Issues of misspecification in long memory models

Misspecification of the short memory dynamics in a long memory model has serious repercussions for the asymptotic properties of any estimator of the long memory parameter. Under misspecification, the estimator converges in probability to a value called the pseudo-true value, which is different from the true value of the parameter. Intuitively, of all the family of spectral densities, the spectral density with the pseudo-true value is the closest spectral density to the true spectral density. Further consequences of misspecification are associated with the rate of convergence and the asymptotic distribution of the estimator of the parameter of the misspecified model. Both the rate of convergence and the asymptotic distribution of the parametric estimator of the misspecified model depends, in turn, on the difference between the true and pseudo-true values. We prove that under misspecification, frequency domain maximum likelihood estimation, Whittle estimation, time domain maximum likelihood estimation and conditional sum of squares estimation are asymptotically equivalent. However, our simulation study demonstrates that in small and medium sized samples, the performance of the parametric estimators of the misspecified model, in terms of bias, mean squared error and the form of the sampling distribution, differs across estimators. Overall, under misspecification, the conditional sum of squares estimator outperforms the other parametric estimators in small and medium sized samples. Further, the approximate frequency domain maximum likelihood estimator is the least efficient of all parametric estimators of the misspecified model, overall. In certain circumstances, where the difference between the true and the pseudo-true value of the long memory parameter is sufficiently large, a clear distinction between the frequency domain and time domain estimators can be observed in small samples. However, as the sample size increases, the behaviour of all of the parametric estimators of the misspecified model is consistent with the theoretical asymptotic results. Whilst misspecified parametric estimators of the long memory parameter are inconsistent for its true value, any semi-parametric estimator is consistent, although very biased in small samples. Thus, we compare the parametric estimators of the long memory parameter in the misspecified model with the semi-parametric Geweke and Porter-Hudak (GPH) estimator, to investigate whether any misspecified parametric estimator is less biased, or more efficient, than this particular semi-parametric estimator to measure the true value of the long memory parameter in finite samples. The CSS estimator under the misspecified model outperforms the GPH estimator in large finite samples in terms of bias and mean squared error, when the misspecified model is close to the true model. If the misspecified model is substantially different from the true model, then the GPH estimator is preferred over the four parametric estimators of the misspecified model in finite samples