Introduction to Econometrics 3rd Edition by James Stock, James Stock

Exercise 1

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 Y t denote the exam score for the i th 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 ß 1 denote the causal effect on test scores of reducing class size from regular to small.
a. Consider the regression Y i = ß 0 + ß 1 X₁i + U i Do you think that E(u i |X₁i ) = 0 Is the OLS estimator of ß 1 unbiased and consistent Explain.
b. Consider the regression Y t = ß 0 + ß 1 X₁ i + ß 2 X₂ i + u i. Do you think that E{u i |X₁ i , X₂i ) depends on X₁ Is the OLS estimator of ß 1 unbiased and consistent Explain. Do you think that E(u i |X₁ i , X₂ i ,) depends on X₂ Will the OLS estimator of ß 2 provide an unbiased and consistent estimate of the causal effect of transferring to a new school (that is, being a newly-enrolled student) Explain.