Microeconometrics Methods and Applications

Introduction

A nonlinear estimator is one that is a nonlinear function of the dependent variable. Most estimators used in microeconometrics, aside from the OLS and IV estimators in the linear regression model presented in Chapter 4, are nonlinear estimators. Nonlinearity can arise in many ways. The conditional mean may be nonlinear in parameters. The loss function may lead to a nonlinear estimator even if the conditional mean is linear in parameters. Censoring and truncation also lead to nonlinear estimators even if the original model has conditional mean that is linear in parameters.

Here we present the essential statistical inference results for nonlinear estimation. Very limited small-sample results are available for nonlinear estimators. Statistical inference is instead based on asymptotic theory that is applicable for large samples. The estimators commonly used in microeconometrics are consistent and asymptotically normal.

The asymptotic theory entails two major departures from the treatment of the linear regression model given in an introductory graduate course. First, alternative methods of proof are needed since there is no direct formula for most nonlinear estimators. Second, the asymptotic distribution is generally obtained under the weakest distributional assumptions possible. This departure was introduced in Section 4.4 to permit heteroskedasticity-robust inference for the OLS estimator. Under such weaker assumptions the default standard errors reported by a simple regression program are invalid. Some care is needed, however, as these weaker assumptions can lead to inconsistency of the estimator itself, a much more fundamental problem.