Limit Theory for an Explosive Autoregressive Process

Large sample properties are studied for a …rst-order autoregression (AR(1)) with a root greater than unity. It is shown that, contrary to the AR coe¢ cient, the leastsquares (LS) estimator of the intercept and its t-statistic are asymptotically normal without requiring the Gaussian error distribution, and hence an invariance principle applies. While the invariance principle does not apply to the asymptotic distribution of the LS estimator of the AR coe¢ cient, we show explicitly how it depends on the initial condition and the intercept. Also established are the asymptotic independence between the LS estimators of the intercept and the AR coe¢ cient and the asymptotic independence between their t-statistics. Asymptotic theory for explosive processes is compared to that for unit root AR(1) processes and stationary AR(1) processes. The coe¢ cient based test and the t test have better power for testing the hypothesis of zero intercept in the explosive process than in the stationary process.