{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

LINREG2

(14 then is called the standard error of also denoted

Info iconThis preview shows pages 9–12. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 t-value (or t-statistic) 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 t-value of converges in probability to + 4 , and if β < 0 and n 6 4 ˆ β then the t-value 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 two-sided t-test 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...
View Full Document

{[ snackBarMessage ]}

Page9 / 29

(14 Then is called the standard error of also denoted by...

This preview shows document pages 9 - 12. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online