Bounds for Bias-Adjusted Treatment Effect in Linear Econometric Models

TL;DR

This paper develops a novel algorithm to compute bounds on treatment effects in linear econometric models with unobserved confounding, using cubic equations derived from model parameters and applying it to maternal behavior and child outcomes.

Contribution

It introduces a new algorithm for bounding treatment effects by analyzing roots of a cubic equation based on selection parameters and R-squared, enhancing bias adjustment methods.

Findings

The algorithm effectively delineates regions with unique and multiple real roots.

Application to maternal behavior shows practical bounds on treatment effects.

Method improves understanding of bias in linear econometric models.

Abstract

In linear econometric models with proportional selection on unobservables, omitted variable bias in estimated treatment effects are real roots of a cubic equation involving estimated parameters from a short and intermediate regression. The roots of the cubic are functions of $\delta$, the degree of selection on unobservables, and $R_{max}$, the R-squared in a hypothetical long regression that includes the unobservable confounder and all observable controls. In this paper I propose and implement a novel algorithm to compute roots of the cubic equation over relevant regions of the $\delta$-$R_{max}$ plane and use the roots to construct bounding sets for the true treatment effect. The algorithm is based on two well-known mathematical results: (a) the discriminant of the cubic equation can be used to demarcate regions of unique real roots from regions of three real roots, and (b) a small change in the coefficients of a polynomial equation will lead to small change in its roots because the latter are continuous functions of the former. I illustrate my method by applying it to the analysis of maternal behavior on child outcomes.