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5 var u jx e u 2 jx 0 all x 2 2 e u var u it

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Unformatted text preview: Var [ui jx ] (xi = M elissa Tartari (Yale) x )2 ¯ i =1 SLR .5 = x ) jx ] ¯ 1 2n σ (xi 4 sx i∑ =1 x )2 ¯ 1 22 σ2 σ sx = 2 . 4 sx sx Econometrics 79 / 93 The Variance of the OLS Estimator - Remarks We do not need SLR.5 to prove unbiasedness of OLS estimators. Melissa Tartari (Yale) Econometrics 80 / 93 The Variance of the OLS Estimator - Remarks We do not need SLR.5 to prove unbiasedness of OLS estimators. We add SLR.5 b/c it simpli…es the variance calculations and, importantly, b/c it implies that OLS has certain e¢ ciency properties. Melissa Tartari (Yale) Econometrics 80 / 93 The Variance of the OLS Estimator - Remarks We do not need SLR.5 to prove unbiasedness of OLS estimators. We add SLR.5 b/c it simpli…es the variance calculations and, importantly, b/c it implies that OLS has certain e¢ ciency properties. We see that σ2 is the unconditional variance of u : Var [u jx ] = E u 2 jx (E [u jx ])2 & E [u jx ] = 0 hence σ2 Melissa Tartari (Yale) by SLR.5 = Var [u jx ] = E u 2 jx 0 all x 2 2 ) σ = E u = Var [u ] Econometrics 80 / 93 The Variance of the OLS Estimator - Remarks We do not need SLR.5 to prove unbiasedness of OLS estimators. We add SLR.5 b/c it simpli…es the variance calculations and, importantly, b/c it implies that OLS has certain e¢ ciency properties. We see that σ2 is the unconditional variance of u : Var [u jx ] = E u 2 jx (E [u jx ])2 & E [u jx ] = 0 hence σ2 by SLR.5 = Var [u jx ] = E u 2 jx 0 all x 2 2 ) σ = E u = Var [u ] it is useful to rewrite SLR.3 and SLR.5 as E [y jx ] = βo + β1 x Var [y jx ] = σ Melissa Tartari (Yale) Econometrics 2 (2.55) (2.56) 80 / 93 The Variance of the OLS Estimator - Remarks We do not need SLR.5 to prove unbiasedness of OLS estimators. We add SLR.5 b/c it simpli…es the variance calculations and, importantly, b/c it implies that OLS has certain e¢ ciency properties. We see that σ2 is the unconditional variance of u : Var [u jx ] = E u 2 jx (E [u jx ])2 & E [u jx ] = 0 hence σ2 by SLR.5 = Var [u jx ] = E u 2 jx 0 all x 2 2 ) σ = E u = Var [u ] it is useful to rewrite SLR.3 and SLR.5 as E [y jx ] = βo + β1 x Var [y jx ] = σ 2 (2.55) (2.56) See Figure 2.8 Melissa Tartari (Yale) Econometrics 80 /...
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