Multiple Regression Coefficient Testing (garp16-p1-09)

[jehwang]

Hi,

In the below example from GARP 2016 Practice exam, the joint significance of the coefficients of the multiple regression is tested with F statistic, not T stat. Please clarify why F stat is appropriate to use in this case.

For a sample of 400 firms, the relationship between corpor ate revenue (Yi) and the average years of experienceper employee (Xi) is modeled as follows:
Yi= β1+ β2Xi+ εi,i = 1, 2,...,400

You wish to test the joint null hypothesis that β1 = 0 and β2 = 0 at the 95% confidence level. The p-value for the t-statistic for β1 is 0.07 , and the p-value for the t-statistic for β2 is 0.06. The p-value for the F-statistic for the regression is 0.045. Which of the following statements is correct?

a. You can reject the null hypothesis because each β is different from 0 at the 95% confidence level.
b. You cannot reject the null hypothesis because neither β is different from 0 at the 95% confidence level.
c. You can reject the null hypothesis because the F-statistic is significant at the 95% confidence level.
d. You cannot reject the null hypothesis because the F-statistic is not significant at the 95% confidence level.

Correct answer: c
Explanation: The t-test would not be sufficient to test the joint hypothesis. In order to test the joint null hypothesis, examine the F-statistic, which in this case is statistically significant at the 95% confidence level. Thus the null can be rejected.

 

[berrymucho]

The F-stat is the appropriate one to test the null hypothesis that all variables are =0 simultaneously, the alternative being that at least one variable is not null. Using the t-stat for each variable in turn would be too loose: for 2 variables at 95% level individually, the chance of having at least one of them non null is 1-0.95^2=9.75% > 5% (assuming independent variables). The F-stat accounts for the level of significance in the joint hypothesis correctly.