Statistics: An Introduction to Quantitative Economic Research

By Daniel B. Suits | Go to book overview

CHAPTER VIII
The Function as an Equation II: Simple Curvilinear Regression by Transformation of Variables

In the preceding chapter we saw how a function can be approximated by a linear regression. For most purposes this is satisfactory, but sometimes the curvature itself is a matter of considerable research interest. The nature of the underlying theory may demand a curvilinear formulation of the equation, or the curvature may be so sharp that the function is not amenable to study by linear approximation. In this chapter we shall measure the apparent curvature in a function and test it for significance. We shall then explore a simple method of curvilinear regression that has wide applicability.

8.1 DEFINITION AND MEASUREMENT OF CURVATURE

Consider a least squares linear regression Y = a + bX, fitted to a set of data. The data are said to show curvature if it is possible to partition the X axis into classes forming a categorical variable X* in such a way that the residuals from the regression are correlated with X*.1

This is illustrated in Figure 8.1, p. 192. When the X axis is divided to define the 6 classes A, B, C, D, E, and F, it is clear that the mean residual υ + ̄ varies significantly among classes. The relationship of υ to X* can be tabulated, and the determination ratio R2 measures the amount of curvature.

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1
If the function is curved, the points at the extremes of the scatter will tend to lie on one side of the straight line, while the points in the middle lie on the other. Taken as a whole, the regression of υ on X will show zero correlation, but if divided in to cells by X* classes, υ will show correlation with X*.

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