Volume I of Advances in Econometrics. Fifth World Congress

Edited by Truman F. Bewley

**CHAPTER 4**

**Identification** **and consistency in
seminonparametric regression**

A. Ronald Gallant

**Abstract.** Nonlinear least squares is the prototypical problem for establishing the consistency of nonlinear econometric estimators in the sense that the
analysis abstracts easily and -the abstraction covers the standard methods of estimation in econometrics: instrumental variables, two- and three-stage least
squares, full information maximum likelihood, seemingly unrelated regression, M-estimators, scale-invariant M-estimators, generalized method of
moments, and so on (Burguete, Gallant, and Souza 1982; Gallant and White 1986). In this chapter, nonlinear least squares is adapted to a function space
setting where the estimator is regarded as a point in a function space rather than a point in a finite-dimensional, Euclidean space. Questions of
identification and consistency are analyzed in this setting. Least squares retains its prototypical status: The analysis transfers directly to both the above
listed methods of inference on a function space and to semi-nonparametric estimation methods. Two semi-nonparainetric examples, the Fourier consumer
demand system (Gallant 1981) and semi-nonparametric maximum likelihood applied to nonlinear regression with sample selection (Gallant and Nychka
1987), are used to illustrate the ideas.