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Cleaned up the math notation in the lead, converted to LaTeX |
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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The compact parameter space condition is imposed to enable the use of [[M-estimator|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 <math>x_{t}\centerdot</math>.<ref name="CameronTrivedi"/> The function ''m'' can, in principle, change over time even though it is often specified as static over time.<ref name="Woolbridge2002">Wooldridge, J. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass.</ref> Note that only the conditional mean function is specified, and we will get consistent estimates of <math>b_{0}</math> 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:<ref name="CameronTrivedi"/><!--not sure if the correct ref is this one or the previous one-->
: <math>\ell_i(b)=\sum[y_{it} \log(m(x_{it},b))-m(x_{it},b)]
A popular choice is <math>m=(x_t,b_0)=\exp(x_t b_0)</math>, as Poisson processes are defined over the positive real line.<ref name="Woolbridge2002"/> This reduces the conditional moment to an exponential index function, where <math>x_t b_0</math> is the linear index and exp is the link function.<ref>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.</ref>
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