On the Role of Jacobian Terms in Maximum Likelihood Estimation

Because of the presence of Jacobian terms, determinants which arose as a result of a transformation of variables, many common likelihood functions have singularities. This fact has several implications for maximum likelihood estimation. The most interesting of these is that singularities often correspond with economically meaningful restrictions, and they can be used to impose the latter. Several applications of this principle are presented. They suggest that maximum likelihood should be preferred to other estimation schemes not only because of its optimal large-sample statistical properties, but also because of its ability to incorporate certain a priori restrictions from economic theory. I would like to thank Charles Beach, Alan Gelb, and Mark Gersovitz for valuable comments on earlier drafts, and Mike Peters for making an illuminating observation. All errors are mine.