Econometrica: Mar, 1987, Volume 55, Issue 2
Semi-Nonparametric Maximum Likelihood Estimation
A. Ronald Gallant, Douglas W. Nychka
Often maximum likelihood is the method of choice for fitting an econometric model to data but cannot be used because the correct specification of the (multivariate) density that defines the likelihood is unknown. Regression with sample selection is an example. In this situation, simply put the density equal to a Hermite series and apply standard finite dimensional maximum likelihood methods. Model parameters and nearly all aspects of the unknown density itself will be estimated consistently provided that the length of the series increases with sample size. The rule for increasing series length can be data dependent. To assure in-range estimates, the Hermite series is in the form of a polynomial squared times a normal density function with the coefficients of the polynomial restricted so that the series integrates to one and has mean zero. If another density is more plausible a priori, it may be substituted for the normal. The paper verifies these claims and applies the method to nonlinear regression with sample selection and to estimation of the Stoker functional.