Hostname: page-component-76c49bb84f-hwk5l Total loading time: 0 Render date: 2025-07-09T07:16:11.164Z Has data issue: false hasContentIssue false
  • English
  • Français

BOOTSTRAP UNIT ROOT TESTS FOR TIME SERIES WITH NONSTATIONARY VOLATILITY

Published online by Cambridge University Press:  06 September 2007

Giuseppe Cavaliere  and
A.M. Robert Taylor
Giuseppe Cavaliere
Affiliation:
University of Bologna
A.M. Robert Taylor
Affiliation:
University of Nottingham
Get access Rights & Permissions [Opens in a new window]

Abstract

The presence of permanent volatility shifts in key macroeconomic and financial variables in developed economies appears to be relatively common. Conventional unit root tests are unreliable in the presence of such behavior, having nonpivotal asymptotic null distributions. In this paper we propose a bootstrap approach to unit root testing that is valid in the presence of a wide class of permanent variance changes that includes single and multiple (abrupt and smooth transition) volatility change processes as special cases. We make use of the so-called wild bootstrap principle, which preserves the heteroskedasticity present in the original shocks. Our proposed method does not require the practitioner to specify any parametric model for the volatility process. Numerical evidence suggests that the bootstrap tests perform well in finite samples against a range of nonstationary volatility processes.We thank two anonymous referees, Paulo Rodrigues, Peter Phillips, and seminar participants at the URCT conference held in Faro, Portugal, September 29 to October 1, 2005, for helpful comments on previous versions of this paper.

Information

Type
Research Article
Information
Econometric Theory , Volume 24 , Issue 1 , February 2008 , pp. 43 - 71
Copyright
© 2008 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

Andrews, D.W.K. & M. Buchinsky (2001) Evaluation of a three-step method for choosing the number of bootstrap repetitions. Journal of Econometrics 103, 345–386.Google Scholar
Berk, K.N. (1974) Consistent autoregressive spectral estimates. Annals of Statistics 2, 489–502.Google Scholar
Busetti, F. & A.M.R. Taylor (2003) Testing against stochastic trend in the presence of variance shifts. Journal of Business & Economic Statistics 21, 510–531.Google Scholar
Cavaliere, G. (2003) Asymptotics for unit root tests under Markov-regime switching. Econometrics Journal 6, 193–216.Google Scholar
Cavaliere, G. (2004) Unit root tests under time-varying variances. Econometric Reviews 23, 259–292.Google Scholar
Cavaliere, G. & A.M.R. Taylor (2004) Testing for Unit Roots in Time Series Models with Non-stationary Volatility. Working paper 04–24, Department of Economics, University of Birmingham.
Cavaliere, G. & A.M.R. Taylor (2007) Testing for unit roots in time series models with non-stationarity volatility. Journal of Econometrics, forthcoming.Google Scholar
Chang, Y. & J.Y. Park (2002) On the asymptotics of ADF tests for unit roots. Econometric Reviews 21, 431–447.Google Scholar
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.
Davidson, R. & E. Flachaire (2001) The Wild Bootstrap, Tamed at Last. Working paper IER 1000, Queens University.
Elliott, G., T.J. Rothenberg, & J.H. Stock (1996) Efficient tests for an autoregressive unit root. Econometrica 64, 813–836.Google Scholar
Engle, R.F. & J. Gonzalo Rangel (2004) The Spline GARCH Model for Unconditional Volatility and Its Global Macroeconomic Causes. Mimeo, New York University.
Giné, E. & J. Zinn (1990) Bootstrapping general empirical measures. Annals of Probability 18, 851–869.Google Scholar
Gonçalves, S. & L. Killian (2004) Bootstrapping autoregressions with conditional heteroskedasticity of unknown form. Journal of Econometrics 123, 89–120.Google Scholar
Hamori, S. & A. Tokihisa (1997) Testing for a unit root in the presence of a variance shift. Economics Letters 57, 245–253.Google Scholar
Hansen, B.E. (1995) Regression with nonstationary volatility. Econometrica 63, 1113–1132.Google Scholar
Hansen, B.E. (1996) Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64, 413–430.Google Scholar
Hansen, B.E. (2000a) Sample splitting and threshold estimation. Econometrica 68, 575–603.Google Scholar
Hansen, B.E. (2000b) Testing for structural change in conditional models. Journal of Econometrics 97, 93–115.Google Scholar
Hansen, E. & A. Rahbek (1998) Stationarity and Asymptotics of Multivariate ARCH Time Series with an Application to Robustness of Cointegration Analysis. Preprint 12/1998, Department of Theoretical Statistics, University of Copenhagen.
Kim, C.-J. & C.R. Nelson (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.Google Scholar
Kim, K. & P. Schmidt (1993) Unit root tests with conditional heteroskedasticity. Journal of Econometrics 59, 287–300.Google Scholar
Kim, T.H., S. Leybourne, & P. Newbold (2002) Unit root tests with a break in innovation variance. Journal of Econometrics 109, 365–387.Google Scholar
Koop, G. & S.M. Potter (2000) Nonlinearity, structural breaks or outliers in economic time series? In W.A. Barnett, D.F. Hendry, S. Hylleberg, T. Terasvirta, D. Tjostheim, & A.H. Wurtz (eds.), Nonlinear Econometric Modelling in Time Series Analysis, pp. 61–78. Cambridge University Press.
Ling, S., W.K. Li, & M. McAleer (2003) Estimation and testing for unit root process with GARCH(1,1) errors: Theory and Monte Carlo evidence. Econometric Reviews 22, 179–202.Google Scholar
Liu, R.Y. (1988) Bootstrap procedures under some non i.i.d. models. Annals of Statistics 16, 1696–1708.Google Scholar
Lopez, J.H. (1997) The power of the ADF test. Economics Letters 57, 5–10.Google Scholar
Loretan, M. & P.C.B. Phillips (1994) Testing the covariance stationarity of heavy-tailed time series: An overview of the theory with applications to several financial datsets. Journal of Empirical Finance 1, 211–248.Google Scholar
Mammen, E. (1993) Bootstrap and wild bootstrap for high dimensional linear models. Annals of Statistics 21, 255–285.Google Scholar
McConnell, M.M. & G. Perez Quiros (2000) Output fluctuations in the United States: What has changed since the early 1980s? American Economic Review 90, 1464–1476.Google Scholar
Nelson, C.R., J. Piger, & E. Zivot (2001) Markov regime-switching and unit root tests. Journal of Business & Economics Statistics 19, 404–415.Google Scholar
Ng, S. & P. Perron (2001) Lag length selection and the construction of unit root tests with good size and power. Econometrica 69, 1519–1554.Google Scholar
Park, J.Y. (2003) Bootstrap unit root tests. Econometrica 71, 1845–1895.Google Scholar
Perron, P. (1989) The great crash, the oil price shock, and the unit root hypothesis. Econometrica 57, 1361–1401.Google Scholar
Perron, P. (1990) Testing for a unit root in a time series with a changing mean. Journal of Business & Economic Statistics 8, 153–162.Google Scholar
Perron, P. & S. Ng (1996) Useful modifications to some unit root tests with dependent errors and their local asymptotic properties. Review of Economic Studies 63, 435–463.Google Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277–301.Google Scholar
Phillips, P.C.B. & K.-L. Xu (2006) Inference in autoregression under heteroskedasticity. Journal of Time Series Analysis 27, 289–308.Google Scholar
Said, S.E. & D.A. Dickey (1984) Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika 89, 1420–1437.Google Scholar
Sensier, M. & D. van Dijk (2004) Testing for volatility changes in U.S. macroeconomic time series. Review of Economics and Statistics 86, 833–839.Google Scholar
Stock, J.H. (1994) Unit roots, structural breaks and trends. In R.F. Engle & D.L. McFadden (eds.), Handbook of Econometrics, vol. 4, pp. 2739–2840. Elsevier Science.
Stock, J.H. (1999) A class of tests for integration and cointegration. In R.F. Engle & H. White (eds.), Cointegration, Causality and Forecasting: A Festschrift in Honour of Clive W.J. Granger, pp. 137–167. Oxford University Press.
Stock, J.H. & M.W. Watson (1999) A comparison of linear and nonlinear univariate models for forecasting macroeconomic time series. In R.F. Engle & H. White (eds.), Cointegration, Causality and Forecasting: A Festschrift in Honour of Clive W.J. Granger, pp. 1–44. Oxford University Press.
van Dijk, D., D.R. Osborn, & M. Sensier (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.Google Scholar