Time-varying Cointegration Relationship between Dividends and Stock Price

Transcription

1 Time-varying Cointegration Relationship between Dividends and Stock Price Cheolbeom Park Korea University Chang-Jin Kim Korea University and University of Washington December 21, 2009 Abstract: We consider the consequence of the existence of firms which are not paying out dividends on the relationship between aggregate dividends and stock price, and have developed a new framework for understanding the extreme persistence of the conventional dividend-price ratio and its sensitive predictive power to the choice of sample periods. Furthermore, we show that the cointegration relationship between dividends and stock price is not fixed as [1, -1] but varies over time due to the movements of the fraction of firms which are paying out dividends. As a result, we have developed and estimated a new variable, which is stationary deviation from the time-varying cointegration relationship between dividends and stock price, and demonstrated that this newly developed variable has stable and strong ability to forecast stock returns from the out-of-sample analysis as well as in-sample analysis. JEL classification: G12, C12, C22 Key Words: Stock returns, Dividend-price ratio, Payout policy, Time-varying cointegration Park: Address: Department of Economics, Korea University, Anam-dong, Seongbuk-gu, Seoul, Korea TEL: cbpark_kjs@korea.ac.kr. Kim: Address: Department of Economics, Korea University, Anam-dong, Seongbuk-gu, Seoul, Korea TEL: cjkim@korea.ac.kr. 1

2 I. Introduction Can the log dividend-price ratio forecast future stock returns? This question has been a long time challenge to economists especially since Campbell and Shiller (1988) derived the log present value relation between the log dividend-price ratio, stock returns, and dividend growth rate. 1 However, the consensus has not been made yet. Some studies provide statistical evidence for the predictive power of the log dividend-price ratio, while others point out econometric problems associated with the statistical evidence and reject the predictive power. More recently, however, a growing body of research is reporting that the forecast ability of the log dividend-price ratio is sensitive to the selection of the sample period. Studies like Wolf (2000), Valkanov (2003), and Campbell and Yogo (2006) have developed new approaches to test the stock return predictability, but found that the log dividend-price ratio evidences strong predictive power during a certain period, whereas it has weak or no predictive power at other times even with those newly developed test statistics. Rapach and Wohar (2006) and Paye and Timmermann (2006) have reported the instability of the predictive regression with the log dividend-price ratio. Lettau and Nieuwerburgh (2008) have argued that the log dividend-price ratio had undergone a shift in its mean and this shift is responsible for the sensitive predictive power. Boudoukh et al. (2007) and Robertson and Wright (2006) have claimed that many US firms had switched dividend payment to share repurchases recently and this change in payout policy had caused the log dividend-price ratio to lose its forecast ability. 1 The claim that stock returns are predictable is a different story from the argument that the investment into the stock market can provide extra profit. The predictable stock returns can be viewed as an equilibrium phenomenon. For example, Campbell and Cochrane (1999) show that the time-varying risk premium can generate predictable stock returns. Cecchetti, Lam and Mark (2000) argue that stock return predictability can be attributable to distorted beliefs. Bansal and Yanron (2004) claim that stock return predictability arises from a small long-run predictable component and fluctuating uncertainty contained in consumption and dividend growth rates. 2

3 This paper suggests that the sensitive predictive power of the log dividend-price ratio may be due to the fluctuations of the fraction of payers (firms which pay out dividends). The fraction of payers in terms of the market value of common equities dropped substantially since the early 1980s because many existing firms had stopped or reduced dividend payment (some of them might have switched dividend payment to share repurchases) and newly listed firms usually do not pay out dividends (see Fama and French (2001)). We show that this fall of the fraction of payers is responsible for the extreme persistence of the log dividend-price ratio and for the loss of forecast ability from the log dividend-price ratio as well. Furthermore, we demonstrate that the cointegration relationship between dividends and stock price is not fixed as [1, -1] but varies over time due to the fluctuations of the fraction of payers. As a result, we construct a modified dividend-price ratio by estimating the time-varying cointegration coefficients. Our argument is similar to that in Boudoukh et al. (2007) and Robertson and Wright (2006) in the sense that changes in firms payout policy have caused measurement errors in dividends data, and our approach to construct a modified dividend-price ratio can be considered as a complementary one to theirs. However, since Fama and French (2001) demonstrate that a substantial portion of share repurchase is done owing to the consideration of employee stock ownership plans or mergers instead of replacing dividend payment, the approach proposed by Boudoukh et al. (2007) and Robertson and Wright (2006) could be quite misleading in modifying the conventional dividend-price ratio. Moreover, as our approach does not require share repurchase data additionally, it can be applied to the case that a portion of firms have never paid out dividends nor repurchased shares for long time. 2 Hence, we think that our approach is more flexible. 2 In fact, many small-size firms have never paid out dividends or repurchased shares for long time. If firms have been non-payers for long time, then the adjustment with the share repurchase data cannot be applied to these firms. 3

4 We also show strong statistical evidence that the cointegration parameters between dividends and stock price vary gradually over time rather than the one or two-time shifts in the mean of the log dividend-price ratio as claimed in Lettau and Nieuwerburgh (2008). Furthermore, we evaluate whether the modified dividend-price ratio using the time-varying cointegration coefficients (TVC dividend-price ratio hereafter) has predictive ability for future stock returns. We present from the in-sample analysis that the TVC dividend-price ratio has stable and significant predictive power for stock returns. The strong predictive power is robust not only for the aggregate or portfolio formed on large-sized or medium-sized firms but also for the portfolio formed on small-sized firms for which the adjustment with the share repurchase data is not expected to work well. This significant predictive power survives at short horizons even when the bootstrapped distribution for the coefficient of the predictive regression is used for the evaluation. We also show from the out-of-sample analysis that the predictive power of the TVC dividend-price ratio is significantly better than that of the random work with the portfolio formed on small-sized firms, and is comparable to that of the random walk with the aggregate stock market data or portfolios formed on large-sized or medium-sized firms. The rest of this paper is organized as follows. Section II discusses the consequence of the existence of non-payers. When non-payers exist in the market, the conventional dividendprice ratio could become nonstationary, and the cointegration relationship between dividends and stock price varies over time. Section III presents the econometric methodology to estimate the time-varying cointegration parameters between dividends and stock price based on the Fourier Flexible Form (FFF). Section IV provides empirical analysis on the timevarying cointegration relationship between dividends and stock price, and shows that the implications in Section II are supported by data. Section V evaluates the predictive power of the TVC dividend-price ratio. Concluding remarks are offered in Section V. Proofs and some 4

5 complementary results are provided in the Appendix. II. Time-Varying Cointegrating Relation between Dividends and Price We address the relationship between the aggregate stock price and dividends when a certain fraction of firms are not paying dividends out in this section. Hence, we assume that there are two types of firms. One is payers, firms which are paying out dividends to stockholders. The other is non-payers, firms which have never paid out dividends and will not pay out dividends in the near future. Although non-payers usually retain and reinvest all the earnings, we can think of a payout policy with which non-payers would issue bonds to finance (a part of) the investment and pay out dividends to stockholders, instead of reinvesting all the earnings. According to the Modigliani-Miller theorem, this assumed dividend policy will not affect the stock price for non-payers. denotes the aggregate dividends by non-payers under the assumed dividend policy. Then, we can safely say that the following log present value relation, derived by Campbell and Shiller (1988), holds for non-payers as well as payers where is the log dividends for payers at t, is the log real stock return for payers at t, is the log aggregate stock price for payers at t, which is the modified log real stock return for non-payers at t, is the log aggregate stock price for non-payers at t, ln, constants which are obtained from the linearization process. and. and are unimportant 3 Note that the log present value relation cannot be defined for non-payers without the assumed payout policy. 5

6 According to the Modigliani-Miller theorem, for non-payers is not necessarily unique but we can pin down so that where is the average fraction of payers in terms of the market value of common equities during the period between t-1 and t. Another assumption for and λ is that both and λ are related with the aggregate economic activities such as and λ where denotes the aggregate economic activities (e.g. the aggregate output). Finally, we assume that, λ,, and are I(1) variables, while and are I(0) variables, as commonly assumed in the literature. Then, we can obtain the following propositions. All the proofs are provided in the Appendix. Proposition 1. The log dividend-price ratio for the aggregate stock market can be written as 1, where and are the aggregate dividends and stock price which are observed in the market. As a result, the log dividend-price ratio for the aggregate stock market contains an I(1) component unless or 1. When there are no non-payers in the market, the present value relation states that the log dividend-price ratio which can be written as the sum of I(0) variables (stock returns and the dividend growth rate) must be an I(0) variable. However, many previous studies report that the log dividend-price ratio is so persistent that many unit root tests fail to reject the unit root null hypothesis for the log dividend-price ratio. Proposition 1 provides a reason for such a puzzling phenomenon. The existence of non-payers can explain the discrepancy between the theoretical model and the empirical findings in previous studies. Furthermore, the persistence of the log dividend-price ratio depends on which is the fraction of payers in the market in terms of market value of common equities. When is high and close to one, 6

7 then the log dividend-price ratio behaves similarly to an I(0) variable and econometricians may not be able to distinguish the small I(1) component in the data. However, as becomes lower, which was observed during the 1980s and the 1990s, the nonstationary component in the log dividend-price ratio dominates its behavior. This consequence also has important implications on the sensitive predictive power of the conventional log dividend-price ratio. When is close to one, the conventional dividend-price ratio could have stable ability to forecast stock returns. However, as falls substantially, the I(1) component in the log dividend-price ratio becomes more dominant so that the conventional dividend-price ratio loses its predictive power. Otherwise, stock returns must inherit the stochastic trend in the log dividend-price ratio when is low. 4 Proposition 2. The aggregate dividends and the aggregate stock price are cointegrated with time-varying cointegration coefficients. That is, there exist and which make be an I(0) variable where 1 and 1. Proposition 1 that has an I(1) component does not necessarily mean that there is no cointegration relation between the aggregate dividends and stock price. In fact, Proposition 2 states that they are cointegrated with time-varying cointegration coefficients and that the time-varying coefficients depend on the fraction of payers in the market in terms of market value of common equities and the sensitivity of dividends to the common macroeconomic factor. Although we cannot directly observe the sensitivity of dividends to the 4 Lo (1991) and Lanne (2002) report that stock returns are I(0). Hence, I(1) dividend-price ratio should not predict stock returns. 7

8 macroeconomic factor, many recent studies show that the fraction of payers is far from being constant and had declined substantially (see Fama and French (2001)). This finding implies that the cointegration coefficient between the aggregate dividends and stock price is varying over time. Furthermore, the time-varying cointegration relationship between dividends and stock price can be interpreted as an underlying reason for seemingly independent explanations proposed in previous studies. First, if an econometrician is not aware of the possibility of the time-varying cointegration relation but fixes the cointegration coefficient between the aggregate stock price and dividends, then the substantial fall in during the 1980s and the 1990s could make the conventional dividend-price ratio based on the fixed cointegration coefficient appear to have undergone a structural break. In fact, Lettau and Nieuwerburgh(2008) emphasize one-time or two-time shifts in, which corresponds to the mean of the conventional dividend-price ratio, assuming that 1. However, we will provide evidence that this adjustment is not enough to account for the gradual time-varying cointegration relation in a later section. Second, if an econometrician mistakenly uses the conventional dividend-price ratio in the predictive regression with stock returns, then the predictive regression is likely to appear to have an unstable coefficient due to the instability in the regressor (the conventional dividend-price ratio) as reported by Paye and Timmermann (2006) and Rapach and Wohar (2006). In addition, the claim in Boudoukh et al. (2007) and Robertson and Wright (2006) that the change in firms payout policy from dividend payment to the repurchase of shares has resulted in a breakdown in the cointegration relationship between the aggregate dividends and stock price sounds similar to the results in propositions 1 and 2. However, Fama and French (2001) demonstrate that the switching of cash dividends into share repurchase can explain only a portion of share repurchase. They show that the other substantial portion of share 8

9 repurchase is done owing to the consideration of employee stock ownership plans or mergers. Hence, the approach proposed by Boudoukh et al. (2007) and Robertson and Wright (2006) could be quite misleading and incomplete in modifying the conventional dividend-price ratio. Furthermore, the approach which is proposed by Boudoukh et al. (2007) and Robertson and Wright (2006) to modify the conventional log dividend-price ratio cannot be applied well to the portfolio formed on small-sized firms. A considerable portion of firms in the portfolio formed on small-sized firms have never paid out dividends nor repurchased shares for long time. As a result, the predictive power of the conventional dividend-price ratio is much weaker than those formed on large-sized or medium-sized firms but the modification using the share repurchase data is not sensible for these long time non-payers. As our approach to estimate the time-varying cointegration parameters does not require share repurchase data, however, our approach seems more flexible and is expected to work even for the portfolio formed on small-sized firms. Finally, we examine whether the stationary deviation of the time-varying cointegration relation can forecast future stock returns or the growth rates of dividends. This issue is addressed in Propostion 3. Proposition 3. The log present value relation states that can have the following relation

10 1 where is the aggregate stock returns which is observable in the market,, and is an unimportant linearization constant. Proposition 3 shows the possibility that the stationary deviation from the time-varying cointengration relation contains information to predict many observable and unobservable variables. If unobservable variables are not varying much, then a high (low) value for predicts high (low) stock returns which is the only observable variable in Proposition 3, in the future. Hence, the question, whether the aggregate stock returns can be predicted by, can be addressed empirically. The implications from the above three propositions will be investigated empirically to distinguish our model from existing studies in susequent sections. III. Estimating the Time-varying Relationship between Dividends and Price Propositions in the previous section state that there is a cointegration relationship between dividends and stock price but the relationship varies smoothly over time when the fraction of payers in terms of market value of common equities ( evolves gradually. 5 Since firms have different dividend-payment frequencies, have different payout dates, and pay dividends out at most four times a year, we cannot have an aggregate measure for at monthly or quarterly frequency. As a result, an approximate measure for can be thought of over a certain period which must be longer than a quarter at least. Figure 1 shows movements of 5 In fact, if the sensitivity of dividends to the macroeconomic factor in the previous section varies over time, the cointegration relationship between dividends and stock price also gradually changes. However, we do not have observable data to examine this channel. 10

11 which is constructed by all firms in the Center for Research in Security Prices (CRSP) valueweighted market portfolio at annual frequency. 6 The aggregate declined gradually from 1946 to 1980, declined sharply from 1981 to 1999, and rose sharply since then. for the portfolio formed on large-sized firms tracks the movements of the aggregate closely. for the portfolio formed on medium-sized firms dropped considerably during the 1960s, rose during the 1970s, and then declined sharply since the early 1980s. for the portfolio formed on small-sized firms has been lower than for any other portfolios, and has the most volatile fluctuations but it also reached its minimum level in 2000 and then rose. 7 Figure 1 strongly suggests that the aggregate and for any portfolios formed on sizes are far from constant but vary over time, which implies the time-varying cointegration relationship between the dividends and stock price. Since we cannot construct at monthly frequency and the number of observations is reduced substantially with annual or quarterly data, we estimate the time-varying cointegration parameter directly from the monthly data. In order to estimate the time-varying cointegration relationship and to evaluate whether the stationary deviation from this relationship has any predictive power for future stock returns, we have employed the Park and Hahn (1999) approach. Thus, we consider the following econometric model. (1) denotes the cointegration coefficient between and, and the gradual changes in can cause to depend on time as well. We denote the sample size by and let 6 If the amount of cash dividend paid by a firm is zero for one year period, the firm is treated as a non-payer for that year. The fraction of payers in terms of market value is computed from the year-end market values of all firms in the CRSP. The cut-off values of market values for each size (large, medium, and small) are taken from French s homepage, 7 Figure 2 in Fama and Frech (2001) shows a similar pattern in the movements of the fraction of payers in terms of the number of firms instead. 11

12 so that is a smooth function defined on [0, 1]. 8 While estimating, no functional form is imposed for. The only assumption required for is that it is sufficiently smooth to be approximated by a series of polynomials, trigonometric functions, or a mixture of both series. That is, assume that 0 as, where is an approximation of given by a combination of a finite series of functions,,. Since, the above econometric model can be expressed as: (2) where,,,, and,,. If and are stationary series, then we can have the asymptotic normality of LS estimator for in equation (2) and chi-square tests (see Andrews (1991)). However, as and are nonstationary, we apply the canonical cointegrating regression (CCR) approach, which was developed by Park (1992), for equation (2) to obtain the asymptotic normality of the LS estimator for and chi-sqaure tests. Hence, we have made CCR transformation for and as follows. and,, 0, (3) (4) where,,, and. is the innovation series in process, for, 1,2 denotes elements of, and denotes 8 When the time-varying cointegration coefficient is approximated by a series of trigonometric functions, it is desirable to scale the data into the interval [0,1] due to the characteristics of trigonometric functions. 12

13 the second row of matrix. Under the CCR transformation, Equation (2) can be written as (5) where,,. We can obtain the asymptotic normality of LS estimator for under this transformation. Once the LS estimator for is obtained, then can be approximated by. We utilize the Fourier Flexible Form (FFF) to approximate nonparametrically. The FFF, which was introduced by Gallant (1981), extends the traditional Fourier theorem. The FFF expansion of can be expressed as,, (6) where 2, and 32. It is worth noting the robustness of the FFF approach. Because economic theories provide few guidelines for, except for the conjecture that might be positive as is fixed at one when all firms are payers, the FFF is ideal, as it approximates under a flexible representation. If only the first term in equation (6) is considered and set as one, then the time-varying cointegration regression based on FFF becomes a cointegration regression with the usual fixed cointegration coefficient [1, -1]. IV. Empirical Analysis on Time-Varying Cointegration Relationship In this section, we examine whether the implications from our model presented in Section II are supported by stock market data using the econometric method described in Section III. Before reporting the results of the empirical analysis, we present a brief description of the data utilized in this study. The U.S. stock market data are constructed from the CRSP valueweighted market portfolio. 9 The aggregate U.S. real stock return is constructed by subtracting the CPI inflation rate from the log returns of the CRSP value-weighted market 9 The CRSP data were obtained from 13

14 portfolio. Following Torous, Valkanov, and Yan (2004), the aggregate stock price index is constructed from monthly returns on the CRSP value-weighted market portfolio without dividends, and the aggregate dividend series is constructed from monthly returns on the CRSP value-weighted market portfolio with and without dividends. Data for portfolios formed on size are obtained from Kenneth French s homepage. 10 These data are constructed from CRSP database by French, and hence consistent with our aggregate stock market data. The stock returns, dividends, stock price indices for portfolios formed on size are constructed in the same way as for the aggregate variables. The sample period is A. Persistence and Predictive Power of Dividend-Price Ratio and Fraction of Payers Proposition 1 states that contains an I(1) component which varies with, the fraction of payers in terms of market value of common equities. This I(1) component is the consequence of the existence of non-payers. When is close to one, the I(1) component may be too small to be distinguished. However, as falls, the I(1) component in becomes more pronounced, which makes become more persistent. This implies that it will be more difficult to reject the unit root null hypothesis when becomes lower. We examine this implication by relating the annual to and the Augmented Dickey-Fuller (ADF) test statistics from the following regression. We compute the first estimate of and the ADF test statistic for using 30-year data since January 1946, and re-compute the estimate of and the ADF test statistic recursively adding one-month at a time. Then, the annual average of the estimates of or the ADF statistics is regressed on a constant and the annual measure of. As shown in Table I, although is 10 The web address for French s homepage is 14

15 somewhat a noisy measure, plays a significant role in explaining the movements of and the ADF statistics for. The coefficient of is highly significant and is about 0.83 and 0.74 with the aggregate stock market data. Since the ADF test statistics are usually negative, the negative coefficient of indicates that as becomes lower, the ADF statistics becomes closer to zero and it becomes more difficult to reject the unit root null hypothesis. The results are also robust to the choice of portfolios formed on any size firms. The coefficient of is always significant at the 5% level and ranges from 0.67 to 0.91 (from 0.67 to 0.86) for the regression of (the ADF statistics). As we find the close relation between the persistence of and movements of, we wonder whether there could be another tight relation between the predictive power of and movements of. We address this question by running the following two regressions. First, we regress one-month ahead stock returns on. That is, is run recursively, in a similar way to the ADF regression in Table I. After obtaining and the t-statistics of, we regress the annual average of and the t- statistics of on. We consider both the Newey-West t-statistics for and the Valkanov t-statistic for as measures of the predictive power of. The former one is the most common measure in the literature while the latter one is a modified t-statistic in the consideration of the extreme persistence of. As shown in Table II, movements of, the Newey-West t-statistics for, and the Valkanov t-statistic for are tightly related with, regardless of portfolio types. For the aggregate stock market data, can explain 75%, 72%, and 85% of the movements of, the Newey-West t-statistics for, and the Valkanov t-statistic for, respectively. This evidence strongly suggests that the inconsistent findings about the forecast ability of in the literature are the consequence of the movements of. 15

16 B. Time-Varying Cointegrating Relationship between Dividends and Price Although the existence of non-payers causes to have an I(1) component, Proposition 2 states that there is a cointegrating relationship between and and that the cointengration coefficients are not fixed as [1, -1] but vary over time. This implication is investigated by testing the constant cointegration null hypothesis against the time varying cointegration between and. We have employed recently developed test methods by Park and Hahn (1999) and Bierens and Martins (2009). and Park and Hahn (1999) propose two types of test statistics which are given by (7) (8) where are the residuals of the regression of on superfluous regressors such as a constant,,, and is a long-run variance estimator of in equation (5). is based on the variable addition approach while is based on the residual-based method. The asymptotic distributions of and under the null hypothesis are discussed in Park and Hahn (1999). The first column of Table III presents the results of these tests with the full sample data. Both tests unanimously reject the null hypothesis of the constant cointegration coefficient for and in favor of the time-varying cointegration relationship for and. The time-varying cointegration relationship between and is evident for the aggregate market portfolio or all types of portfolios considered in this study. We also consider the likelihood ratio test for the time-varying cointegration relationship between and, which is proposed by Bierens and Martins (2009). The null 16

17 hypothesis is again that the cointengration coefficients are fixed as [1, -1] and the limiting distribution of the test statistic is a chi-square distribution. The lower panel of Table III shows the Bierens and Martins (2009) test results when the number of Chebyshev polynomials is four and the lag order is Again, we can find strong evidence that the constant cointegration relation is rejected in favor of time-varying cointegration relation regardless of portfolio types. We also conduct these types of tests, proposed by Park and Hahn (1999) and Bierens and Martins (2009), for sub-samples such as January 1946 December 1990, January 1946 December 1980, January 1961 December 1990, and January 1961 December Lettau and Nieuwerburgh (2008) argue that underwent shifts in its mean during the 1990s or 1950s and 1990s. They further argue that these shifts in the mean are a reason for the sensitive ability of to forecast stock returns. In addition, Boudoukh et al. (2007) and Robertson and Wright (2006) claim that the change in firms payout policy from dividend payment to the repurchase of shares has become considerably important roughly since the 1980s, and has resulted in a breakdown in the cointegration relation between the aggregate dividends and stock price. The sub-sample analyses are conducted to check whether the rejection of the constant cointegration null hypothesis is the consequence of one-time or two-time shifts in the mean of or the change in firm s payout policy. However, we have found that the constant cointegration null hypothesis has been greatly rejected in all of these sub-samples. This evidence suggests that the cointegration relation between and had evolved smoothly throughout the post World War II period. 11 To our best knowledge, no econometric theory for the optimal selection of the number of Chebyshev polynomials or the lag orders in the Bierens and Martins (2009) test has been developed yet. However, our results are qualitatively identical when the number of Chebyshev polynomials changes to three or five or when the lag order changes to 12. These results are available upon request. 17

18 After obtaining strong evidence for time-varying cointegration relationship between and, we estimate the time-varying cointegration parameters based on the FFF. We choose 9 in the FFF representation, implying that is approximated by,,. 12 As the estimates of the individual parameters in the FFF expansion are devoid of economic interpretation, the estimated for the aggregate and is plotted along with the CCR adjusted 95% confidence interval in Figure 2. As shown in the top-left panel in Figure 2, the cointegration coefficient, during the post World War II period, has a few large swings, but generally declines from 0.84 to 0.71 for the aggregate stock market data. These figures are significantly different from one which is the constant cointegration coefficient between and in the literature. These figures imply that 1% increase in dividends would have led to 0.84% (or 0.71%) increase in the longrun equilibrium stock prices at the beginning (or end) of the sample period, while the constant cointegration coefficient would predict 1% increase in stock prices in response to 1% increase in dividends. This decrease in the response of the long-run equilibrium stock price to dividends is due to the increase in the relative proportion of non-payers in the market. Are the movements of the estimated time-varying cointegration coefficient closely related with the fraction of payers (? The correlation between the annual average of the estimated time-varying cointegration coefficient and the fraction of payers is almost 0.85 for the aggregate stock market. Furthermore, when the annual average of the time-varying cointegration coefficient is regressed on a constant and the aggregate as shown in Table IV, has a highly significant coefficient and explains more than 70% of the movements in time-varying cointegration coefficient. The results are robust to the selection of in the FFF 12 To our best knowledge again, no optimal data-dependent rule for the selection of has been developed yet in the nonstationary time series literature. However, our results are similar when we set 7 or 11. The estimated time-varying cointegration parameters and the 95% confidence bands are provided in the Appendix. 18

19 representation. The estimated time-varying cointegrating coefficients and 95% confidence bands for portfolios formed on size are also plotted in Figure 2. The time-varying cointegrating coefficients are flctuating and declining over time as s for corresponding portflios do. The tight relation between and the time-varying cointegration coefficient can be also confirmed via the regression analysis for all types of portfolios, as presented in Table IV. Again, the results are robust to the choice of regardless of portfolio types. We also examine the possibility that the estimated time-varying cointegration relationship between dividends and stock price could make the conventional dividend-price ratio appear to undergo one or two-time shifts in the mean if an econometrician is not aware of the time-varying cointegration relationship. We first conduct the Bai and Perron s (1998) sup-f tests for in Table V to see whether monthly experiences structural breaks in the mean. The Bai and Perron s sup-f test statistic in Table V is denoted by Sup- F(i,j) where i is the number of breaks under the null hypothesis and j is the number of breaks under the alternative hypothesis. As shown in Table V, we can obtain strong statistical evidence for one or two shifts in the mean of regardless of portfolio types. Estimated break points are also provided in the table. The results with the aggregate are qualitatively the same as those with the annual data in Lettau and Nieuwerburgh (2008). We then bootstrap pseudo using the estimated time-varying cointegration coefficients for the aggregate data, and conduct the Bai and Perron s sup-f tests sequentially for the bootstrapped. 13 Figure 4 shows the histogram for the Bai and Perron test results with the pseudo at the 5% significance level. As shown in Figure 4, the Bai and Perron test indicates one or two-time shifts in the mean of the bootstrapped for 13 The bootstrap algorithm is provided in the Appendix. 19

20 more than 9,200 cases out of 10,000 bootstrapped series for pseudo, although they are generated by the estimated time-varying cointegration coefficients. This evidence strongly suggests that the finding by Lettau and Nieuwerburgh (2008) that experiences a few structural breaks in the mean may be the consequence of the time-varying cointegration relationship between dividends and stock price. V. Implications on Stock Return Predictability A. Time-varying Cointegration and Stock Return Predictability As can be written as the present value of observable and unobservable variables together in Proposition 3, we don t know whether has ability to forecast stock returns a priori from Proposition 3. Hence, we investigate whether stationary deviations from the time-varying cointegration relationship between and have any predictive power for future stock returns in this sub-section. Before running the predictive regression with as a regressor, we examine the time-series properties of first. Summary statistics for and are compared in Table VI. As found in the literature, the autocorrelation of is extremely close to one, and the unit root null hypothesis in the ADF test cannot be rejected for not only with the aggregate data but also with portfolios formed on any size. Although the autocorrelations of are also quite high but always lower than those of. Furthermore, the unit root null hypothesis can be greatly rejected in all cases for. Figure 3 plots the movements of along with those of. As shown in Figure 3, all series of appear much more stable and stationary than those of. Additionally, does not seem to undergo a structural break in the mean, while seems to have a shift in its mean as Lettau and Nieuwerburgh (2008) points out. 20

21 Hence, we conduct the Bai and Perron s (1998) sup-f tests for in the lower panel of Table V to see whether any adjustment in the mean is required. Unlike the structural break test results for, the null hypothesis that there is no structural break in the mean has never been rejected for the aggregate and the for any type of portfolios formed on size, which implies that we can safely utilize in the predictive regression without any adjustment in the mean. Table VII shows the results when is used in the predictive regression. 14 When the Newey-West t-statistic is used as a measure of the predictive power, has strong predictive power at all horizons from one-month horizon through 60-month horizon. rises from at one-month horizon to 0.25 at 60-month horizon. 15 This significant predictive power is robust not only to the aggregate data but also to all portfolios formed on size. However, is also persistent as shown in Table V, although is not as persistent as. This persistence of may cause distortions in the asymptotic distribution of the coefficient in the predictive regression and thus affect the results in Table VII, as pointed out in Stambaugh (1999) and Ferson, Sarkissian, and Simin (2003). Hence, we compute bootstrapped p-values for the coefficient of in the predictive regression, using the following regression model. (9) (10) is set equal to zero under the null hypothesis of no predictive power, and the residual vectors, are sampled with replacement to generate bootstrapped and. Then, the predictive regression (9) is run with the bootstrapped and to 14 The results when is used in the predictive regression are provided in the Appendix for the comparison purpose. We can note that from portfolio formed on small-size firms has the weakest predictive power for stock returns, which is probably due to the lowest fraction of payers for this portfolio. 15 with is merely at one-month horizon while it becomes 0.24 at 60-month horizon. 21

22 construct the bootstrapped distribution of which is compared with the actual estimate of. The results in Table VII shows that the possible distortion in the asymptotic distribution has little consequence for the predictive power of for stock returns in short horizons up to five months. The bootstrapped p-values are robustly below 5% over these short horizons for the aggregate data and the portfolio data as well. One concern about the results is that the coefficient in the predictive regression with might be unstable because Paye and Timmermann (2006) and Rapach and Wohar (2006) report that the predictive regression with underwent a structural break. Figure 5 compares the movements of the coefficient of with the movements of the coefficient of in predictive regressions with aggregate data. Predictive regressions of one-month ahead stock returns are first run on a constant and or using data from January 1946 to December Then, we run predictive regressions recursively adding one month observation at a time and re-estimate the coefficient for or. While the coefficients of have declined substantially and appear unstable during our sample period, the coefficients of is remarkably stable during our sample period. We also conduct Bai and Perron s structural break test for the predictive regression where is demeaned log stock returns and is demeaned or. The results are provided in Table VIII. Table VIII shows some evidence for unstable when the conventional is used as a regressor in the predictive regression. However, we cannot find any evidence for unstable when is used as a regressor in the predictive regression. These results are consistent with Figure 5 and suggest that the significant predictive power of is stable. 22

23 B. Out-of-Sample Predictive Power In order to construct in the previous sub-section, the time-varying cointegration parameters are estimated using the full sample. Since the use of full sample observation in the estimation process may cause the so-called look-ahead bias, one may wonder whether an investor at earlier dates would be able to find similarly strong predictive power from. We conduct the out-of-sample test to address this point. That is, the time-varying parameters for is re-estimated every period using the available data at that period of time. Then, the predictive regression of one-month ahead stock returns is run on the estimated recursively. That is, we add one month observations at a time for the estimation of parameters for and make a forecast for one-month ahead stock returns. The first predictive regression is run with the use of 30 year data since January The Diebold-Mariano (1995) test statistics is employed to compare the forecast ability of with that of the random walk model which is reported to have superior predictive power to in Goyal and Welch (2008). The loss function of the Diebold-Mariano test is absolute forecast error to mitigate the effect from outliers when comparing the forecast abilities. Table IX compares the forecast abilities among, and the random walk model. As shown in Table IX, the Diebold-Mariano test statistics are significantly positive in the first column, which indicates the superior predictive ability of the random walk model to that of the conventional regardless of the portfolio types. This is consistent with the results reported in previous studies such as Goyal and Welch (2008). However, when is used as a regressor in the predictive regression instead of, the Diebold-Mariano statistics becomes negative (portfolios formed on the aggregate stock market, medium-sized firms, and small-sized firms) or positive but smaller (portfolio formed on large-sized firms), which implies improvement of predictive power. Although the Diebold- 23

24 Mariano statistics are not significant for the portfolios formed on the aggregate stock market, large-sized and medium-sized firms, we can see significantly better forecast ability from for the portfolio formed on small-sized firms, compared with the random walk model. Furthermore, the strong improvement of the forecast ability compared with or the significantly superior predictive power compared with the random walk model for the portfolio formed on small-sized firms does not result from the recent turmoil in the stock market. Even after excluding the recent three year observations from the analysis, we can obtain qualitatively the same results, which are shown in the last two columns of Table IX. Then, why is the significant improvement of the predictive power from observed particularly well for the portfolio formed on small-sized firms but not for others? That may be the consequence of the weak power of the out-of-sample tests compared with the results from in-sample analyses because much fewer observations are used in the out-ofsample analyses (see Inoue and Kilian (2004)). In addition to this explanation, the change in firms payout policy from dividend payments to share repurchases has probably the weakest impact on the portfolio formed on small-sized firms because many small-sized firms have not paid out dividends nor repurchased shares for long time. Since a portion of firms with share repurchase can be considered as non-payers or payers which have reduced dividend payments, the distinction between payers and non-payers can be contaminated by share repurchases. As many small-sized firms have never paid out dividends, the distinction between payers and non-payers is relatively much clearer for the portfolio formed on small-sized firms than that formed on large-sized or medium-sized firms. As a result, the portfolio formed on small-sized firms has better environment for our model and methodology in this study to be applied, which could be another reason to explain the significantly negative Diebold-Mariano statistics in the case of the portfolio formed on small-sized firms. 24

25 VI. Conclusion In this paper, we have developed a new framework for understanding the extreme persistence of the conventional dividend-price ratio and its sensitive predictive power to the choice of sample periods. We show that the existence of firms which have not paid out dividends can make the conventional dividend-price ratio extremely persistent and have sensitive predictive power to the choice of sample periods. We also show that the cointegration relationship between dividends and stock price is not fixed as [1, -1] but varies over time due to the movements of the fraction of payers, firms which pay out dividends. Furthermore, we have developed a new way to examine empirically the implications presented in this paper. Our method is based on the time-varying cointegration regression approach to take account for the smoothly varying fraction of non-payers, and is more flexible in the sense it does not require additional share repurchase data. We have provided statistical evidence that the time-varying cointegration relationship between dividends and stock price exists, and have demonstrated that stationary deviation from the time-varying cointegration relationship has stable and strong ability to forecast stock returns from the outof-sample analysis as well as the in-sample analysis. 25

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28 Appendix 1. Propositions and Proofs Proposition 1. The log dividend-price ratio for the aggregate stock market can be written as 1. As a result, the log dividend-price ratio for the aggregate stock market contains an I(1) component unless and 1. (Proof) Since non-payers do not payout dividends,. The log aggregate stock price ( may be approximated as 1. Then the log dividend-price ratio can be written as follows. 1 1 If, is I(1). Therefore, is an I(1) variable. Proposition 2. The aggregate dividends and the aggregate stock price are cointegrated with time-varying cointegration coefficients. That is, there exists which makes be an I(0) variable where 1. (Proof) Define 1. Then, 1 λ where 1 and 1. 28