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The article from Hausman, Hall, and Griliches (1984) "Econometric Models for Count Data with an Application to the Patents-R&D Relationship" has become the canonical example for conditional MLE of count panel model with fixed effects. In the Appendix B, Hausman and Newey develop a test for serial correlation for this fixed effects model. I need to implement or find an implementation of this test, and I'm having some trouble figuring out how to calculate $\hat{V}_m$, the asymptotic variance-covariance of this vector $m_i(\beta)$.

The last sentence says that B.3 is analogous to a Lagrange multiplier test and can be computed via a regression. What regression? It's not at all clear to me.

Barring that I'm trying to compute $\hat{V}_m$. The piece I'm missing now is how to compute $U_i{(\hat{\beta})}$. Is this just the score vector of the multinomial likelihood in 2.5? I think I must be mistaken, because it doesn't seem to me that this conformable with $m_i(\hat{\beta})$?

Link to the paper

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    $\begingroup$ The estimator that they propose in that paper doens't control for all time-invariant heterogeneity, according to this paper: www.ssc.upenn.edu/~allison/FENB.pdf Allison et al propose a simple LSDV poisson regression with a correction to the SE's post-estimation. $\endgroup$ Aug 9, 2014 at 9:38

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I still don't see the simple regression to get this LM test, but I just started coding and got through it. It looks like, yes, the $U_i(\hat{\beta})$ are the score function for each firm $i$ from the multinomial likelihood in 2.5. If you assume they're column vectors, then the math goes through. My results make sense at least.

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