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BOOTSTRAP-ASSISTED UNIT ROOT TESTING WITH PIECEWISE LOCALLY STATIONARY ERRORS

Published online by Cambridge University Press:  12 April 2018

Yeonwoo Rho  and
Xiaofeng Shao
Yeonwoo Rho*
Affiliation:
Michigan Technological University
Xiaofeng Shao
Affiliation:
University of Illinois at Urbana-Champaign
*
*Address correspondence to Yeonwoo Rho, Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USA; e-mail: yrho@mtu.edu.
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Abstract

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.

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ARTICLES
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Econometric Theory , Volume 35 , Issue 1 , February 2019 , pp. 142 - 166
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

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.

References

REFERENCES

Adak, S. (1998) Time-dependent spectral analysis of nonstationary time series. Journal of the American Statistical Association 93(444), 1488–1501.CrossRefGoogle Scholar
Andrews, D.W.K. (1984) Non-strong mixing autoregressive processes. Journal of Applied Probability 21(4), 930–934.CrossRefGoogle Scholar
Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59(3), 817–858.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Busetti, F. & Taylor, A.M.R. (2003) Variance shifts, structural breaks, and stationarity tests. Journal of Business and Economic Statistics 21(4), 510–531.CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.M.R. (2007) Testing for unit roots in time series models with non-stationary volatility. Journal of Econometrics 140(2), 919–947.CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.M.R. (2008a) Bootstrap unit root tests for time series with nonstationary volatility. Econometric Theory 24(1), 43–71.CrossRefGoogle Scholar
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), 300–330.CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.M.R. (2009) Bootstrap M unit root tests. Econometric Reviews 28(5), 393–421.CrossRefGoogle Scholar
Chang, Y. & Park, J.Y. (2002) On the asymptotics of adf tests for unit roots. Econometric Reviews 21, 431–447.CrossRefGoogle Scholar
Chang, Y. & Park, J.Y. (2003) A sieve bootstrap for the test of a unit root. Journal of Time Series Analysis 24(4), 379–400.CrossRefGoogle Scholar
Dahlhaus, R. (1997) Fitting time series models to nonstationary processes. The Annals of Statistics 25(1), 1–37.Google Scholar
Dahlhaus, R. & Subba Rao, S. (2006) Statistical inference for time-varying arch processes. The Annals of Statistics 34(3), 1075–1114.CrossRefGoogle Scholar
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), 427–431.Google Scholar
Dickey, D.A. & Fuller, W.A. (1981) Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49(4), 1057–1072.CrossRefGoogle Scholar
Domínguez, M.A. & Lobato, I.N. (2001) Size Corrected Power for Bootstrap Tests. Working papers 102, Centro de Investigacion Economica, ITAM.Google Scholar
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), 1–20.CrossRefGoogle Scholar
Elliott, G., Rothenberg, T.J., & Stock, J.H. (1996) Effcient tests for an autoregressive unit root. Econometrica 64(4), 813–836.CrossRefGoogle Scholar
Fryzlewicz, P., Sapatinas, T., & Subba Rao, S. (2008) Normalized least-squares estimation in time-varying arch models. The Annals of Statistics 36(2), 742–786.CrossRefGoogle Scholar
Fryzlewicz, P. & Subba Rao, S. (2011) Mixing properties of arch and time-varying arch processes. Bernoulli 17(1), 320–346.CrossRefGoogle Scholar
Giurcanu, M. & Spokoiny, V. (2004) Confidence estimation of the covariance function of stationary and locally stationary processes. Statistics and Decisions 22(4), 283–300.CrossRefGoogle Scholar
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), 608–616.CrossRefGoogle Scholar
Kreiss, J.-P. (1988) Asymptotic statistical inference for a class of stochastic processes. Habilitationsschrift, Universität Hamburg.Google Scholar
Künsch, H.R. (1989) The jackknife and the bootstrap for general stationary observations. The Annals of Statistics 17(3), 1217–1241.CrossRefGoogle Scholar
Mallat, S., Papanicolaou, G., & Zhang, Z. (1998) Adaptive covariance estimation of locally stationary processes. The Annals of Statistics 26(1), 1–47.Google Scholar
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), 1464–1476.CrossRefGoogle Scholar
Müller, U.K. & Elliott, G. (2003) Tests for unit roots and the initial condition. Econometrica 71(4), 1269–1286.CrossRefGoogle Scholar
Newey, W. & West, K.D. (1987) A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55(3), 703–708.CrossRefGoogle Scholar
Ng, S. & Perron, P. (2001) Lag length selection and the construction of unit root tests with good size and power. Econometrica 69(6), 1519–1554.CrossRefGoogle Scholar
Palm, F.C., Smeekes, S., & Urbain, J.-P. (2008) Bootstrap unit-root tests: Comparison and extensions. Journal of Time Series Analysis 29(2), 371–401.CrossRefGoogle Scholar
Paparoditis, E. & Politis, D.N. (2002) Local block bootstrap. Comptes Rendus Mathematique 335(11), 959–962.CrossRefGoogle Scholar
Paparoditis, E. & Politis, D.N. (2003) Residual-based block bootstrap for unit root testing. Econometrica 71(3), 813–855.CrossRefGoogle Scholar
Paparoditis, E. & Politis, D.N. (2005) Bootstrapping unit root tests for autoregressive time series. Journal of the American Statistical Association 100(470), 545–553.CrossRefGoogle Scholar
Parker, C., Paparoditis, E., & Politis, D.N. (2006) Unit root testing via the stationary bootstrap. Journal of Econometrics 133(2), 601–638.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(3), 435–463.CrossRefGoogle Scholar
Phillips, P.C.B. (1987a) Time series regression with a unit root. Econometrica 55(2), 277–301.CrossRefGoogle Scholar
Phillips, P.C.B. (1987b) Towards a unified asymptotic theory for autoregression. Biometrika 74(3), 535–547.CrossRefGoogle Scholar
Phillips, P.C.B. & Perron, P. (1988) Testing for a unit root in time series regression. Biometrika 75(2), 335–346.CrossRefGoogle Scholar
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. The Annals of Statistics 20(2), 971–1001.CrossRefGoogle Scholar
Phillips, P.C.B. & Xiao, Z. (1998) A primer on unit root testing. Journal of Economic Surveys 12(5), 423–469.CrossRefGoogle Scholar
Politis, D.N. & Romano, J.P. (1994) The stationary bootstrap. Journal of the American Statistical Association 89(428), 1303–1313.CrossRefGoogle Scholar
Priestley, M.B. (1965) Evolutionary spectra and non-stationary processes. Journal of the Royal Statistical Society: Series B 27(2), 204–237.Google Scholar
Psaradakis, Z. (2001) Bootstrap tests for an autoregressive unit root in the presence of weakly dependent errors. Journal of Time Series Analysis 22(5), 577–594.CrossRefGoogle Scholar
Rho, Y. & Shao, X. (2015) Inference for time series regression models with weakly dependent and heteroscedastic errors. Journal of Business & Economic Statistics 33(3), 444–457.CrossRefGoogle Scholar
Said, S.E. & Dickey, D.A. (1984) Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika 71(3), 599–607.CrossRefGoogle Scholar
Sensier, M. & van Dijk, D. (2004) Testing for volatility changes in U.S. macroeconomic time series. The Review of Economics and Statistics 86(3), 833–839.CrossRefGoogle Scholar
Shao, X. (2010) The dependent wild bootstrap. Journal of the American Statistical Association 105(489), 218–235.CrossRefGoogle Scholar
Shao, X. & Wu, W.B. (2007) Asymptotic spectral theory for nonlinear time series. The Annals of Statistics 35(4), 1773–1801.CrossRefGoogle Scholar
Smeekes, S. (2013) Detrending bootstrap unit root tests. Econometric Reviews 32(8), 869–891.CrossRefGoogle Scholar
Smeekes, S. & Taylor, A.M.R. (2012) Bootstrap union tests for unit roots in the presence of nonstationary volatility. Econometric Theory 28(2), 422–456.CrossRefGoogle Scholar
Smeekes, S. & Urbain, J.-P. (2014) A Multivariate Invariance Principle for Modified Wild Bootstrap Methods with an Application to Unit Root Testing. Technical report.Google Scholar
Stărică, C. & Granger, C. (2005) Nonstationarities in stock returns. Review of Economics and Statistics 87(3), 503–522.CrossRefGoogle Scholar
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. 1–44. Oxford University Press.Google Scholar
Swensen, A.R. (2003) Bootstrapping unit root tests for integrated processes. Journal of Time Series Analysis 24(1), 99–126.CrossRefGoogle Scholar
Wu, C.F.J. (1986) Jackknife, bootstrap and other resampling methods in regression analysis (with discussion). The Annals of Statistics 14(4), 1261–1350.CrossRefGoogle Scholar
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), 14150–14154.CrossRefGoogle Scholar
Wu, W.B. & Zhou, Z. (2011) Gaussian approximations for non-stationary multiple time series. Statistica Sininca 21(3), 1397–1413.CrossRefGoogle Scholar
Zhou, Z. (2013) Heteroscedasticity and autocorrelation robust structural change detection. Journal of the American Statistical Association 108(502), 726–740.CrossRefGoogle Scholar
Zhou, Z. & Wu, W.B. (2009) Local linear quantile estimation for nonstationary time series. The Annals of Statistics 37(5B), 2696–2729.CrossRefGoogle Scholar
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