Financial Risk Manager Part 1

The variance inflation factor (VIF) for a variable $X_j$ in a regression model is given by:

$\text{VIF}_j = \frac{1}{1 - R_j^2}$

where $R_j^2$ is the coefficient of determination from regressing $X_j$ on all other independent variables. In this case with only two independent variables $X_1$ and $X_2$, $R_j^2$ is simply the squared correlation coefficient between $X_1$ and $X_2$, i.e., $R_j^2 = \rho_{X_1 X_2}^2$.

Given that VIF = 15:

$\frac{1}{1 - \rho_{X_1 X_2}^2} = 15$

$`$1` - \rho_{X_1 X_2}^2 = \frac{1}{15}$$

$\rho_{X_1 X_2}^2 = 1 - \frac{1}{15} = \frac{14}{15}$

$\rho_{X_1 X_2} = \sqrt{\frac{14}{15}} = \sqrt{0.9333...} \approx 0.9661$

Therefore, the correlation coefficient between $X_1$ and $X_2$ that yields a VIF of 15 is approximately 0.9661.