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.