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o r k m g r a p e r s e r ie s



Solving Nonlinear Rational Expectations
Models by Parameterized Expectations:
Convergence to Stationary Solutions
Albert Marcet and David A. Marshall

Working Papers Series
Macroeconomic Issues
Research Department
Federal Reserve Bank of Chicago
December 1994 (WP-94-20)

FEDERAL RESERVE BANK
OF CHICAGO

Solving Nonlinear Rational Expectations M o d e l s
b y Parameterized Expectations: C o n v e r g e n c e to Stationary Solutions

Albert Marcet
Universitat Pompeu Fabra, Barcelona

David A. Marshall
Federal Reserve Bank of Chicago and
Northwestern University

March 1994

W e thank Larry Christiano, Wouter den Haan, Joan Ketterer, To m Sargent, Ken Singleton, and
three anonymous referees for helpful suggestions. The first part of this paper was presented at
Stanford University in January 1989; thanks are due to its attendants, including Ken Judd, Rody
Manuelli, John Bryant and Richard Rogerson. Marcet’s research was partially supported by the
NSF and the DGICYT (Spanish Ministry of Education). All errors are our own.




Abstract

This paper develops the Parameterized Expectations Approach (PEA) for solving nonlinear
dynamic stochastic models with rational expectations. The method can be applied to a variety of
models, including models with strong nonlinearities, sub-optimal equilibria, and many continuous
state variables. In this approach, the conditional expectations in the equilibrium conditions are
approximated by flexible functional forms of finite elements. The approach is highly efficient
computationally because it incorporates endogenous oversampling and Monte-Carlo integration,
and it does not impose a discrete grid on the state variables or the stochastic shocks. W e prove
that PEA can approximate the correct solution with arbitrary accuracy on the ergodic set by
increasing the size of the Monte-Carlo simulations and the dimensionality of the approximating
family of functions.










1

Introduction

During the last decade, the use of dynamic stochastic models has extended
to all fields of economics. These models are difficult to analyze because they
typically have no analytic solution. However, recent increases in the power
of computer hardware now allow these models to be studied by numerical
simulation techniques. Simulations can be used to study the model from
both a theoretical and empirical perspective. Theoretically, one can see if
the model reproduces some stylized facts and how it responds to a change in
the environment or in policy. Empirically, one can perform calibration exer­
cises or do more formal testing using estimation by simulation or maximum
likelihood procedures.
The numerical algorithms most widely used for solving dynamic mod­
els with rational expectations in economics are value-function iteration and
linear-quadratic approximation. They are based on dynamic programming,
so they are not well-suited to models in which the equilibrium does not cor­
respond to the solution of a planner’s problem. Furthermore, value-function
iteration is affected by the ’curse of dimensionality’, so is impractical for
models with several continuous endogenous state variables. Linear-quadratic
approximation can handle large models, but does not provide an arbitrarily
good approximation in nonlinear models; this is especially a problem, for
example, when the model has inequality constraints.
This paper introduces the parameterized expectations approach (PEA)
for calculating numerical solutions to stochastic nonlinear models with ratio­
nal expectations. In this approach, the equilibrium conditions of the model
are written as a system of stochastic difference equations, and the conditional
expectations in these equations are parameterized with flexible functional
forms of finite dimensions, such as polynomials or splines. Simulations are
then generated using these functions in place of the conditional expectations
in the equilibrium conditions. A PEA solution corresponds to a parameter­
ization of the conditional expectations that is consistent with the series it
generates.
In this paper, we formally characterize the PEA, and we derive a set
of conditions under which arbitrarily good accuracy can be obtained as the
solution is refined (for example, as the degree of the polynomial goes to
infinity); it contains some of our earlier work in Marcet [1988], Marshall
[1988] and Marcet and Marshall [1992].
3

The PEA algorithm has been successful in delivering solutions to a num­
ber of large and complicated models. Applications in the literature include
monetary asset pricing models, such as Marshall [1988, 1992], and den Haan
[1990, 1991]; models of exchange rates, such as Bansal [1990], Bansal, Gal­
lant, Hussey and Tauchen [1994] and McCurdy and Ricketts [1992]; asset
pricing models with heterogeneous agents, such as Marcet and Singleton
[1989], Ketterer and Marcet [1989], and den Haan [1993]; models of tax
policy, such as Rojas [1991], and Otker [1992]; and nonstandard stochastic
growth models, such as Marcet and Marimon’s [1992] model with participa­
tion constraints and Christiano and Fisher’s [1994] model with investment
constraints. A related method in which the laws of motion of endogenous
variables are parameterized has been used to solve asset-pricing models in
papers by Heaton [1993] and Bekaert [1993]. This non-exhaustive list of
PEA applications includes models with many continuous state variables,
multiple heterogeneous agents, strong non-linearities, inequality constraints,
incentive constraints, participation constraints, non-stationarities, discrete
choice spaces, and suboptimal equilibria. In short, the PEA has been roadtested: its practical applicability to a wide range of economic models has
been demonstrated . 1
We show that the PEA delivers an arbitrarily close approximation to the
true equilibrium as the solution is refined. The set of assumptions we impose
is very general: essentially, our proof applies to most models with continu­
ous laws of motion (including non-differentiable cases). To our knowledge,
no such proof is yet available for other methods that compute approximate
equilibria by solving systems of Euler equations. Our solution procedure is
easily adapted to cases where one has to solve for the transition path to the
stationary distribution starting from an arbitrary initial condition, but in
the current paper we concentrate our discussion on solving for the stationary
and ergodic distribution.
In section 2, we formally introduce the parameterized expectations ap­
proach and discuss how it can be applied to three well-known examples. In
section 3, we relate PEA to two strands of economic literature. We first dis­
cuss PEA’s relationship to models of least-squares learning; we then compare
'This is to be contrasted with proposals to use numerical methods taken from other sci­
ences, without demonstrating their applicability to the problems of concern to economists.
For example, Judd (1993) lists a number of numerical techniques used in engineering and
physics that have not been tried in full scale economic applications.




4

our method to other solution algorithms used in economic applications. In
section 4 , we present the convergence result. Section 5 provides a brief dis­
cussion of certain practical considerations in applying the method. Section
6 contains some conclusions and suggestions for extensions.

2

T h e Parameterized Expectations A p p r o a c h
( P E A )

2.1

A General F r a m e w o r k

We assume that the economy is described by a vector of variables z t € Z C
R n , and a vector of exogenously given shocks u t 6 U C R s . The equilibrium
process {z(,u t} is known to satisfy a stochastic difference equation
<7(£<[<£(z<+i )], z t , z t - i , u t ) =

0

(1 )

for all t, where g : R m x R n x R n x R 3 —» R q and <j> : R n —> R m . Once
the parameters of the model have been fixed, g and <f> are functions known
to the economist. The vector z t includes all endogenous variables, as well as
those exogenous variables that appear inside the expectation. The process
u t is assumed to be Markov of order one. As usual, E t is the conditional
expectation given information up to period t .
The system ( 1 ) will typically include Euler equations, resource constraints,
laws of motion of exogenous processes, market clearing conditions, incentive
constraints, and so forth. Inequality constraints of the form
h { E t [<f>(zt + i ) ] , z t ,

Zt_i, ut) > 0

are incorporated into ( 1 ) by including equations
(A(£t[<£(zt+i)],zt,zt_i,u())" = 0

as part of the system g , where (a:)- is the function ‘negative part of x \ These
inequality constraints may include Kuhn-Tucker conditions, second order op­
timality conditions, or participation constraints in incentive problems.
Often, the sufficient conditions for an equilibrium include, in addition to
equation ( 1 ), a set of transversality conditions. For most models, a process




5

{ z t } that satisfies ( 1 ) also satisfies the transversality conditions if and only if
the process is stationary. Stationarity can be imposed as an additional side
condition of the model.
We assume that the past information that is relevant for predicting <j>(zt+1)
can be summarized in a finite-dimensional vector of state variables x t € X C
R } satisfying
x t = f ( z t- u u t) ,
(2 )

where / is a known function. 2 This implies
Et[4>(z t + 1 )]

= E [< f > {z t+ i)\x t\.
(3)
Furthermore, we assume that the model is recursive, so that the conditional
expectation is given by a time-invariant function £ such that
(4)

£ ( x t ) = E[<f>(zt + i ) \ x t}.

By definition, the function
£ =

£

satisfies

arS {h ™1k»}

E

I* ^ 2 ‘+i)

II2

~

all

(5)

since the conditional expectation is the best predictor in the mean square
sense.
Throughout the paper we refer to a s o l u t i o n as a stochastic process {zt(ut}
satisfying equation (1), (2), and (3), given the exogenous Markov process for
{u<} . Since we will are dealing with systems g that are invertible in their
second argument z t , finding a solution is equivalent to finding a function £
such that, if { z t , u t } satisfies
g ( £ { x t ), z ti z t - u i i t ) =

0,

(6)

then £ satisfies (4) for all t . Alternatively, finding a solution is equivalent to
finding a law of motion H such that the z t generated by
zt -

H ( z t- U u t )

(7)

2We are assuming here that the researcher knows how to select a set of sufficient state
variables i<. Sometimes it is difficult to determine the sufficient state vector z t beforehand;
this is often the case, for example, in models with private information. In principle, it is
possible to apply PEA and find a sufficient set of state variables numerically; we would
then have to incorporate into the algorithm a search over functions of past variables that
summarize past information. We do not pursue this avenue any further in this paper.




6




satisfies (1), (2) and (3).
We now show how three well-known models fit into the above framework.
These examples are chosen for their simplicity; applications of PEA to more
complicated models can be found in the papers cited in the Introduction.
• E xam ple 1.1 (Lucas [1978] asset pricing m odel)
A representative consumer chooses a stochastic consumption process
{ct} of a perishable good in order to maximize E q Y ^ o
° t )• The
single asset in the economy is traded in a competitive market at price p t ,
and pays an exogenous dividend flow {d(}. The consumer also receives
exogenous labor income {u;t} . The Euler equation for the maximization
problem of the consumer is
u ' { c t ) p t = 8 E t [ u ' ( c t+ 1 )

(pi+1 + dt+i)].

(8 )

To map this model into the above framework we let z t = (p<, c t, d t), and
U( = (d t w t); the system of equations corresponding to ( 1 ) is given by
(8 ) and the market clearing condition c t = d t + w t . The function 4>{z)
is given by u ' ( c ) ( p + d ) . A time-invariant solution for the asset price
can be found for which x t = (d t , w t) is a sufficient set of state variables.
• E xam ple 1.2 (Sim ple S tochastic G row th M odel)
Consider the simple growth model where an agent maximizes E o
subject to
it ^
k t = k t- i p

—

8 lu (c t)

/g\

+

it

,

given,

' '

where c< denotes consumption, k t is the capital stock, it is investment,
and 6 t is an exogenous stochastic productivity shock, Markov of order
one. The first order condition for optimality is
u'(ct) =

8 E t[

^(ct+i)

( k t ~ 1a O t + 1

+

p )]

(10)

To map this model into the above framework, set z t = (ct, k t , k t- i , i t , 0t),
and x t = ( k t- i , 0 t ) . The function g is given by the resource con­
straints (9) and the Euler equation (10). The function <f>{z) is given by

ut = Of,

7




ti'(c) (fco - 1 a0 + f i ) . Standard results from dynamic programming guar­
antee that the solution is characterized by a time-invariant conditional
expectation function.
%

• E xam ple 1.3 (Sim ple G row th M odel w ith Lower B ounds on
In v estm en t)
This example shows how inequality constraints can easily be handled
by PEA. Suppose we add a non-negativity constraint to the simple
growth model of example 1 .2 :
t<>0

(11)

With this restriction, the first order condition ( 1 0 ) is replaced by the
Kuhn-Tucker conditions
u'(ct) - A, -

6 E t[

u'(ct+1) (fc“ ‘ q Oi+i + /x) -

fiX t+ i]

= 0

(12)

tjAf = 0

(13)

At > 0,t't > 0,

(14)

where At denotes the Lagrange multiplier associated with constraint
( 1 1 ). Mapping this model into framework ( 1 ) is accomplished by setting
zt =
u t = 0 t . The system g is given by (9), the
Kuhn-Tucker conditions (12) and (13), and the negative parts of the
inequalities in (14). Note that, in this case, <f>(z) = u (c) (A:“- 1 a0 +
fi) —fiX . Again, the model can be shown to be recursive with standard
dynamic programming techniques.3

2.2

Definition a n d Calculation of a n A p p r o x i m a t e P E A
Solution

System ( 1 ) and explicit formulas for g and <j> are easy to find in many mod­
els. Unfortunately, finding the solution analytically in any generality is not
possible. The difficulty is that S cannot be determined unless the process
3The application of PEA with inequality constraints was developed by Marcet and Sin­
gleton [1991]. For a more detailed discussion of how to impose the Kuhn-Tucker conditions
the reader is referred to that paper.
8




for z t is known, but that process can not be backed out from (6 ) unless the
conditional expectation £ is known. The PEA algorithm addresses this dif­
ficulty by replacing £ with an approximating function ip which is chosen to
resemble £ in a manner to be made precise.
Formally, let P denote a class of functions that is dense in
: R l —►f?m|.
It is assumed that each element of P is characterized by a parameter vector /?
with a finite number of nonzero elements. For example, P can be taken as the
set of polynomials, splines, neural networks or finite elements. Then, (3 would
be, respectively, the coefficients of the polynomial, the parameters charac­
terizing each partition for the spline, the elements of the neural net, or the
parameters characterizing the finite elements. The element of P characterized
by parameter vector /? is denoted either by
•) or by the simpler notation
rpp. The s e t o f a d m i s s i b l e p a r a m e t e r s with at most v non-zero elements is
denoted A * That is, A c{/? (E R ° ° : i th element of /? is zero if i > j/|.
In PEA, the function £ is approximated by choosing an element of P
that satisfies a property close to (6 ) and (5). More precisely, for a given
positive integer v , /? € A> and {u<}~0, define the process (zt(/3 ), u t } ~ 0 as
the solution for all t of the system
9

(W , *<(/?)), z t ( 0 ) ,

x t{P )

z t - 1 (/?), ttt)

= 0

(15)

/(z»-i (/?),«<)

=

Now, let
£ „ (£ ) =

argmjn

E

||

<j>(zt + l (/3 ))

-

x t ( 0 ) ) ||2

(16)

We wish to choose € A as close as possible to G v (f3v ) , so the approximate
PEA solution is characterized by a fixed point, denoted /?„> as follows:
0, =

<A(&).

(17)

For the sake of simplicity we will assume in the remainder of this section that
G u is a well-defined mapping, and that a fixed point of G „ exists; these issues
will be addressed formally in section 4. With the appropriate substitutions,
one can see that xppv plays the role of £ in (6 ), so our approximate solution
is the process {zt(^),u< }^0.
9




From this point on, we address the problem of approximating the solution
at the stationary and ergodic set. This is an interesting case, since station­
a r y and ergodicity are the basis of time series econometrics or calibration
exercises. PEA has been applied to solutions outside of the stationary distri­
bution by replacing the long simulation in Step 2 below with many repeated
short-run simulations.4 Therefore, the claim in Judd [1992] (page 447) that
PEA can not solve for equilibria outside the ergodic set and it can not be
used for policy comparisons is incorrect.
We now give an operational algorithm for computing the approximate
PEA solution for a fixed v:
• Step 1 : Write the system g in ( 1 ) so that it is invertible with respect to
its second argument. Find a set of state variables x that satisfies (3). If
a unique solution exists, g and x should be selected so that (1) and (3)
are satisfied only by the unique solution. Replace the true conditional
expectation by the finitely-parameterized function
•) for (3 € D v
to obtain (15).
• Step 2: Fix Z q . For a given yd € D„ and for T large enough, draw
a sample of size T of the exogenous stochastic shock and recursively
calculate{z*(/?), u<}£_0 using (15).
• Step 3: Find the sample version
defined by:
G>,t (£)=

argmin

of G v. More precisely, G v<
t is

Ef=o II ^(*<+i(£)) -

£

^(£»*t(0)) l|2

(18)

This minimization is easy to perform by computing a non-linear least
squares regression with the sample {.?*(/?), ut} ^ 0, taking
as
the dependent variable and V>(*,xt(/?)) as the explanatory function.
• Step 4: Find the fixed point
P v,T =

G „ , t { ^ v ,t )

(19)

We will discuss the numerical calculation of 0„tT in the section 5.

4See Marshall [1988] and [1993], Marcet and Marimon [1992], Rojas [1993] and Christiano and Fisher [1994] for applications. Also, see Marcet and Marshall [1992] for a proof
of convergence when short-run simulations are used.
10




We end this subsection by discussing how this algorithm would be applied
to the three examples presented in section 2 .1 .

• E xam ple 1.1 (Lucas [1978] asset pricing m odel)
The only non-trivial endogenous variable is the stock price p t . In this
and the remaining examples, <f>takes only positive values, so it is ap­
propriate to choose P as a set of functions that can take only positive
values. Step 2 is easily accomplished by solving for the price in each
period from

u'(c«)

P t(P ) =

ip (P ,d u W t)

which is the (15) version of the Euler equation. The fixed point can be
found iteratively5. Notice that our choice of state variables guarantees
that we are approximating the unique stationary (non-bubble) solution.
Bubble solutions could be allowed by adding today’s price to the list
of state variables.

• E xam ple 1.2 (Sim ple S tochastic G row th M odel)
Steps 1 and 2 follow similar considerations to the previous example.
Since the solution is a time-invariant function of ( k t- i , 0 t) and it satisfies
the Euler equation, we can be certain that the solution approximates
the unique solution of the model. Notice that the (suboptimal and
meaningless) solutions to the Euler equation that violate the transversality condition will never be approximated; in those solutions Co is
fixed, and k t is a time-invariant function of (&t_ ! ,0£) and c£, so that
the transversality condition is imposed by our choice of state variables
and the fact that initial consumption is not fixed.

• E xam ple 1.3 (Sim ple G row th M odel w ith Lower B ounds on
In v estm en t)
We show how to use the Kuhn-Tucker conditions to find (ct(/?), *<(/?), k t ( 0 ) }
while imposing inequalities (14). Let us write the parameterized version
of (12) and (14):
u/(ct(/?)) - A,(/?) = 6 *(/?; Jfc*_,0®), 0t)

(20)

5For this model, the mapping G y as well as the fixed point p y can be found analytically.
See Marcet [1988]
11




At(/3) > 0
Notice that, for a fixed value of
increasing in t<(/5).

and *<(/?) > 0

(21)

; kt-i(/3 ),6 t), the left side of (21) is

We can then proceed as follows: for each

t:

(a) compute (ct(/?), *<(/?)) from (20) under the conjecture that At(fi) =
0. If the corresponding investment turns out to be negative, then
(b) set i t ( P ) = 0, find c t ((3) from the feasibility constraint, and then
compute At(/?) from (20).
It is clear that if step (a) delivers a negative investment, consumption
will be lower when we go to step (b), so uf(ct(/3)) will be higher (relative
to (a)). This insures that (20) delivers a positive A, so (21) is satisfied.
This strategy requires that the Kuhn-Tucker condition be written in a
way that the function g is envertible. For example, the Kuhn-Tucker
condition could be expressed as,
u'(ct)ct- Atct = 6 E t [c< u'(ct+i) (k?~l a6t+i + /*) - ^ctAt+1]

(22)

However, if steps (a) and (b) were applied to (22), there may be states
in which the function u'(c)c is decreasing in i. In that case, step (b)
would deliver a negative value for the multiplier A. Therefore, (22)
would be an inappropriate choice for g if PEA is used as a solution
algorithm. In terms of the step-by-step description of the algorithm,
(22) violates step 1 , since it implies a representation for g that may not
be invertible with respect to its second element.

3

Relation

of P E A

3.1 A p p r o x i m a t e P E A
Learning

to t h e

Literature

Solution a n d Least Squares

In section 4, below, we show that the PEA solution can approximate the
rational expectations equilibrium arbitrarily well by letting u and T go to
12




infinity. In this subsection we give an alternative interpretation to the PEA
solution for fixed v as the equilibrium of an economy with boundedly rational
agents.
Suppose it were infeasible for the agents to compute the true conditional
equilibrium function £. For example, agents may not know the correct func­
tional form for 6, or they may face computation constraints. Instead, agents
are constrained to use some function rfip to forecast of <f>(zt+i). In principle,
they can choose any parameter P € D„, but they are restricted to a fixed
degree v. The process {zt(P)} can be interpreted as the vector of endogenous
variables that would be generated by such an economy. If these agents were
to choose an arbitrary parameter vector P € D the economy would not be
in equilibrium: the best forecaster of <f>(zt+1) within the set D u would in fact
be given by i p ( G u( P ) , x t ( P ) ) , rather than by i p ( / 3 , x t ( ( 3 )), the function used
by the agents ex ante. Presumably, agents would eventually recognize their
systematic errors and would update the parameters in their forecast function.
The equilibrium of this boundedly rational economy corresponds to /?„, the
fixed point of G v defined in equation (17): if the boundedly rational agents
use (3V they will eventually realize that this is their best alternative, given
that they are restricted to staying in Dv.
For this model of boundedly rational agents to be meaningful economi­
cally, it should be locally stable: if agents start at t = 0 using some beliefs
Po near the fixed point /?„, and update ft as new information received at each
period (so their best forecast at time t is given by il)(P t,xt-1 )), their forecast
coefficients should converge to /?„. One way to model how agents update their
forecasts is to assume a least squares learning model (LSL), in which p t is
generated by a non-linear least squares recursive algorithm (see Appendix 2 ).
Adapting some results in Ljung [1975], under certain regularity conditions,
it is possible to show the following results:
• For P G £)„, if P 7^ P „ , P r o b ( P t —►/?) = 0. That is, LSL will almost
surely not converge to a parameter vector that is not a fixed point of
• Prob(Pt —►p v) = 1 locally, if and only if the differential equation
0 = G,{P) - P

13

(23)




is locally stable at /?„. That is, local stability of equation (23) is equiv­
alent to local stability of the equilibrium of boundedly rational agents
under least squares learning.
The second result will be used in section 5.1 to suggest some algorithms for
computing the approximate solution.
To prove the convergence result in Proposition 1 , we must impose suf­
ficient conditions to insure that if a law of motion H * is close to the true
solution H in (7), then {z*} (the process generated by H*) is close to {z(}
(This property is implied by assumptions 4* and 4, in section 4.) Not surpris­
ingly, this property is one of the regularity conditions that must be imposed
for least squares learning to converge. Intuitively, if two H 's are close but
generate z ’s very far apart, agents will be chasing a moving target; their
observations on z will not tell them how to update their forecast function. A
condition like this is likely to be needed for any learning scheme to converge,
as long as the learning scheme is only based on observations of the realized
process.

3.2

R e l a t i o n to O t h e r A l g o r i t h m s .

PEA has developed from other solution approaches used previously in eco­
nomics. It is closely related to the backsolving procedure of Sims [1985],
Novales [1991] and Ingram [1990]. In this approach, conditional expectations
are fixed by assuming a process for {<£(zt+i)}. The remaining variables, in­
cluding exogenous processes, are then solved from a system like ( 1 ) using
this assumption. Backsolving can be understood as PEA without Steps 3
and 4, with particular assumptions on the whole process for <j>(zt+1 ). An­
other predecessor to PEA can be found in Townsend [1983] in a linear model
with private information; that paper uses v = 1 and replaces Steps 2 and 3
with an explicit calculation of Gi based on spectral densities, in which the
linearity of the model is exploited. The idea of calculating the mapping G i
with long-run simulations was previously discussed in Marcet and Sargent
[1989].
Other authors have used approximations of Euler equations with flexible
functional forms more recently. Examples can be found in Coleman (1988),
Judd (1992), Baxter (1991) and McGratten (1993); some of these methods
are based on numerical methods used in other sciences such as engineering
14

or physics. We now relate these methods to PEA in terms of the framework
and the notation laid out in section 2. Define £(x; P ) as the analogue to £
for the P process. More precisely,
£(x; P ) =

E[

<j>{zt+l{p))\

x t(P )

=

x

],

(24)

so, if F (• | u) denotes the distribution of ut+i conditional on u t = u, (24)
implies
.£ (/( * ,« );/? ) = I

iF (u '\u ).

Ju

Assuming that
follows:

x

= z, these solution procedures can be summarized as

• Choose a class of functions P and a degree of approximation v . Param­
eterize the law of motion by an element ^t € P ,‘ this is equivalent to
parameterizing expectations as
H P , z) =

(*■(/?» z >

u)

(25)

where g f 1 represents the inverse of g with respect to its first argument,
evaluated at g = 0 .
• Fix a grid of p points (x 1, x 2, ..., xp) in the state space Rf and a set of
weights to1,to2,...,top.
• For a given /?, calculate £ (x J; P ) numerically for each j . As we can see
from (24), this involves calculating p integrals over R 3 (recall that s is
the number of exogenous shocks in ut).
• Find /?„ that solves

P* =

arg {min}

Ejf= i

II £(*J;P)

~ H P , *J)

II* •

(26)

The difference ||£(xJ;/?) —i p ( P , H ) \ \ is often called the ’residual’ of the
Euler equation at grid-point j , and a generic name for these methods is
’Minimum Weighted Residuals’. The methods differ in the way the approxi­
mating class P , the grid-points, and the weights are chosen, how the system




15




is written, the way iterations are performed to find the above minimum,
and in the method for computing the integrals involved in £ ( x ; f l ) .
It is clear that the steps involved in PEA are similar to the steps in MWR
methods. In particular, (26) is analogous to (18). There are, however, several
fundamental differences:

g

1.

In MWR, the grid-points and weights are exogenous in the sense that
they are independent from the approximating parameter /?. Notice that,
in PEA, the sum in (18) is evaluated at the generated series, so that
{ x t ( P ) } J = \ plays the role of the grid, and the weights are given by the
empirical probability of each value of x in the simulated series; hence,
the grid and weights are chosen by the algorithm, as a function of fS.
This is known as e n d o g e n o u s o v e r s a m p l i n g .

2. The conditional expectation involved in S is never calculated explicitly
in PEA. Instead, the approximation in (18) tries to find ^ close to
<f>(zt+ 1 ), so that the calculation of the r e s i d u a l (a fundamental step in
MWR methods), is entirely bypassed.
3. all integrals are calculated by Monte-Carlo instead of quadrature.
The endogenous oversampling feature implies that PEA only pays atten­
tion to those points that actually happen in the solution. This avoids the
problem that researchers face when they must choose grid-points without
knowing which points are likely in equilibrium. With endogenous oversam­
pling, only the economically relevant region of the state space is explored, so
no computer time is spent on states that never happen in equilibrium. Fur­
thermore, the resulting approximation fits more closely at those states that
happen more frequently. This is one reason why PEA has been successful in
models with a large number of state variables. On the other hands, methods
based on exogenous state-space grids are impractical for such models, since
the number of points in the grid increases exponentially with the number of
state variables.
In numerical analysis, the grid (x1, x2, ..., x p) is sometimes adapted in
the course of the algorithm as a result of the iterations on /?. This is a
form of endogenous oversampling. Unfortunately, these procedures have not
yet been used by economists applying MWR. Indeed, some economists have

16

claimed that endogenous oversampling is a bad feature of an algorithm 6 . In
fact, endogenous oversampling is an important feature of many algorithms in
numerical analysis, and it is likely to be an essential element in any algorithm
for solving models with many state variables.
By eliminating the integral computations involved in the calculation of the
residual, PEA further reduces computation time, since standard quadrature
integration is often very costly even in two-dimensional problems. (Notice
that, strictly speaking, this makes PEA a non-MWR method.) The use of
Monte-Carlo integration means that the integrals can be calculated even in
models with a large number of stochastic shocks.
PEA parameterizes the conditional expectations directly, while economists
using MWR often parameterize the laws of motion it. Formally, the two al­
ternatives are equivalent: we showed in section 2 how to find a law of motion
consistent with V’/3j equation (25) shows how to find V’/s consistent with a
proposed law of motion it. However, parameterizing conditional expectations
often has some practical advantages: i ) sometimes we know that x contains
fewer variables than z , so that /? contains fewer parameters; i i ) when the
degree of the polynomial is increased, one can see if some higher-order ele­
ments need to be introduced beforehand by testing the predictive power of
those elements7; H i) if the shocks have a continuous distribution, the condi­
tional expectation is obtained by integrating over H , so that £ is likely to
be smoother than H and easier to approximate with low degree polynomials;
i v ) inequality constraints are often easier to impose when the conditional
expectation function is parameterized8.
The method in Heaton [1993]9 fits into the framework of Section 2 except
that the approximate solution satisfies

y

£ W *+.(fe r» - -

H

A

.

J

=o

Simple algebra shows that this is exactly equivalent to PEA with the pa­
rameterization (25) if the objective function defining G „ in (9) is modified
6See, for example, Judd [1992].
7See den Haan and Marcet (1994) for a full description of this idea.
8Christiano and Fisher (1994) make this point in their comparison of solution techniques
in a model close to our Example 1.3.
9Also used in Bekaert [1993].




17




to E \ \ [<f>(zt+ i ( 0 ) ) - r J > (£ ,X t( 0 ))] h ( z t ( 0 ) ) ||2, where h = |d0 1 1 /d*'|
. In
other words, Heaton’s method is a special case of our algorithm where Step
3 is modified to be a w e i g h t e d non-linear least squares minimization, and
where h- is used as a weighting function. Heaton’s method performs endoge­
nous oversampling, it uses Monte-Carlo integration, and it does not calculate
any residual. A small modification of our convergence proof accounts for this
case, but is not included in this paper. 10

4

C o n v e r g e n c e to the A p p r o x i m a t e Solution
as

v — > oo

Convergence will be proved in the strong sense that the approximate law
of motion H p v , defined in (28) below, converges uniformly to the true law
of motion H on the support of the stationary distribution. Proposition 1
establishes the convergence result when v and T are chosen sufficiently large.
Proposition 2 provides a partial converse to proposition 1 .
A formal proof of convergence is essential for any proposed approximation
method. It is not sufficient to show that typ can, in principle, approximate
any fixed function arbitrarily well. In PEA, the approximating function xjjp
is required to approximate a conditional expectation function £(•; 0 ) which
itself depends on 0 through the simulated process {z*(/3), x t { 0 ) } . In other
words, £(•;/?) is a moving target.
Furthermore, one can easily construct examples of plausible approxima­
tion schemes which fail to converge to the function being approximated. For
example, Judd (1994) points to a standard non-convergence example where,
if a polynomial of degree v was fitted exactly at v equally-spaced grid-points,
the approximation becomes arbitrarily bad as v becomes larger. He suggests
that PEA may suffer from a similar problem of nonconvergence; Proposition
1 formally disposes of Judd’s criticism . 11
10Details on the algebra and the convergence proof for the weighted case are in Marcet
and Marshall [1992].
u In any case, this example does not apply to PEA for two reasons: i) in the example, the
function is evaluated at v points, while PEA evaluates the function at T points, where T is
much larger than v \ ii) the v points in the example cited by Judd are chosen independently
of /?, while in PEA this task is endogenous. If anything, this non-convergence example may
be indicative of convergence problems for algorithms, including some suggested in Judd
18

To demonstrate convergence of the approximate solution, several con­
ceptual issues must be addressed: i ) the class P has to be dense in the
appropriate space, i i ) Compactness of the function space P is needed in
order to:guarantee that a sequence of approximators contains a convergent
subsequence, H i) In general, G vj need not have a fixed point, so the defini­
tion of an approximate solution must be generalized to insure that a solution
exists for all v and T, and that a fixed point exists approximately for large
v and T . i v ) It must be shown that G „ tT is a good approximation to G „ , for
large T . v ) Simulations are generated endogenously in Step 2 , so we must
insure that {zt(/?)} is well behaved. In particular, we must show that the
series does ro t explode, and that the effect of the initial conditions chosen
for z dies out sufficiently rapidly. Point v ) and the choice of P are related
problems since one must insure that P is dense in the set of laws of motion
that generate well-behaved series.
The first set of assumptions are regularity conditions on the functions
defining the equilibrium (1). Let <7,_1 be the implicit function that defines
the i th argument of g ; for example,
(27)

g (a ,g 2 l ( a ,z ,u ) ,z ,u ) = 0

The process { z t ((3), x t ( ( 3 ) } defined by equation (15) is a first-order Markov
process satisfying z o ( /3 ) = zq and
z t ((3)

=

H ( z t„ x ( p ) , u t\ p )

=

For notational convenience we will let
that H p is analogous to H } 2
A ssum ption

1

*<(/?)), Zt-i(/?),«t)
Hp

denote the function

H

(28)

(•, •; /?). Note

( a ) F u n c t i o n s <j>,g, a n d f a r e u n i f o r m l y 13 L i p s c h i t z - c o n t i n u o u s

i n a l l a r g u m e n t s , a n d a r e d i f f e r e n t i a b l e a . e .*
123

(1992), where n = v and where the grid is exogenous. To our knowledge, no convergence
proof for such methods is yet available.
12We use M
H” to denote both the Markov operator defined in (7) and that defined in
(28). It will be clear which usage is intended since the operator defined in (28) has /? as
an argument, while the true solution does not.
13A function with multiple arguments is said to be uniformly Lipschitz-continuous if the
Lipschitz coefficient with respect to the i th argument does not depend on the value of the
j th argument.




19

(b) For all (a ,z ,u ) € <f>{Z) x Z x U, g2 l satisfying (27) exists and is
uniformly Lipschitz-continuous in its second argument.
(c) Tl\e true conditional expectation £ is Lipschitz-continuous a.e.

where ’a.e.’ is with respect to the Lebesgue measure. Notice that parts (a)
and (b) can be checked directly, while part (c) can not be verified directly
from functions <f>, g , and / . It should be kept in mind, however, that in most
models parts (a) and (b) imply part (c) . 14
Assumption 1 (b) insures that H can be derived from knowledge of g and
£; similarly, it insures that Hp is well defined and that it can be derived from
knowledge of g and xpp. The assumption that g fx is Lipschitz-continuous
insures that a small change in zt- 1 does not necessitate an arbitrarily large
change in zt to maintain equilibrium condition (1). If g is differentiable, this
assumption requires the partial derivative of g with respect to its second
argument to be uniformly bounded away from zero. Assumption 1 implies
that H is Lipschitz-continuous.
The next two assumptions insure that the conditional expectation terms
in ( 1 ) can be well approximated by some function xp. Approximating classes
of functions, such as polynomials, exponentiated polynomials, or splines, ap­
proximate a given function arbitrarily well only over a compact set. There­
fore, we impose
A ssum ption

2

Z x U is a compact set.

Assumption 2 implies that both exogenous and endogenous state variables
have compact support. Compact support for endogenous processes can be
implied by economic models in a number of ways. In models with capital
accumulation, depreciation often implies a bounded capital stock if U is
bounded. Asset pricing models are only well defined if a lower bound on
asset-holdings is imposed on agents; this is usually assumed through shortsale constraints. Finally, if the model is stationary, bounded support can be
achieved by directly imposing exogenous constraints on the model’s variables
at levels which will be attained with very small probability. For example,
in the simple growth model with zero depreciation the capital stock can be
HSee, for example, Santos’s [1991] proof that £ is differentiable in dynamic programming
models.




20




arbitrarily large. However, a very high level of the capital stock would be
achieved in equilibrium with extremely low probability, so the technology of
the model can be modified by imposing a very large upper bound on the level
of the capital stock. Presumably, the difference between the behavior of the
original model and this modified model will be negligible.
The next set of assumptions describes the class of approximating func­
tions, { i p : D„ x R p —►R m}, where D „ is a compact subset of the space of
sequences with at most the first u elements non-zero. 15 We will construct the
sequence {A/}£Li such that D „ C A /+ i,V i/, and we define D = (J„>i A / •
A ssum ption 3

(a)

V x <E X ,

f o r each v , th e r e s tr ic tio n o f i p ( - , x ) to D „

s a tis f ie s a L i p s c h i tz c o n d itio n u n i f o r m l y in x .
( b)

|i p ( f i , -)| < M, V/? € A

w here

supz€z <p(z)

< M

< oo.

( c ) ip((3 , •) i s c o n t i n u o u s , d i f f e r e n t i a b l e a l m o s t e v e r y w h e r e , a n d
su ch th a t

|

j <

K y

V/? G D ,

x

GX

: A x R p —►R m s u c h t h a t |
{/?„}£ii,
€ D „ , su ch th at

(d ) F o r a n y co n tin u o u s fu n c tio n q
K ,

th e re e x is ts a se q u e n c e

3

K

< oo

w h ere the d e r iv a tiv e e x ists.
<7

| <

K'

<

dip(/3^,x)
q(x)

= 0.

dx

Assumption 3 (b) is nonrestrictive, since ip is only used to approximate the
conditional expectation of <p. Notice that in assumption 3(d) we assume that
the d e r i v a t i v e s of the approximating function sequence approximate any con­
tinuous function. In Lemma 1 of Appendix 1 we show that assumption 3 (d)
implies that any absolutely continuous function can be uniformly approxi­
mated by some sequence {ip p
In practice, assumption 3 (d) is not very
restrictive. For most commonly-used approximation functions, such as poly­
nomials or splines, the derivatives are themselves a class of approximating
function. (Step functions are one exception.)
lsThe restriction that D v be compact is without loss of generality. In the case that the
parameters may have to be arbitrarily large for obtaining an approximation (as it may be
the case, for example, with polynomials), the {A} sequence is constructed so that there
is a bound on all elements of {A} but this bound goes to infinity as v grows.

21




There is a popular misconception that the derivatives of polynomial ap­
proximators must diverge from the derivatives of the target function when
the approximators become uniformly close to the target. This is true only for
certain ways of constructing polynomial approximators; for example, if the
approximation is required to fit a v th order polynomial to the target function
exactly on v points. It is not true, however, if the approximating sequence is
chosen to minimize other criteria, such as the l? distance, that take into ac­
count the fit of the function at many points. For example, polynomials would
fit Assumption 3, since the derivative of a polynomial is itself a polynomial,
and our Lemma 1 shows that an approximating sequence can be chosen with
bounded derivatives.
In this paper, we restrict our attention to stationary and ergodic pro­
cesses.
D efin itio n {zt°(0 )} and {zt°°} are stationary and ergodic processes sat­
isfying equations (28) and (7), respectively, for t = 0 , ± 1 , ± 2 , ...
That is, { z t} and (z<(/?)} denote processes generated respectively by (25)
and (12) starting from a fixed initial condition zo, while {zf°}t^_oo and
{z(° °(/?)}£-oo denote stationary and ergodic processes. The processes {zt{P)}
and {zt) are, in general, non-stationary, since the initial condition is fixed,
rather than a draw from the stationary distribution.
There is a final set of issues to be addressed. In common with virtually
all solution methods discussed in section 3, PEA delivers an approximation,
Hp, to the equilibrium law of motion H . This approximate law of motion
generates a time series { zt((3)}J= l, starting from some arbitrary initial con­
dition, which is used to obtain inferences about the stochastic properties of
the true stationary equilibrium process zf°. For this procedure to be valid,
three conditions must hold: i) {zf°(/?)} and { z f5} must exist; it) the effect
of the initial condition must decline as t grows; iii) if Hp is close to H , then
zf°(/?) must be close to zf°. These three conditions are closely related; most
processes either satisfy all three conditions or violate all of them. None of
these conditions hold, for example, if { z t} were an explosive process or a
random walk.
Insuring that these conditions hold is not only a problem for PEA. They
must hold for any proposed approximation procedure if the approximator is

22

to converge to the true stationary equilibrium process. For example, if H
were explosive, an approximation method that used a fixed grid (x i, ...jX,,)
could never converge to the true equilibrium, since the endogenous process
would eventually explode out of the pre-specified discrete grid.
These considerations require additional regularity conditions both on the
true equilibrium and on the space of admissible PEA approximators. We
propose two distinct sets of conditions, either of which is sufficient to prove
proposition 1. In the first, more general, approach, we simply assume the
•needed conditions directly. Formally, for an arbitrary law of motion H a
(where a is an index) let { z°} denote the process generated by H a starting
at fixed initial condition z0, and let {zf00} denote the stationary process
associated with H a.
D efin itio n S is a closed set (in the sup norm) of laws of motion such
that, for all H a 6 S
(a) a stationary process

with support in the set Z U ° exists

(b) if initial condition (z0,u i) € Z U °, sup0 |zf — zf00! —* 0 almost surely
as t —►oo, uniformly in the initial condition zq.
(c) for any sequence of functions

such that H k —* H a in the sup norm

as k —» oo, we have zf°k —> zf00 almost surely as k —» oo, uniformly in
t.
Note that condition (b) allows for a process with several disjoint ergodic
sets. All that is required is that, once the process is in the ergodic set, the
effect of the initial conditions disappears.
A ssu m p tio n 4* The law o f motion H is in the interior o f S .
This assumption can not be verified directly from knowledge of system (1).
There is a large literature on how to verify parts (a) and (b) of the definition
of S analytically.16 Part (c) in this definition is essentially a robustness
condition; if it did not hold, one can make the case that the model at hand
is not a particularly interesting one, since the time-series properties of the

16For example, Marimon (1989) shows existence of a stationary and ergodic solution
of the growth model under very general assumptions. Santos [1991] verifies part b) for
dynamic programming models.




23

model’s endogenous variables would be highly sensitive to small deviations
from fully rational behavior. Alternatively, one can verify assumption 4* by
using proposition 2 below: if assumption 4* is the only assumption to fail,
this will;be detected with PEA by the absence of an asymptotic fixed point.
Under assumptions 1, 2, 3, and 4* , the approximate PEA solution con­
verges to the true stationary equilibrium, in the sense of Proposition 1 below,
if the set D„ of admissible parameters is restricted to the subset B f , defined
by

B f = {fi € D„:HP € S ) .

(29)

A potential problem with using assumption 4* is that it is difficult to check
formally if j3 € B f . In principle, (5 €B ^ can be checked infoimally, as follows:
part (a) can be tested by observing if the solution settles around a stationary
distribution; part (b) can be tested by re-doing the calculations with different
z0; finally, if part (c) were not satisfied, the series will not settle down even
with small changes in
and the user will notice that the iterations to find
the fixed point will not converge.
Our second set of regularity conditions avoids this problem: the needed
restrictions on the /?„’s can be checked formally. The cost is that this second
set of regularity conditions is somewhat less general than assumption 4*.
The alternative conditions uses the following version of Duffie and Singleton’s
(1993) asymptotic unit circle (AUC) condition:




A ssu m p tio n 4 (a) H has a stationary and ergodic solution
with support in the set Z U .
(b) There exists a sequence o f positive random variables {/>(««)} satisfying
Elog[p(ut)] = a < 0 a. s.

(30)

such that H (',u ) has Lipschitz coefficient p(u) in Z U .
Assumption 4 is a nonlinear analogue to the unit circle condition in linear
time series models. While this is more restrictive than 4*, it is still satisfied
in most models of interest. First, the Lipschitz constants on H need to be
imposed only on the ergodic set of the true solution. Second, as long as the
expectation in (30)is less than zero, the condition permits Lipschitz constants
greater than unity for a subset of U with positive (possibly substantial)
24




probability measure. Finally, the assumption is silent about the particular
transformation of the endogenous variables used to construct the z t series.
For example, the simple growth model of Example 1.2 for p = 0 is known to
satisfy this condition if we write the law of motion in terms of log(kt), even
though this may not be true if H is written in terms of k t directly.
Under assumption 4, we must impose an analogous AUC condition on
the set of admissible approximators, which will be denoted B ^ u c :
D efin ition : For all i/y 1&*UC is a closed subset of £>„ with the prop­
erty that, V/? € B * u c , there exist psitive constants Sp and positive random
variables {p/?(ut)} satisfying
Elog[p/3(u()] < a < 0 a.s.
such that, for all || (31—/? || <

(31)

H (•, u t; /?') has Lipschitz coefficient pp(ut).

The expectation in (31) is with respect to the stationary distribution of u t.
For any given /?, condition (31) can be verified numerically: set pp(ut) equal
to the maximum derivative of //(•, u t; /?'), and integrate numerically over the
u 's.17
Under either assumption 4* or assumption 4, we must restrict the ap­
proximate solution f3vj to a subset B u € D„, where B„ equals B f or H * uc
depending on which assumption is used. Under this restriction it is not easy
to guarantee that G ^ t has a fixed point, since G u<t now maps B„ into a
larger set D„. One way to guarantee the existence of a fixed point is to re­
strict the minimization that defines G v>
t in (18) to the set B„. The function
G u>t would then map B,. into itself, and existence of a fixed point would
follow from Brouwer’s theorem. It turns out that this strategy does not de­
liver a proof of Proposition 1: we must use the fact that G>tr minimizes the

17Another consequence of using assumption 4 and B^yc is that the proof becomes
much more involved than if B^and assumption 4* were used. The reason is that the proof
needs to show that H can be arbitrarily well approximated by a sequence of H j s with
P € B * u c . In other words, we need that PC\{rpp : P G U ^ xB^yc} is dense in the subset
of { h : R l —* flm} containing S . Now, since U£Lj B AUC has an empty interior, this does
not follow immediately from the fact that P is dense in { h : R 1 - * R m } . On the other
hand, denseness of Pfl [ipp : /? GU^LjBf} in the subset of [ h : R 1 —►
containing 8
follows immediately from assumption 4* and the definition of this set.
25




mean square error among all admissible /? € D„. (See, for example, Lemma
8, equation 62). Instead, we use the following definition of an approximate
solution, which generalizes equation (19):
%

D efin ition : An a p p ro x im a te so lu tio n o f order v, sa m p le size T , is
a parameter vector /?„,t satisfying
Pv,T =

argmin

££f=i I0(/?,*<(/?)) ~ 'P{G„,T(/3),xt(/3))\2

(32)

Under assumptions 1 to 3 an approximate solution always exists by con­
tinuity and boundedness of the objective function. If G„tr had a fixed point
in the set B„, the two definitions would obviously coincide. In practice, a
fixed point usually can be found. Even if no fixed point exists for finite v
and T, lemma 9 in the appendix implies that as v, T —►00

4

IVKA'.t , zt(/?i/,r)) - V ^ . H / W ) ,
“ ♦O*
1 «=i
This makes precise the sense in which and C?„,r(/3) can be made arbitrarily
close. In that sense, the approximate solution (32) represents a fixed point
asymptotically.
The minimization problem in the above definition is over a restricted set.
Finding constrained minima numerically in non-linear setups is often difficult.
In practical applications of PEA, one can avoid imposing this restriction
directly by solving an unconstrained minimization along a homotopy path
for which the constraint does not bind. More details are given at the end of
next section.
Notice that, either under assumption 4 or under 4*, we may have non­
uniqueness. This can happen because system (1) may be satisfied for several
laws of motion H , or because a given law of motion has several ergodic sets
ZU . Our discussion of examples 1.1,1.2 and 1.3 shows how to pick out unique
solutions in some cases.

Now we present the fundamental proposition in this section.18
l8It is understood that the double limits in the paper are defined as:
v

lim C„,r = v lim
(Jim G , t /)
— oo \T - > 0 0

,T —

oq

26




P ro p o sitio n

1

(C onvergence of A pproxim ate Solutions):

A ssum e

th a t th e re is a u n iq u e s o lu tio n w ith a u n iq u e s t a t io n a r y a n d e r g o d ic d i s t r i ­
b u tio n ; d e n o te th e s u p p o r t a s Z U .

either a s s u m p t i o n
B„ = B ^yc, t h e n

a d d itio n ,
and

lim ( sup

U n d e r a s s u m p t i o n s 1, 2 , a n d 3 , i f , i n

4 * h o ld s a n d

|

H (z,

u

B„ = B f or

\(3 ^ t ) — H ( z , u )

\(z,u )e zff

lim I sup

a s s u m p tio n 4 h o ld s

| 1 = 0 , a.s.
)

|

d’ if li'T ,

z) —S { x ) | I = 0, a.s.

\xe/(zU )

)

(All proofs are in Appendix 1 .) Notice that convergence obtains in the strong
sense of uniform convergence. Also notice that convergence is guaranteed
only in the ergodic set ZU . In case that the solution is non-unique or has
several ergodic sets, a trivial modification of the proof of Proposition 1 would
show that the approximate solution becomes arbitrarily close to the set of
ergodic solutions
The following corollary asserts that all properties of the model that are of
interest in time series applications are appropriately approximated by PEA.
C orollary

1

U n d e r t h e c o n d i t i o n s o f P r o p o s i t i o n 1, i f ( z0 ,u i)

€

ZU

,

we

have
(a)

(S im u la te d S o lu tio n P a th s con verge)
Z i {(3v>t )

(b)

—►Zt

a .s., u n if o r m ly in t,

( E q u ilib r iu m C o n d i t i o n s a re s a tis f ie d in th e l i m i t )
g ( E [<j>(zt+ 1 ( p v tT ) ) \ x t ( 0 „tT)\, zt(&,r

(c)

),

( S i m u l a t e d S a m p l e M o m e n t s c o n v e r g e ) : I f d : Z —*

«t) -»

0

R ? is

a n y L ip s c h itz

fu n c tio n , th en

-» £(d(zt°°))
1 t=i

27

a .s.

a .s.




a s u , T —*

oo.

We now provide a partial converse to Proposition 1 . It shows how the
conclusion of Proposition 1 depends on assumption 4* or 4, and provides
a way to check these assumptions in the limit. Specifically, if B„ = B f ,
part ( a) says that if H does not satisfy assumption 4*, this can be detected
because min ||Vv3 —V’G„(/?)| is eventually bounded away from zero; part ( b)
tells us that those equilibria that fail to satisfy assumption 4* will not be
approximated by PEA. (An analogous interpretation holds for B„ = B ^1707.)
For example, this implies that rational expectations bubble equilibria in Ex­
ample 1 .1 will not be approximated by PEA with the choice of state variables
proposed in section 2 .
P ro p o sitio n 2
(a)

U n d e r a s s u m p t i o n s 1, 2 a n d 3,

t/B*, = Bjf

a s s u m p tio n 4 is n o t sa tisfie d , th en th e re ex ist
th a t

W >

or i f B„ = H * u c
7 > 0 and N > 0

a n d a s s u m p tio n 4 * is n o t s a tis fie d

and
such

N

min

E [ r J > ( P ,x ™ ( 0 ) ) -

t{G „ {(3 ),x ? (0 ))]2 > 7

PfcOl/

(b) i f

B„ = B J

a n d th e r e e x ists an eq u ilib riu m H

A s s u m p t i o n 4*> o r i f

B„ =

th a t d o e s n o t s a tis f y

a n d th e re e x ists a n e q u ilib r iu m H

t h a t d o e s n o t s a t i s f y A s s u m p t i o n 4 , t h e n H p vT d o e s n o t c o n v e r g e t o H .

Proposition 2 can be used to verify assumption 4* or 4 by calculating
i f ; 11
1

ii!

t = 1

for v and T arbitrarily large. If B„ = B f , assumption 4* is satisfied if and
only if this infinite sum can be made arbitrarily close to zero. (An analogous
verification of assumption 4 holds for B„ = B ^170.) This is important be­
cause verifying assumption 4* or 4 analytically may be difficult if one simply
inspects the equilibrium conditions ( 1 ).

28




5

S o m e

5.1

P r a c t i c a l Issues.

S i m p l e A l g o r i t h m s for f i n d i n g

(3u>t

In order to calculate the fixed point of Step 4 (or the arg min in equation (32))
one can use standard hill-climbing algorithms for solving non-linear systems
of equations. Nevertheless, this may not always be the best alternative. First,
calculation of the gradient of
can become very expensive in models with
many coefficients. Second,
is only well defined if /3 G B u .
We will discuss the second issue in the subsection on homotopy, below. In
order to avoid the first problem, the following algorithm based on modified
successive approximations has been used successfully in many applications
M

r

+ 1) = (1 - A)fl,,T(r) + AG „ , t

{ M

t

))

(33)

for some A > 0 . It is clear that no gradient has to be calculated, so that
the iterations are extremely easy to program, and each iteration is done very
quickly. This algorithm needs more iterations to converge than a gradient
algorithm, but there is a trade-off between ease of performing each iteration
and number of iterations needed to converge. In practice, the above algorithm
is often at least as fast as gradient algorithms.
Although one can construct examples where iterations on (33) are locally
unstable, this has not been the case in most practical applications up to date.
Furthermore, since the least-squares learning model is locally stable if and
only if the differential equation (23) is stable, iterations on (33) with A small
diverge only if the model is locally unstable under learning. If this were the
case, the model would be uninteresting from an economic standpoint.
Another fast algorithm that avoids calculating the gradient is to simulate
directly the model under least squares learning. More precisely, Steps 2, 3
and 4 are substituted by
• Step 2’: For T large enough, draw a sample of size T of the exoge­
nous stochastic shock and calculate { z t , U t } J =Q using the least squares
learning model of Appendix 2 .
Again, if the model is stable under learning /?t will converge to the fixed
point.

29

5.2

S p e e d o f C o m p u t a t i o n , Initial C o n d i t i o n s a n d H o ­
m o topy.

The design of S t e p s 1 t o 4 in s e c t i o n 2 i s d e s c r i b e s t h e s i m p l e s t p o s s i b l e a p p l i ­
c a t i o n of PEA. A number of elementary modifications can be used to speed
up computations, for example: one iteration on the algorithm for running
the non-linear regressions of Step 3 is sufficient to deliver the fixed point; g
can be often rewritten in a way that solving for z t in Step 2 is very simple;
the class of functions P can be chosen to match some properties of the con­
ditional expectation (for example, it can be set to take only positive values);
and, as suggested by the definition in equation (32), it is better to place the
convergence criterion used to decide that the algorithm has arrived at the
fixed point on the values of 0 ^, instead of on the values of /?.
Suppose a researcher has calculated the solution of order u and wants to
calculate the solution of order j/ + 1. In this case, using (3v>t as initial condition
for the iterations to find the v + 1 approximation is not a good alternative,
since the elements of the higher degree will usually be correlated with those
of a lower degree; instead, it is best to start the algorithm at Gv+i,T(/9i/,T)Furthermore, it is not always necessary to introduce all elements of degree
v + 1. Only those higher-order elements that have some incremental predictive
power for <f>(zt+1(/?)) need be included.
It is likely that the introduction of textbook techniques from numerical
analysis will be useful for finding minima, computing efficient Monte-Carlo
integrals, introducing alternative flexible functional forms, and setting homotopy paths. One has to be careful not to introduce these techniques unless
there is a good reason; oftentimes, the simplest approach will be sufficient
for solving the model of interest. In many cases, a simple algorithm is also
the fastest alternative.
Many successful pplications of PEA make use of homotopy techniques;
this is a simple way of obtaining starting values for the algorithm and of
keeping the simulations in the stable set B„. Along a homotopy, the desired
solution is obtained by moving slowly from a known solution to the solution
of the model we are interested in. For example, den Haan and Marcet (1990)
calculate the simple growth model of example 1 . 2 with partial depreciation
by starting out at the solution for the case p = 0 (a case for which we
know the analytic solution), and then solving a series of models increasing
p gradually. At each step along the homotopy, the solution for the previous




30




step is used as initial condition. In this way, one insures that all iterations
stay close to the true law of motion and, therefore, away from the boundary
of B„. Therefore, there is no need to use c o n s t r a i n e d minimization routines
in order:to solve the minimization problem in (32).
PEA has turned out to be a fast and simple method in many applications.
A comparison of speed of convergence, accuracy and convenience in a highly
non-linear model has been made by Christiano and Fisher (1994); they use a
simple growth model with irreversible investment (our example 1.3) and as­
sume that the shock 6 t is a discrete Markov chain. Their model is particularly
unsuitable to PEA, since it only has one continuous state variable and the
stochastic shock can take only two possible values: in this particular model,
endogenous oversampling is not very useful, and Monte-Carlo integration is
very inefficient, since integrals £ ( x i ] ( 3 ) are given by a simple formula that is
exploited in the other algorithms but not in PEA. The computation times
are higher for PEA but, even for their model, only by factors of five or seven.
Furthermore, of the several algorithms they tried, PEA was the only one
where the computation time increased only slightly when a non-negativity
constraint was introduced. Finally, Christiano and Fisher find that PEA
delivers a very accurate answer for low-degree polynomials and it is, by far,
the easiest method to implement.
The claim in Judd (1992) that MWR methods are ’hundreds of times
faster’ than PEA is based on comparing the computation times of den Haan
and Marcet (1990) with his own computations. These computation times
are not comparable because different starting values were used: den Haan
and Marcet (1990) (who were unaware that they were in a race) deliberately
started the algorithm at a very incorrect initial condition to illustrate the use
of homotopy. The solutions reported in Judd (1992) use the non-stochastic
s t e a d y - s t a t e s o l u t i o n as starting value, which happens to be very close to
the stochastic solution. Despite their model choice, the comparison done in
Christiano and Fisher (1994) is more informative, since they start out all the
algorithms at equivalent initial conditions.

6

Conclusion

We have presented the PEA algorithm for solving dynamic stochastic non­
linear models with rational expectations. This approach is highly flexible,
31




and it has been used successfully in many applications with suboptimal equi­
libria, strong non-linearities and inequality constraints. It is particularly fast
in those models with a large number of state variables and stochastic shocks,
and is quite easy to implement.
We prove that, for models with continuous laws of motion and under
some mild regularity assumptions, the approximate solution converges to the
true solution with arbitrary accuracy as the approximation is refined. Con­
vergence has been proved for the case of the stationary distribution. Similar
results for other techniques solving Euler equations with finite dimensional
approximations are not yet available.
A very general condition requiring, essentially, ergodicity of the true pro­
cess, is enough for convergence. We have also proved convergence under the
more restrictive assumption that Duffie and Singleton’s AUC condition has
to hold in the support of the ergodic distribution in order to guarantee that
the stability condition on the approximate process can be verified formally.
Future research might deliver similar convergence results for PEA under dis­
continuous laws of motion, but certain technical details, such as a condition
for compactness of the class of approximating functions and an equivalent
stability condition, need to be addressed. Our restriction to the support
of the stationary distribution is non-essential: a result that guarantees con­
vergence of the law of motion outside the ergodic set can be found in the
predecessor to this paper when the algorithm uses short-run simulations.
The convergence theorem in this paper represents a first step in studying
the properties of PEA and other numerical methods for solving systems of
Euler equation. Additional work needs to be done. For example, it would be
useful to understand the optimal rate at which u and T should be allowed
to grow. A more formal method of selecting the higher degree elements to
include when increasing u would be valuable. Finally, the use of more sophis­
ticated homotopy techniques can improve the speed of convergence in large
models, and it can provide a basis for a theorem proving that the iterations
to find the approximate solution converge globally from the initial condi­
tions. We believe that this area of research will pay substantial dividends in
expanding the range of economic models accessible to quantitative analysis.

32




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36

Journal

Marcel Dekker




APPENDIX 1
PROOFS AND LEMMAS
For ease of exposition and notation, the propositions and lemmas are
proved for the case where G u and
are well-defined functions (that is, the
least-squares minimizers in (16), (18), (32) and , (54) are unique). This avoids
dealing explicitly with convergence of correspondences. All propositions are
valid for the general case where these objects may be non-unique. First,
suppose G vj { P ) were multi-valued. For any £ € G ^ r i P ) ,
fakes the
same value, so the minimand in (32) is well-defined. If the minimizer in (32)
were non-unique, there would be multiple equivalent PEA approximators for
each value of v and T . In that case, one could still construct a doubly-indexed
sequence {/?k,t } by selecting any one of the equivalent PEA approximators
for each v and T . Any such sequence converges to the true equilibrium, in
the sense of proposition 1. Finally, throughout this appendix, || • || denotes
the L 2 norm, and B = U„>iB„.
The first lemma proves that P is dense in the space of conditional expec­
tations. If € was differentiable, the lemma would follow almost trivially from
the fundamental theorem of integral calculus. Most of the derivations in the
Lemma are to handle nondifferentiabilities in £ .
L em m a 1

The re exist s a sequence

/?„ € D*,

such that

(*)
d£(x)

lim ^ fe l
*/->oo
(jx

Ox

’

a.e. in X .

(b)
Jjm sup |V>(/?„,:r)-£(x)|

= 0

(c )

l i m s u p v-><x>

sup
x€X

dx

37

—sup
xex

d£(x)
dx




Pr oof.

To accommodate points of non-differentiability, our proof strategy in­
volves smoothing the function £ . We first define and characterize kernel
approximator functions. For k = 1,2, • • •, let n k : R p —►R m be a continuous
function for which f Nk Kk{t)dt = 1, and let A C X be a set of Lebesgue
measure zero. Let q : (X — A) —►R m be any arbitrary bounded continuous
function. The kernel approximator function q k is defined as follows:
q k( x ) = J k <l(x

+

t ) n k( t ) d t

In a technical appendix, available upon request, we prove that
limsupsup |
k—>oo x£X

q k( x )

| = sup |
x£X

q(x)

|.

(34)

Property (34) will be used in proving part (c) of the lemma.
To prove the lemma, choose x € X . By theorem (7.29) in Wheeden and
Zygmund (1977)), since £ is absolutely continuous, the derivative of £ exists
a.e., it is uniformly bounded, and we have

for all x € X . Let d £ k denote the kernel approximator to
be defined as follows:

and let

£ k{ x ) = J X
_ d £ k( x ' ) d x ' + M k,

where the scalar sequence
Clearly,

{ M k}

is chosen so that

d £ k( x ) =

so, in particular,

d£k

£ k( x ) = £ ( x )

£k

(35)
for all k.

d £ k( x )

’
is differentiable everywhere and,
lim
k—*oo

dx

d£(x)

d £ k(x)

dx

dx

= 0

pointwise, except on a set of Lebesgue measure zero.
38

(36)




We first prove that
lim sup \ £ k( x ) —£ { x ) \ = 0.
*€* 1
1

(37)

Given our construction (35), it is enough to show that the integrals of d £ k
converge uniformly to the integrals of |£ .
Let K denote the Lipschitz coefficient on £, which exists by virtue of
assumption 1(c). It follows that K is a bound on
and (by virtue of
(34) on 8gg . Using Egorov’s theorem (see Wheeden and Zygmund (1977)
page 57), we know that convergence a.e. of measurable functions implies
uniform convergence except in a set of arbitrarily small measure. Therefore,
given any e > 0, we can find A C X such that | A |< c/(4 K ) (where, | A \
represents the Lebesgue measure of the set A ) and
sup
x£X—A

d £ k{ x )

d£(x)

dx

dx

0 as

k

—►oo

In particular, there exists a k such that, for all k > k , s u p * ^ ^ | d£Q ^ ~
c/(2 |X |). For such a k , Egorov’s theorem implies that,
sup |^ (a:) —£(z)| = sup
xex 1
1 rgX

< sup
x€X

d£(x')
Jx

dx

Jx

UX

dx'

d £ k{x')

d£(x')

d £ k{x')

d£(x')

dx

dx

dx

dx

< sup ( f , , e/{2 | A'
xex \Jlx,x]-A

\)dx'+

| <

dx

I A | 2A') <| A 11/(2 | A |)+2A e/(4
J

■)

K)

This completes the proof of (37).
Parts (a), (b), and (c) of the Lemma can now be demonstrated. Fix
By Assumption 3 there exists a sequence f3k € D v such that
lim sup
xex

di>(ff,,x)

d £ k{ x)

dx

dx

39

0.

= e,

k.

(38)

Equation (38) implies
Jfirn sup |v>(/?*,x) — £fc(x)| = 0.

(39)

and
9 rp (^ l,x )

lim sup

*'-00*6X

8 £ k(x)

— sup

dx

(40)

dx

xex

We know from the properties of kernel approximators that £ k —> £
k —► oo and that, for each k ; from (39) we have that
v

—►oo.

Furthermore,

(34)

implies that

Pi/

as

—* £ k uniformly as

suPa:€A'

sup^g^

|^ f^ |

as k —* oo. Therefore, a sequence {/3 „} can be constructed by taking the
appropriate elements from the doubly indexed sequence {/?*} so that parts
(a) , (b), and (c) of the Lemma are satisfied. Q E D
The next lemma proves that restricting the approximation to stay in the
set of well-behaved laws of motion B
approximation.

does not preclude an arbitrarily good

Formally, the lemma could be stated as saying that P is

dense in B .
L em m a 2 L e t

G B„ f o r v s u f f i c i e n t l y l a r g e .

b e a s i n l e m m a 1, t h e n

Proof.

In the case where assumption 4* is made and B„ = B^, the lemma follows
immediately from part (c) of the definition of 5, since H is in the interior
of S . The remainder of the proof is for the case where assumption 4 is made
and B„ = 3 * u c . We use the following result:
sup

xex

£ { x ) ~

£ (*)

— sup

x —x

xex

d £ (x)
dx

(41)

A proof of equation (41) is in a technical appendix available from the authors.
Assumptions 1(a) and 3(a) imply that H ( z , u \ ‘) is continuous in /?. There­
fore, to prove the lemma, it is sufficient to show that, for each v , the minimal
Lipschitz coefficient of H ( - , u ] P u ) (which we will denote p p (tt)) satisfies




limsuppjj (tt) < p ( u ) ,

1/-+00

40

(42)

where

p(u) is the Lipschitz coefficient on H defined in Assumption 4. Un­

der Assumptions 1(a) and 1(b), the functions <7 , ( 7 2
and / are uniformly
Lipschitz in all arguments. Therefore, the following coefficients are finite:

g{oc,z',z, u ) - g ( a ,z ', z,u)
a —a
Of,Qft2#,Z

K \{u) = max

I<3 ( u ) =

g{<*, z',z,u) - g (a,z',z,u)
z —z
a,7,z',z
max

K ( u ) = max
CXyZyZ

g i l (a,z,u) - g?x{cc, z,u)
z —z

K / ( u ) = max
Z,Z

where

z, z, z

€

Z,

and a

€ <f>(Z).

/(*»“ ) - /(* ,« )
z —z

Assumption 4 then implies that

\l<\(u)I<j(u)K -I- A'3(u)] K ( u ) < p(u)
where

K = supx6^ |d

|. Let

(43)

Kp be defined as follows:

50(/?,x)
dx

Kp = sup
z€A'

It follows that

pp(u) = [K \(u )K j(u )K p + A'3(u)] I<(u)

(44)

Equations (43), (44), and Lemma 1(c) immediately imply (42). QED.
The next Lemma collects a number of results that will be used later. Let
X \ let H* : Z x U —> Z be defined

£ * denote an arbitrary function defined on

by
$(£*(/(*»«))» H ‘ ( z , u), z, u) = 0,
so H mis the law of motion consistent with £*; let
coefficient of //*(•, u).

p’ (u) be the Lipschitz

If assumption 4* is made, we will assume that H* € S. If assumption 4
is made, we will assume that H* satisfies the A U C condition in (31), so that




41

J3[log[/>*(u<)]] < a" < 0

a.s.

(45)

Let zf3'* b e a sta tio n a ry an d ergodic process satisfying zf3'" = H*(
For any 'given zeZ, define {zt(r,z;
recursively as follows:
Z-T(T,Z]0) = z
Zt(T,z-,/3) = t f ( z t_ i ( r , J ; / ? ) , u t; 0 ),

t> -r,

so z< (r,z;/?) is o b tain ed w ith th e law of m otion Hp, sta rtin g a t z a t d ate
—r . Sim ilarly, {zf(T,z)}?l_T is o b tain ed w ith th e law of m otion H * and
initial condition z_T(r, z) = z. N ote th a t zt(r ,z ;/3 ) an d zt* ( r ,z ) depend only
on {uj}j-__T+1, while th e statio n ary processes zf°’’ an d zf°(/?) depend on
Finally, let {/?*} be a sequence such th a t

lim sup

\xj){Pl,x) — £ ' ( x ) \

= 0

xgX

(N ote th a t, according to L em m a

1

, {/?„} plays th e role of th is sequence if

£• = S).
L em m a 3

(a) zf°-* exists, an d lim r _oo |

(b) For all (3 G B ,
0 a.s.

z *{t ,z)

— zf°-* | =

0 a .s.

zf°(/?) exists, and lim T_oo | zt( r , z ; /?) — zf°(/?) | =

(c) For any v and any t, the restriction of zf°((3) to B „ is continuous in /3
a.s.
(d) lim^—oo su p l€ 0 u6t/ | H(z,u;0*) — H*(z,u) | =
(e) lim^_»oo s u p je ^ y T>j | z ,(r, z; ft) - z* (r, z) | =

0.
0

a.s.

Proof.
U nder assum ption 4* w ith B „ = B f , p a rts (a) an d (b) follow im m ediately
from p a rts (a) an d (b) of th e definition of S. U nder assu m p tio n 4 w ith B „ =
B * u c , p a rts (a) an d (b) follow d irectly from Duffie an d S ingleton’s [1993]
L em m a 3. P a r t (c) is proven as follows: Let e > 0 b e given. It is sufficient
to show th a t 35 > 0 such th a t




| /? — 0 | < 5 im plies th a t

| z£°(/?) — z ” (/3 ) | < e

42

a.s.




For any z € Z and any r,

.

I ZV 0 ) ~ ZF(P) I £ I *i°°(0) “ zt(T,z-,P) |
+ I zt{T,z\p) - zt(T,z;(3) | + | zt(T,z;/3) - z f ° 0 ) \

(46)

According to part (b) of this lemma, r can be chosen big enough so the
first and third terms on the right-hand side of (46) are less than e/3 almost
surely. Furthermore, H ( z , w , •) : B —» Z is continuous, so z t ( r , z ; •) : B —* Z
is continuous. It follows that there exists S > 0 such that | /? — /? | < S, and
the second term on the right-hand side of (46) is less than e/3.
Part (d) of the lemma is proved as follows. Assumption 1(b) insures that
g 2 * is uniformly Lipschitz. Since
and

H ’ (z ,u ) = g ^ 1 ( £ ’ ( f ( z , u ) ) , z , u )
H { z , u; P I ) = g ^

f ( z ,«)), z , u)

we have, for some K < oo,
sup | H " ( z , u ) - H { z , u \ 0 ’ ) |
2,U
< - I < sup | £ ‘ ( f ( z , u ) ) - x j > ( 0 ; , f ( z , u ) )

e

| -+ 0

z,u

as v —» oo.
We now turn to part (e) of the Lemma. Under assumption 4* with B„ =
B f , part (e) follows immediately from part (d) of the lemma along with part
(c) of the definition of S . The remainder of this proof treats the case where
assumption 4 is made and where B„ = B * u c . In this part of the proof, it is
useful to notate explicitly the dependence of the z ’s on
the draw from the
sample space, so we use the notation ” zt(r, "z,u>; 0 ) " and ”z*(r, z,u>)” .
According to part (d) of this lemma, for any <5 > 0 3 N ( S ) < o o such that
supI U | H * ( z , u ) — H ( z , u ] (3‘ ) | < S, Vis > N ( 6 ) . Therefore, for date —r + 1,

|z_ T+i(r,J,a > ;/? ;) - z 1 t+1( t , z ,w )|

=

| H ( z , u - t + \ ( u))] P I ) - H m( z , u - T + i ( u j ) ) | < 6,
' i u > N ( S ) . Proceeding recursively forward to date t , one obtains:

<+T-l j

I zt(r,z,u;p*)-z;(T,z,u)

< S

43

h

- 2

n />*(«<-*+!m )

3=1 k=l

W

> N (S).

Since delta can be made arbitrarily small, part (e) of the

lemma follows if M(t,u>,r) =
n i= i />*(u«-/fc+i(w)) is bounded for al­
most all w, with the bound uniform in r. (Notice that M (t,w ,r ) does
not depend on z . ) Since log(/j*(u<(a;)) is stationary and ergodic, the sam­
ple mean of log[p*(u*(u>))] converges almost surely to its population mean.
Therefore, equation (45) there exists a J ( t , u ) < oo such that, for almost all
J E i= i log[?*(ut-Jfc+iH)] < a*,V ; >
<

M (t,u ,T )

and

/>•(«*-*+!(«))

< E / i 'r ’ n i_ , />•(«,-»+.(«))+E 5 , [«*•]'■
=

E /IV * n i_ , />-(«,_*+!(«)) + e » 7 (l - e°‘) = K ( t , w ) < oo,

for almost all u . This proves part (e) of the lemma. Q E D .
The following lemma insures compactness of P in the supnorm.
L e m m a 4 T h e f a m i l y o f f u n c t i o n s P = {V>(/?, •),/? € D } i s e q u i c o n t i n u o u s .
Proof.

Assumption 3 states that the derivative of elements of P is uniformly
bounded, so

I

~ V’(^,x)

g0(/?,(g<)
dx

dx'

< K

X —

X

I

QED.
The next lemma is a uniform strong law of large numbers for the mean
square prediction error. For / 3 , ( e D let h t ( ( 3 , £ ) = <f>(zt + \ { P ) ) — i l > ( £ , x t { ( } ) ) ,
let
= ^(z1
° ?,(^)) - V>(f,x~(/?)) and let 7 „,t be defined by:

it?

L em m a




=

5

sup

P£Bu£(zDt/

{ ifE L .N ^ W -E lW .O F l} -

lim r—oo' f t . T ~ 0 a.s.

Proof

44

(47)

The proof is sketched.
7t,T <

SUP

\ t 'ET=i h t ( P , 0 2 ~

h fiP iO 2

(48)
?

+

sup

If T j = i h ? ( P , 0 2 ~ E hV { P , t Y

h f ’(/?, £) satisfies the conditions of Hansen’s uniform strong law of large num­
bers for stationary and ergodic processes. (See Hansen [1982], Lemma 4.5.)
It follows that the second term on the right-hand side of (48) converges to
zero almost surely as T —►oo. Under assumption 4* with B„ = B^, the first
term on the right-hand side of (48) converges to zero almost surely as an
immediate implication of part (b) of the definition of S. On the other hand,
if assumption 4 is made and B„ = 3 *uc, it can be shown that the first term
on the right-hand side of (48) also converges to zero almost surely by using a
slight modification of the proof of Lemma 4 in Duffie and Singleton (1993).
QED.

The following lemma insures that the non-linear regressions converge to
the population least squares minimizer.
Lemma 6 lim sup | G ut {P) —G„{P) | = 0,
T-°°/3€Bv

a.s.

Proof.

Let e > 0 be given. Let set Q(e, /?) £ D„ be the set of elements of D„
which are bounded away from G U(P). Formally:
Q ( e , 0 ) = { ( z D l, s . t . \ ( - G M \

>0

Define 7r(e,/?) > 0 by
*(£,/?) =

- E [h r(P ,G M ,v )? }

(50)

and let 7r(c) be defined by




7r(e) = inf 7r(e, 3)
/?gB„

(51)

7r(e) > 0

(52)

Compactness of D v implies that

45




Equation (52) and Lemma 5 imply that 3 A 6 fl with prob (A) = 1 and
3 T ( u >, e) such that, for V u t A and V T > T(u>, e),
o < 1;,t M

:

< ^

(53)

where 7 *(T(u>) is defined in (47). Choose some w e A , T >
trary /? 6 B„.
iFrom the definition of

T(u>, e),

and arbi­

0 < E[h?(0,G„,T{i),» ) , * ) ? - E { h ? { 0 , G M , u ) \ 2

1 t=1

T

T

+ ? E IW ,

g ,,t W , » ) } 2 -

^

c .W ) ^ ) ) 2

1 t=1

1 t=l

+ 4 BM /J,G,,( 0 ),")]! -

E \h r (fi,G M ,u )-p

1 t- 1

E[hT(0,

+

G,.r(/>, w), W)]:2-

i; D M A <?,(/»), «)]J -

i £ M l> ,
1 (=1

G,.t W ,

E {h ? (0 ,G M

«)]’

,»)] 2

1 t=l

< 2 7 *t (u>) < 7r(e) <

tt( c, P )

where the third inequality follows from the definition of G ^ t ( P , u>), the fourth
inequality follows from (53), and the fifth inequality follows from (51). This
implies that G v ,t (/?,u ;) ft Q ( P , e), proving the lemma. QED.
Lemma 7 states that the approximate solution for large enough T will be
arbitrarily close to the "population” approximate solution of order u :
/?„ = arg min ||
L em m a 7

Let
g i v e n v, w e h av e

rp(p, * V ( P ) ) ~

xT(P))

II

(54)

he the a p p r o x i m a t e s ol ut ion , as in e q ua t io n ( 3 2 ) . T he n,

lim |

T—
+oo

Pu,T — Pu

46

| = 0, a.s.




Proof.

We first show that the minimand in (54) converges to the minimand in
(18)

uniformly in (3 as T —*■ oo:

2
-E

sup

i

t

sup

1 t = 1PeBv

i!

| < w ,x,(0)) - t ( c , . T(fi),*tW) I2 (55)

+ sup

PzBv

1 <=i

We first show that the first term on the right-hand side of (55) converges
to zero. According to assumption 3(a), tl>(-,x) is Lipschitz, uniformly in x £
X. Furthermore, the xj>: B„ x X —►R m is a continuous function of a compact
set, so is bounded. These conditions imply that |0(/?, xt((3)) — ip(£,xt(P)) |2
is Lipschitz in £ € B„, uniformly in /? 6 B„, so 3A" < oo such that

11 *i>(fi,x,(p)) -

ih g .,t (/3),x,(/3))i

2 - \ < K 0 , x , m - < H G . m , x , m ) i2

< 1< I G„r(/3) - G „ ( t S ) |

(56)

Lemma 6 implies that the right-hand side of (56) converges uniformly to zero
as T —» oo, so, given e > 0,3 T ( e ) such that V T > T(e),
sup

/JeB*

| G„,T { P ) ~ G M

| < £

(57)

Equations (56) and (57) imply that the first term on the right-hand side of
(55) is less than e for all T > T(e), which proves that term converges to zero
as T —» oo.
We now complete the proof. By an argument analogous to that used in
the proof of Lemma 5, | ^(/^ £«(/?)) — 0(Gv(/?),xt(/?)) |2 satisfies a uniform
strong law of large numbers. This implies that the second term on the righthand side of (55) converges to zero as T —► oo. Thus, the left-hand side of

47




(55) converges to zero, so the minimand in (18) converges uniformly to the
minimand of (54). Since the minimization is over the compact set B „, this
implies the conclusion of the lemma. Q E D .

L em m a 8 L e t
lim

1/—+QO

b e a s e q u e n c e s u c h t h a t /?„ G B „,V j/. T h e n

0

£ ( x r ( h ) J v ) - tf(c M & ),* r(& ))

(58)

Proof.

According to Lemma 4, { r p ( ( 3 , - ) , / 3 G B } is an equicontinuous family of
functions. Assumptions 1(a) and 1(b) then imply that {£(•,/?),/? € B } is also
equicontinuous. Furthermore, since <j> is a bounded function, it follows that
{£(•,/?), (5 € 5 } is uniformly bounded and Arzela’s theorem (Kolmogorov
and Fomin (1970, p. 102)) then implies that there exists a subsequence Vk
and a continuous function £ such that
lim sup | £ (x ,/?„ ) — £(x) | = 0.

(59)

k->°° xex

The function £ is the uniform limit of a sequence of uniformly Lipschitz functions, so £ satisfies a Lipschitz condition. By a proof analogous to
Lemma 1(b), there exists a sequence

with

G

D„k

such that

lim sup |£(x) — VK/^n*) I = 0.

(60)

iex

(Notice that this sequence is different from \ P u k ] in the statement of the
lemma)
Equation (58) is now demonstrated along subsequence i/*. The definition
of G implies that

^ A j-a rg n u n

|£ ( * r ( A J » f l J “

| »

(61)

since the expectational error is orthogonal to the least squares predictor.
The following chain of inequalities hold:

48




II £ ( x r ( l k) M - r P ( G Vk( i k) , x r 0 , k)) ii
$ II £ ( * r ( « , - H ^ x r i ^ m
<

supieA- I £ ( x , j 3 y k ) -

I

<

supx6X I £ ( x , j 3 „ k ) - S ( x ) | + supl€JC I S ( x ) - I p 0 „ k , x ) I - * 0

(62)
where the first inequality follows from (61) and the third inequality follows
from the triangle inequality. The first term on the right-hand side of (62)
converges to zero as k —► oo by (59). The second term on the right-hand side
of (62) converges to zero as k —» oo by (60).
Since (62) holds for any arbitrary convergent subsequence, this result
implies the conclusion of the lemma. Q E D .
The next and last lemma shows that /?„ gets arbitrarily close to (?„(/?„)
as v grows. In this sense, we say that an asymptotic fixed point exists.
L em m a 9

Km | min || V>(/?, * W ) ) “ ^ ( G v(fi)t x?(0)) || |

= 0

Proof.

According to Lemma 1(b), a sequence {/?„}£!j exists such that
lim sup | £ ( x ) - ^ ( ^ . x ) | = 0

‘/-,0° xex

(63)

Recall that £ is the true conditional expectation. According to Lemma 2,
/?„ € B„, so the minimand in the statement of the lemma is dominated by
(64)
and it is enough to show that (64) goes to zero. Now,

< | | w , . * r ( « ) - f ( * r ( 0 | | + [5 (* r(A .)) - £ ( * r ( A .) ; /u |
+ | W ( « ; A . ) - * ( G .(/U * r(/U )|
(65)
49

The first term in the right side goes to zero as
The second term is bounded by
| / « » ( * .» ') ) <U VI«) - /

v

—►oo by the choice of

A,)) ^ M « )

where F(*|u) is the conditional distribution of
given u t = u. The above
expression converges to zero by Lemma 3 (d) and the Lipschitz-continuity of
<p. Finally, the third term goes to zero by Lemma 8. QED.
P ro o f of P ro p o sitio n 1
In this proof, convergence of a function of (z, x ) in the supnorm is taken
over the ergodic set of the true process Z U C Z x U.
Let xpp = xp((3, •) and Hp = //(•, •; /?). We first show that xppv converges
to £ pointwise.
Since P is a uniformly bounded, equicontinuous class of functions, it is a
compact set in the sup norm. Therefore, there exists a subsequence (indexed
by k ) and a limiting function xp" such that xppVk —* xp’ in the supnorm. It is
enough if we show that, for any such convergent subsequence, xp* = £ a.e. in
YU.




Let H * be the law of motion consistent with xp’, and £ ’ the true condi­
tional expectation consistent with H ’ . We have Hp —*■ H* uniformly, by
an argument analogous to that used in the proof of Lemma 3(d). Note that
dF{u'\u).

(66)

All the functions inside the integral are continuous and bounded, so Lebesgue
Dominated Convergence implies
(/(*>
as

J v $ (H * ( # “(*« u)>u')] d F { u ' \ u ) = £ ’ { f { z ,

u))

(67)

k —* oo.

We will eventually show that H = H ’ \ to this end, we first have to prove
that H ’ satisfies assumption 4* (if B„ = B f) or assumption 4 (if B„ =
B * u c ). In the former case, H p Vk € S,Vfc, by the definition of B f. Since
S is closed under the sup norm, and H p v
H* uniformly, it follows that
H’ €S.

50

We now turn to the latter case. It is sufficient to show that, if B „ =
B ^ 1707, H * has a Lipschitz coefficient satisfying (45). Let
=

p*{u)

sup

H * (z,u )

z,z£Z

-

z

H -{z,u )

— z

Fix t > 0, and any u € U . By definition of p*(u), there exist z and z such
that

,■(«) < m

-

H "(z, u)

* 'u? ~ ^

H Pl/k( z , u ) | +

\ H 0 „k ( z , u ) - H/}„k ( z , u ) \ +

Iz

"H ‘ { z , u ) -

H p„k ( z , u )|

Iz

’u)l(i + t )

- zI

\z

+
—z

\ H p „k ( z , u )

-

H m( z , u ) \

(1+0

—z I

**»(*■“)-«•(*- “)l' 1
|
+

rt i ( it 1
PPvk \U)

(1+c)
(68)

where p p „ k (u ) is the function defined in (31) Now, fix z, z; take logs on both
sides of (68). Since inequality (68) holds for all
we can take the liminf*
on both sides and use the uniform convergence of H p Uk —> H * to obtain
log p ' ( u )

<

(lirninf

log ftj.,(u )j + lo g(I + e)

(69)

which holds for all e > 0 and all u . Now, given any e, taking expectations
over u we get
B [log />*(“ )] <

E [lhn inf log

+ log(l + e)

< lirninf [lSlog p/j^(u)] + log(l + e) < a + log(l + c)
where the second inequality follows from Fatou’s lemma, and the last in­
equality follows from the fact that (3Uk € B „ . Since the above inequality
holds for all e > 0, we conclude




E

log p * ( u )

51

<

a

This completes the demonstration that H * satisfies (45) when B „ = B £ u c .
We have shown that H * satisfies assumption 4* or 4 (depending on the
definition of B„ being used). It follows that a stationary and ergodic process
{ z f 00} ejcists satisfying z ^ ° ° =
so the L 2 norm of {-2*°°} is well
defined. We can write,

||V >-(*n - £ '(* ;~ )|| <

+ I W A ..,* r to J ) - <HGV.(ft,)>*r(/U)ll
+

aj i i

We can now apply Lemma 3, parts (a), (b), and (e), to conclude
Ixr-zfW I-^ O a.s.

(71)

The limsup of the right side of (70) equals zero, as follows: The limsup of
the first term is equal to zero by the definition of V>*; the second term goes
to zero by (71), along with the continuity of
and Lebesgue Dominated
Convergence; the third term goes to zero by Lemma 9; the fourth term
goes to zero by Lemma 8, the fifth term goes to zero by (71), along with
the continuity of £ ( - , / 3 „ k ) , and Lebesgue Dominated Convergence; the sixth
term goes to zero by (67).
We have proved that the left side of (70) is equal to zero, which implies
V>*(x*°°) = £ m( X f ° ° ) almost surely in the support of x*°°. We can summarize
this derivation as follows:
-

r ( x ) = E (^ (*;~ ) I

= x)

(72)

This is an important result in itself; it says that the limit of the pa­
rameterized expectations V’* >s equal to the true expectation of the process
generated by ij>m.
Finally, since




9 ( ' K 0 » k i x t ( P ' * ) ) * z t ( 0 * k ) i Z t - i ( P » k ) , U t ) = 0, VA:

52




by construction, and g is continuous, (72) and lemma 3(e) imply

s ( £ ( ^ ; +i) I * ;) ,

= °,

with probability one, which implies that H * is a rational expectations equi­
librium or, equivalently, that £ * = € and H = H * . To summarize, we have
shown that any convergent subsequence of H p v goes to H in the supnorm,
which implies that the same holds for the whole sequence.
To complete the proof, we have to show the same type of convergence for
the sample version of the approximate solution. It is enough to show
Lim
T —* oo

i

sup
6A'

^ (x , P v,t ) — V’(~, P v ) \

= 0.

(73)

Lemma 7 and continuity of xpp in /? imply
pointwise. Since P
is equicontinuous, by Lemma 4, and bounded it is compact in the sup norm,
which implies uniform convergence as in (73). Finally, (72), (73), and Lemma
3(d) imply that H p T converges uniformly to //*. This completes the proof
of Proposition 1.

P ro o f of C orollary
a) Follows from lemma 3(e) and the uniform convergence of H p v T —* H
as in Proposition 1.
b) Lemmas 8 and 9 imply || f ( x f ° ( ^ ir),i9„,r) || —> 0.
Part b) of the corollary follows from part a) and continuity of g .
c) Follows from proposition 1 and part (b) of the definition of S (if as­
sumption 4* is imposed and B„ = B^) or from the A U C condition,
Proposition 1 and Lemma 4 in Duffie and Singleton (if assumption 4 is
imposed and B „ = T 5 * u c ).

P ro o f of P ro p o sitio n 2
We give the proof for the case where B„ = B * u c . The proof for the case
where B^ = B f is analogous, and is therefore omitted.
• We first prove that if

min
E \ m * W ) ) - * « ? ,( « , * r ( « ) ] 2 - “
C-Oi/

P

53

(’ 4)

then assumption 4 is satisfied. Consider any convergent subsequence
of {Vvj„r }; as in the proof of proposition 1, this subsequence has a
limit xf>* for which the corresponding law of motion H * satisfies the
AUC condition (30). As in the proof of proposition 1, (70) holds for
this V’* and the corresponding conditional expectation S* (defined as
in (67)) . Now, consider the right side of (70); with the exception of
the third term, all terms go to zero as k —►oo if Assumptions 1, 2 and
3 are satisfied19. The third term goes to zero because of (74). This
proves the existence of an equilibrium H mthat satisfies AUC and that
Assumption 4 is satisfied.
To prove part

(x)

assume, towards a contradiction, that

liminf p £ b v E [ r p ( 0 , x f = { p ) ) - ^ ( ^ ( Z ? ) ,! ^ ^ ) ) ] 2 = 0.
Then there would be a subsequence {/?„*} for which the min goes to
zero and we can find a sub-subsequence {/?„*.} such that Hp„ —* H*.
By the argument given above, this would imply existence of H* that
satisfies Assumption 4, which is a contradiction.
• We now prove part (b). Let Q = { laws of motion for z that satisfy AUC condition}.
By an argument analogous to that used in proving that H * of Propo­
sition 1 satisfies AUC, we can prove that Q is a closed set. Since each
HpvT € Q and H Q , part (b) follows immediately.
QED

19Notice that assumption 4 has only been used in results leading to Lemma 9, which
causes the third term in (70) to go to zero.




54

APPENDIX 2
NON-LINEAR LEAST SQUARES LEARNING
The recursive least squares estimator to predict

<f>(zt+1 )

can be written as

A = A -i + <* R T l
W*<) - iH A ,i.-i)l)
R , = R t-1 + a , *♦<*;*-■>

(75)

Notice that if we fixed /?, replaced 5* by z*(/?), and for the particular choice
a* = (1/0) A ls the non-linear least squares estimator to predict <f>(zt (l3))
with V>(*)£*-i(/?)), where the series z are not affected by /?<, except that the
residuals are calculated recursively.
In the system defined by (75), however, the V s depend on the estimate
/3t . In words, (75) describes an economy where agents use recursive non­
linear least squares to form their expectations about the future, and where
equilibrium is generated by such expectations; the first equation says that
today’s beliefs are updated according with last period’s beliefs and the pre­
diction error made this period, the second equation gives the gradient for
how the prediction error affects the beliefs /?*, and the last equation defines
the equilibrium z t in terms of the expectations of the agents. This is a non­
linear version of economies studied Marcet and Sargent (1989), where the
non-linearities appear both in the equation generating the observations and
in the expectational rules.
Using notation
DMP) =

|W ,

* t - i ( P ) ) / W \ 0=1;

and

R(0) = E{ D M 0 ) W P ) ' )

and adapting results from Ljung (1975], it is possible to show that the only
learning can converge with positive probability are

0 ' s where least squares
0 <= D v that satisfy

R 0 )-'E (D A 0 )

( ? ) ) ] ) = 0.

(76)

This equation is obtained by taking the object multiplying a t in the recursive
algorithm, fixing 0 — 0 , and taking the expectation with respect to the




55




stationary distribution of { z t (/3)}. Now, since the first order conditions for
the maximization problem of (16) imply that
;

E (D M P )

W

t m

- r p ( G M , x t.

,(/?))]) = 0.

it is clear that (76) is satisfied, precisely, at the fixed point /?„.
Further results in Ljung [1975] guarantee that j3t —* /?„ if the differential
equation
W (r),x ,_ ,(/? (r)» ))

? M . = R(0( t ) )-'E { D M 0 ( r ) )

(77)
is stable. To analyze this differential equation, notice that the right side of
(77) is equal to
W > * .-iW )l)

R (0T 'e ( D M 0 )

by the definition of G„. By the mean value theorem, in a neighborhood of /?„
this expression can be written as
m u r 'E (

d m

0) D M 0 ) ' (o ,( 0 ) - 0))

for some /3 in this neighborhood. Therefore, /?t —►/?„ locally if and only if
the differential equation
= R(0( t ) ) - 'E ( D M 0 ( r ) ) D M 0 (t )Y) (G,(0( t ) ) - 0 ( t )) = G„(0( t ) ) - 0 ( t )
is locally stable at /?„.

56

Working Paper Series
A series o f research studies on regional economic issues relating to the Seventh Federal
Reserve District, and on financial and econom ic topics.

REGIONAL ECONOMIC ISSUES
Estimating Monthly Regional Value Added by Combining Regional Input
With National Production Data

WP-92-8

P h ilip R. Is r a ile v ic h a n d K e n n e th N . K u ttn e r

Local Impact o f Foreign Trade Zone

WP-92-9

D a v id D . W e iss

Trends and Prospects for Rural Manufacturing

WP-92-12

W illia m A . T e sta

State and Local Government Spending--The Balance
Between Investment and Consumption

WP-92-14

R ic h a r d H . M a tto o n

Forecasting with Regional Input-Output Tables

WP-92-20

P .R . Is r a ile v ic h , R. M a h id h a r a , a n d G .J .D . H e w in g s

A Primer on Global Auto Markets

WP-93-1

P a u l D . B a lle w a n d R o b e r t H. S c h n o r b u s

Industry Approaches to Environmental Policy
in the Great Lakes Region

WP-93-8

D a v id R. A lla r d ice, R ic h a r d H. M a tto o n a n d W illia m A . T e sta

The Midwest Stock Price Index—Leading Indicator
o f Regional Economic Activity

WP-93-9

W illia m A . S tr a u s s

Lean Manufacturing and the Decision to Vertically Integrate
Some Empirical Evidence From the U.S. Automobile Industry

WP-94-1

T h o m a s H . K lie r

Domestic Consumption Patterns and the Midwest Economy

WP-94-4

R o b e r t S c h n o r b u s a n d P a u l B a lle w




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Workingpaperseriescontinued
To Trade or Not to Trade: Who Participates in RECLAIM?

WP-94-11

T h o m a s H . K lie r a n d R ic h a r d M a tto o n

Restructuring & Worker Displacement in the Midwest

WP-94-18

P a u l D . B a lle w a n d R o b e r t H . S c h n o r b u s

ISSUES IN FINANCIAL REGULATION
Incentive Conflict in Deposit-Institution Regulation: Evidence from Australia

WP-92-5

E d w a r d J. K a n e a n d G e o r g e G. K a u fm a n

Capital Adequacy and the Growth o f U.S. Banks

WP-92-11

H e r b e r t B a e r a n d J o h n M c E lr a v e y

Bank Contagion: Theory and Evidence

WP-92-13

G e o r g e G. K a u fm a n

Trading Activity, Progarm Trading and the Volatility o f Stock Returns

WP-92-16

J a m e s T. M o s e r

Preferred Sources o f Market Discipline: Depositors vs.
Subordinated Debt Holders

WP-92-21

D o u g la s D . E v a n o f f

An Investigation o f Returns Conditional
on Trading Performance

WP-92-24

J a n ie s T. M o s e r a n d J a c k y C. S o

The Effect o f Capital on Portfolio Risk at Life Insurance Companies

WP-92-29

E lija h B r e w e r III, T h o m a s H . M o n d s c h e a n , a n d P h ilip E. S tr a h a n

A Framework for Estimating the Value and
Interest Rate Risk o f Retail Bank Deposits

WP-92-30

D a v id E. H u tc h is o n , G e o r g e G. P e n n a c c h i

Capital Shocks and Bank Growth-1973 to 1991

WP-92-31

H e r b e r t L. B a e r a n d J o h n N . M c E lr a v e y

The Impact o f S&L Failures and Regulatory Changes
on the CD Market 1987-1991

WP-92-33

E lija h B r e w e r a n d T h o m a s H . M o n d s c h e a n




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Workingpaperseriescontinued
Junk Bond Holdings, Premium Tax Offsets, and Risk
Exposure at Life Insurance Companies

WP-93-3

E lija h B r e w e r I I I a n d T h o m a s H . M o n d s c h e a n

Stock Margins and the Conditional Probability o f Price Reversals

WP-93-5

P a u l K o fm a n a n d J a m e s T. M o s e r

Is There Lif(f)e After DTB?
Competitive Aspects o f Cross Listed Futures
Contracts on Synchronous Markets

WP-93-11

P a u l K o fm a n , T o n y B o u w m a n a n d J a m e s T. M o s e r

Opportunity Cost and Prudentiality: A RepresentativeAgent Model o f Futures Clearinghouse Behavior

WP-93-18

H e r b e r t L. B a e r, V ir g in ia G. F r a n c e a n d J a m e s T. M o s e r

The Ownership Structure o f Japanese Financial Institutions

WP-93-19

H esn a G enay

Origins o f the Modern Exchange Clearinghouse: A History o f Early
Clearing and Settlement Methods at Futures Exchanges

WP-94-3

J a m e s T. M o s e r

The Effect o f Bank-Held Derivatives on Credit Accessibility

WP-94-5

E lija h B r e w e r III, B e r n a d e tte A . M in to n a n d J a m e s T. M o s e r

Small Business Investment Companies:
Financial Characteristics and Investments

WP-94-10

E lija h B r e w e r III a n d H e s n a G e n a y

MACROECONOMIC ISSUES
An Examination o f Change in Energy Dependence and Efficiency
in the Six Largest Energy Using Countries--1970-1988

WP-92-2

J a c k L. H e r v e y

Does the Federal Reserve Affect Asset Prices?

WP-92-3

V efa T a rh a n

Investment and Market Imperfections in the U.S. Manufacturing Sector

WP-92-4

P a u la R. W o r th in g to n




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Workingpaperseriescontinued
Business Cycle Durations and Postwar Stabilization o f the U.S. Economy

WP-92-6

M a r k W. W a ts o n

A Procedure for Predicting Recessions with Leading Indicators: Econometric Issues
and Recent Performance

WP-92-7

J a m e s H . S to c k a n d M a r k W. W a tso n

Production and Inventory Control at the General Motors Corporation
During the 1920s and 1930s

WP-92-10

A n i l K. K a s h y a p a n d D a v id W. W ilc o x

Liquidity Effects, Monetary Policy and the Business Cycle

WP-92-15

L a w r e n c e J. C h r is tia n o a n d M a r tin E ic h e n b a u m

Monetary Policy and External Finance: Interpreting the
Behavior o f Financial Flows and Interest Rate Spreads

WP-92-17

K e n n e th N. K u ttn e r

Testing Long Run Neutrality

WP-92-18

R o b e r t G. K in g a n d M a r k W. W a tso n
A Policymaker's Guide to Indicators o f Economic Activity

WP-92-19

C h a r le s E v a n s , S te v e n S tr o n g in , a n d F r a n c e s c a E u g e n i

Barriers to Trade and Union Wage Dynamics

WP-92-22

E lle n R. R is s m a n

Wage Growth and Sectoral Shifts: Phillips Curve Redux

WP-92-23

E lle n R. R is s m a n

Excess Volatility and The Smoothing o f Interest Rates:
An Application Using Money Announcements

WP-92-25

S te v e n S tr o n g in

Market Structure, Technology and the Cyclicality o f Output

WP-92-26

B r u c e P e te r s e n a n d S te v e n S tr o n g in

The Identification o f Monetary Policy Disturbances:
Explaining the Liquidity Puzzle

WP-92-27

S te v e n S tr o n g in




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Workingpaperseriescontinued
Earnings Losses and Displaced Workers

WP-92-28

L o u is S. J a c o b s o n , R o b e r t J. L a L o n d e , a n d D a n ie l G. S u lliv a n

Some Empirical Evidence o f the Effects on Monetary Policy
Shocks on Exchange Rates

WP-92-32

M a r tin E ic h e n b a u m a n d C h a r le s E v a n s

An Unobserved-Components Model of
Constant-Inflation Potential Output

WP-93-2

K e n n e th N . K u ttn e r

Investment, Cash Flow, and Sunk Costs

WP-93-4

P a u la R. W o r th in g to n

Lessons from the Japanese Main Bank System
for Financial System Reform in Poland

WP-93-6

T a k e o H o sh i, A n il K a s h y a p , a n d G a r y L o v e m a n

Credit Conditions and the Cyclical Behavior o f Inventories

WP-93-7

A n il K. K a s h y a p , O w e n A . L a n io n t a n d J e r e m y C. S te in

Labor Productivity During the Great Depression

WP-93-10

M ic h a e l D. B o r d o a n d C h a r le s L. E v a n s

Monetary' Policy Shocks and Productivity Measures
in the G-7 Countries

WP-93-12

C h a r le s L. E v a n s a n d F e r n a n d o S a n to s

Consumer Confidence and Economic Fluctuations

WP-93-13

J o h n G. M a ts u s a k a a n d A r g ia M . S b o r d o n e

Vector Autoregressions and Cointegration

WP-93-14

M a r k W. W a tso n

Testing for Cointegration When Some o f the
Cointegrating Vectors Are Known

WP-93-15

M ic h a e l T. K. H o r v a th a n d M a r k W. W a tso n

Technical Change, Diffusion, and Productivity

WP-93-16

J e ffr e y R. C a m p b e ll




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Workingpaperseriescontinued
Economic Activity and the Short-Term Credit Markets:
An Analysis o f Prices and Quantities

WP-93-17

B e n ja m in M . F r ie d m a n a n d K e n n e th N . K u ttn e r

Cyclical Productivity in a Model o f Labor Hoarding

WP-93-20

A r g ia M . S b o r d o n e

The Effects o f Monetary Policy Shocks: Evidence from the Flow o f Funds

WP-94-2

L a w r e n c e J. C h ris tia n o , M a r tin E ic h e n b a u m a n d C h a r le s E v a n s

Algorithms for Solving Dynamic M odels with Occasionally Binding Constraints

WP-94-6

L a w r e n c e J . C h r is tia n o a n d J o n a s D .M . F is h e r

Identification and the Effects o f Monetary Policy Shocks

WP-94-7

L a w r e n c e J . C h r is tia n o , M a r tin E ic h e n b a u m a n d C h a r le s L. E v a n s

Small Sample Bias in GMM Estimation o f Covariance Structures

WP-94-8

J o s e p h G. A lto n ji a n d L e w is M . S e g a l

Interpreting the Procyclical Productivity o f Manufacturing Sectors:
External Effects o f Labor Hoarding?

WP-94-9

A r g ia M . S b o r d o n e

Evidence on Structural Instability in Macroeconomic Time Series Relations

WP-94-13

J a m e s H. S to c k a n d M a r k W. W a tso n

The Post-War U.S. Phillips Curve: A Revisionist Econometric History

WP-94-14

R o b e r t G. K in g a n d M a r k W. W a tso n

The Post-War U.S. Phillips Curve: A Comment

WP-94-15

C h a r le s L. E v a n s

Identification o f Inflation-Unemployment

WP-94-16

B e n n e tt T. M c C a llu m

The Post-War U.S. Phillips Curve: A Revisionist Econometric History
Response to Evans and McCallum

WP-94-17

R o b e r t G. K in g a n d M a r k W. W a tso n




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Workingpaperseriescontinued
Estimating Deterministic Trends in the
Presence o f Serially Correlated Errors

WP-94-19

E u g e n e C a n je ls a n d M a r k W. W a tso n

Solving Nonlinear Rational Expectations
Models by Parameterized Expectations:
Convergence to Stationary Solutions

WP-94-20

A lb e r t M a r c e t a n d D a v id A . M a r s h a ll




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