MultivariateRegressionInExcel

In general 1 a confidence intervals can be

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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 0.1621-2.604 ⋅ 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 t-statistic in the Excel output with a critical value ta corresponding to a certain confidence level a . If the t-statistic 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 p-value shown in the Excel output. If the p-value 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 p-value 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 p-value 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 t-statistic 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 t-statistic as t * = i ˆ SE (bi ) statistic, we can use either the critical value approach or the p-value approach desc...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.

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