Bootstrapping non-stationary stochastic volatility

In this paper we investigate to what extent the bootstrap can be applied to conditionalmean models, such as regression or time series models, when the volatility of the innovations is random and possibly non-stationary. In fact, the volatility of many economic and financial time series displays persistent changes and possible no-stationarity. However, the theory of the bootstrap for such models has focused on deterministic changes of the unconditional variance and little is known about the performance andthe validity of the bootstrap when the volatility is driven by a non-stationary stochastic process. This includes near-integrated exogenous volatility processes as well as near-integrated GARCH processes, where the conditional variance has a diffusion limit; a further important example is the case where volatility exhibits infrequent jumps. This paper fills this gap in the literature by developing conditions for bootstrap validity in time series and regression models with non-stationary, stochastic volatility. We show that in such cases the distribution of bootstrap statistics (conditional on the data) is random in the limit. Consequently, the conventional approaches to proofs of bootstrap consistency, based on the notion of weak convergence in probability of the bootstrapstatistic, fail to deliver the required validity results. Instead, we use the concept of ‘weak convergence in distribution’ to develop and establish novel conditions for validity of the wild bootstrap, conditional on the volatility process. We apply our results to severaltesting problems in the presence of non-stationary stochastic volatility, including testing in a location model, testing for structural change using CUSUM-type functionals, andtesting for a unit root in autoregressive models. Importantly, we work under sufficient conditions for bootstrap validity that include the absence of statistical leverage effects,i.e.,correlationbetweentheerrorprocessanditsfutureconditionalvariance.Theresultsof the paper are illustrated using Monte Carlo simulations, which indicate that a wildbootstrap approach leads to size control even in small samples.