# When are time series data exchangeable?

E.g. is a linear process exchangeable? What about strict vs weak stationarity?

EDIT: for clarity, I'm asking if/when the sequence of data points $(x_1, x_2, \ldots, x_n)$ in the time series is exchangeable. I'm mainly interested because I am not sure that I've fully understood the concepts of exchangeability and stationarity, and could not find any other questions that address this directly.

• Can you add a bit more information: are you asking about the exchangeability of the sequence of data points in one time-series? Or something else? (It might help to also briefly explain why you are asking this) – Juho Kokkala Jul 2 '18 at 13:28
• Here is a similar question with answer: stats.stackexchange.com/questions/353722/… – kjetil b halvorsen Jul 2 '18 at 17:09

Exchangeability of random variables $$X_1,X_2, \dotsc,X_t$$ means that all permutations of the random variables do have the same (multivariate) distribution. That is, let $$\pi \colon \{1,2,\dotsc,t\} \mapsto \{1,2,\dotsc,t\}$$ be a permutation of the index set. Then $$(X_{\pi 1}, X_{\pi 2}, \dotsc, X_{\pi t}) \stackrel{D}{=} (X_1, X_2, \dotsc, X_t)$$ for all permutations $$\pi$$. This clearly implies (assuming existence) that means and variances are constant, so implies stationarity, but is much stronger than stationarity.
For a time series this really rules out typical models of serial correlation, none of those will be exchangeable, since for example $$X_1$$ must have the same correlation with $$X_2$$ as with $$X_t$$. The only models consistent with exchangeability are equicorrelation models$$^\dagger$$: All pairwise correlations must have the same correlation $$\rho$$. That include independence models with $$\rho=0$$. But, in reality, with only one realization observed of the time series $$X_1,X_2, \dotsc,X_t$$, there is no observable difference between an exchangeable model and an independence model (there might be strange counterexamples with $$\rho=1$$).
$$^\dagger$$ Assuming correlations exist. Otherwise we can assume all pairwise joints are identical. A multivariate Cauchy distribution can be exchangeable, but without correlations.