A combined estimator of regression models with measurement errors

When the regressors are observed with measurement errors, the OLS estimator is inconsistent and typically the use of the IV estimator is recommended. In this paper we place this recommendation under scrutiny, especially (1) when the instruments are weak and (2) when there are many instruments. Following Hansen (Econometric Reviews, http://www.ssc.wisc.edu/~bhansen/papers/shrinkiv.pdf , 2017), we use the Hausman (Econometrica 46(6):1251–1271, 1978) test for errors in variables to combine OLS and IV estimators. The combined estimator has the asymptotic risk strictly less than that of the IV estimator. Then we show some useful findings for small samples based on the Monte Carlo simulations. In terms of the mean squared error risk, we find that (a) typically OLS gets worse as the measurement error gets larger while IV is more robust and better than OLS, (b) OLS can be better than IV when the measurement error is small, and (c) the combined estimator outperforms IV as the asymptotic result predicts. (a) and (b) are true only when the instruments are not weak and when there are not many instruments. However, when the instruments are weak or when there are many instruments: (c) still holds as it is a theorem, but (a), (b) turn out to become quite the opposite, i.e., OLS can be much better than IV even when measurement error is large. This happens because IV is known to be inconsistent with weak instruments and many instruments (Staiger and Stock Econometrica (65(3), 557–5861997, Bekker Econometrica 62, 657–681, 1994)), and can be much worse than OLS, making the combined estimator close to OLS. In that case, the typical recommendation to use IV should be guided by the combined estimator, and IV and the combined estimator need to be regularized for weak instruments and many instruments.