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\f* 2 mattie N(B) - 4- HER) Recall that we can add matrices as follows Let S, (R) be the set of 2 x 2 symmetric matrices with real entries. That is S,( R) - A - HIV-WER&y= a KAR CSR) Prove your answerWe are interested in estimating coefficient 3 = (0, b) and run linear regression (OLS) of y, on r, and a constant for each sub-sample. Thus we obtain estimate pu based on n, observations in the first sub-sample, and estimate 3(2) based on ny observations the second sub-sample. Now let's study the finite sample property of the following estimator of 3, 1. Show that B is an unbiased estimator for 2 Show that S is a linear estimator in y, where y . My. .. yo Um - 1, "Inithas is a column Wefor consisting of all a observations on 7, s. (Hint: the function f(r) = 0 5 is also a linear in r) I Let be the conventional OLS estimator d, obtained via regression of y's on a, and a constant using the full sample of a observations, Suppose now we are interested in comparing the variance of d veraits that of 3. Which estimator has a "smaller" variance" Why4. Consider the general linear regression model Y = XO + e where Y is n x 1, X is n x p and of rank p, B is p x 1, e is n x 1, and & is N(0, o'I). (a) The hat matrix H is given by H = X(X'X) 'X'. Show that (I - H) is idempotent where I is the n x n identity matrix. (b) Using the least squares method, we minimize RSS = e'e = (Y - Xb)'(Y - Xb) to obtain = b = (X'X)-'X'Y. Show that RSS can also be written as Y' (I - H)Y. (c) Obtain an expression for the variance-covariance matrix of the fitted values Y,, i = 1, 2, ..., n, in terms of the hat matrix H. (d) e = Y - Y is the vector of residuals. Are the residuals statistically independent? Justify your answer with an explanation.10. A collection of open sets A = {U) is said to be an open cover of a set X C R if xc ju LEA A set X C R is said to be compact if every open cover of X has a finite subcover. That is, V open covers A of X, 3 a finite subset B C A such that B covers X (a) State what it means for a set Y C R not to be compact. (4) (b) Let Y = (0, 1] be a half-open interval of real numbers. i. Let A = (A, : nc N} where An (, 2). Prove that A is an open cover of Y. (5)\f() Prove that the exkimala is correct. That is, compute T ( ;])- and explain why it be f wwpicture (onto)? Prove your