Skip to main content


  • Yeonwoo Rho (a1) and Xiaofeng Shao (a2)

In unit root testing, a piecewise locally stationary process is adopted to accommodate nonstationary errors that can have both smooth and abrupt changes in second- or higher-order properties. Under this framework, the limiting null distributions of the conventional unit root test statistics are derived and shown to contain a number of unknown parameters. To circumvent the difficulty of direct consistent estimation, we propose to use the dependent wild bootstrap to approximate the nonpivotal limiting null distributions and provide a rigorous theoretical justification for bootstrap consistency. The proposed method is compared through finite sample simulations with the recolored wild bootstrap procedure, which was developed for errors that follow a heteroscedastic linear process. Furthermore, a combination of autoregressive sieve recoloring with the dependent wild bootstrap is shown to perform well. The validity of the dependent wild bootstrap in a nonstationary setting is demonstrated for the first time, showing the possibility of extensions to other inference problems associated with locally stationary processes.

Corresponding author
*Address correspondence to Yeonwoo Rho, Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USA; e-mail:
Hide All

This research was partially supported by NSF grant DMS-1104545. We are grateful to the co-editor and the three referees for their constructive comments and suggestions that led to a substantial improvement of the article. In particular, we are most grateful to Peter C.B. Phillips, who has gone beyond the call of duty for an editor in carefully correcting our English. We also thank Fabrizio Zanello, Mark Gockenbach, Benjamin Ong, and Meghan Campbell for proofreading. Superior, a high performance computing cluster at Michigan Technological University, was used in obtaining results presented in this publication.

Hide All
Adak, S. (1998) Time-dependent spectral analysis of nonstationary time series. Journal of the American Statistical Association 93(444), 14881501.
Andrews, D.W.K. (1984) Non-strong mixing autoregressive processes. Journal of Applied Probability 21(4), 930934.
Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59(3), 817858.
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.
Busetti, F. & Taylor, A.M.R. (2003) Variance shifts, structural breaks, and stationarity tests. Journal of Business and Economic Statistics 21(4), 510531.
Cavaliere, G. & Taylor, A.M.R. (2007) Testing for unit roots in time series models with non-stationary volatility. Journal of Econometrics 140(2), 919947.
Cavaliere, G. & Taylor, A.M.R. (2008a) Bootstrap unit root tests for time series with nonstationary volatility. Econometric Theory 24(1), 4371.
Cavaliere, G. & Taylor, A.M.R. (2008b) Time-transformed unit root tests for models with non-stationary volatility. Journal of Time Series Analysis 29(2), 300330.
Cavaliere, G. & Taylor, A.M.R. (2009) Bootstrap M unit root tests. Econometric Reviews 28(5), 393421.
Chang, Y. & Park, J.Y. (2002) On the asymptotics of adf tests for unit roots. Econometric Reviews 21, 431447.
Chang, Y. & Park, J.Y. (2003) A sieve bootstrap for the test of a unit root. Journal of Time Series Analysis 24(4), 379400.
Dahlhaus, R. (1997) Fitting time series models to nonstationary processes. The Annals of Statistics 25(1), 137.
Dahlhaus, R. & Subba Rao, S. (2006) Statistical inference for time-varying arch processes. The Annals of Statistics 34(3), 10751114.
Dickey, D.A. & Fuller, W.A. (1979) Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74(366), 427431.
Dickey, D.A. & Fuller, W.A. (1981) Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49(4), 10571072.
Domínguez, M.A. & Lobato, I.N. (2001) Size Corrected Power for Bootstrap Tests. Working papers 102, Centro de Investigacion Economica, ITAM.
Draghicescu, D., Guillas, S., & Wu, W.B. (2009) Quantile curve estimation and visualization for nonstationary time series. Journal of Computational and Graphical Statistics 18(1), 120.
Elliott, G., Rothenberg, T.J., & Stock, J.H. (1996) Effcient tests for an autoregressive unit root. Econometrica 64(4), 813836.
Fryzlewicz, P., Sapatinas, T., & Subba Rao, S. (2008) Normalized least-squares estimation in time-varying arch models. The Annals of Statistics 36(2), 742786.
Fryzlewicz, P. & Subba Rao, S. (2011) Mixing properties of arch and time-varying arch processes. Bernoulli 17(1), 320346.
Giurcanu, M. & Spokoiny, V. (2004) Confidence estimation of the covariance function of stationary and locally stationary processes. Statistics and Decisions 22(4), 283300.
Kim, C.-J. & Nelson, C.R. (1999) Has the U.S. economy become more stable? A Bayesian approach based on a Markov-switching model of the business cycle. The Review of Economics and Statistics 81(4), 608616.
Kreiss, J.-P. (1988) Asymptotic statistical inference for a class of stochastic processes. Habilitationsschrift, Universität Hamburg.
Künsch, H.R. (1989) The jackknife and the bootstrap for general stationary observations. The Annals of Statistics 17(3), 12171241.
Mallat, S., Papanicolaou, G., & Zhang, Z. (1998) Adaptive covariance estimation of locally stationary processes. The Annals of Statistics 26(1), 147.
McConnell, M.M. & Perez-Quiros, G. (2000) Output fluctuations in the united states: What has changed since the early 1980’s. American Economic Review 90(5), 14641476.
Müller, U.K. & Elliott, G. (2003) Tests for unit roots and the initial condition. Econometrica 71(4), 12691286.
Newey, W. & West, K.D. (1987) A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55(3), 703708.
Ng, S. & Perron, P. (2001) Lag length selection and the construction of unit root tests with good size and power. Econometrica 69(6), 15191554.
Palm, F.C., Smeekes, S., & Urbain, J.-P. (2008) Bootstrap unit-root tests: Comparison and extensions. Journal of Time Series Analysis 29(2), 371401.
Paparoditis, E. & Politis, D.N. (2002) Local block bootstrap. Comptes Rendus Mathematique 335(11), 959962.
Paparoditis, E. & Politis, D.N. (2003) Residual-based block bootstrap for unit root testing. Econometrica 71(3), 813855.
Paparoditis, E. & Politis, D.N. (2005) Bootstrapping unit root tests for autoregressive time series. Journal of the American Statistical Association 100(470), 545553.
Parker, C., Paparoditis, E., & Politis, D.N. (2006) Unit root testing via the stationary bootstrap. Journal of Econometrics 133(2), 601638.
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(3), 435463.
Phillips, P.C.B. (1987a) Time series regression with a unit root. Econometrica 55(2), 277301.
Phillips, P.C.B. (1987b) Towards a unified asymptotic theory for autoregression. Biometrika 74(3), 535547.
Phillips, P.C.B. & Perron, P. (1988) Testing for a unit root in time series regression. Biometrika 75(2), 335346.
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. The Annals of Statistics 20(2), 9711001.
Phillips, P.C.B. & Xiao, Z. (1998) A primer on unit root testing. Journal of Economic Surveys 12(5), 423469.
Politis, D.N. & Romano, J.P. (1994) The stationary bootstrap. Journal of the American Statistical Association 89(428), 13031313.
Priestley, M.B. (1965) Evolutionary spectra and non-stationary processes. Journal of the Royal Statistical Society: Series B 27(2), 204237.
Psaradakis, Z. (2001) Bootstrap tests for an autoregressive unit root in the presence of weakly dependent errors. Journal of Time Series Analysis 22(5), 577594.
Rho, Y. & Shao, X. (2015) Inference for time series regression models with weakly dependent and heteroscedastic errors. Journal of Business & Economic Statistics 33(3), 444457.
Said, S.E. & Dickey, D.A. (1984) Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika 71(3), 599607.
Sensier, M. & van Dijk, D. (2004) Testing for volatility changes in U.S. macroeconomic time series. The Review of Economics and Statistics 86(3), 833839.
Shao, X. (2010) The dependent wild bootstrap. Journal of the American Statistical Association 105(489), 218235.
Shao, X. & Wu, W.B. (2007) Asymptotic spectral theory for nonlinear time series. The Annals of Statistics 35(4), 17731801.
Smeekes, S. (2013) Detrending bootstrap unit root tests. Econometric Reviews 32(8), 869891.
Smeekes, S. & Taylor, A.M.R. (2012) Bootstrap union tests for unit roots in the presence of nonstationary volatility. Econometric Theory 28(2), 422456.
Smeekes, S. & Urbain, J.-P. (2014) A Multivariate Invariance Principle for Modified Wild Bootstrap Methods with an Application to Unit Root Testing. Technical report.
Stărică, C. & Granger, C. (2005) Nonstationarities in stock returns. Review of Economics and Statistics 87(3), 503522.
Stock, J. & Watson, M. (1999) A comparison of linear and nonlinear univariate models for forecasting macroeconomic time series. In Engle, R. & White, H. (eds.), Cointegration, Causality and Forecasting: A Festschrift for Clive W.J. Granger, pp. 144. Oxford University Press.
Swensen, A.R. (2003) Bootstrapping unit root tests for integrated processes. Journal of Time Series Analysis 24(1), 99126.
Wu, C.F.J. (1986) Jackknife, bootstrap and other resampling methods in regression analysis (with discussion). The Annals of Statistics 14(4), 12611350.
Wu, W.B. (2005) Nonlinear system theory: Another look at dependence. Proceedings of the National Academy of Sciences of the United States of America 102(40), 1415014154.
Wu, W.B. & Zhou, Z. (2011) Gaussian approximations for non-stationary multiple time series. Statistica Sininca 21(3), 13971413.
Zhou, Z. (2013) Heteroscedasticity and autocorrelation robust structural change detection. Journal of the American Statistical Association 108(502), 726740.
Zhou, Z. & Wu, W.B. (2009) Local linear quantile estimation for nonstationary time series. The Annals of Statistics 37(5B), 26962729.
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? *
Type Description Title
Supplementary materials

Rho and Shao supplementary material
Rho and Shao supplementary material 1

 PDF (395 KB)
395 KB


Full text views

Total number of HTML views: 0
Total number of PDF views: 4 *
Loading metrics...

Abstract views

Total abstract views: 30 *
Loading metrics...

* Views captured on Cambridge Core between 12th April 2018 - 20th April 2018. This data will be updated every 24 hours.