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On The Recovery Of Efficiency Losses: Maximum And Quasi-Maximum Likelihood Estimators

In this paper we develop a practical strategy for evaluating eeciency gains when estimating models with dynamic conditional expectations and dynamic conditional variances using the likelihood principle. We prove that the factors which contribute to diierences in eeciency between a maximum likelihood (ML) estimator and a quasi-maximum likelihood (QML) estimator are the Fisher information for location, the Fisher information for scale, and the coeecient of kurtosis of the probability density function of the error term. The quantiication of these three factors can lead to the construction of an ML variance-covariance matrix which recovers eeciency losses arising in QML settings. Furthermore, the researcher can take advantage of the estimation of the Fisher information for location and scale to test for density functional form. We propose both parametric and nonparametric tests for density functional form. We apply our proposed tests in the context of testing the null of a Student-t density and our approach can readily be extended to test for other families of densities. We carry out several Monte-Carlo experiments to examine the nite-sample properties of the proposed tests and to assess departures of the simulated nite-sample sampling distribution from that based upon asymptotic theory. Based on Monte-Carlo performance, we recommend the parametric test based on the Fisher information for location and the nonparametric test based on the Fisher information for scale to test for density functional form. We wish to thank Barry Arnold, Jeerey Wooldridge and Aman Ullah for their reading of the manuscript and their valuable insights. The usual caveat applies.