Financial Risk Manager Part 1
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Carry out a statistical test at the 5% significance level with the null hypothesis that all four of the independent variables are equal to zero. Quote the F-statistic and the conclusion.
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TTanishq
A
F-statistic = 17.40; At least one of the 4 independent variables is significantly different from zero.
B
F-statistic = 2.525; At least one of the 4 independent variables is significantly different from zero.
C
F-statistic = 72.5; All the 4 independent variables are significantly different from zero.
D
F-statistic = 17.40; None of the independent variables is significantly different from zero.
Explanation:
Explanation
This question involves testing the joint significance of all independent variables in a multiple regression model using an F-test.
Hypotheses:
- H₀: B₁ = B₂ = B₃ = B₄ = 0 (All coefficients are zero)
- H₁: At least one Bⱼ ≠ 0 (At least one coefficient is not zero)
F-statistic Calculation:
The F-statistic formula for testing joint significance is:
F = [ESS/k] / [SSR/(n-k-1)]
Where:
- TSS = total sum of squares = 540
- SSR = residual sum of squares = 250
- ESS = explained sum of squares = TSS - SSR = 540 - 250 = 290
- k = number of independent variables = 4
- n = number of observations = 65 (since n-k-1 = 60, so n = 60 + 4 + 1 = 65)
F = [290/4] / [250/60] = 72.5 / 4.167 = 17.4
Decision:
The calculated F-statistic (17.40) is greater than the critical F-value at 5% significance level with 4 and 60 degrees of freedom. Therefore, we reject the null hypothesis.
Conclusion:
At least one of the 4 independent variables is significantly different from zero.
This matches option A: F-statistic = 17.40; At least one of the 4 independent variables is significantly different from zero.
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