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From what I have briefly read, seems like cointegration is used to determine if there is a statistically significant relationship between two unit root processes (as opposed to spurious correlation). If that's the case, would you say when you want to model two variables, you should probably do a cointegration test before you build a model - just to see if it's better to model the untransformed variables vs. transformed variables? In other words, the cointegration test should be one of many pre-modeling tests you do.

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I am trying to get a sense of what cointegration is used for and why not just go ahead and difference stuff.

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    $\begingroup$ There are two distinct questions in your post. I suggest you take the second one out and post it separately. $\endgroup$
    – Richard Hardy
    Jun 21, 2020 at 7:14
  • $\begingroup$ OK I will do that. $\endgroup$
    – confused
    Jun 21, 2020 at 15:08
  • $\begingroup$ Great. The picture no longer makes sense in the edited post (here), though. $\endgroup$
    – Richard Hardy
    Jun 21, 2020 at 15:32

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The most common and important use of cointegration is to decide whether a regression can be used at all or not. Suppose, you have $x$ and $y$ that are non stationary, like in your scribble. Both of them seem to be growing, so they must be nonstationary. In this case regressing $y\sim x$ is problematic, because it can produce a spurious regression. In this case, many would start differencing the variables, until they get stationary variables then only regress: $\Delta^k y\sim \Delta^m x$.

However, if x and y are cointegrated, then it's fine to regress $y\sim x$. That's why the cointegration test can be useful.

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    $\begingroup$ Aksakal said it well but another useful thing about cointegration ( and probably why they got the nobel prize ) is that people were uncomfortable just doing pure differencing because it threw out information about the levels. If you write the cointegrating relationship as an error-correction model, then you retain both how the two series are related in terms of their differences and also how they are related in terms of their levels. By levels, I mean the $ (y_{t-1} - \beta x_{t-1})$ term in the ecm. $\endgroup$
    – mlofton
    Jun 20, 2020 at 22:05
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    $\begingroup$ @confused: In practice, where "practice" means using it to trade pairs, I found it difficult because the cointegration relationships weren't stable. That's not to say that it can't be attempted but there are "news" factors that can turn co-integration into shmo-integration. diversification, position sizing and risk management are just as important if not more. $\endgroup$
    – mlofton
    Jun 20, 2020 at 22:10
  • $\begingroup$ @mlofton, I always hear stories of how cointegration suddenly breaks down on pair trading $\endgroup$
    – Aksakal
    Jun 20, 2020 at 22:16
  • $\begingroup$ yes, because whe company A gets bought by company B, neither company cares that company A used to be cointegrated with company C !!!! :). $\endgroup$
    – mlofton
    Jun 21, 2020 at 2:07
  • $\begingroup$ So basically it breaks down in the presence of other factors essentially. $\endgroup$
    – confused
    Jun 21, 2020 at 15:08

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