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Regression Functions That Are Nonlinear in the Parameters 355

for her value of ln(AHE)? Jane is a 30-year-old female with a high
school degree. What does the regression predict for her value of
ln(AHE)? What is the predicted difference between Alexis’s and
Jane’s earnings? Bob is a 30-year-old male with a bachelor’s degree.
What does the regression predict for his value of ln(AHE)? Jim is a
30-year-old male with a high school degree. What does the regres-
sion predict for his value of ln(AHE)? What is the predicted differ-
ence between Bob’s and Jim’s earnings?
j. Is the effect of Age on earnings different for men than for women?
Specify and estimate a regression that you can use to answer this
question.
k. Is the effect of Age on earnings different for high school graduates
than for college graduates? Specify and estimate a regression that you
can use to answer this question.
l. After running all these regressions (and any others that you want to
run), summarize the effect of age on earnings for young workers.

A p p e n d i x

8.1 Regression Functions That Are Nonlinear

in the Parameters

The nonlinear regression functions considered in Sections 8.2 and 8.3 are nonlinear func-
tions of the X’s but are linear functions of the unknown parameters. Because they are
linear in the unknown parameters, those parameters can be estimated by OLS after defin-
ing new regressors that are nonlinear transformations of the original X’s. This family of
nonlinear regression functions is both rich and convenient to use. In some applications,
however, economic reasoning leads to regression functions that are not linear in the param-
eters. Although such regression functions cannot be estimated by OLS, they can be esti-
mated using an extension of OLS called nonlinear least squares.

Functions That Are Nonlinear in the Parameters

We begin with two examples of functions that are nonlinear in the parameters. We then
provide a general formulation.

Logistic curve. Suppose that you are studying the market penetration of a technology, such
as the adoption of database management software in different industries. The dependent
variable is the fraction of firms in the industry that have adopted the software, a single