Embrace the noise: it is ok to ignore measurement error in a covariate, sometimes

In linear regression models, measurement error in a covariate causes ordinary least squares (OLS) to be biased and inconsistent. Instrumental variables (IV) is a common solution. While IV is also biased, it is consistent. Here, we undertake an asymptotic comparison of OLS and IV in the case where a covariate is mismeasured for ⌊Nδ⌋ of N observations with δ∈[0,1]. We show that OLS is consistent for δ<1 and is asymptotically normal and more efficient than IV for δ<0.5. Simulations and an application to the impact of body mass index on family income demonstrate the practical usefulness of this result.