Question 1 (45 points) Multiple Linear Regression. (You must explain or prove your answers. Answers without reasoning receive no credit.) Consider the multiple linear regression (MLR) model Y = Bo + BX1 + 82X2 +E, X1, X2) = 0. Let {(Y, X1, X2}=1 be a random sample drawn from the distribution of where Ble (Y, X1, X2) X1, X2) as a function of X1, X2. Interpret Bo and in 1. (5 points) Express the conditional mean E(Y equation (1). 2. (5 points) Let & denote the OLS estimator of 31. Derive the formula for 31. 3. (5 points) Is the OLS estimator 1 a random variable? Explain. Is B, a random variable? Explain. 4. (5 points) Which assumptions (among MLR Assumptions 1-5) are missed for model (1)? Under MLR Assumptions 1-5, the OLS estimator is "BLUE". What does "BLUE mcan? 5. (5 points) Prove that Ele) = 0, Cov(X1, c) = 0, and Cou(X1, c) = 0. 6. (5 points) What is the variance of the OLS estimator of B,? Intuitively explain why the numerator and the denominator enter the formula as they do. 7. (5 points) If o' (the variance of c) is unknown, how would you estimate it? What is the distribution of the statistic se(1) under the MLR Assumptions 1-6? 8. (5 points) If B2 = 0, would you use the simple regression model Y = Bo +8X1 + e or the multiple regression model (1)? Explain. 9. (5 points) If 82 +0, is it true that estimating the simple regression model Y = % + 91X1 + always yields a biased OLS estimator for 3, ? Explain. Question 1 (45 points) Multiple Linear Regression. (You must explain or prove your answers. Answers without reasoning receive no credit.) Consider the multiple linear regression (MLR) model Y = Bo + BX1 + 82X2 +E, X1, X2) = 0. Let {(Y, X1, X2}=1 be a random sample drawn from the distribution of where Ble (Y, X1, X2) X1, X2) as a function of X1, X2. Interpret Bo and in 1. (5 points) Express the conditional mean E(Y equation (1). 2. (5 points) Let & denote the OLS estimator of 31. Derive the formula for 31. 3. (5 points) Is the OLS estimator 1 a random variable? Explain. Is B, a random variable? Explain. 4. (5 points) Which assumptions (among MLR Assumptions 1-5) are missed for model (1)? Under MLR Assumptions 1-5, the OLS estimator is "BLUE". What does "BLUE mcan? 5. (5 points) Prove that Ele) = 0, Cov(X1, c) = 0, and Cou(X1, c) = 0. 6. (5 points) What is the variance of the OLS estimator of B,? Intuitively explain why the numerator and the denominator enter the formula as they do. 7. (5 points) If o' (the variance of c) is unknown, how would you estimate it? What is the distribution of the statistic se(1) under the MLR Assumptions 1-6? 8. (5 points) If B2 = 0, would you use the simple regression model Y = Bo +8X1 + e or the multiple regression model (1)? Explain. 9. (5 points) If 82 +0, is it true that estimating the simple regression model Y = % + 91X1 + always yields a biased OLS estimator for 3, ? Explain