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→Pooled QMLE for Poisson models: In QMLE, the first derivative of the quasi-log likelihood has a solution for the parameters when it equals zero. The quasi-log likelihood itself will never equal zero but, instead, is maximized. The change is minimal to reflect that the quasi-log likelihood is correct but should not be set to zero. |
KuyaMoHirowo (talk | contribs) m Fixed lint errors: missing end tag |
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: <math>\max_{\theta\in\Theta} \sum_{i=1}^N\sum_{t=1}^T \log f_t(y_{it} \mid x_{it}; \theta) </math>
In this formulation, the joint conditional density of ''y<sub>i</sub>'' given ''x<sub>i</sub>'' is modeled as ''Π<sub>t</sub>'' ''f<sub>t</sub>'' (''y<sub>it</sub>'' | ''x<sub>it</sub>'' ; θ). We assume that ''f<sub>t</sub> (y<sub>it</sub> |x<sub>it</sub> ; θ)'' is correctly specified for each ''t'' = 1,...,''T'' and that there exists ''θ<sub>0</sub>'' ∈ Θ that uniquely maximizes ''E[f<sub>t</sub> (y<sub>it</sub>│x<sub>it</sub> ; θ)]''.
But, it is not assumed that the joint conditional density is correctly specified. Under some regularity conditions, partial MLE is consistent and asymptotically normal.
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