Issues in the estimation of mis-specified models of fractionally integrated processes

This short paper provides a comprehensive set of new theoretical results on the impact of mis-specifying the short run dynamics in fractionally integrated processes. We show that four alternative parametric estimators – frequency domain maximum likelihood, Whittle, time domain maximum likelihood and conditional sum of squares – converge to the same pseudo-true value under common mis-specification, and that they possess a common asymptotic distribution. The results are derived assuming the true data generating mechanism is a fractional linear process driven by a martingale difference innovation. A completely general parametric specification for the short run dynamics of the estimated (mis-specified) fractional model is considered, and with long memory, short memory and antipersistence in both the model and the data generating mechanism accommodated. The paper can be seen as extending an existing line of research on mis-specification in fractional models, important contributions to which have appeared in Journal of Econometrics. It also complements a range of existing asymptotic results on estimation in correctly specified fractional models. Open problems in the area are the subject of the final discussion.