Financial Risk Manager Part 1

Explanation

When we omit a variable from a regression model, the coefficient of the remaining variable is biased. The formula for the omitted variable bias is:

$\hat{\beta}_1 \rightarrow \beta_1 + \beta_2 \delta$

Where:

• $\beta_1 = 1.5$ (original coefficient of $X_1$)
• $\beta_2 = -2$ (original coefficient of $X_2$)
• $\delta = \frac{\text{Cov}(X_1, X_2)}{\text{Var}(X_1)}$

Step 1: Calculate Covariance

Given:

• $\rho_{X_1X_2} = 0.7$
• $\sigma^2_{X_1} = 25 \Rightarrow \sigma_{X_1} = 5$
• $\sigma^2_{X_2} = 36 \Rightarrow \sigma_{X_2} = 6$

Using the correlation formula: $\rho_{X_1X_2} = \frac{\text{Cov}(X_1, X_2)}{\sigma_{X_1} \sigma_{X_2}}$ $`$0.7` = \frac{\text{Cov}(X_1, X_2)}{5 \times 6}$$\text{Cov}(X_1, X_2) = 0.7 \times 30 = 21$$

Step 2: Calculate $\delta$

$\delta = \frac{\text{Cov}(X_1, X_2)}{\text{Var}(X_1)} = \frac{21}{25} = 0.84$

Step 3: Calculate the Biased Coefficient

$\hat{\beta}_1 = \beta_1 + \beta_2 \delta = 1.5 + (-2) \times 0.84 = 1.5 - 1.68 = -0.18$

Therefore, the value of $\hat{\beta}_1$ in the reduced model is -0.18.

This demonstrates the omitted variable bias: when we omit $X_2$ from the model, the coefficient of $X_1$ changes from its true value of 1.5 to -0.18 due to the correlation between $X_1$ and $X_2$.