In order for predictive regression tests to deliver asymptotically valid inference, account has to be taken of the degree of persistence of the predictors under test. There is also a maintained assumption that any predictability in the variable of interest is purely attributable to the predictors under test. Violation of this assumption by the omission of relevant persistent predictors renders the predictive regression invalid with the result that both the ﬁnite sample and asymptotic size of the predictability tests can be signiﬁcantly inﬂated, with the potential therefore to spuriously indicate predictability. In response we propose a predictive regression invalidity test based on a stationarity testing approach. To allow for an unknown degree of persistence in the putative predictors, and for heteroskedasticity in the data, we implement our proposed test using a ﬁxed regressor wild bootstrap procedure. We demonstrate the asymptotic validity of the proposed bootstrap test. This entails demonstrating that the asymptotic distribution of the bootstrap statistic, conditional on the data, is the same (to ﬁrst-order) as the asymptotic null distribution of the statistic computed on the original data, conditional on the predictor. This corrects a long-standing error in the bootstrap literature whereby it is incorrectly argued that for strongly persistent regressors the validity of the ﬁxed regressor bootstrap obtains through equivalence to an unconditional limit distribution. Our bootstrap results are therefore of interest in their own right and are likely to have important applications beyond the present context. An illustration is given by re-examining the results relating to U.S. stock returns data in Campbell and Yogo (2006).
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Iliyan Georgiev, David I Harvey, Stephen J Leybourne and A M Robert Taylor
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