# Multicollinearity in OLS

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?
• 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. – whuber Dec 26 '12 at 17:26
• @whuber I wonder if that calls into question the methods that the perturb function uses? – Peter Flom Dec 26 '12 at 17:54
• 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). – whuber Dec 26 '12 at 18:03