Financial Risk Manager Part 1

Explanation

The correct answer is B: $\hat{Y} = -1.0799 + 0.5633 X_1$

To calculate the estimated regression equation $\hat{Y} = \hat{\alpha} + \hat{\beta}_1 X_1$, we need to compute:

1. $\hat{\beta}_1$: The slope coefficient
2. $\hat{\alpha}$: The intercept

Step 1: Calculate means

From the data:

• Y values: -2, -0.11, -1.68, -0.36, -0.08, -0.74
• X₁ values: -0.41, 0.40, -0.86, 1.69, 0.46, 1.40

Mean of Y = (-2 - 0.11 - 1.68 - 0.36 - 0.08 - 0.74)/6 = -5.97/6 = -0.995 Mean of X₁ = (-0.41 + 0.40 - 0.86 + 1.69 + 0.46 + 1.40)/6 = 2.68/6 = 0.4467

Step 2: Calculate Cov(Y, X₁) and Var(X₁)

Cov(Y, X₁): First compute deviations:

• For each observation: (Yᵢ - Ȳ)(X₁ᵢ - X̄₁)
• Sum these products and divide by (n-1) = 5

Var(X₁):

• Sum of squared deviations of X₁ from its mean, divided by (n-1)

Step 3: Calculate $\hat{\beta}_1$

$\hat{\beta}_1 = \frac{\text{Cov}(Y, X_1)}{\text{Var}(X_1)}$

Step 4: Calculate $\hat{\alpha}$

$\hat{\alpha} = Ȳ - \hat{\beta}_1 X̄₁$

When performing these calculations with the given data:

• $\hat{\beta}_1 ≈ 0.5633$
• $\hat{\alpha} ≈ -1.0799$

Thus, the estimated regression equation is: $\hat{Y} = -1.0799 + 0.5633 X_1$

This matches option B.