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SOME LIMIT THEORY FOR AUTOCOVARIANCES WHOSE ORDER DEPENDS ON SAMPLE SIZE

Published online by Cambridge University Press:  04 August 2003

David Harris ,
Brendan McCabe  and
Stephen Leybourne
David Harris
Affiliation:
University of Melbourne
Brendan McCabe
Affiliation:
University of Liverpool
Stephen Leybourne
Affiliation:
University of Nottingham
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Abstract

In this paper we provide some weak convergence results for sample statistics of the product of a variable with its kth-order lag. We assume the variable is a stationary vector that can be represented by linear process, and the lag length k is allowed to be a function of the sample size. Employing the Beveridge–Nelson decomposition, we derive a new functional central limit theorem for this situation and establish related stochastic integral convergence results. We then consider the behavior of associated long-run variance estimators and also extend our analysis to the case where the sample statistics are based on regression residuals. We illustrate the potential range of application of these techniques in the context of (i) testing for I(0) versus I(1) behavior and (ii) estimation and testing in a heteroskedastically cointegrated regression model.We thank the co-editor and the referees for helpful comments on earlier drafts.

Type
Research Article
Information
Econometric Theory , Volume 19 , Issue 5 , October 2003 , pp. 829 - 864
Copyright
© 2003 Cambridge University Press

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References

REFERENCES

Davidson, J. (1994) Stochastic Limit Theory. Oxford, UK: Oxford University Press.
Hansen, B.E. (1992a) Heteroscedastic cointegration. Journal of Econometrics 54, 139–158.Google Scholar
Hansen, B.E. (1992b) Convergence to stochastic integrals for dependent heterogeneous processes. Econometric Theory 8, 489–500.Google Scholar
Harris, D., B.P.M. McCabe, & S.J. Leybourne (2002) Stochastic cointegration: Estimation and inference. Journal of Econometrics 111, 363–384. Available at www.economics.unimelb.edu.au/dharris.Google Scholar
Keenan, D.M. (1997) A central limit theorem for m(n) autocovariances. Journal of Time Series Analysis 18, 61–78.Google Scholar
Kwiatkowski, D., P.C.B. Phillips, P. Schmidt, & Y. Shin (1992) Testing the null 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
Leybourne, S.J. & B.P.M. McCabe (1994) A consistent test for a unit root. Journal of Business and Economic Statistics 12, 157–166.Google Scholar
Magnus, J.R. & H. Neudecker (1988) Matrix Differential Calculus with Applications in Statistics and Econometrics. New York: Wiley.
McCabe, B.P.M., S.J. Leybourne, & D. Harris (2002) Stochastic Cointegration: Testing. Mimeo, University of Melbourne. Available at www.economics.unimelb.edu.au/dharris.
Phillips, P.C.B. (1988) Weak convergence of sample covariance matrices to stochastic integrals via martingale approximations. Econometric Theory 4, 528–533.Google Scholar
Phillips, P.C.B. & V. Solo (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.Google Scholar
Saikkonen, P. & R. Luukkonen (1993) Testing for a moving average unit root in autoregressive integrated moving average models. Journal of the American Statistical Association 88, 596–601.Google Scholar