arXiv 23 Mar 2022 · Econometrics
arXiv:2203.12431 · PDF · DOI · OpenAlex · Extracted main text
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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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Oster, E (2019) Unobservable Selection and Coefficient Stability | 0.986 | 50 | 6 | 96% |
| 2 | Alexanderian, A (2013) On continuous dependence of roots of polynomials on coefficients | 0.811 | 4 | 2 | 100% |
| 3 | Altonji, J. G., Elder, T. E., and Taber, C. R (2005) An evaluation of instrumental variable strategies for estimating the effects of catholic schooling | 0.737 | 3 | 2 | 100% |
| 4 | Wooldridge, J. M (2002) Econometric Analysis of Cross Section and Panel Data | 0.511 | 3 | 2 | 33% |
| 5 | Hellesland, J., Challamel, N., Casandjian, C., and Lanos, C (2013) Reinforced Concrete Beams, Columns and Frames: Section and Slender Member Analysis: Mechanics and Design | 0.511 | 2 | 2 | 50% |
| 6 | Altonji, J. G., Elder, T. E., and Taber, C. R (2000) Selection on Observed and Unobserved Variables: Assessing the Effectiveness of Catholic Schools | 0.511 | 2 | 1 | 100% |
| 7 | Clarke, D (2019) A convenient omitted variable bias formula for treatment effect models | 0.000 | 3 | 1 | 0% |
| 8 | Basu, D (2020) Bias of OLS Estimators due to Exclusion of Relevant Variables and Inclusion of Irrelevant Variables self | 0.000 | 1 | 1 | 0% |
| 9 | Najafi, H. S., Edalatpanah, S., and Gravvanis, G (2014) An efficient method for computing the inverse of arrowhead matrices | 0.000 | 1 | 1 | 0% |
Showing the top 9 of 9 scored citations.
arXiv econ.EM papers that cite this one, ranked by how heavily they lean on it.
| Citing paper | Intensity | Mentions | Sections | |
|---|---|---|---|---|
| 1 | Assessing Omitted Variable Bias when the Controls are Endogenous | 0.511 | 2 | 1 |
| 2 | An Axiomatic Approach to Comparing Sensitivity Parameters | 0.405 | 1 | 1 |