N 01 6 these results play a key role in testing

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- N [0,1]. (6) These results play a key-role in testing hypotheses about α and β . The only problem that prevents us from using these results for testing is that σ is unknown. This problem will be addressed in the next section. 4. How to estimate the error variance σ 2 ? If α and β were known then we could estimate σ 2 by # F 2 ' 1 n j n j ' 1 ( Y j & " & $ . X j ) 2 ' 1 n j n j ' 1 U 2 j . (7) However, α and β are not known, but we do have OLS estimators of α and β . This suggests to
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8 replace α and β in (7) by their OLS estimators: # F 2 ' 1 n j n j ' 1 ( Y j & $ " & $ $ . X j ) 2 ' 1 n j n j ' 1 $ U 2 j , (8) where $ U j ' Y j & $ " & $ $ . X j (9) is called the regression residual. However, the estimator (8) is biased, due to the fact that Proposition 5 . Under the assumptions I - V, E [ ' n j ' 1 ˆ U 2 j ] ' ( n & 2) σ 2 . Proof : See the Appendix. This result suggests to use $ F 2 ' 1 n & 2 j n j ' 1 $ U 2 j (10) as an estimator of instead of (8), because then by Proposition 5, is an unbiased estimator σ 2 ˆ σ 2 of : σ 2 E [ $ F 2 ] ' F 2 . (11) The sum is called the Sum of Squares Residuals, shortly SSR , or also called the ' n j ' 1 ˆ U 2 j Residual Sum of Squares (RSS), and is called the Standard Error of the Residuals, ˆ σ ' ˆ σ 2 shortly SER . Thus, SSR ' ' n j ' 1 $ U 2 j , SER ' ' n j ' 1 $ U 2 j n & 2 ' SSR n & 2 ( ' $ F ). (12) Finally, note that the sum of squared residuals can be computed as follows: SSR ' j n j ' 1 ( Y j & ¯ Y ) 2 & $ $ 2 j n j ' 1 ( X j & ¯ X ) 2 . (13) See the Appendix.
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9 5. Standard errors, t-values and p-values of the OLS estimators The variances of and can now be estimated by replacing in (4) by : ˆ α ˆ β σ 2 ˆ σ 2 Estimated var( $ " ) ' $ F 2 ' n j ' 1 X 2 j n ' n j ' 1 ( X j & ¯ X ) 2 ' $ F 2 $ " , say , Estimated var( $ $ ) ' $ F 2 ' n j ' 1 ( X j & ¯ X ) 2 ' $ F 2 $ $ , say . (14) Then is called the standard error of , also denoted by and is ˆ σ ˆ α ' ˆ σ 2 ˆ α ˆ α SE α ), ˆ σ ˆ β ' ˆ σ 2 ˆ β called the standard error of , also denoted by ˆ β SE ( ˆ β ). If we replace σ in Proposition 4 by the SER, , the standard normality results involved ˆ σ change: Proposition 6 . Under the assumptions I - V, $ " & " $ F $ " ' ( $ " & " ) n ' n j ' 1 ( X j & ¯ X ) 2 $ F . ' n j ' 1 X 2 j - t n & 2 , $ $ & $ $ F $ $ ' ( $ $ & $ ) ' n j ' 1 ( X j & ¯ X ) 2 $ F - t n & 2 . (15) The proof of Proposition 6 is based on the fact that under these assumptions, SSR / σ 2 is distributed and is independent of and but the proof involved requires advanced χ 2 n & 2 ˆ α ˆ β , probability theory and is therefore omitted. Because for large degrees of freedom the t distribution is approximately equal to the standard normal distribution, and due to the central limit theorem, Proposition 4 holds if n is large and the errors are not normally distributed, we also have: Proposition 7 . If the sample size n is large then under the assumptions I - IV we have approximately, $ " & " $ F $ " ' ( $ " & " ) n ' n j ' 1 ( X j & ¯ X ) 2 $ F . ' n j ' 1 X 2 j - N (0,1), $ $ & $ $ F $ $ ' ( $ $ & $ ) ' n j ' 1 ( X j & ¯ X ) 2 $ F - N (0,1). (16)
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10 The results in Proposition 6 now enable us to test hypotheses about α and β . In particular the null hypothesis that β = 0 is of importance, because this hypothesis implies that X has no effect on Y . The test statistic for testing this hypothesis is the t-value (or t-statistic) of ˆ β : $ t $ $ ( ' t & value of $ $ ) ' def .
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