st: Re: adjusted r square

[Kit Baum <baum@bc.edu>]

When thinking about how ordinary r^2 is nondecreasing in k, it is  
useful to compare a model with k regressors to one with k+1.  
Regression solves n equations for k unknowns. When you add a k+1'st  
parameter, the data could decide that the optimal value of that  
parameter is exactly zero. That will happen with probability  
approaching zero (as Richard says, due to sampling variation), so the  
k+1'st estimated parameter will generally be nonzero. Forcing it to  
be zero is restricted least squares (-cnsreg-), and the cost of the  
constraint is non-negative (that is, e'e will rise or stay the same,  
and it will stay the same with Pr->0).

A curiosity about adjusted r^2 is that if the k+1'st variable has a  
parameter with a |t| > 1.0, adj r^2 will rise, and vice versa. As 1.0  
is not a commonly used point on Student's t distribution, that  
suggests that if adj r^2 falls at the margin, the k+1'st variable is  
truly uninformative conditional on the k regressors.

Kit Baum, Boston College Economics
http://ideas.repec.org/e/pba1.html
An Introduction to Modern Econometrics Using Stata:
http://www.stata-press.com/books/imeus.html


On Feb 21, 2007, at 2:33 AM, Richard wrote:


> Incidentally, I'm assuming adjusted R^2 does what it purports to
> do.  Its formula is something I've always just taken on blind
> faith.  But regardless of how well it works, I think the rationale
> for it makes sense.
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