Learning of rational expectations using a neural network

called recycling and the number of such repeated training samples is called the number of cycles (or epochs).

It is possible to use the backpropagation algorithm for networks with several output layers and networks with several hidden layers. For instance, if additional layers are added to the approximation function, then errors are ‘propagated’ from layer to layer by repeated application of generalized delta rule.

It should be noted that a neural network model can be identified as a pursuit projection regression (PPR) model (Hastie et al, 2001). In fact, the neural network with one hidden layer has the exactly the same form as the PPR model. The only difference is that the PPR model uses nonparametric functions (

g v

_m( )) while the neural network employs a simpler function which is based on a sigmoid transfer function.

If

α

≠1, there exists a unique rational expectation of

p

_t which is given by the rational expectation function

ϕ ( ) x

_t . If agents do not know the reduced form of the model and the form of

h x

( ), rational expectation may not be reachable. However, they may learn to form RE using the past values of

p

_tand

x

_t. In other words, it is assumed that agents have an auxiliary model showing the relationship between the exogenous variables (

x

_t) and the endogenous variable (

p

_t).

If

h x

( )is linear in

x

_t, the reduced form (1) becomes the linear model

p

_t =

α p

_t^e +

β x

_t +

ε

_t, where

β

is a vector of parameters. If it is assumed agents use the auxiliary model

p

=

δ

^'

x

where

δ

are estimated using recursive least squares, the following results hold (Bray and Savin, 1986; Marcet and Sargent, 1989) (a) If the estimator

δ

ˆfor

δ

converges, this results in rational expectations, i.e.

ˆ '

1 δ β

= α

− .

(b) The estimator for

δ

will converge towards

'

1 β

α

− if and only ifα<1.

If the function

h x

( )is not linear,

ϕ ( ) x

_t is not linear too. In such cases, agents, having no prior knowledge about the functional form of

ϕ ( ) x

_t , may use an auxiliary model such as neural networks which is flexible enough to approximate the rational expectation function

ϕ

( )

x

_t .

The following equations describe the neural network by mapping inputs

x

_jto the output

y

as

_{,0 ,}

1 k

i i i j j

j

n w w x

=

= +

∑

S

_i

= L n ( )

_i ¹ 1

e

⁻ⁿⁱ

= +

₀

1 m

i i i

y q q S

=

= +

∑

=

f x

( , )

θ

, (3)

where

x

^' =( ,..., )

x

₁

x

_k ,

θ

^' =( , ,

q q w

_{0 1 1,0},...,

w q

_1,k, ,...,₂

w

_{m k,} )^and

L n

( )_i shows the log-sigmoid transfer function. A linear combination of the input variables

x

_j, with the coefficient vectors

w

_{i j,} , as well as the constant term,w_i,0, form the variable

n

_i. This variable is squashed by log-sigmoid function, and becomes a neuron

S

_i. The set of m neurons are combined in a linear way with the coefficient vector

q

_i, and taken with a constant term

q

₀ to forecast y.

The model with one layer of hidden units and log-sigmoid transfer function is able to approximate any continuous function if a sufficient number of hidden units are used (Hornik, 1989). The interesting feature of neural networks is their ability to learn.

Therefore, there exists a neural network and a vector of parameters

θ

^*such that

ϕ

( )

x

_t =

f x

( , )

θ

^* . However, since the exact number of hidden units required to obtain a perfect approximation is not known with certainty, a perfect approximation of rational expectation function

ϕ ( ) x

_t can not be guaranteed.

Objectives of learning

Assume agents use the neural network of the form (3) as an auxiliary model. If the expectation of

p

which is given by

p

^e =

f x

( , )

θ

is found to be incorrect, agents will improve the predictive power of their model by changing the values of parameters.

This process, in fact, is called learning.

The mean squared error (MSE) of expectations is a measure for success of learning. It is defined as the expected value of the squared deviation of the agents’

expectation

p

^e =

f x

( , )

θ

from its actual value

p

=

α f x

( , )

θ

+

g x

( )+

ε

. Denoting this MSE as

λ

_θwe obtain

[

^{( , ) ( ) ( , )}

]

^{2 (1 )2 ( ) ( , ) 2}

1

E f x g x f x E g x f x

θ

λ α θ ε θ α ε θ

α

⎡ + ⎤

= + + − = − ⎢⎣ − − ⎥⎦

2

(1 )2 ( ) ( , )

E x

1

ε f x

α ϕ θ

α

⎡ ⎤

= − ⎢⎣ + − − ⎥⎦ (4) The optimal vector of parameters

θ

^*is achieved by minimizing

λ

_θ with respect to

θ

θ

^* =arg min ( )

λ θ

(5)

Using ∇_θ for the gradient vector of

λ θ

( ), the necessary condition for this problem can be written as

( ) 2(1 )² ( , ) ( ) ( , ) 0

E f x x 1 f x

θ θ

λ θ α θ ϕ ε θ

α

⎧ ⎡ ⎤⎫

∇ = − − ⎨⎩∇ ⎢⎣ + − − ⎥⎦⎬⎭= (6) It is clear that equation (6) may have a multiple solutions. In case of existence of a solution

θ

satisfying the necessary condition, a (local) minimum of MSE is obtained if the Jacobian matrix

J

_λ( )

θ

is positive semidefinite.

^J

λ

^{( )}

θ²

^{( ) E}

θ²

^{f x ( , ) ( ) x} ₁ ^{f x ( , ) E} {

θ

^{f x ( , )}

θ

^{f x ( , )}

^'

}

θ λ θ θ ϕ ε θ θ θ

α

⎧ ⎡ ⎤ ⎫

= ∇ = − ∇ ⎨ ⎩ ⎢ ⎣ − − ⎥ ⎦ ⎬ ⎭ + ∇ ∇

⁽⁷⁾

A (local) minimum at

θ

^*is (locally) identified if

J

_λ( )

θ

^* is positive definite. Otherwise, at least one eigenvalue of

J

_λ( )

θ

^* is equal to zero, such that the minimum is not (locally) identified.

Now consider the set Θ^Lwhich includes all vectors of parameters for neural network implying a (local) minimum of MSE

Θ^L={

θ

∈

R

^q|∇_θ

λ θ

( ) 0,=

J

_λ( )

θ

is positive semidefinte}

If a neural network can perfectly approximate the unknown rational expectation function

ϕ

( )

x

, there exist vectors of parameters implying

λ θ

( )=

σ

_ε². Since all hidden units in the neural network stated here employ identical activation functions, there will be no unique vector having this property. To remove this problem, let

ΘG= {

θ

∈

R

^q| ( )

λ θ

=

σ

_ε²} denote the set of all these vectors of parameters. Any

θ

∈ΘGimplies that the expectations formation using the neural network model and rational expectation function

ϕ

( )

x

are identical. This is not true for the remaining

θ

∈Θ Θ^L \ ^G: All

θ

result in (local) minima of the

λ θ

( ), but they do not imply

ϕ

( )

x

=

f

( , )

θ x

. These vectors of parameters result in approximate unknown rational expectation functions and the resulting equilbria are called rational expectation equilibria. (Sargent 1993).

Learnability of the rational expectations

Learning implies that agents estimate the parameters of the neural network model

the agents can learn to form rational expectations or equivalently whether there will result asymptotically correct parameters values. Do the estimated parameter vectors converge to a

θ

∈Θ^G or at least to a

θ

∈Θ^L?

Substituting the expectation function

p

^e =

f x

( , )_t

θ

_t into the reduced form (1), we get the actual value of endogenous variable

p

_t =

α f x

( , )_t

θ

_t +

h x

( )_t +

ε

_t.

If

f x ( , )

_t

θ

_t

≠ ϕ ( ) x

, the agents’ expectation turns out to be incorrect and

p

_tdiverges from the rational expectation equilibrium. Assume the learning algorithm used by agents is the ‘backpropagation’ algorithm. It changes the vector of parameters

θ

taccording to the product of the actual expectation error ( , )

e

t t t t

p

−

p

=

p

−

f x θ

and the gradient of the neural network with respect to

θ

_t:

θ

_t+1= +

θ γ

_{t t+}1

[

∇_θ

f x

( , )(_t

θ

_t

p

_t −

f x

( , )_t

θ

_t

]

(8) Here

γ

_tis a declining learning rule that satisfies

γ

_t =

t

^−k, 0< ≤

k

1. It implies that changes of

θ

_tbecomes smaller over time and this helps us to answer the question whether agents will asymptotically learn to form (approximate) rational expectations or equivalently whether

θ

_t converge to a

θ

∈Θ^L. Since the analysis of the stochastic difference equation (8) is difficult, we follow Ljung (1977) in approximating

θ

_tusing the differential equation

θ τ

&( )=

Q

( ( ))

θ τ

, (9) where

^{Q( )}

θ

=^E

{

∇θ ^{f x( , )}

θ [

^p− ^{f x( , )}

θ ] }

= ^E

{

∇θ ^{f x( , ) ( )}

θ [

^{g x} + − −

ε

⁽¹

α

^{) ( , )f x}

θ ] }

As equation (9) is a deterministic differential equation, all conclusions resulting from (9) about the stochastic difference equation (8) are valid in a probabilistic sense. In other words, the time path of

θ

_taccording to (8) is asymptotically equivalent to the trajectories of

θ

resulting from (9). This means that for t→∞,

θ

_tfrom (8) will- if ever- converge only to stationary points of (9) which are (locally) stable. It will not converge to stationary points that are unstable.

Analyzing the asymptotic properties of the learning algorithm (8) requires to examine the stationary points of the differential equation (9). Since α is constant,

Q

( )

θ

can be written as

^Q

^{( )}

θ

=

^E {

∇θ

^{f x}

^{( , ) ( )}

θ [ ^{g x}

+ − −

ε

⁽¹

α

^{) ( , )}

^{f x} θ ] }

( )

(1 ) ( , ) ( , )

1

E _θ f x g x

ε

f x

α θ θ

α

⎧ ⎡ + ⎤⎫

= − ⎨⎩∇ ⎢⎣ − − ⎥⎦⎬⎭

(1 ) ( , ) ( ) ( , )

E

_θ

f x x

1

ε f x

α θ ϕ θ

α

⎧ ⎡ ⎤⎫

= − ⎨⎩∇ ⎢⎣ + − − ⎥⎦⎬⎭ 1

2(1 ) ^θ

λ θ

( )

= −

α

∇

− (10) According to equation (10), differential equation (9) is a gradient system¹, the potential of which is proportional to

λ θ

( ) from (4). Therefore:

Proposition 1: Any θ implying the mean squared error

λ θ

( )from (4) takes an extreme value is a stationary point of the differential equation (10).

We can state the conditions for (local) stability of a fixed point using the Jacobian matrix of

Q

( )

θ

. Hence, according to (8), we obtain

Proposition 2: let

θ

^* be a stationary point of differential equation (9). The probability that for t→∞,

θ

_taccording to (8), will converge to

θ

^*is positive only if the real parts of all eigenvalues of the following Jacobian matrix are nonpositive

*

*

'

( ) Q ( ) |

J θ θ

_θ

θ

= ∂

∂

Since the equation (9) is a gradient system, we obtain together with (7)

J ( ) ( θ = α − 1) ( ) J

_λ

θ

(11) From equation (11) we get

---

1. A gradient system in is an autonomous ordinary differential equationx&= −gradF x( )^{where F:}

→R. (Hirsch, Smale and Devaney, 2004) For the dynamic system θ&=Q( )θ we have

( ) ( )

F Q

θ θ θ

−∇ = ^{whereF( )θ =}

[

^2(1−α)

]

^{−1λ θ( )}

Proposition 3: let

θ

^* be any element of the set

Θ

^L, i.e.

θ

^* implies a local minimum of the mean-squared error

λ θ ( )

.The probability that

θ

_t from (8) converges to

θ

^*

asymptotically is positive only if

( α − < 1) 0

.

The set

Θ

^Lincludes the rational expectation equilibrium if the neural network can perfectly approximate the unknown rational expectation function. As a result, according to Proposition 3, this rational expectation function will be learnable if

( α − < 1) 0

. This result is similar to that of linear models.

Now consider the learnability of the correct rational expectations graphically. To do so, we need to examine the stability condition

α < 1

. In case the expectation of the endogenous variable

p

^e

= f x ( , )

_t

θ

_t (under) overestimates the actual value

p

, the learning algorithm (8) changes

θ

_tin a way that given

x

_t, there results a lower (higher) expectation. Convergence to the correct rational expectation depends on the value of

α

. Figure 4.7(a) shows that the expectation error

p

_t

− p

_t^ebecomes smaller if

α < 1

. In this case, the learning algorithm may converge. With

α > 1

, figure 4.7(b), this error becomes larger and as a result such an algorithm would never converge. In this case the learning process directs towards (local) maxima of the mean squared error

λ θ ( )

. But there exists no

θ

^*satisfying the sufficient conditions for a maximum.

p

t

e

p

t

e

t t

p =p

e ( )

t t t

p =αp +h x +ε

p*

p*

1e

p

1

pe

p1

a) Stability: <1α

p

t

e

p

t

e

t t

p = p

e ( )

t t t

p =α p +h x +ε

p*

p*

1e

p

1

pe

p1

b) Instability:α >1

Figure 4.7: Learnability of correct expectations

Propositions 2 and 3 provide necessary and sufficient conditions for a parameter vector

θ

^*

∈Θ

^Lto be a locally stable fixed point of differential equation (9). They are conditions for the probability that

θ

_tconverges to an element of

Θ

^Lto be nonzero.

However, this does not mean that

θ

_twill converge almost surely to an element of

Θ

^L. Thus, we need an additional case guaranteeing convergence. This can be done by

augmenting algorithm (8) with a projection facility. But formulating a projection facility in nonlinear models is quite complex task.

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