Bonferroni Type Tests for Return Predictability and the Initial Condition

We develop tests for predictability that are robust to both the magnitude of the initial condition and the degree of persistence of the predictor. While the popular Bonferroni Q test of Campbell and Yogo displays excellent power properties for strongly persistent predictors with an asymptotically negligible initial condition, it can suffer from severe size distortions and power losses when either the initial condition is asymptotically non-negligible or the predictor is weakly persistent. The Bonferroni t test of Elliott, and Stock, although displaying power well below that of the Bonferroni Q test for strongly persistent predictors with an asymptotically negligible initial condition, displays superior size control and power when the initial condition is asymptotically nonnegligible. In the case where the predictor is weakly persistent, a conventional regression t test comparing to standard normal quantiles is known to be asymptotically optimal under Gaussianity. Based on these properties, we propose two asymptotically size controlled hybrid tests that are functions of the Bonferroni Q, Bonferroni t, and conventional t tests. Our proposed hybrid tests exhibit very good power regardless of the magnitude of the initial condition or the persistence degree of the predictor. An empirical application to the data originally analyzed by Campbell and Yogo shows our new hybrid tests are much more likely to find evidence of predictability than the Bonferroni Q test when the initial condition of the predictor is estimated to be large in magnitude.