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.