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SUMS OF EXPONENTIALS OF RANDOM WALKS WITH DRIFT

Published online by Cambridge University Press:  21 May 2012

Xi Qu  and
Robert de Jong
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Abstract

For many time series in empirical macro and finance, it is assumed that thelogarithm of the series is a unit root process. Since we may want to assumea stable growth rate for the macroeconomics time series, it seems natural topotentially model such a series as a unit root process with drift. Thisassumption implies that the level of such a time series is the exponentialof a unit root process with drift and therefore, it is of substantialinterest to investigate analytically the behavior of the exponential of aunit root process with drift. This paper shows that the sum of theexponential of a random walk with drift converges in distribution, afterrescaling by the exponential of the maximum value of the random walkprocess. A similar result was established in earlier work for unit rootprocesses without drift. The results derived here suggest the conjecturethat also in the case when the Dickey-Fuller test or the KPSS statistic isapplied to the exponential of a unit root process with drift, these testswill asymptotically indicate stationarity.

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Type
Brief Report
Information
Econometric Theory , Volume 28 , Issue 4 , August 2012 , pp. 915 - 924
Copyright
Copyright © Cambridge University Press 2012

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References

REFERENCES

Borodin, A.N. & Ibragimov, I.A. (1995) Limit theorems for functionals of random walks. Proceedings of the Steklov Institute of Mathematics 195(2).Google Scholar
Breitung, J. & Gourieroux, C. (1997) Rank tests for unit roots. Journal of Econometrics 81, 7–27.10.1016/S0304-4076(97)00031-6CrossRefGoogle Scholar
Burridge, P. & Guerre, E. (1996) The limit distribution of level crossings of a random walk, and a simple unit root test. Econometric Theory 12, 705–723.10.1017/S026646660000699XCrossRefGoogle Scholar
Corradi, V. (1995) Nonlinear transformation of integrated time series: A reconsideration? Journal of Time Series Analysis 16, 539–549.10.1111/j.1467-9892.1995.tb00253.xCrossRefGoogle Scholar
Davies, P.L. & Krämer, W. (2003) The Dickey-Fuller test for exponential random walks. Econometric Theory 19, 865–877.10.1017/S0266466603195072CrossRefGoogle Scholar
de Jong, R. (2010) Exponentials of Unit Root Processes. Working paper, Ohio State University.Google Scholar
Ermini, L. & Granger, C.W.J. (1993) Some generalizations on the algebra of I(1) processes. Journal of Econometrics 58, 369–384.10.1016/0304-4076(93)90050-FCrossRefGoogle Scholar
Ermini, L. & Hendry, D.F. (2008) Log income vs. linear income: An application of the encompassing principle. Oxford Bulletin of Economics and Statistics 70, 807–827.10.1111/j.1468-0084.2008.00531.xCrossRefGoogle Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Application, vol. 2. Wiley.Google Scholar
Granger, C.W.J. & Hallmann, J. (1991) Nonlinear transformations of the integrated time series. Journal of Time Series Analysis 12, 207–224.10.1111/j.1467-9892.1991.tb00078.xCrossRefGoogle Scholar
Guerre, E. & Jouneau, F. (1998) Geometric versus arithmetic random walk: The case of trended variables. Journal of Statistical Planning and Inference 68, 203–220.10.1016/S0378-3758(97)00142-0CrossRefGoogle Scholar
Park, J.Y. & Phillips, P.C.B. (1999) Asymptotics for nonlinear transformations of integrated time series. Econometric Theory 15, 269–298.10.1017/S0266466699153015CrossRefGoogle Scholar
Ritter, G.A. (1981) Growth of random walks conditioned to stay positive. Annals of Probability 9, 699–704.10.1214/aop/1176994378CrossRefGoogle Scholar