Bias Correction of Semiparametric Long Memory Parameter Estimators via the Prefiltered Sieve Bootstrap

Abstract

This paper investigates bootstrap-based bias correction of semiparametric estimators of the long memory parameter, d, in fractionally integrated processes. The resampling method involves the application of the sieve bootstrap to data prefiltered by a preliminary semiparametric estimate of the long memory parameter. Theoretical justification for using the bootstrap technique to bias adjust log periodogram and semiparametric local Whittle estimators of the memory parameter is provided in the case where the true value of d lies in the range 0 ≤ d < 0.5. That the bootstrap method provides confidence intervals with the correct asymptotic coverage is also proven, with the intervals shown to adjust explicitly for bias, as estimated via the bootstrap. Simulation evidence comparing the performance of the bootstrap bias correction with analytical bias-correction techniques is presented. The bootstrap method is shown to produce notable bias reductions, in particular when applied to an estimator for which some degree of bias reduction has already been accomplished by analytical means.

Higher-order improvements of the sieve bootstrap for fractionally integrated processes

“…an estimated sampling distribution for the mean with a variance that is closer to the theoretical value. As noted in Section 4, the PFSBS algorithm is based on a pre-filtering value of d that is deemed to be "optimal" in the matching experimental design in Poskitt et al (2012). Figure 3 graphs the Monte Carlo distribution of T 1/2−d (ȳ T − µ), the PFSBS distribution of T 1/2−d (ȳ * T −ȳ T ), and the N(0, ω 2 ) distribution, for T = 500, φ = 0.6, and d = 0, 0.2, 0.3, 0.4.…”

Optimal bias correction of the log-periodogram estimator of the fractional parameter: A jackknife approach

“…Andrews and Guggenberger (2003), for example, include additional frequencies, to degree 2r for r ≥ 0, in the log-periodogram regression that defines the LPR estimator, producing an estimator (denoted hereafter by d AG r ) whose bias converges to zero at a faster rate than that of the LPR estimator (recovered by setting r = 0), when r > 1. Alternative analytical procedures appear in Moulines and Soulier (1999), Hurvich and Brodsky (2001) and Robinson and Henry (2003), whilst a method based on the pre-filtered sieve bootstrap has been introduced by Poskitt et al (2016). Critically, all such biascorrection methods come at a cost: namely, an increase in asymptotic variance.…”

“…Notably, Guggenberger and Sun (2006) produce a weighted average of LPR estimators over different bandwidths that achieves the same degree of bias reduction as d AG r for any given r, but with less variance inflation. This estimator, along with that of Poskitt et al (2016), serve as important comparators for the alternative bias-corrected estimator that we develop herein.…”

“…Extensive simulation exercises are conducted in order to compare the finite sample performance of the jackknife estimator with that of alternative approaches, including the bias-adjusted estimators of Guggenberger and Sun (2006) and Poskitt et al (2016). Results show that certain versions of the optimally bias-corrected jackknife estimator outperform the alternative bias-adjusted estimators of Guggenberger and Sun and Poskitt et al, in terms of bias-reduction and root mean squared error (RMSE), with the RMSE being somewhat close to, or even smaller than, that of the LPR in some cases.…”

“…] and the number τ r is as defined in Andrews and Guggenberger (2003). Details regarding the construction of the pre-filtered sieve bootstrap estimator ( d P F SB ) can be found in Poskitt et al (2016). In implementing this method, we set the number of bootstrap samples to B = 1000.…”