Question

In the context of multiple regression, define the n X n matrix M =- X(X'X)-'X'. (i) Show that M is symmetric and idempotent. (ii) Prove that m, the diagonals of the matrix M, satisfy 0 sm s 1 for t = 1, 2, ..., n. (iii) Consider the linear model y = XB + u satisfies the Gauss-Markov Assumptions. Let û be the vector of OLS residuals. Show that Eſûù' x) = oʻM (iv) Conclude that while the errors {u: t = 1, 2, ..., n} are homoskedastic and uncorrelated under the Gauss-Markov Assumptions, the OLS residuals are heteroskedastic and correlated.

Answer 1

solution: Let, Mz In- x(xx)"x"=001 - M= In-xcxx)'x' M = [In - x(xx)"x")' ; In'-In den for L (AB)' = B' A' 21 - [* (x'x)"x) ' - In - XC xx) x1 MOM I do M is a Symmetric matris .) A matrix '' is said to be idempotent ibt A AI in M? = (In - X(X%)" x') (In -* (x'x)\x'). !! Vs In - XCx'x) 'x' - XCx'xJx'+xCx'x5x'x (x'x)x! In - x(x'x)\x2 - x (x'x5%x! + x (x'x x? 1. M² ='M - M is idempotent matrix

: Mis symmetric and idempotent. M = In - x['x)? x , 100 - - .: In = 0 0- -- DOO-- Diagonal Elements of In=1, and We also know that, the diagonal Elements of x(x²x)'x' is oshirci, ki= 1(1)n. lie., hii = ith diagonal Element of. x(x'xJ'x' and oshii si, Hi=160)9). .) The diagonal elements of in VRM=In = x (x'xJ !\' is i-hi. H iz Icon. -) suppose the diagonal Elements of m are a mat → MH = 1- htt, Vt = 100) n. lij os btt si ! 05.1-betsl.., The diagonal Elements of the matrix M Satisfy OS MH sl ; t-11)..

Gauss-markov v Y = xß tu es satisfies the Akscembling, ECU)=0 Varlu) = ? In Bors = (xx) x'y 11 na Now, û ry- 1 = y = x(x'xJ"*'y in û = (In - x (x:x). 'x) y û =my -(i) nikm) :) cov CÂ), El û û lx) ICE CÚ)-0). → ECû û \x)cov (ll) = cov (my) Limia Em cov cy) m? zm. 0² Imm) (cov (4) = 0 In) to2MM ore M EM ECûailx)=²M

iv) and under Gause markave assumptions Y=xß tl, E(M) =0, var( ") = In واد(دعا = امام → ols residuals î=y- ý ro O amy Covců) = 2 in (a) .) The matrix M = In - X (XxJ"%! is generally not diagonal, 1. So, the residuals have different variances and they are correlated 001 in jo Gause markov model the errors (M) are homoskedastic and uncorrelated where as the ols gesiduals (ll) are heteroskedastic and correlated . =