Testing additive versus interactive effects in fixed- panels☆
Introduction
Consider the scalar and vector of variables and , respectively, observable across time periods and 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 , and the problem that this causes when said heterogeneity is correlated with the regressors in . 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): where is a vector of unobservable common factors with and being and matrices of factor loadings, respectively, and and are scalar and vector of idiosyncratic errors, respectively. The IEs are here given by and . Because of the way that these effects enter both (1), (2) the estimation of is nontrivial, as 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 and 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 , and , such that and . 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 and 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 can potentially be very large, 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 is small and only 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 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 , , and , where , , and . Hence, is persistent and heteroskedastic. We further assume that there is a single factor () such that with , and where the loadings are generated as . If , the IEs reduce to cross-section fixed effects, and so
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2020, Journal of Applied Econometrics
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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.