Maximum Likelihood Methods Introduction This chapter contains a general treatment of maximum likelihood estimation (MLE) under random sampling. All the models we considered in Part I could be estimated without making full distributional assumptions about the endogenous variables conditional on the exogenous variables | Maximum Likelihood Methods Introduction This chapter contains a general treatment of maximum likelihood estimation MLE under random sampling. All the models we considered in Part I could be estimated without making full distributional assumptions about the endogenous variables conditional on the exogenous variables maximum likelihood methods were not needed. Instead we focused primarily on zero-covariance and zero-conditional-mean assumptions and secondarily on assumptions about conditional variances and covariances. These assumptions were sufficient for obtaining consistent asymptotically normal estimators some of which were shown to be efficient within certain classes of estimators. Some texts on advanced econometrics take maximum likelihood estimation as the unifying theme and then most models are estimated by maximum likelihood. In addition to providing a unified approach to estimation MLE has some desirable efficiency properties it is generally the most efficient estimation procedure in the class of estimators that use information on the distribution of the endogenous variables given the exogenous variables. We formalize the efficiency of MLE in Section . So why not always use MLE As we saw in Part I efficiency usually comes at the price of nonrobustness and this is certainly the case for maximum likelihood. Maximum likelihood estimators are generally inconsistent if some part of the specified distribution is misspecified. As an example consider from Section a simultaneous equations model that is linear in its parameters but nonlinear in some endogenous variables. There we discussed estimation by instrumental variables methods. We could estimate SEMs nonlinear in endogenous variables by maximum likelihood if we assumed independence between the structural errors and the exogenous variables and if we assumed a particular distribution for the structural errors say multivariate normal. The MLE would be asymptotically more efficient than the best GMM .
