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Chapter

8 Nonlinear Regression Functions

In Chapters 4 through 7, the population regression function was assumed to be
linear. In other words, the slope of the population regression function was
constant, so the effect on Y of a unit change in X does not itself depend on the value
of X. But what if the effect on Y of a change in X does depend on the value of one
or more of the independent variables? If so, the population regression function is
nonlinear.

This chapter develops two groups of methods for detecting and modeling
nonlinear population regression functions. The methods in the first group are useful
when the effect on Y of a change in one independent variable, X1, depends on the
value of X1 itself. For example, reducing class sizes by one student per teacher might
have a greater effect if class sizes are already manageably small than if they are so
large that the teacher can do little more than keep the class under control. If so, the
test score (Y) is a nonlinear function of the student–teacher ratio (X1), where this
function is steeper when X1 is small. An example of a nonlinear regression function
with this feature is shown in Figure 8.1. Whereas the linear population regression
function in Figure 8.1a has a constant slope, the nonlinear population regression
function in Figure 8.1b has a steeper slope when X1 is small than when it is large.
This first group of methods is presented in Section 8.2.

The methods in the second group are useful when the effect on Y of a change
in X1 depends on the value of another independent variable, say X2. For example,
students still learning English might especially benefit from having more one-on-one
attention; if so, the effect on test scores of reducing the student–teacher ratio will be
greater in districts with many students still learning English than in districts with few
English learners. In this example, the effect on test scores (Y) of a reduction in the
student–teacher ratio (X1) depends on the percentage of English learners in the
district (X2). As shown in Figure 8.1c, the slope of this type of population regression
function depends on the value of X2. This second group of methods is presented in
Section 8.3.

In the models of Sections 8.2 and 8.3, the population regression function is a
nonlinear function of the independent variables; that is, the conditional expectation
E(Yi ͉ X1i, c, Xki) is a nonlinear function of one or more of the X’s. Although they are
nonlinear in the X’s, these models are linear functions of the unknown coefficients

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