Mean Reversion and Unit Root Properties of Diffusion Models

Résumé

This paper analyzes the mean reversion and unit root properties of general diffusion models and their discrete samples. In particular, we find that the Dickey-Fuller unit root test applied to discrete samples from a diffusion model becomes a test of mean reversion rather than a unit root, or more generally, nonstationarity in the underlying diffusion. The unit root test has a well defined limit distribution if and only if the underlying diffusion has no mean reversion, and diverges to minus infinity in probability if and only if the underlying diffusion has mean reversion. It is shown, on the other hand, that diffusions are meanreverting as long as their drift terms play the dominant role, and therefore, nonstationary diffusions may well have mean reversion.