Exact geometry of explosive autoregressive models

This paper derives exact expressions for statistical curvature and related geometric quantities in the first order autoregressive models with stable and unit roots, as well as explosive roots larger than unity. We develop a method for deriving exact moments of arbitrary order in general autoregressive models. The covariance of the minimal sufficient statistic is an application of this method. Of particular interest is the Efron curvature which is continuous and bounded in finite samples, but increases rapidly when the autoregressive parameter changes from stable to explosive values, which has important inferential consequences. The initial value effect is also quantified exactly. We also include results for the Efron curvature in the pure stationary case with stochastic initial value for comparison, extending and correcting results from a previous discussion paper.