Provide detailed step-by-step answers

1. We observe a joint i.i.d. sample of two random variables X and Y: {(Xi, 11)}:11". (a) Consider the following model: 311'. = y + 5y,i where pay is a parameter and 8\"; is the error term. What is the OLS estimator, denoted y, for pay? (b) Consider the following model: X1) = \"a: + 83,76 where am is a parameter and 83,1 is the error term. What is the OLS estimator, denoted ET, for #9:? (c) Consider the following model: 3\" = )3 ' 5m + Us where Em and Em are the residuals from part (a) and part (b) regressions respectively, and u: is the error term. What is the OLS estimator, denoted ,8, for 5? Comment. 2. Consider two periods and a treatment intervention (for some subpopulation) between the two periods. Y is the (continuous) outcome variable 1 if the observation is after the treatment A ' ' d ' d' t : A = is a perio in ma or {0 if the observation is before the treatment , , _ 1 if the observation is from the group of treated individuals D is a treated group indicator: D = . _ _ _ , , 0 1f the observation is from the group of untreated ind1v1duals Consider the following model: Y=a+-A+7-D+6-A-D+e where a, ,8, 'y, 6 are parameters and e is the error term. Suppose that the mean of 5 conditional on A and D is zero: 1E [5|A, D] = 0 What is the 5 parameter equal to under the above assumption? (Hint: we are looking for an expression with conditional moments.)