Least Squares and IVX Limit Theory in Systems of Predictive Regressions with GARCH innovations

The paper examines the effect of conditional heteroskedasticity to least squares inference in stochastic regression models. We show that a regressor signal of exact order O^e_p(n^{1+\alpha}) for arbitrary \alpha > 0 is sufficient to eliminate stationary GARCH effects from the limit distributions of least squares based estimators and self-normalised test statistics. The above order dominates the O e p (n) signal of stationary regressors but is dominated by the O e p (n 2 ) signal of I(1) regressors, thereby showing that least squares invariance to GARCH effects is not an exclusively I(1) phenomenon but extends to processes with persistence degree arbitrarily close to stationarity. The theory validates standard inference for self normalised test statistics based on: (i) the OLS estimator when \alpha \in (0,1); (ii) the IVX estimator (Phillips and Magdalinos, 2009; Kostakis, Magdalinos and Stamatogiannis 2015a) when \alpha > 0, when the innovation sequence of the system is a stationary vec-GARCH process. An adjusted version of the IVX testing procedure is shown to also accommodate stationary regressors and produce standard chi-squared inference under conditional heteroskedasticity in the innovations across the full range \alpha \qeq 0.