Size and power in tests of return predictability

Abstract

We study the size-power tradeoff of commonly employed tests of return predictability. For short horizon tests, we show analytically that the indirect dividend test is asymptotically more powerful than the direct return test when dividend growth is less volatile than returns, as appears to be true in the data. The asymptotic power advantages of the dividend test carry over to small samples. Asymptotically, the relative power of the short vs long-horizon return test may depend on size. For empirically relevant parameter values the short-horizon return test is asymptotically more powerful than the long-horizon test at the 1% level but the reverse is true at the 5% and 10% levels. Monte Carlo analysis indicates that, in small samples, the long-horizon return test is more powerful than the short-horizon return test for all sizes. The differences in the relative power of the tests in the small sample case is traced back to the correlation structure of the underlying shocks.

Keywords:

• Power
• Dividend-growth regressions
• Long-horizon regressions
• Return predictability
• Market efficiency

JEL Codes:

• G12
• G14
• C32

Acknowledgments

We would like to thank Sebastian Kripfganz for discussions and comments that helped significantly improve the paper. We would also like to thank Serena Ng, Kieran Walsh and seminar participants at Delhi School of Economics. All errors are our own.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 This view, however, is not without its critics. For example, Boudoukh et al. (2006) showed that, when returns are unforecastable, sampling variation alone can generate estimated return forecasting coefficients that increase proportionally with horizon.

2 Stambaugh (1999) shows how the skewness of the OLS estimator depends on the correlation between the shocks to the regressand and shocks to the predictor variable.

3 Papers that focus on size distortions include Goetzmann and Jorion (1993), Nelson and Kim (1993), Mankiw and Shapiro (1986), Stambaugh (1999), Torous et al. (2004), Boudoukh et al. (2006), Campbell and Yogo (2006), Ang and Bekaert (2007), among others.

4 Hodrick (1992) has some discussion of power under the alternative hypothesis of serially correlated returns.

5 The analysis remains tractable even if ${\text{β}}_d=0.0039$, as in the data. The only difference is that, in that case, one needs to account for the fact that the variance of the short-horizon and long-horizon dividend estimator are not identical under the alternative.

6 Cochrane (2008) obtains the parameter values by running OLS regressions of equations (1(1) $\begin{array}{rl}{r}_{t+1}& ={\alpha }_r+{\text{β}}_r(d_t-p_t)+{ϵ}_{t+1}^r\end{array}$(1) )–(3(3) $\begin{array}{rl}{d}_{t+1}-p_{t+1}& ={\alpha }_{dp}+\varphi (d_t-p_t)+{ϵ}_{t+1}^{dp}.\end{array}$(3) ) involving log real returns, log real dividend growth and log dividend yield on annual CRSP data from 1927-2004. CRSP reports data on total returns and returns without dividends, which are then used to construct the dividend yield; see footnote 5 in Cochrane's paper for more details.

7 We have $\rho =e^{E(p-d)}/(1+e^{E(p-d)})$.

8 These values simply reflect the variance-covariance matrix of the regression errors from the OLS forecasting regressions of equations (1(1) $\begin{array}{rl}{r}_{t+1}& ={\alpha }_r+{\text{β}}_r(d_t-p_t)+{ϵ}_{t+1}^r\end{array}$(1) )–(3(3) $\begin{array}{rl}{d}_{t+1}-p_{t+1}& ={\alpha }_{dp}+\varphi (d_t-p_t)+{ϵ}_{t+1}^{dp}.\end{array}$(3) ) described earlier.

9 The volatility of shocks to returns ${ϵ}_t^r$ and the correlation between ${ϵ}_t^r$ and ${ϵ}_t^{dp}$ follow from (6(6) $\begin{array}{r}{ϵ}_{t+1}^r={ϵ}_{t+1}^d-\rho {ϵ}_{t+1}^{dp}.\end{array}$(6) ), $\begin{array}{rl}{\sigma }_r^2& ={\sigma }_d^2+{\rho }^2{\sigma }_{dp}^2-2\rho {\sigma }_d{\sigma }_{dp}{\eta }_{d,dp},\\ {\eta }_{r,dp}& =\frac{{\eta }_{d,dp}{\sigma }_d-\rho {\sigma }_{dp}}{{\sigma }_r}.\end{array}$ When ${\sigma }_d={\sigma }_{dp}$ and ${\eta }_{d,dp}=\rho$, we have ${\sigma }_r^2=(1-{\rho }^2){\sigma }_d^2$ and ${\eta }_{r,dp}=0$.

10 The standard deviation is given by ${\sigma }_{r,lh}=\sqrt{\frac{{\rho }^2}{1-{\rho }^2}[{\left(\frac{{\text{β}}_r}{1-\rho \varphi }\right)}^2{\sigma }_{dp}^2+\frac{1}{{\rho }^2}{\sigma }_r^2+\frac{2{\text{β}}_r}{\rho (1-\rho \varphi )}{\sigma }_{r,dp}]}$

11 A change in the value of ϕ does not affect the volatilities ${\sigma }_d^2$ and ${\sigma }_r^2$, so Proposition 1 implies the result that the relative power of the ${\widehat{\text{β}}}_r$ and ${\widehat{\text{β}}}_d$ tests in the asymptotic case remains unchanged. The asymptotic power of the short- and long-horizon return tests is determined according to Propositions 2 and 3. Figure 7(a) shows that the numerical values of power for the short- and long-horizon return tests are similar to each other in the asymptotic case.

12 Figure 7(b) shows that an increase in the value of the autocorrelation parameter ϕ leads to an increase in the finite-sample power advantage of the long-horizon return test over the short-horizon return test. This effect can again be traced back to the correlation structure of shocks. As equation (13(13) $\begin{array}{r}{\eta }_{r,dp}^{lh}=\frac{1}{{\sigma }_{r,lh}}[\rho \frac{{\text{β}}_r}{1-\rho \varphi }{\sigma }_{dp}+{\sigma }_r{\eta }_{r,dp}]\end{array}$(13) ) shows, the correlation between shocks to long-horizon returns and shocks to dividend yield depends on ϕ.