CHAPTER 10: MULTICOLLINEARITY: WHAT HAPPENS IF THE REGRESSORS ARE CORRELATED?

ü  The Nature of Multicollinearity

Multicollinearity – the existence of a “perfect”, or exact, linear relationship among some or all explanatory variables for a regression model.

Why does the classical linear regression model assume that there is no multicollinearity among the X’s?

If multicollinearity is perfect, the regression coefficients of the X variables are indeterminate and their standard errors are infinite. If multicollinearity is less than perfect, the regression coefficients, although determinate, possess large standard errors, which means the coefficient cannot be estimated with great passion or accuracy.

Sources of Multicollinearity

1.      The data collection method employed.

2.      Constraints on the model or in the population being sampled.

3.      Model specification.

4.      An overdetermined model.

ü  Practical Consequences of Multicollinearity

1.      Although BLUE, the OLS estimators have large variances and covariances, making precise estimation difficult.

2.      Because of consequence 1, the confidence intervals tend to be much wider, leading to the acceptance of the “zero null hypothesis”.

3.      Also because of consequence 1, the t ratio of one or more coefficients tend to be statistically insignificant.

4.      Although the t ratio of one or more coefficients is statistically insignificant, R², the overall measure of goodness of fit, can be very high.

5.      The OLS estimators and their standard errors can be sensitive to small changes in the data.

ü  Large Variances and Covariances of OLS Estimators

var(β₂) =

                  ∑X_2i(1 – r₂₃)

var(β₃) =

                  ∑X_3i(1 – r₂₃)

cov(β₂, β₃) =         -r₂₃

                        (1 – r₂₃)   ∑X_2i∑X_3i

ü  Detection of Multicollinearity

1.      High R² but few significant t ratios.

2.      High pair-wise correlations among regressors.

3.      Examinations of partial correlations.

4.      Auxiliary regressions.

5.      Eigenvalues and condition index.

CI =  Maximum eigenvalue =    k

         Minimum eigenvalue

6.      Tolerance and variance inflation factor

ü  Remedial Measures

·         Do Nothing

·         Rule-of-Thumb Procedures

1)      A priori information.

2)      Combining cross-sectional and time series data.

3)      Dropping a variable(s) and specification bias.

4)      Transformation of variables.

5)      Additional or new data.

6)      Reducing collinearity in polynomial regressions.

7)      Other methods of remedying multicollinearity.