Piecewise Linear Regression

To illustrate yet another use of dummy variables, consider Figure 9.5, which shows how a hypothetical company remunerates its sales representatives. It pays commissions based on sales in such a manner that up to a certain level, the target, or threshold, level X*, there is one (stochastic) commission structure and beyond that level another. (Note: Besides sales, other factors affect sales commission. Assume that these other factors are represented FIGURE 9.5 Hypothetical relationship between sales commission and sales volume. (Note: The intercept on the Y axis denotes minimum guaranteed commission.)

FIGURE 9.5 Hypothetical relationship between sales commission and sales volume. (Note: The intercept on the Y axis denotes minimum guaranteed commission.)

16For proof, see Adrian C. Darnell, A Dictionary of Econometrics, Edward Elgar, Lyme, U.K., 1995, pp. 150-152.

318 PART ONE: SINGLE-EQUATION REGRESSION MODELS

by the stochastic disturbance term.) More specifically, it is assumed that sales commission increases linearly with sales until the threshold level X* after which also it increases linearly with sales but at a much steeper rate. Thus, we have a piecewise linear regression consisting of two linear pieces or segments, which are labeled I and II in Figure 9.5, and the commission function changes its slope at the threshold value. Given the data on commission, sales, and the value of the threshold level X* the technique of dummy variables can be used to estimate the (differing) slopes of the two segments of the piecewise linear regression shown in Figure 9.5. We proceed as follows:

where Yi = sales commission

Xi = volume of sales generated by the sales person

X* = threshold value of sales also known as a knot (known in advance)17 D = 1 if Xi > X* = 0 if Xi < X*

which gives the mean sales commission up to the target level X~ and

E(Y I Di = 1, Xi, X*) = a1 — P2X~ + (P1 + P2)Xi (9.8.3)

which gives the mean sales commission beyond the target level X~.

Thus, p1 gives the slope of the regression line in segment I, and p1 + p2 gives the slope of the regression line in segment II of the piecewise linear regression shown in Figure 9.5. A test of the hypothesis that there is no break in the regression at the threshold value X~ can be conducted easily by noting the statistical significance of the estimated differential slope coefficient p2 (see Figure 9.6).

Incidentally, the piecewise linear regression we have just discussed is an example of a more general class of functions known as spline functions.18

17The threshold value may not always be apparent, however. An ad hoc approach is to plot the dependent variable against the explanatory variable(s) and observe if there seems to be a sharp change in the relation after a given value of X (i.e., X). An analytical approach to finding the break point can be found in the so-called switching regression models. But this is an advanced topic and a textbook discussion may be found in Thomas Fomby, R. Carter Hill, and Stanley Johnson, Advanced Econometric Methods, Springer-Verlag, New York, 1984, Chap. 14.

18For an accessible discussion on splines (i.e., piecewise polynomials of order k), see Douglas C. Montgomery, Elizabeth A. Peck, and G. Geoffrey Vining, Introduction to Linear Regression Analysis, John Wiley & Sons, 3d ed., New York, 2001, pp. 228-230.

CHAPTER NINE: DUMMY VARIABLE REGRESSION MODELS 319 FIGURE 9.6 Parameters of the piecewise linear regression.

FIGURE 9.6 Parameters of the piecewise linear regression. 