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I understand the basic difference in definition between multicollinearity and autocorrelation. I.e multicollinearity describes a linear relationship between whereas autocorrelation describes correlation of a variable with itself given a time lag.

When should I test for these as part of hypotheses testing? When fitting a model to a time series are the error terms tested for autocorrelation or multicollinearity? Why one over the other?

In a linear regression between Y and X with no time component, I suppose the answer is easy? We fit a linear model and test the residuals for multicollinearity and not autocorrelation because we are not considering time as a factor here. I am sorry for such a naive question.

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  • $\begingroup$ There was a question with almost exactly the same title. Did you search SE? $\endgroup$ – Aksakal Jan 5 '15 at 18:51
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    $\begingroup$ How does one "test residuals for multicollinearity"? What specifically do you mean by "multicollinearity" of a set of residuals (which is just a collection of numbers)? $\endgroup$ – whuber Jan 5 '15 at 19:21
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    $\begingroup$ It's an interesting question: when do residuals become collinear? When two variables are impacted by exactly the same shock. For instance, $y_t=t+\varepsilon$ and $z_t=t^2+\varepsilon$, teh same errors. $\endgroup$ – Aksakal Jan 5 '15 at 19:54
  • $\begingroup$ @Aksakal: Collinearity is a relationship among vectors, not a property of a set of numbers. So what specifically do you mean by "residuals become collinear"? Collinear with what? $\endgroup$ – whuber Jan 5 '15 at 21:35
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    $\begingroup$ @Victor, if it's univariate series, then mutlicollinearity is not even applicable here to residuals. It's for vectors, particularly, explanatory variable vectors. $\endgroup$ – Aksakal Jan 5 '15 at 22:01
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Multicollinearity can't even be defined unless you have multiple explanatory ("X") variables.

Your explanation doesn't suggest you fully understand these two terms. After all, correlation expresses collinearity. Auto-correlation merely means that you will find significant collinearity if you regress the dependent variable against itself with some lag.

Also, before you just start running tests you should assert a model (and hypothesis). I.e., ask yourself whether it is even reasonable to suggest that one explanatory variable is correlated with another, or that the predicted variable have some time interdependence with its own value.

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