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

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

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