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RE: RE: st: Interesting numerical accuracy/collinearity issue
Thanks again Jeff - very helpful.
> -----Original Message-----
> From: email@example.com
> [mailto:firstname.lastname@example.org] On Behalf Of
> Jeff Pitblado, StataCorp LP
> Sent: 12 April 2006 21:30
> To: email@example.com
> 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.
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