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8.1 A General Strategy for Modeling Nonlinear Regression Functions 309
The Expected Change on Y of a Change in X1 Key Concept
in the Nonlinear Regression Model (8.3)
8.1
The expected change in Y, ∆Y, associated with the change in X1, ∆X1, holding
X2, c, Xk constant, is the difference between the value of the population regres-
sion function before and after changing X1, holding X2, c, Xk constant. That is,
the expected change in Y is the difference:
∆Y = f(X1 + ∆X1, X2, c, Xk) - f(X1, X2, c, Xk). (8.4)
The estimator of this unknown population difference is the difference between
the predicted values for these two cases. Let fn(X1, X2, c, Xk) be the predicted
value of Y based on the estimator fn of the population regression function. Then
the predicted change in Y is
∆Yn = fn(X1 + ∆X1, X2, c, Xk) - fn(X1, X2, c, Xk). (8.5)
by fn ; an example of such an estimated function is the estimated quadratic regres-
sion function in Equation (8.2). The estimated effect on Y (denoted ∆Yn ) of the
change in X1 is the difference between the predicted value of Y when the inde-
pendent variables take on the values X1 + ∆X1, X2, c, Xk and the predicted
value of Y when they take on the values X1, X2, c, Xk.
The method for calculating the expected effect on Y of a change in X1 is sum-
marized in Key Concept 8.1. The method in Key Concept 8.1 always works,
whether ∆X1 is large or small and whether the regressors are continuous or dis-
crete. Appendix 8.2 shows how to evaluate the slope using calculus for the special
case of a single continuous regressor when ∆X1 small.
Application to test scores and income. What is the predicted change in test scores
associated with a change in district income of $1000, based on the estimated qua-
dratic regression function in Equation (8.2)? Because that regression function is
quadratic, this effect depends on the initial district income. We therefore consider
two cases: an increase in district income from 10 to 11 (i.e., from $10,000 per
capita to $11,000) and an increase in district income from 40 to 41.
