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.

An annotated reading of rational expectations theory - Part 3

A final comment on Section 3 of Muth's paper

As discussed, this paper now introduces a simple market model (theory?) which is
\begin{align}
D_t &= - \beta P_t\\
S_t &= \gamma P_t^e + U_t\\
D_t &= S_t
\end{align}
where $D_t$ is the demand and $S_t$ is the supply, while $P_t^e$ is the expected market price at time $t$, conditioned on all previous information. Here all $P_t$ is a deviation from the equilibrium, so they are better thought as the amount expected to gain by producing and selling a unit.

The main ambiguity here is  whose expectation that is. It most natural to assume that this is the aggregate expectation of the population, i.e. that the amount produced during the $t$-th period is basically proportional to the price that people expect to get. It's better to make this into a formal assumption about the theory.

Assumption 3. The theory predicts that total production is linear with market expectations, i.e.
\[
P_t^e = \E_\Phi \E_p (P_t \mid x_t),
\]
where $x_t = (P_1, D_1, S_1, \ldots, P_{t-1}, D_{t-1}, S_{t-1})$ is the current information state, which may include some other side-information.

This assumption essentially tells us that the amount of supply is proportional to how much people expect to gain (since the quantities are deviations from the equilibrium) for the goods they produce. At first glance, this appears reasonable, but what decision model does this imply for the producers?

The $U_t$ variable can be taken to be simply zero mean noise in this context, and part of our model $M$. Let us now just equate everything, and write the quantities $U_t, P_t$ in terms of the state $\omega_t$ to obtain:
\begin{equation}
%\gamma P_t^e + U(\omega_t) = - \beta P(\omega_t),
 P(\omega_t) = - \frac{\gamma}{\beta} P_t^e - \frac{1}{\beta} U(\omega_t),
\end{equation}
noting that $P_t^e$ only directly depends on the history of observations and the priors of the population, and not any specific $\omega$.

What is the expected price according to this model? We can take expectations with respect to the model's distribution $M$, to obtain
\begin{align}
\E_M (P_t \mid x_t)
&= \int_{\Omega_t}  P(\omega_t) dM(\omega_t \mid x_t) \\
&=  - \frac{\gamma}{\beta} P_t^e  - \frac{1}{\beta} \int_{\Omega_t}  U(\omega_t) dM(\omega_t \mid x_t) \\
&=  - \frac{\gamma}{\beta} P_t^e  - \frac{1}{\beta}\E_M (U_t \mid x_t)
\end{align}
where $M(\omega_t \mid x_t)$ is the conditional distribution of the model given the information.

Uncorrelated deviations. The paper assumes that we can have $\E_M (U_t \mid x_t) = 0$ for any $x_t$ according to the model $M$. I am not sure how to actually interpret that. This seems to be the case if the actual supply by each provider is proportional to the market price it expects, plus some zero-mean noise due to externalities. This is not necessarily true, but the paper later relaxes this assumption.

Rationality assumption. This boils down to simply
\[
\E_M P_t = P_t^e,
\]
i.e. that the aggregate market prediction agrees with the model's expectation, which actually emplies that the expected price must be the equilibrium price, i.e. $P_t^e = 0$, recalling that quantities are equilibrium deviations. It is unclear whether this is the marginal expectation, or expectations conditioned on something. If the latter, then what?

Finally, can this assumption be relaxed somehow? Though we don't know what it is... precisely. 

Given all of the above assumptions, we can rewrite the  aggregate expectated prediction as
\[
P_t^e = - \frac{1}{\beta + \gamma} \E_M (U_t \mid x_t).
\]
However, I am not sure what this tells us. If we already know $M$, we also must know the beliefs of the agents, and so we already know $P_t^e$. So what is the point?

An annotated reading of rational expectations theory - Part 2

Whereby, having previously guessed a general model from the assertions of Muth's original paper, I now proceed to look at the paper itself.

To summarise, my own take on those assertions would be that they imply:

1. Each agent $i$ has his own prior belief $p_i(\omega)$ about the state of the world.
2. This prior belief is generated independently of those of other agents from some distribution $\Psi$.
3. This belief is conditioned on some information $x$, which is common to all; so each agent has a posterior belief $p_i(\omega \mid x)$.
4. The agents are interested in predicting some quantity $y$, which depends deterministically on the state of the world. Let us call this $f(\omega)$. 
5. Given the posterior $p_i(\omega \mid x)$, each agent can calculate a corresponding posterior distribution for the quantity itself. This can be done by firstly sampling $\omega$ from the posterior and then calculating $f(\omega)$. 
6. A more limited prediction is to just calculate the expectation of $f$ itself. For simplicity, let's call $f_i(x) = \E_{p_i} (f \mid x)$ the expected value of $f$ for the $i$-th agent, given information $x$.
7. There is some "theory" $\theta$ which predicts a particular value for $f$, let us call this $\theta(f, x)$
8. The prior distribution $\Psi$ is such that, for any information $x$, we have that $\theta(f, x) = \E_{\Psi} \E_{p_i} (f \mid x)$.

Section 3 of the paper discusses the problem of price fluctuations. Here we are specifically talking about some kind of time series, although that is not strictly speaking following from the discussion previously.

Here I immediately run into a problem. The paper presents a set of demand supply equations, where it assumes that the demand $D_t$ is equal to the supply $S_t$, i.e. $D_t = S_t$. There is also a market price $P_t$ for a single good, as well as a fluctuation term $U_t$. This is all fine. However, the supply is defined as
\[
S_t = \gamma P_t^e + U_t,
\]
where $P_t^e$ is defined to be "the market price expected to prevail during the t-th period on the basis of information available through the (t-1)'st period". I can't help but feel that this is quite imprecise, so let us turn this back into the basic framework I outlined above.

At time $t$, the state of the world is $\omega_t$. Agents are allowed to make inferences about the complete state of the world $\omega = (\omega_1, \ldots, \omega_t, \omega_{t+1}, \ldots)$.
I assume that the model that we have (again here I am not sure if that describes the actual system or our model of the system) is with respect to the aggregate expectation, so that if $P_t = f(\omega_t)$ then the first interpretation is this.

Interpretation 1. The "price expected to prevail" is the aggregate population expectation
\[
P_t^e = \E_{\Psi} \E_{p_i} (f \mid x).
\]
where $x = (S_1,ldots, S_{t-1}, D_1, \ldots, D_{t-1}, P_1, \ldots, P_{t-1})$ I am not quite sure that's what is intended, though. It is equally likely that the equations represent a "true" model of some sort.

Interpretation 2. The "price expected to prevail" is that of the "true" model
Here there is some other true model which places probability $M$ upon world states, and then
\[
P_t^e = \E_{M} (f \mid x).
\]

Not clear at all what is meant.

An annotated reading of rational expectations theory - Part 1

Following up on this post by Noah, I revisited the rational expectations literature. This is an attempt to do some annotated reading.

Muth's paper from 1961 opens by saying that "the expectations of firms ... tend to be distributed, for the same information set, about the prediction of the theory." 

How can we formalise this more precisely? Adopting a point of view where each economic agent $i$ has some prior belief $p_i$ over the set of possible states of the world $\Omega$, we need to define what we mean by expectations. Let $F$ be a function space of functions $f : \Omega \to Y$. For example, one particular $f$ could be the dollar price of gold per ounce for different world states $\omega$., To take expectations, we need to define a prior belief $p$ over the possible states of the world, with a posterior belief $p(\omega \mid x)$ given some information $x$. Combining those, the expected value of some $f \in F$, given information $x$ is:
\[
\E_p(f \mid x) = \int_\Omega f(\omega) d p(\omega \mid x).
\]
The assertion that economic agents' expectations are distributed about the prediction of the theory, can also be quantified. First we need to identify what we mean by prediction of the theory.

Definition 1. A theory $\theta$ is a function $\theta : F \times X \to Y$ where $F$ is the set of economic quantities we wish to measure, $X$ is the set of information states and $Y$ is the set of possible predictions.

Assumption 1. (a) Each economic agent $i$ has a prior belief $p_i$ drawn independently from a prior distribution $\Psi$. (b) For any information state $x \in X$, the corresponding set of posterior distributions $p_i(\omega \mid x)$ satisfies
\[
\E_{\Psi} \E_p(f \mid x) = \theta(f, x).
\]

This is a quite strong assumption. Note that the outer expectation is with respect to the population distribution of subjective beliefs while the inner one is with respect to the actual beliefs of each agent.
When the number of agents is large, then
\[
\frac{1}{N} \sum_{i=1}^N  \E_{p_i}(f \mid x) \approx \theta(f, x).
\]
I am not sure yet if that's what the paper is really about (I just scanned it once and now going through it) but I am tempted to make guesses. The actual paper never explicitly talks about the above, but that's what I am guessing it assumes behind the scenes. The remaining sections seem to be about a specific market problem, and then about very specific expectations/belief models for the agents.

Assumptions in section 2 of the paper

The paper concludes section 2 by making assumptions about:

1. normal random disturbances. I have no idea what those disturbances are supposed to be. Of what?

2.  Certainly equivalence. This probably means that the optimal investment choice depends only on the posterior expectation of $f$ and not on the complete posterior belief.

3. The system equations are linear. I have no idea how to interpret this.

(to be continued)