<p>As you note in your comment, Wooldridge (1999) addresses your question:</p>
<p>Wooldridge, J.M., 1999, Distribution-free estimation of some nonlinear panel data models. <em>Journal of Econometrics</em>, 90, 77-97.</p>
<p>Interestingly, it was shown by Martin (2017), that despite the impossibility to estimate consistently the <span class="math-container">$c_i$</span> terms (with fixed <span class="math-container">$T$</span>), the marginal partial effects which depends on the <span class="math-container">$c_i$</span>, can be estimated consistently:</p>
<p>Martin, Robert S., 2017, Estimation of average marginal effects in multiplicative unobserved effects panel models, <em>Economics Letters</em>, 160, 16-19.</p>

CC BY-SA 4.0
2021-07-30T20:49:01.823Z
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!
Michael Gmeiner
CC BY-SA 4.0
2021-07-30T22:00:41.617Z
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.
1muflon1
CC BY-SA 4.0
2021-07-31T07:50:48.707Z
@Michael Gmeiner:  I edited my answer to include your point.
Bertrand
CC BY-SA 4.0
2021-07-30T20:05:36.930Z
2021-07-31T07:46:49.467Z
2021-07-31T07:46:49.467Z
Bertrand
<p>Consider:
<span class="math-container">$\ln(E[Y|X])=X_{it}'\beta+\alpha_i$</span> and thus <span class="math-container">$E[Y|X]=e^{X_{it}'\beta+\alpha_i}$</span>.
We can write this regression model as:</p>
<p><span class="math-container">$$Y_{it} =e^{X_{it}'\beta+\alpha_i}\eta_{it} $$</span></p>
<p>For which the contemporaneous exogeneity assumption is <span class="math-container">$E[\eta_{it}|X_{it}]=1$</span>.</p>
<p><a href="https://en.wikipedia.org/wiki/Fixed-effect_Poisson_model" rel="noreferrer">Wikipedia</a> claims this does not suffer from the incidental parameters problem, showing it could be written:</p>
<p><span class="math-container">$$Y_{it} =e^{X_{it}'\beta}\mu_i\eta_{it} $$</span>  where <span class="math-container">$\mu_i=e^{\alpha_i}$</span>. Seeing more of a proof or justification would be useful.</p>
<p>If <span class="math-container">$N\rightarrow \infty$</span> as <span class="math-container">$T$</span> is fixed, how can I show that <span class="math-container">$\hat{\beta}$</span> is consistent even without a consistent estimate for <span class="math-container">$\alpha_i$</span>?</p>

<econometrics><mle>
Incidental Parameters and Poisson Regression
Michael Gmeiner
# Incidental Parameters and Poisson Regression

(Asked by: username_0 on 2021-07-30)

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](https://en.wikipedia.org/wiki/Fixed-effect_Poisson_model) 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 $\alpha\_i$?

**Tags:** econometrics, mle

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## 1 answers

### Answer by username_1 on 2021-07-30. Score: 4

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.

> **Comments:**
> **username_0**: 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!  
> **username_2**: 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.  
> **username_1**: @Michael Gmeiner:  I edited my answer to include your point.  

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