2. Consider a linear regression model without intercept, Yi = BIXil + B2Xiz + ci, (1) where e; ~ N(0, o'). Now suppose that we have the inputs X11 = 5, X12 = 1, X21 = 4, X22 = 3, X31 = -1, X32 = 2. (a) (3) Write the model in the form Y = XB + e and define Y, X, B, and E. (b) (4) Let B = (X X)-1XY be the least square estimator of B. Show that B is unbiased and derive the distribution of B. (c) (3) Suppose we have the following R output and o? = 1, and X is the model matrix in part (a). Compute the variance of the first fitted value (please show your steps of calculation). X/*%solve(t (X)%*%X)%*%t (X) ## [,1] [,2] [,3] ## [1,] 0. 6666667 0.3333333 -0.3333333 ## [2,] 0.3333333 0.6666667 0.3333333 ## [3,] -0.3333333 0.3333333 0.6666667