Re: st: multicollinearity

["Michael S. Hanson" <mshanson@wesleyan.edu>]

Chris:


On Nov 19, 2008, at 3:06 PM, Chris Witte wrote:
On Nov 19, 2008, at 5:01 PM, Chris Witte wrote:
On Nov 19, 2008, at 8:11 PM, Chris Witte wrote:

1) The Statalist FAQ strongly suggests not posting the same message  
multiple times.  You may find it useful to review the FAQ at <http:// 
www.stata.com/support/faqs/res/statalist.html>.


> Is there another way to get the following module (the link isn't  
> working for me)?
>
> Example .  Stata learning module on regression diagnostics:   
> Multicollinearity
>         . . . . . . . . . . . . . . . . . .  UCLA Academic  
> Technology Services
>         12/03   http://www.ats.ucla.edu/stat/stata/modules/reg/ 
> multico.htm

2)  I suspect that page just doesn't exist anymore.  Not too  
surprising:  it is from December 2003 -- almost 5 years ago, which  
was also a few versions of Stata ago.  If you poke around the UCLA  
ATS web site, you might find related materials.  Also, Google is your  
friend (TM).


> Also, I have read that -anova- and -regress- will drop variables  
> that have collinearity problems, but I have never had Stata drop  
> variables on me.  For example:
>
> sysuse auto
> reg price headroom trunk weight length turn displacement gear_ratio

	[snip]

> and the correlation between weight and length is 0.9460.  Why  
> aren't one of these variables dropped?  Does there have to be  
> perfect correlation for dropping variables?

3) In a nutshell, yes.  Multicollinearity and perfect collinearity  
are not the same thing.  Indeed, they are conceptually rather  
different.  (Your (sub)discipline may use slightly different terms  
for these two concepts.)  Kennedy's "A Guide to Econometrics" (for  
example, the 5th edition, MIT Press, 2003) dedicates a whole chapter  
to multicollinearity, and has a decent discussion of this  
distinction.  The explanation I have often given to my students is  
that multicollinearity is a sample problem -- which in many cases  
could conceptually be avoided by collecting more or "better" data --  
whereas perfect collinearity is a model or specification problem --  
in which no amount of additional data will resolve your specification  
error.  Mathematically, with perfect collinearity, the (X'X) matrix  
is rank deficient and therefore not invertible:  the OLS estimator  
simply does not exist in this case.  Stata thus drops each collinear  
variable until (X'X) is of full rank, and the regression then can be  
estimated on the remaining variables.  Other members of Statalist  
suggested to you a few synthetic examples in earlier replies.   
Multicollinearity inflates variances, thereby complicating inference,  
but it does not preclude estimation.

In Wooldridge's "Introductory Econometrics" textbook (for example,  
pp. 102-4 of the 3rd edition, Thomson South-Western, 2006) there is a  
very informative discussion of multicollinearity, which contains the  
following useful insight:

"Worrying about high degrees of correlation among the independent  
variables in the sample is really no different from worrying about a  
small sample size: both work to increase [the variance of beta hat].   
The famous University of Wisconsin econometrician Arthur Goldberger,  
reacting to econometricians' obsession with multicollinearity, has  
(tongue in cheek) coined the term MICRONUMEROSITY, which he defines  
as the 'problem of small sample size.'"

Best,
Mike

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