Classical Measurement Error

Classical ME results in attenuation bias. We all (hopefully) know that. But ...

 The bias depends on the ratio of the variance of the measurement error, μ, to the variance of the observed X not explained by other covariates. Formally, this is shown below where the degree of the attenuation bias depends on the R2 from a regression of X on the remaining covariates in the model.



The implication is that even "small" measurement error (as captured by the variance of μ) can be enormously consequential in multiple regression. It is not sufficient IMHO to dismiss measurement error simply because you believe it to be "small."

On top of that, ignoring measurement error in a covariate because it is not the "regressor of interest" is a common, but costly, mistake. Measurement error in a control variable in your regression model will (in all likelihood) bias your coefficient of interest if the regressor of interest is correlated with the mismeasured regressor.

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