# Omitted Variable Bias

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In social science research, control variables are often included out of concerns about inducing bias into the coefficients of interest [1, 2]. However, short of knowing the true data-generating process—an unlikely situation—the inclusion of even relevant controls may in fact aggravate the problem.

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Contributed by: Alrik Thiem (December 2010)

After work by: Kevin A. Clarke, University of Rochester (USA)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

For the case of OLS, let be the true model, be first misspecified model, and be the second misspecified model.

: , with ,

: , and

: .

If for we have , then .

If for we have , then the bias of the expected values of for and for are given by

(1) ,

(2) .

According to the logic of including controls in order to reduce bias, the following weak inequality should always hold.

For the case of GLM, as before let be the true model, let be the first misspecified model, and let be the second misspecified model.

: , with and ,

: , and

: .

The normalized values of and are given by [1] as

and

.

According to the logic of including controls in order to reduce bias, the following weak inequality should always hold.

.

References

[1] K. A. Clarke, "Return of the Phantom Menace: Omitted Variable Bias in Political Research," *Conflict Management and Peace Science*, 26(1), 2009 pp. 46–66.

[2] K. A. Clarke, "The Phantom Menace: Omitted Variable Bias in Econometric Research," *Conflict Management and Peace Science*, 22(4), 2005 pp. 341–352.

[3] M. H. Gail, S. Wieand, and S. Piantadosi, "Biased Estimates of Treatment Effect in Randomized Experiments with Nonlinear Regressions and Omitted Covariates," *Biometrika*, 71(3), 1984 pp. 431–444.

[4] J. S. Cramer, *Logit Models from Economics and Other Fields*, Cambridge: Cambridge University Press, 2003.

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