Please show your workings clearly

3. Consider simple linear regression under normal error model: Yi = Bot BIX, te, . - 4 N (0, 02) with known o?. (a) Find the Fisher information matrix I (80, 81 ) for (Bo, B1). (b) Use the Fisher information in (a) to obtain the covariance matrix of MLE (B.3)).1. Consider the model y = 1 + zy + /ink + Dot, + e. Explain how you would go about testing the restriction # 30 - 204 - 0 by re writing the model. Explain your method fully. (4 points)consider the linear model Y = XB + e where E ~ N(0:030) and B E RP. Suppose the p columns in the plan X matrix are linearly independent. Either H = X(X'X) 1X' the hat matrix and b = (XX) *X'Y the least squares estimator of B. You will be able to use the fact that H and I-H are symmetrical and idempotent matrixes. 1 - Use the properties of the trace to demonstrate that tr(H) and tr(I-H) = n - p. 2 - Demonstrate that (1-H)X = 0 3 - consider the following quadratic form of Y = (Y1,Y2,Y3)': Q =Y2+2Y2 +3Y2 - 2Y, Y2 - 5 Y1 Y3 + 4Y2 Y3- - Give the matrix A of the quadratic form Q - Calcule the trace of the matrix A of the quadratic from Q.\fConsider the multiple regression model containing three independent variables, under Assumptions I through & as seen in class: You are interested in estimating the sum of the parameters on X and X, call this 9 - By + 82 2 10 points Show that does l dines today's is an unbiased estimator of 9 1 10 points Derive the variance of OOLs.In the following simple regression model: Y = BO + BIX +e How do we produce a more accurate OLS estimate of B1