Efficient Likelihood Inference in Nonstationary Univariate Models

Recent literature shows that embedding fractionally integrated time series models with spectral poles at the long-run and/or seasonal frequencies in autoregressive frameworks leads to estimators and test statistics with nonstandard limiting distributions. However, we demonstrate that when embedding such models in a general I(d) framework the resulting estimators and tests regain desirable properties from standard statistical analysis. We show the existence of a local time domain maximum likelihood estimator and its asymptotic normality, and under Gaussianity asymptotic efficiency. The Wald, likelihood ratio, and Lagrange multiplier tests are asymptotically equivalent and chi-squared distributed under local alternatives. With i.i.d. Gaussian errors and a scalar parameter, we show that the tests in addition achieve the Gaussian power envelope of all invariant unbiased tests, i.e. they are uniformly most powerful invariant unbiased. In a Monte Carlo study we document the finite sample superiority of the likelihood ratio test