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 Y_i denote the exam score for the ith student, X_1i denote a binary variable that equals 1 if the student is assigned to a small class, and X_2i 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 Y_i = β₀ + β₁X_1i + u_i. Do you think that E[u_i[X_1i]]= 0? Is the OLS estimator of β₁ unbiased and consistent? Explain.

b) Consider the regression Y_i = β₀ + β₁X_1i + β₂X_2i + u_i. Do yarn think that E[u_i|X_1i, X_2i]depends oil X₁? Is the OLS estimator of β₁ unbiased and consistent? Explain. Do think that E[u_i|X_1i, X_2i] 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^{^} = (X^TX)^-1X^TY.

Suppose we have just one regressor, i.e.:

y_i = β₀ + β₁x_i + u_i.

We can rewrite the model as:

Y = Xβ + u

Where

and our least squares estimator of β is:

β^{^} = (X^TX)^-1(X^TY)

(1/n X^TX)-1(1/n X^TY)

where 1/n X^TX is a 2 x 2 matrix and 1/n XTX is a 2 x 1 vector.

 a) Write out 1/n X^TX in terms of 1, x^-_n and 1/n _i=1∑ⁿ x_i².

b) Using the formula for the inverse of a 2 x 2 matrix, calculate

(1/n X^TX)^-1

c) Using your answer to b), show that β^ recovers the usual formula for simple linear regression, i.e.: