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RE: RE: st: Interesting numerical accuracy/collinearity issue


From   "Schaffer, Mark E" <M.E.Schaffer@hw.ac.uk>
To   <statalist@hsphsun2.harvard.edu>
Subject   RE: RE: st: Interesting numerical accuracy/collinearity issue
Date   Wed, 12 Apr 2006 21:52:00 +0100

Thanks again Jeff - very helpful.

Cheers,
Mark 

> -----Original Message-----
> From: owner-statalist@hsphsun2.harvard.edu 
> [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of 
> Jeff Pitblado, StataCorp LP
> Sent: 12 April 2006 21:30
> To: statalist@hsphsun2.harvard.edu
> Subject: Re: RE: st: Interesting numerical accuracy/collinearity issue
> 
> Mark Schaffer <M.E.Schaffer@hw.ac.uk> has a follow-up 
> question about -ovtest-:
> 
> > My follow-up question is simple: why does the shifting and scaling 
> > used by Stata's -ovtest- introduce greater accuracy rather 
> than, say, 
> > greater rounding error?  (Either accuracy or error would remove the 
> > numerical collinearity.)  The algebra doesn't help me here, 
> since all 
> > three methods are algebraically equivalent.  I'm guessing 
> that there's 
> > probably a general principle about how best to maintain numerical 
> > precision, but I don't know what it might be.
> 
> Actually, the three methods you describe are not all 
> algebraically equivalent according to -_rmcoll- and 
> -coldiag2-.  The algebra I mentioned only shows us that the 
> regression models yield a statistically equivalent F test.
> 
> The direct approach and your center/rescale method after 
> taking powers are algebraically equivalent to each other, but 
> -ovtest-'s center/rescale then take powers is not.
> 
> Let's just look at x^2 and x^3, if the values of x are not 
> near zero (say they are all positive), then it is easy to see 
> how x^2 and x^3 can become numerically collinear--even if you 
> center/rescale them after taking the powers.
> 
> Now generate z from the centered/rescaled values of x; this 
> results in z^2 always being positive whereas z^3 is negative 
> where z is negative.  There is no mistaking them to be 
> collinear in this case.
> 
> Incidentally, I do not think of this as an accuracy or 
> numerical precision issue.  To me it is more like shifting x 
> into regions where we are better equipped to numerically 
> distinguish between powers of x.
> 
> --Jeff
> jpitblado@stata.com
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