Partial likelihood methods for panel data: Difference between revisions

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Partial (pooled) likelihood estimation for [[panel data]] is a [[Quasi-maximum likelihood estimate|quasi-maximum likelihood]] method for [[panel analysis]] that assumes that density of ''y_it'' given ''x_it'' is correctly specified for each time period but it allows for misspecification in the conditional density of ''y_i≔(y_i1,…...,y_iT) given x_i≔(x_i1,…...,x_iT)''.

==Description==

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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[ ∑Σ_t∇²_θ logf_t (y_it│x_it ; θ)]⁻¹ and B=E[( ∑Σ_t∇_θ logf_t (y_it│x_it ; θ) ) ( ∑Σ_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 probability distribution#Joint density function or mass function|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" />

== Pooled QMLE for Poisson models==