Statistics & Probability Letters

Volume 83, Issue 7, July 2013, Pages 1677–1682

Jackknife estimation with a unit root

• Marcus J. Chambers^{a, 1,} ,
• Maria Kyriacou^{b, ,}

• ^a Department of Economics, University of Essex, Wivenhoe Park, Colchester, Essex CO4 3SQ, England, United Kingdom
• ^b Economics Division, School of Social Sciences, University of Southampton, Highfield, SO17 1BJ, United Kingdom

Received 23 August 2012

Revised 1 February 2013

Accepted 20 March 2013

Available online 30 March 2013

• http://dx.doi.org/10.1016/j.spl.2013.03.016, How to Cite or Link Using DOI

Keywords

• Jackknife;
• Bias reduction;
• Unit root

1. Introduction

A longstanding bias reduction method whose properties have been less widely explored in autoregressive models is the jackknife (Quenouille, 1956 and Tukey, 1958). In recent work, Chambers (2013) investigated jackknife methods in a stationary autoregression and Phillips and Yu (2005) used the jackknife to estimate the parameters of a continuous time model and associated bond option prices. In all the above contributions the properties of the jackknife as a bias reduction device are confirmed by significant bias reductions in simulation experiments. As observed by Quenouille (1956) in his original paper, autoregressive parameter estimators typically suffer from negative bias. The nature of the bias in stationary autoregression has been extensively studied and its properties are well understood. For example, early contributions to this topic can be found in Marriott and Pope (1954), Kendall (1954) and Shenton and Johnson (1965). This bias is large in the unit root case and it is, therefore, particularly interesting to ascertain the extent of bias reduction that can be achieved by jackknife methods in this setting.

The focus of this paper is jackknife estimation in unit root regression. We consider the jackknife proposed by Phillips and Yu (2005) based on non-overlapping sub-samples which was found to perform well by Chambers (2013). In the presence of a unit root, we show that the jackknife in its usual formulation fails to fully eliminate the first-order bias. The source of this failure lies in the different limit distributions of the sub-samples. These distributions motivate a set of optimal jackknife weights that ensure the first-order bias is fully removed. Simulations reveal that the ‘optimal’ jackknife estimator proposed here produces further bias and root mean squared error (RMSE) reductions.