3. (15 points) Suppose a data set consists of N observations. The outcome variable of interests is y, and the explanatory variables are denoted by X = [X17 X2]. There are K1 variables in X1 and K 2 variables in X2. (Put it differently, X1 is a N X K1 matrix, while X2 is a N X K2 matrix.) Do the following four procedures produce the same value for the least squares coefcients on X2? (a) Regress g on both X1 and X2. (b) Regress the residuals from a regression of g on X1 on the residuals (column by column) of regressions of X2 on X1. (c) Same as (b), but do not transform y. (d) Same as (b), but do not transform X2. You may justify your answers based on theoretical proofs or Monte Carlo simulations. For simplicity, you may assume that K1 2 K2 2 1. If you opt for the approach of Monte Carlo simulations, attach the Stata codes and outputs