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Question 18.12 in Wooldridge's Econometric Analysis of Cross Section and Panel Data describes a "fixed effects gamma estimator", which appears to be analogous to a fixed effects Poisson estimator, with the obvious change in distribution assumption.

I am interested in using this estimator in another context, and would like to know something about its properties, but this textbook problem is the only reference I can find. I've tried to find more information about it, with no success at all. Can anybody point me to a book, a published paper, a dissertation, anything?

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  • $\begingroup$ Please add self-study as a tag and provide details about the problem and your issues. $\endgroup$ – Xi'an Mar 30 '18 at 5:58
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    $\begingroup$ @Xi'an: I didn't add the self-study tag because I don't think the self-study tag is appropriate. I am not asking for help with an exercise; I am not even trying to do the exercise. $\endgroup$ – The Laconic Mar 30 '18 at 13:54
  • $\begingroup$ @Xi'an: I have edited the question to make this clearer. $\endgroup$ – The Laconic Mar 30 '18 at 13:57
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I'm going to provide a partial answer to my own question.

It helps if we look at consider the fixed effects Poisson (FEP) estimator first. The following is based on exercise 19.7 in Wooldridge. I'll just sketch it out because I don't want to provide a full answer to someone's homework!

Fixed effects Poisson

The standard FEP assumptions are that, conditional on $(\mathbf{x}_i,c_i)$, $y_1 \ldots y_T$ are independent Poisson variables with means $c_i m(\mathbf{x}_i,\beta)$. Given that, the conditional log likelihood for unit $i$ is

$l_i(c_i,\beta) = \sum_{t=1}^{T} \left[ -c_i m(x_{it},\beta) + y_{it} \log(c_i) + y_{it} \log(m(x_{it},\beta)) \right] $

where we're treating $c_i$ here as a parameter to estimate along with $\beta$, and we omit a term that doesn't depend on $c_i$ or $\beta$.

Note: I think the problem in the textbook has a typo here. In the text, $y_{it}$ does not multiply $\log(c_i m(x_{it},\beta))$.

Now, just to see what happens, let's not do the usual conditional maximum likelihood approach for FEP, and instead treat $c_i$ as parameters to estimate along with $\beta$. If we let $n_i=\sum_{t=1}^{T} y_{it}$, assume that $n_i>0$, and maximize $l_i$ with respect to $c_i$. It's easy to show that we obtain

$c_i(\beta) = \frac{n_i}{\sum_{\tau=1}^{T} m(x_{i \tau},\beta)}$

If we plug this back into the log likelihood, we get log likelihood as a function of $\beta$ alone:

$l_i(c_i(\beta),\beta) = \sum_{t=1}^{T} \left[ -\frac{n_i m(x_{it},\beta)}{\sum_{\tau=1}^{T} m(x_{i \tau},\beta)} + y_{it} \log(\frac{n_i m(x_{it},\beta)}{\sum_{\tau=1}^{T} m(x_{i \tau},\beta)}) \right] $

Defining $p_t(\mathbf{x_i},\beta) = \frac{m(x_{it},\beta)}{\sum_{\tau=1}^{T} m(x_{i \tau},\beta)}$, we have

$l_i(c_i(\beta),\beta) = \sum_{t=1}^{T} \left[ -n_i p_t(\mathbf{x_i},\beta) + y_{it} \log(n_i p_t(\mathbf{x_i},\beta)) \right] $

which simplifies to

$l_i(c_i(\beta),\beta) = \sum_{t=1}^{T} y_{it} \log(p_t(\mathbf{x_i},\beta)) -n_i + n_i \log(n_i)$

Note: Again, I think the text has a typo here. It has $(n_i-1)\log n_i$ instead of $n_i (\log (n_i) -1)$.

Now note that

$l_i(c_i(\beta),\beta) = \sum_{t=1}^{T} y_{it} \log(p_t(\mathbf{x_i},\beta))$

is the conditional log likelihood for unit $i$ under the standard conditional MLE approach to estimating the FEP model. So the log likelihood is the same up to terms that don't depend on $\beta$. In other words, the standard conditional MLE approach is equivalent to (yields the same estimator as) a MLE approach, i.e. just throwing in dummy variables for each unit along with the original explanatory variables $\mathbf{x_i}$.

Fixed effects gamma

This property does not hold for the fixed effects gamma model. The following is based on exercise 19.12 in Wooldridge. Again, I'll just sketch it out.

The fixed effects gamma assumptions are that, conditional on $(\mathbf{x}_i,c_i)$, $y_1 \ldots y_T$ are independent gamma variables with means $c_i m(\mathbf{x}_i,\beta)$ and variances $c_i^2 m(\mathbf{x}_i,\beta)$, i.e.

$y_{it} | \mathbf{x_i}, c_i \sim Gamma(m(\mathbf{x}_i,\beta), 1/c_i)$.

One can show, with a little work, that the density of $y_{i1} \ldots y_{iT}$ conditional on $n_i=\sum_{t=1}^{T} y_{it}$ does not depend on $c_i$. This leads to the fixed effects gamma estimator, which is based on the conditional log likelihood

$l_i(\beta) = \log \Gamma \left( \sum_{t=1}^T m(x_{it},\beta) \right) - \sum_{t=1}^T \log \Gamma \left( m(x_{it},\beta) \right) + \sum_{t=1}^T m(x_{it},\beta) \log y_{it} - \left(\log n_i\right) \sum_{t=1}^T m(x_{it},\beta) $

up to terms that do not depend on $\beta$.

If we try the other approach, where we maximize the full likelihood with respect to $c_i$, and then plug that in, we again obtain

$c_i(\beta) = \frac{n_i}{\sum_{\tau=1}^{T} m(x_{i \tau},\beta)}$

However, the resulting log likelihood with the $c_i$ concentrated out is

$l_i(c_i(\beta),\beta) = \left( \sum_{t=1}^T m(x_{it},\beta) \right) \log \left(\sum_{t=1}^T m(x_{it},\beta)\right) - \sum_{t=1}^T m(x_{it},\beta) + \left( \sum_{t=1}^T m(x_{it},\beta) \right) - \sum_{t=1}^T \log \Gamma \left( m(x_{it},\beta) \right) + \sum_{t=1}^T m(x_{it},\beta) \log y_{it} - \left(\log n_i\right) \sum_{t=1}^T m(x_{it},\beta) $

up to terms that do not depend on $\beta$. This is not the same (up to terms that do not depend on $\beta$) as the conditional ML. If the conditional ML is consistent, then the "full likelihood with estimates for $c_i$ plugged in" is not consistent. However, I think this is in the $N \rightarrow \infty$ limit. The first two terms in the "full + plugin" likelihood look like Stirling's approximation for the first term in the conditional likelihood. So I suspect that, for what it's worth, the second estimator is consistent in the $T \rightarrow \infty$ limit.

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