THE PROPERTIES OF KULLBACK–LEIBLER DIVERGENCE FOR THE UNIT ROOT HYPOTHESIS

Abstract

The fundamental contributions made by Paul Newbold have highlighted how crucial it is to detect when economic time series have unit roots. This paper explores the effects that model specification has on our ability to do that. Asymptotic power, a natural choice to quantify these effects, does not accurately predict finite-sample power. Instead, here the Kullback–Leibler divergence between the unit root null and any alternative is used and its numeric and analytic properties detailed. Numerically it behaves in a similar way to finite-sample power. However, because it is analytically available we are able to prove that it is a minimizable function of the degree of trending in any included deterministic component and of the correlation of the underlying innovations. It is explicitly confirmed, therefore, that it is approximately linear trends and negative unit root moving average innovations that minimize the efficacy of unit root inferential tools. Applied to the Nelson and Plosser macroeconomic series the effect that different types of trends included in the model have on unit root inference is clearly revealed.