stock_watson_ch5 - Stock and Watson Chapter 5: Regression...

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Stock and Watson Chapter 5: Regression with a Single Regressor general form of the t-statistic: t = (estimator – hypothesized value) / (standard error of estimator) what do the distributions of 1 ˆ β and 0 ˆ look like? if the sample is sufficiently large (n=100), then: 1 ˆ ~ N (β 1 , standard deviation given by formula on page 133 SW, 587 MM) 0 ˆ ~ N (β 0 , standard deviation given by formula on page 133 SW, 587 MM) we don’t know the value of these standard deviations (depend on mean and variance of X)- the estimated standard deviations of 1 ˆ and 0 ˆ are the standard errors of 1 ˆ and 0 ˆ : if the sample is sufficiently large (n=100), then: 1 ˆ ~ N (β 1 , standard error given by formula on page 151 SW, 587 MM) 0 ˆ ~ N (β 0 , standard error given by formula on page 180 SW, 587 MM) t = ) ˆ ( _ ˆ 1 1 SE value ed hypothesiz - 1) if the sample is sufficiently large, then 1 ˆ ~ N 2) the t-statistic is calculated assuming that the null is true H 0 : β 1 = 0 vs. H 1 : β 1 0 if the null is true, then the mean ( 1 ˆ ) = 0 3) SE ( 1 ˆ ) is the estimated standard deviation of 1 ˆ 1 ˆ ~ N t-statistic is (approximately) a standardized variable→ (variable – mean) / (estimated sd) t-statistic has (approximately) standard normal distribution: t ~ N (0,1) → can use standard normal table to calculate p-value
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also legitimate to write: if the sample is sufficiently large→ t ~ t n-2 since t n-2 approaches the standard normal distribution as n gets large (return to t distributions in Section 5.6) when will t statistic have exactly a t n-2 distribution? (discuss in Section 5.6) Stock and Watson use: if the sample is sufficiently large→ t ~ N (0,1) we will always use t ~ N (0,1) unless explicitly told otherwise (Section 5.6) Example 1 estScore T ˆ = 700 - 2.32 * STR n= 420 (10.4) (1.12) Calculate the t-statistic for the test of H 0 : β 1 = 0 against the two-sided alternative H a : β 1 0? Calculate the p-value using the standard normal table. Do you reject H 0 at the 5% level? Example 2 estScore T ˆ = 700 - 2.32 * STR n= 420 (10.4) (1.12) Calculate the t-statistic for the test of H 0 : β 1 = 0 against the one-sided alternative H a : β 1 < 0? Calculate the p-value using the standard normal table. Do you reject H 0 at the 5% level? 2
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stock_watson_ch5 - Stock and Watson Chapter 5: Regression...

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