An annotated reading of rational expectations - Conclusion: Where a sleight of hand strengthens the assumptions

Rational agents, or oracles? Concludingf remarks on rational expectations

So far, dear reader, I had been reading the theory of rational expectations under the assumptions I laid out in my first post about this. Most important of those is that agents are rational with respect to their own subjective beliefs. This is what I'd call the classic decision theory assumption about rational agents.

However, economics appears to assign a distinctly different meaning to these words. In particular, Manski, a researcher in econometrics, claims that rational expectations mean that agents' subjective beliefs match the actual probability distribution from which Nature generates events. Is that claim true? 

From my reading of Muth's, it's certainly ambiguous. The central assumption made in the beginning is that merely their aggregate belief matches Nature. However, later on the paper develops a notational obscurity that makes it hard to distinguish about whose expectations we are talking about, and what knowledge each actor has. This includes:

1. The knowledge of each economic agent
2. The knowledge of the person writing up the model (i.e. how much the model corresponds to reality).
3. What the process itself depends on.

Nevertheless, my reading of Manski implies that the rational expectations economics subgroup has actually taken the idea to extremes. For them, "rational agents" are what I call "oracles", because they know everything about the system.

Muth makes things difficult by re-using assumptions without clearly stating them, but his own assumptions are somewhat weaker than that. I will go over this in the next section. 

Correlated deviations.

As the uncorrelated deviations is rather unrealistic, the paper then discusses correlations. In particular, it uses the model of disturbances $\epsilon_k$ to define the disturbance $U_t$ at time $t$:

\begin{equation}
U_t = \sum_{i=0}^\infty w_i \epsilon_{t-i},
\end{equation}

where $\epsilon_k$ are i.i.d standard normal (and so the $\epsilon$ vector has diagonal covariance matrix).

The price is now also a random variable, depending on another parameter vector $v$, so that
\begin{equation}
P_t = \sum_{i=0}^\infty v_i \epsilon_{t-i}
\end{equation}

Now the paper talks about $P_t^e$, the aggregate expected value, which it claims is a linear function of the $\epsilon_{t-i}$ for $i > 1$. However, if the agents don't observe those directly, this is impossible.

I am flummoxed. I believe that this is just clumsy notation and actual expectation of the real process with respect to the filtration generated by $\epsilon_1, \ldots, \epsilon_t$ is meant, and not the aggregate belief!

 In particular, since the agents can't observe $\epsilon$, it must simply denote the expected value of the process given all the values of $\epsilon$ to time $t-1$. So this is another example of the conflation of what information is used within the process itself and what is known by the agents.

Arguably, one could say that if all all the agents observed the previous $\epsilon$ values, then their aggregate conclusion would be equal to the expected value of the process. This is still a point that needs to be spelled out. However, the asymmetry of information between the agents and the process generating the data is critical. At this point, it is still not clear whether the assumption is that given the publicly available information, agents' aggregate predictions are identical to the expected value of the process, even if the latter has hidden state.

Now we can try and plug in the process in the equilibrium model. 

But the way Muth does it just assumes that the expectation of the market will include the sequence of all hidden information $\epsilon_1, \ldots, \epsilon_{t-1}$, as previously discussed. So, in some sense, doing so is not very useful.

Muth's solution is then to first recognise we have to write $P_t^e$ in terms of observables as $P_t^e = \sum_{j=1}^\infty z_j P_{t-j}$. But how does he actually do that? The algebra is straightforward, but what is plugged into what? Muth uses the much stronger "rationality" assumption that \[ \E_M (P_t \mid \epsilon_1, \ldots, \epsilon_{t-1}) = P_t^e\]  to obtain the final result.

This is competely unconvincing. The information that the disturbances $\epsilon$ might carry can be a lot, and the agents should not know it. Thus, from that point on, Muth is subscribing to a very "oracle"-like definition of rationality. The lack of discussion surrounding the trivial algebra, couple with ambiguous notation, creates such a confusion in concepts [who knows what, whose expectations we are talking about], that the paper's claims that its assumptions are mild cannot be supported at all.

The final part of this blog was going to talk about the deviations from rationality that Muth discusses, but I am not sure I have the heart to go into it right now.