It seems intuitive to me that return time series would be ergodic. Is there a test statistic that I can use to check this? Would this be affected by sampling rate?

One way I can think of checking ergodicity is to plot the ACF and check if the autocorrelation decays with sufficiently large lag (which it will).

  • 1
    $\begingroup$ My fear is that someday there will be a big stock market crash from which we will never recover. That would be a failure of ergodicity. That it has never happened yet is no guarantee it might not happen in the future. But maybe I worry too much. $\endgroup$
    – noob2
    May 6 at 16:11
  • $\begingroup$ @noob2 quite a good point, especially as it is showing that any kind of model / assumption is specific to the space and time. $\endgroup$ May 6 at 16:59
  • $\begingroup$ Why would such a thing be intuitive? $\endgroup$ May 6 at 17:18
  • $\begingroup$ @rubikscube09 because it makes sense that autocorrelation decays with time which would make the series ergodic. $\endgroup$
    – s5s
    May 6 at 18:18
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    $\begingroup$ Ergodicity is a (to me) rather tricky property. It asks that average over all states is equal to average over time. It is generally imposed as an assumption, not tested empirically. We cannot see all states and observe the system for all time. $\endgroup$
    – noob2
    May 6 at 20:13

There is a proof that may impinge on this answer. If returns are r(P1,P2) and (P1,P2) follow a Brownian motion, then it is proven that planar Brownian motion is the only Brownian motion that is not ergodic.

It has the interesting property that any point reached $(\alpha,\beta)$ will be returned to an infinite number of times, but also that there will exist many holes that will never be reached.

I don't have the book where I am at, but I believe the proof is in Brownian Motion by Peter Morters and Yuval Peres. ISBN-13: 978-0521760188


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