Regime Shifts in Price-dividend Ratios and Expected Stock Returns: A Present-value Approach

Transcription

1 Regime Shifts in Price-dividend Ratios and Expected Stock Returns: A Present-value Approach by Kwang Hun Choi 1 Korea Institute for Industrial Economics and Trade Chang-Jin Kim University of Washington and Cheolbeom Park Korea University Revised October 2015 Abstract We incorporate regime shifts in the mean of price-dividend ratios into the present value model of Binsbergen and Koijen (2010), who propose a latent variable approach to modeling expected returns and expected dividend growth rates. We assume that economic agents observe current and past regimes but not future regimes. Focusing on a post-war sample, we find that accounting for regime shifts results in i) much lower persistence of expected returns; ii) higher conditional volatility of expected returns, and iii) higher unconditional volatility of expected returns compared to the results from the Binsbergen and Koijen (2010) model. Particularly, the persistence of expected returns are even lower than estimated by Campbell (1991), whose measure of persistence is based on the multivariate variance decomposition framework. All these results contribute to better in-sample predictability of stock returns for our model compared to the Binsbergen and Koijen (2010) model. Moreover, we show that the main source of the increase in the mean of price-dividend ratios in the mid-1990s is a decrease in the mean of expected returns. Keywords: Persistence of Expected Returns, Expected Dividend Growth Rates, Presentvalue Approach, Predictive Regression Approach, Return Predictability, Regime Shifts JEL classification: G12, C12, C32 1 Kwang Hun Choi: Korea Institute for Industrial Economics and Trade ( cogito@korea.ac.kr); Chang-Jin Kim: (Corresponding Author) Dept. of Economics, University of Washington ( changjin@uw.edu); Cheolbeom Park: Dept. of Economics, Korea University ( cbpark kjs@korea.ac.kr). Chang-Jin Kim acknowledges financial support from the Bryan C. Cressey Professorship at the University of Washington. 1

2 1. Introduction In the standard predictive regression approach to modeling the conditional expected return (hereafter, expected return ) or the conditional expected dividend growth rate (hereafter, expected dividend growth rate ), the stock return or dividend growth rate is typically regressed on various predictors, such as the price-dividend ratio, bond yield, and/or the consumption-wealth ratio. In this approach, the predictors are assumed to be perfectly correlated with the expected return, and the expected return is modeled as a linear combination of the predictors. However, the predictors are typically noisy proxies for expected returns, i.e., the predictors are imperfectly correlated with expected returns. Pastor and Stambaugh (2009) noted other shortcomings of the predictive regression approach including its failure to exploit an important property of unexpected returns, e.g., the potential negative correlation between unexpected returns and innovations to the expected return. Thus, attempts have been made to overcome the weaknesses of the predictive regressions approach. For example, by assuming that the expected return is a latent variable with exogenously specified dynamics, Pastor and Stambaugh (2009) propose a version of a state-space model that involves the stock return and the predictive variables. Then, the filtering technique is employed to infer the latent expected return. They show that incorporating a negative correlation between unexpected returns and innovations to expected returns substantially affect the estimates of expected returns and the inference on a predictor s usefulness. Furthermore, their filtering approach typically delivers more precise estimates of expected returns than the predictive regression approach. Additional research along this line includes Rytchkov (2012). Recently, Binsbergen and Koijen (2010) propose a latent variable approach to jointly model expected returns and expected dividend growth rates of the aggregate stock market, based on the present value model of Campbell and Shiller (1988). In particular, by assuming that both the expected return and the expected dividend growth rate are latent variables with pre-specified dynamics, they derive the dynamics of the price-dividend ratio as a function of these latent variables. Then, they employ the Kalman filter to estimate these latent variables from the data on dividend growth rates and price-dividend ratios. They show that the filtered series for expected returns and expected dividend growth rates are good 2

3 predictors of realized returns and dividend growth rates. However, the approach provides superior predictability over the predictive regression approach only for dividend growth rates and not for stock returns. Lettau and van Nieuwerburgh (2008) document one or two structural breaks in the mean of price-dividend ratios for the year 1927 to the year 2004 as the sample period. By employing mean-adjusted price-dividend ratios in the predictive regression model, they achieve improved stock return predictability when accounting for structural breaks. They also argue that a failure to consider shifts in the mean of price-dividend ratios is responsible for the instability in the predictive regressions reported in the literature (Viceira (1996), Goyal and Welch (2004), and Paye and Timmerman (2005)). This paper investigates the nature of regime shifts in expected returns, expected dividend growth rates, and price-dividend ratios, along with their implications on the dynamics of expected returns and return predictability. We use the state-space framework of Binsbergen and Koijen (2010), who propose a latent variable approach to modeling expected returns and expected dividend growth rates, based on the present value model of Campbell and Shiller (1988). We incorporate regime shifts in the Binsbergen and Koijen (2010) model by assuming that economic agents observe the past and current states but not future states. When constructing data on dividends, we consider a cash re-investment strategy. That is, the monthly dividends are assumed to be reinvested in 30-day T-bills. Empirical results from data on annual price-dividend ratios and dividend growth rates (1926 to 2014) are summarized as follows. First, we find that the main source of a recent increase in the mean of price-dividend ratios in the mid-1990s was a decrease in the mean of expected returns. Pastor and Stambaugh (2001) investigate the nature of structural breaks in the equity premium since 1834 and document that the sharpest decline in the equity premium occurred in the 1990s. Lettau et al. (2008) also document a decline in the equity premium in the early 1990s with a fall in macroeconomic risk or the volatility of the aggregate economy. Second, focusing on a post-war sample, we find that accounting for regime shifts in the mean price-dividend ratios, and thus in the mean of expected returns, results in i) lower persistence of expected returns, ii) higher conditional volatility of expected returns, and iii) 3

4 higher unconditional volatility of expected returns when compared to the results from the Binsbergen and Koijen (2010) model without regime shifts. The persistence measure from our model implies that a 1% increase in expected returns would result in a capital loss of 2.93%. Our measure for the persistence of expected returns is much lower than that implied by the model without regime shifts (a capital loss of 11.14% for a 1% increase in expected returns). Our measure is also lower than that of Campbell (1991), who calculates a value of persistence within the vector autoregressive (VAR) framework. Third, incorporating regime shifts results in greater in-sample return predictability in the post-war sample while it results in little improvement on dividend-growth predictability. The difference in return predictability for models with and without regime shifts stems from different dynamics of expected returns implied by the two models. With regime shifts, the persistence measure for the mean-adjusted price-dividend ratio is close to that of the expected returns from our model. We consider their similarity a necessary condition for short-horizon return predictability. This paper is organized as follows. In Section 2, we extend Binsbergen and Koijen s (2010) model to include regime shifts in the mean of stock returns and the mean of dividend growth rates. Section 3 provides the data description and a strategy for estimating the empirical model. Section 4 provides empirical results with discussions on subjects such as the source of the recent regime shift in the mean of price-dividend growth rates, dynamics of expected returns, and return predictability. Section 5 presents concluding remarks. 2. An Extension of the Binsbergen and Koijen (2010) Model to Incorporate Regime Shifts 2.1. Model Specification By denoting r t+1 and d t+1 as the log return in the aggregate stock market and the aggregate log dividend growth rate, respectively, Binsbergen and Koijen (2010) specify that r t+1 = µ t + ɛ r t+1, (1) d t+1 = g t + ɛ d t+1, (2) 4

5 where µ t+j = E[r t+j+1 I t+j ] and g t+j = E[ d t+j+1 I t+j ] are the expected return and the expected dividend growth rate, respectively, formed conditional on I t+j, information up to time t + j. In their specification, economic agents information set I t+j contains all the data up to time t + j. While Binsbergen and Koijen (2010) assume that the latent variables µ t+j and g t+j follow stationary AR(1) processes with time-invariant means, we assume that these means are subject to regime shifts. For this purpose, we introduce a latent regime indicator variable S t, which follows a firstorder Markov-switching process. For a three-state Markov-switching process 2, for example, we define { 1, if St = j, j = 0, 1, 2; S jt = (3) 0, otherwise. Then, we specify the regime-dependent means for stock returns and dividend growth rates (E[r t+j S t+j ] = δ 0,St+j and E[ d t+j S t+j ] = γ 0,St+j ), S t+j = 0, 1, 2, as follows: δ 0,St+j = δ 0,0 + (δ 0,1 δ 0,0 )S 1,t+j + (δ 0,2 δ 0,0 )S 2,t+j = δ 0,0 + S t+jc δ, (4) γ 0,St+j = γ 0,0 + (γ 0,1 γ 0,0 )S 1,t+j + (γ 0,2 γ 0,0 )S 2,t+j = γ 0,0 + S t+jc γ, (5) where C δ = [ (δ 0,1 δ 0,0 ) (δ 0,2 δ 0,0 ) ], C γ = [ (γ 0,1 γ 0,0 ) (γ 0,2 γ 0,0 ) ], and S t = [ S 1t S 2t ]. When regime shifts are introduced in Binsbergen and Koijen s (2010) model, we require additional assumptions about the dynamics of regime shifts and the economic agents information set. The following assumptions apply: Assumption #1: The regime indicator variable S t, which is independent of all the other shocks in our model, follows a three-state, first-order Markov-switching process with the following transition probabilities: 2 Generalization of the model introduced in this section to the general case of N state is straightforward. 5

6 P r[s t = j S t 1 = i] = p ij, 2 p ij = 1, i, j = 0, 1, 2 (6) i=0 Assumption #2: 3 By the end of time t+j or at the beginning of time t+j +1, economic agents observe S t+j but not future states. Thus, economic agents information set at the end of time t + j (i.e., Ψ t+j ) can be specified as: Ψ t+j = { I t+j ; S t+j }, (7) where I t+j consists of observed data up to time t + j. Note that I t+j is the same as the econometricians information set. As a corollary of this assumption, we define µ t+j and g t+j, economic agents conditional expectations as follows: µ t+j = E[r t+j+1 I t+j, S t+j ], (8) g t+j = E[ d t+j+1 I t+j, S t+j ]. (9) Assumption #3: With µ t+j = µ t+j δ 0,St+j+1 and g t+j = g t+j γ 0,St+j+1 representing demeaned conditional expectations of returns and dividend growth rates, we assume that their dynamics are given by the following stationary AR(1) processes: µ t+j = δ 1 µ t+j 1 + ɛ µ t+j, (10) g t+j = γ 1 g t+j 1 + ɛ g t+j, (11) where ɛ µ t+j and ɛ g t+j are innovations to conditional expectations of stock returns and dividend growth rates, respectively. 3 At monthly or quarterly frequencies, this assumption may be debatable. However, at a yearly frequency in our case, we consider this is a reasonable assumption. 6

7 2.2. Imposing the Present-value Constraints on the Model We now consider the following log-linearized return within Campbell and Shiller s (1988) present-value framework, as in Binsbergen and Koijen (2010): r t+1 κ + ρpd t+1 + d t+1 pd t, (12) where pd t is the log of the price-dividend ratio, and by defining pd = E[pd t ] we have κ = log(1 + exp(pd)) ρpd and ρ = exp(pd)/(1 + exp(pd)). By iterating equation (12) in the forward direction conditional on the economic agents information set that consists of I t+1 and S t+1, we obtain the following solution for pd t+1 : pd t+1 = κ 1 ρ + ρ j 1 (E[ d t+j+1 I t+1, S t+1 ] E[r t+j+1 I t+1, S t+1 ]) j=1 = κ 1 ρ + ρ j 1 (E[g t+j I t+1, S t+1 ] E[µ t+j I t+1, S t+1 ]), j=1 assuming no bubble condition. Equations (10) and (11) imply that : (13) E[g t+j+1 I t+1, S t+1 ] = E[γ 0,St+j+1 S t+1 ] + γ j 1 g t+1, (14) As proved in Appendix A, we have: E[µ t+j+1 I t+1, S t+1 ] = E[δ 0,St+j+1 S t+1 ] + δ j 1 µ t+1, (15) E[ S t+j+1 S t+1 ] = π + λ j ( S t+1 π), (16) where S t = [ S 1t S 2t ], π = E[ S t ] is a vector of unconditional probabilities for S t = 1 and [ ] S t = 2, and λ (p11 p 01 ) (p 21 p 01 )) =. Then, from equations (4) and (5), the first (p 12 p 02 ) (p 22 p 02 )) elements of the right-hand-side of equations (14) and (15) are derived as: E[γ 0,St+j+1 S t+1 ] = γ 0,0 + E[ S t+j+1 S t+1 ] C γ = γ 0,0 + ( π + λ j ( S t+1 π)) C γ, (17) 7

8 E[δ 0,St+j+1 S t+1 ] = δ 0,0 + E[ S t+j+1 S t+1 ]C δ = δ 0,0 + ( π + λ j ( S t+1 π)) C δ, (18) where C δ = [ (δ 0,1 δ 0,0 ) (δ 0,2 δ 0,0 ) ] and C γ = [ (γ 0,1 γ 0,0 ) (γ 0,2 γ 0,0 ) ], as defined earlier. We first substitute equations (17) and (18) into equations(14) and (15). Then, by substituting the resulting equations into equation (13), we obtain the following solution: 4 pd t+1 = pd St+1 B 1 µ t+1 + B 2 g t+1, (19) pd St+1 = pd + ( λ(i2 ρ λ) 1 ( S t+1 π) ) (Cγ C δ ), (20) where B 1 = 1 1 ρδ 1, B 2 = 1 1 ργ 1, and pd is the unconditional expectation of pd t+1 given below: pd = Equations (1) and (2) can be rewritten as: κ 1 ρ ρ (γ 00 δ 00 ) ρ π (C γ C δ ). (21) r t+1 = δ 0,St+1 + µ t + ɛ r t+1, (22) d t+1 = γ 0,St+1 + g t + ɛ d t+1. (23) We have three measurement equations given by equations (19), (22) and (23). However, as equation (13) is an identity, one of the three measurement equations is redundant. As in Binsbergen and Koijen (2010), we multiply both sides of equation (19) by (1 δ 1 L), where L is the lag operator. We then substitute equation (11) with j = 1 into the resulting equation to obtain: 4 In the absence of regime switching (i.e., when γ 0,0 = γ 0,1 = γ 0,2 and δ 0,0 = δ 0,1 = δ 0,2 ), we have C γ = C δ = 0. Thus, the derived model collapses to that of Binsbergen and Koijen (2010) with pd t+1 = pd B 1 µ t+1 + B 2 g t+1, (19 ) where pd = κ 1 ρ ρ (γ 00 δ 00 ) (20 ) 8

9 pd t+1 = pd St+1 δ 1 pd St + δ 1 pd t + B 2 (γ 1 δ 1 ) g t + B 2 ɛ g t+1 B 1 ɛ µ t+1. (24) We now have two measurement equations that consist of equations (23) and (24). The model that consists of these two equations and equation (11) can be estimated by the maximum likelihood estimation method. 3. Data and Estimation Procedure The data employed are the annual stock price and the dividend covering the sample period from the year 1926 to the year These data are constructed from the monthly returns (both with-dividend and without-dividend) on the value-weighted portfolio of all NYSE, Amex, Nasdaq, and ARCA stocks in the CRSP files. Following Binsbergen and Koijen (2010), Lettau and Ludvigson (2005), and Fama and French (2002), we employ annual data to avoid seasonality in the monthly or the quarterly dividend payments. The stock price (P t ) is the end-of-year price. For the annual dividend, we consider a cash reinvestment strategy in Binsbergen and Koijen (2010). That is, the monthly dividends are assumed to be reinvested in 30-day T-bills. The corresponding annual series are computed using the 30-day T-bill rate from the CRSP. For the model estimation, we cast the model that consists of equations (11), (23), and (24) into the following state-space model: Measurement equation [ ] dt+1 pd t+1 = [ γ 0,St+1 ] pd St+1 δ 1 pd St [ ] [ ] [ ] 0 0 dt δ 1 pd t B 2 (γ 1 δ 1 ) 0 B 2 B 1 g t ɛ d t+1 ɛ g t+1, (25) ɛ µ t+1 9

10 ( Ỹt+1 = M 0,St,S t+1 + M 1 Ỹ t + M 2 β t+1 ), where pd St+1 is a function of δ 0,St+1 and γ 0,St+1 as given in equations (20) and (21). Transition Equation g t γ g t ɛ d ɛ d t ɛ d t t+1 ɛ g = t ɛ g + ɛ g t t+1 ɛ µ ɛ µ t ɛ µ t+1 t ɛ d t+1 0 σ 2 d σ dg σ dµ ɛ g t+1 i.i.d.n 0, σ gd σg 2 σ gµ ɛ µ t+1 0 σ µd σ µg σµ 2 (26) (β t+1 = F β t + Rɛ t+1, ɛ t+1 i.i.d.n(0, Q) ) The model is then estimated via the maximum likelihood estimation method based on Kim s (1994) approximate Kalman filter. For identification of the model, we follow Binsbergen and Koijen (2010) in imposing the restriction σ gd = 0 on the variance-covariance matrix Q in equation (26). The regime-dependent means of expected returns and those of expected dividend growth rates (δ 0,St and γ 0,St, S t = 0, 1, 2) are estimated directly from the model. Based on these estimates, the regime-dependent means of price-dividend ratios (pd St, S t = 0, 1, 2) are estimated indirectly as in equation (20). Standard errors are estimated using the delta method. Before estimating our model, we first conduct structural break tests for the price-dividend ratio mean. Table 1 shows the results. Univariate test results in the upper panel of Table 1 are obtained by applying Bai and Perron s (1998) test to the data on price-dividend ratios. At a 5% significant level, one structural break is detected around At a 10% significant level, two structural breaks are detected around 1953 and These results are consistent with those reported by Lettau and Van Nieuwerburgh (2008). Multivariate test results in the lower panel of Table 1 are obtained by applying Qu and Perron s (2007) test to the data on the price-dividend ratio, the stock return, and the dividend growth rate. As expected, 10

11 the power of the multivariate test increases compared to the univariate test, and we detect two structural breaks at similar dates detected by the univariate test. Based on these structural break tests, we employ a three-state Markov-switching process for S t for our model, because a structural break with two breaks are nested within a threestate Markov-switching Model. Lettau and Van Nieuwerburgh (2008) also employ a threestate Markov-switching model when modeling regime shifts in the mean of price-dividend ratios. 4. Empirical Results and Discussion 4.1. Preliminary Results Chen (2009) finds a dramatic reversal of predictability from the year 1872 to the year Stock returns are unpredictable in the pre-war period, but become predictable in the post-war period; dividend growth is predictable in the prewar years but this predictability disappears in the post-war years. To account for this reversal in predictability, we incorporate an exogenous dummy variable to the γ 1 and δ 1 coefficients and the variance-covariance matrix of the shocks. For our model with regime switching, a dummy variable is incorporated in the following way: 5 θ = (1 τ t )θ A + τ t θ B, (27) where θ j = [ γ 1,j δ 1,j σ d,j σ g,j σ µ,j σ µg,j σ µd,j ], j = A, B, and the dummy variable τ t is defined as: τ t = { 1 for t 1951, 0 otherwise. For Binsbergen and Koijin s (2010) model without regime switching, we incorporate the same dummy variable in all the parameters. Thus, the post-1951 results reported in this 5 We choose the year 1951 as a break point, because it maximizes the log likelihood function. For the potential years of structural break, we considered the years from 1945 to 1955 as the range. 11 (28)

12 section for the model without regime switching are comparable to those in Binsbergen and Koijin (2010). Table 2 reports parameter estimates of the models both with and without regime shifts. In Figure 1, we depict the filtered probabilities of each regime (P r[s t = 0 I t ], P r[s t = 1 I t ] and P r[s t = 2 I t ]) against the actual price-dividend ratio. We have the following three distinct regimes: Regime 0 (low mean for p/d ratio): 1926 mid-1950s; mid s Regime 1 (medium mean for p/d ratio): late-1950s early 1970s; early 1990s Regime 2 (high mean for p/d ratio): since the mid 1990s Our inferences on regimes are broadly consistent with those of Lettau and van Nieuwerburgh (2008), who report two structural breaks in 1955 and 1995 identifying as the medium-mean regime for the price-dividend ratio. The only difference is that we identify part of their medium-mean regime (i.e., mid-1970s 1980s) as a low-mean regime. In what follows, we discuss the implications of regime shifts in the mean of price-dividend ratios on various issues related to the dynamics of expected returns and return predictability Discussion #1:The Dynamics of Expected Returns: Persistence and Volatility Persistence and volatility of expected returns is important in understanding stock market dynamics and the predictability of returns. Particularly, the persistence of expected returns allows us to calculate the effect of an innovation in the expected return (or news about future returns) on the stock price holding expected future dividends constant. 6 If expected returns are persistent, then movements in expected return will have a large impact on asset prices. Thus, as suggested by Campbell (1990), any attempt to explain the variability of asset prices or returns requires information on the persistence and volatility of movements in expected returns. Table 2 shows that the persistence of expected returns in the post-1951 sample (δ 1,B ) is 6 Refer to Chapter 7 of Campbell, Lo, and MacKinlay (1997) for a discussion on the importance of the expected stock return persistence in understanding stock market dynamics. 12

13 larger than that in the pre-1951 sample (δ 1,A ) while the reverse holds for the persistence of expected dividend growth (γ 1,B and γ 1,A ). Table 3 shows that the difference in the persistence of expected dividend growth is statistically insignificant for both models. However, the difference in the persistence of expected returns over the two periods is statically significant at a 10% significance level for the model without regime shifts (Binsbergen and Koijen s (2010) model), while it is statistically insignificant for the model with regime shifts (our model). A failure to incorporate regime shifts in the Binsbergen and Koijen (2010) model results in overestimation of the persistence of expected returns in the post-1951 sample. Focusing on the post-1951 sample, which is comparable to that of Binsbergen and Koijen (2010), the persistence of expected stock returns (δ 1,B ) for the model without structural breaks is estimated to be in the post-war sample. In this case, a 1% increase in expected returns would result in a capital loss of as much as 11.14%. However, when regime shifts are incorporated into the model, the persistence of expected returns is estimated to be much smaller at δ 1 = This implies that a 1% increase in expected returns would result in a capital loss of only 2.93%. Our measure of persistence is even lower than that estimated by Campbell (1991), whose measure of persistence is based on the multivariate variance decomposition framework. 7 In the OLS predictive regression approach, the persistence of the expected return process should be similar to that of price-dividend ratio dynamics. Thus, it would be interesting to compare the persistence measures for expected returns and price-dividend ratios. Allowing for regime shifts, Lettau and van Nieuwerburgh (2008) estimate the first-order autocorrelation of the price-dividend ratio to be 0.61, which is close to δ 1,B for the expected return from our model. We consider this similarity a necessary condition for the short-horizon return predictability documented in Lettau and van Nieuwerburgh (2008), who employ the mean-adjusted price-dividend ratio as a regressor in their predictive regression approach in the presence of structural breaks. Lettau and Ludvigson (2005) also suggest that the consumption-wealth ratio variable of Lettau and Ludvigson (2001) is much less persistent than the price-dividend ratio, consistent with the finding that the consumption-wealth ratio 7 Campbell (1991) suggests that a typical 1% increase in the expected return is associated with a capital loss of 4 to 5% based on the sample from the year 1927 to the year

14 forecasts stock returns over shorter horizons than the price-dividend ratio. Figure 2 plots and shows a comparison of the estimates of expected returns as implied by the Binsbergen and Koijen (2010) model and our model with regime shifts. The measures of expected returns implied by the two models are different, because the dynamics of expected returns from the two models are different. In addition to the differences in the persistence measures, the conditional volatilities of expected returns from the two models are also very different. For the post-war sample, the conditional volatility of expected returns (or the volatility of innovations to the expected returns) is much bigger for the model with regime shifts (σ µ,b = 0.049) than for the model without regime shifts (σ µ,b = 0.017). The estimates of expected dividend growth rates from the two models plotted in Figure 3 are hardly distinguishable. This is because both the persistence measures and the conditional volatilities of the expected dividend growth rate are almost the same for the two models. 4.3 Discussion #2: The Source of Recent Increases in the Price-Dividend Ratio Equation (13) suggests that an upward shift in the mean of price-dividend ratios in the mid-1990s must have been associated with an upward shift in the mean of expected dividend growth rates and/or a downward shift in the mean of expected returns. Consistent with the first view that the mean of dividend growth rates has increased, Jermann and Quadrini (2007) argue that the prospect of future productivity growth may have been the source of a stock market boom in the 1990s. Financial economists also provide evidence in support of the second view that the mean of expected returns has recently decreased. For example, Lettau et al. (2008) argue that a persistent decline in the macroeconomic uncertainty, or the volatility of aggregate consumption growth in the mid-1990s, resulted in a decline in the equity premium. Abel (2003) and Park (2010) argue that the US stock market boom was caused by a change in the demographic structure in the form of increased savings by the baby boom generation in the 1990s. Lettau and van Nieuwerburgh (2008) document that additional sources of a stock market boom or the decline in the expected return in the 1990s are i) persistent improvements in the degree of risk sharing among households or regions (Lusting and van Nieuwerburgh (2010)); 14

15 ii) persistent changes in the tax code (McGrattan and Prescott (2005)); and iii) an increase in stock market participation and the reduction of investment cost (Heaton and Lucas (1999), Vissing-Jorgensen (2002) and Calvet et al. (2003)). Table 4 compares differences in the mean of price-dividend ratios, the mean of expected returns, and the mean of expected dividend growth across different regimes. When we compare the most recent regime (Regime 2) against Regime 1, we cannot draw any conclusion about the source of an increase in the mean of price-dividend ratios. When we compare the most recent regime against Regime 0, however, we obtain results that are consistent with the second view. The mean of price-dividend ratios since the mid-1990s is higher than the mean of price-dividend ratios for Regime 0. Moreover, the mean of expected returns is lower, and the difference is statistically significant at the 5% significance level. However, the difference in the mean of dividend growth rates between the two regimes is not statistically significant. These pieces of evidence suggest that a shift in the mean of price-dividend ratios in the mid-1990 is related to a decrease in the mean of expected returns, which is consistent with the second view Discussion #3: Stock Return Predictability For the in-sample prediction performance, we follow Binsbergen and Koijen (2010) in calculating the R 2 values for returns and dividend growth rates. They are defined as: RRet 2 = 1 var(r ˆ t+1 E[r t+1 I t )], (29) var(r ˆ t+1 ) RDiv 2 = 1 var( d ˆ t+1 E[ d t+1 I t ]), (30) var( d ˆ t+1 ) where E[r t+1 I t ] and E[ d t+1 I t ] are the filtered estimates of µ t and g t, respectively, conditional on information up to t. Table 5 reports the in-sample R 2 measures from the OLS predictive regression model without regime shifts, the present-value model without regime shifts (Binsbergen and Koijen (2010)), and the present-value model with regime shifts (this paper). The regressor for the 15

16 OLS predictive regression is the price-dividend ratio, whereas the dependent variable is either the stock return or the dividend growth rate. For the sample, stock return predictability across different models is negligible, as measured by the R 2 values that range from -3.98% to 1.88%. However, considerable evidence exists of dividend growth predictability (the R 2 values range from 56.4% to 78.2%), regardless of the models employed. For the sample, the results are somewhat different depending on the models employed. When the OLS procedure is employed, stock returns become predictable (R 2 =8.46%) while the predictability of dividend growth (R 2 =0.00%) disappears, and the results are consistent with the results reported in Chen (2009). For Binsbergen and Koijen s (2010) model, predictability of dividend growth (R 2 =15.29%) is much better than the predictability using the OLS regression (R 2 =0%) while predictability of stock returns (R 2 =8.74%) is almost the same as it is when using the OLS regression. When regime shifts are incorporated to Binsbergen and Koijen s (2010) model, however, there is considerable improvement in the predictability of stock returns (R 2 =13.57%) while predictability of dividend growth (R 2 =15.46%) remains about the same. For the sample, in which the global financial crisis period is included, the predictability of stock returns across different models does not change significantly. For this extended sample, however, the predictability of dividend growth rates for the model with regime shifts is somewhat worse than the predictability of dividend growth rates for the model without regime shifts. The volatility of expected returns estimated from the present-value model with regime shifts is larger than the volatility of expected returns estimated from the model without regime shifts. This, combined with different measures of persistence discussed in Section 4.2, explains the differences in stock return predictability for the two present-value models with and without regime shifts. We also investigated out-of-sample predictability. However, regardless of the out-sample periods considered, we could achieve little improvement in outof-sample predictability for our model over Binsbergen and Koijen s (2010) model or the random walk model. As Lettau and Van Nieuwerburgh (2008) suggest, real-time estimation uncertainty of the magnitude of regime shifts and the date of regime shifts may be responsible 16

17 for this result. 5. Summary and Suggestions for Further Studies We present evidence that an increase in the mean of price-dividend ratios in the early 1990s may be because of a decrease in the mean of expected returns and not an increase in the mean of expected dividend growth rates. Our empirical results also suggest that ignoring regime shifts in the Binsbergen and Koijen (2010) model may result in an overstatement of the persistence of expected returns and an understatement of the conditional volatilities of expected returns. Thus, by appropriately incorporating regime shifts in the model, we considerably improve the in-sample predictability of stock returns, although little improvement is obtained for the in-sample predictability of dividend growth rates. Poor outof-sample predictability of the proposed model may reflect real-time estimation uncertainty of the magnitude of regime shifts and the date of shifts, as suggested by Lettau and Van Nieuwerburgh (2008). Boudoukh et al. (2007) show that there is evidence in the literature (e.g., Fama and French (2001) and Allen and Michaely (2003)) that dividends and share repurchases have substituted for each other, and time-series properties of dividends have changed because of the shift towards repurchases over the past two decades. They argue that asset pricing models that try to fundamentally relate dividends to asset prices need to consider these effects. This is because stock market value depends on investor valuation of all firm cash flows, not just the dividend component, as argued by Miller and Modigliani (1961). Thus, by employing a total payout ratio (that includes dividends and share repurchases) or a new cashflow yield (that aggregates dividend and non-dividend cash flows) in their OLS predictive regressions, Boudoukh et al. (2007) and Robertson and Wright (2006) achieve improved return predictability. We have not considered the effect of non-dividend cash flows or the changing time-series properties of dividends resulting from the changing payout policies by the firms in the current paper. We leave these subjects for future research. 17

18 References [1] Abel, A. B., The effects of a baby boom on stock prices and capital accumulation in the presence of social security, Econometrica 71, [2] Allen, F. and R. Michaely, 2003, Payout Policy in North-Hollan Handbook of Economics, edited by G. Constantinides, M. Harris, and R. Stulz, , Amsterdam: Elsevier/North Holland. [3] Bai, J., and P. Perron, 1998, Estimating and testing linear models with multiple structural changes, Econometrica, 66, [4] Binsbergen, J. H. Van, and R. S. J. Koijen, 2010, Predictive regressions: A present-value approach, Journal of Finance, 65, [5] Boudoukh, J., R. Michaely, M. Richardson, and M. Roberts, 2007, On the importance of measuring payout yield: Implications for empirical asset pricing, Journal of Finance, 62, [6] Campbell, J. Y., 1991, A variance decomposition for stock returns, The Economic Journal, 101, [7] Campbell, J. Y., and R. J. Shiller, 1988, The dividend-price ratio and expectations of future dividends and discount factors, Review of Financial Studies, 1, [8] Campbell, J. Y., 1990, Measuring the persistence of expected returns, American Economic Review (papers and proceedings) 80-2, [9] Calvet, L., M. Gonzalez-Eiras, and P. Sodini, 2004, Financial innovations, market participation, and asset prices, Journal of Financial and Quantitative Analysis, 39, [10] Chen, L., 2009, On the reversal of return and dividend growth predictability: a tale of two periods, Journal of Financial Economics, 92, [11] Fama, E. F. and K. R. French, 2001, Disappearing dividends: Changing from characteristics or lower propensity to pay? Journal of Financial Economics, 60, [12] Fama, E. F., and K. R. French, 2002, The equity premium, Journal of Finance, 57, [13] Goyal, A., and I. Welch, 2008, A comprehensive look at the empirical performance of the equity premium prediction, Review of Financial Studies, 21,

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21 Appendix A. Derivation of Equation (16) Consider the following three-state, first-order Markov-switching process, S t, with the transition probabilities Then, by defining P r[s t = j S t 1 = i] = p ij, 2 p ij = 1, i, j = 0, 1, 2 (A.1) i=0 S jt = we have the following result: { 1, if St = j, j = 0, 1, 2; 0 otherwise. S 0t p 00 p 10 p 20 S 0,t 1 v 0t S 1t = p 01 p 11 p 21 S 1,t 1 + v 1t S 2t p 02 p 12 p 22 S 2,t 1 v 2t ( ) S t = P St 1 + vt (A.2) (A.3) where v t is a vector of martingale difference sequences. Because Σ 2 j=0s j,t 1 = 1, equation (A.3) can be rewritten as: [ ] S1t S 2t = [ ] p10 p 20 [ ] [ ] (p11 p 01 ) (p 21 p 01 )) S1,t (p 12 p 02 ) (p 22 p 02 )) S 2,t 1 ( St = p 0 + λ S ) t 1 + ṽ t [ ] v1t v 2t (A.4) By taking expectations on both sides of equation (A.4), we obtain: π = p 0 + λ π, (A.5) where π is a vector of unconditional probabilities. Then, by subtracting (A.5) from (A.4), we have the following VAR(1) representation for S t : S t = π + λ( S t 1 π) + ṽ t, (A.6) from which we obtain the following results in equation (16): 21

22 E[ S t+j+1 S t+1 ] = π + λ j ( S t+1 π), (A.7) Appendix B. Maximum Likelihood Estimation of the Model The following filter and the procedure for maximum likelihood estimation is based on Kim (1994) and Kim and Nelson (1999). Suppose we know all the hyper-parameters of the state-space Model that consists of equations (25) and (26). Given that S t = i and S t+1 = j, the Kalman filter can be represented as follows: β i t+1 t = F β i t t P i t+1 t = F P i t tf + R QR η i,j t+1 t = Ỹt+1 M 0,i,j M 1 Ỹ t M 2 β i t+1 t H i t+1 t = M 2 P i t+1 tm 2 β (i,j) t+1 t+1 = βi t+1 t + P i t+1 tm 2[H i t+1 t] 1 η i,j t+1 t P (i,j) t+1 t+1 = (I 4 P i t+1 tm 2[H i t+1 t] 1 M 2 )P i t+1 t, (A.8) (A.9) (A.10) (A.11) (A.12) (A.13) where β i t+1 t is an inference of β t+1 conditional on information up to time t given S t = i (i = 0, 1, 2); P i t+1 t is the variance covariance matrix of β t+1 conditional on information up to t given S t = i; η (i,j) t+1 t is the conditional forecast error of Ỹt+1 conditional on information up to time t, given S t = i and S t+1 = j (i, j = 0, 1, 2); H i t+1 t is the conditional variance Ỹt+1 given S t = i, and so on. Note that we have P i,0 t+1 t+1 = P i,1 t+1 t+1 = P i,2 t+1 t+1 in equation (A.13). As in Harrison and Stevens (1976), the (9 9) posteriors (β (i,j) (i,j) t+1 t+1 and P t+1 t+1 ) must be reduced into three 3 to complete the process. The collapsing process employed here is as follows: t+1 t+1 2i=0 β j t+1 t+1 = P r[s t = i, S t+1 = j ψ t+1 ]β (i,j) P r[s t+1 = j ψ t+1 ] 22, (A.14)

23 P j t+1 t+1 = 2i=0 P r[s t = i, S t+1 = j ψ t+1 ]{P (i,j) t+1 t+1 + (βj t+1 t+1 β(i,j) t+1 t+1 )(βj t+1 t+1 β(i,j) P r[s t+1 = j ψ t+1 ] j = 0, 1, 2, t+1 t+1 ) }, (A.15) where ψ t+1 refers to information available at time t+1. β j t+1 t+1, (j=0,1,2), for example, is a weighted average of β (0,j) t+1 t+1, β(1,j) t+1 t+1, and β(2,j) t+1 t+1, where the weights are P r[st=i,s t+1=j ψ t+1 ] P r[s t+1 =j ψ t+1, ] i=0,1,2. The final consideration in completing the modified Kalman filtering is to calculate the P r[s t = i, S t+1 = j ψ t+1 ] and other probability terms, as given below: P r[s t = i, S t+1 = j ψ t+1 ] = P r[s t = i, S t+1 = j ψ t, Ỹt+1] = f(ỹt+1, S t = i, S t+1 = j ψ t ) f(ỹt+1 ψ t ) = f(ỹt+1 S t = i, S t+1 = j, ψ t )P r[s t = i, S t+1 = j ψ t ], f(ỹt+1 ψ t ) (A.16) where f(ỹt+1 S t = i, S t+1 = j, ψ t ) = (2π) 1 H i t+1 t 1 2 exp{ 0.5(η i,j t+1 t ) (H i t+1 t) 1 (η i,j t+1 t )}, f(ỹt+1 ψ t ) = 2 i=0 j=0 2 f(ỹt+1, S t = i, S t+1 = j ψ t ), (A.17) (A.18) P r[s t = i, S t+1 = j ψ t ] = P r[s t+1 = j S t = i]p r[s t = i ψ t ], (A.19) and P r[s t+1 = j ψ t+1 ] = 2 P r[s t = i, S t+1 = j ψ t+1 ]. i=0 (A.20) Thus, equations (A.15) to (A.20) complete the modified Kalman filtering. As a by-product of the above modified Kalman filtering, the conditional log likelihood function can be obtained from (A.17), as given below: T LL = log(f(ỹt, ỸT 1,...)) = log(f(ỹt ψ t 1 )) t=1 (A.21) 23

24 Table 1. Test of Structural Breaks in the Price-Dividend Ratios, Returns, and Dividend Growth Rates [1926~2014] A. Bai & Perron (1998) s SupF Test (Sequential Procedure, Univariate Test) Test Statistic SupF T (1) *** SupF T (2 1) 9.68 ** SupF T (3 2) 2.67 Number of Breaks and Break Dates No. of Breaks Break Dates Significance Level , % % B. Qu & Perron (2007) s SupLR Test (Sequential Procedure, Multivariate Test) Test Statistic SupLR T (1) *** SEQ T (2 1) *** SEQ T (3 2) Number of Breaks and Break Dates No. of Breaks Break Dates , 1993 Note : 1. The table provides the univariate and multivariate structural break tests. The data are annual price and dividend series calculated from the CRSP value-weighted index of all US markets, and span the period from the year 1926 to the year The upper panel reports the results of Bai and Perron s (1998) procedure for the mean of log pricedividend ratio. For the trimming variables, the minimal length of a segment and the maximum number of breaks are chosen as 10% of the full sample period and eight times, respectively. 2. The lower panel indicates Qu and Perron s (2007) multivariate structural break test for the means of log price-dividend ratio (pd t ), stock return (r t ), and dividend growth rate ( d t ). That is, [ pd t r t d t ] = [pd pd i r i d ] i + r [u t u t u d t ], where i represents each regime. The minimal length of a segment and the maximum number of breaks are set to 15% of the full sample 24

25 period and eight times, respectively. 3. Two asterisks (**) indicate significance at the 5% level, and three asterisks (***) indicate significance at 1% level. 25

26 Table 2. Estimation Results: Binsbergen and Koijin (2010) Model With and Without Markov Switching [1926~2014] Parameters B&K With MS Parameters B&K Without MS γ (0.0179)*** γ (0.0187) γ (0.0192)*** δ (0.0176)*** δ (0.0329)* δ (0.0245)** γ 0A (0.0634) γ 0B (0.0114)*** δ 0A (0.0609)* δ 0B (0.0132)*** γ 1A (0.1566)*** γ 1A (0.3346) γ 1B (0.1230)** γ 1B (0.1210)** δ 1A (0.2524) δ 1A (0.2744) δ 1B (0.1205)*** δ 1B (0.0468)*** pd (0.0562)*** pd (0.0695)*** pd A (0.0840)*** pd B (0.2296)*** pd (0.0878)*** p (0.0128)*** p (0.0128)** p (0.0646)*** p (0.0480) p (0.0154)*** p (0.0154) 26

27 Table 2. Continued Parameters B&K With MS Parameters B&K Without MS σ d A (0.0321)* σ d A (0.0391) σ d B (0.0088) σ d B (0.0080) σ g A (0.0311)*** σ g A (0.0304)*** σ g B (0.0057)*** σ g B (0.0058)*** σ μ A (0.0538)*** σ μ A (0.0822) σ μ B (0.0175)*** σ μ B (0.0078)** ρ μg A (0.2105)*** ρ μg A (0.5901) ρ μg B (0.1546) ρ μg B (0.1472) ρ μd A (0.2557)** ρ μd A (0.4449)* ρ μd B (1.0044) ρ μd B (0.0311)*** Note : 1. The data set is constructed using the annual price and dividend series calculated from the CRSP value-weighted index of all US markets, and spanning the period from the year 1926 to the year The B&K Without MS and B&K With MS refer to the Binsbergen and Koijin (2010) model and the proposed model that incorporates Markov-switching mean into that model, respectively. 3. The subscripts A and B indicate the estimates by dummy variables, before the year 1950 and after the year 1951, respectively. 4. The parentheses indicate the standard errors calculated through numerical optimization. The superscripts *, **, and *** denote the significance level at the 10%, 5%, and 1% level, respectively. 27

28 Table 3. Structural Break Tests for Persistence Parameters (t-values) [1926~2014] B&K Without MS B&K With MS H 0 γ 1B = γ 1A H 0 δ 1B = δ 1A * Note : 1. Each number indicates Wald statistics for two models. The B&K Without MS and B&K With MS refer to the Binsbergen and Koijin (2010) model and the proposed model that incorporates the Markov-switching mean into that model, respectively. 2. The superscript, *, denotes that the test statistic is significant at the 10% level. 28

29 Table 4. Estimated Difference in Regime Means of Price-Dividend Ratio, Expected Returns, and Dividend Growths [1926~2014] Between Regime 1 and 2 Between Regime 0 and 2 pd 2 pd (0.0962)*** pd 2 pd (0.0974)*** δ 02 δ (0.0374) δ 02 δ (0.0309)** γ 02 γ (0.0267) γ 02 γ (0.0262) Note : The parentheses indicate the standard errors calculated through the delta method. The superscripts *, **, and *** denote the significance level at the 10%, 5%, and 1% level, respectively. 29

30 Table 5. In-Sample Goodness of Fit (R 2 ) for Alternative Models and Alternative Sample Periods [1926~2014] Stock Return 1926~ ~ ~2014 Dividend Growth Stock Return Dividend Growth Stock Return Dividend Growth OLS 0.68% 56.36% 8.46% 0.00% 9.06% 0.25% B&K Without MS 1.88% 75.32% 8.74% 15.29% 8.85% 18.96% B&K With MS -3.98% 78.17% 13.57% 15.46% 12.19% 11.04% Note : 1. The table reports stock return and dividend growth predictability of the predictive regression and the present-value model, measured by in-sample goodness of fit (R 2 values) using US annual data. 2. OLS refers to the predictive regression based on OLS, that is, r t+1 = α + β pd t + ε t+1 or d t+1 = α + β pd t + ε t+1. The predictive regressions are estimated for each subsample. The R 2 measures are calculated by ) 2 R 2 = 1 (r t+1 r ) 2 for stock return and (r t+1 r t+1 ) 2 R2 = 1 ( d t+1 d t+1 ( d t+1 d t+1 ) 2 for dividend growth. 3 The B&K without MS refers to the Binsbergen and Koijen (2010) model without Markovswitching; and the B&K with MS refers to the proposed model with Markov-switching. These models are estimated for the full sample for the years 1926~2014, and their R 2 s are calculated by each subsample. The R 2 s are measured by R 2 = 1 var( r t+1 μ ) t var( r t+1 ) for stock return and R 2 = 1 var( d t+1 g ) t var( d t+1 ) for dividend growth in the case of the present value model, following Binsbergen and Koijen (2010). 30

31 Figure 1. The Price-dividend Ratios and the Estimated Regime Probabilities Pr [S t+1 = 0 ψ t+1 ] Pr [S t+1 = 1 ψ t+1 ] 31

32 Figure 1. Continued Pr [S t+1 = 2 ψ t+1 ] Note : This figure plots the time series of state probabilities estimated by our model. The thick solid lines show the log price-dividend ratios, and the thin lines represent the state probabilities, Pr[S t+1 = j ψ t+1 ], where j = 0 in the upper panel, j = 1 in the mid panel, and j = 2 in the lower panel. 32

33 Figure 2. Stock Returns and Estimated Expected Returns Note : 1. This figure plots the realized returns and their filtered series from each model. The thin solid line describes the actual returns, and the thick solid line represents the extracted expected returns from the proposed model that incorporates the Markov-switching mean. The dotted line indicates the extracted expected returns from the Binsbergen and Koijin (2010) model. 2. The shaded area refers to the period after the year 1951 when the dummy variables are used in each model. 33

34 Figure 3. Dividend Growth Rates and Estimated Expected Dividend Growth Rates Note : 1. This figure plots the realized dividend growth rates and their filtered series from each model. The thin solid line describes the actual dividend growths, and the thick solid line represents the extracted expected dividend growths from the proposed model that incorporates the Markov-switching mean. The dotted line indicates the extracted expected dividend growths from the Binsbergen and Koijin (2010) model. 2. The shaded area refers to the period after the year 1951 when the dummy variables are used in each model. 34