Please answer the following questions.

Problem 3: We delve into the theory for added variable plots and variance ination factors. Consider a model of the form r = Kai3 + Z? + e where X0 is n x p and full rank, 2 is n X 1 and linearly independent of the columns of X0, E[e) = D, and Corie) = 0'21. The partial regression plot for 2 shows its association with the response y after adjusting for the variables in X\". This plot is obtained by plotting residuals rElf from the LS regression of y on X0 versus residuals r, from the LS regression of s on X0. a) Show that ry = (I Huh.r and rI = [I Hu)z, where Ho = XXEXuTIXE. Answer: b) Under the model assumptions given above1 show that Buy) = rn, so the expected slope of the line is the same as the coefcient 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 tting this \"model\" by LS regression gives estimated slope: .. = thl - Holy = Elite sle- "\" thI - Hm 11e- - a)? where i = Hos and yr, 2,- and z\e) Let R2 denote the multiple R-square statistic for the regression of z on the variables in Xo. It can be shown that R2 = 1 - ELI(Z - Z)2 where z=n > z. Using this fact, show that 02 var(5) = VIF* ELI(z - Zi)2 where VIF, is the "variance inflation factor" given by 1 VIF = 1 - R2