Elsevier

Journal of Econometrics

Volume 188, Issue 1, September 2015, Pages 59-93
Journal of Econometrics

New tools for understanding the local asymptotic power of panel unit root tests

https://doi.org/10.1016/j.jeconom.2015.03.043Get rights and content

Abstract

Motivated by the previously documented discrepancy between actual and predicted power, the present paper provides new tools for analyzing the local asymptotic power of panel unit root tests. These tools are appropriate in general when considering panel data with a dominant autoregressive root of the form ρi=1+ciNκTτ, where i=1,,N indexes the cross-sectional units, T is the number of time periods and ci is a random local-to-unity parameter. A limit theory for the sample moments of such panel data is developed and is shown to involve infinite-order series expansions in the moments of ci, in which existing theories can be seen as mere first-order approximations. The new theory is applied to study the asymptotic local power functions of some known test statistics for a unit root. These functions can be expressed in terms of the expansions in the moments of ci, and include existing local power functions as special cases. Monte Carlo evidence is provided to suggest that the new results go a long way toward bridging the gap between actual and predicted power.

Section snippets

Motivation

Consider the problem of testing for a unit root in the panel data variable {yi,t}t=1T, and assume for simplicity that the data generating process (DGP) is given by yi,t=ρiyi,t1+εi,t, where yi,0=0 and εi,tN(0,1). 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 yi,t=βidtp+ui,t,ui,t=ρiui,t1+εi,t, where ui,0=0, εi,t is independently and identically distributed (iid) with E(εi,t)=0, E(εi,t2)=σε2>0 and E(εi,t4)<. In the derivations we assume that σε2 is known (as in, for example, Moon et al., 2007); hence, we can just as well set σε2=1. Also, dtp=(1,,tp) is a (p+1)-dimensional vector of trends, for which we consider three specifications; (i) no deterministic terms (p=1), (ii)

Main results

Let us introduce the OLS detrending operator Dp, which is such that Dpyi,t=yi,tk=1Tyi,kak,tp, where ak,tp=dkp(n=2Tdnpdnp)1dtp. Define the following sample quantities based on the OLS detrended data: AiT,p=1Tt=2TDpyi,t1DpΔyi,t,BiT,p=1T2t=2T(Dpyi,t1)2. Let A¯NT,p=N1i=1NAiT,p with a similar definition of B¯NT,p.

Almost all (within type) panel unit root tests statistics considered in the literature can be written in terms of A¯NT,p and B¯NT,p. The most common statistic by far is the t

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 A¯NT,p and B¯NT,p. 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.

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