# Multicollinearity/Serial Correlation/Hederoskedasticity Description Summary

I’ve run into a lot of questions regarding the description of characteristics regarding coefficients, standard errors, etc… I’ve created a brief summary below to outline the characteristics of each and get a better grasp of it all.

I’m not confident in my interpretation and I’m looking for some verification/correction. Any and all help is immensely appreciated!

Conditional Heteroskedasticity

Coefficients: Unbiased & Consistent

Standard Errors: Biased, Consistent, and too small

t-Stat: Artificially High

Serial Correlation

Coefficients: Unbiased & Consistent

Standard Errors: Biased, Consistent, and too small

t-Stat: Artificially High

Multicollinearity

Coefficients: Unbiased & Consistent

Standard Errors: Biased, Consistent, and too large

T-Stat: Artficially Low

Model Misspecification

Coefficients are biased and inconsistent

I agree both conditional heteroskedasticity and serial correlation lead to bias standard errors, but t-stat can be either too low or too high in these cases.

In which case t-stat can be too low? As far as I know t-stat in the first 2 cases will be inflated because standard errors will be underestimated.

@OP: You should add solutions for each case because I feel they are likely to be tested.

This is correct. The bias depends on the type of heteroscedasticity and the type of serial correlation.

The OP’s statement about serial correlation will be incorrect (OLS is biased) if we have a lagged dependent variable as an independent variable (a violation of regressor exogeneity when serial correlation is present).

Also, multicollinearity does NOT bias anything in the regression, including the standard errors. It is the standard errors of the coefficients that will be inflated (not biased or inconsistent) by the square root of a quantity known as a Variance Inflation Factor (VIF).

I have to agree with Gebura, I don’t understand your logic regarding the conditional heteroskedasticity and serial correlation standard errors. Please provide an example. Everything I’ve read states the standard errors will be biased downwards which leads to the artificially high t-stats.

Also, multicollinearity leads to artifically low t-stats because the standard errors will be too high. Doesn’t that indicate biased standard errors (biased upwards)?

Hmm. Is it outside the curriculum? I know heteroskedasticity doesn’t affect the consistency of the regression parameter estimates because those estimates are from OLS. But I don’t remember about serial correlation things (i.e OLS is biased)

For serial correlation:

If it’s positive the standard errors will be underestimated, the t-stat overestimated

If it’s negative the opposite -standard errors too high, t-stat too low.

For conditional heteroskedasticity -it’s written in the CFAI text that typically when dealing with financial data the standard errors will be underestimated but that sometimes it will be the opposite. As far as I remember no example was given.

Page 349 - reading 10: in the footnote they say OLS standard errors need not to be underestimates of actual standard errors if negative serial correlation is present in the regression. I am a bit confused with “… need not to be underestimates…”. Does it mean “overestimated”?

The statistics portion of the curriculum, when compared to the entire body of statistical knowledge (even on regression and time series alone), is infinitely dwarfed by the outside world.

It’s loosely in the curriculum. Think about this; the errors are a function of the actual and predicted y-values. If the errors are correlated across time, then we can infer a correlation of Y-values through time. Now, if you remember, we need to have a zero conditional mean-- that is, E(u|x)=0. If we have correlated errors (u or e, whatever your notation), and we use a past value of Y, say Y(t-1), as an independent variable, then we have created an X that can tell us about the error (x-variable correlated with error, loss of exogeneity). We have violated the zero conditional mean assumption, which is needed for unbiasedness.

Let me know if this helps.

Thank you for the clarification,tickersu. It’s true statistics is too huge discipline to be represented correctly within the limited CFAI curriculum.

Remember that negative heteroskedasticity is not common in economic and financial data, so it is highly unlikely that the exam asked that. Focus on positive het by the moment (just being simplistic right now).

_

I had meant to refer to “the Lake House,” not “Gebura.” Mixed up the usernames.

Unfortunately, I’m afraid my post here may have confused me more for the CFA L2 exam. It seems my mistake was trying to simplify a CFA curriculum topic that can’t be because in the broader world of statistics, these are in-depth topics that are barely touched on in CFAI curriculum.

You are obviously at least very knowledgeable regarding stat (whereas I’m not) so please correct me if I’m wrong in the following statements (it would be greatly appreciated):

My takewawy from your statements is that there is cond. het. and serial correlation that lead to biased SEs and incorrect t-scores (in the context of the CFAI curriculum they are honing in on the downard biased SEs that lead to artificially high t-scores). I should just operate under the assumption that the coefficients (when cond. het. and serial correlatio is present) are consistent and unbiased (again, in CFAI curriculum context)

Multicollinearity does not have biased SEs or inconsistent\biased coefficients.

I realize I’m ignoring the underlying logic and still trying to simplify everything. Quite frankly, there are 17 days left and I’m not sure how much room is left in my head.

You’re welcome! My opinion is that the CFA program is geared more towards application, rather than theory. When you boil it down to teaching for application you might not be able to cover all possible scenarios, so you cover the most likely.

Thanks for taking the time tickersu. Good luck on th 6th!

[quote=“tickersu”]

Thanks for your explanation ! I read somewhere in this forum that you are a quantitative analyst? No surprise why you know the materials this detailed ;).

Btw, how is the excel score sheet you mentioned earlier going?