Fixed-effect Poisson model

In statistics, fixed-effect Poisson models are used for static panel data when the outcome variable is count data. Hausman, Hall, and Griliches pioneered the method in the mid 1980s. Their outcome of interest was the number of patents filed by firms, where they wanted to develop methods to control for the firm fixed effects.[1] Linear panel data models use the linear additivity of the fixed effects to difference them out and circumvent the incidental parameter problem. Even though Poisson models are inherently nonlinear, the use of the linear index and the exponential link function lead to multiplicative separability, more specifically [2]

E[y_it ∨ x_i1... x_iT, c_i ] = m(x_it, c_i, b₀ ) = exp(c_i + x_it b₀ ) = a_i exp(x_it b₀ ) = μ_ti (1)

This formula looks very similar to the standard Poisson premultiplied by the term a_i. As the conditioning set includes the observables over all periods, we are in the static panel data world and are imposing strict exogeneity.[3] Hausman, Hall, and Griliches then use Andersen's conditional Maximum Likelihood methodology to estimate b₀. Using n_i = ∑ y_it allows them to obtain the following nice distributional result of y_i

y_i ∨ n_i, x_i, c_i ∼ Multinomial (n_i, p₁ (x_i, b₀), ..., p_T (x_i, b₀ )) (2) where

${\displaystyle p_{t}(x_{i},b_{0})={\frac {m(x_{it},b_{0})}{\sum m(x_{it},b_{0})}}.\quad }$[4]

At this point, the estimation of the fixed-effect Poisson model is transformed in a useful way and can be estimated by maximum-likelihood estimation techniques for multinomial log likelihoods. This is computationally not necessarily very restrictive, but the distributional assumptions up to this point are fairly stringent. Wooldridge provided evidence that these models have nice robustness properties as long as the conditional mean assumption (i.e. equation 1) holds.[5] Chamberlain also provided semi-parametric efficiency bounds for these estimators under slightly weaker exogeneity assumptions. However, these bounds are practically difficult to attain, as the proposed methodology needs high-dimensional nonparametric regressions for attaining these bounds.