Assessing Ergodicity Through Return and Volatility Stability
Summary
The note explains how annual means and variances of index returns can inform a discussion of stationarity and ergodicity. Annual sample means that vary widely do not by themselves disprove a stable long-run mean: the variation may reflect sampling noise. Annual variances that differ substantially raise a stronger question about whether volatility is constant, and the response suggests statistical testing to assess that possibility.
The key distinction is that rejecting a simple constant-mean, constant-variance model does not establish that the process is non-ergodic. A model with volatility switching between regimes can still be ergodic if its regimes and transition behavior can be learned from a sufficiently long trajectory. The annual plots are therefore suggestive rather than decisive: finite-sample charts cannot establish the strong-law conditions in the formal definition, and the proposed variance test does not settle whether a more complex model is ergodic.
Key ideas
- Annual return fluctuations alone do not establish that the mean is nonstationary.
- Large differences in annual variance warrant testing whether a constant-volatility assumption is plausible.
- Rejecting a constant-variance model does not rule out ergodicity under a richer stochastic-volatility model.
- Plots of finite historical samples provide evidence for discussion, not a proof of ergodicity.
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Full text
# A question about stationarity and ergodicity # A question about stationarity and ergodicity 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? ## Answer by Alex C (score 1) https://quant.stackexchange.com/a/41801 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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