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Econ 3120 1st Edition Lecture 14 Outline of Current Lecture I Unbiasedness of Multivariate OLS Estimators Current Lecture I Omitted Variable Bias with Many Regressors 6 3 Omitted Variable Bias with Many Regressors The above discussion only applies to a model with two independent variables When there are more than two independent variables and we leave one out things get much more complicated If this happens all of the estimated 0 s can be biased and the bias of j in the short regression generally depends on the relationship between the omitted variable and all the x s and on the relationship between all the x s But generally speaking if we assume that x j is uncorrelated with the other included x s then we can say something about the bias of the estimated coefficient Suppose the true model is log wage 0 1educ 2exper 3abil u where exper is years of experience and we leave out abil If we assume that education and experience are uncorrelated then the expectation of 1 in the regression log wage 0 1educ 2exper u will be E 1 1 3 Cov educ d abil Vard educ which is essentially the same thing as the formula 7 above Therefore we might expect 1 to be biased upwards if 3 is positive and education and ability are positively correlated Economists often try to sign the bias using this formula regardless of whether x j the variable of interest in the short regression is uncorrelated with the other included x 0 s This is a reasonable shortcut but it s important to know that it isn t exactly right especially if the included x s are highly correlated 2 2The general form for the estimate of j in the short regression when xk is omitted is j j k j where j is the coefficient on xj in the regression of xk on all of the included regressors See Wooldridge Section 3A for more detail 9 7 Variance of OLS Estimators To obtain the variance of OLS estimators we need make an assumption analogous to SLR 5 MLR 5 Homoskedasticity The error term in the OLS equation described by MLR 1 has constant