After addressing plausible confounding through rich controls and fixed effects, accounting for latent heterogeneity, and mitigating simultaneity and measurement error, researchers must assess whether meaningful endogeneity concerns remain with a sensitivity approach.
A common endogeneity critique is that an estimated effect may be driven by an unobserved confound. Frank (2000) reframes this critique as a quantitative question: how strong would an unobserved bias have to be to change the inference? Consider a retailer that deploys personalized coupons through its app. If estimated spending increases following coupon exposure, a reviewer may argue that higher-propensity customers were more likely to receive the coupon, rendering the estimated lift non-causal.
Frank (2000) formalizes this logic using a switch-point framework. Rather than debating whether bias exists, the approach asks how much bias would be required to push an estimate below a specified threshold, which may reflect statistical significance or managerial relevance. The Impact Threshold for a Confounding Variable (ITCV) quantifies the strength an unobserved confound would need, in partial-correlation terms conditional on observed controls, to overturn inference about a focal coefficient.
As Frank (2000, p. 150) explains, the goal is “to calculate a single valued threshold at which the impact of the confound would be great enough to alter an inference with regard to a regression coefficient.” Figure 7A illustrates this logic, and Web Appendix G1 summarizes the implementation steps. The key insight is that an unobserved confound threatens inference only if it is related to both the predictor and the outcome, which ITCV captures as the product of these partial correlations. In this way, the ITCV reframes the debate from whether an unobserved confound exists to how much bias would be required to invalidate the conclusion, providing a concrete benchmark for assessing residual endogeneity risk.
Although ITCV is most directly informative for enhancing confounding that inflates estimated effects and raises Type I error concerns, Frank et al. (2013) show that the same logic applies to suppression. By redefining the inference threshold relative to a substantively meaningful benchmark, researchers can assess how much bias would be required to sustain an incorrect inference that an effect is non-harmful or not meaningful.
A complementary sensitivity metric is the Robustness of Inference to Replacement (RIR), which expresses robustness in terms of the percentage of observations that would need to be affected by endogeneity bias for the inference to fail (Frank et al. 2013). The RIR quantifies the share of treatment cases for which the observed effect would have to be entirely driven by an unobserved confounder to overturn the result. Because this metric captures bias regardless of its source, it encompasses all sources of endogeneity (Frank et al. 2013).
References
Frank, Kenneth A. (2000), “Impact of a confounding variable on a regression coefficient,” Sociological Methods & Research, 29 (2), 147-194.
Frank, Kenneth A, Spiro J. Maroulis, Min Q. Duong, and Benjamin M. Kelcey (2013), “What would it take to change an inference? Using Rubin’s causal model to interpret the robustness of causal inferences,” Educational Evaluation and Policy, 35 (4), 437-460.