Adaptive Wild Bootstrap Tests for a Unit Root With Non‐Stationary Volatility

Abstract

Summary 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. It has been shown how these size distortions can be resolved using the wild bootstrap. In this paper, we first derive the asymptotic power envelope for the unit root testing problem when the non-stationary volatility process is known. Next, we show that under suitable conditions, adaptation with respect to the volatility process is possible, in the sense that non-parametric estimation of the volatility process leads to the same asymptotic power envelope. Implementation of the resulting test involves cross-validation and the wild bootstrap. A Monte Carlo experiment shows that the asymptotic results are reflected in finite sample properties, and an empirical analysis of real exchange rates illustrates the applicability of the proposed procedures.

Adaptive Testing for Cointegration with Nonstationary Volatility

“…The assumption of continuity is made here to facilitate consistent non-parametric estimation of σ(·), although Xu and Phillips (2008) show that this could be relaxed for adaptive estimation. For practical purposes, the continuity assumption can still accommodate large changes in volatility, as discussed in Boswijk and Zu (2018) for the univariate case. The analysis in the present paper could be extended to allow for conditional heteroskedasticity, but this is not considered here to simplify the analysis.…”

“…Alternatively, a two-step approach may be used, where the volatility matrix is estimated based on the residuals from least-squares estimation of the unrestricted VAR model, and the resulting estimator Σ t is then substituted for Σ t in the expressions for the MLE and LR statistic given in the previous section. In this paper we propose to estimate σ t by a nonparametric kernel estimator, generalising the approach of Boswijk and Zu (2018), which in turn is based on Hansen (1995). It should be noted, however, that as analysed by Nelson (1996), multivariate GARCH models (with deterministic parameter sequences instead of estimated parameters) may also be interpreted as nonparametric filters of continuous-time multivariate stochastic volatility processes.…”

“…Following Boswijk and Zu (2018), we propose to use the estimates { Σ t } n t=1 from the original data in the calculation of LR * n (r), i.e., we do not re-estimate {Σ t } n t=1 for each bootstrap sample. The two bootstrap methods we consider differ in the method for drawing the bootstrap errors.…”

Multiple Testing for No Cointegration under Nonstationary Volatility

“…Remark Boswijk and Zu () extend Boswijk's () volatility‐robust unit root test based on non‐parametric estimation of the variance to the cointegration case. Their analysis, however, reveals that such adaptive estimation requires stricter assumption on the volatility pattern, ruling out for instance the practically relevant case of discontinuous volatility shifts.…”

The impact of data frequency on market efficiency tests of commodity futures prices

“…Unit root tests which focus on non‐normal errors can be found in Lucas () and Rothenberg and Stock (). Seo (), Boswijk (), and Ling and Li () present unit root tests when the time series being tested has Gaussian GARCH (Generalized Autoregressive Conditional Heteroskedasticity) stochastics. Although these tests have increased power when testing financial data, a common trait for the existing testing methods is that they all require some specific model specification assumption, either in terms of mean functional form (e.g., the ADF test requires the number of autoregressive lags to be specified) or the error term distribution (ARCH–Autoregressive Conditional Heteroscedasticity–, GARCH, normal, t , etc.).…”