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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 $\alpha_i$?

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As you note in your comment, Wooldridge (1999) addresses your question:

Wooldridge, J.M., 1999, Distribution-free estimation of some nonlinear panel data models. Journal of Econometrics, 90, 77-97.

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:

Martin, Robert S., 2017, Estimation of average marginal effects in multiplicative unobserved effects panel models, Economics Letters, 160, 16-19.

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    $\begingroup$ I think Distribution-free estimation of some nonlinear panel data models by Wooldridge (1999) is what I wanted, but your answer helped me get there. Thank you! $\endgroup$ Jul 30 '21 at 20:49
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    $\begingroup$ could you please also provide an self contained answer? References are strongly encouraged but unless question is just reference request answer should answer the question as well not just provide reference for source with solution. $\endgroup$
    – 1muflon1
    Jul 30 '21 at 22:00
  • $\begingroup$ @Michael Gmeiner: I edited my answer to include your point. $\endgroup$
    – Bertrand
    Jul 31 '21 at 7:50

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