Variation Inflation Factor (VIF)
In a usual least squares regression analysis, the severity of Multicollinearity is quantified by the variance inflation factor (VIF). VIF provides an index which can measure the extent to which a variance of an estimated regression coefficient may get increased due to collinearity.
Variation Inflation Factor - Definition
How much the variance may be inflated is determined by Variance Inflation Factor (VIF).
For the estimated coefficient bk, VIFk —is the factor by which the variance is inflated.
Assume a model in which xk is the only predictor,
Y1 = ß0 + ßk xik + єi
The variance of the estimated coefficient bk will be,
The subscript "min" denotes the smallest the variance can be.
Assume a model with correlated predictors,
yi = ß0 + ß1xik +….+ßp-1xi.p-1 + єi
If a few predictors are correlated with the predictor xk, then the variance of bk will be considered as inflated.
The variance of bk will be,
= R2-value obtained due to regression of the kth predictor on the remaining predictors
The greater the linear dependence between the predictor xk and other predictors, the larger will be the value. The larger the value, the larger will be the variance of bk. How much larger? Let us take the ratio of the two variances.
This quantity shown above is the variance inflation factor for the kth predictor.
= R2-value obtained due to regression of the kth predictor on the remaining predictors.
Note: In a multiple regression model, a variance inflation factor exists for each of the k predictors.
Variance Inflation Factor - Calculation
There are 3 steps to calculate the Variance Inflation Factor. These have been given below.
To begin with, calculate k different VIFs, one for each Xi by applying an ordinary least square regression with Xi as a function of all the other explanatory variables.
If i = 1, the equation would be,
X1 = α2X2 + α3X3 + … + αkXk + c0 + e
Where, c0 = constant and e = error term.
Next, we calculate the VIF factor for ßi by using the given formula,
R2i = coefficient of determination of the regression equation with Xi on the left hand side, and all other predictor variables on the right.
Finally, we analyze the magnitude of Multicollinearity by judging the size of the VIP(ßi). If VIF (ßi)>5 then, Multicollinearity is high.
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