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Heteroskedastic Time Series With A Unit Root

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  • Cavaliere, Giuseppe
  • Taylor, A.M. Robert

Abstract

In this paper we provide a unified theory, and associated invariance principle, for the large-sample distributions of the Dickey–Fuller class of statistics when applied to unit root processes driven by innovations displaying nonstationary stochastic volatility of a very general form. These distributions are shown to depend on both the spot volatility and the integrated variation associated with the innovation process. We propose a partial solution (requiring any leverage effects to be asymptotically negligible) to the identified inference problem using a wild bootstrap–based approach. Results are initially presented in the context of martingale differences and are later generalized to allow for weak dependence. Monte Carlo evidence is also provided that suggests that our proposed bootstrap tests perform very well in finite samples in the presence of a range of innovation processes displaying nonstationary volatility and/or weak dependence.

Suggested Citation

  • Cavaliere, Giuseppe & Taylor, A.M. Robert, 2009. "Heteroskedastic Time Series With A Unit Root," Econometric Theory, Cambridge University Press, vol. 25(05), pages 1228-1276, October.
  • Handle: RePEc:cup:etheor:v:25:y:2009:i:05:p:1228-1276_09
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    Cited by:

    1. Harvey, David I. & Leybourne, Stephen J. & Sollis, Robert & Taylor, A.M. Robert, 2016. "Tests for explosive financial bubbles in the presence of non-stationary volatility," Journal of Empirical Finance, Elsevier, vol. 38(PB), pages 548-574.
    2. Boswijk, H. Peter & Cavaliere, Giuseppe & Rahbek, Anders & Taylor, A.M. Robert, 2016. "Inference on co-integration parameters in heteroskedastic vector autoregressions," Journal of Econometrics, Elsevier, vol. 192(1), pages 64-85.
    3. Hanck, Christoph & Demetrescu, Matei & Kruse, Robinson, 2015. "Fixed-b Asymptotics for t-Statistics in the Presence of Time-Varying Volatility," Annual Conference 2015 (Muenster): Economic Development - Theory and Policy 112916, Verein für Socialpolitik / German Economic Association.
    4. Christoph Hanck & Robert Czudaj, 2013. "Nonstationary-Volatility Robust Panel Unit Root Tests and the Great Moderation," Ruhr Economic Papers 0434, Rheinisch-Westfälisches Institut für Wirtschaftsforschung, Ruhr-Universität Bochum, Universität Dortmund, Universität Duisburg-Essen.
    5. Cavaliere, Giuseppe & Nielsen, Morten Ørregaard & Taylor, A.M. Robert, 2015. "Bootstrap score tests for fractional integration in heteroskedastic ARFIMA models, with an application to price dynamics in commodity spot and futures markets," Journal of Econometrics, Elsevier, vol. 187(2), pages 557-579.
    6. Cavaliere, Giuseppe & Nielsen, Morten Ørregaard & Taylor, A.M. Robert, 2017. "Quasi-maximum likelihood estimation and bootstrap inference in fractional time series models with heteroskedasticity of unknown form," Journal of Econometrics, Elsevier, vol. 198(1), pages 165-188.
    7. Demetrescu, Matei & Kruse, Robinson, 2015. "Testing heteroskedastic time series for normality," Annual Conference 2015 (Muenster): Economic Development - Theory and Policy 113221, Verein für Socialpolitik / German Economic Association.
    8. Cavaliere, Giuseppe & Xu, Fang, 2014. "Testing for unit roots in bounded time series," Journal of Econometrics, Elsevier, vol. 178(P2), pages 259-272.
    9. repec:zbw:rwirep:0434 is not listed on IDEAS
    10. Smeekes, Stephan & Taylor, A.M. Robert, 2012. "Bootstrap Union Tests For Unit Roots In The Presence Of Nonstationary Volatility," Econometric Theory, Cambridge University Press, vol. 28(02), pages 422-456, April.
    11. Skrobotov Anton, 2018. "On Bootstrap Implementation of Likelihood Ratio Test for a Unit Root," Working Papers wpaper-2018-302, Gaidar Institute for Economic Policy, revised 2018.
    12. Christoph Hanck & Robert Czudaj, 2015. "Nonstationary-volatility robust panel unit root tests and the great moderation," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 99(2), pages 161-187, April.
    13. Giuseppe Cavaliere & Peter C. B. Phillips & Stephan Smeekes & A. M. Robert Taylor, 2015. "Lag Length Selection for Unit Root Tests in the Presence of Nonstationary Volatility," Econometric Reviews, Taylor & Francis Journals, vol. 34(4), pages 512-536, April.
    14. Joakim Westerlund, 2013. "A computationally convenient unit root test with covariates, conditional heteroskedasticity and efficient detrending," Journal of Time Series Analysis, Wiley Blackwell, vol. 34(4), pages 477-495, July.
    15. Patrick Marsh, "undated". "Saddlepoint Approximations for Optimal Unit Root Tests," Discussion Papers 09/31, Department of Economics, University of York.
    16. Brendan K. Beare, 2008. "Unit Root Testing with Unstable Volatility," Economics Papers 2008-W06, Economics Group, Nuffield College, University of Oxford.
    17. Matei Demetrescu & Christoph Hanck & Robinson Kruse, 2016. "Fixed-b Inference in the Presence of Time-Varying Volatility," CREATES Research Papers 2016-01, Department of Economics and Business Economics, Aarhus University.
    18. Matei Demetrescu & Christoph Hanck & Adina I. Tarcolea, 2014. "Iv-Based Cointegration Testing In Dependent Panels With Time-Varying Variance," Journal of Time Series Analysis, Wiley Blackwell, vol. 35(5), pages 393-406, August.
    19. Westerlund, Joakim, 2014. "On the choice of test for a unit root when the errors are conditionally heteroskedastic," Computational Statistics & Data Analysis, Elsevier, vol. 69(C), pages 40-53.
    20. Fried, Roland, 2012. "On the online estimation of local constant volatilities," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3080-3090.
    21. Smeekes S. & Urbain J.R.Y.J., 2014. "A multivariate invariance principle for modified wild bootstrap methods with an application to unit root testing," Research Memorandum 008, Maastricht University, Graduate School of Business and Economics (GSBE).
    22. Nikolaos Kourogenis, 2015. "Polynomial Trends, Nonstationary Volatility and the Eicker-White Asymptotic Variance Estimator," Economics Bulletin, AccessEcon, vol. 35(3), pages 1675-1680.
    23. Brandan K. Beare, 2008. "Unit Root Testing with Unstable Volatility," Economics Series Working Papers 2008-WO6, University of Oxford, Department of Economics.
    24. Guodong Li & Chenlei Leng & Chih-Ling Tsai, 2014. "A Hybrid Bootstrap Approach To Unit Root Tests," Journal of Time Series Analysis, Wiley Blackwell, vol. 35(4), pages 299-321, July.

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