Financial Risk Manager Part 1
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What are the estimated values of the parameters (α̂ and β̂₁) in the model: Y = α + β₁X₁
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NNikitesh
A
α̂ = -1.080, β̂₁ = 0.5633
B
α̂ = -1.280, β̂₁ = 0.3433
C
α̂ = -1.5797, β̂₁ = 0.6633
D
α̂ = -1.780, β̂₁ = 0.9933
Explanation:
Explanation
The correct answer is A (α̂ = -1.080, β̂₁ = 0.5633).
Step-by-Step Calculation:
-
Calculate β̂₁: β̂₁ = Cov(Y, X₁) / Var(X₁)
From the table:
- Cov(Y, X₁) = (1/(n-1)) Σ(Y - Ȳ)(X₁ - X̄₁) = (1/5) * 2.759533 = 0.5519066
- Var(X₁) = 0.9797 (given in the explanation)
β̂₁ = 0.5519066 / 0.9797 = 0.5633
-
Calculate α̂: α̂ = Ȳ - β̂₁ * X̄₁
From the table:
- Ȳ = -0.82833
- X̄₁ = 0.446667
α̂ = -0.82833 - (0.5633 * 0.446667) = -0.82833 - 0.2516 = -1.07993 ≈ -1.080
Verification:
The calculations match option A exactly:
- α̂ = -1.080
- β̂₁ = 0.5633
This is a standard linear regression estimation problem using the ordinary least squares (OLS) method. The intercept (α̂) is calculated using the formula α̂ = Ȳ - β̂₁X̄₁, and the slope coefficient (β̂₁) is calculated using β̂₁ = Cov(Y, X₁)/Var(X₁).
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