Bonferroni‐Type Tests for Return Predictability with Possibly Trending Predictors

The Bonferroni Q test of Campbell and Yogo (2006) 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 of Cavanagh, Elliott and Stock (1995), as operationalised for the constant mean case by Campbell and Yogo (2006), under a local-to-zero specification for a linear trend in the predictor, revealing that size and power depends 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 unionof- 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 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 to an updated Welch and Goyal (2008) dataset illustrates the practical relevance of our new approach.