# Incidental Parameters and Poisson Regression

Consider: $$\ln(E[Y|X])=X_{it}'\beta+\alpha_i$$ and thus $$E[Y|X]=e^{X_{it}'\beta+\alpha_i}$$. We can write this regression model as:

$$Y_{it} =e^{X_{it}'\beta+\alpha_i}\eta_{it}$$

For which the contemporaneous exogeneity assumption is $$E[\eta_{it}|X_{it}]=1$$.

Wikipedia claims this does not suffer from the incidental parameters problem, showing it could be written:

$$Y_{it} =e^{X_{it}'\beta}\mu_i\eta_{it}$$ where $$\mu_i=e^{\alpha_i}$$. Seeing more of a proof or justification would be useful.

If $$N\rightarrow \infty$$ as $$T$$ is fixed, how can I show that $$\hat{\beta}$$ is consistent even without a consistent estimate for $$\alpha_i$$?

Interestingly, it was shown by Martin (2017), that despite the impossibility to estimate consistently the $$c_i$$ terms (with fixed $$T$$), the marginal partial effects which depends on the $$c_i$$, can be estimated consistently: