Lecture Notes 8

Parameterized expectations

algorithm

The Parameterized Expectations Algorithm (PEA hereafter) was introduced

by Marcet [1988]. As it will become clear in a moment, this may be viewed as

a generalized method of undetermined coefficients, in which economic agents

learn the decision rule at each step of the algorithm. It will therefore have a

natural interpretation in terms of learning behavior. The basic idea of this

method is to approximate the expectation function of the individuals — rather

than attempting to recover directly the decision rules — by a smooth function,

in general a polynomial function. Implicit in this approach is the fact that the

space spanned by polynomials is dense in the space spanned by all functions

in the sense

limk→∞

infθ∈Rk

supx∈X

|Fθ(x)− F (x)| = 0

where F is the function to be approximated and Fθ is an kth–order interpo-

lating function that is parameterized by θ.

8.1 Basics

The basic idea that underlies this approach is to replace expectations by an a

priori given function of the state variables of the problem in hand, and then

1

reveal the set of parameters that insure that the residuals from the Euler

equations are a martingale difference sequence (Etεt+1 = 0). Note that the

main difficulty when solving the model is to deal with the integral involved by

the expectation. The approach of the basic PEA algorithm is to approximate

it by Monte–Carlo simulations.

PEA algorithm may be implemented to solve a large set of models that

admit the following general representation

F (Et(E (yt+1, xt+1, yt, xt)), yt, xt, εt) = 0 (8.1)

where F : Rm × Rny × Rnx × Rne −→ Rnx+ny describes the model and E :

Rny ×Rnx ×Rny ×Rnx −→ Rm defines the transformed variables on which we

take expectations. Et is the standard conditional expectations operator. εt is

the set of innovations of the structural shocks that affect the economy.

In order to fix notations, let us take the optimal growth model as an

example

λt − βEt[λt+1

(αzt+1k

α−1t+1 + 1− δ

)]= 0

c−σt − λt = 0

kt+1 − ztkαt + ct − (1− δ)kt = 0

zt+1 − ρzt − εt+1 = 0

In this example, we have y = {c, λ}, x = {k, z} and ε = ε, the function E

takes the form

E ({c, λ}t+1, {k, z}t+1, {c, λ}t, {k, z}t) = λt+1(αzt+1k

α−1t+1 + 1− δ

)

while F (.) is given by

F (.) =

λt − βEt [E ({c, λ}t+1, {k, z}t+1, {c, λ}t, {k, z}t)]c−σt − λtkt+1 − ztk

αt + ct − (1− δ)kt

zt+1 − ρzt − εt+1

The idea of the PEA algorithm is then to replace the expectation function

Et(E (yt+1, xt+1, yt, xt)) by an parametric approximation function, Φ(xt; θ), of

2

the current state variables xt and a vector of parameters θ, such that the

approximated model may be restated as

F (Φ(xt, θ), yt, xt, εt) = 0 (8.2)

The problem of the PEA algorithm is then to find a vector θ such that

θ ∈ Argminθ∈Θ

‖Φ(xt, θ)− Et(E (yt+1, xt+1, yt, xt))‖2

that is the solution satisfies the rational expectations hypothesis. At this

point, note that we selected a quadratic norm, but one also may consider

other metrics of the form

θ ∈ Argminθ∈Θ

R(xt, θ)′ΩR(xt, θ)

withR(xt, θ) ≡ Φ(xt, θ)−Et(E (yt+1, xt+1, yt, xt)) and Ω is a weighting matrix.

This would then correspond to a GMM type of estimation. One may also

consider

θ ∈ Argminθ∈Θ

max{|Φ(xt, θ)− Et(E (yt+1, xt+1, yt, xt))|}

which would call for LAD estimation methods. However, the usual practice is

use the standard quadratic norm.

Once, θ and therefore the approximation function has been found, Φ(xt, θ)

and equation (8.2) may be used to generate time series for the variables of the

model.

The algorithm may then be described as follows.

Step 1. Specify a guess for the function Φ(xt, θ), an initial θ. Choose a stopping

criterion η > 0, a sample size T that should be large enough and draw a

sequence {εt}Tt=0 that will be used during all the algorithm.

Step 2. At iteration i, and for the given θi, simulate, recursively, a sequence for

{yt(θi)}Tt=0 and {xt(θ

i)}Tt=0

3

Step 3. Find G(θi) that satisfies

θ̂ ∈ Argminθ∈Θ

1

T

T∑

t=0

‖E (yt+1(θ), xt+1(θ), yt(θ), xt(θ))− Φ(xt(θ), θ)‖2

which just amounts to perform a non–linear least square regression tak-

ing E (yt+1(θ), xt+1(θ), yt(θ), xt(θ)) as the dependent variable, Φ(.) as the

explanatory function and θ as the parameter to be estimated.

Step 4. Set θi+1 to

θi+1 = γθ̂i + (1− γ)θi (8.3)

where γ ∈ (0, 1) is a smoothing parameter. On the one hand, setting

low γ helps convergence, but at the cost of increasing the computational

time. As long as good initial conditions can be found and the model

is not too non–linear, setting γ close to 1 is sufficient, however, when

dealing with strongly non–linear models — with binding constraints for

example — decreasing γ will generally helps a lot.

Step 5. If |θi+1 − θi| < η then stop, otherwise go back to step 2.

Reading this algorithm, it appears that it may easily be given a learning

interpretation. Indeed, each iteration mays be interpreted as a learning step,

in which the individual uses a rule of thumb as a decision rule and reveal

information on the kind of errors he/she does using this rule of thumb. He/she

then corrects the rule — that is find another θ — that will be used during the

next step. But it should be noted that nothing in the algorithm guarantees

that the algorithm always converges and — if it does — delivers a decision

rule that is compatible with the rational expectation hypothesis.1

At this point, several comments stemming from the implementation of the

method are in order. First of all, we need to come with an interpolating

1For a convergence proof in the case of the optimal growth model, see Marcet and Marshall[1994].

4

function, Φ(.). How should it be specified? In fact, we are free to choose any

functional form we may think of, nevertheless economic theory may guide us

as well as some constraints imposed by the method — more particularly in

step 3. A widely used interpolating function combines non–linear aspects of

the exponential function with some polynomials, such that Φj(x, θ) may take

the form (where j ∈ {1, . . . ,m} refers to a particular expectation)

Φj(x, θ) = exp(θ′P (x)

)

where P (x) is a multivariate polynomial.2 One advantage of this interpolating

function is obviously that it guarantees positive values for the expectations,

which turns out to be mostly the case in economics. One potential problem

with such a functional form is precisely related to the fact that it uses simple

polynomials which then may generate multicolinearity problems during step

3. As an example, let us take the simple case in which the state variable is

totally exogenous and is an AR(1) process with log–normal innovations:

log(at) = ρ log(at−1) + εt

with |ρ| < 1 and ε ; N (0, σ). The state variable is then at. If we simulate

the sequence {at}Tt=0 with T = 10000, and compute the correlation matrix of

{at, a2t , a

3t , a

4t } we get, for ρ = 0.95 and σ = 0.01

1.0000 0.9998 0.9991 0.99800.9998 1.0000 0.9998 0.99910.9991 0.9998 1.0000 0.99980.9980 0.9991 0.9998 1.0000

revealing some potential multicolinearity problems to occur. As an illustrative

example, assume that we want to approximate the expectation function in this

model, it will be a function of the capital stock which is a particularly smooth

sequence, therefore if there will be significant differences between the sequence

itself and the sequence taken at the power 2, the difference may then be small

2For instante, let us consider the case nx = 2, P (xt) may then consists of a constantterm, x1t, x2t, x

21t, x

22t, x1tx2t.

5

for the sequence at the power 4. Hence multicolinearity may occur. One way

to circumvent this problem is to rely on orthogonal polynomials instead of

standard polynomials in the interpolating function.

A second problem that arises in this approach is to select initial condi-

tions for θ. Indeed, this step is crucial for at least 3 reasons: (i) the problem

is fundamentally non–linear, (ii) convergence is not always guarantee, (iii)

economic theory imposes a set of restrictions to insure positivity of some vari-

ables for example. Therefore, much attention should be paid when imposing

an initial value to θ.

A third important problem is related to the choice of γ, the smoothing

parameter. A too large value may put too much weight on new values for

θ and therefore reinforce the potential forces that lead to divergence of the

algorithm. On the contrary, setting γ too close to 0 may be costly in terms of

computational CPU time.

It must however be noted that no general rule may be given for these

implementation issues and that in most of the case, one has to guess and try.

Therefore, I shall now report 3 examples of implementation. The first one is

the standard optimal growth model, the other one corresponds to the optimal

growth model with investment irreversibility, the last one will be the problem

of a household facing borrowing constraints.

But before going to the examples, we shall consider a linear example that

will highlight the similarity between this approach and the undetermined co-

efficient approach.

8.2 A linear example

Let us consider the simple model

yt = aEtyt+1 + bxt

xt+1 = (1− ρ)x+ ρxt + εt+1

6

where ε ; N (0, σ2). Finding an expectation function in this model amounts

to find a function Φ(xt, θ) for Et(ayt+1+bxt). Let us make the following guess

for the solution:

Φ(xt, θ) = θ0 + θ1xt

In this case, solving the PEA problem amount to solve

min{θ0,θ1}

1

N

N∑

t=1

(Φ(xt, θ)− ayt+1 − bxt)2

The first order conditions for this problem are

1

N

N∑

t=1

(θ0 + θ1xt − ayt+1 − bxt) = 0 (8.4)

1

N

N∑

t=1

xt (θ0 + θ1xt − ayt+1 − bxt) = 0 (8.5)

Equation (8.4) can be rewritten as

θ0 + θ11

N

N∑

t=1

xt = a1

N

N∑

t=1

yt+1 + b1

N

N∑

t=1

xt

But, since Φ(xt, θ) is an approximate solution for the expectation function,

the model implies that

yt = Et(ayt+1 + bxt) = Φ(xt, θ)

such that the former equation rewrites

θ0 + θ11

N

N∑

t=1

xt = a1

N

N∑

t=1

(θ0 + θ1xt+1) + b1

N

N∑

t=1

xt

Asymptotically, we have

limN→∞

1

N

N∑

t=1

xt = limN→∞

1

N

N∑

t=1

xt+1 = x

such that this first order condition converges to

θ0 + θ1x = aθ0 + aθ1x+ bx

7

therefore, rearranging terms, we have

θ0(1− a) + θ1(1− a)x = bx (8.6)

Now, let us consider equation (8.5), which can be rewritten as

θ01

N

N∑

t=1

xt + θ11

N

N∑

t=1

x2t = a1

N

N∑

t=1

yt+1xt + b1

N

N∑

t=1

x2t

Like for the first condition, we acknowledge that

yt = Et(ayt+1 + bxt) = Φ(xt, θ)

such that the condition rewrites

θ01

N

N∑

t=1

xt + θ11

N

N∑

t=1

x2t = a1

N

N∑

t=1

(θ0 + θ1xt+1)xt + b1

N

N∑

t=1

x2t (8.7)

Asymptotically, we have

limN→∞

1

N

N∑

t=1

xt = x and limN→∞

1

N

N∑

t=1

x2t = E(x2) = σ2x + x

2

finally, we have

limN→∞

1

N

N∑

t=1

xtxt+1 = limN→∞

1

N

N∑

t=1

xt((1− ρ)x+ ρxt + εt+1)

Since ε is the innovation of the process, we have limN→∞1N

∑Nt=1 xtεt+1 = 0,

such that

limN→∞

1

N

N∑

t=1

xtxt+1 = (1− ρ)x2 + ρE(x2) = x2 + ρσ2x

Hence, (8.7) asymptotically rewrites as

x(1− a)θ0 + (1− aρ)(x2 + σ2x)θ1 = b(x

2 + σ2x)

8

We therefore have to solve the system

θ0(1− a) + θ1(1− a)x = bx

x(1− a)θ0 + (1− a)(x2 + ρσ2x)θ1 = b(x

2 + σ2x)

premultiplying the first equation by x, and plugging the result in the second

equation leads to

(1− aρ)θ1σ2x = bσ

2x

such that

θ1 =b

1− aρ

Plugging this result into the first equation, we get

θ0 =ab(1− ρ)x

(1− a)(1− aρ)

Therefore, Asymptotically, the solution is given by

yt =ab(1− ρ)x

(1− a)(1− aρ)+

b

1− aρxt

which corresponds exactly to the solution of the model (see Lecture notes

#1).Therefore, asymptotically, the PEA algorithm is nothing else but an un-

determined coefficient method.

8.3 Standard PEA solution: the Optimal Growth

Model

Let us first recall the type of problem we have in hand. We are about to solve

the set of equations

λt − βEt[λt+1

(αzt+1k

α−1t+1 + 1− δ

)]= 0

c−σt − λt = 0

kt+1 − ztkαt + ct − (1− δ)kt = 0

log(zt+1)− ρ log(zt)− εt+1 = 0

9

Our problem will therefore be to get an approximation for the expectation

function:

βEt[λt+1

(αzt+1k

α−1t+1 + 1− δ

)]

In this problem, we have 2 state variables: kt and zt, such that Φ(.) should be

a function of both kt and zt. We will make the guess

Φ(kt, zt; θ) = exp(θ0 + θ1 log(kt) + θ2 log(zt) + θ3 log(kt)

2 + θ4 log(zt)2 + θ5 log(kt) log(zt)

)

From the first equation of the above system, we have that — for a given

vector θ = {θ0, θ1, θ2, θ3, θ4, θ5} — λt(θ) = Φ(kt(θ), zt(θ); θ), which enables us

to recover

ct(θ) = λt(θ)− 1σ

and therefore get

kt+1(θ) = ztkt(θ)α − ct(θ) + (1− δ)kt(θ)

We then recover a whole sequence for {kt(θ)}Tt=0, {zt}

Tt=0, {λt(θ)}

Tt=0, and

{ct(θ)}Tt=0, which makes it simple to compute a sequence for

ϕt+1(θ) ≡ λt+1(θ)(αzt+1kt+1(θ)

α−1 + 1− δ)

Since Φ(kt, zt; θ) is an exponential function of a polynomial, we may run the

regression

log(ϕt+1(θ)) = θ0 + θ1 log(kt(θ)) + θ2 log(zt) + θ3 log(kt(θ))2

+θ4 log(zt)2 + θ5 log(kt(θ)) log(zt) (8.8)

to get θ̂. We then set a new value for θ according to the updating scheme (8.3)

and restart the process until convergence.

The parameterization we used in the matlab code are given in table 8.1

and is totally standard. γ, the smoothing parameter was set to 1, implying

that in each iteration the new θ vector is totally passed as a new guess in the

10

Table 8.1: Optimal growth: Parameterization

β σ α δ ρ σe0.95 1 0.3 0.1 0.9 0.01

progression of the algorithm. The stopping criterion was set at η=1e-6 and

T=20000 data points were used to compute the OLS regression.

Initial conditions were set as follows. We first solve the model relying on

a log–linear approximation. We then generate a random draw of size T for ε

and generate series using the log–linear approximate solution. We then built

the needed series to recover a draw for {ϕt+1(θ)}∞t=0, {kt(θ)}

∞t=0 and {zt(θ)}

∞t=0

and ran the regression (8.8) to get an initial condition for θ, reported in table

8.2. The algorithm converges after 22 iterations and delivers the final decision

Table 8.2: Decision rule

θ0 θ1 θ2 θ3 θ4 θ5Initial 0.5386 -0.7367 -0.2428 0.1091 0.2152 -0.2934Final 0.5489 -0.7570 -0.3337 0.1191 0.1580 -0.1961

rule reported in table 8.2. When γ is set at 0.75, 31 iterations are needed,

46 for γ = 0.5 and 90 for γ = 0.25. It is worth noting that the final decision

rule does differ from the initial conditions, but not by an as large amount

as one would have expected, meaning that in this setup — and provided the

approximation is good enough3 — certainty equivalence and non–linearities

do not play such a great role. In fact, as illustrated in figure 8.1, the capital

decision rule does not display that much non linearities. Although particularly

simple to implement (see the following matlab code), this method should be

handle with care as it may be difficult to obtain convergence for some models.

Nevertheless it has another attractive feature: it can handle problems with

3Note that for the moment we have not made any evaluation of the accuracy of thedecision rule. We will undertake such an evaluation in the sequel.

11

Figure 8.1: Capital decision rule

2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 32.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3

kt

k t+1

possibly binding contraints. We now provide two examples of such models.

Matlab Code: PEA Algorithm (OGM)

clear all

long = 20000;

init = 500;

slong = init+long;

T = init+1:slong-1;

T1 = init+2:slong;

tol = 1e-6;

crit = 1;

gam = 1;

sigma = 1;

delta = 0.1;

beta = 0.95;

alpha = 0.3;

ab = 0;

rho = 0.9;

se = 0.01;

param = [ab alpha beta delta rho se sigma long init];

ksy =(alpha*beta)/(1-beta*(1-delta));

yss = ksy^(alpha/(1-alpha));

kss = yss^(1/alpha);

12

iss = delta*kss;

css = yss-iss;

csy = css/yss;

lss = css^(-sigma);

randn(’state’,1);

e = se*randn(slong,1);

a = zeros(slong,1);

a(1) = ab+e(1);

for i = 2:slong;

a(i) = rho*a(i-1)+(1-rho)*ab+e(i);

end

b0 = peaoginit(e,param); % Compute initial conditions

%

% Main Loop

%

iter = 1;

while crit>tol;

%

% Simulated path

%

k = zeros(slong+1,1);

lb = zeros(slong,1);

X = zeros(slong,length(b0));

k(1) = kss;

for i = 1:slong;

X(i,:)= [1 log(k(i)) a(i) log(k(i))*log(k(i)) a(i)*a(i) log(k(i))*a(i)];

lb(i) = exp(X(i,:)*b0);

k(i+1)=exp(a(i))*k(i)^alpha+(1-delta)*k(i)-lb(i)^(-1/sigma);

end

y = beta*lb(T1).*(alpha*exp(a(T1)).*k(T1).^(alpha-1)+1-delta);

bt = X(T,:)\log(y);

b = gam*bt+(1-gam)*b0;

crit = max(abs(b-b0));

b0 = b;

disp(sprintf(’Iteration: %d\tConv. crit.: %g’,iter,crit))

iter=iter+1;

end;

13

8.4 PEA and binding constraints: Optimal growth

with irreversible investment

We now consider a variation to the previous model, in the sense that we restrict

gross investment to be positive in each and every period:

it > 0⇐⇒ kt+1 > (1− δ)kt (8.9)

This assumption amounts to assume that there does not exist a second hand

market for capital. In such a case the problem of the central planner is to

determined consumption and capital accumulation, such that utility is maxi-

mum:

max{ct,kt+1}∞t=0

E0

∞∑

t=0

βtc1−σt − 1

1− σ

s.t.

kt+1 = ztkαt − ct + (1− δ)kt

and

kt+1 > (1− δ)kt

Forming the Lagrangean associated to the previous problem, we have

Lt = Et

∞∑

τ=0

βτ

[c1−σt+τ − 1

1− σ+ λt+τ

(zt+τk

αt+τ − ct+τ + (1− δ)kt+τ − kt+τ+1

)

+ µt+τ (kt+1 − (1− δ)kt)

]

which leads to the following set of first order conditions

c−σt = λt (8.10)

λt − µt = βEt[λt+1

(αzt+1k

α−1t+1 + 1− δ

)− µt+1(1− δ)

](8.11)

kt+1 = ztkαt − ct + (1− δ)kt (8.12)

µt (kt+1 − (1− δ)kt) (8.13)

The main difference with the previous example is that now the central planner

faces a constraint that may be binding in each and every period. Therefore,

14

this complicates a little bit the algorithm, and we have to find a rule for both

the expectation function

Et [ϕt+1]

where

ϕt+1 ≡ β(λt+1

(αzt+1k

α−1t+1 + 1− δ

)− µt+1(1− δ)

)

and µt. We then proceed as suggested in Marcet and Lorenzoni [1999]:

1. Compute two sequences for {λt(θ)}∞t=0 and {kt(θ)}

∞t=0 from (8.11) and

(8.12) under the assumption that the constraint is not binding — that

is µt(θ) = 0. In such a case, we just compute the sequences as in the

standard optimal growth model.

2. Test whether, under this assumption, it(β) > 0. If it is the case, then

set µt(θ) = 0, otherwise set kt+1(θ) = (1 − δ)kt(θ), ct(θ) is computed

from the resource constraint and µt(θ) is found from (8.11).

Note that, using this procedure, µt is just treated as an additional variable

which is just used to compute a sequence to solve the model. We therefore

do not need to compute explicitly its interpolating function, as far as ϕt+1 is

concerned we use the same interpolating function as in the previous example

and therefore run a regression of the type

log(ϕt+1(θ)) = θ0 + θ1 log(kt(θ)) + θ2 log(zt) + θ3 log(kt(θ))2

+θ4 log(zt)2 + θ5 log(kt(θ)) log(zt) (8.14)

to get θ̂.

Up to the shock, the parameterization, reported in table 8.3, we used in the

matlab code is essentially the same as the one we used in the optimal growth

model. The shock was artificially assigned a lower persistence and a greater

volatility in order to increase the probability of binding the constraint, and

therefore illustrate the potential of this approach. γ, the smoothing parameter

was set to 1. The stopping criterion was set at η=1e-6 and T=20000 data

points were used to compute the OLS regression.

15

Table 8.3: Optimal growth: Parameterization

β σ α δ ρ σe0.95 1 0.3 0.1 0.8 0.14

Initial conditions were set as in the standard optimal growth model: We

first solve the model relying on a log–linear approximation. We then gener-

ate a random draw of size T for ε and generate series using the log–linear

approximate solution. We then built the needed series to recover a draw for

{ϕt+1(θ)}∞t=0, {kt(θ)}

∞t=0 and {zt(θ)}

∞t=0 and ran the regression (8.14) to get an

initial condition for θ, reported in table 8.4. The algorithm converges after 115

iterations and delivers the final decision rule reported in table 8.4. Contrary

Table 8.4: Decision rule

θ0 θ1 θ2 θ3 θ4 θ5Initial 0.4620 -0.5760 -0.3909 0.0257 0.0307 -0.0524Final 0.3558 -0.3289 -0.7182 -0.1201 -0.2168 0.3126

to the standard optimal growth model, the initial and final rule totally differ

in the sense the coefficient in front of the capital stock in the final rule is half

that on the initial rule, that in front of the shock is double, and the sign in

front of all the quadratic terms are reversed. This should not be surprising as

the initial rule is computed under (i) the certainty equivalence hypothesis and

(ii) the assumption that the constraint never binds, whereas the size of the

shocks we introduce in the model implies that the constraint binds in 2.8% of

the cases. The latter quantity may seem rather small, but this is sufficient to

dramatically alter the decision of the central planner when it acts under ratio-

nal expectations. This is illustrated by figures 8.2 and 8.3 which respectively

report the decision rules for investment, capital and the lagrange multiplier

and a typical path for investment and lagrange multiplier. As reflected in

16

Figure 8.2: Decision rules

0 2 4 6 80

0.5

1

1.5

kt

investment

0 0.5 1 1.50

200

400

600

800Distribution of investment

0 2 4 6 80

2

4

6

8

kt

Capital stock

0 2 4 6 80

0.2

0.4

0.6

0.8

kt

Lagrange multiplier

Figure 8.3: Typical investment path

0 200 400 6000

0.2

0.4

0.6

0.8

1

Time

investment

0 200 400 6000

0.05

0.1

0.15

0.2

0.25

Time

Lagrange multiplier

17

the upper right panel of figure 8.2 which reports the simulated distribution of

investment the distribution is highly skewed and exhibits a mode at it = 0, re-

vealing the fact that the constraint occasionally binds. This is also illustrated

in the lower left panel that reports the decision rule for the capital stock. As

can be seen from this graph, the decision rule is bounded from below by the

line (1 − δ)kt (the grey line on the graph), such situation then correspond to

situations where the Lagrange multiplier is positive as reported in the lower

right panel of the figure.

Matlab Code: PEA Algorithm (Irreversible Investment)

clear all

long = 20000;

init = 500;

slong = init+long;

T = init+1:slong-1;

T1 = init+2:slong;

tol = 1e-6;

crit = 1;

gam = 1;

sigma = 1;

delta = 0.1;

beta = 0.95;

alpha = 0.3;

ab = 0;

rho = 0.8;

se = 0.125;

kss = ((1-beta*(1-delta))/(alpha*beta))^(1/(alpha-1));

css = kss^alpha-delta*kss;

lss = css^(-sigma);

ysk = (1-beta*(1-delta))/(alpha*beta);

csy = 1-delta/ysk;

%

% Simulation of the shock

%

randn(’state’,1);

e = se*randn(slong,1);

a = zeros(slong,1);

a(1) = ab+e(1);

for i = 2:slong;

a(i) = rho*a(i-1)+(1-rho)*ab+e(i);

end

%

% Initial guess

18

%

param = [ab alpha beta delta rho se sigma long init];

b0 = peaoginit(e,param);

%

% Main Loop

%

iter = 1;

while crit>tol;

%

% Simulated path

%

k = zeros(slong+1,1);

lb = zeros(slong,1);

mu = zeros(slong,1);

X = zeros(slong,length(b0));

k(1) = kss;

for i = 1:slong;

X(i,:)= [1 log(k(i)) a(i) log(k(i))*log(k(i)) a(i)*a(i) log(k(i))*a(i)];

lb(i) = exp(X(i,:)*b0);

iv = exp(a(i))*k(i)^alpha-lb(i)^(-1/sigma);

if iv>0;

k(i+1) = (1-delta)*k(i)+iv;

mu(i) = 0;

else

k(i+1) = (1-delta)*k(i);

c = exp(a(i))*k(i)^alpha;

mu(i) = c^(-sigma)-lb(i);

end

end

y = beta*(lb(T1).*(alpha*exp(a(T1)).*k(T1).^(alpha-1)+1-delta) ...

-mu(T1)*(1-delta));

bt = X(T,:)\log(y);

b = gam*bt+(1-gam)*b0;

crit = max(abs(b-b0));

b0 = b;

disp(sprintf(’Iteration: %d\tConv. crit.: %g’,iter,crit))

iter = iter+1;

end;

19

8.5 The Households Problem With Borrowing Con-

straints

As a final example, we now report the example of a consumer that faces

borrowing constraints, such that she solves the program

max{ct}

Et

∞∑

τ=0

βτu(ct+τ )

s.t.

at+1 = (1 + r)at + ωt − ct

at+1 > 0

log(ωt+1) = ρ log(ωt) + (1− ρ) log(ω) + εt+1

Let us first recall the first order conditions that are associated with this prob-

lem:

c−σt = λt (8.15)

λt = µt + β(1 + r)Etλt+1 (8.16)

at+1 = (1 + r)at + ωt − ct (8.17)

log(ωt+1) = ρ log(ωt) + (1− ρ) log(ω) + εt+1 (8.18)

µt(at+1 − a) = 0 (8.19)

µt > 0 (8.20)

In order to solve this model, we have to find a rule for both the expectation

function

Et [ϕt+1]

where

ϕt+1 ≡ βRλt+1

and µt. We propose to follow the same procedure as the previous one:

20

1. Compute two sequences for {λt(θ)}∞t=0 and {at(θ)}

∞t=0 from (8.16) and

(8.17) under the assumption that the constraint is not binding — that

is µt(θ) = 0.

2. Test whether, under this assumption, at+1(β) > a. If it is the case, then

set µt(θ) = 0, otherwise set at+1(θ) = a, ct(θ) is computed from the

resource constraint and µt(θ) is found from (8.16).

Note that, using this procedure, µt is just treated as an additional variable

which is just used to compute a sequence to solve the model. We therefore

do not need to compute explicitly its interpolating function, as far as ϕt+1 is

concerned we use the same interpolating function as in the previous example

and therefore run a regression of the type

log(ϕt+1(θ)) = θ0 + θ1at(θ) + θ2ωt + θ3at(θ)2 + θ4ω

2t + θ5at(θ)ωt (8.21)

to get θ̂.

The parameterization is reported in table 8.5. γ, the smoothing parameter

Table 8.5: Borrowing constraint: Parameterization

a β σ ρ σω R ω

0 0.95 1.5 0.7 0.1 1.04 1

was set to 1. The stopping criterion was set at η=1e-6 and T=20000 data

points were used to compute the OLS regression.

One key issue in this particular problem is related to the initial conditions.

Indeed, it is extremely difficult to find a good initial guess as the only model

for which we might get an analytical solution while being related to the present

model is the standard permanent income model. Unfortunately, this model

exhibits a non–stationary behavior, in the sense it generates an I(1) process

for the level of individual wealth and consumption, and therefore the marginal

utility of wealth. We therefore have to take another route. We propose the

21

following procedure. For a given a0 and a sequence {ωt}Tt=0, we generate

c0 = r̃a0 + ω0 + η0 where r̃ > r and ε0 ; N (0, ση). In practice, we took

r̃ = 0.1 and ση = 0.1. We then compute a1 from the law of motion of wealth.

If a1 < a then a1 is set to a and c0 = Ra0+y0−a, otherwise c0 is not modified.

We then proceed exactly the same way for all t > 0. We then have in hand a

sequence for both at and ct, and therefore for λt. We can then recover easily

ϕt+1 and an initial θ from the regression (8.21) (see table 8.6).

Table 8.6: Decision rule

θ0 θ1 θ2 θ3 θ4 θ5Initial 1.6740 -0.6324 -2.1918 0.0133 0.5438 0.2971Final 1.5046 -0.5719 -2.1792 0.0458 0.7020 0.3159

The algorithm converges after 79 iterations and delivers the final decision

rule reported in table 8.6. Note that if the final decision rule effectively differs

from the initial one, the difference is not huge, meaning that our initialization

procedure is relevant. Figure 8.4 reports the decision rule of consumption in

terms of cash–on–hand — that is the effective amount a household may use to

purchase goods (Rat+ωt− a). Figure 8.5 reports the decision rule for wealth

accumulation as well as the implied distribution, which admits a mode in a,

revealing that the constraints effectively binds (in 13.7% of the cases).

Matlab Code: PEA Algorithm (Borrowing Constraints)

clear

crit = 1;

tol = 1e-6;

gam = 1;

long = 20000;

init = 500;

slong = long+init;

T = init+1:slong-1;

T1 = init+2:slong;

rw = 0.7;

sw = 0.1;

22

Figure 8.4: Consumption decision rule

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Cash−on−hand (R at+ω

t−a)

Consumption

Figure 8.5: Wealth accumulation

0 1 2 3 40

1

2

3

4

at

Wealth

0 1 2 3 40

1000

2000

3000

4000Distribution of wealth

23

wb = 0;

beta = 0.95;

R = 1/(beta+0.01);

sigma = 1.5;

ab = 0;

randn(’state’,1);

e = sw*randn(slong,1);

w = zeros(slong,1);

w(1) = wb+e(1);

for i = 2:slong;

w(i)= rw*w(i-1)+(1-rw)*wb+e(i);

end

w=exp(w);

a = zeros(slong,1);

c = zeros(slong,1);

lb = zeros(slong,1);

X = zeros(slong,6);

a(1) = ass;

rt = 0.2;

sc = 0.1;

randn(’state’,1234567890);

ec = sc*randn(slong,1);

for i=1:slong;

X(i,:) = [1 a(i) w(i) a(i)*a(i) w(i)*w(i) a(i)*w(i)];

c(i) = rt*a(i)+w(i)+ec(i);

a1 = R*a(i)+w(i)-c(i);

if a1>ab;

a(i+1)=a1;

else

a(i+1)= ab;

c(i) = R*a(i)+w(i)-ab;

end

end

lb = c.^(-sigma);

y = log(beta*R*lb(T1));

b0 = X(T,:)\y

iter=1;

while crit>tol;

a = zeros(slong,1);

c = zeros(slong,1);

lb = zeros(slong,1);

X = zeros(slong,length(b0));

a(1) = 0;

for i=1:slong;

X(i,:)= [1 a(i) w(i) a(i)*a(i) w(i)*w(i) a(i)*w(i)];

24

lb(i) = exp(X(i,:)*b0);

a1 = R*a(i)+w(i)-lb(i)^(-1/sigma);

if a1>ab;

a(i+1) = a1;

c(i) = lb(i).^(-1./sigma);

else

a(i+1) = ab;

c(i) = R*a(i)+w(i)-ab;

lb(i) = c(i)^(-sigma);

end

end

y = log(beta*R*lb(T1));

b = X(T,:)\y;

b = gam*b+(1-gam)*b0;

crit = max(abs(b-b0));

b0 = b;

disp(sprintf(’Iteration: %d\tConv. crit.: %g’,iter,crit))

iter=iter+1;

end;

25

26

Bibliography

Marcet, A., Solving Nonlinear Stochastic Models by Parametrizing Expecta-

tions, mimeo, Carnegie–Mellon University 1988.

and D.A. Marshall, Solving Nonlinear Rational Expectations Models

by Parametrized Expectations : Convergence to Stationary Solutions,

Manuscript, Universitat Pompeu Fabra, Barcelone 1994.

and G. Lorenzoni, The Parameterized Expectations Approach: Some

Practical Issues, in M. Marimon and A. Scott, editors, Computational

Methods for the Study of Dynamic Economies, Oxford: Oxford University

Press, 1999.

27

Index

Expectation function, 2

Interpolating function, 1

Orthogonal polynomial, 6

28

Contents

8 Parameterized expectations algorithm 1

8.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

8.2 A linear example . . . . . . . . . . . . . . . . . . . . . . . . . . 6

8.3 Standard PEA solution: the Optimal Growth Model . . . . . . 9

8.4 PEA and binding constraints: Optimal growth with irreversible

investment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

8.5 The Households Problem With Borrowing Constraints . . . . . 20

29

30

List of Figures

8.1 Capital decision rule . . . . . . . . . . . . . . . . . . . . . . . . 12

8.2 Decision rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

8.3 Typical investment path . . . . . . . . . . . . . . . . . . . . . . 17

8.4 Consumption decision rule . . . . . . . . . . . . . . . . . . . . . 23

8.5 Wealth accumulation . . . . . . . . . . . . . . . . . . . . . . . . 23

31

32

List of Tables

8.1 Optimal growth: Parameterization . . . . . . . . . . . . . . . . 11

8.2 Decision rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

8.3 Optimal growth: Parameterization . . . . . . . . . . . . . . . . 16

8.4 Decision rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

8.5 Borrowing constraint: Parameterization . . . . . . . . . . . . . 21

8.6 Decision rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

33