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 max-imum 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 al-ternative 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 sam-ple 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