1 (Total points: 68) 1. This assignment is due by 8:30 am, October 7 (Wednesday). Please put it in my TA's assignment locker on 9/F of KKL Building. 2. You can do this assignment individually or in a group of 2 students. You can work with anyone in the two subclasses (A and B). Your teammates do not need to be the same for all assignments. You are can free to team up with different teammates in different assignments if you like. 3. Please indicate the name, UID, and tutorial session of you and your teammate where applicable on the front page of the assignment. 4. Please type your answers. 5. You are expected to show all your detail derivation or computation steps, and with explanations where needed in all questions. Do not assume the grader understands you have jump some steps. You are not allowed to use STATA to answer the questions in this assignment. 6. You can use the three summation rules if needed without proving them again: a. ( ) = 0 b. ( )( ) = ( ) or ( ) c. ( )2 = ( ) 7. Late or handwritten assignments will not be accepted. ****************************************************************************** Q1. [26 points] Suppose we do not have an intercept parameter in our simple population regression model y = 1 x + u where E(u) = 0 0 (a) (14 points) Please use the method of moment to derive the OLS estimator . (Note: Do not use the method of least squares method) (b) (12 points) Find the expected value of the OLS estimator under the following four assumptions as usual (except E(u)0 in this case): SLR.1 The model is linear in parameter SLR.2 Random sampling SLR.3 Sample variations in x SLR.4 E(ulx) = E(u) Note: we follow the book's approach to take conditional expectation. Q2. [8 points] Please prove that sum of squared total (SST) is indeed equal to sum of squared explained (SSE) and sum of squared residuals (SSR) in OLS estimation. Does this relation still hold if we fit a sample regression line that goes through the sample mean of x and y such that the sum of all residuals is equal to zero? Q3. [24 points] Consider the following population multiple regression model with two explanatory variables y = 0 + 1 x1 + 2 x2 + u (a) (6 points) Please use the method of least squares to derive the three first-order conditions for the global minimum point. (b) A random sample of n = 8 is taken from the population Observation 1 2 3 4 5 6 7 8 y 12.4 11.7 12.4 10.8 9.4 9.5 8.0 7.5 x1 28.0 28.0 32.5 39.0 45.9 57.8 58.1 62.5 x2 18 14 24 22 8 16 1 0 (b1) (4 points) Please solve for the OLS estimates from this sample using the equations you derived in part (a). (b2) (2 points) Compute the residuals and compute their mean. (b3) (2 points) Compute the sample covariance between the residuals and the explanatory variables. (b4) (2 points) Compute the standard error of the regression. (b5) (2 points) Compute the estimated variance of the sampling distribution of the estimator (c) (6 points) Suppose the sample turns out to be Observation 1 2 3 4 5 6 7 8 y 12.4 11.7 12.4 10.8 9.4 9.5 8.0 7.5 x1 28.0 28.0 32.5 39.0 45.9 57.8 58.1 62.5 x2 14.0 14.0 16.25 19.50 22.95 28.90 29.05 31.25 Please solve for the OLS estimates from this sample using the equations you derived in part (a). Discuss the difficulty you may encounter in this case. Q4. [4 points] In Lecture 1 spreadsheet, let X= IQ level and Y = Education level, and we are looking at the population. (a) (2 points) Compute E(X), E(Y), E(XY), COV (X,Y) in sheet 1 (b) 2 points) Compute E(X), E(Y), E(XY), COV (X,Y) in sheet 2 Assume High IQ = 100 Low IQ = 10 University level of education = 20 High school level of education = 10 Primary school level of education = 5 Q5. [6 points] Starting with the basic concept of covariance: COV (X,Y) = E [(X - X) (Y - Y)], where X and Y are random variables and X and Y are their expected values, please prove (a) (4 points) COV (X, V + W) = COV (X, V) + COV (X, W) (b) (2 points) COV (X, aY) = a COV (X, Y) where X, Y, V, and W are random variables and \"a\" is a constant