Mildly Explosive Autoregression Under Stationary Conditional Heteroskedasticity

A limit theory is developed for mildly explosive autoregressions under stationary (weakly or strongly dependent) conditionally heteroskedastic errors. The conditional variance process is allowed to be stationary, integrable and mixingale, thus encompassing general classes of generalized autoregressive conditional heteroskedasticity‐type or stochastic volatility models. No mixing conditions or moments of higher order than 2 are assumed for the innovation process. As in Magdalinos ( ), we find that the asymptotic behaviour of the sample moments is affected by the memory of the innovation process both in the form of the limiting distribution and, in the case of long range dependence, the rate of convergence, while conditional heteroskedasticity affects only the asymptotic variance. These effects are cancelled out in least squares regression theory, and thereby, the Cauchy limit theory of Phillips and Magdalinos ( ) remains invariant to a wide class of stationary conditionally heteroskedastic innovations processes.