Homework 5 (4 Points), due at the beginning of class on October 12, 2016 • Q2 (2 point) please prove M2 M = M on page 17 of the lecture note (Hint, first prove P2 P = P2 ), and β̂1 = β1 + (X′1 M2 X1 )−1 (X′1 M2 U) E((β̂1 − β1 )(β̂1 − β1 )′ |X) = σ 2 (X′1 M2 X1 )−1 on page 20. (Hint: look at page 8 of the lecture note) • Q3 (1 point) Please use House data posted on my webpage, and consider the simple regression baths = β0 + β1 age + u. However, I ask you to apply the FW theorem to obtain β̂1 . You need to construct M2 ≡ I − X2 (X′2 X2 )−1 X′2 explicitly, and interpret M2 X1 . Hint: in this case, X1 is age, and X2 is the intercept term, a variable that is equal to one for all observations. The stata commands may be sca n = 321 gen c = 1 mkmat c, matrix(x2) matrix mx2 = I(n) - x2*invsym(x2’*x2)*x2’ • Q3 (1 point) Please use House data and use command reg to report the regression baths = β0 + β1 age + β2 age2 + β3 log(area) + u. Now consider the null hypothesis H0 : β1 = 0, β2 = 0. Interpret this null hypothesis. Construct R and c (in the codes file it is r) explicitly, and use matrix algebra to find the Wald test and F test. Do not use test command! (Hint: After running the regression, the β̂ and σ 2 (X′ X)−1 can be extracted by 1 matrix betahat = e(b)’ matrix v = e(V) 2