2. Consider the general linear model Y = XA + c, with : ~ N,,(0,o'I,, ), where the first column of X consists of all ones. (a) Using facts about the mean and variance/covariance of random vectors given in lecture, show that the least squares estimate from multiple linear regression satisfies E ( = B and Var (8) = 03(XTX)-1. (b) Let H = X(X X) 'X be the hat matrix, Y = HY be the fitted values and e = (I - H)Y be residuals. Using these matrices and properties derived in class, show that E[( x - P) ( 1 - P)] = 0. im This fact is used to provide the ANOVA decomposition SSTO = SSE + SSR for multiple linear regression.1. Some (More) Math Review a) Let N = 3. Expand out all the terms in this expression: Cov Xi b) Now write out all the terms using the formula from last class for variance of a sum: Var( X:) = _Var(X) + > > Cov(X, X;) i-1 1=1 i-lj=1ifi Verify that (a) and (b) are giving you the same thing. Hint: Cov(X, X) = Var(X). c) Suppose that D is a roulette wheel that takes on the values {1, 2, 3} all with probability 1/3. What is the expected value of this random variable? What about the variance? d) Now suppose that M is another roulette wheel that takes on the values {1, 2,3} all with probability 1/3. Solve for the expected value of 1/M. e) Finally, suppose that D and M are independent. Solve for: E Hint: You do not need to do any new calculations here. Just remember that for independent RVs, E(XY) = E(X)E(Y). f) Does E(D/M) = E(D)/ E(M)?1. Consider the simple regression model: Vi = Bo+ Biri + Hi, for i = 1, ... . n, with E(uilz,) - 0 and let a be a dummy instrumental variable for I, such that we can write: Ci =not matu with E(uilz;) = 0 and E(viz) =0. (c) Denote by no, the number of observations for which = = 0 and by n, the number of observations for which a, = 1. Show that: (a - 2) = =(n-m). 1=1 and that: [( - =)(y: - 9) = -(n - n) (31 - 30) . where to and g are the sample means of y for z equal to 0 and 1 respectively. ( Hint: Use the fact that n = nj + no, and that = = m). (d) Now we regress y on i to obtain an estimator of &. From the standard formula of the slope estimator for an OLS regression and using the result in (c), show that: B - 91 -90 $1 - To This estimator is called the Wald estimator.\f(1) [2 marks] Test Ho: B4 = 0 against B, # 0 at the one percent significance level. Feel free to use the following information: For a random variable u from the standard normal distribution / (0, 1), the following is true: Pr (u