Detecting harmful collinearity
We can determine the number of collinear relations, their severity, and the variables involved using the diagnostics in Section 3. This does not end the diagnostic process, because we must still determine if the collinearity present is actually harmful to our regression. Whether the collinearity matters depends on the magnitude of the regression parameters. The parameters matter in two regards. First, from equations (12.2) and (12.9), it is clear that a small value of the error variance, a2, can offset the effects of high correlation between the regressors or low regressor variability.
Second, the magnitudes of the pk matter. If the variance of bk is a, then 100(1 – a)% interval estimator for pk is bk ± fcch, where tc is a critical value from the f-distribution. Suppose we diagnose severe collinearity affecting (and inflating) the variance of bk and compute fcdh = 3. Is collinearity harmful when pk = 1? What if pk = 1000? If you answered "yes" to the first question, but "no" to the second, you are saying, and rightly so, that the magnitude of the parameter pk also matters when determining if collinearity is harmful or not.
Belsley (1982) addresses these issues by developing tests for adequate "signal – to-noise," abbreviated s/n, in the regression model and data. For a single parameter Belsley defines an s/n parameter,
If t is small, then the error variance a2 is not small enough, and/or pk is not large enough, to offset the effects of collinearity and/or lack of regressor variability. Belsley proposes to test the hypothesis that | t | > t*, where t* is an adequate magnitude. For details of this, and a more general multiparameter test, see Belsley (1982).
In the end, Belsley (1982, p. 225) proposes that investigators (i) examine collinearity using the diagnostics described in Section 3, and (ii) carry out the test for adequate s/n. The conclusions one can draw are summarized in Table 12.2.
The four possible outcomes are these: (I) negligible collinearity and adequate s/n; (II) collinearity present, but not harmful, since adequate s/n is present; (III) negligible collinearity, but inadequate s/n present, caused by lack of regressor variation; (IV) harmful collinearity, the joint occurrence of severe collinearity and inadequate s/n. In the next section we address what remedies are available in cases III and IV.
Table 12.2 Harmful collinearity decision matrix
Collinearity present? no yes