Fixed and random effects in nonlinear panel data model: A discussion of a paper by Manuel Arellano and Jinyong Hahn

INTRODUCTION It was a pleasure to serve as the discussant for this session. The authors have played a major role in developing the area under discussion. The paper is very good so that my role is to comment rather than criticize. I will discuss a connection between reducing the estimation bias in a fixed effects model versus reducing misspecification bias in a random effects model. For this discussion, I use the framework of Woutersen (2002). Woutersen (2002) proposes to use a moment that approximately separates the parameter of interest from the individual parameters. Let zi denote that data on individual i, let αi be the individual parameter and let θ be the common parameter. Then, for any likelihood model, a moment function g (α, θ) = σi g i (α, θ, z i)/N can be constructed with the properties (i) Eg (α0, θ0) = 0 where {α0, θ0} denote the true parameter values and (ii) Eg αi (α0, θ0) = 0 for all i, that is, the partial derivative of the moment function with respect to αi is zero for all i. Condition (ii) means that the moment function depends somewhat less on α. The moment function g (α, θ) is then integrated with respect to the likelihood L i (α, θ) and the prior π(αi;θ), Inference is then based on the integrated moment by minimizing with respect to θ.