# Expectit - β of a linear regression of the form E y | x =...

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D.S.G. POLLOCK: TOPICS IN ECONOMETRICS EXPECTATIONS AND CONDITIONAL EXPECTATIONS The joint density function of x and y is f ( x, y )= f ( x | y ) f ( y )= f ( y | x ) f ( x ) , (1) where f ( x )= Z y f ( x, y ) dx and f ( y )= Z x f ( x, y ) dy (2) are the marginal distributions of x and y respectively and where f ( x | y )= f ( y, x ) f ( y ) and f ( y | x )= f ( y, x ) f ( x ) (3) are the conditional distributions of x given y and of y given x . The unconditional expectation of y f ( y )is E ( y )= Z y yf ( y ) dy. (4) The conditional expectation of y given x is E ( y | x )= Z y yf ( y | x ) dy = Z y y f ( y, x ) f ( x ) dy. (5) The expectation of the conditional expectation is an unconditional expectation: E { E ( y | x ) } = Z x ½ Z y y f ( y, x ) f ( x ) dy ¾ f ( x ) dx = Z x Z y yf ( y, x ) f ( x ) dydx = Z y y ½ Z x f ( y, x ) dx ¾ dy = Z y f ( y ) dy = E ( y ) . (6) The conditional expectation of y given x is the minimum mean squared error predic- tion; and the error in predicting y is uncorrelated with x . The proof of this depends on showing that E yx )= E ( yx ), where ˆ y = E ( y | x ): E yx )= Z x xE ( y | x ) f ( x ) dx = Z x x ½ Z y y f ( y, x ) f ( x ) dy ¾ f ( x ) dx = Z x Z y xyf ( y, x ) dydx = E ( xy ) . (7) 1

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CONDITIONAL EXPECTATIONS The result can be expressed as E { ( y ˆ y ) x } =0. This result can be used in deriving expressions for the parameters α and
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Unformatted text preview: β of a linear regression of the form E ( y | x ) = α + βx, (8) from which and unconditional expectation is derived in the form of E ( y ) = α + βE ( x ) . (9) The orthogonality of the prediction error implies that 0 = E { ( y − ˆ y ) x } = E { ( y − α − βx ) x } = E ( xy ) − αE ( x ) − βE ( x 2 ) . (10) In order to eliminate αE ( x ) from this expression, equation (9) is multiplied by E ( x ) and rearranged to give αE ( x ) = E ( x ) E ( y ) − β { E ( x ) } 2 . (11) This substituted into (10) to give E ( xy ) − E ( x ) E ( y ) = β £ E ( x 2 ) − { E ( x ) } 2 ¤ , (13) whence β = E ( xy ) − E ( x ) E ( y ) E ( x 2 ) − { E ( x ) } 2 = C ( x, y ) V ( x ) . (14) The expression α = E ( y ) − βE ( x ) (15) comes directly from (9). 2...
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## This note was uploaded on 03/02/2012 for the course EC 7087 taught by Professor D.s.g.pollock during the Fall '11 term at Queen Mary, University of London.

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Expectit - β of a linear regression of the form E y | x =...

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