Elsevier

Economics Letters

Volume 174, January 2019, Pages 5-8
Economics Letters

Testing additive versus interactive effects in fixed-T panels

https://doi.org/10.1016/j.econlet.2018.10.016Get rights and content

Highlights

  • The paper considers the very relevant problem of testing additive versus interactive effects.

  • Unlike in Bai (2009), who proposes a similar test, here T is fixed.

  • The asymptotic distribution of the test is derived.

  • Monte Carlo results are provided to show that the tests performs well in small samples.

Abstract

This paper proposes a new test of the null hypothesis of additive fixed effects versus the alternative of general interactive effects. This is not the only test of its kind in the literature; however, it is the only test that is valid in panels where T is fixed. The asymptotic distribution of the new test statistic is derived and simulation results are provided to suggest that it performs well in small samples.

Introduction

Consider the scalar and k×1 vector of variables yi,t and xi,t, respectively, observable across t=1,,T time periods and i=1,,N cross-section units. The use of such panel data variables in regression analysis has attracted considerable attention. A major reason for this is the ability to deal with the presence of unobserved heterogeneity in yi,t, and the problem that this causes when said heterogeneity is correlated with the regressors in xi,t. The state of the art is a so-called “interactive effects” (IEs) model of the following form (see Chudik and Pesaran, 2015 for a recent survey of the literature): yi,t=xi,tβ+γiFt+εi,t,xi,t=ΓiFt+vi,t, where Ft is a m×1 vector of unobservable common factors with γi and Γi being m×1 and m×k matrices of factor loadings, respectively, and εi,t and vi,t are scalar and k×1 vector of idiosyncratic errors, respectively. The IEs are here given by γiFt and ΓiFt. Because of the way that these effects enter both (1), (2) the estimation of β is nontrivial, as xi,t is endogenous. The most common approach by far is to assume the IEs are really additive effects (AEs), which can be accommodated by simply transforming yi,t and xi,t into deviations from means, and there is a huge literature based on this approach. In the above notation, the AEs model is obtained by setting Ft=[1,ηt], γi=[θi,1] and Γi=[Θi,1], such that γiFt=θi+ηt and ΓiFt=Θi+ηt. AEs can therefore be seen as a restriction on the more general IEs model in (1), (2). This is important because while extremely common, in practice the AEs restriction is hardly ever tested, and the little checking that is being done is based on informal residual diagnostics (see, for example, Eberhardt and Presbitero, 2015, Holly et al., 2010).

Bai (2009) is among the first to formally test the hypothesis of AEs versus IEs. Unfortunately, his test is only applicable in panels where both N and T are large, which is rarely the case in practice. Most economic data sets have a peculiar structure. In particular, while the number of time periods for which there is reliable data is limited and cannot be increased other than by the passage of time, statistical agencies keep publishing already existing but previously unavailable time series data for individuals, firms, countries and regions. Thus, while N can potentially be very large, T is usually quite small.

The present paper can be seen as a reaction to the above mentioned problem. The purpose is to develop a test of AEs versus IEs that is applicable even if T is small and only N is large. The test should not only support standard chi-squared inference, but should also be easy to implement, and have good small-sample properties. We propose a test that fits this bill, and study its finite sample and asymptotic properties.

Section snippets

The test and its asymptotic properties

The test statistic that we will consider is based on the Hausman principle, whereby two estimators of β are compared.1

Monte Carlo simulations

In this section, we report the results from a small-scale Monte Carlo simulation exercise. The DGP is given by (1), (2) with k=1, β=1, and ui,t=ρui,t1+ei,t, where ei,tN(02×1,Σe,t), Σe,tdiag[U(1,2),U(1,2)], ei,0=02×1 and ρ{0.5,0.8,1}. Hence, ui,t is persistent and heteroskedastic. We further assume that there is a single factor (m=1) such that FtN(1,σF2) with σF2{0,1,2,4}, and where the loadings are generated as γi=ΓiN(1,1). If σF2=0, the IEs reduce to cross-section fixed effects, and so

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Cited by (6)

The author would like to thank Badi Baltagi (Editor) and one anonymous referee for many useful comments. Thank you also to the Knut and Alice Wallenberg Foundation for financial support through a Wallenberg Academy Fellowship.

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