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308 Chapter 8 Nonlinear Regression Functions

A general formula for a nonlinear population regression function.1 The nonlinear
population regression models considered in this chapter are of the form

Yi = f(X1i, X2i, c, Xki) + ui, i = 1, c, n, (8.3)

where f(X1i, X2i, c, Xki) is the population nonlinear regression function, a pos-
sibly nonlinear function of the independent variables X1i, X2i, c, Xki, and ui is
the error term. For example, in the quadratic regression model in Equation (8.1),
only one independent variable is present, so X1 is Income and the population
regression function is f(Incomei) = b0 + b1Incomei + b2Incomei2.

Because the population regression function is the conditional expectation
of Yi given X1i, X2i, c, Xki, in Equation (8.3) we allow for the possibility that
this conditional expectation is a nonlinear function of X1i, X2i, c, Xki; that is,
E(Yi ͉ X1i, X2i, c, Xki) = f(X1i, X2i, c, Xki), where ƒ can be a nonlinear function.
If the population regression function is linear, then f(X1i, X2i, c, Xki) = b0 +
b1X1i + b2X2i + g + bkXki, and Equation (8.3) becomes the linear regression
model in Key Concept 6.2. However, Equation (8.3) allows for nonlinear regression
functions as well.

The effect on Y of a change in X1. As discussed in Section 6.2, the effect on Y of a
change in X1, ∆X1, holding X2, c, Xk constant, is the difference in the expected
value of Y when the independent variables take on the values X1 + ∆X1, X2, c, Xk
and the expected value of Y when the independent variables take on the values
X1, X2, c, Xk. The difference between these two expected values, say ∆Y, is
what happens to Y on average in the population when X1 changes by an amount
∆X1, holding constant the other variables X2, c, Xk. In the nonlinear
regression model of Equation (8.3), this effect on Y is ∆Y =
f(X1 + ∆X1, X2, c, Xk) - f(X1, X2, c, Xk).

Because the regression function f is unknown, the population effect on Y of a
change in X1 is also unknown. To estimate the population effect, first estimate the
population regression function. At a general level, denote this estimated function

1The term nonlinear regression applies to two conceptually different families of models. In the first
family, the population regression function is a nonlinear function of the X’s but is a linear function
of the unknown parameters (the b’s). In the second family, the population regression function is a
nonlinear function of the unknown parameters and may or may not be a nonlinear function of the
X’s. The models in the body of this chapter are all in the first family. Appendix 8.1 takes up models
from the second family.