Likelihood Based Estimation in a Panel Setting

This paper considers the extension to panel data of models that are specified cross-sectionally in terms of a likelihood. It considers specifically the estimation of stochastic frontier models but the same issue arises in many other models. The model can be estimated for any single value of the time-index t by maximizing a likelihood that depends on the distribution of yit given xit. Estimation in the panel could be based on the joint distribution of yi1,...,yiT given xi1,...,xiT. Many different joint distributions may exist that imply the given marginal distributions of the yit separately, however, and except in the normal case none is "obviously" correct. The paper observes the well-known fact that maximizing a quasi-likelihood that assumes independence yields consistent estimates, and it shows how to obtain asymptotically correct standard errors. It shows how to use GMM methods to improve on the quasi-MLE, without assuming any specific form of the joint distribution, and derives the condition under which there is or is not an improvement. Finally, it shows how copulas can be used to construct joint distributions. It addresses the question of whether or not there are any copulas with the robustness property that the quasi-MLE is consistent even if the assumed copula is incorrect.