Financial Risk Manager Part 1

This is a tricky question that needs application of the omitted variables formula. Recall that if the regression model is stated as:

$Y_i = \alpha + \beta_1 X_{1i} + \beta_2 X_{2i} + e_i$

If we omit $X_2$ from the estimated model, then the model is given by:

$Y_i = \alpha + \beta_1 X_{1i} + e_i$

Now, in large sample sizes, the OLS estimator $\hat{\beta}_1$ converges to:

$\beta_1 + \beta_2 \delta_1$

Where:

$\delta_1 = \frac{\text{Cov}(X_1, X_2)}{\text{Var}(X_1)}$

Maintaining this line of thought, the first regression equation suggests that:

$\beta_1 + \beta_2 \delta_1 = 0.5633$

And the second equation suggests that:

$\beta_2 + \beta_1 \delta_2 = -0.7633$

Where $\delta_2 = \frac{\text{Cov}(X_1, X_2)}{\text{Var}(X_2)}$

Given:

• $\text{Cov}(X_1, X_2) = 0.603$
• $\text{Var}(X_1) = 0.874$
• $\text{Var}(X_2) = 0.75$

We can calculate: $\delta_1 = \frac{0.603}{0.874} = 0.6899$ $\delta_2 = \frac{0.603}{0.75} = 0.804$

Now we have two equations:

1. $\beta_1 + 0.6899\beta_2 = 0.5633`$2. $0.80`4\beta_1 + \beta_2 = -0.7633$

Solving these simultaneous equations: From equation 1: $\beta_1 = 0.5633 - 0.6899\beta_2$ Substitute into equation 2: $0.804(0.5633 - 0.6899\beta_2) + \beta_2 = -0.7633$`$0.4529 - 0.5547\beta_2 + \beta_2 = -0.7633$ $0.45`29 + 0.4453\beta_2 = -0.7633$ $0.4453\beta_2 = -1.2162$$\beta_2 = -2.7312$

Then $\beta_1 = 0.5633 - 0.6899(-2.7312) = 0.5633 + 1.884 = 2.4473$

Now for the intercept, we need to consider that the intercept from the first regression (0.5767) is actually $\alpha + \beta_2 \times \text{mean}(X_2)$ when $X_1$ is omitted, but since we don't have means, we can use the fact that the intercept should satisfy:

From the first regression: $\hat{Y} = 0.5767 + 0.5633X_1$ From the second regression: $\hat{Y} = 0.6767 - 0.7633X_2$

When we have both variables, the intercept $\hat{\alpha}$ should be such that: $\hat{Y} = \hat{\alpha} + \beta_1 X_1 + \beta_2 X_2$

Given the calculated coefficients $\beta_1 = 2.4473$ and $\beta_2 = -2.7312$, and matching the intercept from either equation, we get the expression in option D: $\hat{\alpha} = \hat{Y}_i - 2.4476X_{1i} + 2.7312X_{2i}$ (note the sign change for $\beta_2$ since it's negative).