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ng 3 slr2 e u jx 0 2 slr1 slr3 random

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Unformatted text preview: f β, ˆ namely on E β . Melissa Tartari (Yale) Econometrics 49 / 93 Unbiasedness of the OLS Estimator I - the Roadmap ˆ We start by focusing on the …rst moment of the distribution of β, ˆ namely on E β . ˆ A natural question to ask is whether we have E β = β; more precisely, under what assumptions is the OLS estimator unbiased. Melissa Tartari (Yale) Econometrics 49 / 93 Unbiasedness of the OLS Estimator I - the Roadmap ˆ We start by focusing on the …rst moment of the distribution of β, ˆ namely on E β . ˆ A natural question to ask is whether we have E β = β; more precisely, under what assumptions is the OLS estimator unbiased. We next state four assumptions and then verify that they are su¢ cient for unbiasedness of OLS. Melissa Tartari (Yale) Econometrics 49 / 93 Unbiasedness of the OLS Estimator II - Assumptions 1 bivariate linear model y = βo + β1 x + u Melissa Tartari (Yale) Econometrics (SLR.1) 50 / 93 Unbiasedness of the OLS Estimator II - Assumptions 1 bivariate linear model y = βo + β1 x + u 2 (SLR.1) f(yi , xi ) ji = 1, ..., ng (SLR.2) random sampling Melissa Tartari (Yale) Econometrics 50 / 93 Unbiasedness of the OLS Estimator II - Assumptions 1 bivariate linear model y = βo + β1 x + u f(yi , xi ) ji = 1, ..., ng 2 3 (SLR.1) (SLR.2) E [u jx ] = 0 (SLR.3) random sampling zero conditional mean Notice that 2. and 3. imply that nothing is lost (in derivations) in treating xi as non-random or …xed in repeated samples. Melissa Tartari (Yale) Econometrics 50 / 93 Unbiasedness of the OLS Estimator II - Assumptions 1 bivariate linear model y = βo + β1 x + u f(yi , xi ) ji = 1, ..., ng 3 (SLR.2) E [u jx ] = 0 2 (SLR.1) (SLR.3) random sampling zero conditional mean Notice that 2. and 3. imply that nothing is lost (in derivations) in treating xi as non-random or …xed in repeated samples. 4 sample variation in the eVar xi for i = 1, ..., n are not all equal Melissa Tartari (Yale) Econometrics (SLR.4) 50 / 93 Unbiasedness of the OLS Estimator III Recall ¯ ¯ ¯ ∑n= yi (xi x ) ∑n (yi y ) (...
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