Question

Consider the following multiple linear regression modely=β0 +β1x1 +···+βkxk +ε.

(a) What is multicollinearity?

(b) How can multicollinearity be detected?

(c) What effect does multicollinearity have on your ability to make inferences about the coef- ficients?

Answer 1

a)

Multicollinearity occurs when independent variables in a regression model are correlated. This correlation is a problem because independent variables should be independent. If the degree of correlation between variables is high enough, it can cause problems when you fit the model and interpret the results.

b)

1. Review the correlation matrix and find the correlations between the independent variables. If the correlation between two independent variables is > 0.70 then the multicollinearity exists.

2. Calculate the variance inflation factor( VIF). if it is > 5 for two variables then he multicollinearity exists.

3. Look for the instability of coefficients of regression. If the coefficient has unstable or different than the theoretical the multicollinearity can exist.

C)

Effect of multicollinearity has on your ability to make inferences about the coefficients.

1. The coefficient estimates can swing wildly based on which other independent variables are in the model. The coefficients become very sensitive to small changes in the model.

2. Multicollinearity reduces the precision of the estimate coefficients, which weakens the statistical power of your regression model. You might not be able to trust the p-values to identify independent variables that are statistically significant.

3. Multicollinearity has no impact on the overall regression model and associated statistics such as R2, F ratios and p values. It also should not generally have an impact on predictions made using the overall model. (The latter might not be true if the predictor correlations in the sample don’t reflect the correlations in the situation you are making predictions for – but that isn’t really a multicollinearity issue, but a consequence of having an unrepresentative sample).

4. Multicollinearity is a problem if you are interested in the effects of individual predictors. Multicollinearity therefore reduces the effective amount of information available to assess the unique effects of a predictor. The fundamental statistical impact of multicollinearity is to reduce effective sample size and thus statistical power for estimates of individual predictors.