QUESTION 1: A school district undertakes an experiment to estimate the effect of class size on test scores in second-grade classics. The district assigns 50% of its previous year's first grades 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 casual effect on teat 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 yarn think that E[ui|X1i, X2i]depends oil X1? Is the OLS estimator of β1 unbiased and consistent? Explain. Do think that E[ui|X1i, X2i] depends on X2? Will the OL-S 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,
QUESTION 2: In class we showed that the formula for the OLS estimator in multiple regressions is:
β^ = (XTX)-1XTY.
Suppose we have just one regressor, i.e.:
yi = β0 + β1xi + ui.
We can rewrite the model as:
Y = Xβ + u
Where
and our least squares estimator of β is:
β^ = (XTX)-1(XTY)
(1/n XTX)-1(1/n XTY)
where 1/n XTX is a 2 x 2 matrix and 1/n XTX is a 2 x 1 vector.
a) Write out 1/n XTX in terms of 1, x-n and 1/n i=1∑n xi2.
b) Using the formula for the inverse of a 2 x 2 matrix, calculate
(1/n XTX)-1
c) Using your answer to b), show that β^ recovers the usual formula for simple linear regression, i.e.:
