The last hypothesis is usually what is meant by

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The last hypothesis is usually what is meant by “gender discrimination.” A test for the null hypothesis β = 0 against one of these alternative hypotheses can be based on the estimate of β , ˆ β provided that we know how is related to β . ˆ β It will be shown below that and are indeed reasonable approximations of α and β , ˆ α ˆ β respectively, possessing particular desirable properties. In general an estimator of an unknown parameter is a function of the data that serves as an approximation of the parameter involved. It follows from (1) that and are functions of ˆ α ˆ β the data, Because and will be used as approximations of α and β , ( Y 1 , X 1 ),...,( Y n , X n ). ˆ α ˆ β respectively, and were obtained by minimizing the squared errors, we will call and the ˆ α ˆ β Ordinary 2 Least Squares (OLS) estimators of α and β , respectively. 3.1 Unbiasedness The first property of and is that they are unbiased estimators of α and β : ˆ α ˆ β Proposition 1 . Under Assumptions II and IV the OLS estimators and are unbiased , which ˆ α ˆ β means that = E α ] α and E [ ˆ β ] ' β . This result follows from the fact that we can write $ " ' " % j n j ' 1 1 n & ¯ X ( X j & ¯ X ) ' n i ' 1 ( X i & ¯ X ) 2 . U j , $ $ ' $ % ' n j ' 1 ( X j & ¯ X ) U j ' n i ' 1 ( X i & ¯ X ) 2 . (3) See the Appendix.
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6 3.2 The variances of and ˆ α ˆ β . Our next issue concerns the variances of and . For deriving these variances the ˆ α ˆ β following two lemmas are convenient. Lemma 1 . Let U 1 , U 2 ,...,U n be independent random variables with zero mathematical expectation ( thus E ( U j ) = 0) and variance σ 2 . ( Thus E [( U j - E ( U j )) 2 ] = E ( U j 2 ) = σ 2 ). Let v 1 , v 2 ,..., v n and w 1 , w 2 ,..., w n be given constants. Then E [( ' n j ' 1 v j U j )( ' n j ' 1 w j U j )] ' σ 2 ' n j ' 1 v j w j . Proof . See the Appendix. Note that if we choose in Lemma 1 then it reads: v j ' w j for j ' 1,2,..., n Lemma 2 . Let U 1 , U 2 ,...,U n be independent random variables with zero mathematical expectation and variance σ 2 . Let w 1 , w 2 ,..., w n be given constants. Then E [( ' n j ' 1 w j U j ) 2 ] ' σ 2 ' n j ' 1 w 2 j . Using (3) and Lemmas 1 and 2 it can be shown that Proposition 2 . Under the assumptions I - IV, var( $ " ) ' F 2 ' n j ' 1 X 2 j n ' n j ' 1 ( X j & ¯ X ) 2 ' F 2 $ " , say , var( $ $ ) ' F 2 ' n j ' 1 ( X j & ¯ X ) 2 ' F 2 $ $ , say , and cov( $ " , $ $ ) ' & F 2 ¯ X ' n j ' 1 ( X j & ¯ X ) 2 . (4) Proof . See the Appendix 3.3 Normality of and ˆ α ˆ β . If we also assume normality of the error terms then and are also normally U j ˆ α ˆ β distributed. This result follows from the following lemma.
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7 Lemma 3 . Let Z 1 , Z 2 ,... Z m be independent N (μ, σ 2 ) distributed random variables and let w 1 ,.., w m be constants. Then ' m j ' 1 w j Z j is distributed N [( ' m j ' 1 w j )μ ,( ' m j ' 1 w 2 j ) σ 2 ]. The proof of this lemma requires advanced probability theory and is therefore omitted. It follows now straightforwardly from Proposition 2, Lemma 3, and (3) that: Proposition 3 . Under the assumptions I - V, $ " & " - N 0, F 2 ' n j ' 1 X 2 j n ' n j ' 1 ( X j & ¯ X ) 2 , $ $ & $ - N 0, F 2 ' n j ' 1 ( X j & ¯ X ) 2 , (5) where is the symbol for is distributed as. - Moreover, applying Lemma 3 again for m = 1 it follows from (5) (Exercise: Why?) that Proposition 4 . Under the assumptions I - V, ( $ " & " ) n ' n j ' 1 ( X j & ¯ X ) 2 F . ' n j ' 1 X 2 j - N [0,1], ( $ $ & $ ) ' n j ' 1 ( X j & ¯ X ) 2 F
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