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

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.

Least Squares Bias in Time Series with Moderate Deviations from a Unit Root

“…As such, Theorem 1 depicts why it would be impossible to construct the jackknife for mildly explosive series for positive values of c that are close to zero. The issue can be tackled by either substituting what should be the correct explosive weights by the ones applied to stationary series (Kaufmann and Kruse, ) provide an interesting comparison between a number of estimators) or by considering local to unit root alternatives (Chambers and Kyriacou, ; Stoykov, ), where the approximate bias function has been shown to be continuous at c =0 (Phillips, ).…”