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melissa tartari yale econometrics 89 93 the variance

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Unformatted text preview: or that is not unbiased. The unbiased estimator makes a degrees-of-freedom adjustment ˆ and we denote it by σ2 : SSR ˆ σ2 = . (2.61) n2 Melissa Tartari (Yale) Econometrics 85 / 93 The Variance of the OLS Estimator - Theorem 2.3 Theorem 2.3: under assumptions SLR.1-SLR.5 ˆ E σ 2 jx = σ 2 . Melissa Tartari (Yale) Econometrics 86 / 93 The Variance of the OLS Estimator - Theorem 2.3 Theorem 2.3: under assumptions SLR.1-SLR.5 ˆ E σ 2 jx = σ 2 . ˆ ˆ ˆ Thus, if we plug σ2 into Var β1 jx and Var βo jx we have unbiased \ \ ˆ ˆ estimators, which we denote Var β1 jx and Var βo jx . (Incidentally: How can we say that they are unbiased? Because they are linear functions of unbiased estimators and the expected value is a linear operator). Melissa Tartari (Yale) Econometrics 86 / 93 The Variance of the OLS Estimator - Theorem 2.3 - proof ˆ We are interested in E σ2 jx and we know that (n ˆ 2) σ 2 = n ˆ ∑ ui2 , ) E (n i =1 n ˆ ˆ 2) σ2 = E [ ∑ ui2 jx ] i =1 therefore we need to work out the RHS. Melissa Tartari (Yale) Econometrics 87 / 93 The Variance of the OLS Estimator - Theorem 2.3 - proof ˆ We are interested in E σ2 jx and we know that (n ˆ 2) σ 2 = n ˆ ∑ ui2 , ) E (n i =1 n ˆ ˆ 2) σ2 = E [ ∑ ui2 jx ] i =1 therefore we need to work out the RHS. Consider (2.59) and (a) its averaged-form, (b) di¤erence from its averaged form: ui ˆ = ui 0=u ui ˆ Melissa Tartari (Yale) = (ui ˆ βo ˆ β o u) βo βo ˆ β Econometrics ˆ β1 ˆ β 1 1 β1 xi β1 x β1 (xi x) (2.59) (a) (b) 87 / 93 The Variance of the OLS Estimator - Theorem 2.3 - proof Next we (c) square both sides and then (d) sum both sides: ˆ u )2 + β1 ˆ 2 (ui u ) β ui2 = (ui ˆ 1 n ˆ ∑ ui2 i =1 n = ∑ (ui β1 ˆ u )2 + β1 x )2 (xi β1 (xi β1 x) 2 (c) n ∑ (xi x )2 i =1 i =1 ˆ 2 β1 2 n β1 ∑ (ui u ) (xi x) (d) i =1 Melissa Tartari (Yale) Econometrics 88 / 93 The Variance of the OLS Estimator - Theorem 2.3 - proof We are now in the position of taking expectations of both sides and write (( ) on 2...
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