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8.1 A General Strategy for Modeling Nonlinear Regression Functions 311

Equation (8.6), which is ∆Yn = bn1 * (11 - 10) + bn2 * (112 - 102) = bn1 + 21bn2.
The standard error of the predicted change therefore is

SE(∆Yn ) = SE(bn1 + 21bn2). (8.7)

Thus, if we can compute the standard error of bn1 + 21bn2, then we have computed
the standard error of ∆Yn . There are two methods for doing this using standard
regression software, which correspond to the two approaches in Section 7.3 for
testing a single restriction on multiple coefficients.

The first method is to use Approach #1 of Section 7.3, which is to compute the
F-statistic testing the hypothesis that b1 + 21b2 = 0. The standard error of ∆Yn is
then given by2

SE(∆Yn ) = ͉ ∆Yn ͉ . (8.8)
2F

When applied to the quadratic regression in Equation (8.2), the F-statistic testing
the hypothesis that b1 + 21b2 = 0 is F = 299.94. Because ∆Yn = 2.96, applying
Equation (8.8) gives SE(∆Yn ) = 2.96 > 2299.94 = 0.17. Thus a 95% confidence
interval for the change in the expected value of Y is 2.96 { 1.96 * 0.17 or
(2.63, 3.29).

The second method is to use Approach #2 of Section 7.3, which entails
transforming the regressors so that, in the transformed regression, one of the
coefficients is b1 + 21b2. Doing this transformation is left as an exercise
(Exercise 8.9).

A comment on interpreting coefficients in nonlinear specifications. In the mul-
tiple regression model of Chapters 6 and 7, the regression coefficients had a
natural interpretation. For example, b1 is the expected change in Y associated
with a change in X1, holding the other regressors constant. But, as we have
seen, this is not generally the case in a nonlinear model. That is, it is not very
helpful to think of b1 in Equation (8.1) as being the effect of changing the dis-
trict’s income, holding the square of the district’s income constant. In nonlinear
models the regression function is best interpreted by graphing it and by calcu-
lating the predicted effect on Y of changing one or more of the independent
variables.

2Equation (8.8) is derived by noting that the F-statistic is the square of the t-statistic testing this
hypothesis—that is, F = t2 = [(bn1 + 21bn2)>SE(bn1 + 21bn1)]2 = [∆Yn >SE(∆Yn )]2—and solving for
SE( ∆Yn ).