Bootstrapping Non-Stationary Stochastic Volatility

Tinbergen Institute Discussion Paper 2019-083/III

38 Pages Posted: 17 Dec 2019

See all articles by H. Peter Boswijk

H. Peter Boswijk

Amsterdam School of Economics; Tinbergen Institute

Giuseppe Cavaliere

University of Bologna - Department of Economics

Iliyan Georgiev

University of Bologna

Anders Rahbek

University of Copenhagen - Department of Statistics and Operations Research; University of Copenhagen - Department of Economics

Date Written: November 29, 2019


To what extent can the bootstrap be applied to conditional mean 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 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 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 bootstrap statistic, 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 several testing problems in the presence of non-stationary stochastic volatility, including testing in a location model, testing for structural change using CUSUM-type functionals, and testing for a unit root in autoregressive models. Importantly, we show that sufficient conditions for conditional wild bootstrap validity include the absence of statistical leverage effects, i.e., correlation between the error process and its future conditional variance. The results of the paper are illustrated using Monte Carlo simulations, which indicate that a wild bootstrap approach leads to size control even in small samples.

Keywords: Bootstrap, Non-stationary stochastic volatility, Random limit measures, Weak convergence in Distribution

JEL Classification: C32

Suggested Citation

Boswijk, H. Peter and Cavaliere, Giuseppe and Georgiev, Iliyan and Rahbek, Anders, Bootstrapping Non-Stationary Stochastic Volatility (November 29, 2019). Tinbergen Institute Discussion Paper 2019-083/III, Available at SSRN: or

H. Peter Boswijk (Contact Author)

Amsterdam School of Economics ( email )

Roetersstraat 11
Amsterdam, North Holland 1018 WB


Tinbergen Institute ( email )

Burg. Oudlaan 50
Rotterdam, 3062 PA

Giuseppe Cavaliere

University of Bologna - Department of Economics ( email )

+390512098489 (Phone)

Iliyan Georgiev

University of Bologna

Anders Rahbek

University of Copenhagen - Department of Statistics and Operations Research

Universitetsparken 5
+45 3532 0682 (Phone)

University of Copenhagen - Department of Economics

Øster Farimagsgade 5
Bygning 26
1353 Copenhagen K.

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