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I am studying the time series and only kind of understand correlation vs cointegration. Can someone provide an example of two series that are correlated but not cointegrated, and two that are cointegrated but not correlated?

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Correlation is a statistical measure of how two variables, $X$ and $Y$, move in relation to each other. If the two variables are correlated, they move either in tandem, when they are positively correlated, or opposing directions, when negatively correlated. For example, the prices of equity stocks and fixed interest bonds often move in opposite directions; when investors sell stocks, they often use the proceeds to buy bonds and vice versa. In this case, stocks and bonds are (negatively) correlated, not cointegrated.
Cointegration only measures whether or not the distance between the two variables remains stable over time. It doesn’t say anything about the movement of $Y$ given a change, increase or decrease, in $X$. What this means is that $X$ and $Y$, provided they are cointegrated, cannot move in opposite directions for very long without coming back to an average distance eventually. Think about a person walking their dog; the entities don't necessarily move in a predictable manner/direction, but their distance remains more or less same (long run equilibrium) over time. This scenario is an example of cointegration, not correlation.

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    $\begingroup$ Some extra points: (1) cointegrated series will be correlated; (2) your description of cointegration allows $X$ and $Y$ to be stationary, which should not be; $X$ and $Y$ need to be integrated of order 1 or greater. $\endgroup$ Commented Jun 16, 2016 at 19:58
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    $\begingroup$ Richard, so there are no cointegrated pairs that are not correlated? Could you expand on that (or some kind of proof)? Thanks. $\endgroup$
    – user34829
    Commented Jun 23, 2016 at 1:21
  • $\begingroup$ Cointegrated series will be correlated because of the common trend, but might well be uncorrelated when the common trend is removed. But need not happen, there might be a contemporaneous correlation that is not removed by removing the trend. $\endgroup$ Commented Mar 9, 2019 at 17:06

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