What is collinearity and what to do with it? How to remove multicollinearity?

What is collinearity and what to do with it? How to remove multicollinearity?



Collinearity/Multicollinearity:
- In multiple regression: when two or more variables are highly correlated
- They provide redundant information
- In case of perfect multicollinearity: β=(XTX)−1XTyβ=(XTX)−1XTy doesn't exist, the design matrix isn't invertible
- It doesn't affect the model as a whole, doesn't bias results
- The standard errors of the regression coefficients of the affected variables tend to be large
- The test of hypothesis that the coefficient is equal to zero may lead to a failure to reject a false null hypothesis of no effect of the explanatory (Type II error)
- Leads to overfitting

Remove multicollinearity:
- Drop some of affected variables
- Principal component regression: gives uncorrelated predictors
- Combine the affected variables
- Ridge regression
- Partial least square regression

Detection of multicollinearity:
- Large changes in the individual coefficients when a predictor variable is added or deleted
- Insignificant regression coefficients for the affected predictors but a rejection of the joint
hypothesis that those coefficients are all zero (F-test)
- VIF: the ratio of variances of the coefficient when fitting the full model divided by the variance of the coefficient when fitted on its own
- rule of thumb: VIF>5VIF>5 indicates multicollinearity
- Correlation matrix, but correlation is a bivariate relationship whereas multicollinearity is multivariate

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