An Omitted Variable Bias Framework for Sensitivity Analysis of Instrumental Variables

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Abstract We develop an omitted variable bias framework for sensitivity analysis of instrumental variable estimates that naturally handles multiple side-effects (violations of the exclusion restriction assumption) and confounders (violations of the ignorability of the instrument assumption) of the instrument, exploits expert knowledge to bound sensitivity parameters, and can be easily implemented with standard software. Specifically, we introduce sensitivity statistics for routine reporting, such as (extreme) robustness values for instrumental variables, describing the minimum strength that omitted variables need to have to change the conclusions of a study. Next we provide visual displays that fully characterize the sensitivity of point estimates and confidence intervals to violations of the standard instrumental variable assumptions. Finally, we offer formal bounds on the worst possible bias under the assumption that the maximum explanatory power of omitted variables is no stronger than a multiple of the explanatory power of observed variables. Conveniently, many pivotal conclusions regarding the sensitivity of the instrumental variable estimate (e.g. tests against the null hypothesis of zero causal effect) can be reached simply through separate sensitivity analyses of the effect of the instrument on the treatment (the first stage) and the effect of the instrument on the outcome (the reduced form). We apply our methods in a running example that uses proximity to college as an instrumental variable to estimate the returns to schooling.