That is not my dog: why doesn t the log dividend-price ratio seem to predict future log returns or log dividend growth?

Transcription

1 That is not my dog: why doesn t the log dividend-price ratio seem to predict future log returns or log dividend growth? Philip H. Dybvig Washington University in Saint Louis Huacheng Zhang Southwest University of Finance and Economics First draft: August, 2016 Current version: June, 2018

2 That is not my dog: why doesn t the log dividend-price ratio seem to predict future log returns or log dividend growth? 1 By Philip H. Dybvig and Huacheng Zhang Abstract Campbell and Shiller s accounting identity implies that changes in the log dividend-price ratio must predict either future returns or future log dividend growth. However, neither quantity seems to be predictable a well-known puzzle in the literature. We examine this puzzle step-by-step from theoretical derivation through empirical testing. Stationarity of the log dividend-price ratio is an important assumption behind the accounting identity, but Campbell and Shiller s test justifying this assumption does not make sense, and a corrected test does not reject non-stationarity. Nonetheless, a truncated accounting identity works reasonably well in the existing sample, and we find that the log dividend-price ratio predicts log dividend growth, not returns. Unfortunately, this result does not seem to be robust to subsamples. Also, it seems unwise to rely too much on asymptotic properties of estimators when the entire sample includes only five non-overlapping observations. Key words: return predictability, dividend-price ratio, stationarity test. [JEL G12 G17] 1 Dybvig: Olin School of Business, Washington University in St. Louis, One Brookings Drive, St. Louis, MO 63130, USA, (dybvig@wustl.edu; Zhang: Institute of Financial Studies, Southwestern University of Finance and Economics, 55 Guanghuacun Street, Chengdu, , China (zhanghuacheng1@gmail.com. We thank to especially Jason (Huafeng Chen. We also thank to Michael Brennan, Pingyang Gao, Jun Liu, Zhongzhi Song, Rossen Valkanov, Russ Wermers, and participants at the SFS Cavalcade Asia-Pacific 2017 Annual Meeting (Beijing, the GCFC 2018 Conference (Xiamen, Southwestern University of Finance and Economics, University of Alabama, and Washington University at Saint Louis for helpful comments.

3 Clouseau: Does your dog bite? Innkeeper: No. Clouseau: Nice doggy. (Clouseau tries to pet the dog on the floor and is bitten Clouseau (angry: I thought you said your dog did not bite. Innkeeper: That is not my dog. 2 Often, like Clouseau, we get into trouble because we ask the wrong question. This paper examines the failure of the log dividend-price ratio (hereafter LDPR to predict either future log returns or future log dividend growth. Campbell and Shiller s (1988 accounting identity asserts that the current LDPR is approximately equal to a constant plus the sum of present values of future log returns minus the sum of present values of future log dividend growths. This implies that current LDPR should be able to predict future returns, or dividend growth rates, or both. Empirically, however, the LDPR seems to predict neither. This puzzle was examined by Cochrane (2008, who side-stepped the puzzle by assuming a just-identified model, using the analogy of the dog that didn t bark from Sherlock Holmes, but that is not our dog. Since, like Clouseau, we do not know which question to ask, we go step-by-step through the entire argument to uncover where the problem is. We find that: (1 The log-linear approximation works very well both in a single period and over 30 years. (2 Campbell and Shiller make a theoretical assumption that the long-term mean LDPR exists. They justify this assumption by an empirical test that unfortunately does not actually test this at all. Our correctly specified test fails to reject the null that the LDPR does not exist. (3 Even though the long-term mean of the LDPR seems not to exist, the approximation works reasonably well in the sample to date. In particular, discarding the final term has an impact, but the relationship is still strong without it. (4 Using a test with many lags as in the theory, 3 future log dividend growth is significantly predictable but expected returns are not. We show that this result is hard to uncover because the large prediction error introduced by the unpredictable part of future log returns is a common factor in future 2 Blake, Edwards (Producer & Director. (1976. The Pink Panther Strikes Again. United Kingdom: Amjo Productions Limited. 3 The test uses the appropriate Newey and West (1987 adjustment for heteroskedasticity and serially correlated errors and the Stambaugh (1999 adjustment for spurious regression bias. 1

4 log dividend growth but cancels in the accounting identity. (5 Although these results are significant, they do not seem to be very robust. In particular, the relationship is reversed significantly in the second half-sample (consistent with Chen (2009. Also, the estimation asks a lot of the Newey-West adjustment, since there are only about five nonoverlapping observations in the whole sample (and even fewer in subperiods. The lack of evidence for stationarity of the LDPR is also troubling for estimation. approximation in the accounting identity should get worse and worse over time. If the LDPR is not stationary, the The Campbell-Shiller accounting identity can be derived by starting with the single period definition of returns as the sum of dividends and capital gains. Using algebra, taking logs, and doing a Taylor series expansion around some typical value for the LDPR, 4 we obtain a one-period approximation formula linking log returns and log dividend growth with beginning- and end-ofperiod LDPR. By telescoping this approximation over many periods, current LDPR can be linearly approximated by the sum of weighted log returns and the sum of weighted dividend growth rates over future periods plus the LDPR in the final period. Using the annual nominal dividend payments and prices of the S&P500 index (with Shiller s backfill 5 using data from Cowles (1939 from 1871 to 2015, and performing a regression of the LDPR on the sums over future 30 years and the final log dividend-price ratio, all coefficients on independent variables are close to the theoretical value one (or minus one and the R 2 is close to 100% (98.91%. The approximation is worse but still acceptable if we drop the final LDPR in year 30 (the R 2 drops to 82.25%. Therefore, the source of the puzzle is the lack of power in previous tests rather than any intrinsic problem with the theory, at least in the truncated identity. Campbell and Shiller (1988 purport to show existence of the long-term mean of the LDPR using a flawed test of stationarity. They use an augmented Dickey-Fuller test in which the null is nonstationarity plus a trend, which implies nonstationarity and the long-term mean doesn t exist, and the alternative is stationarity plus a trend, which also implies the process is nonstationary and the long-term mean does not exist. Obviously, we cannot test for stationarity if both the null and alternative hypotheses implies nonstationarity. Instead, we conduct a conventional Dickey-Fuller stationarity test in which the null hypothesis is nonstationarity without a trend, which implies the 4 Campbell and Shiller expand around the long-term mean, but that is not necessary. 5 The backfill is described in Shiller (1981 and Campbell and Shiller (

5 long-term mean doesn t exist, and the alternative hypothesis is stationary without a trend, which implies the long-term mean does exist. Rejection of the null would be evidence that the long-term mean exists. We cannot reject the null of nonstationarity, which may mean the long-term mean does not exist, when we use the entire sample. In principle, this is a big problem for the Campbell-Shiller approximation, which is based on an expansion around the long-term mean, but the truncated version of the approximation still works well in our sample looking 30 years out. It is also a big problem for any asymptotic interpretations of the statistical tests, since nonstationarity of LDPR would mean that the approximation error in the derivation will become unbounded over time. We test the predictability of stock returns and dividend growth rate using an equation similar to the equation in the model rather than using one or a few lags as is common in the literature. Our estimation uses statistical corrections for the correlation in error terms and for spurious regression bias. The Campbell-Shiller approximation implies that the current LDPR is able to predict either future returns or dividend growth or both; we find that future log dividend growth is significantly predictable, but future returns are not. The results are robust when we expand the log dividend price ratio around alternative points rather than the sample mean. As noted by Cochrane (2008, dividends are smooth. He concludes that log dividend growth is not predictable (implying under the model restriction that returns are predictable. However, it is more accurate to assert that the predictability of log dividend growth is spread over many maturities and that nearby dividends are not very predictable because dividends are smooth. What is happening is that there is small predictability of dividend growth spread over many periods, which is buried by noise in conventional simple regression or vector-autoregressive (VAR estimation. This issue could be exaggerated in tests with one or few lags if the log dividend-price ratio is not stationary. The limitation inherent in using small lags to search for predictability of dividend growth seems to be a solution of the puzzle of why the theory (based on an accounting identity and an approximation that is not so bad in the current sample is hard to verify. In general, the predictability of log dividend growth is difficult to find because of the large prediction error introduced by the unpredictable part of future log returns, which is a common factor with future log dividend growth that cancels in the accounting identity. Although the best evidence (based on our whole sample suggests that the LDPR predicts log 3

6 dividend growth but not log returns, this result seems fragile. The result is not robust to subperiods, consistent with Chen (2009 and explaining an apparent inconsistentcy with a similar regression of Cochrane (2008, Section 7.2. We also worry about the statistical properties of the estimators, both because the whole sample has only about five non-overlapping observations (and subsamples even fewer and because of the apparent instability over time. If the LDPR is not, in fact, stationary, the problems will become bigger as the sample size gets larger because the Taylor series expansion will become much less accurate as the range of values increases over time. One interesting aspect of the accounting identity is that it is not an economic model since its derivation uses only manipulation of identities and approximations. If we think about the economics, Modigliani-Miller suggests that to first order dividends are irrelevant, which is consistent with instability of these relationships over time. Therefore, both statistical and economic arguments suggest that asymptotic justification of estimators do not apply. The rest of this paper is organized as follows. We review the approximation leading to the accounting identity in Section 1 and test the quality of this approximation in Section 2. We use a model-implied approach to test the predictability of returns and dividend growth in Section 3. In Section 4, we analyze whether the failure of the LDPR to predict stock returns is caused by noise in itself or noise introduced by the modeling procedure. Section 5 conducts robustness analysis and Section 6 concludes. 1 Dividend-Price Decomposition We begin by specifying the standard definition relating return, current and future prices, and dividend payment. Define gross investment return over one period as: (1 1 + R t+1 = +1 + D t+1 = +1 ( 1 + D t+1, +1 4

7 where and +1 denote start-of-period and end-of-period stock prices, R t+1 denotes the net return over the period, and D t+1 denotes the end-of-period dividend. 6 This may seem like a strange way to write the return, since we would normally look at gross capital gains +1 / and dividend yield D t /, with information known at the beginning of the period in the denominator. For our purpose, we simply manipulate accounting identities to express returns in this unconventional manner because placing in both denominators would give us a telescoping series in which the final term cannot vanish. Taking logs on both sides, (1 becomes: (2 log(1 + R t+1 = log ( Pt+1 + log(1 + δ t+1, where δ t+1 log(d t+1 /+1. We approximate (2 by a first-order Taylor series expansion around δ t+1 = δ. Traditionally, the constant δ is taken to be the long-term mean of the log dividend-price ratio, log(d t /, but we will take a broader view and view δ as taking on some reasonable value. This distinction could be important, given the evidence later in this section that the long-term mean may not exist. Letting ρ 1/(1 + exp(δ, then d log(1 + exp(δ t+1 dδ t+1 δt+1 =δ = exp(δ t exp(δ t+1 = 1 ρ. δt+1 =δ Therefore, letting κ log(1 + exp(δ (1 ρδ, the Taylor approximation is: (3 log(1 + R t+1 = log ( Pt+1 + log(1 + exp(δ t+1 log( +1 + log(1 + exp(δ + (1 ρ(δ t+1 δ = log(+1 log( + log(1 exp(δ +(1 ρ(log(d t+1 log(+1 (1 ρδ = κ + ρ log(+1 + (1 ρlog(d t+1 log(. 6 For empirical tests, we might treat all dividend payments during the period as coming at the end of the period. Alternatively, we could try to construct a more accurate return calculation that takes into account the timing of the dividends and the returns within each period. In practice, these approaches are likely to yield very similar results. Empirical work must also account for splits and distributions other than dividends, although the data we use has already made these adjustments and including them in our notation here would not change the substance of the analysis. 5

8 It is possible to take an theoretical approach to analyze this approximation, but instead we will take an empirical approach to determine how well the approximation works in the available data. We can rewrite (3 as (4 log ( Dt κ + log(1 + R t+1 + ρ log ( Dt+1 +1 log(d t+1. We can use (4 itself for t = t +1 to substitute in for the term ρ log(d t+1 /+1 on the right-hand side. Doing this repeatedly (for t = t + 1,t + 2, T 1, the expression telescopes to become: (5 log ( Dt κ 1 ρ (1 ρt t + T s=t+1 ( ρ s t 1 (log((1 + R s log(d s + ρ T t DT log. P T This is the essential formula that we will work with. Campbell and Shiller argue that, if the dividendprice ratio is stationary and ρ < 1, the final term should vanish as T increases, and we have the asymptotic expression: (6 log ( Dt κ 1 ρ + ρ s t 1 (log((1 + R s log(d s, s=t+1 often referred to in the literature as the accounting identity. This identity states that, subject to the quality of the approximation, today s log dividend-price ratio log(d t / is identically equal to a constant plus a linear combination of future log returns log(1+r s and future changes in log dividend log(d s. This implies that the log dividend-price ratio must predict one or both of them. The justifications of the LDPR approximation and the predictability of return and dividend growths rely on the existence of the long term mean of the log-dividend price ratio series. literature typically assumes that the LDPR is stationary and can be expanded around its long-term mean. Figure 1 shows that the time series of both the annual dividend-price ratio for the S&P 500 index (dash line and the corresponding log ratio (solid line between 1871 and 2015 are much smaller towards the end than in the first half of the sample. 7 The Specifically, the dividend-price ratio varies around 5% during the Campbell-Shiller period ( , but declines to around 2% during the post 7 The data was obtained from Robert Shiller s website at The early observations are backfilled using data collected by Cowles (1939; see Campbell and Shiller (1988 for details. We focus on annual data because monthly dividend payments are linearly interpolated from annual and quarterly dividend payments, and we do not want to deal with the approximation error this might entail. 6

9 Campbell-Shiller period ( In short, Figure 1 suggests that the long-term means of the dividend-price and log dividend-price ratios may not exist. We conduct an appropriate stationarity analysis to formally test this. Figure 1: Time Series of Dividend-Price Ratio and Log Dividend-Price Ratio Here we call our test appropriate because Campbell and Shiller do not correctly test for the existence of the long-term mean: the long-term mean does not exist under either their null or their alternative hypothesis because both hypotheses include trends. To correctly test the existence of the LDPR s long-term mean, we use the conventional Dickey-Fuller stationarity test, in which the null is that the LDPR is a non-stationary process. The stationarity test is specified as log(d t / = α + β log(d t 1 / 1 + ε t and the Campbell-Shiller trend is absent. The results are reported in Table 1. The Dicky-Fuller statistic is over the whole sample period, over the Campbell- Shiller sample period, and -4.8 over the post Campbell-Shiller period, and the corresponding critical values at the 5% level are -16.3, and -14.6, respectively. In short, we fail to reject the hypothesis that the LDPR is a non-stationary series, and nonstationarity would imply the long-term mean LDPR does not exist. It will certainly be a problem over time if the long-term mean does not exist and the LDPR gets more and more dispersion that will make the Taylor approximation disintegrate. This certainly weakens the interpretation of the sample mean as the long-term mean, but it doesn t necessarily invalidate the analysis using the current data. In the following analyses, we investigate 8 Over the sample period, the average dividend-price ratio is 4.47% (corresponding to a ρ of 0.95 with a standard deviation of 1.52% while the average log dividend price ratio is with a standard deviation of

10 what we can learn about the approximation and predictability with the nonstationary LPDR data series. Table 1: Stationarity Tests This table reports the results of three regressions testing whether the annual series of the log dividend-price ratio of the S&P 500 index is stationary over the whole sample period, the Campbell-Shiller period, and the post Campbell-Shiller period, respectively. The stationarity test is specified as log(d t / = α +β log(d t 1 / 1 + ε t. The whole sample period is from 1871 to 2015 and the Campbell-Shiller period is from 1871 to log( D P t (Entire sample Campbell-Shiller Post Campbell-Shiller α (0.06 (0.09 (0.15 β (0.04 (0.07 (0.09 Dicky-Fuller stat Dicky-Fuller critical N Adj R Reject unit root No Yes No 2 Approximation Test In this section, we test the quality of the LDPR approximation in (5 using the current data. Campbell and Shiller (1988 suggest a vector autoregression (VAR approach to test (5 without the final term. They find that the LDPR series is persistent and able to predict both future stock returns and future dividend growth, but the associated R 2 s in their tests are small. We replicate and confirm their results. Unfortunately, the VAR approach suffers several shortcomings. A VAR procedure with a limited number of lags imposes a restriction that does not sufficiently capture the long-term relationship among current dividend-price ratio, future returns and future dividend growth rates. Cochrane (2011 also shows that VAR estimates can be biased and significantly different from those in the true linear regressions. In general, the analysis in Campbell and Shiller (1988 does not tell us whether (5 holds empirically. 8

11 Table 2: Approximation of the Log Dividend Price Ratio (LDPR This table reports the results of a regression testing the accuracy of the approximation of log dividend-price ratio in (5. The results are based on the annual prices of and dividend payments on the S&P 500 index from 1871 to (T t is set to be 30 years. The associated Newey and West (1987 standard errors, computed using four lags, are in parentheses. denotes statistical significance at the 1% level. Model 1 Model 2 α (0.05 (0.08 β (0.02 (0.08 β (0.02 (0.08 β (0.05 N Adj-R 2 (% An improved approach that avoids such shortcomings is to conduct a true linear regression of log dividend-price ratio on 2(T t terms of discounted log return and dividend growth plus one final term. This procedure, however, is burdensome and may not be implementable when (T t is large and the sample period is not sufficiently long. In this study, we propose a parsimonious regression in the form of (5, that is, regressing the current LDPR on the sum of weighted future returns, the sum of weighted dividend growth rates and the log dividend-price ratio in the last period: (7 log ( Dt = α + β 1 ( T s=t+1 ρ s t 1 (log(1 + R s ( + β 3 ρ T t log( D T + ε t. P T + β 2 ( T s=t+1 ρ s t 1 log(d s By construction, this regression overcomes the shortcomings of both conventional linear and VAR estimations and is more parsimonious. If the approximation of (5 is effective, we should expect that the estimated β 1 and β 3 in (7 to have values close to one and β 2 close to minus one. The corresponding R 2 should be close to 100%. 9

12 We take (T t to be 30 years, which is reasonably long and gives us 115 overlapping observations (years for analysis. The results are reported in the first column of Table 2 and suggest that the LDPR approximation in (5 is effective. The coefficient on the sum of discounted returns is positive and close to one, and the coefficient on the sum of discounted dividend growth is approximately equal to minus one. All coefficients are statistically significant at the 1% level and the corresponding R 2 is as high as 99%. This regression uses Newey-West standard errors to adjust for serial correlation (including that due to overlapping observations and heteroscedasticity. 9 The high R 2 suggests that current LDPR predicts at least one regressor but does not indicate which one(s. Note that we have not adjusted for spurious regression bias (caused by low frequency series on both sides. For now, it suffices to note that the fit is very good. We will correct for spurious regression bias when we conduct predictive regressions. To address the concern raised by Kleidon (1986, Marsh and Merton (1986, and Merton (1987, we further test whether the final term in (5 is small by repeating the analysis on (7 after dropping this term. The results are reported in the second column of Table 2 and suggest that the log dividend-price ratio in the final period (30 years from now is neither trivial nor particularly large. The coefficients on the sum of discounted returns and dividend growth are still significant but are still close to the theoretical values (1 or -1, and the R 2 drops significantly by 17%, from 99% to 82%, evidence that (7 is a good specification for empirically estimating the log dividend-price ratio. 3 Predictability Test Swapping sides of (5 suggests a predictive relationship between the current LDPR and cumulative future log returns or cumulative log dividend growth rates (see, for example, Campbell and Shiller, 1988; Cochrane, Furthermore, (5 suggests that the true predictive tests should be conducted by reversing the dependent and independent variables in (7: (8 T ρ s t 1 (log(1 + R s = α + β 1 log s=t+1 ( Dt + µ T. 9 We report the Newey-West standard errors with 4 lags. In an untabulated analysis, we find that the Newey-West standard errors based on 10, 20 or 30 lags are similar. 10

13 This type of regression is presented in Cochrane (2008, although his regression is based on a subset of our sample period ( rather than ; Cochrane s sample period is similar to the second half of ours; we thus also conduct analyses over our half-samples. A significant β 1 suggests that the LDPR predicts future returns. A similar specification with the weighted average of future log dividend growths (resp. final term on the LHS tests whether the LDPR predicts future log dividend growth rates (resp. final term. The use of cumulative present values of the predicted variable in future periods has advantages over a conventional predictive specification, in which one-period leading predicted variable is mostly used, in that it can capture the total predictability of future returns or dividend growth. In other words, (8 captures both short-run and long-run return predictabilities (if any. Given that dividends change slowly, it makes sense that any predictability of log dividend growth would be mostly at long lags, which is consistent with our findings. We use two adjustments to the inference in the predictability tests: a Newey and White (1987 adjustment of the standard errors for heteroskasticity and serial correlation of the errors, and a Stambaugh (1999 adjustment of the coefficients for spurious regression bias (SRB. Serial correlation of the errors is likely to be present given the moving averages used in the estimation, and might be present even without the moving averages. We report the Newey-West standard errors with 4 lags in Table 3. The Newey-West standard errors based on 10, 20 or 30 lags are similar. SRB, studied by Granger and Newbold (1974, Stambaugh (1999, and Ferson, Sarkissian, and Simin (2003, is a small sample bias for linear regressions with lagged stochastic regressors. Stambaugh shows that this bias is pronounced in the predictive coefficient but not in the standard error of the predictive coefficient or the R 2. By assuming log( Dt to be a first-order autoregressive process ( as log( Dt = c + τ log Dt ν t, Stambaugh shows that the magnitude of SRB in the predictive coefficient in (8 equals σ µν ( 1+3τ σν 2 N, where σ µν is the covariance of µ t and ν t, σν 2 the variance of ν t, and N the number of observations of the sample. Table 3 contains the predictability tests using the Newey-West and Stambaugh adjustments. Interestingly, the Stambaugh adjustment does not seem very important for these tests. Equation (5 implies that the sum of the coefficients on the log dividend-price ratio (β 1 s across all three predictive tests (i.e. predictability tests of T s=t+1 ρs t 1 (log(1 + R s, T s=t+1 ρs t 1 log(d s (with 11

14 Table 3: Predictability Test This table reports the results of three regressions testing whether current log-dividend-price ratio is able to predict the sum of discounted future returns, the sum of discounted future dividend growths, or the discounted log dividend-price 30 years from now. Panel A contains the results based on the annual S&P 500 index data from 1871 to Panels B and C report the results over equal non-overlapping periods, respectively. The spurious regression bias (SRB is estimated following Stambaugh (1999. The associated Newey-West standard errors with four lags are in parentheses., and denote statistical significance at the 1%, 5% and 10% levels, respectively. Predicted variable α β 1 SRB-adjusted β 1 Adj-R 2 (% Panel A: Whole Same Period T s=t+1 ρs t 1 (log(1 + R s (0.76 (0.24 (0.24 T s=t+1 ρs t 1 log(d s (0.66 (0.20 (0.20 ρ T t log( D T P T (0.19 (0.07 (0.07 Panel B: First Subsample Period ( T s=t+1 ρs t 1 (log(1 + R s (0.86 (0.29 (0.29 T s=t+1 ρs t 1 log(d s (0.91 (0.31 (0.31 ρ log( T t DT P T (0.16 (0.05 (0.05 Panel C: Second Subsample Period ( T s=t+1 ρs t 1 (log(1 + R s (0.39 (0.13 (0.13 T s=t+1 ρs t 1 log(d s (0.23 (0.07 (0.07 ρ T t log( D T P T (0.20 (0.07 (

15 flipped sign and ρ T t log( D T P T should be one if our predictive specification is exact. Over the sample period from to 2015, Panel A in Table 3 shows that the sum of the LDPR coefficients from the three predictability tests is indeed close to one, (0.19 ( = 0.96, and the corresponding sum of SRB-adjusted predictor coefficients is similar (0.90. More interestingly, Panel A shows that the LDPR does not predict returns significantly, but it does predict log dividend growth significantly and the final term. It may be surprising that the LDPR s predictive ability 30 years out in Panel A of Table 3 is both economically and statistically significant, leading us to ask what we know now about what will happen 30 years in the future. 10 This view might be compelling if we took the dividend process as exogenous, but as Modigliani and Miller (1958 point out, dividends are somewhat arbitrary (and in their model almost completely arbitrary. Although our new information today may be primarily about cash flows in the coming ten years, this cash may go into repurchasing shares rather than paying dividends, with the actual dividend increase spread slowly over decades. The predictability of the LDPR 30 years out only depends on (1 the predictability of cash flows over a short horizon, and (2 firm policies implying that it takes a very long time for these increased cash flows to appear in dividends. All of this is consistent with the smoothness of dividends as noted by Lintner (1956 and others. We further split the whole sample period ( into two equal-long subsample periods and repeat our predictability tests for each. Panels B and C in Table 3 present the results for both subsample periods, which correspond to the results in Table 3 for the whole sample. In the first subsample, covering the period from 1871 to 1928 (Panel B, we see that cumulative discounted dividend growth rates are significantly predictable by the current LDPR but cumulative discounted returns are not, consistent with the results over the whole sample period. In this subperiod, the coefficient of the LDPR for the predictability of the LDPR in 30 years is negative and insignificant, in contrast to the significant positive coefficient in the whole sample. The results for the second subsample from 1929 to 1985 (Panel C are almost contradictory to the results for the whole sample. In this subsample, the cumulative discounted dividend growths are not predictable by the LDPR while the cumulative discounted log returns are significantly predictable. We increase the Newey- West correction substantially from 5 to 30 lags and the standard errors are almost unchanged. These 10 We must, of course, grant that the entire economic system may change over 30 years and the stock market may not exist. 13

16 results are consistent with a similar test in Cochrane (2008, Section 7.2 on a similar sample period. Unfortunately, our subperiods have even fewer nonoverlapping observations (about 2 1/2 instead of about 5 than the whole sample period regressions presented in Section 3. It seems unrealistic to think that economic relationships will remain stable over 30 years (Chen, 2009 let alone 100+ years, which is a weakness of the whole literature. Certainly dividend policy has changed over time, we found evidence of decreasing dividend payouts in the test of stationarity of the LDPR in Section 2 and disappearing dividends have also been documented by DeAngelo, DeAngelo and Skinner (2004, and Brav, Gramham, Harvey and Michaely (2005. This could be why the final term is significant in the second period but not the first. As Modigliani and Miller (1958 emphasize, in a frictionless world, the mix between dividends and share repurchases would be irrelevant, and probably we should not expect dividend policy to be stable in the actual economy. However, it seems unlikely that markets were very efficient during the unstable periods in the first half of the sample but inefficient later. The reversal of the results in the subperiods leads us to wonder about the size of our tests (both on the whole sample and on the subperiods. Of our whole sample of 145 years and discounted weighted sums over 30 years, we only have about five non-overlapping observations in the whole sample and even fewer over each subperiod. The lack of robustness to subperiods may also be due to model instability over time. 3.1 Why is Predictability Much Weaker than the Approximation? It is worth exploring why the coefficient on stock returns in Table 2 is consistently close to one and statistically significant, while the predictor coefficient (on the LDPR in the stock return predictability test in Table 3 is small and insignificant. After all, in a univariate regression, the standard algebra implies the R 2 is unchanged if you interchange dependent and independent variables. Furthermore, since the univariate and multivariate regression coefficients are the same when the independent variables are uncorrelated, we know the explanation must come from correlation between the independent variables. The explanation lies in the high correlation between the weighted sum of future stock returns and the weighted sum of future dividend growth rates rather than any information in the LDPR about future stock returns. We can think of the significant coefficient on future returns in Table 2 as a correction to the future log dividends (by removing common noise rather than any 14

17 correlation between today s LDPR and future returns. To illustrate this argument, let s start with the assumption that the weighted sum of future log returns is just equal to some noise Z that is uncorrelated with the LDPR: (9 T ρ s t 1 (log(1 + R s Z t. s=t+1 Now, use this expression and (5 without the final term and ignoring the constant to approximate the weighted sum of future dividend growth rates: (10 T ρ s t 1 log(d s log( D t + Z t. s=t+1 Then the covariance matrix between the LDPR and sum of discounted future dividend growth rates is: (11 var ( log ( Dt, T s=t+1 ρ s t 1 log(d s = σ δ 2 σδ 2 σδ 2 σδ 2 + σ Z 2 When we regress current LDPR on the sum of discounted future dividend growth rates alone, the regression coefficient (ignoring estimation error is β = σδ 2/(σ δ 2 + σ Z 2. The coefficient is biased towards zero compared to what it would be without the noise Z t (the standard errors-in-variables result, and when the noise σ 2 Z in stock returns is large compared to σ 2 δ. (which is consistent with the data, then the bias is large. However, if we run the LDPR on both the weighted sum of future returns and the weighted sum of future dividends, we obtain a coefficient of 1 on the dividend sum and a weight of 1 on the return sum (a perfect fit given our approximations (9 and (10. Including returns allows the regression to cancel the noise in the dividend sum. To confirm the common factor in returns and dividends, let s look at this both theoretically and empirically. Given (9 and (10, we have that: (12 var( T s=t+1 ρ s t 1 (log(1 + R s, T s=t+1 ρ s t 1 log(d s = σ δ 2 σ 2 Z σ 2 Z σ 2 δ + σ 2 Z. 15

18 (13 corr( T s=t+1 ρ s t 1 (log(1 + R s, T s=t+1 ρ s t 1 log(d s = σ Z 2 σz 2 + σ δ 2. When σz 2 is large compared to σ δ 2, the correlation between the return sum and the dividend sum should be large. Figure 2 shows that the evolutions of cumulative log returns and dividend growth rates are closely correlated. In fact, the correlation between the log return and the log dividend growth is 0.63, and the correlation between ( T s=t+1 ρs t 1 (log(1 + R s and ( T s=t+1 ρs t 1 log(d s is Moreover, the standard deviations of the cumulative returns and log dividend growth rates are respectively 38.1% and 34.3% which implies that the sum of the two terms variances is as high as 37.2%, or 61.0% in terms of standard deviation, while the standard deviation of the log dividendprice ratio over the same period is 23.6%. This relationship is also observed by Ferson, Sarkissian and Simin (2003 and Valkanov (2003 with simulated data. Figure 2: Time Series of Cumulative Discounted Returns and Dividend Growth Rates One concern about the results in this section is that our sample includes only about five nonoverlapping observations of the weighted average of log differenced dividends. Although it is impressive that the estimates (with Newey-West and Stambaugh corrections are significant in spite of this, this puts a lot of demand on the Newey-West adjustment and we are far from its asymptotic

19 justification. 4 Alternative Expansion Points In the log linear approximation in (3, we approximate log(1+exp(δ t+1 around some value δ using a first-order Taylor expansion. After telescoping this expression and dropping the final term, we arrive at (6 or its finite horizon version. In this section, we examine the impact of the approximation error and dropping the final term on the approximation and predictability tests, with a special focus on how the error depends on the expansion point δ. 4.1 Single Period Taylor Approximation Error We first look at the possible magnitude of the approximation error in a single period. Note that (3 is based on a first order Taylor expansion of δ t+1 around some constant, δ. The approximation error in the Taylor expansion is (14 ξ t = (1 ρ(δ t+1 δ (log(1 + exp(δ t+1 log(1 + exp(δ, where ρ 1/(1 + exp(δ. The approximation error ξ t is zero when δ t+1 = δ and negative everywhere else. Given δ, ξ t is a concave function of δ t+1 that gets more negative as δ t+1 moves away from δ. 11 To see how ξ t is influenced by the selection of expanding point δ as well as the LDPR (δ t+1, we plot the relationship of ξ t as a function of δ t+1 for different δ, theoretically as well as using the data. The data plot on the theoretical curves because we have an exactly expression for the error. Specifically, we consider expanding δ t+1 around its sample mean, which is around 3.18 (the corresponding dividend-price ratio is 4.47%, as well as four alternative expanding points to take into account the declining trend in the dividend-price ratio: 2%, 3%, 7% and 8%, which correspond to the expanding points of LDPRs of -3.91, -3.51, and The results are illustrated in Figure 3. Regardless of expanding points, Figure 3 shows that the approximation error ξ t is close to zero when the LDPR is close to the 11 When δ t+1 is far from δ, ξ t is roughly affine, with slope 1 ρ if δ t+1 δ and slope ρ when δ t+1 δ. 17

20 expanding point. However, Figure 3 also shows that ξ t is far from zero if the LDPR deviates too much from the expanding point. Figure 3: Time Series of Approximation Error 4.2 Multiple Period Approximation Error and Dropping the Final Term In the case of approximation over multiple periods as shown in (5 or (6, we define the approximation error as the following: (15 ζ t = κ T 1 ρ + ρ s t 1 (log((1 + R s log(d s log( D t, s=t+1 where ρ 1/(1 + exp(δ and κ log(1 + exp(δ (1 ρδ. (15 and (6 suggest that ζ t contains two sources of errors: the cumulative approximation error in the sense of (14, and the omission of the final term in (5, ρ T t log( D T P T. When δ approaches negative infinite and ρ approaches one, the weight on the final term, ρ T t, tends to be one, which may lead to a pretty big ζ t. Table 4 gives us some idea how much deterioration we can expect in the approximation if the LDPR continues to wander away from the past values in the next one or two hundred years. Overall, the approximations still seem useful. The most striking problem is that as δ falls, omitting 18

21 the final term has more and more impact. In Table 5, we see that changing the expansion point δ has little impact on the predictability regressions. 5 Conclusion Whether stock returns are predictable is an important and challenging question for both academia and industry. Campbell and Shiller (1988 argue, based on accounting definitions and some approximations, that the log dividend-price ratio must predict future returns, future log dividend growth, or both. However, in past literature neither prediction has been found to be economically or statistically significant, creating a well-known puzzle. We check each step of Campbell and Shiller s argument, from the accounting definition through the approximation to the statistical tests. Our findings show that the source of the failure to find a significant relationship arises from a mismatch between the small lags in the traditional tests and the many terms in the theoretical expression. When we conduct a test closer to the theoretical expression, with appropriate correction for serial correlation due to overlapping data, possible heteroscedasticity, and spurious regression bias, we find that future log dividend growth is significantly predictable but future returns are not, thus resolving the puzzle. While this is the best conclusion given the data currently available, this result does not seem to be robust for several reasons. For one, there are only a few (about five non-overlapping observations of the truncated identity for the whole period, so we are asking a lot of the Newey-West adjustment. Also, the results are different over two subperiods, which calls into question our reliance on asymptotic properties of the statistical estimates. Perhaps we should not expect stability of the dividend process over time, since, according to Modigliani and Miller, dividends are irrelevant. Even if Modigliani- Miller s arguments should not be taken too literally, they do mean that seemingly small changes in taxes or transaction cost can have a big impact on dividend policy and affect the time series properties of log returns, log dividend growth, and log dividend-price rations. Possible nonstationarity, which we cannot reject for the whole sample or for the second half of the sample, is a serious problem for the theory because the Taylor series approximation worsens as the range of the LDPR increases. For these reasons, it seems that the limitations of this approach may be intrinsic, and the accounting identity may never tell us much about return predictability, even as we collect more and more data. 19

22 Table 4: Approximation Test: Alternative Expansion Points This table reports the empirical results of the Taylor expansion of log dividend-price ratio around alternative points. The regression is specified as: log(d t / = α + β 1 ( T s=t+1 ρs t 1 (log(1 + R s + β 2 ( T s=t+1 ρs t 1 log(d s + β 3 (ρ T t log(d T /P T + ε t. The results are based on annual data from the S&P 500 index from 1871 to The (T t is set to be 30 years. The associated Newey-West standard errors with four lags are in parentheses. denotes statistical significance at the 1% level. α β 1 β 2 β 3 Adj-R 2 (% Panel A: Expanding point: (ρ 0.98, D/P = 2% Model (0.08 (0.04 (0.04 (0.05 Model (0.07 (0.08 (0.08 Panel B: Expanding point: (ρ 0.97, D/P = 3% Model (0.04 (0.02 (0.02 (0.05 Model (0.09 (0.07 (0.08 Panel C: Expanding point:-2.66 (ρ 0.94, D/P = 7% Model (0.08 (0.04 (0.04 (0.14 Model (0.07 (0.09 (0.08 Panel D: Expanding point:-2.53 (ρ 0.93, D/P = 8% Model (0.09 (0.05 (0.05 (0.22 Model (0.06 (0.09 (

23 Table 5: Predictability Test: Alternative Expansion Point This table reports the empirical results of whether current log dividend-price ratio is able to predict sums of discounted future returns or discounted dividend growth rates, or the discounted final-period log dividend-price ratio. The results are based on annual data from the S&P 500 index from 1871 to The spurious regression bias (SRB is estimated following Stambaugh (1999. The associated Newey-West standard errors with four lags are in parentheses.,, and denote statistical significance at the 1%, 5%, and 10% levels, respectively. Predicted variable α β 1 SRB-adjusted β 1 Adj-R 2 (% Panel A: Expanding point: (ρ 0.98, D/P =2% T s=t+1 ρs t 1 (log(1 + R s (0.98 (0.32 (0.32 T s=t+1 ρs t 1 log(d s (0.86 (0.27 (0.27 ρ log( T t DT P T (0.47 (0.17 (0.17 Panel B: Expanding point: (ρ 0.97, D/P=3% T s=t+1 ρs t 1 (log(1 + R s (0.90 (0.29 (0.29 T s=t+1 ρs t 1 log(d s (0.79 (0.25 (0.25 ρ T t log( D T P T (0.35 (0.12 (0.12 Panel C: Expanding point: (ρ 0.94, D/P=7% T s=t+1 ρs t 1 (log(1 + R s (0.67 (0.21 (0.21 T s=t+1 ρs t 1 log(d s (0.58 (0.18 (0.18 ρ log( T t DT P T (0.18 (0.04 (0.04 Panel D: Expanding point: (ρ 0.93, D/P=8% T s=t+1 ρs t 1 (log(1 + R s (0.64 (0.20 (0.20 T s=t+1 ρs t 1 log(d s (0.54 (0.17 (0.17 ρ log( T t DT P T (0.08 (0.03 (

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