## Non-ergodic stationarity (wonkish)

19 May, 2017 at 08:21 | Posted in Economics | 4 CommentsLet’s say we have a stationary process. That does not guarantee that it is also ergodic. The long-run time average of a single output function of the stationary process may not converge to the expectation of the corresponding variables — and so the long-run time average may not equal the probabilistic (expectational) average. Say we have two coins, where coin A has a probability of 1/2 of coming up heads, and coin B has a probability of 1/4 of coming up heads. We pick either of these coins with a probability of 1/2 and then toss the chosen coin over and over again. Now let H1, H2, … be either one or zero as the coin comes up heads or tales. This process is obviously stationary, but the time averages — [H1 + … + Hn]/n — converges to 1/2 if coin A is chosen, and 1/4 if coin B is chosen. Both these time averages have a probability of 1/2 and so their expectational average is 1/2 x 1/2 + 1/2 x 1/4 = 3/8, which obviously is not equal to 1/2 or 1/4. The time averages depend on which coin you happen to choose, while the probabilistic (expectational) average is calculated for the whole “system” consisting of both coin A and coin B.

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How is this case related to non-ergodic?

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It may not converge to the expectation in the long-run since

Var(X̅) = σ²/n assuming equal variance σ² and uncorrelated variables.

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Thus, Var(X̅) ≠ 0 if n is finite, but σ²/(n-1) > σ²/n and Var(X̅) = 0 if n –> ∞.

Var(X̅) is approaching to 0. As long as Var(X̅) ≠ 0, it will not converge to the expectation (X̅).

Comment by Peiya Liu— 19 May, 2017 #

A random process — in mathematical statistics — is ergodic if the time average of a sequence of observations is the same as the time average of the entire phase space of the system for ‘long enough’ sequences. Check out e.g. Borovkov’s “Ergodicity and Stability of Stochastic Processes” if you’re rusty on ergodic theory!

Comment by Lars Syll— 19 May, 2017 #

Prof Syll, Thank you for your reference and some additional information.

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In general, economic accounting data are based on valid-time interpretation on time-series objects as logical assertions about real economies. They are not based on transaction-time interpretation as stochastic processes.

Comment by Peiya Liu— 22 May, 2017 #

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