We welcome the comments on our paper by Ackerberg and Devereux (AD) and Blomquist and Dahlberg (BD), and we are happy to take this opportunity to respond briefly to them.
We agree wholeheartedly with both AD and BD that is it unprofitable to try to obtain meaningful empirical results when all available instruments are very weak. At least in our experiments, however, it is only in this case that JIVE1 ever outperforms LIML in terms of dispersion; see Figure 5 and the top left panel of Figure 6. Thus we disagree with BD that our 'categorical rejection of JIVE' is not in accord with our simulation results. BD are, however, absolutely correct to point out that our experiments deal with a very simple case and do not tell the whole story.
AD discuss two quite new estimators, IJIVE and UIJIVE, which apparently improve upon the JIVE1 estimator. Their comment suggests that our conclusion that JIVE1 is inferior to LIML does not necessarily apply to these estimators. In order to investigate the properties of these two estimators, we performed a new set of simulation experiments, with the same design as those we describe in the paper. The results of the new experiments are presented graphically in Figures 1a-6a, which, like Figures 1-13 from the paper itself, are available from the JAE Data Archive website at www.econ.queensu.ca/jae/. The new figures deal with the same cases as Figures 1-6 of the paper. Results for IJIVE and UIJIVE are added, and, for readability, results for 2SLS are removed.
The superiority of IJIVE and, especially, UIJIVE relative to JIVE1 emerges quite clearly from the new figures. What is particularly interesting is that UIJIVE tends to be substantially less dispersed than the other JIVE estimators. When the instruments are weak, it is often much less dispersed than LIML. Our tentative conclusion is that UIJIVE is the best JIVE estimator to date and may well be worth using in practice.
The simulation results suggest that IJIVE and UIJIVE, like JIVE1 and LIML, have no moments, and we have confirmed this analytically. This leads us to question the interpretation of the results in Phillips and Hale (1977) and in Ackerberg and Devereux (2003) that purport to yield approximate biases for these estimators. These results are based on stochastic expansions, which can be represented schematically as