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BOOTSTRAP UNION TESTS FOR UNIT ROOTS IN THE PRESENCE OF NONSTATIONARY VOLATILITY

Published online by Cambridge University Press:  13 September 2011

Stephan Smeekes  and
A.M. Robert Taylor
Stephan Smeekes
Affiliation:
Maastricht University
A.M. Robert Taylor*
Affiliation:
University of Nottingham
*
*Address correspondence to Robert Taylor, School of Economics, University of Nottingham, Nottingham, NG7 2RD, U.K. e-mail: [email protected].
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Abstract

Three important issues surround testing for a unit root in practice: uncertainty as to whether or not a linear deterministic trend is present in the data; uncertainty as to whether the initial condition of the process is (asymptotically) negligible or not, and the possible presence of nonstationary volatility in the data. Assuming homoskedasticity, Harvey, Leybourne, and Taylor (2011, Journal of Econometrics, forthcoming) propose decision rules based on a four-way union of rejections of quasi-differenced (QD) and ordinary least squares (OLS) detrended tests, both with and without a linear trend, to deal with the first two problems. In this paper we first discuss, again under homoskedasticity, how these union tests may be validly bootstrapped using the sieve bootstrap principle combined with either the independent and identically distributed (i.i.d.) or wild bootstrap resampling schemes. This serves to highlight the complications that arise when attempting to bootstrap the union tests. We then demonstrate that in the presence of nonstationary volatility the union test statistics have limit distributions that depend on the form of the volatility process, making tests based on the standard asymptotic critical values or, indeed, the i.i.d. bootstrap principle invalid. We show that wild bootstrap union tests are, however, asymptotically valid in the presence of nonstationary volatility. The wild bootstrap union tests therefore allow for a joint treatment of all three of the aforementioned issues in practice.

Type
ARTICLES
Information
Econometric Theory , Volume 28 , Issue 2 , April 2012 , pp. 422 - 456
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Basawa, I.V., Mallik, A.K., McCormick, W.P., Reeves, J.H., & Taylor, R.L. (1991) Bootstrapping unstable first-order autoregressive processes. Annals of Statistics 19, 1098–1101.CrossRefGoogle Scholar
Bayer, C. & Hanck, C. (2009) Combining Non-cointegration Tests. METEOR Research Memorandum 09/012, Maastricht University.Google Scholar
Busetti, F. & Taylor, A.M.R. (2003) Testing against stochastic trend in the presence of variance shifts. Journal of Business and Economic Statistics 21, 510–531.CrossRefGoogle Scholar
Cavaliere, G., Harvey, D.I., Leybourne, S.J., & Taylor, A.M.R. (2011) Testing for unit roots in the pres-ence of a possible break in trend and nonstationary volatility. Econometric Theory 27, 957–991.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, 919–947.CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.M.R. (2008) Bootstrap unit root tests for time series with nonstationary volatility. Econometric Theory 24, 43–71.Google Scholar
Cavaliere, G. & Taylor, A.M.R. (2009a) Bootstrap M unit root tests. Econometric Reviews 28, 393–421.Google Scholar
Cavaliere, G. & Taylor, A.M.R. (2009b) Heteroskedastic time series with a unit root. Econometric Theory 25, 1228–1276.CrossRefGoogle Scholar
Chang, Y. & Park, J.Y. (2002) On the asymptotics of ADF tests for unit roots. Econometric Reviews 21, 431–447.Google Scholar
Chang, Y. & Park, J.Y. (2003) A sieve bootstrap for the test of a unit root. Journal of Time Series Analysis 24, 379–400.CrossRefGoogle Scholar
Elliott, G. & Müller, U.K. (2006) Minimizing the impact of the initial condition on testing for unit roots. Journal of Econometrics 135, 285–310.CrossRefGoogle Scholar
Elliott, G., Rothenberg, T.J., & Stock, J.H. (1996) Efficient tests for an autoregressive unit root. Econometrica 64, 813–836.CrossRefGoogle Scholar
Giné, E. & Zinn, J. (1990) Bootstrapping general empirical measures. Annals of Probability 18, 851–869.CrossRefGoogle Scholar
Hannan, E.J. & Kavalieris, L. (1986) Regression, autoregression models. Journal of Time Series Analysis 7, 27–49.CrossRefGoogle Scholar
Harvey, D.I., Leybourne, S.J., & Taylor, A.M.R. (2009a) Unit root testing in practice: Dealing with uncertainty over the trend and initial condition. Econometric Theory 25, 587–636.Google Scholar
Harvey, D.I., Leybourne, S.J., & Taylor, A.M.R. (2009b) Unit root testing in practice: Dealing with uncertainty over the trend and initial condition — Rejoinder. Econometric Theory 25, 658–667.Google Scholar
Harvey, D.I., Leybourne, S.J., & Taylor, A.M.R. (2011) Testing for unit roots in the presence of uncertainty over both the trend and initial condition. Journal of Econometrics, forthcoming.Google Scholar
Kim, C.-J. & Nelson, C.R. (1999) Has the US economy become more stable? A Bayesian approach based on a Markov-switching model of the business cycle. Review of Economics and Statistics 81, 608–616.CrossRefGoogle Scholar
Marsh, P. (2007) The available information for invariant tests of a unit root. Econometric Theory 23, 686–710.Google Scholar
McConnell, M.M. & Perez Quiros, G. (2000) Output fluctuations in the United States: What has changed since the early 1980s? American Economic Review 90, 1464–1476.Google Scholar
Müller, U.K. & Elliott, G. (2003) Tests for unit roots and the initial condition. Econometrica 71, 1269–1286.Google Scholar
Ng, S. & Perron, P. (2001) Lag length selection and the construction of unit root tests with good size and power. Econometrica 69, 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, 371–401.CrossRefGoogle Scholar
Palm, F.C., Smeekes, S., & Urbain, J.-P. (2010) A sieve bootstrap test for cointegration in a conditional error correction model. Econometric Theory 26, 647–681.CrossRefGoogle Scholar
Paparoditis, E. & Politis, D.N. (2003) Residual-based block bootstrap for unit root testing. Econometrica 71, 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, 545–553.CrossRefGoogle Scholar
Park, J.Y. (2002) An invariance principle for sieve bootstrap in time series. Econometric Theory 18, 469–490.CrossRefGoogle Scholar
Parker, C., Paparoditis, E., & Politis, D.N. (2006) Unit root testing via the stationary bootstrap. Journal of Econometrics 133, 601–638.CrossRefGoogle Scholar
Perron, P. & Qu, Z. (2007) A simple modification to improve the finite sample properties of Ng and Perron’s unit root tests. Economics Letters 94, 12–19.Google Scholar
Phillips, P.C.B. & Magdalinos, T. (2009) Unit root and cointegrating limit theory when initialization is in the infinite past. Econometric Theory 25, 1682–1715.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
Romano, J.P. & Wolf, M. (2005) Stepwise multiple testing as formalized data snooping. Econometrica 73, 1237–1282.Google Scholar
Sensier, M. & Van Dijk, D. (2004) Testing for volatility changes in U.S. macroeconomic time series. Review of Economics and Statistics 86, 833–839.Google Scholar
Smeekes, S. (2009) Detrending Bootstrap Unit Root Tests. METEOR Research Memorandum 09/056, Maastricht University.Google Scholar
Smeekes, S. & Taylor, A.M.R. (2010) Bootstrap Union Tests for Unit Roots in the Presence of Nonstationary Volatility. Granger Centre Discussion Paper 10/03, University of Nottingham.Google Scholar
Stock, J.H. (1994) Unit roots, structural breaks and trends. In Engle, R.F. & McFadden, D.L. (eds.), Handbook of Econometrics, vol. 4, Ch. 46, pp. 2739–2841. North Holland.Google Scholar
Stock, J.H. & Watson, M.W. (1999) A comparison of linear and nonlinear univariate models for forecasting macroeconomic time series. In Engle, R.F. & White, H. (eds.), Cointegration, Causality and Forecasting: A Festschrift in Honour of 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, 99–126.CrossRefGoogle Scholar
Van Dijk, D., Osborn, D.R., & Sensier, M. (2002) Changes in Variability of the Business Cycle in the G7 Countries. Econometric Institute Report EI 2002-28, Erasmus University Rotterdam.Google Scholar
Watson, M.W. (1999) Explaining the increased variability in long-term interest rates. Federal Reserve Bank of Richmond Economic Quarterly 85, 71–96.Google Scholar
White, H. (2000) A reality check for data snooping. Econometrica 68, 1097–1126.Google Scholar