Bounding sets for treatment effects with proportional selection

In linear econometric models with proportional selection on unobservables, omitted variable bias in estimated treatment effects are roots of a cubic equation involving estimated parameters from a short and intermediate regression, the former excluding and the latter including all observable controls. The roots of the cubic are functions of ffi, the degree of proportional selection on unobservables, and Rmax, the R-squared in a hypothetical long regression that includes the unobservable confounder and all observable controls. In this paper a simple method is proposed to compute roots of the cubic over meaningful regions of the ffi-Rmax plane and use the roots to construct bounding sets for the true treatment effect. The proposed method is illustrated with both a simulated and an observational data set.