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 years 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, Xi denote a binary

View complete question » 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 years 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, Xi denote a binary variable that equals 1 if the student is assigned to a small class, and Wi denote a binary variable that equals 1 if the student is newly enrolled. Let 1 denote the causal effect on test scores of reducing class size from regular to small. a. Consider the regression Yi = 0 + 1Xi + ui. Do you think that E1ui _Xi2 = 0? Is the OLS estimator of 1 unbiased and consistent? Explain. b. Consider the regression Yi = 0 + 1Xi + b2Wi + ui. Do you think that E1ui _Xi,Wi2 depends on Xi? Is the OLS estimator of 1 unbiased and consistent? Explain. Do you think that E1ui _Xi,Wi2 depends on Wi? Will the OLS estimator of 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.

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