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Slopes and Elasticities for Nonlinear Regression Functions 359

Equation (8.39) using the California test score data yields bn0 = 703.2 (heteroskedasticity-
robust standard error = 4.44), bn1 = 0.0552 (SE = 0.0068), and bn2 = - 34.0 (SE = 4.48).
Thus the estimated nonlinear regression function (with standard errors reported below the
parameter estimates) is

TestScore = 703.231 - e-0.0552(Income +(344..40)84). (8.42)
(4.44) (0.0068)

This estimated regression function is plotted in Figure 8.13, along with the logarithmic
regression function and a scatterplot of the data. The two specifications are, in this case,
quite similar. One difference is that the negative exponential growth curve flattens out at
the highest levels of income, consistent with having an asymptote.

A p p e n d i x

8.2 Slopes and Elasticities for Nonlinear

Regression Functions

This appendix uses calculus to evaluate slopes and elasticities of nonlinear regression func-

tions with continuous regressors. We focus on the case of Section 8.2, in which there is a

single X. This approach extends to multiple X’s, using partial derivatives.
Consider the nonlinear regression model, Yi = f (Xi) + ui, with E(ui͉ Xi) = 0. The

slope of the population regression function, f(X), evaluated at the point X = x, is the
derivative of f, that is, df (X)>dX͉ X=x. For the polynomial regression function in Equation
(8.9), f (X) = b0 + b1X + b2X2 + g + brXr and dXa>dX = aXa - 1 for any constant a, so
df (X)>dX ͉ X=x = b1 + 2b2x + g + rbr xr-1. The estimated slope at x is dfn(X)> dX ͉ X=x =
bn1 + 2bn2x + g + rbnr xr-1. The standard error of the estimated slope is SE(bn1 + 2bn2x +
g+ rbnr xr - 1); for a given value of x, this is the standard error of a weighted sum of regression
coefficients, which can be computed using the methods of Section 7.3 and Equation (8.8).

The elasticity of Y with respect to X is the percentage change in Y for a given percent-

age change in X. Formally, this definition applies in the limit that the percentage change in

X goes to zero, so the slope appearing in the definition in Equation (8.22) is replaced by

the derivative and the elasticity is

elasticity of Y with respect to X = dY * X = d ln Y .
dX Y d ln X