Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Home
  • Home
Article
  • Cited by 13

  • Get access
    Add to cart USD25.00
    Check if you have access via personal or institutional login
    Log in Register

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

  • Stephan Smeekes (a1) and A.M. Robert Taylor (a2)

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.

Copyright

COPYRIGHT: © Cambridge University Press 2011

Corresponding author

*Address correspondence to Robert Taylor, School of Economics, University of Nottingham, Nottingham, NG7 2RD, U.K. e-mail: robert.taylor@nottingham.ac.uk.

References

Hide All
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.
Bayer, C. & Hanck, C. (2009) Combining Non-cointegration Tests. METEOR Research Memorandum 09/012, Maastricht University.
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.
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.
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.
Cavaliere, G. & Taylor, A.M.R. (2008) Bootstrap unit root tests for time series with nonstationary volatility. Econometric Theory 24, 43–71.
Cavaliere, G. & Taylor, A.M.R. (2009a) Bootstrap M unit root tests. Econometric Reviews 28, 393–421.
Cavaliere, G. & Taylor, A.M.R. (2009b) Heteroskedastic time series with a unit root. Econometric Theory 25, 1228–1276.
Chang, Y. & Park, J.Y. (2002) On the asymptotics of ADF tests for unit roots. Econometric Reviews 21, 431–447.
Chang, Y. & Park, J.Y. (2003) A sieve bootstrap for the test of a unit root. Journal of Time Series Analysis 24, 379–400.
Elliott, G. & Müller, U.K. (2006) Minimizing the impact of the initial condition on testing for unit roots. Journal of Econometrics 135, 285–310.
Elliott, G., Rothenberg, T.J., & Stock, J.H. (1996) Efficient tests for an autoregressive unit root. Econometrica 64, 813–836.
Giné, E. & Zinn, J. (1990) Bootstrapping general empirical measures. Annals of Probability 18, 851–869.
Hannan, E.J. & Kavalieris, L. (1986) Regression, autoregression models. Journal of Time Series Analysis 7, 27–49.
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.
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.
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.
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.
Marsh, P. (2007) The available information for invariant tests of a unit root. Econometric Theory 23, 686–710.
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.
Müller, U.K. & Elliott, G. (2003) Tests for unit roots and the initial condition. Econometrica 71, 1269–1286.
Ng, S. & Perron, P. (2001) Lag length selection and the construction of unit root tests with good size and power. Econometrica 69, 1519–1554.
Palm, F.C., Smeekes, S., & Urbain, J.-P. (2008) Bootstrap unit root tests: Comparison and extensions. Journal of Time Series Analysis 29, 371–401.
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.
Paparoditis, E. & Politis, D.N. (2003) Residual-based block bootstrap for unit root testing. Econometrica 71, 813–855.
Paparoditis, E. & Politis, D.N. (2005) Bootstrapping unit root tests for autoregressive time series. Journal of the American Statistical Association 100, 545–553.
Park, J.Y. (2002) An invariance principle for sieve bootstrap in time series. Econometric Theory 18, 469–490.
Parker, C., Paparoditis, E., & Politis, D.N. (2006) Unit root testing via the stationary bootstrap. Journal of Econometrics 133, 601–638.
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.
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.
Phillips, P.C.B. & Perron, P. (1988) Testing for a unit root in time series regression. Biometrika 75, 335–346.
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.
Romano, J.P. & Wolf, M. (2005) Stepwise multiple testing as formalized data snooping. Econometrica 73, 1237–1282.
Sensier, M. & Van Dijk, D. (2004) Testing for volatility changes in U.S. macroeconomic time series. Review of Economics and Statistics 86, 833–839.
Smeekes, S. (2009) Detrending Bootstrap Unit Root Tests. METEOR Research Memorandum 09/056, Maastricht University.
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.
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.
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.
Swensen, A.R. (2003) Bootstrapping unit root tests for integrated processes. Journal of Time Series Analysis 24, 99–126.
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.
Watson, M.W. (1999) Explaining the increased variability in long-term interest rates. Federal Reserve Bank of Richmond Economic Quarterly 85, 71–96.
White, H. (2000) A reality check for data snooping. Econometrica 68, 1097–1126.

Related content

Powered by UNSILO
  • Cited by 13

  • Get access
    Add to cart USD25.00
    Check if you have access via personal or institutional login
    Log in Register

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

  • Stephan Smeekes (a1) and A.M. Robert Taylor (a2)

Metrics

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.