pea algorithm

Decentralised economies II

Martin Ellison

1 Motivation

In the last lecture we described how to simulate a decentralised economy

using a log-linear approximation to the first order conditions of maximising

behaviour. Such a procedure will be acceptable as long as the underlying

structure of the economy is close to log-linearity. However, if the model is

not log-linear then approximation errors are likely to be significant. For ex-

ample, log-linearisation of models with time-varying risk premia is likely to

cancel out variance effects. A method based on second order log-linearisation

(such Michel Juillard’s DYNARE at http://www.cepremap.cnrs.fr/dynare/)

would go some way to alleviating these concerns. Rather than proceeding

to such second-order approximations, our preferred route in this lecture is

to eliminate approximation error and describe a technique which leads to an

exact solution to the model. The technique is known as the Parameterised

Expectations Algorithm (PEA). Whilst most applications of PEA are macro-

economic in nature, the technique is more general and can be applied to any

model, for example models of nonlinear pricing in microeconomics.

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2 Key reading

The Parameterised Expectations Algorithm was first presented in den Haan,

W.J. and A. Marcet, 1990, “Solving the Stochastic Growth Model by Pa-

rameterized Expectations”, Journal of Business and Economic Statistics 8,

31-34. The accuracy test - equally applicable to testing the accuracy of any

simulation method - is published as “Accuracy in Simulations”, Review of

Economic Studies 61, 3-17.

3 Other reading

The asymptotic learning convergence results on which PEA is based are

taken fromMarcet, A. and T.J. Sargent, 1989, “Convergence of Least-Squares

Learning Mechanisms in Self-Referential Linear Stochastic Models, Journal

of Economic Theory 48, 337-368. More advance PEA techniques such as

grids are discussed in L.J. Christiano and J.D.M. Fisher, 1997. “Algorithms

for solving dynamic models with occasionally binding constraints”, Working

Paper 9711, Federal Reserve Bank of Cleveland. A special issue of the Journal

of Economic and Business Statistics in 1991 compared alternative solution

techniques.

4 Parameterised expectations algorithm

This section tries to give in a general setting the basic approach of the PEA.

For more details on the intuition see den Haan and Marcet (1990). For those

of you who prefer concrete examples the next section gives a fully worked

out example.

Most economic models are based around certain key Euler equations.

These Euler equations link together a function of current dated variables

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to a conditional expectation of a function of future dated variables. For

instance, let �� denote a vector of endogenous variables at time �, �� a vector

of state variables and let �� denote a vector of random disturbances. Then

an economic model will be characterised by an Euler equation of the form

�(��) = ��(��+1� ��+1� ��)

where �� denotes the expectations operator conditional on information

at time �. For instance, � might be consumption, �(·) the marginal utility ofconsumption and �(·) = �(·)� where is the interest rate. Notice that for

plausible assumptions about the utility function and the production function

�(·) and �(·) are likely to be highly nonlinear. For example, as risk aversionincreases then the marginal utility function becomes increasingly nonlinear

and if the variance of shocks is large then the interest rate term will reflect a

large risk premium. The result is that the Euler equation will contain large

and important nonlinearities which will be both hard to solve and poorly

approximated via quadratic approximations.

The PEA focuses on the Euler equation. Notice that we can re-write this

Euler equation as

�� = �−1(��(��+1� ��+1� ��))

Given that we know the function �−1, if we knew what ��(��+1� ��+1� ��)

was then we could solve for ��. In other words, if we knew the value of ��+1then we could solve backwards for ��. By contrast, the PEA approach solves

forward for the solution - that is it uses ��−1 to solve for ��. It does this

by noting that although ��(��+1� ��+1� ��) is a function of ��+1 it must by

definition be a function of information available at time � when the consumer

is choosing ��. Therefore it must be possible to calculate �� using information

recursively - that is by solving the model forward over time. More specifically,

the PEA assumes that

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��(��+1� ��+1� ��) = (�� 1� )

where �1� is a subset of state variables (potentially all of them). In other

words PEA assumes that the expectational term is a function of variables

known at time � (the predetermined variables and the current realisations

of the shocks). From this it is obvious where the name Parameterised Ex-

pectations comes from - the PEA approach assumes that it is possible to

model any expectational term as a polynomial in the information set. Of

course an important decision is of the form of (·) and here den Haan andMarcet assume that it is a polynomial in the variables �1� . The success of

the PEA approach depends on how well the polynomial (·) approximatesthe expectational term ��(·). This depends on two factors (a) the degreeof the polynomial (·) and (b) the validity of the parameter vector �. Byincreasing the degree of the polynomial (·) it is possible to get ever moreaccurate approximations to ��(·). However, the key thing is to arrive atestimates of � which ensure that ��(·) is well approximated by (·). In thenext section we explain how the PEA arrives at the final estimates of the �.

5 An example

Consider the standard stochastic growth model where a representative agent

seeks to

max{��}

��

∞X�=0

�� 1−��

1− �

subject to

� + �� − ��−1 = ��−1

where �� denotes a random productivity shock and

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ln��+1 = � ln�� + ��

and � is the discount factor. � is one less the depreciation rate and � the

coefficient of relative risk aversion. The first order condition for this model

is

�� = ��+1��+1

where �� is the marginal utility of consumption at time � or

−�� = ��[ −��+1(��+1�

�−1� + �)]

The PEA approach is then to replace this expectation term on the right

hand side with a polynomial in the state variables. In this simple model the

state variables are the capital stock inherited from the last period and the

current productivity shock. This we re-write the Euler equation as

−�� = �(��−1� ��; �)

where (·) denotes the polynomial that the PEA approach uses to sub-stitute for the conditional expectation in the Euler equation. In what follows

we assume that

(��−1� ��; �) = �1��2�−1�

�3�

that is that (·) is a first order polynomial in the state variables. How-ever, we could use any order polynomial, i.e. we could use instead

(��−1� ��; �) = �1��2�−1�

�3� (��−1��)

�4(�2�−1)�5(�2� )

�6

Notice that we have not yet specified anything about the coefficients of

the polynomial, the �’s. For the moment we shall just set them equal to some

5

arbitrary starting values. Given these starting values the PEA then iterates

to solve for the “correct” values as we shall explain below.

If we go back to the first order polynomial we now have

−�� = �1��2�−1�

�3�

Therefore if we make an assumption about the initial capital stock (�0)

and if we take a random draw from the computer for the first period produc-

tivity term (�1) then for our given values of �1� �2 and �3 we can calculate

first period consumption 1. If we then look at the budget constraint we

see we have all the information required to calculate �1. We can therefore

calculate values for all relevant variables in the first period.

However, we have now solved for �1 and �1 and so if we take a random

draw from the computer for the value of �2 the productivity shock in time

2 then we can use the Euler equation for the second period

−�2 = ��2[ −�2 (��3�

�−12 + �)]

to solve for 2. Once again, the trick is to use the PEA to write this Euler

equation as

−�2 = �1��21 �

�32

as we have values for �1 and �2 we can once again (for given �’s) solve

for 2. Given 2 we can from the budget constraint then calculate �2. By

repeating this process for all time periods we can therefore arrive at a time

series for consumption, capital and output for any arbitrary number of peri-

ods. However, as can be seen the resulting { �}� {��} and {��} series dependon the coefficients in the PEA polynomial, that is the �’s. Different choices

of these �’s will produce different sequences for consumption, output and

capital. While we can always solve for { �}� {��} and {��} for given �’s the

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resulting equilibrium sequences need not satisfy the Euler equation. That

is agents’ expectations may not generate actions which are consistent with

these expectations. The solution of the PEA approach is to arrive at the

“correct” �’s - that is those expectations which produce behaviour which is

consistent with those expectations.

What criteria should we use in choosing the correct parameters? If the

PEA approach is to be correct then it must be the case that��[ −��+1(��+1�

�−1� +

�)] is approximately equal to �1��2�−1�

�3� or

��[ −��+1(��+1�

�−1� + �)]− �1�

�2�−1�

�3� = �� ≈ 0

The closer to zero is �� the better the approximation being used by PEA.

this immediately suggests one criteria to be used in choosing the �’s for

the Parameterised Expectations term - that is minimisingP

�2� . But this

is of course exactly what a regression would do. In other words, if you

regress ��[ −��+1(��+1�

�−1� + �)] on �1�

�2�−1�

�3� then the resulting estimates of

�’s will minimiseP

�2� . Obviously this regression is non-linear so needs to be

estimated using non-linear least squares.

Notice a circularity in what we have outlined so far. In order to calculate

{ �}� {��} and {��} we needed to assume values for �1� �2 and �3. Different

choices of �1� �2 and �3 will lead to different sequences of { �}� {��} and {��}.Having made an initial choice for �1� �2 and �3 we then construct a sequence

for { �}� {��} and {��} and run a nonlinear regression1 to estimate the �1� �2and �3 that minimises

P�2� . The �’s from the nonlinear regression will not

necessarily be the same as the �’s assumed in the construction of { �}� {��}and {��}. That is the formula used to form expectations of the future may

not produce outcomes which are consistent with these forecasts. In this

case agents need to change their forecasting equation. The point of the

1Like all nonlinear regressions a certain amount of “art” is required in choosing sensible

starting values and nudging the program to a solution.

7

PEA procedure is to continue until the �’s from the nonlinear regression are

the same as the �’s assumed in constructing { �}� {��} and {��} - in otherwords a fixed point has been reached. This fixed point is the solution to the

model as at this point the forecasting function used by consumers produces

consumption demands which lead to outcomes for future consumption which

prove the original forecasting equations correct.

As an example of the end product of the PEA approach, consider the

above stochastic growth model calibrated on UK data. In this case we arrive

at estimates for the coefficients of �1 = 1�7931� �2 = −1�9249 and �3 =

−1�6701. Given that � = 0�99 and � = 5 that means that the solution to

our model

−�� = �1��2�−1�

�3�

is that

� = exp

µ−15[ln 0�99 + ln 1�7931− 1�9249 ln ��−1 − 1�6701 ln��

¶therefore the solution is a simple expression relating current consumption

to the current state variables.

What is the intuition behind the fixed point which underlies PEA? Prob-

ably the best way to rationalise the PEA approach is in terms of learning.

Imagine a world where agents know their first order conditions but do not

know how to form expectations that the Euler equations entail. That is

they have to learn how the economy works. One simple way of forming an

expectation would be to try and forecast the expectational terms using a

forecasting rule involving only the state variables. At first the consumer has

very few observations with which to construct a forecasting rule so essentially

makes a guess as to the coefficients in the forecasting equation. However, as

more data becomes available they can estimate a more precise forecasting

8

rule and need to rely less on intuition. At first the consumer will make mis-

takes. They do not as yet know how the economy operates and will form

expectations on the basis of an incorrect forecast of the forward-looking term

in the Euler equation. In other words [ −��+1(��+1��−1� +�)] will not be equal

in expectation to ��[ −��+1(��+1�

�−1� +�)]. Therefore agents expectations will

be inconsistent with actual outcomes. However, under certain convergence

assumptions on the economy and beliefs it is possible to show (see the work

of Marcet and Sargent) that eventually agents expectations and the equilib-

rium of the economy will coincide so that [ −��+1(��+1��−1� +�)] will equal the

prediction made by the consumer’s forecasting equation. That is the actions

which consumers make on the basis of their expectations lead to economic

realisations which confirm those expectations are correct.

The PEA approach essentially mimics this learning algorithm and the

solution to the PEA algorithm is the fixed point of the learning process. In

other words when agents use �1��2�−1�

�3� to form their expectations this leads

to consumption and capital choices at time � which ensure that in expectation

��[ −��+1(��+1�

�−1� + �)]− �1�

�2�−1�

�3� = 0, i.e. expectations are correct.

We can therefore write down the following plan for how to use the PEA

for this simple growth model.

STEP 1 : Simulate a sequence for the productivity shock, {��}.STEP 2 : Use guess of �1� �2 and �3 and {��} and PEA expression to solve

for { �}.STEP 3: Use results from STEP 2 and budget constraint to solve for {��}

and {��}.STEP 4: Run nonlinear regression of [ −��+1(��+1�

�−1� + �)] on �1�

�2�−1�

�3�

and record new estimates of �1� �2 and �3.

STEP 5: If new estimates of �1� �2 and �3 equal those of STEP 2 then

STOP, model solution found.

STEP 6: If new estimates of �1� �2 and �3 differ from those of STEP 2

9

then repeat STEPS 2-5 using new �1� �2 and �3.

6 Some practical issues

6.1 Will the model always converge to a fixed point?

Marcet and Marshall (1994) have written in detail on this issue. In theory the

answer is that for most economic model we are interested in the model will

converge. Only in the case of models with multiple equilibria or increasing

returns will we found problems of convergence. However, in practice obtain-

ing covergence may not be a trivial process. For instance, if the user chooses

a first order polynomial but this is a poor approximation then the model

may not converge. In this case it is worth experimenting with increasing the

order of the polynomial and/or trying alternative combinations of the state

variables.

6.2 What about non-stationary variables?

A crucial part of the solution algorithm is the nonlinear least squares regres-

sion. Notice that the data used in this regression is a function of the previous

estimates of the �’s. Because of this normal super-convergence results con-

cerning nonstationary variables do not hold and the case of nonstationary

variables you will not get convergence on a fixed point. In our case the data

changes with every set of parameter estimates so there is no reason for para-

meters to converge. Therefore in the case where variables are nonstationary

it is important to detrend all the variables to induce stationarity. This in-

volves replacing all variables �� which contain a trend with their detrended

values �∗� . For instance, assume that ln��+1 = �+ ln�� + ��. In this case all

the variables share a common trend �1�(1−�). In this case we therefore need

to transform everything by dividing through by this trend. Thus instead

10

of the utility function being ln � it becomes ln ∗� and the budget constraint

changes from �+��−��−1 = ��−1 to

∗�+�

∗� −��∗−1�(1−�)�−1 (�+��)

−1�(1−�) =

�∗�−1(�+��)−��(1−�) by dividing both sides through by �. Having transformed

the model to stationarity in this way we then simply derive the first order

conditions for this transformed model and solve using PEA as in the previous

example.

6.3 What convergence criteria should I use on the �’s?

This partly depends on the problem but I have found 0.00001 to be fairly

reliable.

6.4 When I construct {��}� {��} and {��} to run the non-linear regression how many observations should I

use?

The convergence results to do with the learning intuition are all asymptotic.

In other words, it takes a long time to hit the fixed point. As a result you

may need to do simulations/regressions with lots of data. In my experience

I have found this involves using between 40000 and 100000 observations for

each simulation/regression.

7 Testing the accuracy of solutions

When we arrive at a fixed point to the PEA we have found a set of coefficients

such that

��[ −��+1(��+1�

�−1� + �)]− �1�

�2�−1�

�3� = �� ≈ 0

but is this a good solution? In other words, just because we have found a

11

fixed point which minimises a metric of squared residuals this does not mean

that we have an accurate solution to the model. We merely have a solution.

In order to gauge the accuracy of the solution, den Haan and Marcet

(1994) propose a very simple test. This test can be used to gauge the accuracy

of any solution technique and is not dependent on the use of the PEA.

−�� = ��[ −��+1(��+1�

�−1� + �)]

We can therefore define an expectational error

��+1 = −�� − ��[ −��+1(��+1�

�−1� + �)]

According to Rational Expectations this error term ��+1 should be un-

predictable at time �. That is agents do not make predictable forecasting

mistakes. Therefore to assess the accuracy of the model solution all one needs

to do is regress ��+1 on a variety of variables dated at time � or previously. If

these variables have no predictive power then the solution is accurate. If the

variables predict the expectational error then the solution is inaccurate and

the order of the polynomial needs to be increased and the process repeated.

7.1 What variables should I use to try and predict

��+1?

The obvious choices are lags of the state variables. In the case of our model the

obvious candidates would be {��−1� ��−2� ��−3� ��−4� �� −1� ��−2� ��−3� ��−4}.

7.2 How do I assess the ability of these variables at

predicting ��+1?

The details are given in den Haan and Marcet but essentially if the model

solution is accurate then the test statistic for whether these variables have

12

any predictive power follows a chi-squared distribution centred around zero.

Therefore you examine the distribution of the test statistic at its upper and

lower tails and see if it accords with a chi-squared distribution.

7.3 How should I use the accuracy test?

the accuracy test is incredibly powerful at distinguishing between alternative

models. For instance, comparing the stylised facts of the business cycle

form solutions produced by quadratic approximations and from PEA show

relatively small differences. However, the PEA solution comfortably passes

the accuracy test whereas the quadratic approximation fails. Passing the

accuracy test is therefore a very demanding criteria.

8 Example computer code

8.1 PEA.PRG

The main program used for applying the den Haan and Marcet solution tech-

nique. The program searches for the rational expectations solution by first

simulating and then updating the parameterised expectation until conver-

gence.

CLS;

PRINT ” –––––––––––––––––––—”;

PRINT ” PEA.PRG ”;

PRINT ” This program replicates the solution developed by ”;

PRINT ” den Haan and Marcet for the Stochastic growth

economy ”;

PRINT ” ”;

PRINT ” created by Martin Ellison 22.VIII.1997 ”;

13

PRINT ” ”;

PRINT ” –––––––––––––––––––—”;

PRINT;

PRINT ”Intialising variables ...”;;

#include params.prg;

PRINT ” done. Press any key to continue”;WAIT;

PRINT;

PRINT ”STEP 1 : simulate a sequence for the productivity

shock ...”;;

#include shock.prg;

PRINT ”done.”;

PRINT;

PRINT ”Example values of productivity term (t=100:111)”;

PRINT theta[100:111]’;

PRINT ”Press any key to continue”;WAIT;

PRINT;

/* begin the PEA loop. continue until the updated delta esti-

mates (nd)

are close to the previous estimates (od) */

DO UNTIL maxc(abs(nd-od))�0.000001;

od=nd;

/* set the random number generator seed - not essential */

RNDSEED 1;

PRINT ”STEPS 2 & 3: use guess of deltas and productivity

term”;

PRINT ” to solve for consumption and capital.”;

PRINT;

PRINT ”Current deltas:”;

PRINT nd’;

14

PRINT ”Press any key to begin simulating ... ”;;WAIT;

z=streams(nd);

PRINT ”done.”;

PRINT;

PRINT ”Example values of consumption (t=100:111)”;

PRINT z[100:111,1]’;

PRINT;

PRINT ”Example vales of capital stock (t=100:111)”;

PRINT z[100:111,2]’;

PRINT;

ng=onl2;

nl2=0;

PRINT ”STEP 4 : run non-linear regression”;

PRINT ”Press any key to begin regressing.”;WAIT;

PRINT;

PRINT ”Convergence of deltas:”;

PRINT;

PRINT nd’;

DO UNTIL maxc(abs(nl2-nl1))�0.000001;

nl1=ng;

nl2=choose(nl1);

PRINT nl2’;

ng=nl2;

ENDO;

PRINT ”Converged”;

PRINT;

PRINT ”Previous estimate of deltas”;

PRINT nd’;

PRINT ”New estimate of deltas”;

15

PRINT nl2’;

PRINT;

PRINT ”Do you want to test the (A)ccuracy of this solution”;

PRINT ”or (C) continue iterating?”;

PRINT;

ans=cons;

IF ans $==”a” OR ans $==”A”;

fd=nl2;

#include accuracy.prg;

PRINT ”Press any key to continue”;WAIT;

PRINT;

ENDIF;

onl2=ng;

ND=nl2;

ENDO;

z=streams(nD); /* calculate final c and k*/

PRINT ”Final solution:”;

PRINT nd’;

PRINT;

PRINT ”Do you want to test the accuracy of the final solu-

tion?”;

ans=cons;

IF ans $== ”y” OR ans $== ”Y”;

fd=nd;

#include accuracy.prg;

ENDIF;

end;

PROC streams(D);

16

/* calculates simulated time series of c and k given parame-

terisation D */

n=1;

DO UNTIL n�T;

/* define lagged capital lkn */

if n�2;

lkn=k0;

else;

lkn=k[n-1];

endif;

/* calculate the parameterised expectation

c comes from the first order condition

k from the resource constraint */

@ — replace the following with the first order conditions of the

model — @

lw[n]=ln(d[1,1])+d[2,1]*ln(lkn)+d[3,1]*theta[n]

+d[4,1]*(theta[n])^2;

c[n]=exp((-1/u)*(ln(beta)+lw[n]));

k[n]=(exp(theta[n]))*(lkn^a)+m*lkn-exp(lc[n]);

@ ––––––––––––––––––––––––@

n=n+1;

ENDO;

/* return the simulated c and k series */

z=c~k;

RETP(z);

ENDP;

PROC choose(gam);

local tx;

/* fits a weighted regression to the expectations stream */

17

n=1;

DO UNTIL n�T-1;

@ –– replace the following with the outcome of the expecta-

tion ––- @

phi[n]=(c[n+1]^(-u))*(a*exp(theta[n+1])*(z[n,2]^(a-1))+m);

@ ––––––––––––––––––––––––@

/* construct the weighting function */

IF n�2; /* initial conditions */

f[1,n]=ln(k0);

lkn=k0;

ELSE; /* lagged values */

lkn=z[n-1,2];

f[1,n]=ln(z[n-1,2]);

ENDIF;

f[2,n]=theta[n];

f[3,n]=theta[n]^2;

mlt=(1/gam[1])|f[.,n];/* calculate approximation to the expectation */

lw[n]=ln(gam[1,1])+gam[2,1]*ln(lkn)+gam[3]*theta[n]

+gam[4]*theta[n]^2;

/* dependant and independant variables */

dep[n]=(phi[n]-exp(lw[n])+gam’(exp(lw[n]))*mlt);

indep[n,.]=(exp(lw[n])*mlt’);

n=n+1;

ENDO;

/* perform the regression using the standard formula b=(X’X)^-

1(X’Y) */

/* the first tx observations are discarded */

tx=101;

18

inden=invpd(iNdep[tx:t-2,.]’indep[tx:t-2,.])*(indep[tx:t-2,.]’dep[tx:t-

2,.]);

/* return the new estimate */

RETP(inden);

ENDP;

8.2 PARAMS.PRG

Initialises all variables and calibrated parameters needed by PEA.PRG.

@ –––––––––––––––––––––— @

@ PARAMS.PRG @

@ This program initialises all the parameters used @

@ in PEA.PRG to values calibrated from UK data @

@ @

@ created by Martin Ellison 22.VIII.1997 @

@ @

@ –––––––––––––––––––––— @

/* calibrated parameters */

a=0.4436; /* steady-state capital share of income */

beta=0.99; /* discount factor */

u=5; /* risk aversion */

m=0.975; /* depreciation rate */

T=4000; /* number of time periods of interest */

k0=95; /* steady-state capital stock (guess) */

l=1; /* step length for updating deltas */

iter=1; n=1; /* counting variables */

/* initial guess of deltas in parameterised expectation. change

if

algorithm does not converge */

19

delta=ones(4,1);

delta[1,1]=1.1703263;

delta[2,1]=-1.8305086;

delta[3,1]=-1.6100586;

delta[4,1]=-0.26105643;

/* initialise delta vectors */

OD=ones(4,1);

ND=delta;

OG=ones(4,1);

NG=delta;

gam=ones(4,1);

onl2=delta;

/* Variables and parameters used in the procedure to calculate

the

streams of consumption and capital */

theta=zeros(t,1); /* productivity term */

lw=ones(T,1)*0.5; /* approximate of the Euler eqn */

lk=ones(T,1)*k0;lkn=0; /* lagged capital stock */

k=(lk); /* capital stock */

lc=ones(T,1)*0.93; /* logarithm of consumption */

c=exp(lc); /* consumption */

z=c~k; /* z = [ c | k ] contains simulations *//* Variables and parameters used in calculating the optimal

gamma via Non-Linear Least Squares Regression */

phi=ones(T-1,1); /* outcome of expectation */

f=ones(3,T); /* used in construction of mlt */

mlt=ones(4,1); /* weighting factor */

dep=zeros(T-1,1); /* dependant variable Y */

num=zeros(4,1); /* numerator = X’Y */

20

den=zeros(4,4); /* denominator = (X’X) */

inden=zeros(4,4); /* inverse of denominator */

indep=zeros(T-1,4); /* independant variable X */

nl1=10;nl2=0;

8.3 SHOCK.PRG

Generates a random draw of the productivity term.

@ –––––––––––––––––––––— @

@ SHOCK.PRG @

@ This program generates a sequence of length t for @

@ the productivity term with variance v and AR(1) @

@ persistence of rho @

@ @

@ created by Martin Ellison 22.VIII.1997 @

@ @

@ –––––––––––––––––––––— @

v=0.009253;

rho=0.95;

epsilom=v*rndn(T,1); /* calculate the productivity shocks */

theta[1,1]=0; /* initial condition of productivity term */

/* generate productivity term */

count=2;

DO UNTIL count+1�t;

theta[count,1]=rho*theta[count-1,1]+epsilom[count,1];

count=count+1;

ENDO;

21

8.4 ACCURACY.PRG

Tests the accuracy of the parameterisation of the expectation according to

the Marcet (1994) Review of Economic Studies test statistic. This requires

making repeated simulations of the model economy and testing whether t

dated or lagged values of state variables have predictive power for the rational

expectations error. If not, the solution passes the accuracy test.

PRINT ” –––––––––––––––––––—”;

PRINT ” ACCURACY.PRG ”;

PRINT ” This program tests the accuracy of the solution ”;

PRINT ” for the deltas contained in nd ”;

PRINT ” ”;


PRINT ” ”;

PRINT ” –––––––––––––––––––—”;

PRINT;

draw=1;lcount=0;ucount=0; /* initialise variables */

ndraw=10; /* number of draws */

t=500; /* length of each draw */

PRINT ”Proposed solution is”;

PRINT fd’;

PRINT;

PRINT ”Press any key to begin accuracy test.”;WAIT;

PRINT;

DO UNTIL draw�ndraw;

/* make a draw of the productivity term for t periods */

#include shock.prg;

/* use guess of deltas and productivity term to solve for

consumption and capital */

22

loop=1;

z=tstreams(fd);

/* calculate outcome of the expectation */

phi=tchoose;

/* define matrix of regressors. here uses three lags of the

capital

and productivity term */

ah=ONES(1,T-103)|k[103:T-1]’|theta[103:T-1]’|k[102:t-2]’;ah=ah|theta[102:t-2]’|k[101:t-3]’|theta[101:t-3]’;/* calculate the rational expectations error */

res=beta*phi[103:t-1]-c[103:t-1]^(-u);

/* calculate test statistic.

see Marcet Review of Economic Studies 1994 */

ihh=invpd(Ah*Ah’);

ahat=ihh*Ah*res;

mdhm=ahat’*(Ah*Ah’)*invpd((Ah*(Ah’.*(res^2))))*(Ah*Ah’)*ahat;

kkkk=rows(Ah);

format 1,3;

print ”Test statistic is : ” mdhm ”P-value ” cdfchinc(mdhm,kkkk,0);

@dmdhm[1,draw]=mdhm; dmdhm[2,draw]=cdfchinc(mdhm,kkkk,0);@

/* update counts if in upper or lower tail */

IF cdfchinc(mdhm,kkkk,0)�0.05;

lcount=lcount+1;

ENDIF;

IF cdfchinc(mdhm,kkkk,0)�0.95;

ucount=ucount+1;

ENDIF;

FORMAT 2,2;

PRINT ”t=” draw;;

23

PRINT ” lower tail % ” lcount/draw ” upper tail % ” ucount/draw;

PRINT;

draw=draw+1;

ENDO;

PRINT ”Final accuracy results”;

PRINT;

FORMAT 16,8;

PRINT ”Delta tested:”;

PRINT fd’;

PRINT;

FORMAT 3,2;

PRINT ”Number of draws ” draw;

PRINT ” lower tail % ” lcount/draw ” upper tail % ” ucount/draw;

PRINT;

FORMAT 16,8;

PROC tchoose;

/* calculate actual outcome of expectation */

phi=ones(t,1);

n=1;

DO UNTIL n�T-1;

@ –– replace the following with the outcome of the expecta-

tion ––- @

phi[n]=(c[n+1]^(-u))*(a*exp(theta[n+1])*(z[n,2]^(a-1))+m);

@ ––––––––––––––––––––––––@

n=n+1;

ENDO;

retp(phi);

24

ENDP;

PROC tstreams(D);

/* calculate sequence of c and k given delta */

n=1;

z=ones(t,2);k=ones(t,1);lw=ones(t,1);

lc=ones(t,1);c=ones(t,1);

DO UNTIL n�T;

if n�2;lkn=k0;theta[1]=epsilom[1];

else;lkn=k[n-1];

endif;

lk[n]=lkn;


model — @

lw[n]=ln(d[1,1])+d[2,1]*ln(lkn)+d[3,1]*theta[n]+d[4,1]*(theta[n])^2;

lc[n]=(-1/u)*(ln(beta)+lw[n]);

k[n]=(exp(theta[n]))*(lkn^a)+m*lkn-exp(lc[n]);

@––––––––––––––––––––––––-@

n=n+1;

ENDO;

c=exp(lc);

z=c~k;

RETP(z);

ENDP;

8.5 SIMS.PRG

Simulates the model economy. This program can be easily adapted to give

any stylised facts desired. At present it calculates cross correlations and

volatilities by averaging across 50 simulations of 130 periods each. Data is

25

Hodrick-Prescott filtered but this can be omitted if desired. It is also simple

to use this program to give impulse response functions by defining epsilon in

the first simulation to follow a process of 0, 0, 0, 0, 0, , 0, 0, 0, 0, 0, 0 ...

CLS;

PRINT ” –––––––––––––––––––—”;

PRINT ” SIMS.PRG ”;

PRINT ” This program simulates the stochastic growth econ-

omy ”;

PRINT ” ”;


PRINT ” ”;

PRINT ” –––––––––––––––––––—”;

PRINT;

PRINT ”Press any key to begin simulations”;WAIT;

/* define calibrated parameters */

beta=.99;

alpha=.4436;

mu=.975;

sigma=.009253;

tau=5;

/* parameters of the parameterised expectation */

nd = {1.1703263,-1.8305086,-1.6100586,-0.26105643};

ks = 95.499;

/* simulate the model. this program produces 50 simulations,

each of

130 periods in length */

output file=results.out reset;

epsilon = sigma*rndn(131,50);

26

theta = zeros(131,50);

theta[1,.] = epsilon[1,.];

k = zeros(131,50);

c = zeros(130,50);

y = zeros(130,50);

invest = zeros(130,50);

k[1,.] = ks*ones(1,50);

ii = 1;

do while ii�=130;


model — @

theta[ii+1,.] = rho*theta[ii,.]+epsilon[ii+1,.];

c[ii,.] = ln(nd[1])*ones(1,50)+nd[2]*ln(k[ii,.])+nd[3]*theta[ii,.];

c[ii,.] = c[ii,.]+nd[4]*theta[ii,.].^2;

c[ii,.] = exp((-1/tau)*(ln(beta)+c[ii,.]));

y[ii,.] = (k[ii,.].^alpha).*exp(theta[ii,.]);

invest[ii,.] = y[ii,.]-c[ii,.];

k[ii+1,.] = invest[ii,.]+mu*k[ii,.];

@ ––––––––––––––––––––––––@

ii = ii + 1;

endo;

/* take logarithms of the series in preparation for HP filtering

*/

lc = ln(c);

ly = ln(y);

li = ln(invest);

/* Hodrick-Prescott filter the generated series */

n=rows(c);

vv=lc~li~ly;

27

lamda=1600;

format /rdn 10,4;

aa=3*eye(n);

i=1;

DO UNTIL i�(n-2);

aa[i,i+1]=-4;

aa[i,i+2]=1;

i=i+ 1;

ENDO;

aa=aa+aa’;

aa[1,1]=1;

aa[2,1]=-2;

aa[1,2]=-2;

aa[2,2]=5;

aa[n-1,n-1]=5;

aa[n-1,n]=-2;

aa[n,n-1]=-2;

aa[n,n]=1;

screen off;

W=eye(n)+lamda*aa;

IW=invpd(W);

trend=IW*vv;

cp=vv[.,1:50]-trend[.,1:50];

ip=vv[.,51:100]-trend[.,51:100];

yp=vv[.,101:150]-trend[.,101:150];

screen on;

S = zeros(50,20);

S[.,1] = stdc(cp);

S[.,2] = stdc(ip);

28

S[.,3] = stdc(yp);

j = 1;

do while j �= 50;

aa = zeros(5,26);

aa[1:2,1:2] = corrx(cp[2:130,j]~cp[1:129,j]);

aa[1:2,3:4] = corrx(ip[2:130,j]~ip[1:129,j]);

aa[1:2,5:6] = corrx(yp[2:130,j]~yp[1:129,j]);

aa[.,7:11]=corrx(cp[4:130,j]~yp[4:130,j]~yp[3:129,j]~yp[2:128,j]~yp[1:127,j]);

aa[.,12:16] = corrx(ip[4:130,j]~yp[4:130,j]~yp[3:129,j]

~yp[2:128,j]~yp[1:127,j]);

aa[1:4,17:20] = corrx(yp[4:130,j]~cp[3:129,j]~cp[2:128,j]~cp[1:127,j]);

aa[1:4,21:24] = corrx(yp[4:130,j]~ip[3:129,j]~ip[2:128,j]~ip[1:127,j]);

S[j,4] = aa[1,2];

S[j,5] = aa[1,4];

S[j,6] = aa[1,6];

S[j,7:13] = (rev(aa[1,18:20]’)’)~aa[1,8:11];

S[j,14:20] = (rev(aa[1,22:24]’)’)~aa[1,13:16];

j = j+1;

endo;

/* print results */

R = zeros(20,2);

R = meanc(S)~stdc(S);

PRINT ” –––––––––––––––––––—”;

PRINT ” SIMS.PRG ”;

PRINT ” Simulation of the the Stochastic growth economy ”;

PRINT ” ”;


PRINT ” ”;

PRINT ” –––––––––––––––––––—”;

29

PRINT;

PRINT ” StDev(Consumption) ” 100*R[1,.]’;

PRINT ” StDev(Investment) ” 100*R[2,.]’;

PRINT ” StDev(Output) ” 100*R[3,.]’;

PRINT ” Rho(Consumption) ” R[4,.]’;

PRINT ” Rho(Investment) ” R[5,.]’;

PRINT ” Rho(Output) ” R[6,.]’;

PRINT ” Corr(C,Y) ” R[7:13,1]’|R[7:13,2]’;PRINT ” Corr(I,Y) ” R[14:20,1]’|R[14:20,2]’;PRINT ” –––––––––––––––––––—”;

output off;

9 Example printouts

9.1 PEA.GSS

–––––––––––––––––––––—

PEA.PRG

This program replicates the solution developed by

den Haan and Marcet for the Stochastic growth economy

created by Martin Ellison 22.VIII.1997

–––––––––––––––––––––—

Intialising variables ... done. Press any key to continue

STEP 1 : simulate a sequence for the productivity shock ...done.

Example values of productivity term (t=100:111)

-0.034809919 -0.027160032 -0.030133135 -0.018852850

-0.025530091 -0.010038216 -0.013528491 0.0067613673

0.0026377245 0.0099787851 -9.2593289e-005 0.0099178908

Press any key to continue

30

STEPS 2 & 3: use guess of deltas and productivity term

to solve for consumption and capital.

Current deltas:

1.1703263 -1.8305086 -1.6100586 -0.26105643

Press any key to begin simulating ... done.

Example values of consumption (t=100:111)

8.2850372 8.3188592 8.3252443 8.3691233

8.3661706 8.4218103 8.4279932 8.4983040

8.5042629 8.5410426 8.5305513 8.5740324

Example vales of capital stock (t=100:111)

361.74087 363.44014 365.08506 366.86624

368.54229 370.41288 372.21977 374.29027

376.28590 378.36701 380.28957 382.33571

STEP 4 : run non-linear regression

Press any key to begin regressing.

Convergence of deltas:

1.1703263 -1.8305086 -1.6100586 -0.26105643

1.0265682 -1.8145534 -1.5758951 -0.14321741

1.0323259 -1.8141734 -1.5751578 -0.14134240

1.0323314 -1.8141768 -1.5751638 -0.14135557

1.0323314 -1.8141768 -1.5751638 -0.14135559

Converged

Previous estimate of deltas

1.1703263 -1.8305086 -1.6100586 -0.26105643

New estimate of deltas

1.0323314 -1.8141768 -1.5751638 -0.14135559

Do you want to test the (A)ccuracy of this solution

or (C) continue iterating?

c

31

STEPS 2 & 3: use guess of deltas and productivity term

to solve for consumption and capital.

Current deltas:

1.0323314 -1.8141768 -1.5751638 -0.14135559

Press any key to begin simulating ...

9.2 ACCURACY.GSS

–––––––––––––––––––––—

ACCURACY.PRG

This program tests the accuracy of the solution

for the deltas contained in nd


–––––––––––––––––––––—

Proposed solution is

1.0323314 -1.8141768 -1.5751638 -0.14135559

Press any key to begin accuracy test.

Test statistic is : 5.39 P-value 0.388

t=1.0 lower tail % 0.00 upper tail % 0.00











32








t=10. lower tail % 0.00 upper tail % 0.10

Final accuracy results

Delta tested:

1.0323314 -1.8141768 -1.5751638 -0.14135559

Number of draws 11.

lower tail % 0.00 upper tail % 0.091

Press any key to continue

9.3 SIMS.GSS

–––––––––––––––––––––—

SIMS.PRG

Simulation of the the Stochastic growth economy


–––––––––––––––––––––—

StDev(Consumption)

0.3846

0.0459

StDev(Investment)

2.8964

0.3203

33

StDev(Output)

1.1796

0.1324

Rho(Consumption)

0.7182

0.0524

Rho(Investment)

0.6990

0.0533

Rho(Output)

0.7022

0.0529

Corr(C,Y)

0.1696 0.3882 0.6618 0.9885 0.7359 0.5129 0.3246

0.0994 0.0849 0.0501 0.0034 0.0513 0.0910 0.1105

Corr(I,Y)

0.2601 0.4625 0.7096 0.9986 0.6886 0.4272 0.2161

0.1063 0.0900 0.0529 0.0005 0.0526 0.0886 0.1037

–––––––––––––––––––––—

10 Speeding up the PEA

The paper by Christiano and Fisher contains numerous ways of speeding

up the PEA approach. Here we focus on only one such trick. The main

computational cost of the method we have outlined above is the need to run

repeatedly a nonlinear least squares regression involving more than 40000

observations. We shall call this the time series approach to PEA. Essentially

what we are doing is estimating the parameterised expectations polynomial

34

by using lots of data. However, that is not necessary as all we need to do is

solve the model so that

��[ −��+1(��+1�

�−1� + �)] ≈ �1�

�2�−1�

�3�

Solving this equation for the �’s does not necessarily require lots of ob-

servations. This is the motivation behind the grid approach to using the

PEA technique. Rather than generate long time series and solve the model

that way, the grid approach chooses certain values for the state variables and

then chooses �’s so that the PEA is a good approximation to the expectations

term at these values. If the grid values are chosen well (i.e. values which

frequently occur in the model) then these should give good answers to the

model.

To understand more fully this approach go back to our simple stochastic

growth model. In this case we have two state variables - the capital stock

and the productivity term. The point about using a grid is to select a vector

of potential values for the capital stock and the productivity term and then

solve for a fixed point using all combinations of these chosen capital and

productivity terms. For instance, assume that we choose three capital values:

a low, medium and high capital stock and three values of the productivity

term: a low, medium and high value. then we have 9 potential combinations

of capital and the productivity term (all the way from high capital, low

productivity to high productivity/low capital). For each of these grid points

we can calculate � and �� just as we did in the time series approach. However,

by focusing only on 9 grid points we only have to calculate � and �� 9 times

- not the 40000 times we did in the time series approach. The next thing

we have to do is calculate ��[ −��+1(��+1�

�−1� + �)] for each of these 9 grid

points. We already have �� calculated and by using the PEA formula we

can calculate �+1 or at least a guess of it. It only needs a drawing of ��

35

and we can calculate −��+1(��+1��−1� + �). However, we are interested in

the expectation of this term so we have to make lots of different drawings

for �� (in practice I use 500) and average across all of them to arrive at

��[ −��+1(��+1�

�−1� + �)]. At this point we can perform the same nonlinear

regression and iterative procedure to arrive at the fixed point for the �’s.

However, the major advantage is that the regression is now performed over

only 9 observations rather than 40000. In other words, there is an enormous

time saving produced via the grid approach.

10.1 What grid points should I use?

Christiano and Fisher contain a detailed discussion of this point. The most

obvious choice for the productivity shock is to choose its mean and also plus

and minus one standard deviation. For each of these values we then calculate

the steady-state capital stock attached to each of these shocks. This will give

us a lower and upper bound on capital (the worst and best case scenario)

and a mean level. If all we want is three gridpoints for capital this is enough.

However, the more gridpoints used then the more accurate the solution so

we would probably want some additional capital values to be considered in

the grid. In choosing these values we also want to put a heavier weight

on capital stocks close to its mean. These values are likely to occur more

frequently and so it is important that the grid method solves for the values

in this range well. Therefore a range of capital values is chosen between the

lower and upper bounds calculated earlier and via means of a cosine function

these values tend to be clumped together around the mean value.

36

11 Hints and tips

1. The non-linear regression is highly sensitive if the independent variables,

i.e. the state variables in the parameterised expectation, have different orders

of magnitude. To avoid this problem divide the state variables by their

steady state variables so that each variable has an order of magnitude of

approximately one.

2. Obtaining an accurate non-linear regression can be difficult if the

initial PEA simulation is poor so it is best to set the accuracy tolerance of

the non-linear regression loop relatively high, say to 0.001. This can then be

decreased as the PEA continues and the simulation improves.

3. Finding satisfactory starting values can be problematic. The starting

values should be such that the initial simulation has some economic plau-

sibility. The simplest solution is to use the starting values from a similar

model, e.g. use flexible price model parameters for the starting values of an

inflexible price model. If this is not possible then try setting 1 to the long

run value of the parameterised expectation. In simple macromodels this is

usually , the marginal utility of income. In simple macromodels the on the

capital stock, divided by its steady-state value as suggested above, should

be set initially at -1. This is very important. By setting it to -1 we are

introducing some basic economic intuition into the model. If capital stock

is above its steady-state value then the choice of -1 ensures that will fall

and consequently consumption will rise. This is what we would expect the

representative consumer to do and leads to simulations roughly in line with

economic theory. The PEA approach will tend to converge much faster from

this type of simulation.

4. When choosing which state variables to use when parameterising an

expectation it is preferable to use a Specific to General approach. This has

the advantage initially that computational time is less. In addition, the

37

algorithm tends to converge at greater speed without complication quadratic

and cubic terms which may be insignificant. The most parsimonious linear

approximation may be the best starting point. It can always be tested using

the accuracy test and higher order terms incorporated.

5. When adding extra state variable terms to the set of dependent vari-

ables it is advisable to use some economic intuition. For example, real shocks

would tend to be crossed with real variables and nominal shocks with nomi-

nal variables. Again the accuracy test will show whether these are desirable

or necessary.

6. When de-bugging it is useful to set the variance of all shocks to zero

and start off the economy from its steady state. Look then at the first

couple of simulations. If the economy is started from its steady state and

no shocks applied then it would be expected that the economy will still be

at its steady state in the next period. The capital stock in period 2 should

then be almost identical to that in period 1. This provides a useful check

on the programming up of the model. If the steady state is not maintained

then there is an error in the original mathematics, programming or steady

state calculations. It is usually easy to proceed to PEA when the model

is properly programmed. In my experience most of the problems relate to

human error than the algorithm. Do not worry though if this procedure

does not give exactly the same capital stock value as before. It may be that

the way the model has been set up via the parameterised expectation means

that the steady-state equilibrium is initially unstable. If for some reason the

initial steady-state calculations are infinitesimally away the steady state then

the economy may move away from steady state even if there are no shocks

occurring. This is not a cause for concern, in general the PEA will correct

this and update the parameterised expectation in such a way that the final

equilibrium is stable.

7. In connection with the above it has been suggested that the model can

38

be solved by first solving for a very small variance of any shocks so that the

model is almost deterministic. The values of the parameterised expectation

can then be used as starting values as the variance is increased.

8. It is useful to set the number of simulations to a low value when first

estimating the model. This saves time whilst debugging. A rough parame-

terisation can be obtained with a low number of simulations and then this

used as starting values for the full estimation.

9. To enable results to be replicated it is always advisable to set the

random number generator seed to a specific value. This ensures that the

random draw of any shocks is identical from estimation to estimation and

helps the PEA algorithm to converge. It creates no problems for the algo-

rithm because the parameterised values still have to pass the accuracy test.

The point is that the parameterisation will inevitably be slightly different for

different draws of the random shocks. Setting the random number generator

seed eliminates this unnecessary variation.

10. If there is more than one expectation in the model to parameterise

then initially hold one of the parameterisations constant whilst optimising

the other. Then hold the other constant and estimate the other and so

on. Only after a number of iterations such as these is it worth attempting

to estimate both parameterisations simultaneously. This ensures that the

simulations remain sensible and the algorithm converges.

11. Stylised facts can obviously be constructed from inaccurate solutions

that have not passed the accuracy test. Whilst these facts cannot be inter-

preted rigorously they do give some idea as to whether the model is working

correctly. For example, is consumption smoother than output as expected in

most of these types of model?

12. Problems tend to arise if the expectation to be parameterised also

contains known values of the state variables. In this case it is preferable to

remove the known variable and parameterise the rest. For example rather

39

than parameterise � = �� −1, where is � unknown but ��−1 is already

known, parameterise 0� = ��−��−1 = �� . This avoids some multicollinear-

ity problems.

40

pea algorithm

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