This paper studies standard predictive regressions in economic systems governed by persistent vector autoregressive dynamics for the state variables. In particular, all -- or a subset -- of the variables may be fractionally integrated, which induces a spurious regression problem. We propose a new inference and testing procedure -- the local spectrum (LCM) approach -- for joint significance of the regressors, that is robust against the variables having different integration orders. The LCM procedure is based on fractional filtering and band spectrum regression using a suitably selected set of frequency ordinates. Contrary to existing procedures, we establish a uniform Gaussian limit theory and a standard $\chi^2$-distributed test statistic. Using the LCM inference and testing techniques, we explore predictive regressions for the realized return variation. Standard least squares inference indicates that popular financial and macroeconomic variables convey valuable information about future return volatility. In contrast, we find no significant evidence using our robust LCM procedure. If anything, our tests support a reverse chain of causality, with rising financial volatility predating adverse innovations to key macroeconomic variables. Simulations are employed to illustrate the relevance of the theoretical arguments for finite-sample inference.
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