Econometric take home APPS_Part_18

Econometric take home APPS_Part_18 - 1 1 11q22 plim = 12...

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71 plim 1 1 22 11 2 12 12 3 12 13 11 1 2 2 1 2 1 2 1 12 12 2 11 22 3 11 23 12 12 22 11 2 ˆ 1 ˆ qqq qq ⎛⎞ ⎡⎤ β σ β σ β σ βσ σ = ⎜⎟ ⎢⎥ −β σ + β σ + β σ σσ Δ β ⎝⎠ ⎣⎦ . Taking the first coefficient separately and collecting terms, plim β 1 = β 1 [ σ 11 σ 22 q 11 q 22 -( σ 12 q 12 ) 2 ]/ Δ + β 2 [ σ 11 q 22 σ 12 q 12 + σ 12 q 12 σ 11 q 22 ]/ Δ + β 3 [ σ 11 q 22 σ 12 q 13 + σ 12 q 12 σ 11 q 23 ]/ Δ The first term in brackets equals Δ while the second equals 0. That leaves plim β 1 = β 1 - β 3 [ σ 11 σ 12 ( q 22 q 13 - q 12 q 23 )]/ Δ which is not equal to β 1 . There are two special cases worthy of note, though. The right hand side does equal β 1 if either (1) σ 12 = 0; the regressions are actually unrelated, or (2) q 12 = q 13 = 0; the regressors in the two equations are uncorrelated. The second of these is similar to our finding for omitted variables in the classical regression model. ± 6. The model is y y 0 ix0 0i 1 2 1 2 1 2 = + α β α ε ε . The GLS estimator of the full coefficient vector, θ , is () θ = ⎟ + + σ σ 11 22 1 11 1 1 12 2 2 12 1 22 2 nn x nx n x n ny x'x x'y σ σ 12 12 . Let q xx equal x x / n , q x 1 = x y 1 / n and, q x 2 = x y 2 / n . The n s in the inverse and in the vector cancel. Also, as suggested, we assume that x = 0. As in the previous exercise, we replace elements of the inverse with elements from the original matrix and cancel the determinant which multiplies the matrix (after inversion) and divides the vector. Thus, θ = −+ σ 11 12 22 12 11 1 22 1 12 2 11 1 12 2 12 1 11 2 0 00 0 q yy xx x x . The inverse of the matrix is straightforward. Proceeding directly, we obtain θ = 1 0 0 22 11 22 12 2 11 22 12 22 11 22 12 2 12 22 22 1 22 1 12 2 11 1 12 2 12 1 11 2 σ σ σ σ q xx xx xx xx xx
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Econometric take home APPS_Part_18 - 1 1 11q22 plim = 12...

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