The Restricted Likelihood Ratio Test at the Boundary in Autoregressive Series

Abstract. The restricted likelihood ratio test, RLRT, for the autoregressive coefficient in autoregressive models has recently been shown to be second order pivotal when the autoregressive coefficient is in the interior of the parameter space and so is very well approximated by the χ21 distribution. In this paper, the non-standard asymptotic distribution of the RLRT for the unit root boundary value is obtained and is found to be almost identical to that of the χ21 in the right tail. Together, the above two results imply that the χ21 distribution approximates the RLRT distribution very well even for near unit root series and transitions smoothly to the unit root distribution