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# 1 slr4 fails if slr1 fails see chp 13 for methods that

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Unformatted text preview: xi x ) ˆ = in 1 β 1 = i =1 n 2 ¯ ¯2 ∑i =1 (xi x ) ∑i =1 (xi x ) Melissa Tartari (Yale) Econometrics (2.49) 51 / 93 Unbiasedness of the OLS Estimator III Recall ¯ ¯ ¯ ∑n= yi (xi x ) ∑n (yi y ) (xi x ) ˆ = in 1 β 1 = i =1 n 2 ¯ ¯2 ∑i =1 (xi x ) ∑i =1 (xi x ) (2.49) ˆ We can then write β1 in terms of ( βo , β1 ), the u 0 s , and the x 0 s : ∑n ( β + β xi + ui ) (xi ˆ β 1 = i =1 o n 1 ¯2 ∑i =1 (xi x ) x) ¯ = β1 + ¯ ∑n=1 ui (xi x ) i n ¯2 ∑ (xi x ) | i =1 {z } generally 6=0 Melissa Tartari (Yale) Econometrics 51 / 93 Unbiasedness of the OLS Estimator IV - Theorem Theorem 2.1: under assumptions SLR.1 to SLR.4 ˆ ˆ E β1 jx = β1 and E βo jx = βo . Melissa Tartari (Yale) Econometrics (2.53) 52 / 93 Unbiasedness of the OLS Estimator IV - Theorem Theorem 2.1: under assumptions SLR.1 to SLR.4 ˆ ˆ E β1 jx = β1 and E βo jx = βo . (2.53) ˆ This means that the distribution of β1 is centered about β1 and the ˆ is centered about β . distribution of βo o Melissa Tartari (Yale) Econometrics 52 / 93 Unbiasedness of the OLS Estimator V - Proof Proof of Theorem 2.1: ˆ E β1 jx = E [ β1 jx ] + E " ¯ ∑n=1 ui (xi x ) i jx n ¯2 ∑i =1 (xi x ) # ¯ ∑n=1 E [ui jx ] (xi x ) i 2 n ¯ ∑i =1 (xi x ) n ¯ ∑ = 0 (xi x ) = β1 + in 1 = β1 + 0 = β1 ¯2 ∑i =1 (xi x ) = β1 + and ˆ E βo jx = E y ¯ M elissa Tartari (Yale) ˆ¯ β1 x jx = y ¯ ˆ E β1 jx x = y ¯ ¯ Econometrics β1 x = βo . ¯ 53 / 93 Unbiasedness of the OLS Estimator: Failures I Unbiasedness generally fails if any of SLR.1-SLR.4 fails: if SLR.1 fails =) see Chp 13 for methods that handle non-linear models, Melissa Tartari (Yale) Econometrics 54 / 93 Unbiasedness of the OLS Estimator: Failures I Unbiasedness generally fails if any of SLR.1-SLR.4 fails: if SLR.1 fails =) see Chp 13 for methods that handle non-linear models, if SLR.2 fails (e.g. when samples are not representative of the population because the data is collected by over-sampling some sub-populations)...
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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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