Partial likelihood methods for panel data: Difference between revisions

Revision as of 00:20, 26 July 2020

Revision as of 02:23, 26 July 2020

Line 12:

Writing the conditional density of y_it given ''x_it'' as ''f_t'' (''y_it'' | ''x_it'';θ), the partial maximum likelihood estimator solves:

: <math>\undersetmax_{\theta\in\Theta}{\operatorname{max}} \sum_{i=1}^N\sum_{t=1}^T \log f_t(y_{it} \mid x_{it}; \theta) </math>

: <math>

\underset{\theta\in\Theta}{\operatorname{max}}\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_i'' given ''x_i'' is modeled as ''Π_t'' ''f_t'' (''y_it'' | ''x_it'' ; θ). We assume that ''f_t (y_it |x_it ; θ)'' is correctly specified for each ''t'' = 1,...,''T'' and that there exists ''θ₀'' ∈ Θ that uniquely maximizes ''E[f_t (y_it│x_it ; 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.

By the usual argument for [[M-estimator]]s (details in Wooldridge <ref name= "Woolridge" />), the asymptotic variance of {{radic|''N''}} ''(θ_MLE- θ₀) is A⁻¹ BA⁻¹'' where ''A⁻¹ = E[ ∑_t∇²_θ logf_t (y_it│x_it ; θ)]⁻¹ and B=E[( ∑_t∇_θ logf_t (y_it│x_it ; θ) ) ( ∑_t∇_θ logf_t (y_it│x_it; θ ) )^T]''. If the joint conditional density of y_i given x_i 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. Therefore, for valid inference, the above formula for asymptotic variance should be used. 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.<ref name= "Woolridge" />

==References==