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352 Chapter 8 Nonlinear Regression Functions

and X are independent, as is done in Appendix 8.2 for the log-linear
model.)
8.12 The discussion following Equation (8.28) interprets the coefficient on
interacted binary variables using the conditional mean zero assump-
tion. This exercise shows that interpretation also applies under con-
ditional mean independence. Consider the hypothetical experiment
in Exercise 7.11.

a. Suppose that you estimate the regression Yi = g0 + g1X1i + ui using
only the data on returning students. Show that g1 is the class size effect
for returning students—that is, that g1 = E(Yi ͉ X1i = 1, X2i = 0) -
E(Yi ͉ X1i = 0, X2i = 0). Explain why gn1 is an unbiased estimator of g1.

b. Suppose that you estimate the regression Yi = d0 + d1X1i + ui using
only the data on new students. Show that d1 is the class size effect for new
students—that is, that d1 = E(Yi ͉ X1i = 1, X2i = 1) - E(Yi ͉ X1i = 0,
X2i = 1). Explain why dn1 is an unbiased estimator of d1.

c. Consider the regression for both returning and new students,
Yi = b0 + b1X1i + b2X2i + b3(X1i * X2i) + ui. Use the conditional
mean independence assumption E(ui ͉ X1i, X2i) = E(ui ͉ X2i) to show
that b1 = g1, b1 + b3 = d1, and b3 = d1 - g1 (the difference in the
class size effects).

d. Suppose that you estimate the interaction regression in (c) using the
combined data and that E(ui ͉ X1i, X2i) = E(ui ͉ X2i). Show that bn1 and
bn3 are unbiased but that bn2 is in general biased.

Empirical Exercises

(Only two empirical exercises for this chapter are given in the text, but you can
find more on the text website www.pearsonglobaleditions.com/Stock_Watson.)

E8.1 Lead is toxic, particularly for young children, and for this reason govern-
ment regulations severely restrict the amount of lead in our environment.
But this was not always the case. In the early part of the 20th century, the
underground water pipes in many U.S. cities contained lead, and lead from
these pipes leached into drinking water. In this exercise you will investigate
the effect of these lead water pipes on infant mortality. On the text website
www.pearsonglobaleditions.com/Stock_Watson, you will find the data file
Lead_Mortality, which contains data on infant mortality, type of water pipes
(lead or non-lead), water acidity (pH), and several demographic variables