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Melissa tartari yale econometrics 29 93 the ordinary

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Unformatted text preview: ce: Assume ZCMA but allow E [U ] = λ 6= 0. Show that the moment condition E [(u E (u )) x ] = 0, implied by the ZCMA, is su¢ cient to identify β1 . Derive the implied sample analogue-based MoM estimator of β1 . ˆ MM in equation 2.19? How does it relate to β 1 Melissa Tartari (Yale) Econometrics 28 / 93 The Ordinary Least Squares Approach I Consider …gure 2.4.: we represent 2 data points (x1 , y1 ) and (xi , yi ) for some i of choice. Melissa Tartari (Yale) Econometrics 29 / 93 The Ordinary Least Squares Approach I Consider …gure 2.4.: we represent 2 data points (x1 , y1 ) and (xi , yi ) for some i of choice. ˆ ˆ Let βo and β1 be some value for ( βo , β1 ) - think of these as a guess for the unknown parameters ( βo , β1 ). Melissa Tartari (Yale) Econometrics 29 / 93 The Ordinary Least Squares Approach I Consider …gure 2.4.: we represent 2 data points (x1 , y1 ) and (xi , yi ) for some i of choice. ˆ ˆ Let βo and β1 be some value for ( βo , β1 ) - think of these as a guess for the unknown parameters ( βo , β1 ). Let ui denote the vertical distance between the points (yi , xi ) and ˆ ˆ ˆ βo + β1 xi , xi - what is often called residual : ui = yi ˆ Melissa Tartari (Yale) ˆ βo Econometrics ˆ β1 xi (2.21) 29 / 93 The Ordinary Least Squares Approach I - …gure 2.4 y (xi , yi ) yi ˆˆˆ y = β o + β1 x ˆ ui ˆ yi x1 Melissa Tartari (Yale) ˆˆˆ yi = β o + β1 xi (x1 , y1 ) y1 xi Econometrics x 30 / 93 The Ordinary Least Squares Approach II ˆˆ Suppose we choose βo , β1 to make the sum of the squares of the residuals SSR as small as possible: 1n 2 1n ui , min ˆ ∑ ∑ yi ˆˆ ˆˆ ( βo , β1 ) n i =1 ( βo , β1 ) n i =1 min Melissa Tartari (Yale) Econometrics ˆ βo ˆ β1 xi 2 (2.22) 31 / 93 The Ordinary Least Squares Approach II ˆˆ Suppose we choose βo , β1 to make the sum of the squares of the residuals SSR as small as possible: 1n 2 1n ui , min ˆ ∑ ∑ yi ˆˆ ˆˆ ( βo , β1 ) n i =1 ( βo , β1 ) n i =1 min ˆ βo ˆ β1 xi 2 The FOCs of the optimization problem (2.22) are: 8 ˆˆ > ∂SSR ( βo ,β1 ) = 0 < ˆ ˆ ˆ 2 ∑n=1 yi βo β1 xi = 0 ∂ βo i ) n ˆˆ ˆ ˆ ∂SSR ( βo , β1...
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