Partial (pooled) likelihood estimation for panel data is a quasi-maximum likelihood method for panel analysis that assumes that density of
is correctly specified for each time period but it allows for misspecification in the conditional density of
This generality facilitates maximum likelihood methods in panel data setting because fully specifying conditional distribution of yi can be computationally demanding.
[1] On the other hand, allowing for misspecification generally results in violation of information equality and thus requires robust standard error estimator for inference.
Writing the conditional density of yit given xit as ft (yit | xit;θ), the partial maximum likelihood estimator solves: In this formulation, the joint conditional density of yi given xi is modeled as Πt ft (yit | xit ; θ).
Under some regularity conditions, partial MLE is consistent and asymptotically normal.
If the joint conditional density of yi given xi is correctly specified, the above formula for asymptotic variance simplifies because information equality says B=A.
Yet, except for special circumstances, the joint density modeled by partial MLE is not correct.
For information equality to hold, one sufficient condition is that scores of the densities for each time period are uncorrelated.
In dynamically complete models, the condition holds and thus simplified asymptotic variance is valid.
[1] Pooled QMLE is a technique that allows estimating parameters when panel data is available with Poisson outcomes.
The computational requirements are less stringent, especially compared to fixed-effect Poisson models, but the trade off is the possibly strong assumption of no unobserved heterogeneity.
is specified as follows:[2] the starting point for Poisson pooled QMLE is the conditional mean assumption.
in a compact parameter space B, the conditional mean is given by[2] The compact parameter space condition is imposed to enable the use of 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
[3] Note that only the conditional mean function is specified, and we will get consistent estimates of
This leads to the following first order condition, which represents the quasi-log likelihood for the pooled Poisson estimation:[2] A popular choice is
, as Poisson processes are defined over the positive real line.
[3] This reduces the conditional moment to an exponential index function, where
is the linear index and exp is the link function.