20.4.1 Empirical Application

20.4.1 Empirical Application

The question of whether the dividend yield predicts stock returns has generated considerable interest in the finance literature. Intuitively, we would expect the dividend–price ratio to mean revert at least at longer horizons and so this variable could act as a measure of whether stocks are expensive (small dividend–price ratio) or cheap (large dividend–price ratio). Valkanov (2003) is one study that focuses on return predictability from the dividend–price ratio.

The return prediction model considered by Valkanov is a special case of (20.10):

$$\begin{array}{r}{y}_{t+1}=\beta z_t+u_{1t+1},\\ z_{t+1}=\varphi z_t+u_{2t+1},\end{array}\text{\hspace{1em}(20.14)}$$

where $\varphi =1-\gamma /T$ and $\gamma$ is an unobserved nuisance parameter with $\gamma =0$ in the unit root case, while $\gamma <0$ for persistent regressors.

Define long-horizon variables $\begin{array}{r}{y}_t^h={\sum }_{i=0}^{h-1}{y}_{t+i},z_t^h={\sum }_{i=0}^{h-1}{z}_{t+i};\end{array}$ Valkanov considers long-run regressions on either “short” predictors $\left({\mathbf{z}}_t\right)$ or “long” predictors, $z_t^h$ , which cumulate the predictor over the previous $h$ periods:

$$y_{t+1}^h=\beta z_t+u_{1ht+1},\text{\hspace{1em}(20.15)}$$

$$y_{t+1}^h=\beta z_t^h+u_{2ht+1}.\text{\hspace{1em}(20.16)}$$

Valkanov assumes that the overlap, $h,$ is a constant fraction of the sample size, $h=$ $\lfloor \lambda T\rfloor ,\lambda \in (0,1)$ , and that the innovations $u_t$ are martingale difference sequences.

Under these assumptions, Valkanov shows that the estimate $\hat{\mathbf{\beta }}$ in regressions (20.15) of cumulated h-period returns on the single-period regressor, $z_t.$ , is not consistent and has a limiting distribution that depends on the unknown nuisance parameter, $\gamma .$ . Moreover, the t-statistic for $\beta$ does not converge to a well-defined distribution; rather, it diverges at rate $T^{1/2}$ . In practice, this means that longer horizons tend to produce higher t-statistics and that the $R^2$ -value from the predictive regression (20.15) does not converge in probability under the null.

TABLE 20.3:
Test statistics for predictive regressions undertaken for monthly stock returns. The table reports the slope coefficient on the dividend yield (column 2) in a regression of monthly stock returns on an intercept and the dividend yield. Also reported is the t-statistic for the slope coefficient using Newey–West HAC standard errors (column 3) along with the t-statistic scaled by the square root of the sample size, T. The sample uses monthly data over the period 1926–2013.

Conversely, the regression (20.16) that uses the long-run predictor $z_t^h$ leads to a consistent estimator of $\beta .$ Although the $t_{\hat{\mathbf{\beta }}}$ -statistic still does not converge to a welldefined distribution, the scaled statistic ${\dot{t}}_{\hat{\beta }}/T^{1/2}$ converges weakly, so tests can be based on simulated critical values for this test statistic. In fact, under the alternative that $\beta \ne 0,\hat{\mathbf{\beta }}$ is super-consistent and converges at rate T and the $R^2$ converges in probability to 1. These findings suggest that it is preferable to regress long-run variables (e.g., cumulative returns) on long-horizon predictors (cumulative dividend yields) to obtain consistent estimates of $\beta .$ .

As an illustration, we next consider results for an empirical application to monthly US stock returns. We use a linear model to predict stock returns by means of the dividend–price ratio at horizons $h=\{1,2,3,6,12,24,36,48,60\}$ months. Table 20.3 reports the test statistic recommended by Valkanov (2003) along with conventional t-statistics which, as we have seen, can be misleading. The highly persistent behavior for the dividend yield, together with the strongly negative correlation between innovations to stock returns and the dividend suggests that care should be exercised when interpreting the predictive regressions for this data. Indeed, although the conventional t-test appears to be highly significant at longer horizons,√ there is little evidence of predictability from the scaled t-test, $t_{\hat{\beta }}/\sqrt{T}$

20.4.2 Results for Multivariate Long-Run Forecasts

A small literature has extended the univariate results to multivariate models. Stock (1996) and Phillips (1998) provide results for vector autoregressions which may or may not have unit roots—or roots close to unity—with cointegrated or nearcointegrated variables. The main result is that the forecasts and forecast errors have distributions that depend on the local-to-unity parameters, as they do in the univariate problem shown above. This makes it extremely difficult to construct uniformly consistent forecasts and confidence intervals that uniformly cover the outcome of the predicted variable. Rossi (2005b) employs results for the multivariate model to set up tests based on the difference between random walk and estimated models. This involves taking the unknown values for the local-to-unity parameters into account. Corradi, Swanson, and Olivetti (2001) study the Diebold–Mariano test in settings with cointegrated variables.