{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

LINREG2

¯ x(64 so that we can write $ u j $ u j& 1 n j n

Info iconThis preview shows pages 27–29. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ¯ X ' (64) so that we can write $ U j ' $ U j & 1 n j n i ' 1 $ U i ' ( Y j & ¯ Y ) & $ $ .( X j & ¯ X ). (65) Next, observe from (2) that where Y j & ¯ Y ' U j & ¯ U % β .( X j & ¯ X ), ¯ U ' (1/ n ) ' n j ' 1 U j . Substituting the former equation in (65) yields $ U j ' ( U j & ¯ U ) & ( $ $ & $ )( X j & ¯ X ), (66) hence j n j ' 1 $ U 2 j ' j n j ' 1 ( U j & ¯ U ) & ( $ $ & $ )( X j & ¯ X ) 2 ' j n j ' 1 ( U j & ¯ U ) 2 & 2( $ $ & $ ) j n j ' 1 ( X j & ¯ X )( U j & ¯ U ) % ( $ $ & $ ) 2 j n j ' 1 ( X j & ¯ X ) 2 ' j n j ' 1 ( U j & ¯ U ) 2 & 2( $ $ & $ ) j n j ' 1 ( X j & ¯ X ) U j % ( $ $ & $ ) 2 j n j ' 1 ( X j & ¯ X ) 2 , (67) 28 where the last equality follows from the fact that It follows from (52), (67) ' n j ' 1 ( X j & ¯ X ) ¯ U ' 0. and the equality that ' n j ' 1 ( U j & ¯ U ) 2 ' ' n j ' 1 U 2 j & n ¯ U 2 j n j ' 1 $ U 2 j ' j n j ' 1 ( U j & ¯ U ) 2 & ( $ $ & $ ) 2 j n j ' 1 ( X j & ¯ X ) 2 ' j n j ' 1 U 2 j & n ¯ U 2 & ( $ $ & $ ) 2 j n j ' 1 ( X j & ¯ X ) 2 . ' j n j ' 1 U 2 j & 1 n ' n i ' 1 U i 2 & ( $ $ & $ ) 2 j n j ' 1 ( X j & ¯ X ) 2 . (68) Taking expectations and using Lemma 2 and Proposition 2 it follows now from (68) that E [ ' n j ' 1 $ U 2 j ] ' ' n j ' 1 E [ U 2 j ] & 1 n E ' n i ' 1 U i 2 & E ( $ $ & $ ) 2 ' n j ' 1 ( X j & ¯ X ) 2 ' n F 2 & F 2 & F 2 ' ( n & 2) F 2 . (69) Proof of (13): SSR ' j n j ' 1 $ U 2 j ' j n j ' 1 ( Y j & $ " & $ $ . X j ) 2 ' j n j ' 1 ( Y j & ( ¯ Y & $ $ . ¯ X ) & $ $ . X j ) 2 ' j n j ' 1 ( Y j & ¯ Y ) & $ $ .( X j & ¯ X ) 2 ' j n j ' 1 ( Y j & ¯ Y ) 2 & 2 $ $ j n j ' 1 ( Y j & ¯ Y )( X j & ¯ X ) % $ $ 2 j n j ' 1 ( X j & ¯ X ) 2 ' j n j ' 1 ( Y j & ¯ Y ) 2 & $ $ 2 j n j ' 1 ( X j & ¯ X ) 2 . (70) Proof of (28): It follows from (3) that Y n % 1 & $ Y n % 1 ' U n % 1 & ( $ " & " ) & ( $ $ & $ ). X n % 1 ' U n % 1 & j n j ' 1 1 n & ¯ X ( X j & ¯ X ) ' n i ' 1 ( X i & ¯ X ) 2 . U j & j n j ' 1 X n % 1 ( X j & ¯ X ) ' n i ' 1 ( X i & ¯ X ) 2 U j ' U n % 1 & j n j ' 1 1 n % ( X n % 1 & ¯ X )( X j & ¯ X ) ' n i ' 1 ( X i & ¯ X ) 2 . U j . (71) 29 Proof of (29): It follows from (28) and Lemma 3 that F 2 Y n % 1 & $ Y n % 1 ' F 2 % j n j ' 1 1 n % ( X n % 1 & ¯ X )( X j & ¯ X ) ' n i ' 1 ( X i & ¯ X ) 2 2 . F 2 ' F 2 1 % 1 n % 2 n . ( X n % 1 & ¯ X ) ' n j ' 1 ( X j & ¯ X ) ' n i ' 1 ( X i & ¯ X ) 2 % ( X n % 1 & ¯ X ) 2 ' n j ' 1 ( X j & ¯ X ) 2 ( ' n i ' 1 ( X i & ¯ X ) 2 ) 2 ' F 2 n % 1 n % ( X n % 1 & ¯ X ) 2 ' n j ' 1 ( X j & ¯ X ) 2 . (72)...
View Full Document

{[ snackBarMessage ]}

Page27 / 29

¯ X(64 so that we can write $ U j $ U j& 1 n j n i 1 $...

This preview shows document pages 27 - 29. Sign up to view the full document.

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