FIGURE 3.11 Correlation patterns (adapted from Henri Theil, Introduction to Econometrics, Prentice-Hall, Englewood Cliffs, N.J., 1978, p. 86).

Some of the properties of r are as follows (see Figure 3.11):

1. It can be positive or negative, the sign depending on the sign of the term in the numerator of (3.5.13), which measures the sample covariation of two variables.

2. It lies between the limits of -1 and +1; that is, -1 < r < 1.

3. It is symmetrical in nature; that is, the coefficient of correlation between X and Y(rXY) is the same as that between Y and X(rYX).

4. It is independent of the origin and scale; that is, if we define X* = aXi + C and Y* = bYi + d, where a > 0, b > 0, and c and d are constants,


then r between X~ and Y~ is the same as that between the original variables X and Y.

5. If X and Y are statistically independent (see Appendix A for the definition), the correlation coefficient between them is zero; but if r = 0, it does not mean that two variables are independent. In other words, zero correlation does not necessarily imply independence. [See Figure 3.11(h).]

6. It is a measure of linear association or linear dependence only; it has no meaning for describing nonlinear relations. Thus in Figure 3.11(h), Y = X2 is an exact relationship yet r is zero. (Why?)

7. Although it is a measure of linear association between two variables, it does not necessarily imply any cause-and-effect relationship, as noted in Chapter 1.

In the regression context, r2 is a more meaningful measure than r, for the former tells us the proportion of variation in the dependent variable explained by the explanatory variable(s) and therefore provides an overall measure of the extent to which the variation in one variable determines the variation in the other. The latter does not have such value.25 Moreover, as we shall see, the interpretation of r (= R) in a multiple regression model is of dubious value. However, we will have more to say about r2 in Chapter 7.

In passing, note that the r2 defined previously can also be computed as the squared coefficient of correlation between actual Yi and the estimated Yi, namely, Yi. That is, using (3.5.13), we can write

r2 _ [£<Y

- Y)(Y -

£(Y -

Y)2 £(Y

where Yi = actual Y, Yi = estimated Y, and Y = Y = the mean of Y. For proof, see exercise 3.15. Expression (3.5.14) justifies the description of r2 as a measure of goodness of fit, for it tells how close the estimated Y values are to their actual values.

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