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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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 Fall '10
 DonaldBrown
 Econometrics

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