In this paper it is argued that maximizing the complete data (log) likelihood function with respect to the missing data and the unknown parameters will not improve the efficiency of the estimators but may affect consistency instead. If no auxiliary relations are available or additional assumptions are made, the maximum likelihood estimator based on the observed data is (asymptotically) the most efficient estimator.

econometric models
Bayesian Analysis (jel C11),
Economics Letters
Erasmus School of Economics

Verbeek, M.J.C.M. (1993). Missing measurements in econometric models with no auxiliary relations. Economics Letters, 125–128. doi:10.1016/0165-1765(93)90024-7