We want to estimate the treatment effect of D on y via a linear regression model y equals b subscript 0 plus b subscript 1 x plus r D plus u, where x is an exogenous control variable. If the value of the treatment variable D is randomly assigned, then: a. the OLS estimate of r is unbiased. b. the OLS estimate of r is biased due to sample selection. c. the OLS estimator is BLUE. d. the error u is homoskedastic.

(a) Consider the classical linear regression model: y = Xβ + u under the Gauss Markov assumptions, such that E uu = σ 2 I . Consider the estimator for σ 2 given by s2∗ = e e/(T − 10) where e is the OLS residual vector: is this an unbiased estimator? If not, how can we correct it to get an unbiased estimator of σ 2 .

A group of scientists wants to determine the efficacy of a vaccine. To do so they recruited 1000 subjects. Two months ago 300 random subjects received the vaccine T=1 . The other 700 received a placebo T=0 . The scientist also measured if the subjects became infected with the virus Y=1 or not Y=0 in the last 2 months. They plan to test the efficacy of the vaccine by running the following regression: Y=β0+β1T+u Which of the following statements is FALSE: Question 6Select one: a. This model will generate biased estimates because it does not control for many omitted variables b. The fact that more people receive the placebo than the vaccine will not generate bias c. Scientists do not need to observe the initial health status of the subjects to get unbiased estimates because the assignment of the vaccine was random d. A negative β1 would indicate that the vaccine is effective

A group of scientists wants to determine the efficacy of a vaccine. To do so they recruited 1000 subjects. Two months ago 300 random subjects received the vaccine T=1 . The other 700 received a placebo T=0 . The scientist also measured if the subjects became infected with the virus Y=1 or not Y=0 in the last 2 months. They plan to test the efficacy of the vaccine by running the following regression: Y=β0+β1T+u Which of the following statements is FALSE: Question 6Select one: a. This model will generate biased estimates because it does not control for many omitted variables b. The fact that more people receive the placebo than the vaccine will not generate bias c. Scientists do not need to observe the initial health status of the subjects to get unbiased estimates because the assignment of the vaccine was random d. A negative β1 would indicate that the vaccine is effective Question 7

In the model Wage = B0 + B1 Education + U, if U is correlated with Educatiom and 8, > 0 the oLS estimators areSelect one: a.biased and consistentb,unbiased and inconsistentc,positively biased and consistentd,biased and inconsistent e.negatively biased and inconsistentf,unbiased and consistent

pset 6.1 Prove that the OLS estimator of $\boldsymbol{\beta}$ in the linear regression model with individualspecific intercepts, that is$$y_{i t}=\boldsymbol{x}_{i t}^{\prime} \boldsymbol{\beta}+\alpha_{i}+u_{i t},$$where $\alpha_{i}$ is a parameter to be estimated, for $i=1, \ldots, n$ is equal to $\hat{\boldsymbol{\beta}}_{W G}$.