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I am struggling with two examples where I wanted to identify whether it is a problem of perfect or rather imperfect multicollinearity. First, a variable "expenditure" is calculated by expenditure = age - education - 6, while age and education are also variables that are included in the regression. I would think that there is then perfect multicollinearity between expenditure, education and age as they are a linear combination of each other (inkl. -6). However in the solutions, it is stated that this is only a case of imperfect multicollinearity. However in another example where the variable "government expenditure G" is calculated by the sum of consumption and investment (G = C + I) we said that this is a case of perfect MC if consumption and investment are also included in the regression. I struggle to see the difference between these two cases. Any help is appreciated very much!

All the best!

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    $\begingroup$ The solutions are wrong (possibly a typo). Both are cases of perfect multicollinearity. $\endgroup$
    – smcc
    Jul 15, 2023 at 19:04

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Perfect multicollinearity means that one explanatory variable is a linear combination of other explanatory variables. Now, expenditure as given there is not a linear combination of age and education (it is an affine function of these variables, not a linear one). Indeed, there are no numbers $\alpha$ and $\lambda$ such that $$\text{expenditure}=\alpha\cdot\text{age}+\lambda\cdot\text{education}.$$ In that sense, there is no perfect multicollinearity. However, one usually adds to the explanatory variables a variable that always has the value $1$. Let's write it as $\mathbf{1}$. The regression coefficient of that explanatory variable would be your constant. If you have this constant included, as one usually does, expenditure can be expressed as a linear combination of age, education, and $\mathbf{1}$. Indeed,

$$\text{expenditure}=1\cdot\text{age} +(-1)\cdot\text{education}+(-6)\mathbf{1}.$$ So in the usual setting with a regression constant, there is perfect multicollinearity.

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