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] This formula looks very similar to the standard Poisson premultiplied by the term ai.
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.
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.