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Do high VIF values for a a particular variable $x$ just indicate that it is highly correlated with at least one of the other variables in the model? Does it specify which variables and how many variables that $x$ is correlated with?

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No. That's one reason it is better to look at condition indices and the variance proportion matrix. In SAS PROC REG you can get this with the /collin option on the model statement.

See Regression Diagnostics by Belsley, Kuh & Welsch or Conditioning Diagnostics: Collinearity and Weak Data in Regression by Belsley (out of print, but more thorough on the problem of collinearity specifically). Or you could even look at my dissertation: Multicollinearity Diagnostics for Multiple Regression: A Monte Carlo Study (haven't really looked much at this since then, though).

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    $\begingroup$ Hi Peter. I'm unfamiliar with some of these aspects of reg. diagnostics. Could you say just briefly why such output is more useful than, say, the combination of the tolerance statistic and a correlation matrix? $\endgroup$ – rolando2 Feb 1 '12 at 17:23
  • $\begingroup$ The condition index has better ability to tell about collinearity that will have severe effects on the regression model, and the variance proportion matrix lets you see collinearity among multiple variables, rather than just pairs of variables High correlation is neither necessary nor sufficient for collinearity. As the number of independent variables increases, the minimum correlation for there to be collinearity also goes down; these days, with data sets that often have many many variables, you could have very modest correlation indeed. $\endgroup$ – Peter Flom Feb 1 '12 at 18:14
  • $\begingroup$ @PeterFlom: would you elaborate more on the difference between collinearity and correlation? Thanks. $\endgroup$ – FMZ Feb 2 '12 at 5:56
  • $\begingroup$ @PeterFlom - Thanks for your answer. It sounds as if you don't rely on the tolerance statistics for each predictor. I've been relying on them--I like their simplicity-- and am trying to figure out in what way they might be deficient as compared to the statistics you're using. So far I have to admit I just don't see it, though. Tolerance results do show collinearity among multiple variables, and they aren't subject to one's sense of a minimum correlation that signals a problem. See what I mean? $\endgroup$ – rolando2 Feb 2 '12 at 18:51
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    $\begingroup$ I see. I guess that'd be helpful when the corr. matrix is just too big to study. $\endgroup$ – rolando2 Feb 3 '12 at 22:59

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