2 Suppose that you estimate the following relationship using OLS: Y = Bo + BX +...

 

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2 Suppose that you estimate the following relationship using OLS: Y₁ = Bo + B₁X₁ + B₂ Zi + ei However, the true data generating process for Y is: Y₁ = Bo + B₁X₁ + B₂Z₁ + B3W₁ + U₁ where cov(X, u) = 0, cov(W, u) = 0 and cov(Z,u) = 0. You are told that the OLS estimator for B₁ when estimating equation (1) is given by: B₁ = plim (₁) Σ1(Yi - Y) (Xi − X) Σ}=₁(Z; - Z)² – Σ=1(Yk – Y) (Zk — Z) Σï-1(Z₁ – Z) (x₁ - x) Σm=1(Xm - X)² Σñ=1(Zp - Z)² – (Zª-1(Za − Z) (Xq − x)) ² By taking probability limits, it is trivial to show that: cov (Y, X)var (Z) - cov(Y,Z)cov (Z, X) var (X)var (Z) - (cov(X, Z))² You are not required to show the probability limit above. a) Using the probability limit specified in (3), and the true data generating process for Y (2), show that = (1) plim (B₁) =B₁ + B3 (2) cov(W, X)var (Z) - cov(W, Z)cov (Z, X) var (X)var (Z) - (cov(X, Z))² (3) Note: There is no need to take plims in this question. b) Hence, under what circumstances will the OLS estimator yield consistent estimates of ₁ when estimating (1)? 2 Suppose that you estimate the following relationship using OLS: Y₁ = Bo + B₁X₁ + B₂ Zi + ei However, the true data generating process for Y is: Y₁ = Bo + B₁X₁ + B₂Z₁ + B3W₁ + U₁ where cov(X, u) = 0, cov(W, u) = 0 and cov(Z,u) = 0. You are told that the OLS estimator for B₁ when estimating equation (1) is given by: B₁ = plim (₁) Σ1(Yi - Y) (Xi − X) Σ}=₁(Z; - Z)² – Σ=1(Yk – Y) (Zk — Z) Σï-1(Z₁ – Z) (x₁ - x) Σm=1(Xm - X)² Σñ=1(Zp - Z)² – (Zª-1(Za − Z) (Xq − x)) ² By taking probability limits, it is trivial to show that: cov (Y, X)var (Z) - cov(Y,Z)cov (Z, X) var (X)var (Z) - (cov(X, Z))² You are not required to show the probability limit above. a) Using the probability limit specified in (3), and the true data generating process for Y (2), show that = (1) plim (B₁) =B₁ + B3 (2) cov(W, X)var (Z) - cov(W, Z)cov (Z, X) var (X)var (Z) - (cov(X, Z))² (3) Note: There is no need to take plims in this question. b) Hence, under what circumstances will the OLS estimator yield consistent estimates of ₁ when estimating (1)?