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The Limit Distribution of level Crossings of a Random Walk, and a Simple Unit Root Test

Published online by Cambridge University Press:  11 February 2009

Peter Burridge  and
Emmanuel Guerre
Peter Burridge
Affiliation:
University of Birmingham
Emmanuel Guerre
Affiliation:
Université P et M Curie
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Abstract

We derive the limit distribution of the number of crossings of a level by a random walk with continuously distributed increments, using a Brownian motion local time approximation. This complements the well-known result for the random walk on the integers. Use of the frequency of level crossings to test for a unit root is examined.

Type
Research Article
Information
Econometric Theory , Volume 12 , Issue 4 , October 1996 , pp. 705 - 723
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

Abrahams, J. (1986) A survey of recent progress on level-crossing problems for random processes. Communications and Networks, 6–25.CrossRefGoogle Scholar
Akonom, J. (1988) Processus transformed d'un ARMA ou d'un processus de Wiener. Problemes d'estimation. These d'Etat, Universite des Sciences et Techniques de Lille Flandres Artois, No. 767.Google Scholar
Akonom, J. (1993) Comportement asymptotique du temps d'occupation du processus des sommes partielles. Annales de I'Institut Henri Poincari 29 (1), 57–81.Google Scholar
Anderson, T.W. (1948) On the theory of testing serial correlation. Skandinavisk Aktuarietid-skrift 31, 88–116.Google Scholar
Blake, I.F. & Lindsey, W.C. (1973) Level-crossing problems for random processes. IEEE Transactions on Information Theory IT-19 (3), 295–315.CrossRefGoogle Scholar
Csaki, E., Csorgo, M.Foldes, A., & Revesz, P. (1983) How big are the increments of the local time of a Wiener process? Annals of Probability 11 (3), 593–608.Google Scholar
Csdrgo, M. & Horvath, L. (1993) Weighted Approximations in Probability and Statistics. New York: John Wiley.Google Scholar
Csorgo, M. & Revesz, P. (1981) Strong Approximations in Probability and Statistics. New York: Academic Press.Google Scholar
Csorgo, M. & Revesz, P. (1985) On strong invariance for local times of partial sums. Stochastic Processes and Their Applications 20, 59–84.CrossRefGoogle Scholar
Dickey, D.A. & Fuller, W.A. (1979) Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427–431.Google Scholar
Dickey, D.A. & Fuller, W.A. (1981) Likelihood ratio tests for autoregressive time series with a unit root. Econometrica 49, 1057–1072.CrossRefGoogle Scholar
Erdos, P. & Kac, M. (1947) On the number of positive sums of independent random variables. Bulletin of the American Mathematical Society 53, 1011–1020.Google Scholar
Feller, W. (1968) Introduction to Probability Theory and Its Applications, vol. I. New York: John Wiley.Google Scholar
Feuerverger, A., Hall, P., & Wood, A.T.A. (1994) Estimation of fractal index and fractal dimension of a Gaussian process by counting the number of level crossings. Journal of Time Series Analysis 15 (6), 587–606.CrossRefGoogle Scholar
Granger, C.W.J. & Hallman, J. (1991) Nonlinear transformations of integrated time series. Journal of Time Series Analysis 12 (3), 207–224.CrossRefGoogle Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Its Application. London: Academic Press.Google Scholar
Komlos, J., Major, P., & Tusnady, G. (1975/1976) An approximation of partial sums of independent random variables and sample distribution function. Zeitschrjft fur Wahrscheinlich-keitstheorie und Verwandte Cebeite 32, 111–131; 34, 33–58.Google Scholar
Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., & Shin, Y. (1992) Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics 54, 159–178.CrossRefGoogle Scholar
Levy, P. (1948) Processus Stochastiques et Mouvement Brownien. Paris: Gauthier-Villars.Google Scholar
Lo, A.W. & MacKinlay, A.C. (1989) The size and power of the variance ratio test in finite samples. Journal of Econometrics 40, 203–238.Google Scholar
Marie, CM. (1974) Mesure et probability. Paris: Hermann.Google Scholar
Petrov, V.V. (1975) Sums of Independent Random Variables. New York: Springer-Verlag.Google Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277–301.CrossRefGoogle Scholar
Revuz, D. & Yor, M. (1991) Continuous Martingales and Brownian Motion. Berlin: Springer-Verlag.Google Scholar
Schwert, G.W. (1989) Tests for unit roots: A Monte Carlo investigation. Journal of Business and Economic Statistics 7 (2), 147–159.Google Scholar
Stock, J.H. (1994) Unit roots and trend breaks in econometrics. In Engle, R.F. (ed.), Handbook of Econometrics, vol. 4, pp. 2739–2841. Amsterdam: Elsevier.Google Scholar