We provide a comprehensive set of new 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 a completely general parametric specification for the short run dynamics of the estimated (mis-specified) fractional model, and with long memory, short memory and antipersistence in both the model and the true data generating process accommodated. As well as providing new theoretical insights, we undertake an extensive set of numerical explorations, beginning with the numerical evaluation, and implementation, of the (common) asymptotic distribution that holds under the most extreme form of mis-specification. Simulation experiments are then conducted to assess the relative finite sample performance of all four mis-specified estimators, initially under the assumption of a known mean, as accords with the theoretical derivations. The importance of the known mean assumption is illustrated via the production of an alternative set of bias and mean squared error results, in which the estimators are applied to demeaned data. The paper concludes with a discussion of open problems.
We use the jackknife to bias correct the log-periodogram regression (LPR) estimator of the fractional parameter in a stationary fractionally integrated model. The weights for the jackknife estimator are chosen in such a way that bias reduction is achieved without the usual increase in asymptotic variance, with the estimator viewed as 'optimal' in this sense. The theoretical results are valid under both the non-overlapping and moving-block sub-sampling schemes that can be used in the jackknife technique, and do not require the assumption of Gaussianity for the data generating process. A Monte Carlo study explores the finite sample performance of different versions of the jackknife estimator, under a variety of scenarios. The simulation experiments reveal that when the weights are constructed using the parameter values of the true data generating process, a version of the optimal jackknife estimator almost always out-performs alternative semi-parametric bias-corrected estimators. A feasible version of the jackknife estimator, in which the weights are constructed using estimates of the unknown parameters, whilst not dominant overall, is still the least biased estimator in some cases. Even when misspecified short run dynamics are assumed in the construction of the weights, the feasible jackknife still shows significant reduction in bias under certain designs. As is not surprising, parametric maximum likelihood estimation out-performs all semi-parametric methods when the true values of the short memory parameters are known, but is dominated by the semi-parametric methods (in terms of bias) when the short memory parameters need to be estimated, and in particular when the model is misspecified.
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