I started a thread a few weeks ago about an error in a QM subject test. The original thread can be found here: http://www.analystforum.com/forums/cfa-forums/cfa-level-ii-forum/91340506 (Note: the question has been changed by the CFAI as a result of me approaching them, but they have yet to change the error in the explanation for that question. The change should be reflected soon, I would think. Read on for an explanation).
My assertion was that the presence of multicollinearity neither under nor overstates the F-statistic or R-squared, because multicollinearity doesn’t impact model fit and OLS is still unbiased with MC. In plain terms, the F-statistic (and R-squared) is unaffected by multicollinearity.
Didn’t get many hits on the initial thread, but I went ahead and emailed the Institute in April regarding the issue. Long story short (and it’s a long, long story), after many back-and-forth emails with an extremely helpful rep, who relayed the info to the QM question writer(s), the issue was eventually pushed to the curriculum author. The rep just informed me that the curriculum author agrees with the assertion I have made-- multicollinearity does not affect (under or overstate) the F-statistic (and it follows for R-squared as well). In terms of standard errors, it is only the standard errors on the affected coefficients that are “inflated”. The SER (and MSE) is unaffected.
I decided I would give a bump on the topic, since I’m not sure how or if they will make any errata adjustments for the question’s explanation. It’s also not mentioned in the official curriculum (directly). I don’t think this will have too big of an impact on test day, but I figured it would be worth passing on, just in case it comes up exam day.
This is definitely very helpful in many ways, but I think the real value (to me at least) is that it indicates this concept, “multicollinearity does not affect (under or overstate) the F-statistic (and it follows for R-squared as well,” might not be tested on the exam. It’s more likely than not that there will only be only 6 quant problems, so I don’t think they’d want to create any controversies. Just a thought.
I would suggest grabbing a plate of food (or a drink), before reading on. Really (too short) summary: IF you have significant F-test, a relatively high r-squared, but nonsignificant t-tests (need not be all when you have many variables, but that’s a stronger indication), MC is quite possibly and issue-- investigate it. The standard errors on the coefficients can become “large”, making t-tests lack statistical significance.
So, I do think it is difficult for the curriculum to fully cover this topic (and many other topics), since we already are responsible for a large amount of information (imagine a full textbook just for QM…).
Multicollinearity isn’t a black and white issue as much as it is a matter of degree. Typically, though, when people say no MC is preset, they mean it’s not a problem. There is no “test” for MC, but there are several indicators that can suggest it has become an issue (others are Variance Inflation Factors, checking the signs on the estimated coefficients, and more). They are correct in that statement because it does imply that MC is an issue.
Here is why:
If R-squared (preferrably adjusted R-squared) is relatively high, it indicates that the group of independent variables has explained a relatively high proportion of the total sample varition coming from the dependent variable. These independent variables are doing a “decent” job at predicting the DV. Here, we don’t actually know if they are statistically significant, though (we need a hypothesis test).
Moving to the F-test for total model utility, we should recall that this test has a null hypothesis that none of the predictor variables is statistically useful for predicting the DV (as a group, they’re not “good”). Ho: B1=B2=…=Bn=0
–The alternative hypothesis is that at least one of these predictor variables is useful for predicting the DV (we don’t know which one or how many [yet], but we know that something in the model is statistically useful). Ha: at least one Bn not equal to zero (as a group they are “good”).
–So, if we have a significant F-test, that means we have evidence of Ha, at least one independent variable is useful for predicting the DV (as a whole group, these variables are useful).
Recap at this point: R-squared (no measure of reliability) and the F-test (with a measure of reliability) both indicate the variables we have are somewhat useful for predicting the DV. MC won’t “make” either of these statistics larger or more significant.
Now, if we move on to the t-tests, we would expect a similar story-- right? However, if we look at the t-test and see that none of the variables are significant on their own, then we can say MC is probably an issue (since we have conflicting results of the F-test and t-tests).
The conclusion of these points is that we can say MC is likely a problem if we have all of these things together-- a (statistically) useful model, as denoted by the F-test; the R-squared says the independent variables are useful for explaining variation in the DV, yet the t-tests are giving us the exact opposite picture.
As hard as it is to believe, the t-test and the F-tests are both correct-- but it’s because they’re answering different questions. Think of the simplest case: X1 and X2 used to predict Y.
—The F-test answers the following question: are these variables (X1 and X2) useful, as a group? If we reject Ho, then yes, they are useful as a group.
—The t-test on X1s coefficient answers a slightly different question: After accounting for X2, is X1 statistically useful for predicting Y? Well, if X1 is (very) similar to X2 (highly collinear), then we would probably say X1 isn’t useful when we have X2-- they give us the same information about Y!
-------Then, we look at the t-test for X2: After accounting for X1, is X2 statistically significant? Again, they’re very similar, so we would probably say X2 isn’t significant if we have X1.
This is how both t-tests would say each variable isn’t significant (individually, after accounting for the others), but the F-test says at least one is significant for predicting Y.
If you’re following so far, let’s go a little deeper in the rabbit hole: MC can “inflate” the standard errors on the estimated coefficients (so you see “small” t-stats, referring to one coefficient), but it will not affect the SER (which is why R-squared and the F-test are OK—they refer to the whole model). The SER deals with the error term uncertainty (model fit), whereas the standard errors on the coefficients tell us about uncertainty (imprecision) in the estimated coefficients. There is more uncertainty (imprecision) in the estimated coefficients when we have less “unique” data points to estimate the coefficients with-- in other words, if X1 and X2 are collinear (in a problematic way they “share” a lot of information to predict Y), it becomes very difficult to truly see the stand-alone effect X1 has on Y and the stand-alone effect X2 has on Y. Hence, we have less precision in our estimated coefficients for X1 and X2 (another way of looking at this puzzle).
As I’m sure you’re noticing, this isn’t a small topic. There’s quite a bit to it and always more to explore. Hopefully, this helps (and I know it’s a bit more than the original question).