Ols Equal Blue

Econometrics Assignment OLS proving

Written by Bernard Lesmana Vilia Clarissa Priska Anggrita Dwangkoro

Assumptions: 1. Let Y be a random scalar such that E(Y ) < ∞, and let X be a random k × 1 vector,k ∈ N+, such that for some β∗ ∈ Rk, E[Y | X] = X0β∗.

2. E(Y^2) < ∞, E(X0X) < ∞, and for some σ^2∗ > 0, E(U2| X) = σ2∗, where U := Y − X’β∗; 3. For n ∈ N+, {(Yt, Xt’)’: t = 1, 2, . . . , n} is independently and identically distributed (IID), (Yt, Xt’)’ ∼(Y, X’)’; and 4. Pnt=1 XtXt’ is invertible. Let Y(n×1) := [Yt] and X(n×k):= [Xt’]. The following facts are standard. The linear estimators of β∗have the form β˜n:= B(X)Y, where B : Rn×k → Rk×n. Given linearity, A.1, and A.3, E(β˜n| X) =β∗, i.e., β˜n is (conditionally) unbiased, if and only if B(X)X = Ik. In what follows, β˜nis linear unbiased.

Given A.3 and A.4, the OLS estimator βˆn:= (X’X)^−1X’Y exists and is unique; it is a linear estimator of β∗ with A(X) := (X’X)−1X’. Given A.1, A.3, and A.4, βˆn is linear unbiased.

Observe that for all B(X) such that B(X)X = Ik, A(X)B(X)’ = (X’X)−1X’B(X)’ = (X’X)−1Ik= A(X)A(X)’ = B(X)A(X)’.

Given A.1 - A.4, cov(β˜n|X) := E([β˜n − β∗][β˜n − β∗]’| X) =E(B(X)UU’B(X)’| X) = σ^2∗ B(X)B(X)’, as E(UU’| X) =σ^2∗In, where U := Y − Xβ∗. In particular, cov(βˆn|X) = σ^2∗ A(X)A(X)’ =σ^2∗(X’X)−1.

With this foundation, we have: Theorem. Given A.1 - A.4, βˆnis the best linear unbiased estimator. That is, for all X and B such thatB(X)X = Ik, cov(β˜n|X)−cov(βˆn|X) = σ^2∗[B(X)B(X)’ − A(X)A(X)’] is positive semi-definite (psd).

Proof: We show that B(X)B(X)’ − A(X)A(X)’is psd. As A(X)B(X)’ = A(X)A(X)0’,

B(X)B(X)0’ − A(X)A(X)’ = B(X)B(X)’ − A(X)B(X)’

(1)

As B(X)A(X)’ = A(X)A(X)’,B(X)B(X)’ − A(X)A(X)’ = B(X)B(X)’ − A(X)B(X)’ − B(X)A(X)’ + A(X)A(X)’ (2)

Collecting terms gives B(X)B(X)’ − A(X)A(X)’ = [B(X) − A(X)][B(X) − A(X)]’, a positive semi-definite matrix. (3)

References: DAVIDSON, R. AND MCKINNON, J. (2003): Econometric Theory and Methods. NY: Oxford...