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Bounds for Bias-Adjusted Treatment Effect in Linear Econometric Models


  • Deepankar Basu


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.

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  • Deepankar Basu, 2022. "Bounds for Bias-Adjusted Treatment Effect in Linear Econometric Models," Papers 2203.12431,
  • Handle: RePEc:arx:papers:2203.12431

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    References listed on IDEAS

    1. Deepankar Basu, 2020. "Bias of OLS Estimators due to Exclusion of Relevant Variables and Inclusion of Irrelevant Variables," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 82(1), pages 209-234, February.
    2. Joseph G. Altonji & Todd E. Elder & Christopher R. Taber, 2005. "Selection on Observed and Unobserved Variables: Assessing the Effectiveness of Catholic Schools," Journal of Political Economy, University of Chicago Press, vol. 113(1), pages 151-184, February.
    3. Joseph G. Altonji & Todd E. Elder & Christopher R. Taber, 2005. "An Evaluation of Instrumental Variable Strategies for Estimating the Effects of Catholic Schooling," Journal of Human Resources, University of Wisconsin Press, vol. 40(4), pages 791-821.
    4. Emily Oster, 2019. "Unobservable Selection and Coefficient Stability: Theory and Evidence," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 37(2), pages 187-204, April.
    5. Clarke, Damian, 2019. "A convenient omitted variable bias formula for treatment effect models," Economics Letters, Elsevier, vol. 174(C), pages 84-88.
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