## Semi-Nonparametric Maximum Likelihood Estimation

A. Ronald Gallant, Douglas W. Nychka

*Econometrica*, Vol. 55, No. 2. (Mar., 1987),
pp. 363-390.

### Abstract

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.
**Keywords: **Semi-nonparametric, nonparametric, semi-parametric,
maximum likelihood, sample selection, Stoker functional