You are trying to model the effect of class size on students' performance in schools. You hypothesize that larger class size (number of students per teacher) should decrease test scores in school. But you forgot to control for median income in the community, that is income in the community is part of the error term. Would you expect coefficient for class size to be biased? Discuss directionality of bias. TestScore_i=B0+B1*ClassSize_i+ε_i. You must show your work.

 

Failing to control for median income in the community is equal to omitting it in the regression model. This will lead to an overstate of the effect of class size on test score. This change in the magnitude of the coefficient of class size is typical of the effect of incorrectly omitting a relevant variable(median income) in a regression model. Omission of a relevant variable, i.e., one whose coefficient is non-zero/significant in the model leads to an estimator that is biased but a reduced variance. This bias is known as omitted variable bias. Hence, omitting median income from the regression is equivalent to imposing the restriction B2=0 which is imposing an incorrect constraint on the parameter. Let b1 be the least squares estimator for B1 when X2(median income) is omitted from the regression.

Bias(b1) = E(b1) - B1 = B2Cov(X1,X2)/Var(X1)

To know the directionality of this bias, we need to know the sign of B2 and the covariance between X1 and X2. If the covariance between X1 and X2 is positive and the sign of B2 is negative, the direction of the bias will be negative but if the covariance is negative and the sign of B2 is negative, the direction of the bias will be positive. The opposite is true.