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From |
Mark Schaffer <[email protected]> |

To |
[email protected], Mike Hollis <[email protected]> |

Subject |
RE: st: RE: Re: errors in outcome variables regression |

Date |
Sat, 05 Jul 2003 22:52:41 +0100 (BST) |

Mike et al., Quoting Mike Hollis <[email protected]>: > 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: [email protected] > [mailto:[email protected]]On Behalf Of Mark > Schaffer > Sent: Saturday, July 05, 2003 9:26 AM > To: [email protected]; Mike Hollis > Subject: Re: st: RE: Re: errors in outcome variables regression > > > Mike et al., > > Quoting Mike Hollis <[email protected]>: > > > 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: [email protected] > > [mailto:[email protected]]On Behalf Of Scott > > Merryman > > Sent: Friday, July 04, 2003 5:58 AM > > To: [email protected] > > Subject: st: Re: errors in outcome variables regression > > > > > > ----- Original Message ----- > > From: "Margaret May" <[email protected]> > > To: <[email protected]> > > 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. > > > > My reply was: > > > > 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 > > > > > > > > * > > * For searches and help try: > > * http://www.stata.com/support/faqs/res/findit.html > > * http://www.stata.com/support/statalist/faq > > * http://www.ats.ucla.edu/stat/stata/ > > > > > > * > > * For searches and help try: > > * http://www.stata.com/support/faqs/res/findit.html > > * http://www.stata.com/support/statalist/faq > > * http://www.ats.ucla.edu/stat/stata/ > > > > > > 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: [email protected] > web: http://www.sml.hw.ac.uk/ecomes > ________________________________________________________________ > > DISCLAIMER: > > This e-mail and any files transmitted with it are confidential > and intended solely for the use of the individual or entity to > whom it is addressed. If you are not the intended recipient > you are prohibited from using any of the information contained > in this e-mail. In such a case, please destroy all copies in > your possession and notify the sender by reply e-mail. Heriot > Watt University does not accept liability or responsibility > for changes made to this e-mail after it was sent, or for > viruses transmitted through this e-mail. Opinions, comments, > conclusions and other information in this e-mail that do not > relate to the official business of Heriot Watt University are > not endorsed by it. > ________________________________________________________________ > * > * For searches and help try: > * http://www.stata.com/support/faqs/res/findit.html > * http://www.stata.com/support/statalist/faq > * http://www.ats.ucla.edu/stat/stata/ > > > * > * For searches and help try: > * http://www.stata.com/support/faqs/res/findit.html > * http://www.stata.com/support/statalist/faq > * http://www.ats.ucla.edu/stat/stata/ > 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: [email protected] web: http://www.sml.hw.ac.uk/ecomes ________________________________________________________________ DISCLAIMER: This e-mail and any files transmitted with it are confidential and intended solely for the use of the individual or entity to whom it is addressed. If you are not the intended recipient you are prohibited from using any of the information contained in this e-mail. In such a case, please destroy all copies in your possession and notify the sender by reply e-mail. Heriot Watt University does not accept liability or responsibility for changes made to this e-mail after it was sent, or for viruses transmitted through this e-mail. Opinions, comments, conclusions and other information in this e-mail that do not relate to the official business of Heriot Watt University are not endorsed by it. ________________________________________________________________ * * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

**References**:**RE: st: RE: Re: errors in outcome variables regression***From:*"Mike Hollis" <[email protected]>

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