intro-ects-formula

intro-ects-formula - R 2 ): R 2 = ESS TSS = 1 − SSR TSS =...

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Econometrics 0701 (2009) 1. Probability and statistics variance = expected value of the squared deviations around the mean of X var ( X )= E [ X E ( X )] 2 2. Simple linear regression model the model: y = β 0 + β 1 x + ε the ordinary least-squares (OLS) estimators of β 0 and β 1 , denoted by b β 0 and b β 1 ,areg ivenby : b β 1 = n ( P x i y i ) ( P x i )( P y i ) n ( P x 2 i ) ( P x i ) 2 = P ( x i x )( y i y ) P ( x i x ) 2 b β 0 = P y i n b β 1 P x i n = y b β 1 x under the Gauss-Markov assumptions, the variance of b β 1 is given by: var ³ b β 1 ´ = σ 2 " 1 P ( x i x ) 2 # an unbiased estimator of σ 2 is b σ 2 = P n i =1 ( b ε i ) 2 n 2 where b ε i is the OLS residual b ε i = y i b β 0 b β 1 x i 3. Multiple linear regression model the model: y = β 0 + β 1 x 1 + β 2 x 2 + ... + β k x k + ε coe
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Unformatted text preview: R 2 ): R 2 = ESS TSS = 1 − SSR TSS = 1 − P b ε 2 i P ( y i − y ) 2 where b ε i = y i − b β − b β 1 x i 1 − ... − b β k x ik , SSR is sum of squared residuals, ESS is explained sum of squares and TSS (= ESS + SSR ) is total sum of squares • adjusted R 2 (denoted by R 2 ): R 2 = 1 − SSR/ [ n − ( k + 1)] TSS/ ( n − 1) • an unbiased estimator of σ 2 is b σ 2 = P n i =1 ( b ε i ) 2 n − ( k + 1) 1...
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