The Properties of Entropy for the Unit Root Hypothesis1

This paper details the differential and numeric properties of two measures of entropy, Shannon entropy and Kullback-Leibler distance, applicable for the unit root hypothesis. It is found that they are differentiable functions of the degree of trending in any included deterministic component and of the correlation of the underlying innovations. Moreover, Shannon entropy is concave in these, and thus maximisable. Kullback-Leibler is instead convex, and thus minimizable. It is explicitly confirmed, therefore, that it is approximately linear trends and negative unit root moving average innovations which minimize the efficacy of unit root inferential tools. Moreover, applied to the Nelson and Plosser macroeconomic series the effect that the inclusion, or not, of a linear trend, for example, is explicitly quantified.