Improving Oster’s δ*: Exact Calculation for the Coefficient of Proportionality Without Subjective Specification of a Baseline Model

Abstract

Sensitivity analyses characterizing the hypothetical unobserved conditions that can alter estimates and statistical inferences are increasingly being applied in the social and health sciences. One of the most ascendant techniques is Oster’s (2019) coefficient of proportionality, which builds on Altonji, Elder and Tabor (2005) to frame sensitivity in terms of how strong selection on unobservables must be compared to selection on observables to change an inference. In this paper, we derive an alternative expression for the coefficient of proportionality that satisfies Oster’s constraints of reducing the estimated effect to a specified threshold with a corresponding coefficient of determination (R2). The derivation is exact instead of Oster’s approximation and does not depend on the subjective choice of baseline model. The derivation suggests an intuition of observed covariates potentially pre-empting, rather than representing, selection on unobserved covariates. Simulations demonstrate the improved performance of the exact expression relative to Oster’s approximation. Specifically, while Oster’s coefficient can overstate or understate the robustness of an inference, it is especially likely to overstate for strong designs in which observed covariates account for a large portion of an effect estimated in a baseline model. An application shows Oster’s approximation overstates the robustness of the estimated effect of low birth weight and preterm birth on IQ by more than 50%. From a practical standpoint, the quantities required for the exact expression are conventionally reported in published studies.

Keywords: omitted variable bias, coefficient of proportionality, exact

JEL Classification: C01,C18