How measurement error affects inference in linear regression

Measurement error biases OLS results. When the measurement error variance in absolute or relative (reliability) form is known, adjustment is simple. We link the (known) estimators for these cases to GMM theory and provide simple derivations of their standard errors. Our focus is on the test statistics. We show monotonic relations between the t-statistics and R2s of the (infeasible) estimator if there was no measurement error, the inconsistent OLS estimator, and the consistent estimator that corrects for measurement error and show the relation between the t-value and the magnitude of the assumed measurement error variance or reliability. We also discuss how standard errors can be computed when the measurement error variance or reliability is estimated, rather than known, and we indicate how the estimators generalize to the panel data context, where we have to deal with dependency among observations. By way of illustration, we estimate a hedonic wine price function for different values of the reliability of the proxy used for the wine quality variable.