Re: st: cmp and condition numbers

[David Hoaglin <dchoaglin@gmail.com>]

In a least-squares regression problem, the condition number is the
ratio of the largest and smallest singular values of the matrix of
regressors (X).  The singular values are the eigenvalues of
(X-transpose)X.  For a nonsingular square matrix A, the eigenvalues of
A-inverse are the reciprocals of the eigenvalues of A.  Centering and
rescaling the regressors changes the condition number, usually making
it smaller.  The correlation matrix should have a smaller condition
number than (X-transpose)X, but that need not be smaller than the
condition number of X.  As condition numbers go, 1000 is not terribly
high.

David Hoaglin

On Fri, May 4, 2012 at 7:45 AM, David Roodman (droodman@cgdev.org)
<DRoodman@cgdev.org> wrote:
> Philip the two issues you raise may be unrelated.
>
> cmp is not designed for true simultaneous systems, by which I mean ones in the matrix of coefficients of the dependent variables in each other's equations is not triangular.
>
> As for the condition number, there is more than one way to compute this. For each equation, cmp applies Mata's built-in cond() function to the correlation matrix of the non-constant regressors, taking cond()'s defaults. This amounts to computing the product of the maximum eigenvalues of the correlation matrix and its inverse. I think I got this formulation from Greene, but I can't check today because I am travelling.
>
> --David
>
> ---------------
> From      "Bromiley, Philip" <bromiley@uci.edu>
> To        "statalist@hsphsun2.harvard.edu" <statalist@hsphsun2.harvard.edu>
> Subject   st: cmp and condition numbers
> Date      Mon, 30 Apr 2012 23:30:28 +0000
>  I'm trying to estimate a simultaneous system with three continuous and one discrete variable using cmp.  I have been unable to get it to estimate properly - lots of not concave and backed up messages and then it crashes saying it has hit a discontinuous or flat region.
>
> Cmp warns me that I have an ill-conditioned regressor matrix and reports high condition numbers for each of the equations (40 to 1000).  However, when I run the equation with regress, I don't get high VIF's, and get a much lower condition number.
>
> Would someone know the reason for such a discrepancy?  Any suggestions would be welcome.
>
> Phil

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