# 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?

• 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)? – noob2 Sep 18 '18 at 15:58
• I know how to judge stationarity, it's the ergodicity that has me confused. – Dasum Sep 18 '18 at 16:43
• @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. – Matthew Gunn Sep 18 '18 at 16:56
• 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. – Alex C Sep 18 '18 at 17:01

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.