25 algebraic properties of ols the sum of the ols

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25 Algebraic Properties of OLS The sum of the OLS residuals is zero Thus, the sample average of the OLS residuals is zero as well The sample covariance between the regressors and the OLS residuals is zero The OLS regression line always goes through the mean of the sample Fall 2008 under Econometrics Prof. Keunkwan Ryu
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26 Algebraic Properties (precise) x y u x n u u n i i i n i i n i i 1 0 1 1 1 ˆ ˆ 0 ˆ 0 ˆ thus, and 0 ˆ β β + = = = = = = = Fall 2008 under Econometrics Prof. Keunkwan Ryu
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27 More terminology ( 29 ( 29 SSR SSE SST Then (SSR) squares of sum residual the is ˆ (SSE) squares of sum explained the is ˆ (SST) squares of sum total the is : following the define then We ˆ ˆ part, d unexplaine an and part, explained an of up made being as n observatio each of can think We 2 2 2 + = - - + = i i i i i i u y y y y u y y Fall 2008 under Econometrics Prof. Keunkwan Ryu
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28 Proof that SST = SSE + SSR ( 29 ( 29 ( 29 [ ] ( 29 [ ] ( 29 ( 29 ( 29 ( 29 = - + - + = - + - + = - + = - + - = - 0 ˆ ˆ that know we and SSE ˆ ˆ 2 SSR ˆ ˆ ˆ 2 ˆ ˆ ˆ ˆ ˆ 2 2 2 2 2 y y u y y u y y y y u u y y u y y y y y y i i i i i i i i i i i i i i Fall 2008 under Econometrics Prof. Keunkwan Ryu
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29 Goodness-of-Fit How do we think about how well our sample regression line fits our sample data? Can compute the fraction of the total sum of squares (SST) that is explained by the model, call this the R-squared of regression R 2 = SSE/SST = 1 – SSR/SST Fall 2008 under Econometrics Prof. Keunkwan Ryu
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30 Unbiasedness of OLS Assume the population model is linear in parameters as y = β 0 + β 1 x + u Assume we can use a random sample of size n , {( x i , y i ): i =1, 2, …, n }, from the population model. Thus we can write the sample model y i = β 0 + β 1 x i + u i Assume E( u|x ) = 0 and thus E( u i |x i ) = 0 Assume there is variation in the x i Fall 2008 under Econometrics Prof. Keunkwan Ryu
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31 Unbiasedness of OLS (cont) In order to think about unbiasedness, we need to rewrite our estimator in terms of the population parameter Start with a simple rewrite of the formula as ( 29 ( 29 - - = 2 2 2 1 where , ˆ x x s s y x x i x x i i β Fall 2008 under Econometrics Prof. Keunkwan Ryu
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32 Unbiasedness of OLS (cont) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 i i i i i i i i i i i i i i i u x x x x x x x u x x x x x x x u x x x y x x - + - + - = - + - + - = + + - = - 1 0 1 0 1 0 β β β β β β Fall 2008 under Econometrics Prof. Keunkwan Ryu
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33 Unbiasedness of OLS (cont) ( 29 ( 29 ( 29 ( 29 ( 29 2 1 1 2 1 2 ˆ thus and , as rewritten be can numerator the , so , 0 x i i i i x i i i i s u x x u x x s x x x x x x x - + = - + - = - = - β β β Fall 2008 under Econometrics Prof. Keunkwan Ryu
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34 Unbiasedness of OLS (cont) ( 29 ( 29 ( 29 1 2 1 1 2 1 1 ˆ then , 1 ˆ that so , let β β β β β = + = + = - = i i x i i x i i i u E d s E u d s x x d Fall 2008 under Econometrics Prof. Keunkwan Ryu
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