Estimation with weak instruments: Accuracy of higher‐order bias and MSE approximations

Abstract

In this paper, we consider parameter estimation in a linear simultaneous equations model. It is well known that two‐stage least squares (2SLS) estimators may perform poorly when the instruments are weak. In this case 2SLS tends to suffer from the substantial small sample biases. It is also known that LIML and Nagar‐type estimators are less biased than 2SLS but suffer from large small sample variability. We construct a bias‐corrected version of 2SLS based on the Jackknife principle. Using higher‐order expansions we show that the MSE of our Jackknife 2SLS estimator is approximately the same as the MSE of the Nagar‐type estimator. We also compare the Jackknife 2SLS with an estimator suggested by Fuller (Econometrica 45, 933–54) that significantly decreases the small sample variability of LIML. Monte Carlo simulations show that even in relatively large samples the MSE of LIML and Nagar can be substantially larger than for Jackknife 2SLS. The Jackknife 2SLS estimator and Fuller's estimator give the best overall performance. Based on our Monte Carlo experiments we conduct informal statistical tests of the accuracy of approximate bias and MSE formulas. We find that higher‐order expansions traditionally used to rank LIML, 2SLS and other IV estimators are unreliable when identification of the model is weak. Overall, our results show that only estimators with well‐defined finite sample moments should be used when identification of the model is weak.