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I wonder whether one can judge strength of coupling between fluctuations of two time series by looking at correlation between residuals of ARIMA models for these two series.

Let's say I have two series, one provides daily air temperature and the second provides water temperatures of a river. Both series are strongly periodic and stationary. I fit, let's say, ARIMA(2, 0, 0) to both of them and both models are pretty good. Then I check that residuals of both models are also stationary and have no significant autocorrelations. And finally I correlate them using standard Pearson's $r$ and get a correlation coefficient of about 0.20. Can I say that random fluctuations of one series "explain" about 4% of random fluctuations of another series?

EDIT. The series was periodic (so no stationary). What I mean is that the ARIMA model residuals of both series were stationary.

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You have stated "both series are strongly periodic and stationary". I guess by "periodic" you meant seasonal.

If a series is seasonal, then it can not be stationary.

Your series and (probably) your residuals are therefore non-stationary and so you can not rely on the correlation coefficient at all.

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    $\begingroup$ That was quite a long time ago, but as fat as I recall I should have said that the residuals were stationary (constant mean and variance + no significant autocorrelations). I edited the question. $\endgroup$ – sztal Feb 14 '17 at 22:40
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Yes, you could look at the correlation between the two residual series, but you should also see if it is really significant. A correlation of 0.2 might well not be, without details (like length of the series) we cannot say. Better, you should look at the cross-correlation function between the residual series, for instance Cross-correlation significance in R or Do we need to detrend when do Cross-Correlation between two time series?.

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