Asymptotic Efficiency of Joint Estimator Relative to Two-Stage Estimator Under Misspecified Likelihoods

The two-stage estimator is often more tractable when there are nuisance parameters that can be separately estimated and plugged into an objective function. The joint estimator tends to bear the higher computational cost since it estimates all parameters in one stage by optimizing the sum of objective functions used in the two stages. It is well-known that the joint estimator is asymptotically more efficient than the two-stage estimator if the objective function is the true log-likelihood. When the objective function is not the true log-likelihood, I show that the relative asymptotic efficiency of the joint estimator still holds under a finite number of testable moment conditions. The implications of the main result on models based on quasi-limited information likelihoods are discussed.