Partial likelihood methods for panel data

Partial (pooled) likelihood estimation for panel data 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). Concretely, partial likelihood estimation uses the product of conditional densities as the density of the joint conditional distribution. This generality facilitates maximum likelihood methods in panel data setting because fully specifying conditional distribution of y_i can be computationally demanding. On the other hand, allowing for misspecification generally results in violation of information equality and thus requires robust standard error estimator for inference.

In the following exposition, we follow the treatment in Wooldridge. Particularly, the asymptotic derivation is done under fixed-T, growing-N setting.

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

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 (details in Wooldridge ), the asymptotic variance of √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.