While Davidson and MacKinnon (2006) (DM) present an interesting analysis of recently advocated estimators for overidentified instrumental variables models, we disagree with their conclusion that the LIML estimator should almost always be prefered to the JIVE estimators of Phillips and Hale (1977) (PH), Angrist et al. (1999) (AIK) and Blomquist and Dahlberg (1999) (BD). Our disagreement stems from three general points. First, we show that many of the advantages of LIML over JIVE that DM illustrate are significantly reduced when one uses the 'improved' JIVE estimators proposed by Ackerberg and Devereux (2003) (AD). Second, we argue that the resulting advantages of LIML in terms of median bias and dispersion are small except in cases where the instruments are so weak that meaningful or relevant inference is likely not possible with any estimator. Lastly, we discuss some important advantages of JIVE estimators over the LIML estimator-in particular, citing new results from AD regarding the robustness of JIVE estimators to heteroskedasticity.
AD suggest two new JIVE estimators that improve the small sample properties of the original JIVE estimators proposed by PH, AIK and BD, and studied in DM. The first of these, the IJIVE ((Improved) JIVE) estimator, removes small sample bias by partialling out exogenous explanatory variables (including the constant term) before running the JIVE procedure. AD show that this eliminates a bias term in JIVE that is proportional to the number of such variables. The second of these, the UIJIVE ((Unbiased) IJIVE) estimator, removes an additional bias term. Monte Carlo results also show that IJIVE and UIJIVE, particularly UIJIVE, also appear to be more precise estimators than JIVE. The first two panels of Figure replicate the middle two panels of DM's Figure , with additional series for IJIVE and UIJIVE. The first panel indicates that in the region where R 2 > 0.2 (we discuss values of R 2 < 0.2 in a moment), the IJIVE estimator removes almost all of the median bias of the JIVE estimator. The second panel shows that IJIVE improves on the 9 decile range of JIVE by about 20%, although this is still considerably larger than that of LIML. Interestingly, UIJIVE does not do as well on the median bias, with levels about equal to that of JIVE (except in the opposite direction). However, the UIJIVE adjustment is designed to eliminate mean bias, not median bias. This is verified in panel 6, which looks at bias in the trimmed mean of the estimators. 1 The trimmed mean bias of UIJIVE is very small, in most cases significantly