Consistent Inference for Predictive Regressions in Persistent VAR Economies

This paper studies the properties of standard predictive regressions in model economies, characterized through persistent vector autoregressive dynamics for the state variables and the associated series of interest. In particular, we consider a setting where all, or a subset, of the variables may be fractionally integrated, and note that this induces a spurious regression problem. We then propose a new inference and testing procedure – the local spectrum (LCM) approach – for the joint significance of the regressors, which is robust against the variables having different integration orders. The LCM procedure is based on (semi-)parametric fractional-filtering and band spectrum regression using a suitably selected set of frequency ordinates. We establish the asymptotic properties and explain how they differ from and extend existing procedures. Using these new inference and testing techniques, we explore the implications of assuming VAR dynamics in predictive regressions for the realized return variation. Standard least squares predictive regressions indicate that popular financial and macroeconomic variables carry valuable information about return volatility. In contrast, we find no significant evidence using our robust LCM procedure, indicating that prior
conclusions may be premature. In fact, if anything, our results suggest the reverse causality, i.e., rising volatility predates adverse innovations to key macroeconomic variables. Simulations are employed to illustrate the relevance of the theoretical arguments for finite-sample inference.