Financial Risk Manager Part 1
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Peter Bridge, FRM, runs a regression of monthly stock returns on four independent variables over 65 months. The total sum of squares is 540, and the sum of squared residuals is 250. 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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NNikitesh
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
Given Data:
- Total Sum of Squares (TSS) = 540
- Sum of Squared Residuals (SSR) = 250
- Number of independent variables (k) = 4
- Number of observations (n) = 65
- Significance level = 5%
Step 1: Calculate Explained Sum of Squares (ESS)
ESS = TSS - SSR = 540 - 250 = 290
Step 2: Calculate F-statistic
F-statistic = [ESS/k] / [SSR/(n-k-1)]
Where:
- ESS/k = 290/4 = 72.5
- SSR/(n-k-1) = 250/(65-4-1) = 250/60 = 4.1667
F = 72.5 / 4.1667 = 17.4
Step 3: Determine Critical F-value
Degrees of freedom:
- Numerator: k = 4
- Denominator: n-k-1 = 65-4-1 = 60
At 5% significance level, the critical F-value with (4, 60) degrees of freedom is approximately 2.525.
Step 4: Decision Rule
Since calculated F-statistic (17.4) > critical F-value (2.525), we reject the null hypothesis.
Step 5: Conclusion
Reject H₀: B₁ = B₂ = B₃ = B₄ = 0 Accept H₁: At least one Bⱼ ≠ 0
Therefore, the correct answer is A with F-statistic = 17.40 and conclusion that at least one of the 4 independent variables is significantly different from zero.
Why other options are incorrect:
- B: Uses the critical F-value (2.525) as the calculated F-statistic
- C: Uses ESS/k (72.5) as the F-statistic instead of the complete calculation
- D: Has the correct F-statistic but wrong conclusion (fails to reject null when we should reject it)
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