Multicollinearity

     Multicollinearity is the violation of classical assumption that no independent variable is a linear function of one or more other independent variables. Multicollinearity can be either perfect or imperfect.

Perfect versus Imperfect Multicollinearity

  Perfect Multicollinearity

      Perfect multicollinearity violates the classical assumption that no explanatory variable is perfectly linear function of any other explanatory variables. the word perfect in this context implies that the changes in one explanatory variable can be completely explained by movements in another explanatory variable. such a perfect linear relationship between two independent variable would be

     Xi = a0 + a1 X2i                        (1)

 Where the 'a' s are constants and the Xs are independent variables in 

     Yi = b0 + b1X1i + b2X2i + ui.  (2)

Notice that there is no error term in equation 1. This implies that X1 can be exactly calculated given X2 and the equation. Examples of such a perfect linear relationship would be

     X1i = 3X2i.                  (3)

     X1i = 2 + 4X2i.            (4)

When there is perfect multicollinearity, OLS is in capable of generating estimates of the regression coefficients, and most OLS computer programs will print out an error message in such situations. Thus perfect multicollinearity ruins our ability to estimate the coefficients because that two variables cannot be distinguished. 

      

      Fortunately instances in which one explanatory variable is a perfect linear function of another are there. Besides perfect multicollinearity should be fairly easy to discover before a regression is run. If identified then we can drop the variable and run the regression as there is no essential difference between the two.

   

Imperfect Multicollinearity

    Sims perfect multicollinearity is fairly easy to avoid economic relations almost never talk about it. Instead when we use the word multicollinearity we really are talking about several I'm perfect multicollinearity. imperfect multicollinearity can be defined as a linear functional relationship between the two or more independent variables that is so strong that it can significantly affect the estimation of the coefficients of the variables.

     In other words imperfect multicollinearity occurs when two explanatory variables are imperfectly linearly related as in.

    X1i = a0 + a1x2i + ui.           (5)

     Comparing the first equation and the equation (5) includes a ,  stochastic error term. This implies that all the the relationship between X1 and X2 might be fairly strong (it is strong enough to Alo x12 be completely explained by X2) some unexplained and variables still remain.

    imperfect multicollinearity is a strong linear relationship between the explanatory variables. the stronger the relationship between the two explanatory variables, the mire likely it is that they will be considered significantly multicollinear.