Bonferroni-Type Tests for Return Predictability With Possibly Trending Predictors

Abstract

The Bonferroni $Q$ test is widely used in empirical studies investigating predictability in asset returns by strongly persistent and endogenous predictors. Its formulation, however, only allows for a constant mean in the predictor, seemingly at odds with many of the predictors used in practice. We establish the asymptotic size and local power properties of the $Q$ test, and the corresponding Bonferroni $t$-test, under a local-to-zero specification for a linear trend in the predictor, revealing that size and power depend on the magnitude of the trend for both. To rectify this, we develop with-trend variants of the operational Bonferroni $Q$ and $t$ tests. However, where a trend is not present in the predictor, we show that these tests lose (both finite sample and asymptotic local) power relative to the extant constant-only versions of the tests. In practice, uncertainty will necessarily exist over whether a linear trend is genuinely present in the predictor or not. To deal with this, we also develop hybrid tests based on union-of-rejections and switching mechanisms to capitalise on the relative power advantages of the constant-only tests when a trend is absent (or very weak) and the with-trend tests otherwise. A further extension allows the use of a conventional $t$-test where the predictor appears to be weakly persistent. We show that, overall, our recommended hybrid test can offer excellent size and power properties regardless of whether or not a linear trend is present in the predictor, or the predictor's degrees of persistence and endogeneity. An empirical application illustrates the practical relevance of our new approach.

JEL Classifications: C22, C12, G14