# Optimal Bias Correction of the Log-periodogram Estimator of the Fractional Parameter: A Jackknife Approach

## TL;DR

This paper introduces an optimal jackknife bias correction method for the log-periodogram estimator of the fractional parameter, which reduces bias without increasing variance, validated through theoretical analysis and simulations.

## Contribution

It develops a bias correction technique using the jackknife that is optimal in bias reduction without variance inflation, applicable under non-Gaussian data and various sampling schemes.

## Key findings

- Optimal jackknife estimator outperforms alternatives when true parameters are known.
- Feasible jackknife reduces bias even with estimated parameters.
- Parametric MLE performs best with known parameters but is sensitive to misspecification.

## Abstract

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.