We proceed to evaluation of in-sample performance of different models for dividend growth. We consider all three suggested adjustments and vary s (number of years used to estimate the model for dividend growth). Also in each case we consider the model with all available data used to estimate future dividend growth (denoted by s=*). All predictive regressions estimated with OLS in EViews. The results are summarized in Exhibit 1. It should be noted that t-statistics were computed with OLS standard errors without any adjustments.
However, it was reasonable, because neither heteroscedasticity nor autocorrelation was detected in fitted residuals in all regressions at the conventional significance levels (1%, 5%, 10%). The results for AR(1) and ARMA(1,1) models of dividend growth are presented only for s=*, because when we use rolling window approach, values and t-statistics are essentially zero (even when s=30).
Moreover, there is no evidence so far to use AR and ARMA models with time-varying coefficients and such poor performance confirms that we simply create noise when allow for time-varying.
At first glance, we can conclude three things. First of all, we see that and exhibit weak performance when we start the sample from 1937, but fit changes drastically if we shift starting point of predictive regressions to 1947. It is related to the fact that 10-20 observations are very low numbers to deliver stable estimates of AR and ARMA models. Therefore, between 1937 and 1947 both and show enormous variation, but then they stabilize. In turn, stabilizes much faster and performs good even in the long subsample.
Second, if we ignore the exception above, all three models clearly outperform simple predictive regression with regular log price-dividend ratio. Therefore, we confirm results of Lacerda and Santa-Clara (2010) and Koijen and van Binsbergen (2010). Finally, there is no model, which is obviously more parsimonious than others. Still it is worth mentioning that for s=* all three models almost replicate each other when we consider the short subsamples (from 1947).
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This table summarizes the results of estimation of different predictive regressions.
Each regression includes next period log return as a dependent variable, while four different subsamples are considered for each predictor (2007, 1937-2012, 1947- 2007 and 1947-2012). The detailed description of different adjustments for price-dividend ratio is presented in Section 2. "s" denotes a number of previous observations used for estimation of next period log dividend growth. "s=*" corresponds to the case when all available by the current moment observations are used for estimation of next period log dividend growth. For each predictive regression three values are presented: , slope and t-statistic. "*" inside the t-statistic's field reflects the significance at the 5%-level, while "**" denotes the significance at the 1%-level. The best predictor in terms of either or t-statistic is highlighted for each subsample (bold font).
Predictor s 1937-2007 1937-2012
Slope t-stat Slope t-stat
10 14.29% -0.0785 -3.3913** 11.94% -0.0775 -3.1682**
20 - - - - - -
* 7.60% -0.1102 -2.3821* 7.29% -0.1169 -2.4121*
* 1.70% -0.0408 -1.0929 1.77% -0.0454 -1.1546 * 0.38% 0.0126 0.5118 0.38% 0.0139 0.535
4.98% -0.0768 -1.9009 6.73% -0.0932 -2.3116*
Predictor s 1947-2007 1947-2012
Slope t-stat Slope t-stat
10 16.18% -0.0871 -3.3751** 13.27% -0.0865 -3.1290**
20 15.07% -0.1283 -3.2355** 12.20% -0.1237 -2.9819**
* 17.32% -0.1721 -3.5159** 15.78% -0.182 -3.4624**
* 16.88% -0.1673 -3.4610** 15.26% -0.1763 -3.3950**
* 17.50% -0.161 -3.5373** 12.73% -0.1514 -3.0556**
9.05% -0.0986 -2.4235* 10.90% -0.1165 -2.7981**