Testing for Mean Reversion in Processes of Ornstein-Uhlenbeck Type

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In this paper, we analyse processes of Ornstein-Uhlenbeck (OU) type, driven by Lévy processes. This class is designed to capture mean reverting behaviour if it exists; but the data may in fact be adequately described by a pure Lévy process with no OU (autoregressive) effect. For an appropriate discretised version of the model, we utilise likelihood methods to test for such a reduction of the OU process to Lévy motion, deriving the distribution of the relevant pseudo-log-likelihood ratio statistics, asymptotically, both for a refining sequence of partitions on a fixed time interval with mesh size tending to zero, and as the length of the observation window grows large. These analyses are non-standard in that the mean reversion parameter vanishes under the null of a pure Lévy process for the data. Despite this we are able to give a very general analysis with no technical restrictions on the underlying processes or parameter sets, other than a finite variance assumption for the Lévy process. As a special case, for Brownian motion as driving process, we deduce the limiting distribution in a quite explicit way, finding results which generalise the well-known Dickey-Fuller (‘unit-root’) theory.

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