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CHAPTER TEN: MULTICOLLINEARITY 353

will depend on the three ingredients: (1) a2, (2) VIF, and (3) Yxf- The last one, which ties in with Assumption 8 of the classical model, states that the larger the variability in a regressor, the smaller the variance of the coefficient of that regressor, assuming the other two ingredients are constant, and therefore the greater the precision with which that coefficient can be estimated.

Before proceeding further, it may be noted that the inverse of the VIF is called tolerance (TOL). That is, tol/=VF=(1 - R2 ) aa5.5)

When R2 = 1 (i.e., perfect collinearity), TOL = 0 and when R = 0 (i.e., no collinearity whatsoever), TOLj is 1. Because of the intimate connection between VIF and TOL, one can use them interchangeably.

Wider Confidence Intervals

Because of the large standard errors, the confidence intervals for the relevant population parameters tend to be larger, as can be seen from Table 10.2. For example, when r23 = 0.95, the confidence interval for p2 is larger than when r23 = 0 by a factor of V10.26, or about 3.

Therefore, in cases of high multicollinearity, the sample data may be compatible with a diverse set of hypotheses. Hence, the probability of accepting a false hypothesis (i.e., type II error) increases.

TABLE 10.2 THE EFFECT OF INCREASING COLLINEARITY ON THE 95% CONFIDENCE INTERVAL FOR ft: & ± 1-96 se(ft)

Value of r23 95% confidence interval for ft