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356 Chapter 8 Nonlinear Regression Functions

Figure 8.12 Two Functions That Are Nonlinear in Their Parameters
YY
1 b0

0 0 X
(a) A logistic curve X

(b) A negative exponential growth curve

Part (a) plots the logistic function of Equation (8.38), which has predicted values that lie between 0 and 1. Part (b) plots
the negative exponential growth function of Equation (8.39), which has a slope that is always positive and decreases as
X increases, and an asymptote at b0 as X tends to infinity.

independent variable X describes an industry characteristic, and you have data on n indus-
tries. The dependent variable is between 0 (no adopters) and 1 (100% adoption). Because
a linear regression model could produce predicted values less than 0 or greater than 1, it
makes sense to use instead a function that produces predicted values between 0 and 1.

The logistic function smoothly increases from a minimum of 0 to a maximum of 1. The
logistic regression model with a single X is

Yi = 1 + 1 + ui. (8.38)
e-(b0 + b1Xi)

The logistic function with a single X is graphed in Figure 8.12a. As can be seen in the graph,
the logistic function has an elongated “S” shape. For small values of X, the value of the
function is nearly 0 and the slope is flat; the curve is steeper for moderate values of X; and
for large values of X, the function approaches 1 and the slope is flat again.

Negative exponential growth. The functions used in Section 8.2 to model the relation
between test scores and income have some deficiencies. For example, the polynomial mod-
els can produce a negative slope for some values of income, which is implausible. The
logarithmic specification has a positive slope for all values of income; however, as income
gets very large, the predicted values increase without bound, so for some incomes the pre-
dicted value for a district will exceed the maximum possible score on the test.

The negative exponential growth model provides a nonlinear specification that has a
positive slope for all values of income, has a slope that is greatest at low values of income