Econometric Problem Solving

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Econometrics Problem-solving
(a) Prove that E(bR) = β1 + Pβ2.
First, it is essential to express bR in terms of ∈:
bR = (X’1 X1) -1 X’1y
= (X’1 X1) -1 X’1 (X1 bI + X2b2 + e)
= (X’1 X1) -1 X’1 X1 bI + (X’1 X1) -1 X’1 X2 β2 + (X’1 X1) -1 X’1 e
=I β1 + (X’1 X1) -1 X’1 X2 β2 + (X’1 X1) -1 X’1 ∈
= β1 + P β2 + (X’1 X1) -1 X’1 ∈
By definition
y =X1 β1 + X2 β1 + ∈
By matrix algebra
(X’1 X1) -1 X’1 X1 = I
(X’1 X1) -1 X’1 X2 = P
Therefore, E(bR) is:
E(bR) = E( β1 + P β2 + (X’1 X1) -1 X’1 ∈)
= β1 + P β2 + (X’1 X1) -1 X’1 E(∈) β, X fixed (A2, A6)
= β1 + P β2 E(∈) = 0 (A3)
(b) Prove that var(bR) = σ 2 (X 0 1X1) −1
First, we define the br – E(br) expression in terms of X1, with part (a) findings.
bR – E(br) = [β1 + Pβ2 + (Xi X1) 1Xie]  – [β1 + Pβ2 ]  by (a)
= (Xi’ X1) 1Xie
Then we establish var (bR):
var (bR): = E([bR – E(bR)] [bR – E(bR)]1 ) by Probability theory
= E( [(X’1 X1) -1 X’1 e] [(X1’ X1 ) -1 X1’ ∈]’)
= E( [(X’1 X1) -1 X’1 e] [∈’(X1’)’((Xi X1)-1)’] ) (X’1 X1)-1 Symmetric matrix
= E( [(X’1 X1) -1 X’1 e] [∈’XI)( X’1 X1)-1…