To calculate the Variance Inflation Factor (VIF) for $x_1$, we use the formula:

$\text{VIF}(x_1) = \frac{1}{1 - R_1^2}$

where $R_1^2$ is the coefficient of determination from the regression of $x_1$ on the other predictors (in this case, $x_2$).

In multiple linear regression, the VIF can also be directly calculated from the diagonal elements of the inverse of the $X'X$ matrix. The diagonal element corresponding to $x_1$ gives the necessary information for this.

Step-by-Step Approach

The inverse of the $X'X$ matrix is provided:

\begin{pmatrix} 25.0487 & -0.8457 & 0.2864 \\ -0.8457 & 0.0294 & -0.0104 \\ 0.2864 & -0.0104 & 0.0040 \end{pmatrix}$$ The diagonal element corresponding to $$x_1$$ is 0.0294. The VIF for $$x_1$$ is given by: $$\text{VIF}(x_1) = \frac{1}{\text{Diagonal element corresponding to } x_1}$$ Thus: $$\text{VIF}(x_1) = \frac{1}{0.0294} = 34.01$$ ### Final Answer: The Variance Inflation Factor for $$x_1$$ is approximately **34.01**. --- Would you like more details or any further clarification? ### Related Questions: 1. How is the Variance Inflation Factor interpreted in multiple regression? 2. What does a high VIF indicate about multicollinearity? 3. Can we compute the VIF for $$x_2$$ using the same method? 4. How is the $$R^2$$ value related to VIF in regression analysis? 5. What are the typical threshold values for VIF to signal multicollinearity? ### Tip: A VIF greater than 10 is often considered a sign of serious multicollinearity, which can distort the results of a regression model.