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Given daily returns of a stock index over 50+ years, a homework question asks:

Plot the annual sample mean and variances of the returns and their absolute values. Are these estimates in agreement with the assumption of an ergodic time series?

Having done so, I am unsure how to proceed. How do I judge whether my time series are ergodic based on the annual sample mean and variances of returns and their absolute values?

That seems like a strange question given the complicated definition of what it means to be ergodic. I know that a time series is ergodic if all "nice" functions of the time series satisfy the strong law of large numbers. How in the world would I be able to judge that by the plots I made?

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    $\begingroup$ Look at the variance of return during recession years (1960,1970,1974-1975,1980,1990,2001,2008-2009). Are these the same as in the other years? What do you conclude about the hypothesis that variance is constant over time (stationary)? $\endgroup$ – noob2 Sep 18 '18 at 15:58
  • $\begingroup$ I know how to judge stationarity, it's the ergodicity that has me confused. $\endgroup$ – Dasum Sep 18 '18 at 16:43
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    $\begingroup$ @noob2 Different conditional variance during recession years doesn't violate stationarity. I can easily write a stationary, ergodic, Markov regime switching model with two states and different conditional volatility in each state. $\endgroup$ – Matthew Gunn Sep 18 '18 at 16:56
  • $\begingroup$ Yes. We can reject a model where variance is constant, but we can accept a more complicated model where it varies. The parameters of this model might be stationary. $\endgroup$ – Alex C Sep 18 '18 at 17:01
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I think this should be handled as an essay type question, rather than a math problem. Ultimately you will conclude that "there is no reason why it cannot be ergodic" but you will have a few well written paragraphs before this to review some issues and show your understanding of the subtle concepts involved.

The intuition of ergodicity is that a statistical model is ergodic if constant parameters exist (stationarity) and they can be estimated from any single (random) trajectory observed a sufficiently long time. (Reasons for non-ergodicity include: non-stationarity, or the occurrence of pathological (trapped) trajectories that do not yield valid statistical information even if observed for a long time).

As a first step, to check the stationarity, we draw the charts mentioned. The average annual returns are all over the place, but they are not inconsistent with an average return of 9% a year and a standard deviation of 17 or 18% around this value. They are very volatile, but they are mostly in a 2 std dev band around the mean, so are not necessarily non-stationary. The annual variances are more problematic, and they are markedly higher in some years (eg. 2007-2008) than others. Statistical tests (Chi Square test) can be applied to check if these are explained as sampling error around a fixed variance.

If variances are not stationary, this raises the possibility that the process is non-ergodic. However all is not lost, as there are models in which variance can switch between a higher and a lower value according to a random Markov process; if the transition probabilities and $\sigma_H,\sigma_L$ are themselves ergodic (can be estimated from a historical sample) then the process is ergodic.

In conclusion, although a simple model of constant expected return and constant variance (which would be the simplest ergodic model) can be rejected, more sophisticated stochastic volatility models (such as Markov switching between two values of vol, a lower one during economic expansions and a higher one during recessions) could restore the assumption of ergodicity.

(Which is good news, because if the stock market is not ergodic then it would make no sense to study it quantitatively).

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