Large Sample Properties of the Three-Step Euclidean Likelihood Estimators Under Model Misspecification

This paper studies the three-step Euclidean likelihood (3S) estimator and its corrected version as proposed by Antoine, Bonnal and Renault (2007) in globally misspecified models. We establish that the 3S estimator stays sqrt-consistent and asymptotically Gaussian. The discontinuity in the shrinkage factor makes the analysis of the corrected-3S estimator harder to carry out in misspecified models. We propose a slight modification to this factor to control its rate of divergence in case of misspecification. We show that the resulting modified-3S estimator is also higher order equivalent to the maximum empirical likelihood (EL) estimator in well specified models and sqrt-consistent and asymptotically Gaussian in misspecified models. Its asymptotic distribution robust to misspecification is also provided. Because of these properties, both the 3S and the modified-3S estimators could be considered as computationally attractive alternatives to the exponentially tilted empirical likelihood estimator proposed by Schennach (2007) which also is higher order equivalent to EL in well specified models and sqrt-consistent in misspecified models