Problem 4. Consider the model: Yi = B. +3,21,i + B222, i + B3X3,i + B424,i + 3525,i + Ui, i=1,2,...,150. Suppose that we have the following null hypotheses. Write out R and c in the form of Ho : Rp. = c. And write down the null distribution. = 0. 1. HP) : B0,1 =B0,2 = 30,5 2. H(2) : 30,0 + 3B0,3 630,4 = 4. 3. HP) : B0,2 +420,4 +1 = 230,1 4. HX") : 30,0 + 3B0,1 + 30,3 + 530,5 = 7. Remark 1 For the null hypothesis H. : 30,1 = Bo,2 consider the following statistics: Bop 0, another way to test the null is to RP / P FE (TSS ESS)/P (12) ESS/(n - p-1) (1 R4)/(n- p - 1) This F statistic in equation (12) is numerically identical to the statistic in equation (11). We skip the details. Therefore, for any null hypothesis Ho : RB. = c, we have the following test statistics: F (11) (RBOLS c)' (R BOLS R') (ROLS c) (Rols -c) (R (XX) R') (R$ous-e) k F(k,n - p-1). ko2 For example, the null hypothesis, Ho : B0,1 = 30,2 Bo,p = 0, can be written as H : RB, =c, where 1 0 0 0 0 0 0 1 0 0 222 R = [0px1, Ip] = 0 0 0 1 and C= 0px1 0 0 0 0 0 1 Problem 4. Consider the model: Yi = B. +3,21,i + B222, i + B3X3,i + B424,i + 3525,i + Ui, i=1,2,...,150. Suppose that we have the following null hypotheses. Write out R and c in the form of Ho : Rp. = c. And write down the null distribution. = 0. 1. HP) : B0,1 =B0,2 = 30,5 2. H(2) : 30,0 + 3B0,3 630,4 = 4. 3. HP) : B0,2 +420,4 +1 = 230,1 4. HX") : 30,0 + 3B0,1 + 30,3 + 530,5 = 7. Remark 1 For the null hypothesis H. : 30,1 = Bo,2 consider the following statistics: Bop 0, another way to test the null is to RP / P FE (TSS ESS)/P (12) ESS/(n - p-1) (1 R4)/(n- p - 1) This F statistic in equation (12) is numerically identical to the statistic in equation (11). We skip the details. Therefore, for any null hypothesis Ho : RB. = c, we have the following test statistics: F (11) (RBOLS c)' (R BOLS R') (ROLS c) (Rols -c) (R (XX) R') (R$ous-e) k F(k,n - p-1). ko2 For example, the null hypothesis, Ho : B0,1 = 30,2 Bo,p = 0, can be written as H : RB, =c, where 1 0 0 0 0 0 0 1 0 0 222 R = [0px1, Ip] = 0 0 0 1 and C= 0px1 0 0 0 0 0 1