The Annals of Applied Statistics

Instrumental variable analysis with censored data in the presence of many weak instruments: Application to the effect of being sentenced to prison on time to employment

Ashkan Ertefaie, Anh Nguyen, David J. Harding, Jeffrey D. Morenoff, and Wei Yang

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This article discusses an instrumental variable approach for analyzing censored data that includes many instruments that are weakly associated with the endogenous variable. We study the effect of imprisonment on time to employment using an administrative data on all individuals sentenced for felony in Michigan in the years 2003–2006. Despite the large body of research on the effect of prison on employment, this is still a controversial topic, especially since some of the studies could have been affected by unmeasured confounding. We take advantage of a natural experiment based on the random assignment of judges to felony cases and construct a vector of instruments based on judges’ ID that can avoid the confounding bias. However, some of the constructed instruments are weakly associated with the sentence type, that is, the endogenous variable, which can potentially lead to misleading results. Using a dimension reduction technique, we propose a novel semi-parametric estimation procedure in a survival context that is robust to the presence of many weak instruments. Specifically, we construct a test statistic based on the structural failure time model and provide inference by inverting the testing procedure. Under some assumptions, the optimal choice of the test statistic has also been derived. Analyses show a significant negative impact of imprisonment on time to employment which is consistent with some of the previous results. Our simulation studies highlight the importance of accounting for weak instruments in the analyses in terms of both bias and inflated type-I error rates.

Article information

Ann. Appl. Stat., Volume 12, Number 4 (2018), 2647-2673.

Received: May 2017
Revised: March 2018
First available in Project Euclid: 13 November 2018

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Instrumental variables survival time test statistics two-stage least squares unmeasured confounders


Ertefaie, Ashkan; Nguyen, Anh; Harding, David J.; Morenoff, Jeffrey D.; Yang, Wei. Instrumental variable analysis with censored data in the presence of many weak instruments: Application to the effect of being sentenced to prison on time to employment. Ann. Appl. Stat. 12 (2018), no. 4, 2647--2673. doi:10.1214/18-AOAS1174.

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Supplemental materials

  • Supplement to “Instrumental variable analysis with censored data in the presence of many weak instruments: Application to the effect of being sentenced to prison on time to employment”. The supplementary material contains: the proof of Proposition 1, additional simulation studies, and a discussion on the heterogeneous treatment effects.