Saturday, November 20, 2010

Ergodic Theory: A Verbal Monte-Carlo...

An interesting verbal example, a verbal monte-carlo if you will based on a post by Robert Vienneau.  I suspect one of my macro profs Sevi will like about ergodic and non-ergodic processes goes as follows:

Suppose I observe the consumption sample paths of 10,000 individuals over 10,000 units of time.  Let us also now make the completely implausible assumption that these 10,000 consumption paths were generated by the sample process.  Suppose that I pluck one sample path and look at the distribution across time.  Now suppose that I pluck out the observation at t=350 from each of the 10,000 sample paths and look at the distribution.  If this process was ergodic, then these two distributions should converge to one another in large enough samples.

With ergodic processes, distributions across time and distributions across people should be the statistically the same (in large samples).  If the process is non-ergodic, then the distribution across people at a given moment in time and the distribution across time will not converge.  Sevi always comments about how summing (aggregating) across people, and summing across time are not always equivalent statements...is this the same as saying that economic processes in such cases are non-ergodic?  I don't know.

Now let's go back and relax the ridiculous assumption that all agents have the same process that determines there consumption.  With heterogeneous agents even if each agent individually is following an ergodic process, the aggregate distributions across agents and across time will be a mixture of ergodic processes and therefore must be non-ergodic (I think).

Whether or not the above mentioned processes are stationary or non-stationary and why is still a mystery to me.  In Vienneau's example, the process he used in his actual monte-carlo was stationary but non-ergodic.