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{{Английская Википедия/Панель перехода}}
{{Short description|Statistical models used for static panel data}}
In [[statistics]], a '''fixed-effect Poisson model''' is a [[Poisson regression]] model used for static [[panel data]] when the outcome variable is [[count data]]. Hausman, Hall, and Griliches pioneered the method in the mid 1980s. Their outcome of interest was the number of patents filed by firms, where they wanted to develop methods to control for the firm [[fixed effects model|fixed effects]].<ref>Hausman, J. A., B. H. Hall, and Z. Griliches (1984): "Econometric Models for Count Data with an Application to the Patents-R&D Relationship." ''Econometrica'' (46), pp. 909–938</ref>  Linear panel data models use the linear additivity of the fixed effects to difference them out and circumvent the [[incidental parameter problem]]. Even though Poisson models are inherently nonlinear, the use of the linear index and the exponential link function lead to multiplicative [[Separation of variables|separability]], more specifically <ref>Cameron, C. A. and P. K. Trivedi (2015) "Count Panel Data," ''Oxford Handbook of Panel Data'', ed. by B. Baltagi, Oxford University Press, pp. 233–256</ref>

: E[''y''<sub>''it''</sub> ∨ ''x''<sub>''i''1</sub>... ''x''<sub>''iT''</sub>, ''c''<sub>''i''</sub> ] = ''m''(''x''<sub>''it''</sub>, ''c''<sub>''i''</sub>, ''b''<sub>0</sub> ) = exp(''c''<sub>''i''</sub> + ''x''<sub>''it''</sub> ''b''<sub>0</sub> ) = ''a''<sub>''i''</sub> exp(''x''<sub>''it''</sub> ''b''<sub>0</sub> ) = ''μ''<sub>''ti''</sub>				(1)

This formula looks very similar to the standard Poisson premultiplied by the term ''a<sub>i</sub>''. As the conditioning set includes the observables over all periods, we are in the static panel data world and are imposing [[strict exogeneity]].<ref>Wooldridge, J. (2002): ''Econometric Analysis of Cross Section and Panel Data,'' MIT Press, Cambridge, Mass.</ref>  Hausman, Hall, and Griliches then use Andersen's conditional Maximum Likelihood methodology to estimate ''b<sub>0</sub>''. Using ''n''<sub>''i''</sub> = Σ ''y''<sub>''it''</sub> allows them to obtain the following nice distributional result of  ''y<sub>i</sub>''

: ''y''<sub>''i''</sub> ∨ ''n''<sub>''i''</sub>, ''x''<sub>''i''</sub>, ''c''<sub>''i''</sub> ~ Multinomial (''n''<sub>''i''</sub>, ''p''<sub>1</sub> (''x''<sub>''i''</sub>, ''b''<sub>0</sub>), ..., ''p''<sub>''T''</sub> (''x''<sub>''i''</sub>, ''b''<sub>0</sub> ))						(2) where

: <math> p_t(x_i, b_0) = \frac{m(x_{it}, b_0)}{\sum m(x_{it}, b_0)}. \quad </math><ref>Andersen, E. B. (1970): "Asymptotic Properties of Conditional Maximum Likelihood Estimators." ''Journal of the Royal Statistical Society'', Series B, 32, pp. 283–301</ref>

At this point, the estimation of the fixed-effect Poisson model is transformed in a useful way and can be estimated by maximum-likelihood estimation techniques for [[multinomial distribution|multinomial]] log likelihoods. This is computationally not necessarily very restrictive, but the distributional assumptions up to this point are fairly stringent. Wooldridge provided evidence that these models have nice robustness properties as long as the conditional mean assumption (i.e. equation 1) holds.<ref>Wooldridge, J. M. (1999): "Distribution-Free Estimation of Some Nonlinear Panel Data Models." ''Journal of Econometrics'' (90), pp. 77–97</ref>  Chamberlain also provided [[semi-parametric efficiency bounds]] for these estimators under slightly weaker exogeneity assumptions. However, these bounds are practically difficult to attain, as the proposed methodology needs [[high-dimensional]] [[nonparametric regression]]s for attaining these bounds.

==See also==
*{{sectionlink|Partial likelihood methods for panel data#Example: pooled QMLE for Poisson models}}

==References==
{{Reflist}}

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