Adaptive testing for a unit root with nonstationary volatility

Recent research has emphasized that permanent changes in the innovation variance (caused by structural shifts or an integrated volatility process) lead to size distortions in conventional unit root tests. Cavaliere and Taylor (2004) and Beare (2004) propose nonparametrically corrected versions of unit root tests that have the same asymptotic null distribution as the uncorrected versions in case of homoskedasticity. In this paper, we first derive the asymptotic power envelope for the unit root testing problem when the nonstationary volatility process is known. Next, we show that under suitable conditions, adaptation with respect to the volatility process is possible, in the sense that nonparametric estimation of the volatility process leads to the same asymptotic power envelope. A Monte Carlo experiment shows that these asymptotic results are reflected in finite sample properties, although fairly large sample sizes are needed to fully obtain the asymptotic local power gains.