Issues in the Estimation of Mis-Specified Models of Fractionally Integrated Processes

In this paper we quantify the impact of model mis-specification on the properties of parameter estimators applied to fractionally integrated processes. We demonstrate the asymptotic equivalence of four alternative parametric methods: frequency domain maximum likelihood, Whittle estimation, time domain maximum likelihood and conditional sum of squares. We show that all four estimators converge to the same pseudo-true value and provide an analytical representation of their (common) asymptotic distribution. As well as providing theoretical insights, we explore the finite sample properties of the alternative estimators when used to fit mis-specified models. In particular we demonstrate that when the difference between the true and pseudo-true values of the long memory parameter is sufficiently large, a clear distinction between the frequency domain and time domain estimators can be observed - in terms of the accuracy with which the finite sample distributions replicate the common asymptotic distribution - with the time domain estimators exhibiting a closer match overall. Simulation experiments also demonstrate that the two time-domain estimators have the smallest bias and mean squared error as estimators of the pseudo-true value of the long memory parameter, with conditional sum of squares being the most accurate estimator overall and having a relative efficiency that is approximately double that of frequency domain maximum likelihood, across a range of mis-specification designs.