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358 Chapter 8 Nonlinear Regression Functions

Figure 8.13 The Negative Exponential Growth and Linear-Log Regression Functions

The negative exponential growth regression Test score Linear-log regression
function [Equation (8.42)] and the linear-log Negative exponential
growth regression
regression function [Equation (8.18)] both 700
capture the nonlinear relation between test

scores and district income. One difference

between the two functions is that the

negative exponential growth model has

an asymptote as Income increases to

infinity, but the linear-log regression 650

function does not.

600 20 40 60
0 District income

In linear regression, a relatively simple formula expresses the OLS estimator as a function
of the data. Unfortunately, no such general formula exists for nonlinear least squares, so the
nonlinear least squares estimator must be found numerically using a computer. Regression
software incorporates algorithms for solving the nonlinear least squares minimization problem,
which simplifies the task of computing the nonlinear least squares estimator in practice.

Under general conditions on the function f and the X’s, the nonlinear least squares estima-
tor shares two key properties with the OLS estimator in the linear regression model: It is con-
sistent, and it is normally distributed in large samples. In regression software that supports
nonlinear least squares estimation, the output typically reports standard errors for the esti-
mated parameters. As a consequence, inference concerning the parameters can proceed as
usual; in particular, t-statistics can be constructed using the general approach in Key Concept
5.1, and a 95% confidence interval can be constructed as the estimated coefficient, plus or
minus 1.96 standard errors. Just as in linear regression, the error term in the nonlinear regres-
sion model can be heteroskedastic, so heteroskedasticity-robust standard errors should be used.

Application to the Test Score–Income Relation

A negative exponential growth model, fit to district income (X) and test scores (Y), has the
desirable features of a slope that is always positive [if b1 in Equation (8.39) is positive] and
an asymptote of b0 as income increases to infinity. The result of estimating b0, b1, and b2 in