Nonstationary Time Series with a Close Alternative Hypothesis: Locally Asymptotic Distribution of the Likelihood Ratio

Within the framework of the autoregressive model, we solve the problem of testing the hypothesis of the existence of the unit root of the characteristic polynomial against a close alternative hypothesis of this root to come infinitesimally close to unity. The asymptotic distribution of the likelihood ratio is calculated. This distribution is found to be substantially dependent on the information amount of the innovation noise distribution.