Financial Risk Manager Part 1
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A sample of 200 firms reveals the following relationship between the annual stock return (Yᵢ) and the average years of experience per employee, Xᵢ:
Yᵢ = β₁ + β₂Xᵢ + εᵢ i = 1, 2, ..., 200
An analyst wishes to test the joint null hypothesis that β₁ = 0 and β₂ = 0 at the 10% level of significance. The p-value for the t-statistics for β₁ and β₂ are 0.12 and 0.11 respectively. The p-value for the F-statistic for the regression is 0.09. This implies that the analyst:
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NNikitesh
A
Can reject the null hypothesis since β₁ and β₂ are different from zero at the 10% level of significance
B
Can reject the null hypothesis because the F-statistic is significant at the 10% level of significance
C
Cannot reject the null hypothesis because we have insufficient evidence to prove both β₁ and β₂ are different from zero at the 10% level of significance
D
Cannot reject the null hypothesis because the F-statistic is not significant at the 10% level of significance
Explanation:
Explanation
When testing a joint null hypothesis (β₁ = 0 AND β₂ = 0), we should use the F-statistic rather than individual t-statistics. Here's why:
Key Points:
-
Individual t-tests vs. Joint F-test:
- Individual t-tests examine each coefficient separately
- F-test examines whether ALL coefficients in the model are jointly significant
- For joint hypothesis testing, the F-test is appropriate
-
P-value Interpretation:
- p-value = 0.09 for the F-statistic
- Since 0.09 < 0.10 (10% significance level), we can reject the joint null hypothesis
- The F-statistic is statistically significant at the 10% level
-
Individual t-statistics:
- β₁ p-value = 0.12 (> 0.10) → not individually significant at 10%
- β₂ p-value = 0.11 (> 0.10) → not individually significant at 10%
- However, this doesn't matter for the joint hypothesis test
Why Option B is Correct:
- The F-statistic p-value (0.09) is less than the significance level (0.10)
- This means we have sufficient evidence to reject the joint null hypothesis
- The F-test properly accounts for testing multiple coefficients simultaneously
Why Other Options are Incorrect:
- A: Incorrect because individual t-statistics are not appropriate for joint hypothesis testing
- C: Incorrect because the F-test provides sufficient evidence to reject the null
- D: Incorrect because the F-statistic IS significant (p-value 0.09 < 0.10)
Statistical Insight:
The F-test for overall regression significance tests whether at least one of the coefficients is different from zero. Even if individual coefficients are not statistically significant, their joint effect might be significant due to correlation between variables or the combined explanatory power.
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