Liyu Dou, Pengjin Min, Wenjie Wang, Yichong Zhang
arXiv 30 Jun 2025 · Econometrics
arXiv:2506.23816 · PDF · DOI · OpenAlex · Extracted main text
Researchers often report empirical results that are based on low-dimensional IVs, such as the shift-share IV, together with many IVs. Could we combine these results in an efficient way and take advantage of the information from both sides? In this paper, we propose a combination inference procedure to solve the problem. Specifically, we consider a linear combination of three test statistics: a standard cluster-robust Wald statistic based on the low-dimensional IVs, a leave-one-cluster-out Lagrangian Multiplier (LM) statistic, and a leave-one-cluster-out Anderson-Rubin (AR) statistic. We first establish the joint asymptotic normality of the Wald, LM, and AR statistics and derive the corresponding limit experiment under local alternatives. Then, under the assumption that at least the low-dimensional IVs can strongly identify the parameter of interest, we derive the optimal combination test based on the three statistics and establish that our procedure leads to the uniformly most powerful (UMP) unbiased test among the class of tests considered. In particular, the efficiency gain from the combined test is of “free lunch" in the sense that it is always at least as powerful as the test that is only based on the low-dimensional IVs or many IVs.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Goldsmith-Pinkham, P., I. Sorkin, and H. Swift (2020) Bartik instruments: What, when, why, and how | 1.000 | 9 | 4 | 100% |
| 2 | Card, D (2009) Immigration and Inequality | 1.000 | 8 | 4 | 100% |
| 3 | Hausman, J. A., W. K. Newey, T. Woutersen, J. C. Chao, and N. R. Swa… (2012) Instrumental variable estimation with heteroskedasticity and many instruments | 1.000 | 6 | 4 | 100% |
| 4 | Chao, J. C., N. R. Swanson, J. A. Hausman, W. K. Newey, and T. Woute… (2012) Asymptotic Distribution Of JIVE In A Heteroskedastic IV Regression With Many Instruments | 0.899 | 11 | 4 | 73% |
| 5 | Angrist, J. D. and A. B. Krueger (1991) Does Compulsory School Attendance Affect Schooling and Earning? | 0.894 | 7 | 4 | 71% |
| 6 | Lim, D., W. Wang, and Y. Zhang (2024) a): A conditional linear combination test with many weak instruments | 0.843 | 10 | 6 | 60% |
| 7 | Mikusheva, A. and L. Sun (2022) Inference with many weak instruments | 0.794 | 6 | 5 | 50% |
| 8 | Müller, U. K (2011) Efficient tests under a weak convergence assumption | 0.737 | 3 | 3 | 67% |
| 9 | Belloni, A., D. Chen, V. Chernozhukov, and C. Hansen (2012) Sparse models and methods for optimal instruments with an application to eminent domain | 0.737 | 3 | 2 | 100% |
| 10 | Andrews, I (2016) Conditional linear combination tests for weakly identified models | 0.644 | 2 | 2 | 100% |
Showing the top 10 of 63 scored citations.