Wild bootstrap seasonal unit root tests for time series with periodic nonstationary volatility

ABSTRACT

We investigate the behavior of the well-known Hylleberg, Engle, Granger and Yoo (HEGY) regression-based seasonal unit root tests in cases where the driving shocks can display periodic nonstationary volatility and conditional heteroskedasticity. Our set up allows for periodic heteroskedasticity, nonstationary volatility and (seasonal) generalized autoregressive-conditional heteroskedasticity 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 oversized in the presence of such effects. As a consequence, we propose wild bootstrap implementations of the HEGY tests. Two possible wild bootstrap resampling schemes are discussed, both of which are 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.

KEYWORDS:

• Conditional heteroscedasticity
• (periodic) nonstationary volatility
• seasonal unit roots
• wild bootstrap

JEL CLASSIFICATION:

• C12
• C22

Acknowledgments

We thank the Editor, Essie Massoumi, an Associate Editor and two anonymous referees for their helpful and constructive comments on earlier versions of this paper.

Notes

¹The condition imposed by Assumption 1 that E(h_Sn+s) = 1, implies that $E\left(E_n{E}_n^{\prime }\right)=I_S$, where I_k denotes the k×k identity matrix. This restriction entails no loss of generality, however, because the leading diagonal elements of Ω_n are unrestricted, and is made only to simplify notation. In particular, any ${\mathcal{ℰ}}_n={\Omega }_n{E}_n$ satisfying Assumption 1 with $E\left(E_n{E}_n^{\prime }\right)=\Xi$, where Ξ is diagonal, can also be expressed as ${\mathcal{ℰ}}_n={\tilde{\Omega }}_n{\tilde{E}}_n$ with $E\left({\tilde{E}}_n{\tilde{E}}_n^{\prime }\right)=I_S$ and ${\tilde{\Omega }}_n:={\Omega }_n{\Xi }^{-1∕2}$, where both ${\tilde{E}}_n$ and ${\tilde{\Omega }}_n$ satisfy Assumption 1.

²In what follows, it is understood that terms relating to frequency π are to be omitted when S is odd and that where reference is made to the Nyquist frequency this is understood only to apply where S is even.

³It should be noted that the idea of a block wild bootstrap is not new. Shao (2011) proposes a block wild bootstrap in the context of bootstrap tests for white noise. However, the block wild bootstrap scheme in Step 2b differs from that in Shao (2011) in the important regard that while our block length is fixed and equal to the number of seasons, in Shao (2011) the block length is an increasing function of the sample size.

⁴Corresponding results for the wild bootstrap HEGY tests using the seasonal block wild resampling device in Step 2b of Algorithm 1 can be found in the accompanying online Supplementary appendix. A comparison of these results suggests that overall the two wild bootstrap schemes give qualitatively very similar finite sample behavior.

⁵We also considered a variety of conditionally heteroskedastic specifications for e_4n+s, including stationary GARCH and autoregressive stochastic volatility models. Consistent with the findings of Cavaliere and Taylor (2009), the results for the wild bootstrap HEGY tests were little different to those reported here for the case of e_4n+s IID standard normal.

⁶We also considered the cases of a double volatility shift and trending volatility but found the results to be qualitatively similar for the results reported for Model 3.