GENERAL APPROACHES TO NONLINEAR ESTIMATION In this part we begin our study of nonlinear econometric methods. What we mean by nonlinear needs some explanation because it does not necessarily mean that the underlying model is what we would think of as nonlinear. | GENERAL APPROACHES TO NONLINEAR ESTIMATION In this part we begin our study of nonlinear econometric methods. What we mean by nonlinear needs some explanation because it does not necessarily mean that the underlying model is what we would think of as nonlinear. For example suppose the population model of interest can be written as y xfi u but rather than assuming E u x 0 we assume that the median of u given x is zero for all x. This assumption implies Med y x xb which is a linear model for the conditional median of y given x. The conditional mean E y x may or may not be linear in x. The standard estimator for a conditional median turns out to be least absolute deviations LAD not ordinary least squares. Like OLS the LAD estimator solves a minimization problem it minimizes the sum of absolute residuals. However there is a key difference between LAD and OLS the LAD estimator cannot be obtained in closed form. The lack of a closed-form expression for LAD has implications not only for obtaining the LAD estimates from a sample of data but also for the asymptotic theory of LAD. All the estimators we studied in Part II were obtained in closed form a fact which greatly facilitates asymptotic analysis we needed nothing more than the weak law of large numbers the central limit theorem and the basic algebra of probability limits. When an estimation method does not deliver closed-form solutions we need to use more advanced asymptotic theory. In what follows nonlinear describes any problem in which the estimators cannot be obtained in closed form. The three chapters in this part provide the foundation for asymptotic analysis of most nonlinear models encountered in applications with cross section or panel data. We will make certain assumptions concerning continuity and differentiability and so problems violating these conditions will not be covered. In the general development of M-estimators in Chapter 12 we will mention some of the applications that are ruled out and provide .
