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Unformatted text preview: Econometrics 710 Answer Key to Problem Set 2 4. The linear projection coe¢ cients are given by @ & & 1 1 A = @ E 2 4 1 x i x i x 2 i 3 5 1 A & 1 E 2 4 y i x i y i 3 5 : (1) By some calculations, we will &nd out E [ x i ] = E [ y i ] = 5 = 8 , E & x 2 i ¡ = 7 = 15 , and E [ x i y i ] = 3 = 8 . Therefore, & = 55 = 73 and & 1 = & 15 = 73 are obtained. On the other hand, the conditional mean E [ y i j x i = x ] is given by E [ y i j x i = x ] = Z 1 yf ( y j x ) dy where f ( y j x ) = f ( x;y ) f ( x ) = 3 2 ( x 2 + y 2 ) 3 2 x 2 + 1 2 : Therefore, E [ y i j x i = x ] = Z 1 y ¡ 3 2 ( x 2 + y 2 ) 3 2 x 2 + 1 2 dy = 3 4 ¡ 2 x 2 + 1 3 x 2 + 1 ; (2) which is a nonlinear function of x . The following &gure shows graphs of y = E ( y i j x i = x ) and y = & + & 1 x . They are very similar but di/erent. 1 0.75 0.5 0.25 0.75 0.7 0.65 0.6 0.55 x y x y 1 5. Note that we need both of E [ e i ] = 0 and E [ x i e i ] = 0 ....
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This note was uploaded on 10/08/2010 for the course ECON 710 taught by Professor Staff during the Spring '08 term at University of Wisconsin.
 Spring '08
 Staff
 Econometrics

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