Financial Risk Manager Part 1
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Tom Well, FRM, works for a trading company. Using historical data, he has computed the following variables considering one independent and one dependent variable.
- Explained Sum of Squares (ESS) = 60
- Sum of Squared Residuals (SSR) = 15
If we're dealing with a sample size of 62 observations, determine the coefficient of determination and the standard error of the estimate, respectively.
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NNikitesh
A
Coefficient of Determination = 0.50, Standard Error of the Estimate = 0.50
B
Coefficient of Determination = 0.80, Standard Error of the Estimate = 0.80
C
Coefficient of Determination = 0.80, Standard Error of the Estimate = 0.50
D
Coefficient of Determination = 0.25, Standard Error of the Estimate = 0.25
Explanation:
Step-by-step calculation:
-
Calculate Total Sum of Squares (TSS): TSS = ESS + SSR = 60 + 15 = 75
-
Calculate Coefficient of Determination (R²): R² = ESS / TSS = 60 / 75 = 0.80
-
Calculate Standard Error of the Estimate (SEE): SEE = √[SSR / (n - 2)] = √[15 / (62 - 2)] = √[15 / 60] = √0.25 = 0.50
Key formulas:
- TSS = Σ(Y - Ȳ)² (Total variation)
- ESS = Σ(Ŷ - Ȳ)² (Explained variation)
- SSR = Σ(Y - Ŷ)² = Σ(ê)² (Unexplained variation/residuals)
- R² = ESS / TSS = 1 - (SSR / TSS)
- SEE = √[SSR / (n - k - 1)] where k = number of independent variables (here k = 1, so denominator = n - 2)
Verification:
- Option C matches our calculations: R² = 0.80 and SEE = 0.50
- The sample size of 62 observations gives us 60 degrees of freedom (n - 2 = 62 - 2 = 60) for the SEE calculation.
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