Suppose that you want to calculate the VIF of variable X1 in the following model:
Yi =β0 + β1 X1i + β2 X2i + β3 X3i + ε
and remember that the equation for calculating the VIF is the following: VIFj=1/(1-Rj2)
How is the Rj2 that is used for getting the VIF calculated?
To calculate the Rj2 used for getting the VIF (Variance Inflation Factor), you need to perform a separate regression analysis for each independent variable in your model.
In the case of calculating the VIF for variable X1, follow these steps:
1. Run a regression analysis with X1 as the dependent variable and all other independent variables (X2, X3, etc.) as predictors.
- The regression equation would be: X1 = β0 + β2 X2 + β3 X3 + ε.
2. Calculate the coefficient of determination (R-squared, denoted as Rj2) for the regression model. This value represents the proportion of the variance in X1 that is explained by the other independent variables (X2, X3, etc.).
3. Plug the value of Rj2 into the VIF formula: VIFj = 1 / (1 - Rj2), where j refers to the specific independent variable (in this case, X1).
The resulting VIFj value represents the extent of multicollinearity between X1 and the other independent variables in the model. A VIFj of 1 suggests no multicollinearity, while larger values indicate the presence of multicollinearity (with higher values indicating stronger multicollinearity).
The Rj2 used for calculating the VIF is obtained by regressing the predictor variable (Xj) of interest on all the other predictor variables (X1, X2, X3, etc.) in the model.
Specifically, the Rj2 is the coefficient of determination (also known as the squared multiple correlation coefficient) of the regression of Xj on all other predictors. It represents the proportion of variance in Xj that is explained by the other predictor variables in the model.
To calculate the Rj2, you would perform the following steps:
1. Regress Xj on all other predictors (X1, X2, X3, etc.) in the model by estimating the coefficients β1, β2, β3, and so on. The resulting regression equation would be of the form: Xj = γ0 + γ1X1 + γ2X2 + γ3X3 + ... + γnXn + εj, where γ0, γ1, γ2, γ3, and so on are the estimated coefficients.
2. Calculate the residual sum of squares (RSS) for this regression, which represents the unexplained variation in Xj after accounting for the other predictors.
3. Calculate the total sum of squares (TSS) for Xj, which represents the total variation in Xj.
4. Compute the coefficient of determination (Rj2) by taking the ratio of the explained variation (TSS - RSS) to the total variation: Rj2 = (TSS - RSS) / TSS.
The resulting Rj2 value is then used to calculate the VIF for variable Xj using the formula VIFj = 1 / (1 - Rj2).
To calculate the Rj2 (R-squared) value used to obtain the VIF (Variance Inflation Factor), you need to perform a linear regression for each independent variable in your model.
Here's how you can calculate the Rj2 for the variable X1:
1. Run a multiple linear regression, using all the independent variables (X1, X2, X3) to predict the dependent variable (Y).
2. Calculate the coefficient of determination or R-squared (R2) for this regression model. R2 represents the proportion of the variance in the dependent variable (Y) that can be explained by the independent variables (X1, X2, X3) in the model.
3. Exclude the variable X1 from the model and run another linear regression using the remaining independent variables (X2, X3) to predict Y.
4. Calculate the R-squared value for this new regression without X1.
5. Calculate the VIF (Variance Inflation Factor) for X1 using the formula VIFj = 1 / (1 - Rj2). In this case, Rj2 refers to the R-squared value obtained from Step 4.
The VIF measures the extent to which the variance of the estimated regression coefficient for X1 is inflated due to the correlation with other independent variables (X2, X3) in the model. A VIF value greater than 1 indicates a correlation between X1 and the other variables, and a higher VIF suggests that multicollinearity may be present.
Repeat this process for each independent variable in your model to calculate the VIFs for all variables.