Monday, March 03, 2008

Goodbye to "Rational Expectations"

Consider an economic model in which the agents within the model act on decisions based on their understanding of the model. For ease of exposition, assume the models within the heads of the agents all have the same form, and that that form matches the actual model. The agents must estimate the parameters of the model.

Suppose the agents have made some estimate of the model parameter. Their decisions result in the parameters being set in the actual model. And the agents use the data generated from the actual model to make their estimates. A rational expectations equilibrium is said to result when the agents' estimates match the model parameters. A rational expectations equilibrium can be thought of as a fixed point of a function from the agents' estimated parameters to the actual parameters.

Rational estimations is often applied to models of economic time series considered as stochastic processes. An important parameter for a stochastic process is the population mean at a given point in time. One can conceptually describe two types of sample means for a stochastic process:
  • At a single point in time across many realizations of a stochastic process
  • Across time samples for a single realization of a stochastic process
If and only if a stochastic process is ergodic, these two types of sample means converge as the number of realizations and the number of time samples increase.

Some stochastic processes observed in real world economies are non-stationary, for example, if they have a component growing at a constant rate. Non-stationary is sufficient for non-ergodicity, but not necessary. (For an example of a non-ergodic stationary process, consider a Spherically Invariant Random Process (SIRP).) Hence, some real-world processes are non-ergodic.

Agents only have access to a single realization of some processes. They therefore cannot form a sample spatial average for such a process. They only can take statistics, such as a time average, for a time series. And, if that process is non-ergodic, such a sample average will have no tendency to converge to the true model parameter, which is an average across the population of all realizations.

So much for "rational expectations".

Reference
  • Paul Davidson (1982-1983). "Rational Expectations: A Fallacious Foundation for Studying Crucial Decision-Making Processes", Journal of Post Keynesian Economics, 5 (Winter): 182-197.

4 comments:

YouNotSneaky! said...

True, true. But then that still leaves the question of how to model expectations. Adaptive expectations are equally undesirable being the opposite extreme case.

Gabriel Mihalache said...

R.E. is not about learning.

R.E. assumes that all agents know the true structure of the economy and its structural parameters, including distributions' parameters.

Sure, that's a tough restriction, but so if the one where each commodity can only be produced by a single activity, for example.

YouNotSneaky! said...

R.E. could be about learning. If you don't want to assume that agents already know the model you could postulate that they learn the "true" model through repeated interactions in/with the market over time. But if the economic processes are non-stationary then no learning can occur.

So here you can sort of order the assumptions in terms of how restrictive they are:
1. Agents know the model, have RE - very restrictive. Model is fairly straightforward.
2. Agents don't know the model, but the processes are stationary, so the model can be learned over time - still pretty restrictive but less so. Model becomes a lot more complicated but can still be handled by presently available mathematical tools.
3. Agents don't know the model and the processes are non-stationary so the model cannot be learned. Much closer to reality. But pretty much cannot be modeled at all since it leaves the question 'how are expectations' completely open.

13.72. The agents always expect it to be 13.72. Or the agents always expect the future to be exactly like the past. These are also clearly very unrealistic assumptions (and each one makes modeling easy). Are they more or less unrealistic than 1) and 2) above? A comparison with 3) isn't meaningful.

Robert Vienneau said...

Assumption (3) is a challenge. But I can cite plenty of economists. For example, Paul Davidson, Johnd Maynard Keynes, Jan Kregel (e.g., his 1976 Economic Journal article), and G. L. S. Shackle. Brian Arthur and Paul David have done some interesting work. And what about Kirman's work with agent-based modeling?