Some curious power properties of long-horizon tests

Based on simulations and asymptotic results, I highlight three distinct properties of long-horizon predictive tests. (i) The asymptotic power of long-horizon tests increases only with the sample size relative to the forecasting horizon. Keeping this ratio fixed as the sample size increases does not lead to any power gains asymptotically. (ii) Although the power of long-horizon tests increases with the magnitude of the slope coefficient for alternatives close to the null hypothesis, there are no gains in power as the slope coefficient grows large. That is, the power curve is asymptotically horizontal when viewed as a function of the slope coefficient. (iii) For endogenous regressors—i.e., when the innovations to the regressand are contemporaneously correlated with the innovations to the regressor—traditional tests based on the standard long-run OLS estimator result in power curves that are sometimes decreasing in the magnitude of the slope coefficient.