > > ¼ qSt > : ij;St 4.3. A bivariate joint model for Loser and Winner Stocks’ expected returns 4.3.1. Model specification So far the Markov switching regression models for excess returns have been estimated separately (i.e., univariately) for each of the 10 momentum portfolios. That is, the condition that the recession and expansion states occur simultaneously for all test portfolios is not imposed in the estimation. The joint framework allows us to impose a common state process that drives all excess return series. Since there are difficulties in estimating a multivariate joint model when the excess returns of all 10 portfolios and the loser and winner portfolios are our main target portfolios, we consider a bivariate framework that simultaneously estimates the conditional mean equations for both loser and winner portfolios. This bivariate framework can model the time-varying momentum profit and test its procyclical variations. As in ð17Þ 0 is a (2 1) coefficient vector with elements bLk;St ; bW k;St ii;St for i ¼ j ð18Þ for i – j jj;St In other words, we assume that the diagonal elements of this P variance–covariance matrix, ii;St , depend only on the state of economy, as in the univariate case of Eq. (11). The off-diagonal elements, P ij;St , assume a state-dependent correlation between the residuals, denoted qSt . We also do not include ARCH effects in the conditional volatility equation. As in the univariate case, we use the relative three-month Treasury bill rate (RREL) as the instrumental variable in modeling state transition probabilities to compare the results with those of the univariate case. As an alternative instrumental variable, we also use the one-month-lagged monthly growth rate of industrial production (MPt1), defined as MPt = logIPt logIPt1, where IPt is the index level of industrial production at month t. The purpose of using MP as an alternative instrumental variable is twofold. The first is to obtain the estimates of the state transition probabilities based on the real economic variable rather than the financial economic variable. These estimates would be more Table 4 Estimation results of the Bivariate Markov switching model for excess returns of the loser and winner portfolios. Parameter In State 1 Loser (L) Parameter Winner (W) WML Panel A: Instrumental variable in modeling state transition probabilities = Relative three-month T-bill rate (RREL) Mean equation: ^0;1 : 0.090 0.023 0.067 Constant, b (1.71) (0.63) ^0;2 b (1.94) In State 2 Loser (L) Winner (W) WML 0.007 0.017 0.010 (0.51) (1.41) (1.63) 0.181 0.697 0.516 (0.58) (2.49) (3.20) 0.173 0.148 0.026 (0.34) (0.30) (0.10) 0.014 0.001 0.015 (0.27) (0.02) (0.63) 0.248 0.042 0.205 (0.86) (0.16) (1.42) 0.052 (21.5) 0.049 (20.6) W Test for Hb0 : bL0;1 bL0;2 ¼ bW 0;1 b0;2 , Log-likelihood value = 1,957 [p-value = 0.11] ^1;1 RREL, b 1.709 0.709 1.000 (1.44) (0.88) (1.23) ^1;2 b W Test for Hb1 : bL1;1 bL1;2 ¼ bW 1;1 b1;2 , Log-likelihood value = 1,957 [p-value = 0.07] ^2;1 DEF, b 1.428 0.117 1.546 (0.78) (0.09) (1.30) ^2;2 b W Test for Hb2 : bL2;1 bL2;2 ¼ bW 2;1 b2;2 , Log-likelihood value = 1,958 [p-value = 0.20] ^3;1 MB, b 0.015 0.158 (1.53) (0.20) (2.08) ^3;2 b W Test for Hb3 : bL3;1 bL3;2 ¼ bW 3;1 b3;2 , Log-likelihood value = 1,957 [p-value = 0.06] ^4;1 DIV, b 2.114 1.095 1.019 (2.02) (1.59) (1.45) ^4;2 b W Test for Hb4 : bL4;1 bL4;2 ¼ bW 4;1 b4;2 , Log-likelihood value = 1,957 [p-value = 0.09] Error term volatility: ^1 Std. dev, r 0.139 (15.9) Transition probability parameters: ^0 Constant, p ^1 RREL, p r^ 2 0.092 (17.1) 1.251 (10.5) 66.32 (4.04) ¼ p2 , Log-likelihood value = 1,957 [p-value = 0.05] p^ 2 31.90 (2.48) 0.748 q^ 2 (22.1) Panel B: Instrumental variable in modelling state transition probabilities = Monthly growth rate of industrial production (MP) 0.861 (58.5) Test for Hp1 : p1 Correlation coefficients: ^1 q Mean equation: ^0;1 : Constant, b 0.068 0.015 0.053 (1.19) (0.37) (1.38) ^0;2 b 0.009 0.018 0.009 (0.52) (1.15) (1.45) 0.068 0.443 0.374 (0.19) (1.44) (1.84) 0.301 0.220 0.080 (0.49) (0.39) (0.32) 0.008 0.008 0.016 (0.15) (0.17) (0.66) 0.249 0.080 0.209 (0.79) (0.23) (1.43) D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 0.172 W Test for Hb0 : bL0;1 bL0;2 ¼ bW 0;1 b0;2 , Log-likelihood value = 1,957 [p-value = 0.11] ^1;1 RREL, b 1.253 1.093 0.160 (1.04) (1.29) (0.19) ^1;2 b W Test for Hb1 : bL1;1 bL1;2 ¼ bW 1;1 b1;2 , Log-likelihood value = 1,958 [p-value = 0.56] ^2;1 DEF, b 0.992 0.419 1.411 (0.50) (0.30) (1.12) ^2;2 b W Test for Hb2 : bL2;1 bL2;2 ¼ bW 2;1 b2;2 , Log-likelihood value = 1,958 [p-value = 0.25] ^3;1 MB, b 0.181 0.022 0.159 (1.63) (0.30) (2.15) ^3;2 b W Test for Hb3 : bL3;1 bL3;2 ¼ bW 3;1 b3;2 , Log-likelihood value = 1,957 [p-value = 0.08] ^4;1 DIV, b 1.657 1.092 0.564 (1.51) (1.42) (0.70) ^4;2 b W Test for Hb4 : bL4;1 bL4;2 ¼ bW 4;1 b4;2 , Log-likelihood value = 1,958 [p-value = 0.33] 199 (continued on next page) 200 Table 4 (continued) Parameter Error term volatility: ^1 Std. dev, r In State 1 Loser (L) Winner (W) 0.141 (12.9) 0.091 (14.7) Transition probability parameters ^0 Constant, p ^1 MP, p Test for Hp1 : p1 Correlation coefficients: ^1 q Parameter WML 1.180 (9.32) 108.47 (3.65) ¼ p2 , Log-likelihood value = 1,951 [p-value = 0.00] 0.731 (22.9) r^ 2 In State 2 Loser (L) Winner (W) 0.053 (17.4) 0.051 (17.9) p^ 2 20.29 (1.33) ^2 q 0.868 (88.9) WML The bivariate Markov switching regression model for the loser and winner portfolios’ excess returns is specified as follows: 0 0 P L W where r t ¼ rLt ; r W be a (2 1) vector consisting of excess returns on the loser and winner portfolios, r Lt and r W for k = 1,2,3,4, and et Nð0; St Þ; St ¼ f1; 2g. t t , respectively, bk;St is a (2 1) coefficient vector with elements bk;St ; bk;St P P 1=2 P 1=2 P P The conditional variance–covariance matrix, St , has the following form: ln ¼ kiSt , for i = j and ij;St ¼ qSt for i – j. The transition probabilities are defined as ii;St ii;St jj;St pt ¼ PðSt ¼ 1jSt1 ¼ 1Þ ¼ Uðp0 þ p1 IV t1 Þ and qt ¼ PðSt ¼ 2jSt1 ¼ 2Þ ¼ Uðp0 þ p2 IV t1 Þ; where U() is the cumulative density function of a standard normal variable, and IV is the instrumental variable in the state transition probabilities. Two instrumental variables are used. The first one is the relative three-month Treasury-bill rate calculated as the difference between the current T-bill rate and its 12-month backward moving average (RRELt1) (Panel A), and the second one is the monthly growth rate of industrial production ( MPt1) (Panel B). DEF is the spread between Moody’s seasoned Baa-rated corporate bond yield and 10-year treasury bond yield, MB is defined as the 12-month log-difference in the monetary base reported by the St. Louis Federal Reserve, and DIV is the sum of dividend payments accruing to the CRSP value-weighted market portfolio over the previous 12 months divided by the contemporaneous level of the index. Numbers in parentheses are t-statistics (parameter estimates divided by standard errors). The sample period is from March 1960 to December 2012. D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 rt ¼ b0;St þ b1;St RRELt1 þ b2;St DEF t1 þ b3;St MBt2 þ b4;St DIV t1 þ et ; D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 informative to identify the states in relating to actual business cycles such as the NBER cycles, since MP is more directly related to real economic activities than is a financial variable such as the Treasury rate. The second is to conduct a robustness check for the sensitivity of the estimation results of the conditional mean equation to the choice of the instrumental variable. The state transition probabilities for the bivariate model are modeled as follows: pt ¼ PðSt ¼ 1jSt1 ¼ 1Þ ¼ Uðp0 þ p1 IV t1 Þ; ð19Þ qt ¼ PðSt ¼ 2jSt1 ¼ 2Þ ¼ Uðp0 þ p2 IV t1 Þ; ð20Þ 201 where IVt1 is the instrumental variable which is defined as either RRELt1 or MPt1. 4.3.2. Estimation results Table 4 presents the estimation results of the bivariate Markov switching regression model when the instrumental variables in modeling state transition probabilities are the relative threemonth Treasury bill rate (Panel A) and the monthly growth rate of industrial production (Panel B), respectively. When the instrumental variable is the relative Treasury bill rate, the coefficient estimates on the conditioning variables for both loser and winner Fig. 1. Expected excess returns from univariate and bivariate markov switching models. The expected excess returns for the winner portfolio (Panel A), the loser portfolio (Panel B), and the winner-minus-loser (Panel C) obtained from the univariate and bivariate Markov switching models are plotted over time, when the instrumental variable in the state transition probabilities is the relative three-month Treasury bill rate. The solid lines denote the expected excess returns obtained from the bivariate Markov switching model, and the dashed lines denote the expected excess returns obtained from the univariate model. Shaded areas indicate NBER recession periods. 202 D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 portfolios are qualitatively similar to those from the univariate model specification in Table 2, implying that imposing a common state process changes little. In particular, the asymmetries in the coefficients on the macroeconomic variables in the conditional mean equation are very similar to those found in the univariate model specification. For instance, in the recession state, the coefficient on RREL for loser stocks is greater in absolute value than that for winner stocks. The difference in the coefficient estimate between winner and loser stocks is 1.000 (with t-statistic of 1.23). The opposite pattern is found for the expansion state; the coefficient on RREL for winner stocks is greater in absolute value than that for loser stocks. The difference in the coefficient estimates between the winner and loser portfolios is statistically significant: 0.516 (with t-statistic of 3.20). The estimation results for the conditional mean equation are also similar, when the instrumental variable in the state transition probabilities is the monthly growth rate of industrial production. That is, the coefficient estimates on the macroeconomic variables in the mean equation show that loser (winner) stocks are more greatly affected in the recession (expansion) state by the conditioning variables. Table 4 also reports testing results for the proposition that the asymmetry in the coefficients observed for the loser portfolio across recession and expansion states equals the asymmetry for the winner portfolio. For each set of the coefficients, we test the null hypothesis that L W for k ¼ 1; 2; 3; and 4: bk;1 bLk;2 ¼ bW k;1 bk;2 ð21Þ The nulls of identical asymmetries across states for loser and winner stocks are modestly rejected at standard significance levels for the slope coefficients on the conditioning variables, RREL, MB, and DIV, in the conditional mean equation, when the instrumental variable in the state transition probabilities is the relative Treasury bill rate. These results are somewhat consistent with those reported by Perez-Quiros and Timmermann (2000) for size portfolios and with those of Gulen et al. (2011) for book-to-market portfolios. These authors report that the null hypotheses of identical asymmetries for small and large stocks and for growth and value stocks are strongly rejected using the conditioning variables similar to ours. Therefore, the state-based asymmetries in the coefficients between the winner and loser portfolios are prominent, although the degree of the asymmetries is weaker than in the cases of large-small and value-growth stocks. Fig. 1 plots the expected excess returns obtained from the univariate and bivariate Markov switching models for the winner portfolio (Panel A), the loser portfolio (Panel B), and the WML (Panel C), when the instrumental variable in the state transition probabilities is the relative Treasury bill rate. The solid line denotes the expected excess returns obtained from the univariate Markov switching model and the dashed lines denote the expected excess returns obtained from the bivariate model. The shaded areas indicate NBER recession periods. All panels of Fig. 1 show that the expected excess returns of the loser and winner portfolios and the expected momentum profit (WML) display time-variations across the states of the economy. Panels A and B of Fig. 1 show that the series obtained from the univariate and joint bivariate models for the loser and winner portfolios are approximately similar. The expected returns of both loser and winner portfolios tend to increase during recession periods but decrease during expansion periods. However, the loser portfolio tends to display this pattern more strongly than the winner portfolio. As a result, as seen in Panel C, the expected momentum profit (WML) tends to decrease sharply and have a negative value during recessions but to increase and have a positive value just Fig. 2. Time-series of the state transition probabilities of being in high and low volatility states. The figure plots the time-series of the state transition probabilities of being in state 1 (high volatility; Panel A) and state 2 (low volatility; Panel B), which are estimated from the bivariate Markov switching model using the monthly growth rate of industrial production as an instrumental variable in the state transition probability. Shaded areas indicate NBER recession periods. D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 203 Fig. 3. Predicted excess returns from the bivariate Markov switching model. The predicted excess returns for the loser portfolio (Panel A), the winner portfolio (Panel B), and the winner-minus-loser portfolio (Panel C) are obtained from the bivariate Markov switching regression model. The solid lines plot the in-sample predicted excess returns, and the dotted lines plot the out-of-sample predicted excess returns. The out-of-sample forecasts are from January 1977 to December 2012. after recessions and during expansions. It tends to be lower during recessions than during expansions. For example, during the whole sample period, the expected momentum profit is 1.2% per month during expansions (i.e., low volatility states), while it is 0.4% per month during recessions (i.e., high volatility states) and the difference in expected momentum profits between the two states is highly significant (with t-statistic of 4.18). Although there is a little discrepancy in the expected excess returns of WML obtained from the univariate and bivariate models, this pattern is overall very similar. This procyclical behavior of the expected momentum profit is the opposite of the counter-cyclical behavior of the value premium shown by Gulen et al. (2011). These authors illustrate that the value premium increases sharply during the later stages of recessions but decreases just after recessions and that it tends to be higher during recessions than during expansions. To further investigate the opposite behavior of the momentum profit and the value premium across business cycles, we examine the correlation coefficients between the expected excess returns of WML and the growth rates of procyclical macroeconomic variables such as the gross domestic product (GDP) and industrial production. They are positive: 0.30 and 0.25 for the GDP and industrial production, respectively. On the other hand, the correlation 204 D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 Table 5 Trading results based on the out-of-sample prediction from the Bivariate Markov switching regression model. Treasury bill Loser portfolio Winner portfolio Switching portfolio Buy-and-hold Switching portfolio Panel A: The whole period (January 1977 to December 2012) Average return 5.07 10.62 Std dev of return 1.00 31.41 Sharpe ratio 0.177 Buy-and-hold 14.21 24.38 0.375 19.39 22.16 0.646 13.57 17.78 0.478 Panel B: Recession states – NBER Average return 6.35 Std dev of return 1.41 Sharpe ratio 8.35 50.68 0.039 26.22 43.39 0.345 2.15 29.59 0.142 12.18 25.85 0.226 Panel C: Expansion states – NBER Average return 4.86 Std dev of return 0.90 Sharpe ratio 10.99 27.07 0.227 12.24 19.61 0.377 22.23 20.62 0.842 13.80 16.11 0.555 The buy-and-hold strategy reinvests all funds in a given momentum under consideration (the loser or winner) portfolio. The switching portfolios take a long position in the momentum portfolio if the excess return recursively predicted from the bivariate Markov switching regression model is positive; otherwise, the position switches into the one-month Treasury bill. Average returns and standard deviations are annualized. The sample period is from January 1977 to December 2012. coefficients between the expected excess returns of HML and these two macroeconomic variables are negative: 0.11 and 0.17, respectively.11 4.3.3. Further identifying the states To further identify the states, we plot the state transition probabilities over time. Fig. 2 shows the time-series state transition probabilities estimates of being in state 1 (Panel A) and state 2 (Panel B) at time t conditional on the information set at time t 1; Prob Sit ¼ 1jXt1 ; h and Prob Sit ¼ 2jXt1 ; h , respectively. These probabilities estimates are obtained when the instrumental variable in the state transition probabilities is the monthly growth rate of industrial production. As mentioned earlier, these estimates would be more informative to identify the states in relating to actual business cycles such as the NBER cycles, since MP is more directly related to real economic activities than is a financial variable such as the Treasury bill rate. As shown in Fig. 2, during the NBER recession periods (shaded areas), the state transition probabilities of being in state 1 tend to increase and are relatively high, while the state transition probabilities of being in state 2 tend to decrease and are low. On the other hand, during the NBER expansion periods, an opposite pattern is observed. The results suggest that state 1 can be regarded as the recession state and state 2 can be regarded as the expansion state. To examine more specifically whether the estimated state transitional probabilities are associated with actual business cycles (i.e., the NBER classification of business cycles), we conduct some formal statistical tests. First, we compute the averages of the state transition probabilities estimates of being in state 1 across NBER expansionary and recessionary periods. The averages are 0.264 and 0.580, respectively. The difference between these two averages is highly statistically significant (t-statistic of 11.57).12 Second, to examine the extent to which the state transition probabilities are correlated with the NBER business cycle, we estimate the following probit regression model: Prob½Dt ¼ 1jpt ¼ Fða þ bpt Þ; ð22Þ 11 Since the GDP is of quarterly frequency, the monthly excess returns are transformed into quarterly returns by compounding monthly returns over each quarter. 12 The averages of the state transition probabilities of being in state 2 across NBER expansionary and recessionary periods are 0.736 and 0.237, respectively. The tstatistic of the difference in the averages is 11.57. where F is the cumulative normal distribution, Dt is a binary variable that takes a value of one during the NBER recessionary periods and zero otherwise, pt is the state transition probability estimate of ^ is positive being in state 1. The estimate of the slope coefficient, b, and statistically significant; it is 2.000, with t-statistic of 9.53. This suggests that the state transition probability of being in state 1 is strongly positively associated with the probability of being a NBER recession period. Finally, we compute the point-biserial correlation coefficient, which is used to measure the degree of the association when one variable is dichotomous (discrete) and the other is continuous.13 The computed point-biserial correlation coefficient between the NBER recession binary variable (Dt) and the state transitional probabilities estimates of being in state 1 is also positive and strongly statistically significant; it is 0.418, with t-statistic of 11.57. These above results suggest that state 1 can be regarded as the recession state and state 2 can be regarded as the expansion state. Further evidence supporting this is the estimation results of the slope coefficients in the state transition probabilities of Eqs. (19) and (20). The slope coefficient estimate of the transition probabil^ 1 , is negative and statistically significant; it is ity in state 1, p 108.47, with t-statistic of 3.65. However, the slope coefficient ^ 2 , is positive estimate of the transition probability in state 2, p and statistically moderately significant; it is 20.29, with t-statistic of 1.33. These results indicates that an increase in the industrial production growth is associated with a decrease in the probability of being in state 1 but an increase in the probability of being in state 2. This implies that states 1 and 2 can be regarded as the recession and expansion states, respectively. 4.4. Trading rules based on out-of-sample predictions To avoid potential problems from over-fitting a complex nonlinear model with a large number of parameters being estimated as in this paper, it is necessary to examine out-of-sample predictability of the model. Since the conditional mean Eq. (11) uses one-month– lagged predictive macroeconomic variables, it can be used to predict the current month’s returns using conditioning information available up to the previous month. We follow Perez-Quiros and Timmermann (2000) and Gulen et al. (2011) to do a recursive out-of-sample prediction of excess returns for the loser and winner portfolios. Specifically, we first start to estimate our bivariate 13 The point-biserial correlation coefficient is a special case of Pearson in which one variable is quantitative and the other variable is dichotomous (see Glass and Hopkins, 1995). It has been often used in accounting and finance literature (Hagerman and Zmijewski, 1979; Clinch et al., 2012). Table 6 Estimation results of the univariate Markov switching model for excess returns on momentum portfolios after estimating state transition probabilities exogenously. Parameters In State 1 Parameters Loser P2 P4 P6 P8 Winner WML HML Mean equation: ^i0;1 Constant, b 0.058 0.043 0.044 0.034 0.026 0.015 0.043 0.005 ^i1;1 RREL, b (2.06) 1.017 (2.08) 0.669 (2.60) 0.622 (2.17) 0.594 (1.62) 0.565 (0.77) 0.721 (0.95) 0.296 (0.17) 0.877 ^i2;1 DEF, b (1.52) 0.824 (1.35) 0.582 (1.55) 0.577 (1.61) 0.272 (1.50) 0.066 (1.51) 0.437 (0.27) 1.261 ^i3;1 MB, b (0.93) 0.142 (0.88) 0.090 (1.07) 0.086 (0.56) 0.069 (0.13) 0.050 (0.72) 0.031 ^i4;1 DIV, b (2.18) 1.114 (1.89) 0.830 (2.23) 0.911 (1.97) 0.781 (1.36) 0.718 In State 2 P2 P4 P6 P8 Winner WML HML ^i0;2 b 0.005 0.001 0.011 0.007 0.012 0.015 0.020 0.034 ^i1;2 b (0.24) 0.431 (0.07) 0.547 (0.87) 0.527 (0.56) 0.546 (0.92) 0.619 (0.82) 0.963 (0.57) 0.532 (1.35) 0.487 (0.91) 0.241 ^i2;2 b (0.94) 0.052 (1.64) 0.036 (1.75) 0.330 (1.84) 0.475 (1.92) 0.221 (2.02) 0.017 (0.61) 0.069 (0.75) 1.094 (0.89) 0.111 (0.26) 0.399 ^i3;2 b (0.06) 0.169 (0.06) 0.009 (0.58) 0.233 (0.88) 0.229 (0.39) 0.199 (0.02) 0.062 (0.04) 0.107 (0.95) 0.044 (0.72) 0.787 (1.06) 0.327 (0.53) 0.075 ^i4;2 b (0.52) 0.168 (0.07) 0.260 (3.94) 0.384 (3.90) 0.496 (2.55) 0.466 (0.61) 0.012 (0.30) 0.156 (0.10) 0.485 (0.41) (0.90) (1.48) (1.97) (1.67) (0.03) (0.21) (1.45) 0.053 (17.5) 0.038 (16.5) 0.034 (16.3) 0.033 (18.1) 0.036 (18.2) 0.048 (18.1) 0.005 0.007 (1.68) (1.70) (2.28) (2.15) (1.92) (1.67) (0.30) (0.70) Error term volatility: ^ i;1 0.112 Std dev, r (23.1) 0.084 (23.3) 0.067 (23.0) 0.061 (25.6) 0.062 (22.8) 0.076 (22.9) 0.036 0.016 Log-likelihood value: 683.2 865.5 978.1 1027.0 1006.9 854.9 r^ i;2 The two-state Markov switching model is estimated in a two-step approach. In the first step, we estimate the following two-state Markov switching model for the consumption growth rates. Dct ¼ a0;St þ a1;St DEF t1 þ zt zt N 0; r2c;St ; St ¼ f1; 2g pt ¼ PðSt ¼ 1jSt1 ¼ 1Þ ¼ Uðp0 þ p1 RRELt1 Þ; qt ¼ PðSt ¼ 2jSt1 ¼ 2Þ ¼ Uðp0 þ p2 RRELt1 Þ; 1 pt ¼ PðSt ¼ 2jSt1 ¼ 1Þ 1 qt ¼ PðSt ¼ 1jSt1 ¼ 2Þ; where Dct = ln(Ct) ln(Ct3) is the quarter-on-quarter consumption growth rate, and Ct is the real personal consumption expenditures of nondurable goods and services obtained from the Bureau of Economic Analysis. In the second ^t ; qt Þ are used as given probabilities in the following Markov switching model for excess returns on decile momentum portfolios. step, the state transition probabilities estimated in the first step ðp rit ¼ bi0;St þ bi1;St RRELt1 þ bi2;St DEF t1 þ bi3;St MBt2 þ bi4;St DIV t1 þ eit eit N 0; r2i;St ; St ¼ f1; 2g pt ¼ PðSt ¼ 1jSt1 ¼ 1Þ; 1 pt ¼ PðSt ¼ 2jSt1 ¼ 1Þ qt ¼ PðSt ¼ 2jSt1 ¼ 2Þ; 1 qt ¼ PðSt ¼ 1jSt1 ¼ 2Þ: Numbers in parentheses are t-statistics (parameter estimates divided by standard errors). ‘WML’ indicates ‘Winner’ portfolio minus ‘Loser’ portfolio, and ‘HML’ is the highest book-to-market decile portfolio minus the lowest book-tomarket decile portfolio. The sample period is from March 1960 to December 2012. D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 Loser 205 206 D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 Fig. 4. Expected momentum profits from the two-step Markov switching model. The figure displays time-series of the expected momentum profits (WML) obtained from the two-step Markov switching model in which state transition probabilities are estimated separately in the Markov switching model for consumption growth rates and these estimated state transition probabilities are used as given probabilities in the Markov switching model for excess returns on decile momentum portfolios. Shaded areas indicate NBER recession periods. Markov switching regression model by using observations from March 1960 to December 1976 and predict the return for the next month (January 1977) based on the estimated parameters and the values of the conditioning information on the most recent month (December 1976).14 In this way, we re-estimate the nonlinear model by adding one new month to the previous estimation window (starting from March 1960) to ensure that we have enough in-sample observations to precisely estimate the model and compute the predicted returns for each of the loser and winner portfolios. Consequently, we obtain the predicted returns from January 1977 to December 2012. Fig. 3 plots the predicted excess returns obtained from the bivariate Markov switching regression model for the loser portfolio (Panel A), the winner portfolio (Panel B), and the WML portfolio (Panel C). For comparison, the out-of-sample predicted excess returns (with the dotted line) are overlaid with the in-sample predicted excess returns (with the solid line). The in-sample predictions are obtained from the one-time estimation of the same bivariate Markov switching regression model using the wholeperiod observations from March 1960 to December 2012. The out-of-sample predictions are highly correlated with the in-sample predictions. Their correlation coefficients are 0.35 for the winner portfolio, 0.73 for the loser portfolio, and 0.84 for the winnerminus-loser portfolio. For the portfolios, the out-of-sample and in-sample predictions have similar average returns and standard deviations. However, the out-of-sample predictions have slightly 14 The initial sample period from 1960 to 1976 follows from Perez-Quiros and Timmermann (2000) and Gulen et al. (2011). lower average excess returns but slightly higher standard deviation than the in-sample predictions. For the winner portfolio, the average returns are 1.00% and 1.05% for the out-of-sample and in-sample predictions, respectively, while the standard deviations are 2.00% and 0.96%, respectively. For the loser portfolio, the average returns are 0.79% and 0.57% for the out-of-sample and in-sample predictions, respectively, while the standard deviations are 3.32% and 2.26%, respectively. For the winner-minus-loser portfolio, the average returns are 0.21% and 0.49% for the out-of-sample and in-sample predictions, respectively, while the standard deviations are 2.45% and 1.89%, respectively. The economic significance of the out-of-sample prediction can be measured by the performance of a simple stylized trading rule based on the prediction. We follow the trading rule of PerezQuiros and Timmermann (2000), under which, if the predicted excess return is positive, we take a long position in a given momentum portfolio under consideration (the loser or winner portfolio) and otherwise switch the position into the Treasury bill. Table 5 presents the average returns, standard deviations, and Sharpe ratios over the whole period (Panel A) and the NBER recession states (Panel B) and NBER expansion states (Panel C) for such switching portfolios when the trading rule is based on the loser and winner portfolios, respectively. This table also presents these return and risk characteristics for Treasury bills and the buy-and-hold strategy that reinvests all funds in the portfolio under consideration. Table 5 shows that the economic significance of out-of-sample predictability is particularly significant for the switching portfolio based on the loser portfolio and during the recession state. Table 7 Parameter estimates for the univariate Markov Switching model of excess returns on ten decile momentum portfolios using alternative instrumental variables in modeling state transition probabilities. Parameters In State 1 Loser Parameters P2 P4 P6 P8 Winner WML HML Panel A: Instrumental variable (IV) in modeling state transition probabilities = Monthly growth rate of industrial production (MP) Mean equation: ^i0;1 ^i0;2 0.097 0.071 0.084 0.07 0.073 0.002 0.099 0.064 Constant, b b In State 2 Loser P2 P4 P6 P8 Winner WML HML 0.014 0.011 0.010 0.003 0.007 0.002 0.016 0.042 (0.90) 0.189 (0.89) 0.126 (0.28) 0.283 (0.76) 0.525 (0.17) 1.068 (0.58) 1.225 (1.95) 0.056 (2.27) 1.103 (2.60) 0.874 (2.46) 0.798 (2.47) 0.567 (0.10) 0.160 (1.24) 1.180 (1.26) 0.124 ^i1;2 b (1.06) 0.157 ^i2;1 DEF, b (1.30) 1.822 (1.59) 1.283 (1.40) 0.967 (1.47) 0.313 (0.94) 0.150 (0.27) 1.027 (0.69) 2.849 (0.11) 0.782 ^i2;2 b (0.47) 0.543 (0.60) 0.303 (0.52) 0.018 (1.20) 0.159 (2.38) 0.035 (3.34) 0.590 (1.96) 1.133 (0.11) 0.654 ^i3;1 MB, b (1.09) 0.140 (1.16) 0.074 (0.93) 0.049 (0.34) 0.057 (0.16) 0.052 (1.21) 0.036 (1.03) 0.104 (0.46) 0.006 ^i3;2 b (1.00) 0.021 (0.62) 0.032 (0.04) 0.031 (0.40) 0.022 (0.11) 0.008 (1.19) 0.005 (1.05) 0.025 (1.90) 0.044 ^i4;1 DIV, b (1.42) 1.669 (1.11) 1.413 (0.76) 1.807 (1.00) 1.672 (0.93) 1.836 (0.49) 0.351 (0.63) 1.318 (0.06) 1.456 ^i4;2 b (0.30) 0.261 (0.68) 0.097 (0.92) 0.078 (0.69) 0.178 (0.21) 0.223 (0.14) 0.548 (0.25) 0.810 (0.38) 1.416 (2.44) (2.61) (2.88) (0.69) (0.85) (1.40) (0.88) (0.37) (0.31) (0.58) (1.03) (1.73) (1.26) (2.60) 1.554 1.571 1.714 1.160 0.239 0.041 (8.28) 86.38 (7.88) 79.54 (6.40) 104.7 (7.49) 11.10 113.0 108.9 19.76 21.44 10.09 0.410 25.76 0.999 20.76 62.32 (0.77) (0.82) (0.36) (0.01) (0.64) (0.04) 0.051 (18.5) 0.037 (15.5) 0.034 (16.4) 0.032 (16.4) 0.035 (17.3) 0.038 (14.1) 0.013 0.014 0.003 0.007 0.001 0.015 0.042 (0.32) 0.324 (0.77) 0.500 (0.07) 1.047 (0.60) 1.338 (1.80) 0.106 (1.70) (2.05) Transition probability parameters: 1.400 1.364 ^ i0 Constant, p (8.57) (7.88) 101.9 84.23 ^ i1 MP,p (3.78) Error term volatility: ^ i;1 0.135 Std dev, r (15.3) (3.04) (3.04) (2.29) (2.34) (0.65) 0.096 (15.6) 0.080 (14.6) 0.071 (14.7) 0.073 (13.5) 0.080 (18.7) Log-likelihood value: 731.0 907.2 1013.0 1054.3 1033.2 876.5 0.055 0.025 p^ i2 r^ i;2 Panel B: Instrumental variable (IV) in modeling state transition probabilities = two-month lagged value of the log difference in the Composite Leading Indicator (DCLI) Mean equation: ^i0;1 ^i0;2 0.084 0.071 0.061 0.055 0.049 0.001 0.085 0.054 0.014 0.016 0.001 Constant, b b (1.95) (2.44) (2.39) (2.39) (1.77) (0.05) (1.12) (1.32) (1.02) (1.41) (0.13) ^i1;2 ^i1;1 1.400 1.144 0.796 0.862 0.270 0.037 1.363 0.033 0.291 0.327 0.256 b RREL, b ^i2;1 DEF, b (1.53) 1.627 (1.86) 1.187 (1.40) 0.605 (1.64) 0.013 (0.37) 0.083 (0.06) 0.854 (0.78) 2.481 (0.03) 0.721 ^i2;2 b (0.77) 0.627 (1.16) 0.497 (1.01) 0.435 (1.40) 0.425 (2.20) 0.486 (3.16) 0.519 (2.11) 1.146 (0.20) 0.654 ^i3;1 MB, b (1.13) 0.130 (1.22) 0.070 (0.68) 0.069 (0.02) 0.069 (0.07) 0.036 (0.93) 0.012 (0.94) 0.118 (0.52) 0.005 ^i3;2 b (1.10) 0.041 (0.96) 0.028 (1.11) 0.009 (1.28) 0.015 (1.42) 0.017 (0.97) 0.001 (1.14) 0.042 (0.84) 0.035 ^i4;1 DIV, b (1.56) 1.433 (1.21) 1.367 (1.36) 1.262 (1.37) 1.434 (0.26) 1.310 (0.14) 0.307 (0.71) 1.126 (0.06) 1.231 ^i4;2 b (0.45) 0.262 (0.36) 0.172 (0.11) 0.241 (0.43) 0.319 (0.78) 0.545 (0.02) 0.561 (0.36) 0.824 (0.21) 1.363 (1.70) (2.31) (2.37) (2.91) (2.31) (0.60) (0.74) (1.37) (0.78) (0.63) (0.91) (1.25) (2.16) (1.77) (1.41) (2.18) Transition probability parameters: 1.225 1.263 ^ i0 Constant, p 3.824 2.352 0.280 8.278 10.97 7.029 10.85 35.94 (0.44) (0.27) (0.03) (0.82) (1.23) (0.74) ^ i1 CLI,p 1.493 1.530 1.464 1.176 0.048 0.155 (6.39) 13.04 (7.82) 12.84 (10.0) 6.163 (10.1) 5.459 (9.43) 6.309 (7.25) 7.521 20.56 4.206 (1.43) (1.63) (0.81) (0.57) (0.81) (1.21) p^ i2 D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 ^i1;1 RREL, b (1.94) 1.339 (continued on next page) 207 Table 7 (continued) where rit is the monthly excess return for a given decile portfolio and Sit is the regime indicator. RREL is the relative three-month Treasury-bill rate, DEF is the spread between Moody’s seasoned Baa-rated corporate bond yield and 10year treasury bond yield, MB is defined as the 12-month log-difference in the monetary base, DIV is the sum of dividend payments accruing to the CRSP value-weighted market portfolio over the previous 12 months divided by the contemporaneous level of the index. IV is the instrumental variable in the state transition probabilities. Two instrumental variables are used. The first one is the monthly growth rate of industrial production (MPt1) (Panel A), and the second one is the two-month lagged value of the year-on-year log difference in the Composite Leading Indicator (DCLIt2) (Panel B). Numbers in parentheses are t-statistics (parameter estimates divided by standard errors). ‘WML’ indicates ‘Winner’ portfolio minus ‘Loser’ portfolio, and ‘HML’ is the highest book-to-market decile portfolio minus the lowest book-to-market decile portfolio. The sample period is from March 1960 to December 2012. 0.010 0.038 (14.4) 0.033 (20.0) 0.032 (15.4) r it ¼ bi0;St þ bi1;St RRELt1 þ bi2;St DEF t1 þ bi3;St MBt2 þ bi4;St DIV t1 þ eit eit N 0; r2i;St ; St ¼ f1; 2g pit ¼ P Sit ¼ 1jSit1 ¼ 1 ¼ U pi0 þ pi1 IV t1 ; 1 pit ¼ P Sit ¼ 2jSit1 ¼ 1 qit ¼ P Sit ¼ 2jSit1 ¼ 2 ¼ U pi0 þ pi2 IV t1 ; 1 qit ¼ P Sit ¼ 1jSit1 ¼ 2 ; 877.8 1032.3 1052.4 1009.0 901.8 Log-likelihood value: 723.9 The following univariate Markov switching model was estimated for excess returns on each momentum decile portfolio i: 0.068 (16.5) 0.091 (17.8) 0.075 (16.0) 0.067 (16.8) 0.080 (19.6) 0.047 0.020 r^ i;2 0.048 (14.9) 0.035 (16.5) 0.031 (18.4) WML Winner P8 P6 P4 Loser P2 Error term volatility: ^ i;1 0.127 Std dev, r (15.9) In State 2 Parameters HML WML Winner P8 P6 P4 P2 Loser In State 1 Parameters 0.013 D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 HML 208 In the whole period (Panel A of Table 5), the switching portfolio based on the loser portfolio outperforms the buy-and-hold strategy of the loser portfolio in terms of risk-return characteristics (higher average return of 14.21% versus 10.62%, lower standard deviation of 24.38% versus 31.41%, and thus a higher Sharpe ratio of 0.375 versus 0.177). However, the switching portfolio does not outperform the buy-and-hold strategy for the winner portfolio. In the recession states (Panel B of Table 5), the outperformance of the switching portfolio over the buy-and-hold strategy is conspicuous for both loser and winner portfolios. Panel B of Table 5 shows that the switching portfolio outperforms the buy-and-hold strategy applied to both portfolios. For the loser portfolio, the average return, standard deviation, and Sharpe ratio of the switching portfolio are 26.22%, 43.39%, and 0.345, respectively, and the corresponding statistics of the buy-and-hold strategy are 8.35%, 50.68%, and 0.039, respectively. For the winner portfolio, the switching portfolio similarly outperforms the buy-and-hold strategy. In the expansion states (Panel C), however, the switching portfolio outperforms the buy-and-hold strategy only for the loser portfolio. These two trading strategies perform similarly for the winner portfolio. This switching trading strategy based on the loser portfolio performs better than that based on the winner portfolio. 4.5. Robustness tests 4.5.1. Estimating the state transition probabilities exogenously to the model We have interpreted the high return volatility state as recession and the low return volatility state as expansion economy. However, this interpretation should be drawn with caution, since the state variables may not coincide with actual business cycles.15 The states identified by stock returns could be somewhat inconsistent with the real economic conditions. Our interpretation could mean that the returns of momentum portfolios are endogenous to the recession/expansion states. As a robustness check for the validity of our interpretation of the identified states, we therefore identify exogenously the states by using a real macroeconomic variable rather than using stock returns. As such macroeconomic variable, we choose real consumption growth, since consumption growth reflects the underlying fundamentals of the aggregate economy, and thus, high and low consumption growth volatility states can coincide with expansion and recession states. To identify exogenously the states and the state transition probabilities, we employ a two-step estimation approach. In the first step, we estimate the following two-state Markov switching model for the quarter-on-quarter consumption growth rates to obtain the state transition probabilities and the high and low consumption growth volatility states. Dct ¼ a0;St þ a1;St DEF t1 þ zt ; zt N 0; r2c;St ð23Þ pt ¼ PðSt ¼ 1jSt1 ¼ 1Þ ¼ Uðp0 þ p1 RRELt1 Þ; qt ¼ PðSt ¼ 2jSt1 ¼ 2Þ ¼ Uðp0 þ p2 RRELt1 Þ; where Dct = ln(Ct) ln(Ct3) is the quarter-on-quarter consumption growth rate, Ct is the real personal consumption expenditures of nondurable goods and services, and St is an unobservable state variable St = {1,2} that follows a two-state Markov chain with time-varying transition probability matrix, as characterized in Eqs. (13)–(16).16 In the second step, the state transition probabilities 15 For instance, stock return volatilities reached exceptionally high level in October 1987, but this period is not considered as a recession, according to the NBER classification of the business cycle. 16 The similar specification has been used in the literature (e.g., Whitelaw, 2000; Ozoguz, 2009). Table 8 Parameter estimates for the univariate Markov switching model of excess returns on ten decile momentum portfolios: modelling error terms with the student t distribution. Parameters In State 1 Parameters P2 P4 P6 P8 Winner WML HML Mean equation: ^i0;1 Constant, b 0.218 0.141 0.112 0.048 0.031 0.005 0.223 0.080 ^i1;1 RREL, b (4.66) 0.828 (4.65) 0.665 (3.70) 0.540 (1.77) 0.039 (1.15) 0.140 (0.23) 0.205 (1.87) 1.033 (2.13) 1.069 ^i2;1 DEF, b (1.08) 3.555 (1.25) 2.486 (1.21) 1.987 (0.07) 0.351 (0.21) 1.284 (0.33) 1.380 (1.04) 4.935 ^i3;1 MB, b (2.10) 0.241 (2.62) 0.104 (2.32) 0.070 (0.34) 0.153 (1.11) 0.042 (1.47) 0.028 ^i4;1 DIV, b (2.48) 3.673 (2.12) 2.481 (1.70) 2.072 (1.78) 1.310 (0.43) 1.474 (4.68) (4.24) (3.54) (2.42) 1.554 (10.3) 66.43 Loser P2 P4 P6 P8 Winner WML HML ^i0;2 b 0.031 0.032 0.030 0.009 0.003 0.002 0.030 0.068 ^i1;2 b (1.41) 0.353 (2.53) 0.330 (2.49) 0.485 (0.83) 0.666 (0.58) 0.549 (0.94) 1.158 (1.57) 0.805 (3.73) 0.533 (1.38) 2.030 ^i2;2 b (1.10) 0.217 (1.26) 0.107 (1.99) 0.091 (2.37) 0.571 (2.17) 0.413 (3.71) 0.574 (1.80) 0.791 (1.02) 1.470 (2.55) 0.213 (1.59) 0.381 ^i3;2 b (0.45) 0.027 (0.26) 0.190 (0.23) 0.196 (1.61) 0.031 (1.11) 0.026 (1.18) 0.003 (1.15) 0.023 (2.43) 0.346 (0.20) 0.457 (1.25) 3.216 (3.94) 1.875 ^i4;2 b (0.34) 0.859 (3.31) 0.391 (4.34) 0.189 (1.36) 0.576 (1.13) 0.416 (0.09) 0.455 (0.28) 1.315 (6.38) 2.497 (2.39) (0.91) (3.45) (1.36) (2.28) (1.48) (0.74) (1.32) (1.28) (1.39) (2.64) (4.98) 1.456 1.418 1.253 0.173 0.034 (10.0) 57.62 (10.1) 103.7 (5.02) 41.02 (3.24) 41.91 55.90 55.07 60.29 70.20 38.51 27.37 12.68 42.39 59.08 (3.08) (1.83) (2.57) (2.63) (3.15) (3.04) (2.64) (1.42) (1.41) (0.79) 0.090 (9.22) 0.070 (11.4) 0.063 (9.18) 0.068 (11.0) 0.078 (16.0) 0.058 0.013 r^ i;2 0.051 (14.5) 0.038 (13.4) 0.033 (11.6) 0.033 (11.8) 0.033 (18.0) 0.038 (12.5) 0.013 0.019 3.917 (3.24) 4.777 (3.12) 4.576 (3.19) 4.882 (2.89) 6.488 (2.71) 3.453 8.940 ni,2 10.87 (1.05) 12.60 (0.83) 8.263 (1.60) 9.721 (1.54) 145.8 (0.38) 26.76 (0.46) 15.89 31.44 916.7 1025.6 1060.6 1044.5 886.0 Transition probability parameters: 1.426 1.473 ^ i0 Constant, p (10.5) (9.76) 82.93 68.77 ^ i1 RREL,p (3.57) Error term volatility: ^ i;1 0.136 Std dev, r (4.99) Degree of freedom: ni,1 3.035 (3.38) Log-likelihood value: 742.4 In State 2 p^ i2 The following univariate two-state Markov switching model is estimated for excess returns on each momentum decile portfolio i: r it ¼ bi0;St þ bi1;St RRELt1 þ bi2;St DEF t1 þ bi3;St MBt2 þ bi4;St DIV t1 þ eit ; eit ¼ ri;St wt ; wt Student t with degrees of freedom ni;St > 2; St ¼ f1; 2g pit ¼ P Sit ¼ 1jSit1 ¼ 1 ¼ U pi0 þ pi1 RRELt1 ; qit ¼ P Sit ¼ 2jSit1 ¼ 2 ¼ U pi0 þ pi2 RRELt1 ; D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 Loser 1 pit ¼ P Sit ¼ 2jSit1 ¼ 1 1 qit ¼ P Sit ¼ 1jSit1 ¼ 2 ; where rit is the monthly excess return for a given decile momentum portfolio, and Sit is the regime indicator. RREL is the relative 3-month T-bill rate calculated as the difference between the current T-bill rate and its 12-month backward moving average, DEF is the spread between Moody’s seasoned Baa-rated corporate bond yield and 10-year treasury bond yield from the FRED, MB is defined as the 12-month log-difference in the monetary base reported by the St. Louis Federal Reserve, and DIV is the sum of dividend payments accruing to the CRSP value-weighted market portfolio over the previous 12 months divided by the contemporaneous level of the index. Numbers in parentheses are t-statistics (parameter estimates divided by standard errors). ‘WML’ indicates ‘Winner’ portfolio minus ‘Loser’ portfolio, and ‘HML’ is the highest book-to-market decile portfolio minus the lowest book-to-market decile portfolio. The sample period is from March 1960 to December 2012. 209 210 D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 Table 9 Averages of the financial ratios proxying for leverage and growth options for ten decile momentum portfolios. Leverage Loser 2 3 4 5 6 7 8 9 Winner WML (t-statistic) Growth option Debt to Equity Asset to Equity Market-to-book equity Market-to-book asset 0.742 0.527 0.462 0.413 0.397 0.372 0.353 0.338 0.323 0.291 0.451 (26.54) 2.572 2.038 1.862 1.713 1.642 1.563 1.493 1.426 1.349 1.206 1.366 (30.88) 1.181 1.276 1.338 1.419 1.480 1.547 1.638 1.729 1.879 2.294 1.113 (42.92) 1.079 1.127 1.160 1.197 1.226 1.255 1.294 1.331 1.390 1.552 0.473 (38.38) This table presents the time series averages of the financial ratios proxying for leverage (the asset-to-equity and debt-to-equity ratios) and for growth options (the market-to-book equity and market-to-book asset ratios) of each momentum portfolio over the whole period from 1963 to 2012. The asset-to-equity ratio is defined as the ratio of the book value of assets to the market value of equity. The debt-to-equity ratio is defined as the ratio of total assets minus book equity to market equity. The market-to-book equity is defined as the ratio of market equity to book equity, and the market-to-book asset is defined as the ratio of the sum of book debt and market equity to the book value of asset. Each ratio (A/B) of the portfolio is computed as the median value of the ratios of accounting variable A to accounting variable B of the firms included in the portfolio, every month when momentum portfolios are rebalanced. ‘WML’ indicates ‘Winner’ portfolio minus ‘Loser’ portfolio. ^t Þ are used as given probabilities in ^t ; q estimated in the first step ðp estimating the coefficients on the conditional mean equation of (17) through the likelihood function of (7) for excess returns on ten decile momentum portfolios. This two-step approach can estimate the state transition probabilities and identify the states exogenously from the returns of momentum portfolios. Table 6 reports the two-step estimation results of the two-state Markov switching model. The results are similar in the pattern of the coefficient estimates across momentum portfolios to those obtained from the (one-step) simultaneous estimation for the parameters of the two-state Markov switching model which are reported in Table 2. That is, the results obtained from the two-step estimation also lead to the same conclusions that momentum profits are procyclical and that loser stocks are riskier in recession periods than winner stocks, while winner stocks are riskier in expansion periods than loser stocks. To further examine the similarity of the expected momentum profits (WML) obtained from the two-step estimation and the simultaneous estimation, we plot these two sets of time-series of the expected momentum profits (WML) in Panels A and B of Fig. 4 for the univariate and bivariate Markov switching models, respectively. Fig. 4 shows that these two sets of the expected momentum profits co-move very closely. Specifically, in the univariate case (Panel A), the correlation coefficients between the two sets of the expected momentum profits are 0.561, 0.465, and 0.580 for the whole, expansion, and recession periods, respectively.17 The averages (in percent) of the two sets of the expected momentum profits (the two-step vs. the simultaneous estimation) are 0.76 vs. 0.92, 1.02 vs. 1.17, and 0.71 vs. 0.40 for the whole, expansion, and recession periods, respectively. In the bivariate case (Panel B), the correlation coefficients are 0.813, 0.738, and 0.862 for the whole, expansion, and recession periods, respectively. The averages (in percent) of the expected momentum profits are 0.72 vs. 0.70, 0.95 vs. 0.89, and 0.53 vs. 0.27, respectively. In summary, there is no significant difference in the results between the two-step estimation and the (one-step) simultaneous estimation. Therefore, our interpretation of the high return volatility state as recession and the low return volatility state as expansion economy is valid. 17 The p-values of the correlation coefficients are all 0.0000. 4.5.2. Using alternative instruments in modeling state transition probabilities Following Gray (1996), we have used the relative three-month Treasury bill rate as the instrumental variable in the state transition probabilities for the univariate Markov switching model. Since the estimation results of the conditional mean equation may be sensitive across the states to the choice of the instrument variable in the state transition probabilities, it would be necessary to conduct robustness tests by using alternative instrument variables. We choose two alternative instrument variables in the state transition probabilities: The first one is the one-month-lagged monthly growth rate of industrial production (MPt1), and the second one is the two-month-lagged value of the year-on-year log difference in the Composite Leading Indicator (DCLIt2), by following PerezQuiros and Timmermann (2000). As in Tables 2 and 7 presents the estimation results of the univariate Markov switching model for portfolios P1 (loser), P2, P4, P6, P8, and P10 (winner) when the instrumental variables in the state transition probabilities are MPt1 (Panel A) and DCLIt2 (Panel B), respectively.18 The overall results from using these new instrumental variables are similar to those from Table 2, which uses the relative three-month Treasury bill rate as the instrumental variable. Therefore, the inferences from Table 2 are robust to changes in the specification of the state transition probabilities. Specifically, in the recession state, the loser portfolio has a greater sensitivity to all four conditioning macroeconomic variables than does the winner portfolio. Contrary to the case in the recession state, however, the winner portfolio tends to have a greater sensitivity to the variables in the expansion state than does the loser portfolio. These results indicate that loser stocks are riskier in recession periods than winner stocks, while winner stocks are riskier in expansion periods than loser stocks. The null hypotheses of identical asymmetries across states for loser and winner stocks are also modestly rejected at standard significance levels in the slope coefficients on the four conditioning variables (not reported). 4.5.3. Incorporating the fat-tailed nature of returns into the model It has been well documented that the empirical distribution of stock returns exhibits excess kurtosis. It is worthwhile, therefore, 18 The estimation results for portfolios P3, P5, P7, and P9 are not reported. The results are available upon request. D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 211 Fig. 5. Leverage and growth options of loser and Winner Stocks before and after portfolio formation. For the loser and winner portfolios, we take the values of two ratios proxying for leverage (the asset-to-equity and debt-to-equity ratios) and two ratios proxying for growth options (the market-to-book equity (ME/BE) and market-to-book asset ratios (MA/BA)) over the period from t 36 months to t + 36 months, where t is the portfolio formation month and it varies from January 1966 to December 2009. Then, we compute the averages over the period (t 36, t + 36). ‘Month 0’ is the portfolio formation month. to conduct a robustness check by incorporating the fat-tailed nature of returns in estimating the Markov switching model. To do this, we assume that the error term, eit, in the conditional mean Eq. (11) follows a Student t distribution. Specifically, the error term is assumed to have the following form. eit ri;St wt ; ð24Þ where wt is a Student-t variate with degrees of freedom ni;St . In this specification, the return volatility and degrees of freedom are allowed to switch regimes. Table 8 reports the estimation results of the two-state Markov switching model with Student t error terms. The results are quite similar to those with normal error terms reported in Table 2. In other words, our conclusions are robust to the fat-tailed nature of stock returns. 5. A plausible explanation for time-varying momentum profits We have shown that during the expansion state winner stocks are riskier than loser stocks, while during the recession state loser stocks are riskier than winner stocks. Consequently, the expected momentum profits display strong procyclical variations. We now examine the potential driving sources of time-variations in expected momentum profits. Other things being equal, firms with large recent positive price moves (winners) are more likely to decrease their (financial) leverage than firms with large recent negative price moves (losers). Hence, a momentum sort will tend to sort firms by recent leverage changes. Since higher leverage implies higher systematic risk (Mandelker and Rhee, 1984), losers are riskier than winners; hence momentum trading should have lower expected returns. With the presence of growth options, however, winner stocks become riskier than loser stocks, as discussed in Section 2. Winner stocks that have had recent good performance are more likely to increase the value of growth options than loser stocks that have had recent bad performance. Since growth options are riskier than assets in place, winners are riskier than losers and hence momentum trading should have higher expected returns. Therefore, the riskiness and expected return of momentum portfolios result from the relative importance of the leverage and growth options effect. During expansions, when growth options have a higher effect and leverage has a lower effect, winners are riskier than losers. Likewise, during recessions when growth options have a lower effect and leverage has a higher effect, losers are riskier than winners. To provide a plausible explanation for the time-varying momentum profits observed in the previous section, it is necessary to show that the degree of growth options and leverage differ across momentum portfolios and that macro-level leverage and growth options covary with the business cycle. According to the above arguments, we expect winner stocks to have higher growth options and lower leverage than loser stocks and aggregate leverage to be lower during expansions than recessions, while aggregate growth options are expected to be higher during expansions than recessions. 212 D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 Fig. 6. Aggregate leverage and growth options across business cycles. These figures plot the time-series averages of the aggregate financial ratios proxying for leverage (the asset-to-equity and debt-to-equity ratios) and for growth options (the market-to-book equity (ME/BE) and market-to-book asset (MA/BA) ratios) across business cycles over the whole period from 1963 to 2012. Shaded areas indicate NBER recession period. 5.1. Momentum, leverage, and growth options This section examines how leverage and growth options differ across momentum portfolios. We use the asset-to-equity and debt-to-equity ratios as proxies for leverage and the market-tobook equity and market-to-book asset ratios as proxies for growth options.19 The asset-to-equity ratio of a portfolio is computed as the median value of the asset-to-equity ratios of the firms included in the portfolio, every month when momentum portfolios are rebalanced. Likewise, we compute the other ratios of the portfolio. Table 9 presents the time-series averages of the asset-to-equity, debt-to-equity, market-to-book equity, and market-to-book asset ratios over the whole period from 1963 to 2012.20 Moving from the loser portfolio to the winner portfolio, we observe a nearly monotonically decreasing relation between past stock returns and the measures of leverage. The asset-to-equity ratio decreases from 2.572 for loser stocks to 1.206 for winner stocks. We also observe a similar pattern in the debt-to-equity ratio. In contrast, we observe an opposite pattern in the variables proxying for growth options. The market-to19 The asset-to-equity ratio is defined as the ratio of the book value of assets (Compustat annual item AT) to the market value of equity. The debt-to-equity ratio is defined as the ratio of total assets minus book equity (Compustat annual item CEQ) to market equity, following Bhandari (1988). Following Sagi and Seasholes (2007), the market-to-book equity is defined as the ratio of market equity to book equity, and the market-to-book asset is defined as the ratio of the sum of book debt and market equity to the book value of assets, as in Goyal et al. (2002). 20 Since some of Compustat items are available from 1963, our firm characteristic analysis begins from 1963. book equity ratio monotonically increases across momentum portfolios from 1.181 (the loser portfolio) to 2.294 (the winner portfolio). The market-to-book asset ratio also monotonically increases across portfolios from 1.079 (the loser portfolio) to 1.552 (the winner portfolio). The differences in the values of all four ratios between the loser and winner portfolios are statistically significant at the 1% level. To shed further light on the role of leverage and growth options in sorting momentum portfolios, we examine how leverage and growth options evolve before and after portfolio formation. To do this, we take the values of the four ratios proxying for leverage and growth options over the period from t 36 months to t + 36 months, where t is the portfolio formation month and varies from January 1966 to December 2009, and compute the averages over the period (t 36,t + 36). Fig. 5 illustrates the values of the four proxy ratios of the loser and winner portfolios over the period (t 36,t + 36). It shows that the winner portfolio has lower values of the leverage proxy variables (asset-to-equity and debt-to-equity ratios) and greater values of the growth option proxy variables (market-to-book equity and market-to-book asset ratios) than the loser portfolio does over the portfolio formation period (six months before portfolio formation). In fact, the spread in the value of each proxy variable between the winner and loser portfolios sharply increases over the portfolio formation period and peaks at the portfolio formation month (month 0). The spread begins to decrease after the portfolio formation month but remains positive. Overall, the results in Table 9 and Fig. 5 show that sorting firms on past stock returns is related to sorting firms on leverage and growth options. 213 D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 Table 10 Averages of the financial ratios proxying for leverage and growth options across business cycles. Leverage Expansion Recession Difference (E - R) (t-statistic) Growth option Debt-to-Equity Asset-to-Equity Market-to-book equity Market-to-book asset 0.282 0.433 0.151 (6.51) 1.431 2.068 0.637 (8.14) 1.686 1.203 0.483 (12.12) 1.271 1.075 0.196 (12.38) This table presents the time-series averages of the aggregate financial ratios proxying for leverage (the asset-to-equity and debt-to-equity ratios) and for growth options (the market-to-book equity and market-to-book asset ratios) across business cycles over the whole period from 1963 to 2012. The asset-to-equity ratio is defined as the ratio of the book value of assets to the market value of equity. The debt-to-equity ratio is defined as the ratio of total assets minus book equity to the market equity. The market-tobook equity is defined as the ratio of market equity to book equity, and the market-to-book asset is defined as the ratio of the sum of book debt and market equity to the book value of asset. Each ratio (A/B) of the portfolio is computed as the median value of the ratios of accounting variable A to accounting variable B of the firms included in the portfolio, every month when momentum portfolios are rebalanced. Recession and expansion periods are based on historical NBER business cycle dates. 5.2. Leverage and growth options across business cycles To provide a plausible explanation for time-varying momentum profits over business cycles, it is necessary to show that (macrolevel) leverage and growth options covary with business cycles, since leverage and growth options are implicit driving forces in sorting momentum portfolios. Fig. 6 plots the aggregate values of the two proxy variables for leverage (the asset-to-equity ratio in Panel A and the debt-toequity ratio in Panel B) and two other proxy variables for growth options (the market-to-book ratio in Panel C and the market-tobook ratio in Panel D) along with the NBER contraction period over the period from January 1963 to December 2012. The aggregate leverage exhibits strong countercyclical variation. The two leverage proxy variables (in Panels A and B) sharply increase during recessions and tend to decrease during expansions. On the contrary, the aggregate growth options exhibit strong procyclical variation. The two growth option proxy variables (in Panels C and D) sharply decrease during recessions and tend to increase during expansions. Table 10 shows that the averages of the aggregate leverage variables are higher during recessions than during expansions (2.068 versus 1.431 for the asset-to-equity ratio and 0.433 versus 0.282 for the debt-to-equity ratio), while the averages of the aggregate growth option variables are higher during expansions than during recessions (1.686 versus 1.203 for the market-to-book equity ratio and 1.271 versus 1.075 for the market-to-book asset ratio). The differences in the averages between expansions and recessions are all statistically significant at the 1% level. The results in Fig. 6 and Table 10 indicate that winner stocks are riskier during expansions, since these stocks tend to have greater growth options and lower leverage during expansions when growth options have a higher effect and leverage has a lower effect. Conversely, loser stocks are riskier during recessions, since these stocks tend to have lower growth options and greater leverage during recessions when growth options have a lower effect and leverage has a higher effect. 6. Conclusions We examine the procyclicality of momentum profits using the two-state Markov switching regression framework of PerezQuiros and Timmermann (2000) and find that momentum profits display strong procyclical variation. Our results show that in the recession state loser stocks tend to have greater loadings on the conditioning macroeconomic variables than winner stocks, while in the expansion state winner stocks tend to have greater loadings on those variables than loser stocks. In other words, in recessions loser (winner) stocks are most (least) strongly affected by aggregate economic conditions, whereas in expansions winner (loser) stocks are most (least) strongly affected. This indicates that returns on momentum portfolios react asymmetrically to the aggregate economic conditions in recession and expansion states. This asymmetry across recession and expansion states for loser stocks is identical to the asymmetry for winner stocks. This identical asymmetry for winner and loser stocks is contrasted with the results reported by Perez-Quiros and Timmermann (2000) for size portfolios and by Gulen et al. (2011) for book-to-market portfolios. Using conditioning variables similar to ours, these authors report that identical asymmetries for small and large stocks and for growth and value stocks are strongly rejected. To further confirm the procyclicality of momentum profits, we plot the momentum profit estimated from the Markov switching regression model with NBER recession dates. The momentum profit (or winner-minus-loser) tends to sharply decrease and have a negative value during recessions but to increase and have a positive value just after recessions and during expansions. It is higher in expansion periods and lower in recession periods. This procyclical time-varying behavior of the expected momentum profit is the opposite of the counter-cyclical behavior of the value premium shown by Gulen et al. (2011). The above results are robust to using alternative instrumental variables in modeling state transition probabilities. 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. Under this trading rule, if the predicted excess return is positive, we take a long position in the loser or winner portfolio and otherwise we switch the position into the Treasury bill. The results show that the economic significance of out-of-sample predictability is particularly significant for the switching portfolio based on the loser portfolio and during the recession state. The overall results indicate that the expected returns of winner stocks co-move more with aggregate economic variables in expansion states than those of loser stocks and the expected momentum profits display procyclical time-variations. The possible reason that winner stocks do well in expansions is that they tend to have higher exposure to growth rate risk and more valuable growth options in expansions than in recessions and thus should have higher expected returns in expansions. We argue, therefore, that momentum profits are the realizations of such expected returns and can be interpreted as the procyclicality premium. Acknowledgements The authors would like to thank conference participants at the 2014 World Finance Conference in Venice, the 2013 11th International Paris Finance Meeting, the 2012 Financial Management Association Annual Meeting in Atlanta, the 2012 7th International Conference on Asia-Pacific Financial Markets in Seoul, and the 2012 Joint Conference Allied Korea Finance Associations. The paper 214 D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 received the outstanding paper award at the 2012 7th International Conference on Asia-Pacific Financial Markets and the best paper award at the 2012 Joint Conference Allied Korea Finance Associations. Kim is supported by the Korea Research Foundation Grant funded by the Government of Korea (NRF-2012S1A5A2A01014413). The authors thank the editor (Ike Mathur) and an anonymous reviewer for their constructive comments and suggestions that greatly improved the paper. Any remaining errors are our own responsibilities. Appendix A. Estimation methodologies of the Markov switching regression model Three different estimation methodologies of the two-state Markov switching regression model are employed in this paper. We describe each of these estimation methodologies below. A.1. A univariate Markov switching regression model We first specify the conditional mean Eq. (1) for the excess return of each momentum portfolio. As exogenous variables that affect the excess returns of the momentum portfolio, we choose the four macroeconomic variables for the conditional mean equation as follows: r it ¼ bi0;St þ bi1;St RRELt1 þ bi2;St DEF t1 þ bi3;St MBt2 þ bi4;St DIV t1 þ eit ; ðA1Þ where rit is the monthly excess return for the ith decile momentum portfolio at time t,eit is the normally distributed random error term with mean zero and variance r2i;St , and St = {1,2}. The conditional variance of excess returns, r2i;St , is allowed to depend only on the state of economy: ln r2i;St ¼ ki;St : ðA2Þ The time-varying state transition probabilities are dependent on the level of an instrumental variable (IV) and are specified as follows: pit ¼ Prob Sit ¼ 1jSit1 ¼ 1; yt1 ¼ U pi0 þ pi1 IV t1 ; 1 pit ¼ Prob Sit ¼ 2jSit1 ¼ 1; yt1 ; qit ¼ Prob Sit ¼ 2jSit1 ¼ 2; yt1 ¼ U pi0 þ pi2 IV t1 ; 1 qit ¼ Prob Sit ¼ 1jSit1 ¼ 2; yt1 ; ðA3Þ ðA4Þ ðA5Þ ðA6Þ where U() is the cumulative probability density function of a standard normal variable. We use the relative three-month Treasury bill rate and the monthly growth rate of industrial production as the instrumental variable. A.2. A Bivariate Markov switching regression model 0 Let r t ¼ r Lt ; rW be a (2 1) vector consisting of excess returns t on the loser and winner portfolios, r Lt and rW t , respectively. The joint conditional mean equation for the excess return of the loser and winner portfolios is specified as follows: convenience, we assume the form of the conditional covariance matrix as follows: ! 8 X > > > ln ¼ ki;St > > < ii;St !1=2 !1=2 > X X X > > > ¼ q > St : ij;St ðA9Þ qt ¼ PðSt ¼ 2jSt1 ¼ 2Þ ¼ Uðp0 þ p2 IV t1 Þ; ðA10Þ where IVt1 is the one-month lagged instrumental variable for which the relative three-month Treasury bill rate or the monthly growth rate of industrial production is used. A.3. A Two-step approach to estimate a Markov switching regression model In addition to the simultaneous estimation of the conditional mean equation and the state transition probabilities as in the previous cases, we also estimate the two-state Markov switching regression model in a two-step approach. In the first step, we identify exogenously the states and the state transition probabilities as follows: we estimate the following univariate two-state Markov switching model for the consumption growth rates and obtain the state transition probabilities and the high and low consumption growth volatility states. Dct ¼ a0;St þ a1;St DEF t1 þ zt ; ðA11Þ ðA12Þ qt ¼ PðSt ¼ 2jSt1 ¼ 2Þ ¼ Uðp0 þ p2 RRELt1 Þ; ðA13Þ where Dct = ln(Ct) ln(Ct3) is the quarter-on-quarter consumption growth rate, Ct is the real personal consumption expenditures of nondurable goods and services, and St is an unobservable state variable St = {1,2} that follows a two-state Markov chain with timevarying transition probability matrix. In the second step, the state transition probabilities estimated ^t Þ are used as given probabilities in estimating ^t ; q in the first step (p the coefficients on the conditional mean equation for the excess returns of each momentum portfolio. This two-step approach estimates the state transition probabilities and identifies the states exogenously from the returns of momentum portfolios. Appendix B . Definition of the variables The variables used in the Markov switching regression model are defined below. Variable Definition Source DEF the spread between Moody’s seasoned Baa-rated corporate bond yield (BAA) and 10-year treasury bond yield (GS10) the monthly growth rate of industrial production (INDPRO) the 12-month log-difference in the monetary base (AMBNS) the relative three-month Treasury-bill rate calculated as the difference between the current T-bill rate (TB3MS) and its 12-month backward moving average FRED MP MB loser and winner portfolios’ excess returns in state St. For estimation zt N 0; r2c;St pt ¼ PðSt ¼ 1jSt1 ¼ 1Þ ¼ Uðp0 þ p1 RRELt1 Þ; ðA7Þ tive semidefinite (2 2) covariance matrix of the residuals from the for i – j jj;St pt ¼ PðSt ¼ 1jSt1 ¼ 1Þ ¼ Uðp0 þ p1 IV t1 Þ; 0 is a (2 1) coefficient vector with elements bLk;St ; bW k;St for k = 1,2,3,4, and et is a (2 1) vector of normal residuals with P P mean zero and covariance matrix St , St = {1,2}. Here St is a posi- ðA8Þ As in the univariate case, the time-varying transition probabilities for the bivariate model are as follows: rt ¼ b0;St þ b1;St RRELt1 þ b2;St DEF t1 þ b3;St MBt2 þ b4;St DIV t1 þ et ; where bk;St ii;St for i ¼ j RREL FRED FRED FRED D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215 Appendix B. (continued) Variable Definition Source DIV the sum of dividend payments accruing to the CRSP value-weighted market portfolio over the previous 12 months divided by the contemporaneous level of the index CRSP References Ahn, D.-H., Conrad, J., Dittmar, R.F., 2002. Risk adjustment and trading strategies. Rev. Financial Stud. 16, 459–485. Ang, A., Bekaert, G., 2007. Stock return predictability: is it there? Rev. Financial Stud. 20, 651–707. Avramov, D., Chordia, T., 2006. Asset pricing models and financial market anomalies. Rev. Financial Stud. 19, 1001–1040. Bakshi, G.S., Chen, Z., 1996. 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