Financial Risk Manager Part 1

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Explanation:

Using the omitted variable formula, we know that:

\hat{\beta}_1 = \beta_1 + \beta_2 \delta

Where:

\sigma = \frac{\text{Cov (X1, X2)}}{\text{Var (X1)}}

\hat{\beta}_1 = \beta_1 + \beta_2 \delta = 2.4

= 2.2 + 1.1 \left( \frac{\rho_{X_1 X_2} \sigma_{X_1} \sigma_{X_2}}{\text{Var (X1)}} \right)

= 2.4

= 2.2 + 1.1 \left( \frac{\rho_{X_1 X_2} \times \sqrt{16} \times \sqrt{49}}{16} \right) = 2.4

\Rightarrow \rho_{X_1 X_2} = 0.1039

Note that we have used the analogy:

\rho_{XY} = \frac{\text{Cov (X, Y)}}{\sigma_X \sigma_Y} \Rightarrow \text{Cov (X, Y)} = \sigma_X \sigma_Y \rho_{XY}

$\rho_{XY}$ = the correlation between X and Y
$\sigma_X$ = standard deviation of X
$\sigma_Y$ = standard deviation of Y
$\text{Cov(X,Y)}$ = covariance between X and Y

Explanation:

Using the omitted variable formula, we know that:

\hat{\beta}_1 = \beta_1 + \beta_2 \delta

Where:

\sigma = \frac{\text{Cov (X1, X2)}}{\text{Var (X1)}}

\hat{\beta}_1 = \beta_1 + \beta_2 \delta = 2.4

= 2.2 + 1.1 \left( \frac{\rho_{X_1 X_2} \sigma_{X_1} \sigma_{X_2}}{\text{Var (X1)}} \right)

= 2.4

= 2.2 + 1.1 \left( \frac{\rho_{X_1 X_2} \times \sqrt{16} \times \sqrt{49}}{16} \right) = 2.4

\Rightarrow \rho_{X_1 X_2} = 0.1039

Note that we have used the analogy:

\rho_{XY} = \frac{\text{Cov (X, Y)}}{\sigma_X \sigma_Y} \Rightarrow \text{Cov (X, Y)} = \sigma_X \sigma_Y \rho_{XY}

$\rho_{XY}$ = the correlation between X and Y
$\sigma_X$ = standard deviation of X
$\sigma_Y$ = standard deviation of Y
$\text{Cov(X,Y)}$ = covariance between X and Y

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Q.76 A model is estimated as:
$Y_i = 2.8 + 2.2X_{1i} + 1.1X_{2i} + e_i$
Where:
$\sigma_{X_1}^2 = 16, \text{ and } \sigma_{X_2}^2 = 49$
What is the value of $\rho_{X_1 X_2}$ given that $\hat{\beta}_1 = 2.4$ such that the model is reduced to
$Y_i = \alpha + \hat{\beta}_1 X_{1i} + e_i$

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RRohit

Last updated: June 22, 2026 at 14:25

-0.1981.

-0.2078.

0.2078.

0.1039.