# What does that mean that two time series are colinear?

I am familiar with the concept of cointegration.

But I hear sometimes people talking about colinearity (or collinearity) for time series. A set of points is collinear if they are on the same line. But what does that mean for time series?

Is it exactly the same as cointegration of order 1? Or is there something stronger/different in the concept of collinearity?

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The wikipedia gives a good example:

Two variables are perfectly collinear if there is an exact linear relationship between the two. For example, $X_1$ and $X_2$ are perfectly collinear if there exist parameters $\lambda_0$ and $\lambda_1$ such that, for all observations $i$, we have

$$X_{2i}=\lambda_0+\lambda_1X_{1i}$$

This applies directly to time series, just change $i$ to $t$.

Two times series $X_1$ and $X_2$ are cointegrated of order 1 if they are

1. Integrated of order 1, meaning that first differences of $X_1$ and $X_2$ are stationary processes.

2. There exists parameters $\alpha_1$ and $\alpha_2$ such that linear combination

$$\alpha_1X_{1t}+\alpha_2X_{2t}$$

is a stationary process.

The answer to your second question can be yes. The colinearity is in a sense stronger, i.e. more restricting property. From statistical point of view colinear time series are the same, i.e. if you know distributional properties of one series you immediately know the distributional properties of the other. In fact if your data have time series which are perfectly colinear it usually means that one of the series was artificialy created, for example: profit equals income minus expenses.

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I guess that people may extend the colinearity wikipedia definition for time series by adding a noise: $$X_{2t}=\lambda_0+\lambda_1X_{1t}+e_t$$ Then $$X_{2t}-\lambda_1X_{1t}=\lambda_0+e_t$$ which is stationary. If $X_1$ and $X_2$ are $I(1)$, you have the definition of cointegration of order 1 ... Do you think this makes sense? That is actually precisely the purpose of my question. –  RockScience Jul 30 '12 at 3:34
If you add noise, you lose colinearity. The definitions usually serve some purpose, I see no benefits of extending colinearity definition specifically for time series. –  mpiktas Jul 31 '12 at 12:50
Sure, I agree we should follow the definitions. But it seems that here everybody do not agree on the definition. It is not unusual to see slightly different definitions in different context (that's the reason to start a paper with a list of conventions and definitions). I am just trying to understand the street interpretation and not just wikipedia's opinion. –  RockScience Oct 31 at 3:42
I can assure you that in this case, street intepretation coincides with wikipedia. –  mpiktas Oct 31 at 7:20