Post-Instrument Bias in Linear Models

Adam Glynn, Miguel Rueda, Julian Schüssler

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Post-instrument covariates are often included as controls in instrumental variable (IV) analyses to address a violation of the exclusion restriction. However, we show that such analyses are subject to biases unless strong assumptions hold. Using linear constant-effects models, we present asymptotic bias formulas for three estimators (with and without measurement error): IV with post-instrument covariates, IV without post-instrument covariates, and ordinary least squares. In large samples and when the model provides a reasonable approximation, these formulas sometimes allow the analyst to bracket the parameter of interest with two estimators and allow the analyst to choose the estimator with the least asymptotic bias. We illustrate these points with a discussion of the settler mortality IV used by Acemoglu, Johnson, and Robinson.
Original languageEnglish
JournalSociological Methods & Research
Number of pages17
Publication statusE-pub ahead of print - 2024


  • causal inference
  • covariate adjustment
  • instrumental variables
  • linear models
  • measurement error


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