Abstract
A unified theory of estimation and inference is developed for an autoregressive process with root in (-1; ∞ that includes the stable, unstable, explosive and all intermediate regions. The discontinuity of the limit distribution of the t-statistic along autoregressive regions and its dependence on the distribution of the innovations in the explosive region (1; ∞) are ad- dressed simultaneously. A novel estimation procedure, based on a data-driven combination of a near-stationary and a mildly explosive endogenously constructed instrument, delivers an asymptotic mixed-Gaussian theory of estimation and gives rise to an asymptotically standard normal t-statistic across all autoregressive regions independently of the distribution of the innovations. The resulting hypothesis tests and confidence intervals are shown to have correct asymptotic size (uniformly over the parameter space) both in autoregressive and in predictive regression models, thereby establishing a general and unified framework for inference with autoregressive processes. Extensive Monte Carlo experimentation shows that the proposed methodology exhibits very good infinite sample properties over the entire autoregressive parameter space (-1; ∞) and compares favourably to existing methods within their parametric (-1; 1] validity range. We demonstrate that a first-order difference equation for the number of infections with an explosive/stable root results naturally after linearisation of an SIR model at the outbreak and apply our procedure to Covid-19 infections to construct confidence intervals on the model's parameters, including the epidemic's basic reproduction number, across a panel of countries without a priori knowledge of the model's stability/explosivity properties.