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Unformatted text preview: (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 (ˆ 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) 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 tvalue (or tstatistic) of ˆ β : $ t $ $ ( ' t & value of $ $ ) ' def . $ $ $ F $ $ ' $ $ ' n j ' 1 ( X j & ¯ X ) 2 $ F t n & 2 if $ ' 0. (17) If β > 0 and then the tvalue of converges in probability to + 4 , and if β < 0 and n 6 4 ˆ β then the tvalue of converges in probability to !4 . Moreover, if the sample size n is n 6 4 ˆ β large then by Proposition 7 we may use the standard normal distribution instead of the t distribution to find critical values of the test. Similarly, $ t $ " ( ' t & value of $ " ) ' def . $ " $ F $ " t n & 2 if " ' 0. (18) However, the hypothesis α = 0 is often of no interest. In the ice cream example, ' n j ' 1 ( X j & ¯ X ) 2 ' 18 Y ' n j ' 1 ( X j & ¯ X ) 2 ' 18 . 4.24264, ' n j ' 1 ( Y j & ¯ Y ) 2 ' ' n j ' 1 Y 2 j & n . ¯ Y 2 ' 1020 & 8×11 2 ' 52 and by (13), ˆ σ 2 ' 1 n & 2 j n j ' 1 ˆ U 2 j ' 1 n & 2 j n j ' 1 ( Y j & ¯ Y ) 2 & ˆ β 2 1 n & 2 j n j ' 1 ( X j & ¯ X ) 2 ' 52 & (1.5) 2 .18 8 & 2 ' 11.5 6 . 1.916667 Y ˆ σ . 1.384437 Hence, $ t $ $ ' $ $ ' n j ' 1 ( X j & ¯ X ) 2 $ F ' 1.5×4.24264 1.384437 . 4.597 (19) Assuming that the conditions of Proposition 6 hold, the null hypothesis can be tested H : β ' 11 against the alternative hypothesis using the twosided ttest at say the 5% H 1 : β … significance level, as follows. Under the null hypothesis, (19) is a random drawing from the t distribution with n ! 2 = 6 degrees of freedom. Look up in the table of the t distribution the value...
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 Fall '10
 H.Bierens
 Economics, Econometrics, Normal Distribution, Regression Analysis, Null hypothesis, Statistical hypothesis testing, j Uj, j'1

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