Expectational stability: adaptive learning of RE equilibria in dynamic systems

Question

There are two steps in the explanation of the expectational stability concept by Evans and Honkapohja (2001) (see below) that I don't understand.

Step 1.

What does this formula below mean, intuitively, and why is the unique rational expectations solution the unique fixed point of the T-map. What is a mapping actually? It's formula 2.7, but, it's just an equation or relations or something special?

$$ \frac{d}{d \tau} \begin{pmatrix} a \\ b \\ \end{pmatrix} = T \begin{pmatrix} a \\ b \\ \end{pmatrix} - \begin{pmatrix} a \\ b \\ \end{pmatrix} $$

---

Step 2. They write the differential equation component by component

$$ \frac{da}{d \tau} = \mu+ (\alpha - 1)a $$

$$ \frac{db_i}{d \tau} = \delta_i + (\alpha - 1)b_i $$

I get how to arrive at this assuming I understand 2.8 (which I don't). What do these derivations mean: da/dtau? But, how do you define that the rational expectations solution is stable. They say: It follows that the REE is E-stable if and only if α <1. How can you see that?

Thanks!

---

The whole excerpt from Evans and Honkapohja (2001) can be read below.

To see all the less important math parts too, check the following google books link from page 30 on (section 2.4 and onwards): https://books.google.be/books?id=A7rWCwAAQBAJ&pg=PA40&hl=nl&source=gbs_selected_pages&cad=3#v=onepage&q&f=false

2.4 Expectational Stability

The condition α < 1 can be interpreted in terms of a general stability principle, known as “expectational stability” or “E-stability.” Since, as we will see, this principle works quite generally to provide the condition for the stability of an REE under adaptive learning, we introduce the concept now. The basic required concept is the map from the perceived law of motion (PLM) to the actual law of motion (ALM). The E-stability principle stated in its most comprehensive form is that the mapping from the PLM to the ALM governs the stability of equilibria under learning. More specifically, E-stability conditions obtained from this mapping provide the conditions for asymptotic stability of an REE under least squares learning. We focus here on obtaining this condition for the cobweb model. We begin with the assumption that agents have a PLM which they use to make forecasts of the variables of interest. Usually we take the form of the PLM to correspond to the REE of interest. Thus in the current case we take the PLMto be of the form

(2.2), pt = a+b

wt−1+ηt . F

For a = ¯a and b = ¯b, the PLMwould be the REE, but we allow for the possibility that agents have “nonrational” expectations. For any given values of a and b, the appropriate time-(t −1) forecast of pt is given by pe t = a +b wt−1. (2.5)

Inserting equation (2.5) into equation (2.1), one can solve for the actual law of motion, or ALM, implied by the PLM:

pt = (μ+αa)+(δ +αb)

wt−1 +ηt . (2.6)

This implicitly defines the mapping from the PLM to the ALM

(2.7)

$$ T \begin{pmatrix} a \\ b \\ \end{pmatrix} = T \begin{pmatrix} \mu+\alpha a \\ \delta + \alpha b \\ \end{pmatrix} $$

The interpretation of the ALM is that it describes the stochastic process followed by the economy if forecasts are made under the fixed rule given by the PLM. We can now define E-stability in the form appropriate for determining the stability of the REE under least squares learning. Note first that the unique REE for our model is the unique fixed point of the T -map (2.7). Consider the differential equation

(2.8)

$$ \frac{d}{d \tau} \begin{pmatrix} a \\ b \\ \end{pmatrix} = T \begin{pmatrix} a \\ b \\ \end{pmatrix} - \begin{pmatrix} a \\ b \\ \end{pmatrix} $$ (2.8)

where τ denotes “notional” or “artificial” time. We say that the REE is expectationally stable, or E-stable, if the REE is locally asymptotically stable under equation (2.8). Intuitively, E-stability determines the stability of the REE under a stylized learning rule in which the PLM parameters a and b are adjusted slowly in the direction of the implied ALM parameters. The REE ( ¯ a, ¯b

) is E-stable if small displacements from ( ¯ a, ¯b

) are returned to ( ¯ a, ¯b

) under this rule. Expectational stability in this form was introduced in Evans (1989) and Evans and Honkapohja (1992). The closely related notion of iterative expectational stability, which appeared earlier in the literature, will be discussed below. To determine E-stability in our example, combine equations (2.7) and (2.8) and write the differential equation component by component to obtain

$$ \frac{da}{d \tau} = \mu+ (\alpha - 1)a $$

$$ \frac{db_i}{d \tau} = \delta_i + (\alpha - 1)b_i $$

where n is the dimension of w. It follows that the REE is E-stable if and only if α <1. Note that this is precisely the condition obtained by Bray and Savin for convergence of least squares learning. The connection between E-stability and the convergence of least squares learning turns out to be quite general, applying in a very wide range of models. This is a great advantage since E-stability conditions are often easy to work out, while the technical analysis of the convergence of econometric learning is substantially more involved.

Answer 1

These are many questions. O.k., so let's go step by step:

(Q1) What is a mapping actually?

A map is just another term for a function. Here, every "law of motion", the actual one (ALM) and the perceived one (PLM), is characterized by its parameters $a$ and $b$. The ALM depends on the PLM, and the function mapping the PLM-parameters to the ALM-parameters is given by $T(.)$ in (2.7).

(Q2) Why is the unique rational expectations solution the unique fixed point of the $T$-map?

If expectations are rational, i.e. in an REE, the PLM is identical to the ALM, let's denote them by $\begin{pmatrix} a^* \\ b^* \end{pmatrix}$. This means that $T \begin{pmatrix} a^* \\ b^* \end{pmatrix} = \begin{pmatrix} a^* \\ b^* \end{pmatrix}$. Thus, $\begin{pmatrix} a^* \\ b^* \end{pmatrix}$ is a fixed point of $T$.

(Q3) What does this formula below [i.e., (2.8)] mean, intuitively?

If you start with a PLM that is not identical to the ALM, you "learn" the real law of motion over time $\tau$. That is, you slowly adjust the parameters of your PLM in the direction of those of the current ALM. In parameter space this means that your parameter point $\begin{pmatrix} a \\ b \end{pmatrix}$ moves towards the current ALM, which is given by $T \begin{pmatrix} a \\ b \end{pmatrix}$. Thus, the direction of movement (i.e. the velocity vector $\frac{d}{d\tau}\begin{pmatrix} a \\ b \end{pmatrix}$) in parameter space is given by the vector $T \begin{pmatrix} a \\ b \end{pmatrix} - \begin{pmatrix} a \\ b \end{pmatrix}$. Equation (2.8) just states this formally; it is the learning dynamics of the parameters.

(Q4): What do these derivations mean: da/dtau?

$\frac{da}{d\tau}$ is just the velocity of the parameter $a(\tau)$ in state space while learning.

(Q5): How do you define that the rational expectations solution is stable?

They define E-stability of the REE as local asymptotic stability of the REE under the learning dynamics (2.8).

(Q6): They say: It follows that the REE is E-stable if and only if α <1. How can you see that?

In general, a fixed point $X^*$ of a dynamics of the form $\frac{d}{d\tau}X(\tau)=F(X(\tau))$ is locally asymptotically stable if the linearization of the dynamics (i.e., the Jacobian matrix) at $X^*$ has all eigenvalues having negative real part. In the case given by (2.7) and (2.8), the Jacobian matrix is a simple diagonal matrix with diagonal elements $\alpha-1$, which are therefore also the eigenvalues. Those are negative if and only if $\alpha < 1$.

Intuitively, each component of the system of differential equations has the form $\frac{dx}{d\tau} = f(x)-x$. This is a simple 1-dimensional dynamics: A fixed point $x^*$ is locally asymptotically stable if $x(\tau)$ falls ($\frac{dx}{d\tau} < 0$, i.e. $f(x)<x$) when it is above $x^*$ and increases ($\frac{dx}{d\tau} > 0$, i.e. $f(x)>x$) if it is below $x^*$. This just means that at the fixed point $x^*$, $f(x)$ is locally decreasing in $x$, i.e. its slope is negative. In the case at hand, for each component this slope is $\alpha - 1$, so the slope being negative means $\alpha < 1$.