Estimation for Unit Root Testing

TL;DR

This paper revisits estimation methods for unit root testing, clarifies the correct two-step approach, introduces a new efficient two-step DF autoregression, and improves robustness by incorporating missing observations.

Contribution

It introduces a new two-step DF autoregression that is correctly specified and efficient, addressing shortcomings of existing methods in unit root testing.

Findings

The usual one-step approach is prone to misspecification.

The new two-step method is always correctly specified and efficient.

Incorporating missing observations enhances robustness.

Abstract

We revisit estimation and computation of the Dickey Fuller (DF) and DF-type tests. Firstly, we show that the usual one step approach, based on the "DF autoregression", is likely to be subject to misspecification. Secondly, we clarify a neglected two step approach for estimation of the DF test. (In fact, we introduce a new two step DF autoregression.) This method is always correctly specified and efficient under the circumstances. However, it is either neglected or misused in unit root testing literature. The commonly employed hybrid of the (correct) two step method is shown to be inefficient, even asymptotically. Finally, we further improve/robustify the proposed two step method by employing the missing initial observations. Our finally proposed method is to be used in unit root testing, since it is a new DF autoregression that retains the missing observations.