Wild Bootstrap Seasonal Unit Root Tests for Time Series with Periodic Non-Stationary Volatility

This paper investigates the behaviour of the well-known HEGY (Hylleberg, Engle, Granger and Yoo, 1990, Journal of Econometrics, vol.44, pp.215-238) regression-based seasonal unit root tests in cases where the driving shocks are allowed to display periodic non-stationary volatility and conditional heteroskedasticity. Our set up allows for periodic heteroskedasticity, non-stationary volatility and (seasonal) GARCH as special cases. We show that the limiting null distributions of the HEGY tests depend, in general, on nuisance parameters which derive from the underlying volatility process. Monte Carlo simulations show that the standard HEGY tests can be substantially over-sized in the presence of such effects. As a consequence, we propose bootstrap implementations of the HEGY tests, based around a seasonal block wild bootstrap principle. This is shown to deliver asymptotically pivotal inference under our general conditions on the shocks. Simulation evidence is presented which suggests that our proposed bootstrap tests perform well in practice, largely correcting the size problems seen with the standard HEGY tests even under extreme patterns of heteroskedasticity, yet not losing finite sample relative to the standard HEGY tests.