Much of the analysis of panel data has been based on an assumption of strict exogeneity. Distributions are speciﬁed for outcome variables conditional on a latent individual effect and conditional on observed predictor variables at all dates, with the future values of the predictor variables assumed to have no effect on the conditional distribution. The paper relaxes this assumption in order to allow for lagged dependent variables and, more generally, for feedback from lagged dependent variables to current values of the predictor variables. Such feedback would arise in an evaluation study if the treatment variable is randomly assigned only conditional on the individual effect and on previous outcomes.
An information bound is derived for a semiparametric regression model with sequential moment restrictions, with the information set increasing over time. The bound is then applied to a model with a (scalar) multiplicative random effect. The mean of the random effect conditional on the predictor variables is not restricted, so that the random effect can control for various omitted variables. This conditional mean is the nonparametric component of the semiparametric regression model. There is a transformation that eliminates the random effect and leads to a set of sequential moment restrictions in which the moment function depends on only a ﬁnite-dimensional parameter. The information bound for this simpler problem coincides with that of the original problem. The form of the optimal instrumental variables is derived.
The paper also considers the identiﬁcation problems that arise when the random effect is a vector with two or more components.
This paper by Gary Chamberlain was initially released, in paper form only, in 1993 as a Harvard Institute for Economics Research (HIER) working paper. It became an underground classic but was not readily available online. Before his passing, Gary submitted this paper to a special issue of the Journal of Econometrics in his honor (where it is now forthcoming). We thank Gary’s daughter, Laura Gehl, for her support and encouragement in making some of Gary’s unpublished work more widely available to the economics community.
– Bryan Graham and Kei Hirano