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Instrumental Variables Regression with Weak Instruments
Open accessAbstract
This paper develops asymptotic distribution theory for instrumental variable regression when the partial correlation between the instruments and a single included endogenous variable is weak, here modeled as local to zero. Asymptotic representations are provided for various instrumental variable statistics, including the two-stage least squares (TSLS) and limited information maximum- likelihood (LIML) estimators and their t-statistics. The asymptotic distributions are found to provide good approximations to sampling distributions with just 20 observations per instrument. Even in large samples, TSLS can be badly biased, but LIML is, in many cases, approximately median unbiased. The theory suggests concrete quantitative guidelines for applied work. These guidelines help to interpret Angrist and Krueger's (1991) estimates of the returns to education: whereas TSLS estimates with many instruments approach the OLS estimate of 6%, the more reliable LIML and TSLS estimates with fewer instruments fall between 8% and 10%, with a typical confidence interval of (6%, 14%).
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“In either of these situations, controlling for such a covariate could bias the MR effect estimate.55–57”
From Mendelian randomization (2022) · cites this paper as [57]
In the citing paper
“Another type of bias that can arise in MR studies that cannot be easily corrected for is selection bias64 In an MR study, an example of selection bias would be if an individual’s exposure and outcome values affected their participation.57”
From Mendelian randomization (2022) · cites this paper as [57]
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Vermeulen, J. M., Wootton, R. E., Treur, J. L., Sallis, H. M., Jones, H. J., Zammit, S., van den Brink, W., Goodwin, G. M., de Haan, L., & Munafò, M. R. (2021) · The British Journal of Psychiatry · doi:10.1192/bjp.2019.202 · Cited by 78
Loyle, C. E., & Appel, B. J. (2017) · International Studies Quarterly · doi:10.1093/isq/sqx045 · Cited by 78
An analysis of the pricing of traits in the U.S. corn seed market
Shi, G., Chavas, J.-P., & Stiegert, K. (2010) · American Journal of Agricultural Economics · doi:10.1093/ajae/aaq063 · Cited by 52
“Following Staiger and Stock (1997), this means that there is no statistical evidence that our instruments are weak.”
Taylor, P. N., Richmond, R., Davies, N., Sayers, A., Stevenson, K., Woltersdorf, W., Taylor, A., Groom, A., Northstone, K., Ring, S., et al. (2016) · The Journal of Clinical Endocrinology & Metabolism · doi:10.1210/jc.2015-3505 · Cited by 48
Russell, A. E., Ford, T., Gunnell, D., Heron, J., Joinson, C., Moran, P., Relton, C., Suderman, M., Hemani, G., & Mars, B. (2020) · Brain, Behavior, and Immunity · doi:10.1016/j.bbi.2020.05.065 · Cited by 39
“Potential global pleiotropy was assessed through examining consistency of the causal effect estimates, and F statistics were generated to assess weak instrument bias (Hartwig et al., 2017; Khandaker et al., 2019; Georgakis et al., 2019; Staiger and Stock, 1994).”
MethodEfficient Tests Under a Weak Convergence Assumption
(2011) · Econometrica · doi:10.3982/ecta7793 · Cited by 30
Inference with a single treated cluster
Hagemann, A. (2020) · arXiv · doi:10.48550/arxiv.2010.04076 · Cited by 18
Liu, Z., Zhang, Y., Graham, S., Pique-Regi, R., Dong, X. C., Chen, Y. E., Willer, C., & Liu, W. (2019) · doi:10.1101/657734 · Cited by 250
Wu, C., Pu, Q., Zou, Y., Jin, J., Lei, Y., Hu, Y., Chen, Z., & Yu, L. (2025) · Scientific Reports · doi:10.1038/s41598-025-15525-4 · Cited by 3
Metadata
- Authors
- Staiger, D., & Stock, J.
- ORCID
- Staiger, Douglas
- Venue
- National Bureau of Economic Research
- Publisher
- National Bureau of Economic Research
- Published
- 1994-01
- OpenAlex
- W3122531453
- Cited by
- 5,302
- References
- 0
- Citations by year
- 2026: 39 · 2025: 51 · 2024: 42 · 2023: 41 · 2022: 73 · 2021: 242 · 2020: 254 · 2019: 246 · 2018: 248 · 2017: 286 · 2016: 328 · 2015: 343 · 2014: 345 · 2013: 336 · 2012: 337
- Type
- report
- Language
- en
- Open access
- green
- doi.org
- Locations
- National Bureau of Economic Research (publishedVersion)
- RePEc: Research Papers in Economics (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Causal Inference Techniques
- Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Spatial and Panel Data Analysis
- Social Sciences › Economics, Econometrics and Finance › General Economics, Econometrics and Finance › Monetary Policy and Economic Impact
- Keywords
- weak instruments, instrumental variable regression, asymptotic distribution, two-stage least squares, limited information maximum likelihood, endogeneity, returns to education, t-statistic
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