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Unformatted text preview: ope parameters are shown in the Excel output
columns “Lower 95%” and “Upper 95%”. In general, 1  a confidence intervals can be
constructed as
P r(t £ ta ) = 1  éb  t ⋅ SE (b ), b + t ⋅ SE (b )ù , where t
ˆ
ˆˆ
ˆ
a
a
i
i
a
iú
êë i
û is chosen such that a
for a t distribution with n  p  1 degrees of freedom.
2 For example, suppose we want to find the 99% confidence interval of the regression
coefficient for log(sales ) . We know ta = 2.604 when df = 174 and a = 1% , and
ˆ
ˆ
so
the
confidence
interval
is
SE (blog(sales ) ) = 0.03967,
blog(sales ) = 0.1621, é 0.16212.604 ⋅ 0.03967, 0.1621+2.604 ⋅ 0.03967 ù =
êë
úû 6. é 0.059, 0.27 ù .
êë
úû Testing for Statistical Significance of Coefficients For a given regression coefficient, we can test the null hypothesis H 0 : bi = 0
against the alternative H 1 : bi ¹ 0 VER. 2/2/2012. © P. KOLM 9 There are two ways of performing the test. The first way is to compare the tstatistic in
the Excel output with a critical value ta corresponding to a certain confidence level a .
If the tstatistic is larger in absolute value than the critical value ta , we can reject the
null hypothesis and conclude the regression coefficient is statistically significant.
The second way is to use the pvalue shown in the Excel output. If the pvalue is
smaller than a specific threshold, then we can reject the null hypothesis and conclude
the regression coefficient is statistically significant.
The following two examples illustrate the critical value approach and the pvalue
approach for significance testing.
Example 1: Suppose we want to test the significance of the regression coefficient log(mktval ) at 1% level of significance. We know ta = 2.604 when df = 174 and
a = 1% . Comparing ta with the t statistic for log(mktval ) , which is t * = 2.129 , we conclude that log(mktval ) is not statistically significant since ta > t * .
Example 2: Suppose we want to test the significance of the intercept at 5% level of significance. We note that the pvalue of the intercept is 0 up to the 7th decimal place,
which is much smaller than 5%. So we reject the null hypothesis H 0 : intercept = 0 and
conclude that the intercept is statistically significant at the 5% level. 7. Testing Hypothesis on a Slope Parameter The procedure of hypothesis testing is similar to the procedure of significance testing
explained in the previous section, except that the tstatistic needs to be recalculated
according to the specific null hypothesis. In general, if we want to test H 0 : bi = b *
ˆ
b  b*
. Once we have the test
against H 1 : bi ¹ b * , we compute the tstatistic as t * = i
ˆ
SE (bi )
statistic, we can use either the critical value approach or the pvalue approach desc...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.
 Fall '14

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