Page 311 -

310 Chapter 8 Nonlinear Regression Functions

To compute ∆Yn associated with the change in income from 10 to 11, we can
apply the general formula in Equation (8.5) to the quadratic regression model.
Doing so yields

∆Yn = (bn0 + bn1 * 11 + bn2 * 112) - (bn0 + bn1 * 10 + bn2 * 102), (8.6)

where bn0, bn1, and bn2 are the OLS estimators.
The term in the first set of parentheses in Equation (8.6) is the predicted value

of Y when Income = 11, and the term in the second set of parentheses is the
predicted value of Y when Income = 10. These predicted values are calculated
using the OLS estimates of the coefficients in Equation (8.2). Accordingly, when
Income = 10, the predicted value of test scores is 607.3 + 3.85 * 10 - 0.0423 *
102 = 641.57. When Income = 11, the predicted value is 607.3 + 3.85 * 11 -
0.0423 * 112 = 644.53. The difference in these two predicted values is
∆Yn = 644.53 - 641.57 = 2.96 points; that is, the predicted difference in test
scores between a district with average income of $11,000 and one with average
income of $10,000 is 2.96 points.

In the second case, when income changes from $40,000 to $41,000, the difference
in the predicted values in Equation (8.6) is ∆Yn = (607.3 + 3.85 * 41 - 0.0423 *
412) - (607.3 + 3.85 * 40 - 0.0423 * 402) = 694.04 - 693.62 = 0.42 points.
Thus a change of income of $1000 is associated with a larger change in predicted
test scores if the initial income is $10,000 than if it is $40,000 (the predicted changes
are 2.96 points versus 0.42 point). Said differently, the slope of the estimated qua-
dratic regression function in Figure 8.3 is steeper at low values of income (like
$10,000) than at the higher values of income (like $40,000).

Standard errors of estimated effects. The estimator of the effect on Y of changing
X1 depends on the estimator of the population regression function, fn, which varies
from one sample to the next. Therefore, the estimated effect contains a sampling
error. One way to quantify the sampling uncertainty associated with the estimated
effect is to compute a confidence interval for the true population effect. To do so,
we need to compute the standard error of ∆Yn in Equation (8.5).

It is easy to compute a standard error for ∆Yn when the regression function is
linear. The estimated effect of a change in X1 is bn1∆X1, so the standard error of
∆Yn is SE(∆Yn ) = SE(bn1)∆X1 and a 95% confidence interval for the estimated
change is bn1∆X1 { 1.96 SE(bn1)∆X1.

In the nonlinear regression models of this chapter, the standard error of ∆Yn can
be computed using the tools introduced in Section 7.3 for testing a single restriction
involving multiple coefficients. To illustrate this method, consider the estimated
change in test scores associated with a change in income from 10 to 11 in