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lecture3_140b_2011

# lecture3_140b_2011 - Lecture 3 Omitted Variables Bias and...

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Olivier Deschenes, UCSB, Econ 140B, Winter 2011 Lecture 3: Omitted Variables Bias and Multivariate Regression

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Olivier Deschenes, UCSB, Econ 140B, Winter 2011 Omitted Variables Bias Consider the simple model Y i = 0 + 1 X i + u i If the regressor X i is correlated with a variable that (i) has been omitted from the model, and (ii) that is also a predictor of the dependent variable Y i , then the OLS estimator will have omitted variable bias (i.e. OLS is not consistent) In the test score example, omitted variable bias will arise if class size is correlated (for example) with average family income, average student ability, etc OVB occurs when LSA#1 [E(u|X) is not satisfied]
Olivier Deschenes, UCSB, Econ 140B, Winter 2011 Another Example of OVB: Recall the study of the impact of veteran status on post-service health. We want to estimate the following regression: Y i = 0 + 1 X i + u i Y i = a measure of health X i = indicator of veteran status (=1 if person i is veteran, =0 otherwise) u i = unmeasured determinants of health But veterans are screened by the military administration, and applicants to military are subject to mental and physical aptitude tests Individuals admitted in military have better health than those rejected Thus: Corr(X i ,u i )>0, and the OLS estimator of 1 will be biased due to omitted variable bias

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Olivier Deschenes, UCSB, Econ 140B, Winter 2011 Key Result: Let corr(X i ,u i ) = Xu (Note: If LSA #1 satisfied then Xu =0) Then, we can prove that the OLS estimator has the following probability limit: This says that as the sample size increases, does not get close to the population 1 with high probability X u Xu 1 p 1 σ σ ρ β β ˆ 1 β ˆ
Olivier Deschenes, UCSB, Econ 140B, Winter 2011 Conclusion on OVB OVB occurs when: an omitted variable is both : (1) a determinant of Y (that is, it is contained in u ); and (2) correlated with X Omitted variable bias is a problem whether the sample size is small or large. Even in the limit experiment when n , the OLS estimator remains inconsistent Whether this “bias” is large or small depends on the magnitude of the correlation between X i and u i . The larger | Xu |, the larger is the bias The direction of the bias depends on the sign of Xu . If Xu >0, then the OLS estimator overstates 1

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Olivier Deschenes, UCSB, Econ 140B, Winter 2011 Solutions to problem of omitted variables bias: 1. Add more variables to the regression model Effectively, this improves the credibility of the assumption E[u i |X i ]=0 (LSA#1) Why: the more variables you include, the more potential relevant predictors of Y i you include However: there is a bias/variance tradeoff in finite samples (more regressors lower precision of estimator) 2. Later : instrumental variables regression and panel data models (also controlled random experiments)
Olivier Deschenes, UCSB, Econ 140B, Winter 2011 The Population Multiple Regression Model (S&W Section 6.2) Consider the case of two regressors: Y i = 0 + 1 X 1 i + 2 X 2 i + u i , i = 1,…, n

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lecture3_140b_2011 - Lecture 3 Omitted Variables Bias and...

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