  In this paper we investigate how the bootstrap can be applied to time series
regressions when the volatility of the innovations is random and
non-stationary. The volatility of many economic and financial time series
displays persistent changes and possible non-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 and the
validity of the bootstrap when the volatility is driven by a non-stationary
stochastic process. This includes near-integrated volatility processes as well
as near-integrated GARCH processes. This paper develops conditions for
bootstrap validity in time series regressions 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 proving bootstrap validity, involving weak
convergence in probability of the bootstrap statistic, fail to deliver the
required 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
several testing problems in the presence of non-stationary stochastic
volatility, including testing in a location model, testing for structural
change and testing for an autoregressive unit root. Sufficient conditions for
bootstrap validity include the absence of statistical leverage effects, i.e.,
correlation between the error process and its future conditional variance. The
results are illustrated using Monte Carlo simulations, which indicate that the
wild bootstrap leads to size control even in small samples.
