Adaptive Testing for Cointegration with Nonstationary Volatility

Tinbergen Institute Discussion Paper 2019-043/III

37 Pages Posted: 26 Jun 2019

See all articles by H. Peter Boswijk

H. Peter Boswijk

Amsterdam School of Economics; Tinbergen Institute

Yang Zu

University of Nottingham - School of Economics

Date Written: June 21, 2019


This paper generalises Boswijk and Zu (2018)'s adaptive unit root test for time series with nonstationary volatility to a multivariate context. Persistent changes in the innovation variance matrix of a vector autoregressive model lead to size distortions in conventional cointegration tests, which may be resolved using the wild bootstrap, as shown by Cavaliere et al. (2010, 2014). We show that it also leads to the possibility of constructing tests with higher power, by taking the time-varying volatilities and correlations into account in the formulation of the likelihood function and the resulting likelihood ratio test statistic. We find that under suitable conditions, adaptation with respect to the volatility process is possible, in the sense that nonparametric volatility matrix estimation does not lead to a loss of asymptotic local power relative to the case where the volatilities are observed. The asymptotic null distribution of the test is nonstandard and depends on the volatility process; we show that various bootstrap implementations may be used to conduct asymptotically valid inference. Monte Carlo simulations show that the resulting test has good size properties, and higher power than existing tests. Two empirical examples illustrate the applicability of the tests.

Keywords: Adaptive estimation, Nonparametric volatility estimation, Wild bootstrap

JEL Classification: C32, C12

Suggested Citation

Boswijk, H. Peter and Zu, Yang, Adaptive Testing for Cointegration with Nonstationary Volatility (June 21, 2019). Tinbergen Institute Discussion Paper 2019-043/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

Yang Zu

University of Nottingham - School of Economics ( email )

University Park
Nottingham, NG7 2RD
United Kingdom

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