This short course deals with recent developments in nonparametric maximum likelihood estimation (NPMLE) methods, including the growing literature of empirical likelihood (EL). The NPMLE procedure is a direct application of the maximum likelihood estimation (MLE) method to nonparametric estimation. An interesting fact is that the MLE remains valid in many models that have nonparametric components, provided that the 'likelihood function' is formulated appropriately. This is an extension of great interest, since econometric theory rarely suggests a parametric form of the probability law of data. A typical NPMLE employs an approximating distribution function whose support is determined by data values to estimate the underlying distribution nonparametrically. This approach has intuitive appeal and demands little restriction about the smoothness and other unknown characteristics of the true distribution, and also sidesteps the vexing problem of tuning parameter choice. Moreover, an NPMLE often achieves efficiency properties akin to those of parametric likelihood procedures.

Applications to be covered include (conditional) moment restriction models, semiparametric discrete choice models, stratified/biased samples, mixtures, and missing data models. Computational algorithms and issues associated with practical implementation are discussed in detail.