Bootstrapping non-stationary stochastic volatility (with Giuseppe Cavaliere, Iliyan Georgiev and Anders Rahbek
Abstract: Recent research has shown that the wild bootstrap delivers consistent inference in time-series models with persistent changes in the unconditional error variance. Consistency means that bootstrap p-values are asymptotically uniformly distributed under the null hypothesis. This paper addresses the question whether this consistency result can be extended to models with non-stationary stochastic volatility. This includes near-integrated exogenous volatility processes (as analysed by Hansen, 1995), as well as near-integrated GARCH processes, where the conditional variance has a diffusion limit (Nelson, 1990). We show that the conventional approach, based on weak convergence in probability of the bootstrap test statistic, fails to deliver the required result. Instead, we introduce the concept of weak convergence in distribution, closely related to the notion of weak convergence to random measures as used by Basawa et al. (1991). Using this concept, we develop conditions for consistency of the wild bootstrap for testing problems with non-pivotal test statistics. Examples are the sample average of a martingale difference sequence, and unit root test statistics with martingale difference errors, both in the presence of non-stationary stochastic volatility. An important condition for wild bootstrap validity is the absence of statistical leverage effects, i.e., correlation between the error process and its conditional variance. The results of the paper are illustrated using Monte Carlo simulations and an empirical application.
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