Suppose that we are interested in the regression of Y on two predictors, X1 and X2,...

 

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Suppose that we are interested in the regression of Y on two predictors, X1 and X2, that is, Y; = Bo + B1 Xii + B2X2 + , i = 1, ..., n. %3D Let Y = EY:/n, SyY SyyΣΥ- Υ', i=1 i=1 X1 = EXu/n, Sx,x, = E(Xn - X1)°, i=1 i=1 X2 = E X2/n, Sx2X2 = E(X2 - X2)*. – X2)³. i=1 i=1 After centering and scaling the variables, we have Y; - Ỹ X;1 - X1 X2 - X2 | Y* %3D s/2 PYY s!/2 Sx,X1 SX2X2 SyY The model for the transformed variables is Y = a1 X+ a2X + Ni, i = 1,..., n, Suppose we observe n = 10 observations that yield the following statistics 10 10 10 r12 = Eh = 0.99, r1y = Ev = 0.85, r2y = Ey = 0.78, * %3D %3D %3D i=1 i=1 Syy = 2.75, Sx,X1 1.50, Sx,X2 = 2.50. Derive the least squares estimates for ai and a2. b) Find the estimates for B1 and B2 based on the results in part (1). Suppose that we are interested in the regression of Y on two predictors, X1 and X2, that is, Y; = Bo + B1 Xii + B2X2 + , i = 1, ..., n. %3D Let Y = EY:/n, SyY SyyΣΥ- Υ', i=1 i=1 X1 = EXu/n, Sx,x, = E(Xn - X1)°, i=1 i=1 X2 = E X2/n, Sx2X2 = E(X2 - X2)*. – X2)³. i=1 i=1 After centering and scaling the variables, we have Y; - Ỹ X;1 - X1 X2 - X2 | Y* %3D s/2 PYY s!/2 Sx,X1 SX2X2 SyY The model for the transformed variables is Y = a1 X+ a2X + Ni, i = 1,..., n, Suppose we observe n = 10 observations that yield the following statistics 10 10 10 r12 = Eh = 0.99, r1y = Ev = 0.85, r2y = Ey = 0.78, * %3D %3D %3D i=1 i=1 Syy = 2.75, Sx,X1 1.50, Sx,X2 = 2.50. Derive the least squares estimates for ai and a2. b) Find the estimates for B1 and B2 based on the results in part (1).