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A FIXED-b PERSPECTIVE ON THE PHILLIPS–PERRON UNIT ROOT TESTS

Published online by Cambridge University Press:  12 November 2012

Timothy J. Vogelsang  and
Martin Wagner
Timothy J. Vogelsang*
Affiliation:
Michigan State University
Martin Wagner
Affiliation:
Institute for Advanced Studies, Vienna and Frisch Centre for Economic Research, Oslo
*
*Address correspondence to Timothy J. Vogelsang, Dept. of Economics, 486 W. Circle Drive, 110 Marshall-Adams Hall, East Lansing, MI 48224-1038; e-mail: tjv@msu.edu.
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Abstract

In this paper we extend fixed-b asymptotic theory to the nonparametric Phillips–Perron (PP) unit root tests. We show that the fixed-b limits depend on nuisance parameters in a complicated way. These nonpivotal limits provide an alternative theoretical explanation for the well-known finite-sample problems of the PP tests. We also show that the fixed-b limits depend on whether deterministic trends are removed using one-step or two-step detrending approaches. This is in contrast to the asymptotic equivalence of the one- and two-step approaches under a consistency approximation for the long-run variance estimator. Based on these results we introduce modified PP tests that allow for asymptotically pivotal fixed-b inference. The theoretical analysis is cast in the framework of near-integrated processes, which allows us to study the asymptotic behavior both under the unit root null hypothesis and for local alternatives. The performance of the original and modified PP tests is compared by means of local asymptotic power and a small finite-sample simulation study.

Type
Miscellanea
Information
Econometric Theory , Volume 29 , Issue 3 , June 2013 , pp. 609 - 628
Copyright
Copyright © Cambridge University Press 2012 

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Footnotes

The authors thank a co-editor (A.M. Robert Taylor), two referees, conference participants at the DAGStat 2010 meeting in Dortmund, the 5th CSDA International Conference on Computational and Financial Econometrics in London, and seminar participants at the Institute for Advanced Studies in Vienna for helpful comments and suggestions. The authors acknowledge financial support from the Jubiläumsfonds of the Oesterreichische Nationalbank (grant 13398). The usual disclaimer applies.

References

Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–858.CrossRefGoogle Scholar
Berk, K.N. (1974) Consistent autoregressive spectral estimates. Annals of Statistics 2, 489–502.CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.M.R. (2009) Heteroskedastic time series with a unit root. Econometric Theory 25, 1228–1276.CrossRefGoogle Scholar
Gonçalves, S. & Vogelsang, T.J. (2011) Block bootstrap HAC robust tests: The sophistication of the naive bootstrap. Econometric Theory 27, 745–791.CrossRefGoogle Scholar
Hashimzade, N. & Vogelsang, T.J. (2008) Fixed-b asymptotic approximation of the sampling behavior of nonparametric spectral density estimators. Journal of Time Series Analysis 29, 142–162.CrossRefGoogle Scholar
Jansson, M. (2002) Consistent covariance matrix estimation for linear processes. Econometric Theory 18, 1449–1459.CrossRefGoogle Scholar
Kiefer, N.M. & Vogelsang, T.J. (2005) A new asymptotic theory for heteroskedasticity-autocorrelation robust tests. Econometric Theory 21, 1130–1164.CrossRefGoogle Scholar
Newey, W. & West, K. (1994) Automatic lag selection in covariance matrix estimation. Review of Economic Studies 61, 631–653.CrossRefGoogle Scholar
Perron, P. & Ng, S. (1996) Useful modifications to some unit root tests with dependent errors and their local asymptotic properties. Review of Economic Studies 63, 435–463.CrossRefGoogle Scholar
Perron, P. & Vogelsang, T.J. (1992) Testing for a unit root in a time series with a changing mean: Corrections and extensions. Journal of Business & Economic Statistics 10, 467–470.Google Scholar
Phillips, P.C.B. (1987) Towards a unified asymptotic theory for autoregression. Biometrika 74, 535–547.CrossRefGoogle Scholar
Phillips, P.C.B. & Perron, P. (1988) Testing for a unit root in time series regression. Biometrika 75, 335–346.CrossRefGoogle Scholar
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.CrossRefGoogle Scholar
Said, S.E. & Dickey, D.A. (1984) Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika 71, 599–607.Google Scholar
Schwert, W. (1989) Tests for unit roots: A Monte Carlo investigation. Journal of Business & Economic Statistics 7, 147–159.Google Scholar
Sims, C.A., Stock, J.H., & Watson, M.W. (1990) Inference in linear time series models with some unit roots. Econometrica 58, 113–144.CrossRefGoogle Scholar