The main points discussed in this chapter can be summarized as follows:
1. Although linear regression models predominate theory and practice, there are occasions where nonlinear-in-the-parameter regression models (NLRM) are useful.
2. The mathematics underlying linear regression models is comparatively simple in that one can obtain explicit, or analytical, solutions of the coefficients of such models. The small-sample and large-sample theory of inference of such models is well established.
3. In contrast, for intrinsically nonlinear regression models, parameter values cannot be obtained explicitly. They have to be estimated numerically, that is, by iterative procedures.
4. There are several methods of obtaining estimates of NLRMs, such as (1) trial and error, (2) nonlinear least squares (NLLS), and (3) linearization through Taylor series expansion.
5. Computer packages now have built-in routines, such as Gauss-Newton, Newton-Raphson, and Marquard. These are all iterative routines.
6. NLLS estimators do not possess optimal properties in finite samples, but in large samples they do have such properties. Therefore, the results of NLLS in small samples must be interpreted carefully.
7. Autocorrelation, heteroscedasticity, and model specification problems can plague NLRM, as they do linear regression models.
8. We illustrated the NLLS with several examples. With the ready availability of user-friendly software packages, estimation of NLRM should no longer be a mystery. Therefore, the reader should not shy away from such models whenever theoretical or practical reasons dictate their use. As a matter of fact, if you refer to exercise 12.10, you will see from Eq. (1) that there is intrinsically a nonlinear regression model that should be estimated as such.
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