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Depends how you calculating correlation, but probably you have rolling window from what you get high and low for calculation, when you adding samples to window then some samples must exit the window, when sample that exiting are not equal to high and low then it's don't matter, but when high or low is exiting then suddenly everything changes in your ...

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I would say that you can use Johansens methods to test for rank of co-integration matrix. There are tests for that. If there is no co-integration vector present and both series are I(0) then there is no co-integration. Series still might have some short-run dynamics. If series are I(1) and no con-integration vector is present then modeling these series by ...

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My opinion is that using rolling correlations of returns which themselves are computed over rolling windows is not reliable. Taking rolling windows smothers information. Instead, I would specify a simple EWMA filter for the variances and the covariance, which would give me a value for the spot correlation. For example something like  \begin{align} ...

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I don't think there is a correct answer to this question. If you're trying to study short-term correlations (e.g., to construct short-term trading signals), then 1-month or 3-month rolling correlation of daily returns is a feasible option. These short-term stock/bond correlations are quite unstable though. On the other hand, if you're studying long term ...

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