A school district undertakes an experiment to estimate the effect of class size on test scores in second grade classes. The district assigns 50% of its previous year's first graders to small second-grade classes (18 students per classroom) and 50% to regular-size classes (21 students per classroom). Students new to the district are handled differently: 20% are randomly assigned to small classes and 80% to regular-size classes. At the end of the second-grade school year, each student is given a standardized exam. Let Yi denote the exam score for the ith student, X1i denote a binary variable that equals 1 if the student is assigned to a small class, and X2i denote a binary variable that equals 1 if the student is newly enrolled. Let β1 denote the causal effect on test scores of reducing class size from regular to small.
(a) Consider the regression Yi = β0 + β1X1i + ui. Do you think that E(ui|X1i) = 0? Is the OLS estimator of β1 unbiased and consistent? Explain.
(b) Consider the regression Yi = β0 + β1X1i + β2X2i + ui. Do you think that E(ui|Xli, X2i) depends on X1? Is the OLS estimator of β1 unbiased and consistent? Explain. Do you think that E(ui|X1i, X2i) depends on X2? Will the OLS estimator of β2 provide an unbiased and consistent estimate of the causal effect of transferring to a new school (that is, being a newly-enrolled student)? Explain.