Nearly efficient likelihood ratio tests of a unit root in an autoregressive model of arbitrary order

We study large-sample properties of likelihood ratio tests of the unit root hypothesis in an autoregressive model of arbitrary, finite order. Earlier research on this testing problem has developed likelihood ratio tests in the autoregressive model of order one, but resorted to a plug-in approach when dealing with higher-order models. In contrast, we consider the full model and derive the relevant large-sample properties of likelihood ratio tests under a local-to-unity asymptotic framework. As in the simpler model, we show that the full likelihood ratio tests are nearly efficient, in the sense that their asymptotic local power functions are virtually indistinguishable from the Gaussian power envelopes.