Next consider we have already established that

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Next consider . We have already established that Substituting the right- ˆ α ˆ α ' ¯ Y & ˆ β . ¯ X . hand side of (52) for in this equation yields ˆ β $ " ' ¯ Y & $ % ' n j ' 1 ( X j & ¯ X ) U j ' n j ' 1 ( X j & ¯ X ) 2 . ¯ X ' ¯ Y & $ . ¯ X & ' n j ' 1 ¯ X ( X j & ¯ X ) U j ' n j ' 1 ( X j & ¯ X ) 2 . (55) Substituting ¯ Y ' 1 n j n j ' 1 Y j ' 1 n j n j ' 1 ( α % β X j % U j ) ' α % β . ¯ X % 1 n j n j ' 1 U j in (55) yields $ " ' " % 1 n j n j ' 1 U j & ' n j ' 1 ¯ X ( X j & ¯ X ) U j ' n i ' 1 ( X i & ¯ X ) 2 ' " % j n j ' 1 1 n & ¯ X ( X j & ¯ X ) ' n i ' 1 ( X i & ¯ X ) 2 . U j . (56) Similar as for we therefore have: ˆ β E [ $ " ] ' " % j n j ' 1 1 n & ¯ X ( X j & ¯ X ) ' n i ' 1 ( X i & ¯ X ) 2 E [ U j ] ' " . (57) This completes the proof of Proposition 1. Proof of Lemma 1: We have E ' n j ' 1 v j U j ' n j ' 1 w j U j ' E ' n i ' 1 ' n j ' 1 v i w j U i U j ' j n i ' 1 j n j ' 1 v i w j E ( U i U j ) ' j n j ' 1 v j w j F 2 , (58) where the last equality in (58) follows from E ( U i U j ) ' E ( U i ) E ( U j ) ' 0 if i j , ' E ( U 2 j ) ' F 2 if i ' j . (59)
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26 Proof of Proposition 2: It follows from formula (52) and Lemma 2 that var( $ $ ) ' E [( $ $ & $ ) 2 ] ' E j n j ' 1 X j & ¯ X ' n i ' 1 ( X i & ¯ X ) 2 U j 2 ' F 2 j n j ' 1 X j & ¯ X ' n i ' 1 ( X i & ¯ X ) 2 2 ' F 2 ' n j ' 1 ( X j & ¯ X ) 2 ' n i ' 1 ( X i & ¯ X ) 2 2 ' F 2 ' n j ' 1 ( X j & ¯ X ) 2 ' n j ' 1 ( X j & ¯ X ) 2 2 ' F 2 ' n j ' 1 ( X j & ¯ X ) 2 . (60) Similarly, it follows from formula (56) and Lemma 2 that var( $ " ) ' E [( $ " & " ) 2 ] ' E j n j ' 1 1 n & ¯ X ( X j & ¯ X ) ' n i ' 1 ( X i & ¯ X ) 2 U j 2 ' F 2 j n j ' 1 1 n & ¯ X ( X j & ¯ X ) ' n i ' 1 ( X i & ¯ X ) 2 2 ' F 2 j n j ' 1 1 n 2 & 2 1 n ¯ X ( X j & ¯ X ) ' n i ' 1 ( X i & ¯ X ) 2 % ¯ X 2 ( X j & ¯ X ) 2 ' n i ' 1 ( X i & ¯ X ) 2 2 ' F 2 1 n & 2 ¯ X (1/ n ) ' n j ' 1 ( X j & ¯ X ) ' n i ' 1 ( X i & ¯ X ) 2 % ¯ X 2 ' n j ' 1 ( X j & ¯ X ) 2 ' n i ' 1 ( X i & ¯ X ) 2 2 ' F 2 1 n % ¯ X 2 ' n j ' 1 ( X j & ¯ X ) 2 ' F 2 (1/ n ) ' n j ' 1 ( X j & ¯ X ) 2 % ¯ X 2 ' n j ' 1 ( X j & ¯ X ) 2 ' F 2 ' n j ' 1 X 2 j n ' n j ' 1 ( X j & ¯ X ) 2 , (61) where the last equality follows from the fact that (1/ n ) ' n j ' 1 ( X j & ¯ X ) 2 ' (1/ n ) ' n j ' 1 X 2 j & ¯ X 2 . Finally, it follows from Lemma 1 and the formulas (52) and (56) that
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27 ov( $ " , $ $ ) ' E [( $ " & " )( $ $ & $ )] ' E j n j ' 1 1 n & ¯ X ( X j & ¯ X ) ' n i ' 1 ( X i & ¯ X ) 2 U j j n j ' 1 X j & ¯ X ' n i ' 1 ( X i & ¯ X ) 2 U j ' F 2 j n j ' 1 1 n & ¯ X ( X j & ¯ X ) ' n i ' 1 ( X i & ¯ X ) 2 ( X j & ¯ X ) ' n i ' 1 ( X i & ¯ X ) 2 (62) which can be rewritten as cov( $ " , $ $ ) ' F 2 (1/ n ) ' n j ' 1 ( X j & ¯ X ) & ¯ X ' n j ' 1 ( X j & ¯ X ) 2 ' n i ' 1 ( X i & ¯ X ) 2 2 ' & F 2 . ¯ X ' n j ' 1 ( X j & ¯ X ) 2 . (63) Proof of Proposition 5. Observe first from (44) and (9) that 1 n j n j ' 1 $ U j ' ¯ Y & $ " & $ $ . ¯ X ' 0 (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)
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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 .
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