ˆ β k β k σ n j 1 w 2 k j n 01 18 the error

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ˆ β k & β k σ ' n j ' 1 w 2 k , j - N [0,1] (18) The error variance can be estimated similar to the case of the two-variable linear σ 2 regression model, namely using the sum of squared residuals SSR ' ' n j ' 1 ˆ U 2 j , (19) where ˆ U j ' Y j & ' k i ' 1 ˆ β i X i , j (20) is the OLS residual. It can be shown that under Assumptions 1-3, ' n j ' 1 ˆ U 2 j σ 2 - χ 2 n & k . (21) Since the expected value of a distributed random variable is n ! k , the result (21) suggests to χ 2 n & k estimate by σ 2 ˆ σ 2 ' 1 n & k j n j ' 1 ˆ U 2 j . (22) Due to (21), this estimator is unbiased: Moreover, it can be shown that under E σ 2 ] ' σ 2 . Assumptions 1-3, is independent of the ‘s, hence it follows from (18) and (21) and ' n j ' 1 ˆ U 2 j ˆ β i the definition of the t distribution that under Assumptions 1 and 2,
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