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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: of a linear regression of the form E ( y | x ) = + x, (8) from which and unconditional expectation is derived in the form of E ( y ) = + E ( x ) . (9) The orthogonality of the prediction error implies that 0 = E { ( y y ) x } = E { ( y x ) x } = E ( xy ) E ( x ) E ( x 2 ) . (10) In order to eliminate E ( x ) from this expression, equation (9) is multiplied by E ( x ) and rearranged to give E ( x ) = E ( x ) E ( y ) { E ( x ) } 2 . (11) This substituted into (10) to give E ( xy ) E ( x ) E ( y ) = E ( x 2 ) { E ( x ) } 2 , (13) whence = E ( xy ) E ( x ) E ( y ) E ( x 2 ) { E ( x ) } 2 = C ( x, y ) V ( x ) . (14) The expression = E ( y ) E ( x ) (15) comes directly from (9). 2...
View Full Document

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online