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STATIONARITY TESTS UNDER TIME-VARYING SECOND MOMENTS

Published online by Cambridge University Press:  23 September 2005

Giuseppe Cavaliere  and
A.M. Robert Taylor
Giuseppe Cavaliere
Affiliation:
University of Bologna
A.M. Robert Taylor
Affiliation:
University of Birmingham
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Abstract

In this paper we analyze the effects of a very general class of time-varying variances on well-known “stationarity” tests of the I(0) null hypothesis. Our setup allows, among other things, for both single and multiple breaks in variance, smooth transition variance breaks, and (piecewise-) linear trending variances. We derive representations for the limiting distributions of the test statistics under variance breaks in the errors of I(0), I(1), and near-I(1) data generating processes, demonstrating the dependence of these representations on the precise pattern followed by the variance processes. Monte Carlo methods are used to quantify the effects of fixed and smooth transition single breaks and trending variances on the size and power properties of the tests. Finally, bootstrap versions of the tests are proposed that provide a solution to the inference problem.We are grateful to Peter Phillips, a co-editor, and two anonymous referees whose comments on an earlier draft have led to a considerable improvement in the paper.

Type
Research Article
Information
Econometric Theory , Volume 21 , Issue 6 , December 2005 , pp. 1112 - 1129
Copyright
© 2005 Cambridge University Press

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References

REFERENCES

Busetti, F. & A.C. Harvey (2001) Testing for the presence of a random walk in series with structural breaks. Journal of Time Series Analysis 22, 127–150.Google Scholar
Busetti, F. & A.M.R. Taylor (2003) Variance shifts, structural breaks and stationarity tests. Journal of Business & Economic Statistics 21, 510–531.Google Scholar
Cavaliere, G. (2004a) Testing stationarity under a permanent variance shift. Economics Letters 82, 403–408.Google Scholar
Cavaliere, G. (2004b) Unit root tests under time-varying variances. Econometric Reviews 23, 259–292.Google Scholar
Cavaliere, G. & A.M.R. Taylor (2004) Stationarity Tests under Time-Varying Second Moments. Discussion Papers in Economics 04-12, University of Birmingham.
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.
de Jong, R. (2000) A strong consistency proof for heteroskedasticity and autocorrelation consistent covariance matrix estimators. Econometric Theory 16, 262–268.Google Scholar
Giné, E. & J. Zinn (1990) Bootstrapping general empirical measures. Annals of Probability 18, 851–869.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. (1992a) Consistent covariance matrix estimation for dependent heterogeneous processes. Econometrica 60, 967–972.Google Scholar
Hansen, B.E. (1992b) Convergence to stochastic integrals for dependent heterogeneous processes. Econometric Theory 8, 489–500.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. (2000) Testing for structural change in conditional models. Journal of Econometrics 97, 93–115Google Scholar
Jansson, M. (2002) Consistent covariance matrix estimation for linear processes. Econometric Theory 18, 1449–1459.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
Kwiatkowski, D., P.C.B. Phillips, P. Schmidt, & Y. Shin (1992) Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? Journal of Econometrics 54, 159–178.Google Scholar
Lo, A.W. (1991) Long-term memory in stock market prices. Econometrica 59, 1279–1314.Google Scholar
Nyblom, J. (1989) Testing the constancy of parameters over time. Journal of the American Statistical Association 84, 223–230.Google Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277–301.Google Scholar
Phillips, P.C.B. & Y. Sun (2001) Nonorthogonal Hilbert projections in trend regression. Econometric Theory, Problem, 17, 854; Solution, 18, 1011–1015.Google Scholar
Phillips, P.C.B. & Z. Xiao (1998) A primer on unit root testing. Journal of Economic Surveys 12, 423–470.Google Scholar
Revuz, D. & M. Yor (1991) Continuous Martingales and Brownian Motion. Springer-Verlag.
Shin, Y. (1994) A residual-based test of the null of cointegration against the alternative of no cointegration. Econometric Theory 10, 91–115.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.
Xiao, Z. (2001) Testing the null hypothesis of stationarity against an autoregressive unit root alternative. Journal of Time Series Analysis 22, 87–105.Google Scholar