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You have multicollinearity when you have 2 variables $(X_1, X_2)$ that have a relationship, $X_1=a+X_2$ where $a$ is constant. My question is: is there still a multicollinearity issue if $a$ is not constant, $X_1=X_3+X_2$? For example your testing children and you have their birthday and the date of the test, so date of test $-$ birthday$=$age at test. Could you use age at test and birthday in a regression model? I ask because there would be a perfect correlation for one test interval, but as you add test intervals the correlation decreases.

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Strictly speaking, whenever you can express one variable in terms of a linear combination of one or many other variables (which are included in your regression model), then you have perfect multicollinearity.

If you are unsure, you can always try it. Any good software will automatically exclude one variable from the regression when there is perfect multicollinearity, simply because it will be unable to produce coefficients in linear models otherwise (the design matrix won't have full rank, which means that it is not invertible). If the software doesn't exclude a variable when you think it should do so, then there is probably a data issue somewhere (miscoding or sth) and you should look into that.

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  • $\begingroup$ would there still be an issue if you only use 2 of the three variables? $\endgroup$ – Thomas Apr 30 '14 at 23:50

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