Pooled QMLE for Poisson Models
Pooled QLME for Poisson Models
Pooled QMLE is a technique that allows estimating parameters when panel data is available with Poisson outcomes. For instance, one might have information on the number of patents files by a number of different firms over time. Pooled QMLE does not necessarily contain unobserved effects (which can be either Random Effects or Fixed Effects), and the estimation method is mainly proposed for these purposes. The computational requirements are less stringent, especially compared to Fixed Effects Poisson models, but the tradeoff is the possibly strong assumption of no unobserved heterogeneity. Pooled refers to pooling the data over the different time periods T, while QMLE refers to the Quasi-Maximum Likelihood Technique.
The Poisson distribution of yi given xi is specified as follows:
the starting point for Poisson pooled QMLE is the conditional mean assumption. Specifically, we assume that for some b0 in a compact parameter space B, the conditional mean is given by:
The compact parameter space condition is imposed to enable the use M-estimation techniques, while the conditional mean reflects the fact that the population mean of a Poisson process is the parameter of interest. In this particular case, the parameter governing the Poisson process is allowed to vary with respect to the vector xt. The function m(.) can, in principle, change over time even though it is often specified as static over time. Note that only the conditional mean function is specified, and we will get consistent estimates of b0 as long as this mean condition is correctly specified. This leads to the following first order condition, which represents the quasi-log likelihood for the pooled Poisson estimation:
A popular choice is m(xt,b0)=exp(xt b0), as Poisson processes are defined over the positive real line. This reduces the conditional moment to an exponential index function, where xt b0 is the linear index and exp(.) is the link function.
References
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
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
Wooldridge, J. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass.
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
Wooldridge, J. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass.
McCullagh, P. and J. A. Nelder (1989): Generalized Linear Models, CRC Monographs on Statistics and Applied Probability (Book 37), 2nd Edition, Chapman and Hall, London.