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8.1 A General Strategy for Modeling Nonlinear Regression Functions 305

Figure 8.2 Scatterplot of Test Score vs. District Income with a Linear OLS Regression Function

There is a positive correlation Test score
between test scores and district 740
income (correlation = 0.71), 720
but the linear OLS regression line
does not adequately describe the 700
relationship between these variables.
680

660

640

620

600 10 20 30 40 50 60
0 District income

(thousands of dollars)

In short, it seems that the relationship between district income and test scores
is not a straight line. Rather, it is nonlinear. A nonlinear function is a function
with a slope that is not constant: The function ƒ(X) is linear if the slope of ƒ(X) is
the same for all values of X, but if the slope depends on the value of X, then ƒ(X)
is nonlinear.

If a straight line is not an adequate description of the relationship between
district income and test scores, what is? Imagine drawing a curve that fits the points
in Figure 8.2. This curve would be steep for low values of district income and
then would flatten out as district income gets higher. One way to approximate
such a curve mathematically is to model the relationship as a quadratic function.
That is, we could model test scores as a function of income and the square of
income.

A quadrEatlieccptroopnuiclaPtiuobnlisrehginrgesSsieornvimceosdeInlcr.elating test scores and income is
written mathSemtoactki/cWalalytsaosn, Econometrics 1e

STOC.ITEM.0022
FigT. 0e6st.S0c2orei = b0 + b1Incomei + b2Incomei2 + ui, (8.1)

where b0, b1, and 1st aPrreoocof efficien2tns,dIPncrooomfei is the3irndcPomroeofin the itFh indaisltrict,
b2
Incomei2 is the square of income in the ith district, and ui is an error term that, as

usual, represents all the other factors that determine test scores. Equation (8.1) is

called the quadratic regression model because the population regression function,