Adaptive Wild Bootstrap Tests for a Unit Root with Non�?Stationary Volatility

Recent research has emphasised 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 (2008) show how these size distortions may be resolved using the wild bootstrap. 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. 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.

[1]  Valentin Patilea,et al.  Adaptive estimation of vector autoregressive models with time-varying variance: Application to testing linear causality in mean , 2010, 1007.1193.

[2]  Michael Jansson,et al.  Nearly Efficient Likelihood Ratio Tests of the Unit Root Hypothesis , 2009 .

[3]  Brendan K. Beare.  Unit Root Testing with Unstable Volatility , 2018 .

[4]  Giuseppe Cavaliere,et al.  Time‐Transformed Unit Root Tests for Models with Non‐Stationary Volatility , 2008 .

[5]  A. M. Robert Taylor,et al.  BOOTSTRAP UNIT ROOT TESTS FOR TIME SERIES WITH NONSTATIONARY VOLATILITY , 2007, Econometric Theory.

[6]  Ali M. Kutan,et al.  Testing PPP in the non-linear STAR framework , 2007 .

[7]  Ke-Li Xu,et al.  Adaptive Estimation of Autoregressive Models with Time-Varying Variances , 2006 .

[8]  P. Phillips,et al.  Inference in Autoregression under Heteroskedasticity , 2006 .

[9]  L. Wasserman.  All of Nonparametric Statistics , 2005 .

[10]  H. P. Boswijk.  Adaptive Testing for a Unit Root with Nonstationary Volatility , 2005 .

[11]  Giuseppe Cavaliere,et al.  Unit Root Tests under Time-Varying Variances , 2005 .

[12]  A. Z. Baharumshah,et al.  Nonlinear Mean Reversion In Real Exchange Rates: Evidence From The ASEAN-5§ , 2003 .

[13]  Michael McAleer,et al.  Estimation and Testing for Unit Root Processes with GARCH (1, 1) Errors: Theory and Monte Carlo Evidence , 2003 .

[14]  Paul Newbold,et al.  Unit root tests with a break in innovation variance , 2002 .

[15]  T. N. Sriram.  Asymptotics in Statistics–Some Basic Concepts , 2002 .

[16]  Mark P. Taylor,et al.  Nonlinear Mean‐Reversion in Real Exchange Rates: Toward a Solution To the Purchasing Power Parity Puzzles , 2001 .

[17]  H. P. Boswijk,et al.  Testing for a Unit Root with Near-Integrated Volatility , 2001 .

[18]  Byeongseon Seo,et al.  Distribution theory for unit root tests with conditional heteroskedasticity 1 1 I wrote this paper w , 1999 .

[19]  W. Li,et al.  Limiting Distributions of Maximum Likelihood Estimators for Unstable Autoregressive Moving-Average Time Series with General Autoregressive Heteroscedastic Errors , 1998 .

[20]  A. V. D. Vaart,et al.  Asymptotic Statistics: Frontmatter , 1998 .

[21]  P. Jeganathan.  Some Aspects of Asymptotic Theory with Applications to Time Series Models , 1995, Econometric Theory.

[22]  B. Hansen.  REGRESSION WITH NONSTATIONARY VOLATILITY , 1995 .

[23]  Kenneth Rogoff,et al.  Chapter 32 Perspectives on PPP and long-run real exchange rates , 1995 .

[24]  Peter Schmidt,et al.  Unit root tests with conditional heteroskedasticity , 1993 .

[25]  E. Mammen.  Bootstrap and Wild Bootstrap for High Dimensional Linear Models , 1993 .

[26]  J. Stock,et al.  Efficient Tests for an Autoregressive Unit Root , 1992 .

[27]  Bruce E. Hansen,et al.  Convergence to Stochastic Integrals for Dependent Heterogeneous Processes , 1992, Econometric Theory.

[28]  E. Giné,et al.  Bootstrapping General Empirical Measures , 1990 .

[29]  Regina Y. Liu.  Bootstrap Procedures under some Non-I.I.D. Models , 1988 .