Please do it asap.

Problem 3: We delve into the theory for added variable plots and variance inflation factors. Consider a model of the form y = XoB + zy+ e where Xo is n x p and full rank, z is n x 1 and linearly independent of the columns of Xo, E(e) = 0, and Cov(e) = 021. The partial regression plot for z shows its association with the response y after adjusting for the variables in Xo. This plot is obtained by plotting residuals ry from the LS regression of y on Xo versus residuals r, from the LS regression of z on Xo. a) Show that ry = (I - Holy and Iz = (1 - Ho)z, where Ho = Xo(XJXo)-'XJ. Answer: b) Under the model assumptions given above, show that E(ry) = ry, so the expected slope of the line is the same as the coefficient of z in the full model. Answer: c) The conditional expectation model in b) has the form of simple linear regression through the origin (no intercept). Show that fitting this "model" by LS regression gives estimated slope: 7 = z"(I - Holy _El(z - 2.)yi zT(I - Ho)Z Ell(z - 2;)2 where z = Hoz and yi, z; and ; are the ith components of y, z and z respectively. Answer: d) Using c) and the model assumptions, show E(5) = y and 02 var(6) = Answer:e) Let R. denote the multiple R-square statistic for the regression of z on the variables in Xo. It can be shown that R2 = 1 - ELI(zi - 2;)2 ELI(z - Z;)2 where z=n )_ z. Using this fact, show that 02 var(5) = VIF* ELI(zi - Zi)2 where VIF, is the "variance inflation factor" given by 1 VIF= = 1 - R2