ECONOMETRIC REVIEWS, 1(2), 151-190 (1982)
ON UNIFICATION OF THE ASYMPTOTIC THEORY OF NONLINEAR ECONOMETRIC MODELS
Jose F. Burguete, A. Ronald Gallant, and Geraldo Souza
Institute of Statistics, North Carolina State University,
Raleigh, NC 27650 USA
Key Words and Phrases: nonlinear models; asymptotic theory.
After reading a few articles in the nonlinear econometric literature one begins to notice that each discussion follows roughly the same lines as the classical treatment of maximum likelihood estimation. There are some technical problems having to do with simultaneously conditioning on the exogenous variables and subjecting the true parameter to a Pittman drift which prevent the use of the classical methods of proof but the basic impression of similarity is correct. An estimator - be it nonlinear least squares, three-stage nonlinear least squares, or whatever - is the solution of an optimization problem. And the objective function of the optimization problem can be treated as if it were the likelihood to derive the Wald test statistic, the likelihood ratio test statistic, and Rao's efficient score statistic. Their asymptotic null and non-null distributions can be found using arguments fairly similar to the classical maximum likelihood arguments. In this article we exploit these observations and unity much of the nonlinear econometric literature. That which escapes this unification is that which has an objective function which is not twice continuously differentiable with respect to the parameters - minimum absolute deviations regression for example.
The model which generates the data need not be the same as the model which was presumed to define the optimization problem. Thus, these results can be used to obtain the asymptotic behavior of inference procedures under specification error. We think that this will prove to be the most useful feature of the paper. For example, it is not necessary to resort to Monte Carlo simulation to determine if a Translog estimate of an elasticity of substitution obtained by nonlinear three-stage least squares is robust against a CES true state of nature. The asymptotic approximations we give here will provide an analytic answer to the question, sufficiently accurate for most purposes.