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Sums Of Exponentials Of Random Walks With Drift

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  • Qu, Xi
  • de Jong, Robert

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.

Suggested Citation

  • Qu, Xi & de Jong, Robert, 2012. "Sums Of Exponentials Of Random Walks With Drift," Econometric Theory, Cambridge University Press, vol. 28(04), pages 915-924, August.
  • Handle: RePEc:cup:etheor:v:28:y:2012:i:04:p:915-924_00
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