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The Problem:

Given two time series, I want to determine if the (Pearson) correlation between the two time series is constant throughout time.

Example:

For example, I have the following two time series:enter image description here

Clearly, there is correlation between them. If I calculate the Pearson correlation coefficient I get $\rho = 0.960$. However, I want to test whether this correlation coefficient remains the same throughout time. In principle, the correlation of two time series could change through time right?

My current 'solution':

I partition the two time series into sub-time series of equal length. In this case the total length of the time series is $n \approx 2600$, so I split each time series into 52 time series of length 50. Then I calculate the Pearson coefficient for each of the 52 pairs. I then plot the coefficients:

enter image description here

Apart from the three earlier points, the correlation looks to be fairly constant. However, this method isn't very mathematical. Is there a more rigorous test that I can do to determine if the correlation remains constant?

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    $\begingroup$ Hi: You're dealing in a very dangerous-subtle area here because the correlation of two time series should only be defined after the series have been filtered so that the residuals are white noise. I can't explain this here ( too long and I wouldn't do it justice anyway ) but I would google for "pre-whitening a time series" or "transfer response" because what you're doing can lead to incorrect inference if not done correctly. If you google for those terms, you will see what I am referring to. Apologies for not providing all the details but it's quite involved. $\endgroup$ Commented Aug 6, 2020 at 16:29
  • $\begingroup$ This gives some idea of the issue. online.stat.psu.edu/stat510/lesson/9/9.1 $\endgroup$ Commented Aug 6, 2020 at 16:31

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Wouldn't you want to start with if they are cointegrated? If they are not I would think the correlation is besides the point (since there is no real relationship anyway in the long run).

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  • $\begingroup$ Hi. The OP is looking at correlations ( hopefully at various lags but he wasn't clear on that ) which is a very different thing from testing for cointegration. Cointegration tests whether there is a relationship between the levels of the series so that the spread between the levels tends to revert. That's not what I'm suggesting when I point to that link. $\endgroup$ Commented Aug 6, 2020 at 22:52
  • $\begingroup$ I understand. I was questioning if correlation made sense for time series. I think with data that moves over time the danger of spurious regression or changing relations in the time series is so high that it is dangerous to use correlations. But I am still learning time series so maybe this is not the case. $\endgroup$ Commented Aug 7, 2020 at 17:21
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    $\begingroup$ yes, the stability of correlation measurements can an issue.but a more funadamental issue is that any ccf analysis requires pre-whitening or a transfer response type approach. Also, note that there are many correlations between two series because there is one at each non-zero lag and zero lag so you should always specify what lag correlation you are referring to. I think you are referring to lag zero but it's better to be explicit. $\endgroup$ Commented Aug 8, 2020 at 10:16
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A slightly modified version of your question can be answered using the tools in the following paper: Aue, Alexander, et al. "Break detection in the covariance structure of multivariate time series models." (2009): 4046-4087. This paper can be found here: https://projecteuclid.org/journals/annals-of-statistics/volume-37/issue-6B/Break-detection-in-the-covariance-structure-of-multivariate-time-series/10.1214/09-AOS707.full.

Essentially this paper allows you to test for a changepoint in the covariance matrix of the two series.

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