Estimation Of Linear And Nonlinear Regression Models

To see the difference in estimating linear and nonlinear regression models, consider the following two models:

By now you know that (14.2.1) is a linear regression model, whereas (14.2.2) is a nonlinear regression model. Regression (14.2.2) is known as the exponential regression model and is often used to measure the growth of a variable, such as population, GDP, or money supply.

2If you try to log-transform the model, it will not work because ln (A + B) = ln A + ln B.

3For properties of the CES production function, see Michael D. Intriligator, Ronald Bodkin, and Cheng Hsiao, Econometric Models, Techniques, and Applications, 2d ed., Prentice Hall, 1996, pp. 294-295.


Suppose we consider estimating the parameters of the two models by OLS. In OLS we minimize the residual sum of squares (RSS), which for model (14.2.1) is:

where as usual p1 and 02 are the OLS estimators of the true j's. Differentiating the preceding expression with respect to the two unknowns, we obtain the normal equations shown in (3.1.4) and (3.1.5). Solving these equations simultaneously, we obtain the OLS estimators given in Eqs. (3.1.6) and (3.1.7). Observe very carefully that in these equations the unknowns (j's) are on the left-hand side and the knowns (X and Y) are on the right-hand side. As a result we get explicit solutions of the two unknowns in terms of our data.

Now see what happens if we try to minimize the RSS of (14.2.2). As shown in Appendix 14A, Section 14A.1, the normal equations corresponding to (3.1.4) and (3.1.5) are as follows:

Unlike the normal equations in the case of the linear regression model, the normal equations for nonlinear regression have the unknowns (the ยก3's) both on the left- and right-hand sides of the equations. As a consequence, we cannot obtain explicit solutions of the unknowns in terms of the known quantities. To put it differently, the unknowns are expressed in terms of themselves and the data! Therefore, although we can apply the method of least squares to estimate the parameters of the nonlinear regression models, we cannot obtain explicit solutions of the unknowns. Incidentally, OLS applied to a nonlinear regression model is called nonlinear least squares (NLLS). So, what is the solution? We take this question up next.


To set the stage, let us consider a concrete example. The data in Table 14.1 relates to the management fees that a leading mutual fund in the United States pays to its investment advisors to manage its assets. The fees paid depend on the net asset value of the fund. As you can see, the higher the net asset value of the fund, the lower are the advisory fees, which can be seen clearly from Figure 14.1.

To see how the exponential regression model in (14.2.2) fits the data given in Table 14.1, we can proceed by trial and error. Suppose we assume that


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