Jackknife Bias Reduction in the Presence of a Near-Unit Root

Marcus Chambers,Maria Kyriacou

doi:10.3390/econometrics6010011

Marcus Chambers, Maria Kyriacou

Open Access

https://doi.org/10.3390/econometrics6010011

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Journal: Econometrics	Publication Date: Mar 5, 2018
Citations: 8	License type: CC BY 4.0

Affiliation: University of Essex, University of Southampton

Abstract

This paper considers the specification and performance of jackknife estimators of the autoregressive coefficient in a model with a near-unit root. The limit distributions of sub-sample estimators that are used in the construction of the jackknife estimator are derived, and the joint moment generating function (MGF) of two components of these distributions is obtained and its properties explored. The MGF can be used to derive the weights for an optimal jackknife estimator that removes fully the first-order finite sample bias from the estimator. The resulting jackknife estimator is shown to perform well in finite samples and, with a suitable choice of the number of sub-samples, is shown to reduce the overall finite sample root mean squared error, as well as bias. However, the optimal jackknife weights rely on knowledge of the near-unit root parameter and a quantity that is related to the long-run variance of the disturbance process, which are typically unknown in practice, and so, this dependence is characterised fully and a discussion provided of the issues that arise in practice in the most general settings.

Highlights

Throughout his career, Peter Phillips has made important contributions to knowledge across the broad spectrum of econometrics and statistics, providing inspiration to many other researchers along the way
We explore the construction of jackknife estimators that eliminate fully the first-order bias in the near-unit root setting
We find that jackknife estimators can be constructed in the presence of a near-unit root that achieve this aim of bias reduction

Summary

Introduction

Throughout his career, Peter Phillips has made important contributions to knowledge across the broad spectrum of econometrics and statistics, providing inspiration to many other researchers along the way. With a near-unit root, the autoregressive parameter plays an important role, and it is necessary to derive the appropriate asymptotic expansion of sub-sample estimator bias for this more general case, as well as the MGFs of the relevant limiting distributions that can be used to construct the appropriate jackknife weights. The derivation of such results is challenging in itself and is a major reason why we focus on the bias-minimising jackknife, rather than attempting to derive results for the variance-minimising jackknife of Chen and Yu (2015).

The Model and the Standard Jackknife Estimator

Sub-Sample Properties

A Moment Generating Function and Its Properties

An Optimal Jackknife Estimator

Conclusions