New tools for understanding the local asymptotic power of panel unit root tests☆
Section snippets
Motivation
Consider the problem of testing for a unit root in the panel data variable , and assume for simplicity that the data generating process (DGP) is given by , where and . The analysis of the local power of various unit root test statistics when applied to such variables has attracted much attention in recent years (see Westerlund and Breitung, 2013, Section 2, for a review of this literature). The limit theory makes extensive use of the laws of large
Model
The DGP is similar to the one considered in Section 1 and is given by where , is independently and identically distributed (iid) with , and . In the derivations we assume that is known (as in, for example, Moon et al., 2007); hence, we can just as well set . Also, is a -dimensional vector of trends, for which we consider three specifications; (i) no deterministic terms (), (ii)
Main results
Let us introduce the OLS detrending operator , which is such that where . Define the following sample quantities based on the OLS detrended data: Let with a similar definition of .
Almost all (within type) panel unit root tests statistics considered in the literature can be written in terms of and . The most common statistic by far is the
Illustrations
The purpose of this section is to illustrate how the results reported in Section 3 can be used in deriving IO asymptotic distributions of panel unit root test statistics, and also to show how these compare to the corresponding FO distributions reported in the literature. The test statistics that we will consider are all based on and . In this section we therefore begin by deriving the asymptotic distributions of some useful transformations of these quantities. We then show how the
Conclusion
Recently, much effort has been directed toward the analysis of the local power of panel unit root tests. The main thrust of this paper is that the conventional FO asymptotic analysis, in which only the leading term in the local power function is considered, can be a rather unreliable guide to what happens in practice. While this observation is in itself nothing new but has been made in several Monte Carlo studies (see, for example, Moon and Perron, 2008, Moon et al., 2007), as far as we are
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A previous version of the paper was presented at a seminar at University of Barcelona. The authors would like to thank seminar participants, and in particular Cheng Hsiao (Editor) Peter Phillips, Giuseppe Cavaliere, Josep Carrion-i-Silvestre, David Harris, Vasilis Sarafidis, an Associate Editor, and two anonymous referees for many valuable comments and suggestions.