slides_Ch2_W[1]

The unbiased estimator makes a degrees of freedom

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Unformatted text preview: unknown, they are not estimators. Melissa Tartari (Yale) Econometrics 84 / 93 The Variance of the OLS Estimator - Estimation I ˆ ˆ Notice that Var β1 jx and Var βo jx depend on σ2 and σ2 is ˆ ˆ unknown; thus, Var β1 jx and Var βo jx are unknown, they are not estimators. Can we estimate σ2 ? Melissa Tartari (Yale) Econometrics 84 / 93 The Variance of the OLS Estimator - Estimation I ˆ ˆ Notice that Var β1 jx and Var βo jx depend on σ2 and σ2 is ˆ ˆ unknown; thus, Var β1 jx and Var βo jx are unknown, they are not estimators. Can we estimate σ2 ? Hint: recall the LLN: 1n 2 u n i∑ i =1 Melissa Tartari (Yale) ! E ui2 = σ2 Econometrics (12) 84 / 93 The Variance of the OLS Estimator - Estimation I ˆ ˆ Notice that Var β1 jx and Var βo jx depend on σ2 and σ2 is ˆ ˆ unknown; thus, Var β1 jx and Var βo jx are unknown, they are not estimators. Can we estimate σ2 ? Hint: recall the LLN: 1n 2 u n i∑ i =1 ! E ui2 = σ2 (12) 1 Based on (12) it is natural to ask whether n ∑n=1 ui2 is an unbiased i 2 ? No, it is not ... in fact, it is not an estimator estimator for σ altogether because it depends on unobservables ui0 s . Still we may be on the right path ... Melissa Tartari (Yale) Econometrics 84 / 93 The Variance of the OLS Estimator - Estimation II The right answer entails using ui in place of ui in (12) to obtain a ˆ proper estimator: 1n 2 SSR ˆ ∑ ui = n n i =1 M elissa Tartari (Yale) Econometrics 85 / 93 The Variance of the OLS Estimator - Estimation II The right answer entails using ui in place of ui in (12) to obtain a ˆ proper estimator: 1n 2 SSR ˆ ∑ ui = n n i =1 There is one more hurdle to deal with: while the expression above provides a computable rule for any sample of data, it happens to de…ne an estimator that is not unbiased. Melissa Tartari (Yale) Econometrics 85 / 93 The Variance of the OLS Estimator - Estimation II The right answer entails using ui in place of ui in (12) to obtain a ˆ proper estimator: 1n 2 SSR ˆ ∑ ui = n n i =1 There is one more hurdle to deal with: while the expression above provides a computable rule for any sample of data, it happens to de…ne an estimat...
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This note was uploaded on 02/13/2014 for the course ECON 350 taught by Professor Donaldbrown during the Fall '10 term at Yale.

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