Expansions for approximate maximum likelihood estimators of the fractional difference parameter

Summary  This paper derives second-order expansions for the distributions of the Whittle and profile plug-in maximum likelihood estimators of the fractional difference parameter in the ‘Autoregressive Fractionally Integrated Moving Average of order (0, d, 0)' with unknown mean and variance. Both estimators are shown to be second-order pivotal. This extends earlier findings of Lieberman and Phillips (2004a, Econometric Theory, 20, 464–84), who derived expansions for the Gaussian maximum likelihood estimator under the assumption that the mean and variance are known. One implication of the results is that the parametric bootstrap upper one-sided confidence interval provides an o(n−1 ln n) improvement over the delta method. For statistics that are not second-order pivotal, the improvement is generally only of the order o(n−1/2 ln n).