A multiple regression stumper
- From: rick.deshon@xxxxxxxxx
- Date: 29 Sep 2005 08:00:00 -0700
Hi All.
I can't figure out the solution to what should be a fairly
straightforward regression problem.
Assume you have a set of variables (X) that you use to predict a single
variable (Y) in a standard multiple regression model. X is nxp and Y
is nx1.
In this model, Y = Xb + e, where e is a nx1 vector of residuals.
The OLS estimate of b is inv(X`X)*X'Y. Consider b to be a (px1) vector
of optimal weights that minimize the variance of e.
One way to examine the quality of the fitted regression is to compute
R^2 (the coefficient of variation or determination). R2 = (b'*
cov_XY)/var(Y) where cov_XY is a px1 vector of covariances between the
columns of X with Y (cov_XY = (X'Y)/(n-1)) and var(Y) is the variance
of the vector Y. Conceptually, R2 is the ratio of predictable variance
in Y to total variance in Y.
I would like to compute R2 for non-optimal sets of weights. What
happens to R2 as you use less and less optimal weights?
This would be simple under normal circumstances but I'd like to do it
for a special case where you don't know Y. In other words, you have X,
b, and you know the R2 for the optimal model. Further, using knowledge
of X, b, and the optimal R2 you can compute the variance of Y so you
know that quantity also.
Is it possible to estimate R2 for non-optimal weights if you know b, X,
and var(y)? The only missing quantity is cov_XY but b clearly has
information on these missing covariances of X and Y. I have not been
able to determine a unique solution to this apparently simple problem.
Perhaps an orthogonal projection?
Thanks for any insights you can provide!
Rick
.
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