An instrument is a haphazard push to accept a treatment in a context in which an individual may refuse to accept the treatment despite the push. To be an instrument, the push must have no effect on the outcome if the push fails to prompt acceptance of the treatment -- this is called the "exclusion restriction." A perfect instrument is not merely haphazard, but rather truly randomized; however, there are typically doubts about whether the instrument is actually randomized and whether it actually satisfies the exclusion restriction. As indicated in the first item below, my interest in instruments was stimulated by the important paper of Angist, Imbens and Rubin, published in JASA in 1996. For an informal discussion of instruments, see Chapter 7 of my book Causal Inference (MIT Press 2023) or Chapter 13 of my book Observation and Experiment.
Rosenbaum PR. Comment on "Identification of causal effects using instrumental variables" by Angist, Imbens and Rubin. Journal of the American Statistical Association. 1996 Jun 1;91(434):465-8. In JSTOR: https://www.jstor.org/stable/2291633 https://doi.org/10.2307/2291633
Rosenbaum PR. Covariance adjustment in randomized experiments and observational studies. Statistical Science. 2002 Aug;17(3):286-327. (Section 6 discusses instruments.) In JSTOR: www.jstor.org/stable/3182790 Open access at ProjectEuclid: https://doi.org/10.1214/ss/1042727942
Greevy R, Silber JH, Cnaan A, Rosenbaum PR. Randomization inference with imperfect compliance in the ACE-inhibitor after anthracycline randomized trial. Journal of the American Statistical Association. 2004 Mar 1;99(465):7-15. (Noncompliance in randomized trials can come close to the perfect instrument.) In JSTOR: www.jstor.org/stable/27590348 https://doi.org/10.1198/016214504000000025
Imbens GW, Rosenbaum PR. Robust, accurate confidence intervals with a weak instrument: quarter of birth and education. Journal of the Royal Statistical Society A. 2005 Jan;168(1):109-26. In JSTOR: www.jstor.org/stable/3559712 https://doi.org/10.1111/j.1467-985X.2004.00339.x
Small DS, Rosenbaum PR. War and wages: the strength of instrumental variables and their sensitivity to unobserved biases. Journal of the American Statistical Association. 2008 Sep 1;103(483):924-33. In JSTOR: www.jstor.org/stable/27640133 https://doi.org/10.1198/016214507000001247
Baiocchi M, Small DS, Lorch S, Rosenbaum PR. Building a stronger instrument in an observational study of perinatal care for premature infants. Journal of the American Statistical Association. 2010 Dec 1;105(492):1285-96. In JSTOR: www.jstor.org/stable/27920165 https://doi.org/10.1198/jasa.2010.ap09490
Ertefaie A, Small DS, Rosenbaum PR. Quantitative evaluation of the trade-off of strengthened instruments and sample size in observational studies. Journal of the American Statistical Association. 2018 Jul 3;113(523):1122-34. https://doi.org/10.1080/01621459.2017.1305275
Heng S, Small DS, Rosenbaum PR. Finding the strength in a weak instrument in a study of cognitive outcomes produced by Catholic high schools. Journal of the Royal Statistical Society A. 2020 Jun;183(3):935-58. https://doi.org/10.1111/rssa.12559
Karmakar B, Small DS, Rosenbaum PR. Reinforced designs: Multiple instruments plus control groups as evidence factors in an observational study of the effectiveness of Catholic schools. Journal of the American Statistical Association. 2021 Jan 2;116(533):82-92. https://doi.org/10.1080/01621459.2020.1745811