However the r 2 is only a descriptive statistic it

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However, the R 2 is only a descriptive statistic; it does not play a role in hypotheses testing, so that the unreliability of the R 2 in small samples is harmless. Notice the quite dramatic increase of the t-values. Recall that these t-values are the test statistics of the null hypotheses that the corresponding parameters are zero. Because the true parameters are equal to 1, what you see in Table 2 is the increase of the power of the t-test with the sample size.
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22 APPENDIX Proof of (1): The first-order conditions for a minimum of are: Q α , ˆ β ) ' ' n j ' 1 ( Y j & ˆ α & ˆ β X j ) 2 dQ ( $ " , $ $ )/ d $ " ' 0 ] j n j ' 1 2( Y j & $ " & $ $ X j )( & 1) ' 0 ] j n j ' 1 ( Y j & $ " & $ $ X j ) ' 0 ] j n j ' 1 Y j & j n j ' 1 $ " & j n j ' 1 ( $ $ X j ) ' 0 ] j n j ' 1 Y j ' n $ " % $ $ j n j ' 1 X j ' 0 ] ¯ Y ' $ " % $ $ . ¯ X , (42) and dQ ( $ " , $ $ )/ d $ $ ' 0 ] j n j ' 1 2( Y j & $ " & $ $ X j )( & X j ) ' 0 ] j n j ' 1 ( Y j X j & $ " X j & $ $ X 2 j ) ' 0 ] j n j ' 1 X j Y j & $ " j n j ' 1 X j & $ $ j n j ' 1 X 2 j ' 0 ] j n j ' 1 X j Y j ' $ " j n j ' 1 X j % $ $ j n j ' 1 X 2 j ] 1 n j n j ' 1 X j Y j ' $ " ¯ X % $ $ 1 n j n j ' 1 X 2 j (43) where are the sample means of the X j 's and Y j 's, ¯ X ' (1/ n ) ' n j ' 1 X j and ¯ Y ' (1/ n ) ' n j ' 1 Y j respectively. The last equations in (42) and (43) are called the normal equations : ¯ Y ' $ " % $ $ . ¯ X , (44) 1 n j n j ' 1 X j Y j ' $ " . ¯ X % $ $ 1 n j n j ' 1 X 2 j . (45) To solve these normal equations, substitute in (45). Then we get ˆ α ' ¯ Y & ˆ β . ¯ X
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23 1 n j n j ' 1 X j Y j ' ( ¯ Y & ˆ β ¯ X ) 1 n j n j ' 1 X j % ˆ β 1 n j n j ' 1 X 2 j ' ¯ Y . ¯ X & ˆ β ¯ X 2 % ˆ β 1 n j n j ' 1 X 2 j ' ¯ X . ¯ Y % ˆ β 1 n j n j ' 1 X 2 j & ¯ X 2 hence 1 n j n j ' 1 X j Y j & ¯ X . ¯ Y ' $ $ 1 n j n j ' 1 X 2 j & ¯ X 2 . (46) Equation (46) can also be written as 1 n j n j ' 1 ( X j & ¯ X )( Y j & ¯ Y ) ' $ $ 1 n j n j ' 1 ( X j & ¯ X ) 2 , (47) because 1 n j n j ' 1 ( X j & ¯ X )( Y j & ¯ Y ) ' 1 n j n j ' 1 X j Y j & ¯ X . Y j & X j . ¯ Y % ¯ X . ¯ Y ' 1 n j n j ' 1 X j Y j & ¯ X . 1 n j n j ' 1 Y j & ¯ Y . 1 n j n j ' 1 X j % ¯ X . ¯ Y ' 1 n j n j ' 1 X j Y j & ¯ X . ¯ Y (48) and similarly 1 n j n j ' 1 ( X j & ¯ X ) 2 ' 1 n j n j ' 1 X 2 j & ¯ X 2 . (49) Moreover, 1 n j n j ' 1 ( X j & ¯ X )( Y j & ¯ Y ) ' 1 n j n j ' 1 ( X j & ¯ X ) Y j & 1 n j n j ' 1 ( X j & ¯ X ) ¯ Y ' 1 n j n j ' 1 ( X j & ¯ X ) Y j & ( ¯ X & ¯ X ) ¯ Y ' 1 n j n j ' 1 ( X j & ¯ X ) Y j (50)
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24 The result (1) now follows from (44) and (46) through (50). Proof of Proposition 1. Recall from (1) that $ $ ' ' n j ' 1 ( X j & ¯ X ) Y j ' n j ' 1 ( X j & ¯ X ) 2 . (51) Substitute model (2) in (51). Then $ $ ' ' n j ' 1 ( X j & ¯ X )( " % $ X j % U j ) ' n j ' 1 ( X j & ¯ X ) 2 ' " ' n j ' 1 ( X j & ¯ X ) % $ ' n j ' 1 ( X j & ¯ X ) X j % ' n j ' 1 ( X j & ¯ X ) U j ' n j ' 1 ( X j & ¯ X ) 2 ' $ . ' n j ' 1 ( X j & ¯ X ) X j ' n j ' 1 ( X j & ¯ X ) 2 % ' n j ' 1 ( X j & ¯ X ) U j ' n j ' 1 ( X j & ¯ X ) 2 ' $ % ' n j ' 1 ( X j & ¯ X ) U j ' n j ' 1 ( X j & ¯ X ) 2 , (52) where the last step follows from the fact that similar to (50), j n j ' 1 ( X j & ¯ X ) 2 ' j n j ' 1 ( X j & ¯ X )( X j & ¯ X ) ' j n j ' 1 ( X j & ¯ X ) X j . (53) Now take the mathematical expectation at both sides of (52). Then, E [ $ $ ] ' $ % E ' n j ' 1 ( X j & ¯ X ) U j ' n j ' 1 ( X j & ¯ X ) 2 ' $ % ' n j ' 1 ( X j & ¯ X ) E ( U j ) ' n j ' 1 ( X j & ¯ X ) 2 ' $ , (54) because taking the mathematical expectation of a constant ( β ) does not effect that constant, and taking the mathematical expectation of a linear function of random variables is equal to taking the linear function of the mathematical expectation of these random variables. The last conclusion in (54) follows from assumption II, and the second step in (54) can be taken because
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25 we have assumed that the X j 's are non-random (assumption IV).
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