Skip to main content Accessibility help


  • Timothy J. Vogelsang (a1) and Martin Wagner (a2)

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.

Corresponding author
*Address correspondence to Timothy J. Vogelsang, Dept. of Economics, 486 W. Circle Drive, 110 Marshall-Adams Hall, East Lansing, MI 48224-1038; e-mail:
Hide All
Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817858.
Berk, K.N. (1974) Consistent autoregressive spectral estimates. Annals of Statistics 2, 489502.
Cavaliere, G. & Taylor, A.M.R. (2009) Heteroskedastic time series with a unit root. Econometric Theory 25, 12281276.
Gonçalves, S. & Vogelsang, T.J. (2011) Block bootstrap HAC robust tests: The sophistication of the naive bootstrap. Econometric Theory 27, 745791.
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, 142162.
Jansson, M. (2002) Consistent covariance matrix estimation for linear processes. Econometric Theory 18, 14491459.
Kiefer, N.M. & Vogelsang, T.J. (2005) A new asymptotic theory for heteroskedasticity-autocorrelation robust tests. Econometric Theory 21, 11301164.
Newey, W. & West, K. (1994) Automatic lag selection in covariance matrix estimation. Review of Economic Studies 61, 631653.
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, 435463.
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, 467470.
Phillips, P.C.B. (1987) Towards a unified asymptotic theory for autoregression. Biometrika 74, 535547.
Phillips, P.C.B. & Perron, P. (1988) Testing for a unit root in time series regression. Biometrika 75, 335346.
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 9711001.
Said, S.E. & Dickey, D.A. (1984) Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika 71, 599607.
Schwert, W. (1989) Tests for unit roots: A Monte Carlo investigation. Journal of Business & Economic Statistics 7, 147159.
Sims, C.A., Stock, J.H., & Watson, M.W. (1990) Inference in linear time series models with some unit roots. Econometrica 58, 113144.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed