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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 the logarithm of the series is a unit root process. Since we may want to assume a stable growth rate for the macroeconomics time series, it seems natural to potentially model such a series as a unit root process with drift. This assumption implies that the level of such a time series is the exponential of a unit root process with drift and therefore, it is of substantial interest to investigate analytically the behavior of the exponential of a unit root process with drift. This paper shows that the sum of the exponential of a random walk with drift converges in distribution, after rescaling by the exponential of the maximum value of the random walk process. A similar result was established in earlier work for unit root processes without drift. The results derived here suggest the conjecture that also in the case when the Dickey-Fuller test or the KPSS statistic is applied to the exponential of a unit root process with drift, these tests will asymptotically indicate stationarity.

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

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