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This table gives what is called the correlation matrix. In this table the entries on the main diagonal (those running from the upper left-hand corner to the lower right-hand corner) give the correlation of one variable with itself, which is always 1 by definition, and the entries off the main diagonal are the pair-wise correlations among the X variables. If you take the first row of this table, this gives the correlation of X1 with the other X variables. For example, 0.991589 is the correlation between X1 and X2, 0.620633 is the correlation between X1 and X3, and so on.

As you can see, several of these pair-wise correlations are quite high, suggesting that there may be a severe collinearity problem. Of course, remember the warning given earlier that such pair-wise correlations may be a sufficient but not a necessary condition for the existence of multicollinearity.

To shed further light on the nature of the multicollinearity problem, let us run the auxiliary regressions, that is the regression of each X variable on the remaining X variables. To save space, we will present only the R2 values obtained from these regressions, which are given in Table 10.9. Since the R2 values in the auxiliary regressions are very high (with the possible exception of the regression of X4) on the remaining X variables, it seems that we do have a serious collinearity problem. The same information is obtained from the tolerance factors. As noted previously, the closer the tolerance factor is to zero, the greater is the evidence of collinearity.

Applying Klein's rule of thumb, we see that the R2 values obtained from the auxiliary regressions exceed the overall R2 value (that is the one obtained from the regression of Y on all the X variables) of 0.9954 in 3 out of

TABLE 10.9 R2 VALUES FROM THE AUXILIARY REGRESSIONS

Dependent variable

R2 value

Tolerance (TOL) = 1 - R2

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