# RE: st: RE: Re: errors in outcome variables regression

 From Mark Schaffer To statalist@hsphsun2.harvard.edu, Mike Hollis Subject RE: st: RE: Re: errors in outcome variables regression Date Sat, 05 Jul 2003 22:52:41 +0100 (BST)

```Mike et al.,

> Assuming "well behaved" measurement error in the dependent variable,
> the OLS
> regression coefficient(s) will be unbiased.  But the standard errors
> for the
> coefficients (and other statistics involving the variance of the
> dependent
> variable) will be wrong.
>
> Consider an extension of Mark's model where we add a single
> explanatory
> variable and his error term distinguishing between the "true" and
> "measurement" components of the error term:
>
>   y = bo + b1X1 + u + u_m.
>
> Under standard assuptions, the residual variance of y is then V(u +
> u_m) =
> V(u) + V(u_m).  If y is measured without error, the standard error
> of b1
> would be sqrt( V(u) / SSx ), where u_m=0 and SSx is the usual
> mean-corrected
> sum of squares for X.  With measurement error, the standard error
> for b1
> would be sqrt ( V(u) + V (u_m) / SSx).  Accordingly, in this case,
> measurement error causes the true standard error of b1 to be
> overstated by 1
> + v(u-m)/v(m).

I think this is mixing up the true coefficient (b1 above) with estimates of
the true coefficient.  For inference what we need is the standard error of
our estimate of b1 - call it b1_hat.  Mike notes above that as the
measurement error of y goes up, so does the standard error of b1_hat.  But
this is natural - the greater the noisiness with which we observe y, the
noisier will be our estimate of b1, b1_hat, and hence the larger will be
the SE of b1_hat.

It's a standard result.  Here's Greene, "Econometric Analysis" (2000), p.
376: "This result conforms completely to the assumptions of the classical
regression model.  As long as the regressor is measured properly, the
measurement error on the dependent variable [our u_m] can be absorbed in
the disturbance of the regression [our u] and ignored."

And here's Woodridge, "Introductory Econometrics", p. 292:  "The usual
assumption is that the measurement error in y [our u_m] is statistically
independent of each explanatory variable.  If this is true, then the OLS
estimators are unbiased and consistent.  Further, the usual OLS inference
procedures (t, F and LM statistics) are valid."

--Mark

>  Other statistics involving either the variance of y
> or the
> residual variance of y|x (e.g., simple correlation coefficient, R**2
> for the
> equation, standardized regression coefficients) will likewise be
> incorrect.
>
> -----Original Message-----
> From: owner-statalist@hsphsun2.harvard.edu
> [mailto:owner-statalist@hsphsun2.harvard.edu]On Behalf Of Mark
> Schaffer
> Sent: Saturday, July 05, 2003 9:26 AM
> To: statalist@hsphsun2.harvard.edu; Mike Hollis
> Subject: Re: st: RE: Re: errors in outcome variables regression
>
>
> Mike et al.,
>
>
> > Measurement error in the endogeneous variable will, however,
> cause
> > the
> > residual variance for the equation to be overstated, meaning, in
> > general,
> > that the standard errors for the regression coefficients will be
> too
> > large
> > and the estimated t- and F-statistics will be too small.
>
> Scott re-replied to Margaret's original post, so I'll re-reply to
> Mike's.
>
> I'm pretty sure Mike's point above isn't correct.   So long as the
> measurement error satisfies the usual distributional assumptions
> that make
> OLS kosher (homoskedasticity, orthogonality etc.), and so long as
> the
> regressions error (the "non-measurement-error" error) also satisfies
> these
> assumptions, then OLS is fine.
>
> Intuitively, the reason is the following.  Say the measurement error
> is u_m
> and the regression error is u.  Define a new combined error term
> u_c = u + u_m.  Now rewrite the regression equation with this
> single
> combined error term.  It's not hard to see that so long as u_c
> satisfies
> the usual distributional assumptions (and it should if both u and
> u_m do
> so) then OLS is fine.
>
> For more details, see Scott's cite of Greene.
>
> That said, there will often be times that measurement error in the
> endogenous variable will not satisfy the usual assumptions and OLS
> will not
> be kosher.  In particular, if the measurement error is
> heteroskedastic,
> then the SEs and the F-stat will not be consistent.  But this is a
> heteroskedasticity problem, not a measurement error problem per
> se.
>
> Hope this helps.
>
> --Mark
>
>
> >
> > If you have a estimate of the reliability of the outcome
> variable,
> > you could
> > conceivable use this to adjust the standard errors and
> associated
> > statistics, although the quality of this adjustment obviously
> > depends on the
> > quality of your reliability estimate.  (Note, however, that the
> > intra-class
> > correlation coefficient is a measure of non-independence.
> > Correcting for
> > measurement error in your case requires something like
> Chronbach's
> > alpha or,
> > if you're lucky enough to have them, multiple indicators for the
> > outcome
> > variable.  See Ken Bollen's _Structural Equations with Latent
> > Variables_ for
> > a discussion of different strategies.)
> >
> > If the regression coefficients in your current model are
> > statitically
> > significant (i.e., you're not in a situation where you're trying
> to
> > correct
> > for measurement error to reduce standard errors in an attempt to
> > cause
> > statistically non-significant to become significant), you might
> > simply note
> > the fact that you suspect your outcome variable is affected by
> > measurement
> > error and that this will cause the significance level of the
> > regression
> > coefficients in your model to be underestimated.
> >
> > -----Original Message-----
> > From: owner-statalist@hsphsun2.harvard.edu
> > [mailto:owner-statalist@hsphsun2.harvard.edu]On Behalf Of Scott
> > Merryman
> > Sent: Friday, July 04, 2003 5:58 AM
> > To: statalist@hsphsun2.harvard.edu
> > Subject: st: Re: errors in outcome variables regression
> >
> >
> > ----- Original Message -----
> > From: "Margaret May" <M.T.May@bristol.ac.uk>
> > To: <statalist@hsphsun2.harvard.edu>
> > Sent: Friday, July 04, 2003 5:32 AM
> > Subject: st: errors in outcome variables regression
> >
> >
> > > I have been looking at the command eivreg (errors in variables
> > regression)
> > > which corrects the effect estimate when independent variables
> are
> > measured
> > > with error. The problem I have is looking at differences in a
> > continuous
> > > outcome between exposure groups where the outcome variable is
> > measured
> > with
> > > error. I can estimate the reliability of the outcome measure as
> I
> > have
> > data
> > > from a validity study so can estimate the intra-class
> > correlation
> > > coefficient. Is there a method for correcting for measurement
> > error in
> > > outcome variables?
> > >
> > > Margaret May
> > >
> >
> >
> > A question concerning errors in the dependent variable came up
> on
> > March 6th
> > by Charlie Trevor with replies by myself and Mark Schaffer on
> March
> > 6th and
> > 7th.
> >
> >
> > Is this necessary?
> > >From Greene (4th ed. page 376):
> > "...assuming for the moment that only y* is measured with
> error...
> > this
> > result conforms completely to the assumption of the classical
> > regression
> > model.  As long as the regressor is measured properly,
> measurement
> > error on
> > the dependent variable can be absorbed in the disturbance of the
> > regression
> > and ignored."
> >
> > Hope this helps,
> > Scott
> >
> >
> >
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>
>
>
> Prof. Mark Schaffer
> Director, CERT
> Department of Economics
> School of Management & Languages
> Heriot-Watt University, Edinburgh EH14 4AS
> tel +44-131-451-3494 / fax +44-131-451-3008
> email: m.e.schaffer@hw.ac.uk
> web: http://www.sml.hw.ac.uk/ecomes
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Prof. Mark Schaffer
Director, CERT
Department of Economics
School of Management & Languages
Heriot-Watt University, Edinburgh EH14 4AS
tel +44-131-451-3494 / fax +44-131-451-3008
email: m.e.schaffer@hw.ac.uk
web: http://www.sml.hw.ac.uk/ecomes
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