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


From   jpitblado@stata.com (Jeff Pitblado, StataCorp LP)
To   statalist@hsphsun2.harvard.edu
Subject   Re: RE: st: Interesting numerical accuracy/collinearity issue
Date   Wed, 12 Apr 2006 15:30:29 -0500

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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