No, not at all. OLS is remarkably robust for deviations in the
distribution of the residuals. I think this is interesting
theoretically, but in applied research this would probably be the very
last thing I would care about.
-- Maarten
--- Chris Witte <[email protected]> wrote:
> so for the layman... Is the most important measure of fit for a
> model the distribution of residuals?
>
>
>
> ----- Original Message ----
> From: Stas Kolenikov <[email protected]>
> To: [email protected]
> Sent: Tuesday, August 12, 2008 10:32:01 AM
> Subject: Re: RE : st: tobit?
>
> The distribution of the standard errors will depend on both the
> distribution of the error terms and the distribution of the
> explanatory variables (design measure, to wit). But in terms of
> working with just the first two moments (means and variances),
> nothing
> says error must be Gaussian, and the explanatory variables have to be
> uniform, to ensure that the estimates are unbiased, and that s^2
> (X'X)^{-1} is an unbiased estimator of variance. In your simulation
> example, if you looked at two-sided coverage (and a sample size of
> 100), you will probably see that rejections outside the nominal 90%
> CI
> will be 3% on one side and 12% on the other.
>
> The distribution of the residuals is closer to normality than that of
> the errors. In each residual, all other errors are added up (through
> e
> = (I-H)errors formula), although with unequal weights. For points of
> low leverage, when no such weight dominates too much, some sort of
> the
> CLT argument will show that the residuals will be approximately
> normal. So to see notable non-normality in residuals, you need to
> make
> quite big departures from normality in errors, and/or points of high
> leverage (that would most likely produce small residuals for the
> leverage points themselves, but will also skew the distribution of
> all
> other terms a little bit).
>
> On Tue, Aug 12, 2008 at 10:17 AM, Maarten buis
> <[email protected]> wrote:
> > --- Gaul� Patrick <[email protected]> wrote:
> >> >You should be careful however that
> >> >the assumption behind -regress- is not that BMI is normally
> >> >distributed, but that the residuals are normally distributed.
> >>
> >> My understanding is that the desirable properties of ordinary
> least
> >> squares hold without the normality assumption. Moreover, the
> >> assumption would be that the error term, not the residuals, is
> >> normally distributed.
> >
> > -regress- will always give you the line/(hyper)plane that minimizes
> the
> > sum of squared errors, regardless of the distrubtion of the error
> term.
> > In that sense you are correct. I have always learned that the
> standard
> > errors depend on the distribution of the error term. However, when
> I
> > simulated this with a skewed error term (log-normal with mean
> zero),
> > the p values seem ok: approximately uniformly distributed and
> > approximately 500 rejections of the true null hypothesis out of
> 10,000
> > draws. Regarding your second comment: The distribution of the
> residuals
> > gives you an estimate of the distribution of the error term.
> >
> > -- Maarten
> >
> > *-------------------- begin simulation -------------------------
> > capture program drop sim
> > program sim, rclass
> > drop _all
> > set obs 1000
> > gen x = invnorm(uniform())
> > gen y = 1 + x + exp(invnormal(uniform())) - exp(.5)
> > reg y x
> > tempname t
> > scalar `t' = (_b[x]-1)/_se[x]
> > return scalar p = 2*ttail(`e(df_r)', abs(`t'))
> > end
> >
> > simulate p=r(p), reps(10000) : sim
> > hist p
> > count if p < .05
> > *----------------------- end simulation ------------------------
> >
> >
> > -----------------------------------------
> > Maarten L. Buis
> > Department of Social Research Methodology
> > Vrije Universiteit Amsterdam
> > Boelelaan 1081
> > 1081 HV Amsterdam
> > The Netherlands
> >
> > visiting address:
> > Buitenveldertselaan 3 (Metropolitan), room Z434
> >
> > +31 20 5986715
> >
> > http://home.fsw.vu.nl/m.buis/
> > -----------------------------------------
> >
> > Send instant messages to your online friends
> http://uk.messenger.yahoo.com
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> > * For searches and help try:
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> >
>
>
>
> --
> Stas Kolenikov, also found at http://stas.kolenikov.name
> Small print: I use this email account for mailing lists only.
>
> *
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>
>
>
>
>
> *
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>
-----------------------------------------
Maarten L. Buis
Department of Social Research Methodology
Vrije Universiteit Amsterdam
Boelelaan 1081
1081 HV Amsterdam
The Netherlands
visiting address:
Buitenveldertselaan 3 (Metropolitan), room Z434
+31 20 5986715
http://home.fsw.vu.nl/m.buis/
-----------------------------------------
Send instant messages to your online friends http://uk.messenger.yahoo.com
*
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* http://www.ats.ucla.edu/stat/stata/
