The Chi-Square Approximation of the Restricted Likelihood Ratio Test for the Sum of Autoregressive Coefficients with Interval Estimation

The restricted likelihood (RL) of an autoregressive (AR) process of order one with intercept/trend possesses enormous advantages, such as yielding estimates with significantly reduced bias, powerful unit root tests, small curvature and a well-behaved likelihood ratio test (RLRT ) near the unit root. We consider the likelihood ratio test based on the Restricted Likelihood (RLRT ) for the sum of the coefficients in AR(p) processes with intercept/trend. The limit of the leading error term in the chi-square approximation to the RLRT distribution is shown to be finite as the unit root is approached, suggesting a good approximation over the entire parameter space and well-behaved interval inference for nearly integrated processes. Our result is stronger than the pointwise first order result currently available for competing confidence interval procedures in the intercept/trend model. We extend the correspondence between the AR coefficients and the partial autocorrelations from the stationary to the unit root case. The resulting parameter space is the bounded p-dimensional hypercube (−1, 1] × (−1, 1)p−1, which greatly simplifies both computation and optimisation of the Restricted Likelihood (RL) as well as the computation of confidence intervals for the sum of the AR coefficients. In simulations, we show that the RLRT intervals have almost exact coverage and also shorter lengths and significantly higher power against stationary alternatives than other competing intervals. An application to the Nelson-Plosser data yields intervals that can be markedly different from those in the literature.