Friday, December 31, 2010

Rational Expectations

I've recently read in a macroeconomics textbook a comment that the development of rational expectations models was necessary because other expectations models are ad hoc and not well grounded in theory. This is, as a historical matter, largely true of the models that Lucas critiqued 34 years ago; I'm not sure it's necessarily the case of any model that eschews expectations that are fully statistically accurate in the sense that "rational expectations" means to modern economists.*

By way of illustration, I was playing, a month ago, with a rational expectations model rather like the common modern Dynamic Stochastic General Equilibrium models; it linearized to a set of equations equating linear combinations of variables at a time with linear combinations of the agent-expected values of the same variables at the next time. As is the case in modern DSGE models, the coefficients of these linear equations were somewhat complicated functions of underlying parameters. Using rational expectations, agent-expected values were set to statistically-expected values, given the underlying parameters and the distribution of exogenous shocks. The linearized system, with this rational expectations assumption, could be fairly easily solved by finding stable and unstable modes and associating control variables with convergent expressions in expectations for the future, with state variables as convergent expressions in shocks from the past. This is all standard in modern macroeconomics.

Because state variables are expressed as linear combinations of one-period-before state variables plus shocks, however, it is the case in this model that the vector of state variables follows a VAR(1) process. If the state variables are directly observed, it's not remotely "ad hoc" for agents to form expectations based on a VAR(1) regression, especially if the relationship between the coefficients is such that, for any VAR(1) coefficient, there will be a set of underlying parameters that supports that coefficient. More generally, it would seem reasonable to expect that agents would infer the underlying parameters econometrically from past data. It is my understanding — though I am not perfectly clear on this — that rational expectations rejects this.

It may be that this is rejected because, with data going back far enough, agents in a model of this sort would have arbitrarily close estimates for the parameters. In this case, it's reasonable to note that the model, not being perfect or perfectly comprehensive, is, at best, an approximation that works well over finite periods of time, similar to what high-energy physicists would call an "effective-field theory". The underlying parameters may be robust to the Lucas critique, at least within a reasonable domain, and yet not perfectly stationary. It can be useful, even without bounded rationality, to suppose that expectations would be formed over a finite window, or one that weights more recent observations more heavily; with bounded rationality, of course, such an alteration to the model requires no other justification. In any case, asking agents to form expectations in a situation in which they are uncertain about the deep parameters of the model, and infer it only through the observation of macroeconomic variables, reintroduces nonlinearities that are very different from those that were linearized away in the first place, and it seems likely to me they would give behavior that would be interesting, whether or not it actually proved to fit the data better than the rational expectations models.

* I don't want to get sucked into recounting a full history of macroeconomics and macroeconometrics over the last 50 years; I will say that there are nice attributes of the assumption of rational expectations, and, as is so often the case, I tend to feel that its most ardent proponents understate its shortcomings, but its most ardent opponents underappreciate its benefits, and almost always fail both to appreciate its history and to actually understand the somewhat limited scope of what it is used to mean.

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