Problem 1 (50pt) Suppose that the daily log return of a pair of securities follows the following model: T1,; 0.05 + 0.1T1';_1 + 0.05r215_1 + (11'; T2,; = 0]. 0.1T1';_1 + 0.3T2';_1 + 0.2.; where (i; denotes a bivariate normal distribution with mean 0 and covariance 1 ._ 0.4 0.1 L '_ ( (}.1 0.2 ) Any matrix operations can be computed in R. or with the programming language of your choice if desired. If any formulas require innite series, you may approximate using the rst I" 0 terms. (i) Verify that the return series {an} is a weakly stationary process. Hint: The \"polyroot\" function can be used to nd all roots of a polynomial in R. and the \"eigen\" function can be used to nd all eigenwalues of a matrix in 11. (ii) (a) What is the mean vector of the return series n? (b) What is the commence matrix of the return series rs? (c) What are the lag-1, lag-21 and lag-5 crosseorrelation matrices of the return series T1? (iii) Assume that r\" = (0.02,0.08)T and on = {0.08,0.1)T. Compute the 1-, 2-, and 3-step ahead forecasts of the return series at the forecast origin 3 = 1. What are the comriance matrices of the associated forecast errors