This F value is obviously highly significant.

It is interesting to look at this result geometrically. (See Figure 10.3.) Based on the regression (10.6.1), we have established the individual 95% confidence intervals for 2 and 3 following the usual procedure discussed in Chapter 8. As these intervals show, individually each of them includes the value of zero. Therefore, individually we can accept the hypothesis that the

FIGURE 10.3 Individual confidence intervals for fa2 and fa3 and joint confidence interval (ellipse) for fa2 and fa3.


two partial slopes are zero. But, when we establish the joint confidence interval to test the hypothesis that = = 0, that hypothesis cannot be accepted since the joint confidence interval, actually an ellipse, does not include the origin.16 As already pointed out, when collinearity is high, tests on individual regressors are not reliable; in such cases it is the overall F test that will show if Y is related to the various regressors.

Our example shows dramatically what multicollinearity does. The fact that the F test is significant but the t values of X2 and X3 are individually insignificant means that the two variables are so highly correlated that it is impossible to isolate the individual impact of either income or wealth on consumption. As a matter of fact, if we regress X3 on X2, we obtain

which shows that there is almost perfect collinearity between X3 and X2.

Now let us see what happens if we regress Y on X2 only:

In (10.6.1) the income variable was statistically insignificant, whereas now it is highly significant. If instead of regressing Y on X2, we regress it on X3, we obtain

We see that wealth has now a significant impact on consumption expenditure, whereas in (10.6.1) it had no effect on consumption expenditure.

Regressions (10.6.4) and (10.6.5) show very clearly that in situations of extreme multicollinearity dropping the highly collinear variable will often make the other X variable statistically significant. This result would suggest that a way out of extreme collinearity is to drop the collinear variable, but we shall have more to say about it in Section 10.8.

16As noted in Sec. 5.3, the topic of joint confidence interval is rather involved. The interested reader may consult the reference cited there.


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