I am reading Greene's textbook Econometric Analysis where he says that, if there's multicollinearity, then:

  • Small changes in data lead to large swings in parameter estimates.
  • Coefficients have high standard errors even though they're jointly significant.
  • Coefficients have the "wrong" sign or implausible magnitudes.

I have three question:

  • What are the consequences for the unbiasedness and consistency of the OLS estimators in the presence of multicollinearity?
  • Is the efficiency of the estimators reduced in the presence of multicollinearity?
  • Do Greene's points hold (yet to a lesser extent) for slightly correlated independent variables? For example, would all three points hold (to a small extent) if the correlation between the regressors is $\rho = 0.1$, for example?
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    $\begingroup$ I hope and trust that Greene is more nuanced: small changes in data can lead to large changes in estimates--but they do not necessarily do so. Coefficients can have implausible signs or magnitudes--but they do not necessarily do so. He's (apparently) just trying to list some of the possible consequences of high standard errors: multicollinearity tends to produce high SEs (but does not necessarily do so!) When multicollinearity is viewed as an issue concerning how the variables are encoded, rather than about the model, the answers to the first two questions are clear. $\endgroup$ – whuber Dec 26 '12 at 17:26
  • $\begingroup$ @whuber I wonder if that calls into question the methods that the perturb function uses? $\endgroup$ – Peter Flom Dec 26 '12 at 17:54
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    $\begingroup$ As far as I can tell, perturb is merely computing directional derivatives--which are just linear combinations of partial derivatives--in random directions. I see no point to this, given that those partials are already computed using standard regression diagnostics. It sounds merely like a computationally expensive way to obtain the same information provided by the differential alternative to DFBETA discussed by Belsley, Kuh, and Welsch, 1980 (formula 2.42). $\endgroup$ – whuber Dec 26 '12 at 18:03

Re your 1st question Collinearity does not make the estimators biased or inconsistent, it just makes them subject to the problems Greene lists (with @whuber 's comments for clarification).

Re your 3rd question: High collinearity can exist with moderate correlations; e.g. if we have 9 iid variables and one that is the sum of the other 9, no pairwise correlation will be high but there is perfect collinearity.

Collinearity is a property of sets of independent variables, not just pairs of them.

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    $\begingroup$ +1: That's a good point about how sets of variables can be highly collinear without exhibiting any large correlations. $\endgroup$ – whuber Dec 26 '12 at 17:51
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    $\begingroup$ Thanks! I did my dissertation on collinearity - it's been a while (since 1999), but I still remember some stuff :-). $\endgroup$ – Peter Flom Dec 26 '12 at 17:53
  • $\begingroup$ So what I should take away from Greene's 3 points is that the estimator variance becomes very high, meaning that we are more likely to have estimates that are far away from the population value, and we're likely to find it difficult to get statistical significance? $\endgroup$ – Jase Dec 27 '12 at 2:53
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    $\begingroup$ I think the really key point is that small changes in the input data can yield big changes in the output. That's a bad thing. Belsley presents one example where changes in the third significant digit of the input reversed the signs of the coefficients. $\endgroup$ – Peter Flom Dec 27 '12 at 12:54

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