Problem 3 A matrix A ( R"XP is said to have orthonormal columns if ATA = Ipxp. This implies that the columns of A are orthogonal to one another, and further that It_ja;; = 1 for j = 1, ..., p. Let X be an n x p matrix containing the p predictor variables. Suppose that the columns of X are orthonormal and have mean zero (so they are orthogonal to the intercept column), and suppose the weaker linear model holds. y = PotXB+, with E(E) = 0, Var(e) = 0 Inxn, and B = (81, ..., Bp). a. (3 pts) Under the setup of this problem, find expressions for the ordinary least squares intercept Bo and slope coefficients = (81, ..., Bp) . Be sure to simplify given the structure of the matrix X in this problem. You may use results on the general closed-form expressions for the OLS coefficients from lecture without proof. b. (3 pts) Consider running ridge regression for a fixed value of A. As a reminder, ridge considers the following minimization problem: 2 arg min yi - (Bo + ) Bjxu) + 1 Bo,...;8p j=1 j=1 Under the setup of this problem, find expressions for the intercept S , and slope coefficients Bf = (BR, .... P) . Be sure to simplify given the structure of the matrix X in this problem. You may use results on the general closed-form expressions for the ridge coefficients from lecture without proof.c. (3 pts) For j = 1, .., p, calculate the ratio BR This ratio shows by what proportion the ridge coefficients have been shrunken towards zero relative to OLS. d. (3 pts) For j = 1, ..., p, calculate the ratio Var(BR,) Var(B;) This ratio shows the reduction in variance attained by using ridge regression relative to OLS. You may use results from lecture without proof