%Paper: ewp-mac/9309002
%From: jensen@wuecona.wustl.edu (Mark J. Jensen)
%Date: Tue, 21 Sep 1993 16:50:44 -0500 (CDT)

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\begin{document}
\begin{titlepage}
\title{The Tracking Ability of the Divisia \\ Monetary Aggregate
Under Risk}
\author{Mark J. Jensen \\Department of
Economics, Washington University \\ St.\ Louis, MO 63130}\thanks{The
author wishes to thank William A.\ Barnett for motivating this
paper's topic and Robert P.\ Parks for the many helpful technical
discussions that occurred during the work of this paper.}
\date{\today}
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\section{Introduction}
Over the last decade the area of monetary aggregation has been
receiving considerably more and more attention both from economists
and from monetary-policy decision makers.  This literature of monetary
aggregation came about as a result of the Federal Reserves' simple-sum
index lacking a strong microeconomic foundation.  In Barnett's (1980)
seminal paper, Barnett presented the weaknesses
of the simple-sum index as a measure of consumer's demand for money, and in
the process established a link between Diewert's (1978)
``superlative'' quantity index and monetary aggregation
theory.  This integration of monetary aggregation with micro-demand
analysis established the superlative-Divisia monetary index, a
nonparametric index that provides a second-order approximation of the
theoretical monetary aggregate, as the most theoretically correct
method of measuring the demand for money.

Both Barnett's original paper and the subsequent work by him and
others has been based on the assumption that the representative
consumer enters her utility maximization problem with either perfect
certainty or risk neutral preferences.\footnote{See Barnett, Fisher
and Serletis (1992) for an excellent literature review on the area of
monetary aggregation.} However, if the representative consumer is risk
averse the current theorems on economic aggregates
and statistical indices no longer hold.  Currently there is
very little known about aggregation and index theory
under risk, and even less about the connection between the two.
Fortunately, Barnett and Yue (1991) have already proven that a
theoretical monetary aggregate does exist under risk aversion.

With the existence of a theoretical monetary aggregate already
established the next step is to determine if the link between monetary
aggregates and statistical index theory continues to exist under risk
averse preferences.  One way of tackling this question is to
investigate how the approximating ability of the Divisia monetary
aggregate for a specific theoretical monetary aggregate is affected as
risk aversion increases.

The tracking ability of the Divisia monetary aggregate under risk
aversion is critical to both economist studying monetary aggregates
and those who utilize aggregates in setting monetary policy.  Since
Barnett (1980) showed that the theoretical aggregate is burdened by
estimation and model specification, the empirically calculated
aggregate will be dependent on the econometrician's method of
estimation and on the model he selects.  Whereas with the Divisia
aggregate every econometrician will obtain the same result regardless,
since it is a nonparametric approximation of the theoretical aggregate
and is easily calculated from known prices and quantities.

To show how well the Divisia monetary aggregate tracks the `true'
theoretical aggregate we numerically solve for the individual monetary
assets from the Euler equations associated with a dynamic optimization
problem.  We use the calculated rational expectation equilibrium
to determine both the Divisia and the theoretical monetary aggregate
and compare how well the Divisia tracks the movement of the
theoretical aggregate.

Although there are exists a number of numerical methods that solve for
the rational expectation equilibrium of such problems, for this paper
we have selected the approach advocated by Den Haan and Marcet
(1990).\footnote{See Taylor and Uhlig (1990) for a list and comparison
of the known methodologies to solving dynamic optimization problems.}
The Den Haan and Marcet approach has been shown to provide good
results for a number of complex optimization problems.\footnote{See
Den Haan and Marcet (1989), Marshall (1992), and Bansal et. al.
(1992).} Furthermore, the complexity of the dynamic programming
problem found in this paper is an important factor in choosing the Den
Haan and Marcet algorithm since this solution method does not require
discretizing the state space; a formidable task for any computer when
the number of state variables is as plentiful as they are in our
model.  Hence, in addition to determining if the link between the
Divisia and the theoretical monetary aggregate exists under risk
aversion, this paper also provides a complex dynamic optimization
problem that serves to test the Den Haan and Marcet algorithm's
ability to find an approximate rational expectation equilibrium.

\section{Modeling Consumer Demand for Monetary Assets}
\subsection{Preferences}

In this section we formulate a discrete time period optimization
problem for a maximizing representative consumer in an economy that
consists of a quantity aggregate for consumption goods and a quantity
aggregate for monetary goods.  We have chosen to comprise the monetary
aggregate with three monetary assets so that we can use the empirical
results found by Barnett and Yue (1991) to set the unknown parameters
of the model.  In addition we have tried to use the same notation that
Barnett and Yue (1991) used in an attempt to keep some consistency in
the literature.

We assume that the individual consumer maximizes her utility over the
finite planning horizon $t, t+1, t+2,\ldots,t+T$, where $t$ is the
beginning time period and $t+T$ is the terminal time period.  The
representative consumer's utility function is assumed to have the form
\begin{eqnarray}
V(M(\bld{m}_{t})^{1-\beta} X_{t}^{\beta}) & + & E_{t}\left[
\sum_{s=t+1}^{t+T-1} \rho^{s-t}
V(M(\bld{m}_{s})^{1-\beta} X_{s}^{\beta}) \right. \nonumber \\
& + & \left. \rho^{T}
V_{T}(M(\bld{m}_{t+T})^{1-\beta} X_{t+T}^{\beta},A_{t+T}) \right]
\end{eqnarray}
where $V$ is the constant relative risk aversion utility function
\begin{eqnarray}
V(M(\bld{m}_{s}),X_{s}) = \frac{1}{\sigma} \left[X_{s}^{\beta}
M_{s}^{1-\beta}
\right]^{\sigma}
\end{eqnarray}
with $\sigma \in (-\infty,0) \cup (0,1)$, $X_{s}$ and $M_{s}$ are
respectively the consumption good and monetary good quantity
aggregates,
and $\bld{m}_{s}$ is a $3 \times 1$ quantity vector of monetary
assets. If $\sigma \rightarrow 0$ the consumer's utility function
is $V(M_{s},X_{s}) =
\ln(X_{s}^{\beta} M_{s}^{1-\beta})$. In Eq. (1) the allocation
between the two aggregates is determined by $\beta$ which upon
closer
observation is the parameter of a subnested Cobb-Douglas utility
function, hence, $\beta \in (0,1)$.  The parameter $\rho$ is the
subjective discount factor with the restriction $\rho \in (0,1)$.

The variable $A_{s}$ represents the quantity of the benchmark asset
that the consumer plans on holding in period $s$. Because the yield on
$A_{s}$ only serves to transfer wealth across time periods and does
not yield liquidity nor any other services during the current period,
the benchmark asset provides nothing to the consumer and hence only
enters her utility function in the final time period of the planning
horizon. Furthermore, the benchmark asset's period yield is defined to
contain all the premiums of the market for foregoing the services
provided by a monetary asset.  Hence, the probability of the benchmark
asset's yield exceeding the yields of all the other assets is nonzero,
otherwise there would be no reason for the agent to hold the benchmark
asset.

Lastly, we assume that the `true' monetary aggregate,
$M(\bld{m}_{s})$, is defined as the CES function
\begin{eqnarray}
M(\bld{m}_{s}) = \left( \sum_{i=1}^{3} \delta_{i} m_{is}^{\alpha}
\right)^{1/\alpha}
\end{eqnarray}
with $\sum_{i=1}^{3} \delta_{i} = 1$, and $\alpha \in (0,1]$. Even
though the theoretical monetary aggregate is an explicit function of
the individual monetary assets, we drop the function's arguments to
simplify the notation and represent the monetary aggregate as $M_{s}$.
The monetary aggregate will equal the Cobb-Douglas aggregator function
if $\alpha \rightarrow 0$ and the linear aggregator function if
$\alpha$ equals one.

\subsection{Optimization Problem}

The representative consumer's dynamic optimization problem consists of
choosing the
deterministic point $(\bld{m}_{t}',X_{t},A_{t})$ and the stochastic
process $(\bld{m}_{s}',X_{s},A_{s})$ for $s=t+1,t+2,\ldots,t+T$
that
maximizes Eq. (1) subject to the budget constraints
\begin{eqnarray}
I_{s} \geq \sum_{i=1}^{3} \left[ (1+r_{i,s-1}) p_{s-1}^{*}
m_{i,s-1} -
p_{s}^{*} m_{i,s} \right] + (1+R_{s-1}) p_{s-1}^{*} A_{s-1} -
p_{s}^{*} A_{s} - p_{s}^{*} X_{s}
\end{eqnarray}
for $s=t,t+1,\ldots,t+T$.  The budget constraint is comprised of
the
expected nominal holding period yields of the monetary assets,
$r_{is}$, for $i=1,2,3$, and the expected one-period holding yield
on the
benchmark asset during period $s$, $R_{s}$.  Under the assumption
of
rational expectation the distribution of each asset's yield is
known
to the consumer.  However, since the payment received on a interest
accruing good does not occur until the end of the period the
consumer
will not know the actual value of either the $r_{is}$'s nor $R_{s}$
during period $s$.  On the other hand, the consumption good's price
aggregate (or true cost of living index) $p_{s}^{*}$ is determined
and
fully known by the representative agent at the beginning of period
$s$.  $I_{s}$ represents the sum of all other sources of income
during
period $s$.

Following Section 2 of Barnett and Yue (1991) the Bellman equation
for
the above dynamic optimization problem provides the following
first-order necessary condition Euler equations\footnote{Under more
restrictive assumptions Poterba and Rotemberg (1987) also find
these
same first-order conditions.}
\begin{eqnarray}
X_{s}^{\sigma \beta-1} M_{s}^{\sigma(\beta-1)} & = & E_{s} \left[
\left( \rho \frac{p_{s}^{*}}{p_{s+1}^{*}} (1+R_{s}) \right)^{-1}
X_{s+1}^{1-\sigma \beta} M_{s+1}^{\sigma (\beta-1)} \right] \\
X_{s}^{-\sigma \beta} M_{s}^{\sigma (\beta-1)+\alpha}
m_{1s}^{1-\alpha} & = & E_{s} \left[ \left( \frac{1}{\delta_{1}}
\frac{\beta \rho}{1-\beta} \frac{p_{s}^{*} R_{s}}{p_{s+1}^{*}}
\right)^{-1} X_{s+1}^{1-\sigma \beta} M_{s+1}^{\sigma (\beta-1)}
\right] \\
X_{s}^{-\sigma \beta} M_{s}^{\sigma (\beta-1)+\alpha}
m_{2s}^{1-\alpha} & = & E_{s} \left[ \left( \frac{1}{\delta_{2}}
\frac{\beta \rho}{1-\beta} \frac{p_{s}^{*}
(R_{s}-r_{2s})}{p_{s+1}^{*}}
\right)^{-1} X_{s+1}^{1-\sigma \beta} M_{s+1}^{\sigma (\beta-1)}
\right] \\
X_{s}^{-\sigma \beta} M_{s}^{\sigma (\beta-1)+\alpha}
m_{3s}^{1-\alpha} & = & E_{s} \left[ \left( \frac{1}{\delta_{3}}
\frac{\beta \rho}{1-\beta} \frac{p_{s}^{*}
(R_{s}-r_{3s})}{p_{s+1}^{*}}
\right)^{-1} X_{s+1}^{1-\sigma \beta} M_{s+1}^{\sigma (\beta-1)}
\right]
\end{eqnarray}
In the above dynamic programming problem the consumer faces the
exogenous stochastic state vector $\bld{\phi}_{s} = (R_{s-1},
r_{2,s-1}, r_{3,s-1}, p_{s}^{*}/p_{s-1}^{*}, I_{s}/I_{s-1})'$,
along
with the endogenous state vector $\bld{m}_{s-1}$.  Together they
define the state vector $\bld{\sigma}_{s}' = (\bld{m}_{s-1}',
\bld{\phi}_{s}')$, while the representative consumer's control
vector
for the optimization problem is $\bld{z}_{s}' =
(\bld{m}_{s}',A_{s},X_{s})$.

We assume that the stochastic process $\{ \bld{\phi}_{s} \}$ behaves
as two independent Markovian processes that have the behavior
\begin{eqnarray}
\bld{\phi}_{1s} & = & \bld{a}_{1} + \bld{A}_{1} \bld{\phi}_{1,s-1}
+ \bld{u}_{1s} \\
\ln(\bld{\phi}_{2s}) & = & \bld{a}_{2} + \bld{A}_{2}
\ln(\bld{\phi}_{2,s-1}) + \bld{u}_{2s}
\end{eqnarray}
where $\bld{\phi}_{1s} = (R_{s-1}, r_{2,s-1}, r_{3,s-1})'$ and
$\bld{\phi}_{2s} = (p_{s}^{*}/p_{s-1}^{*},
I_{s}/I_{s-1})'$.\footnote{A more complex stochastic processes for the
exogenous state process can be used if the economist so desires.}
$\bld{u}_{1s}$ and $\bld{u}_{2s}$ are both distributed $i.i.d.\; {\cal
N}({\bf 0},\bld{\Omega}_{i}),\;\mbox{for}\; i=1,2$, with $\bld{A}_{1}$
and $\bld{\Omega}_{1}$ being $3 \times 3$ matrices, whereas
$\bld{A}_{2}$ and $\bld{\Omega}_{2}$ are $2 \times 2$ matrices. The
processes' intercept terms $\bld{a}_{1}$ and $\bld{a}_{2}$ are $3
\times 1$ and $2 \times 1 $ vectors, respectively.

Each Markovian process is a first-order vector autoregressive process
(VAR(1)), the only difference between them being the natural
logarithmic transformation of $\bld{\phi}_{2s}$.  This transformation
insures that the process is stationary.  Since $\bld{\phi}_{1s}$ is
only comprised of interest rates there is no reason to transform this
process.

To insure that the benchmark asset's yield exceeds $r_{2s}$ and
$r_{3s}$ for all $s$ , $R_{s}$ is generated in the following manner $$
R_{s} = \cases{\max(r_{2s},r_{3s}) + \frac{1}{4}\max(r_{2s},r_{3s}) +
u_{1s_{1}}, & if $\frac{(R_{s}-\max(r_{2s},r_{3s}))}{R_{s}} \leq
0.05$\cr R_{s}, & otherwise\cr} $$ where $u_{1s_{1}}$ is the first
component of $u_{1s}$.

We have now completed the construction of the dynamic optimization
problem.  The representative agent makes her choice of the
deterministic point $\bld{z}_{t}$ and the stochastic points
$\bld{z}_{s}$ for $s=t+1,t+2,\ldots,t+T$ for a given parameter
vector
describing the economy.  If we define the economy's parameter
vector as
\begin{eqnarray}
\bld{\lambda} =
(\rho,\sigma,\alpha,\beta,\delta_{1},\delta_{2},\delta_{3},\bld{a
}_{1}',\bld{a}_{2}',vec(\bld{A}_{1})',vec(\bld{A}_{2})',vec(\bld{
\Omega}_{1})',vec(\bld{\Omega}_{2})')
\nonumber
\end{eqnarray}
then for each value of $\bld{\lambda}$ the above optimization
problem defines a nonlinear mapping from the exogenous state
process $\{
\bld{\phi}_{s} \}$ to $\{ \bld{z}_{s} \}$.  The theoretical
monetary
aggregate, $\{ M_{s} \}$, and the Divisa monetary aggregate
\begin{eqnarray}
Q_{s} = Q_{s-1} \prod_{i=1}^{3} \left( \frac{m_{is}}{m_{i,s-1}}
\right)^{s_{is}^{*}}
\end{eqnarray}
where
\begin{eqnarray}
s_{is}^{*} & = & \frac{1}{2}(s_{is} + s_{i,s-1}) \nonumber \\ s_{is} &
= & \frac{\pi_{is} m_{is}}{\sum_{j=1}^{3} \pi_{js} m_{js}},
\nonumber \\
\pi_{is} & = & \frac{p_{s}^{*} (R_{s} - r_{is})}{(1 + R_{s})}
\nonumber
\end{eqnarray}
can then be calculated from $\{ \bld{\phi}_{s} \}$ and $\{ \bld{z}_{s}
\}$.

\section{Solving the Optimization Problem}
\subsection{Parameterized Expectation Approach}
Although there has been a large demand for numerical methods that
solve a dynamic optimization problem like that found in Section 2 only
recently have such algorithms been devised.\footnote{See Taylor and
Uhlig (1990) for excellent source on the available methods and a
comparison on their performance record.} Of those currently available
to economists the Parameterized Expectation Approach (PEA) of Den Haan
and Marcet (1990) has preformed well in head-to-head tests with
other algorithms.  In addition, the PEA approach has been applied to a
number of different economic areas, including growth models (den Haan
and Marcet (1990)), asset markets with heterogenous agents (Ketterer
and Marcet (1989), Marcet and Singleton (1990)), and monetary
economies (den Haan (1990a, 1990b), Marshall (1992)).

The PEA method simply approximates the expectation operators found in
the Euler equations (5)-(8) by parameterizing them with a basis
function that spans the set of expectation operators.  The basis
function is a globally flexible functional form who's arguments are
the state variables $\bld{\sigma}_{s}$.\footnote{See Barnett and Jonas
(1982), Barnett and Yue (1988), and Gallant (1982) for the properties
and examples of globally flexible functions.} In this paper we have
chosen the first-order polynomial function, however, in future
research we plan to utilize the recent advances in wavelet theory to
provide a best wavelet basis.  One then iterates over the parameters
of these flexible functions until a convergence criterion is met.

Intuitively, each iteration of the PEA method can be viewed as a
nonlinear least-square learning behavior by the consumer.\footnote{See
Marcet and Sargent (1999a, 1989b) for examples of a linear
least-square learning model that have a locally stable equilibrium.}
Once learning is no longer occurring, the representative agent selects
the stochastic solutions $\{
\bld{z}_{s} \}$ that satisfies the Euler equations, given her
learned prediction of the expectation operators.  Hence, it can be
argued that the numerical solution found with the PEA approach is
an
equilibrium for a representative agent restricted to the learning
associated with a specific globally flexible functional form.

To apply the PEA method to the optimization problem in Section 2 we
write the system of Euler equations as
\begin{eqnarray}
\Phi( \bld{z}_{s} ) = E \left[ \Gamma( \bld{z}_{s+1},
\bld{\phi}_{s+1})
| \bld{\sigma}_{s} \right]
\end{eqnarray}
where $\Phi: \Re^{5} \rightarrow \Re^{4}$ and $\Gamma: \Re^{10}
\rightarrow \Re^{4}$. By defining each vector component of
$\Gamma$'s
range as $g_{i}( \bld{z}_{s+1}, \bld{\phi}_{s+1}),\;\mbox{for}
\;i=1,2,3,4$, we can write the PEA parameterization of each Euler
equation's
expectation operator as
\begin{eqnarray}
E [ g_{i}(\bld{z}_{s+1}, \bld{\phi}_{s+1}) | \bld{\sigma}_{s} ] =
\exp [
poly(\tilde{\bld{\sigma}}_{s},\bld{v}_{i}) ] \hspace{.25in}
i=1,2,3,4 \nonumber
\end{eqnarray}
where $\tilde{\bld{\sigma}}_{s}' = (1,\bld{\sigma}_{s}')$, and the
$\bld{v}_{i}$'s are the polynomial's coefficient vector.

An economist using the PEA approach first generates a single
realization of the random variables $\{ \bld{u}_{1s} \}$ and $\{
\bld{u}_{2s} \}$ from their known distributions.  These series along
with the two VAR(1) functions, (9) and (10), are then used by the
economist to generate the stochastic exogenous state processes, $\{
\bld{\phi}_{1s} \}$ and $\{
\bld{\phi}_{2s} \}$.  To initiate the two Markovian processes we
chose to let $\bld{\phi}_{10}$ equal $(0.098, 0.039, 0.071)'$, the
sample mean of the interest rates from an empirical data source, and
$\bld{\phi}_{20}$ equal to a vector of ones.\footnote{To reduce the
dependence of the exogenous state variables on these initial values we
exclude the first five periods from the simulation.}

To begin the PEA's iteration process an initial vector is selected
by
the economist for each of the polynomial's coefficients, which we
define as $\bld{v}_{i}^{0}$ for $i=1,2,3,4$.  Like all numerical
algorithms the choice of the $\bld{v}_{i}^{0}$'s is critical, but
even
more so with the PEA approach since each iteration does not
calculate
a directional update.  Hence, the PEA algorithm only has good local
convergence properties and thus, considerable attention should be
made
by the economist in selecting $\bld{v}_{i}^{0}$.\footnote{Homotopy
theory is helpful in insuring that the initial coefficients are
within
the set which converges to a solution.  We will present the
essentials of
homotopy theory in the next subsection.}

After the $\bld{v}_{i}^{0}$'s are determined the economist
calculates
each polynomial
$\exp[poly(\tilde{\bld{\sigma}}_{s}^{0},\bld{v}_{i}^{0})]$ and
substitutes them into Eq.\ (12) for
$E[\Gamma(\bld{z}_{s+1},\bld{\phi}_{s+1})|\bld{\sigma}_{s}]$.  Eq.\
(12) is a nonlinear system of equations which can be solved
explicitly
for $\{ \bld{z}_{s} \}$ for a special limiting case within the
parameter space,
but for most values of $\bld{\lambda}$ Eq.\ (12) is an implicit
system
of equations that requires the economist to use a Newton-like
algorithm to solve for $\{ \bld{z}_{s}^{1} \}$.\footnote{In our
actual
calculations we use a modified Powell hybrid algorithm based on the
MINPACK subroutine {\em HYBRD1} to solve the implicit system of
equations.}

To insure that the Newton-like algorithm converges to the correct
solution we rely on homotopy theory to provide initial guesses for the
solutions.  Later in the paper we will discuss the intricate details
of homotopy theory and how it is applied to the current problem of
selecting initial solutions for a Newton-like algorithm and the
earlier problem of choosing the starting vectors, $\bld{v}_{i}^{0}$.
However, for now we will only state that homotopy theory suggests
first solving the optimization problem for the special parameters case
where Eq.\ (12) can be explicitly solved for $\bld{z}_{s}$.  The
PEA solution to the optimization problem for that value of
$\bld{\lambda}$ provides the economist with a reasonably good choice
for the starting values of the Newton-like algorithm when the
optimization problem has a $\bld{\lambda}$ close to the special case.

Once $\{ \bld{z}^{1}_{s} \}$ is calculated, the economist takes the
solution and calculates $g_{i}(\bld{z}_{s+1}^{1}, \bld{\phi}_{s+1})$
for $i=1,2,3,4$. She then regresses each of them on a linearized
version of $\exp[poly(\tilde{\bld{\sigma}}_{s}^{1},\bld{v}_{i})]$.
This linearization is preformed around $\bld{v}_{i}^{0}$, which allows
the economist to use ordinary least-squares rather than a nonlinear
regression at each step of the iteration.

Because of the instability involved in the above iteration process for
complex optimization problems, Den Haan and Marcet (1990) suggest that
the updated coefficients for the polynomials be a convex combination
of the estimated and previous iterations coefficients. In other words,
the updated coefficients are calculated as
\begin{eqnarray}
\bld{v}_{i}^{1} = \eta \bld{b}_{i}^{1} + (1-\eta)\bld{v}_{i}^{0}
\end{eqnarray}
where $\eta \in [0,1]$ and $\bld{b}_{i}^{1}$ is the estimated
coefficients vector found in the previous paragraph.  The convex
combination of the coefficient estimates has a stabilizing affect on
the PEA's iterations for models that are normally explosive.  By
choosing smaller values of $\eta$ the updating of the polynomial's
coefficients is more gradual and hence, more resistant to an explosive
path.

The above steps are then repeated by the economist until the solutions
$\{ \bld{z}_{s} \}$ converge.  We use the convergence criterion
recommended by Bansal et.  al. (1992) which is
\begin{eqnarray}
\max_{i} \max_{s} \left|
\frac{(z_{is}^{k}-z_{is}^{k-1})}{(z_{is}^{k-1}+\epsilon)} \right|
\leq \xi
\end{eqnarray}
where $\epsilon$ and $\xi$ are small positive number, and $k$ is an
index for the number of the iterations performed.  Den Haan and Marcet
(1990) originally suggested that the estimated coefficients vectors,
$\bld{v}_{i}^{k}$, be tested for convergence.  However, since the
optimization problem contained in this paper has eight state variables
that are possibly correlated, testing convergence with the coefficients
of the polynomials would most likely suffer from the oscillating
nature associated with the estimated parameters of a multicollinear
model.\footnote{Later we explain and use Marshall's (1992) sum of
squared residuals convergence criterion.}

\subsection{Homotopy Theory}
Homotopy theory although very simple is a very powerful tool in
finding a solution to a complex fixed point algorithm like the ones
found in this paper.  The basic idea behind homotopy is to slowly
move
from a known solution to the solution that the
economist desires to find.  As a result of this methodology many
previously unstable and explosive algorithms are much more well
behaved and quickly converge to the correct solution.  For a
pedagogical example of homotopy theory we apply the theory to
solving the
nonlinear system Eq.\ (12).\footnote{See Garcia and Zangwill for
a
more extensive study on homotopy theory being applied to fixed
point
problems.}

Suppose that the economist has substituted
$\exp[poly(\tilde{\bld{\sigma}}_{s},\bld{v}_{i})]$ for $E[\Gamma
(\bld{z}_{s+1},\bld{\phi}_{s+1})|\bld{\sigma}_{s}]$, as we
instructed
in Subsection 3.1, and subtracts the appropriate exponentiated
polynomial scalar from both sides so that the system of equations
found in Eq.\ (12) can be written as
\begin{eqnarray}
F(\bld{z},\tilde{\bld{\sigma}}_{s},\bld{\lambda}) = {\bf 0}
\end{eqnarray}
where $F: \Re^{4} \times \Re^{10} \time \Re \rightarrow \Re^{4}$.  The
economist desires to solve Eq.\ (15) for the parameter value
$\bld{\lambda}^{*}$, but she does not have a good starting solution
for the Newton-like algorithm.  Fortunately, the economist happens to
know that the solution to Eq.\ (15) is $\bld{z}_{0}$ when the
parameter vector is $\bld{\lambda}_{0}$, i.e.\
$F(\bld{z}_{0},\tilde{\bld{\sigma}}_{s},\bld{\lambda}_{0}) = {\bf 0}$.

With this information the economist defines the homotopy function,
$H: \Re^{4}
\times [0,1] \rightarrow \Re^{4}$, where $H(\bld{z}_{0},0) = 0$,
$H(\bld{z}^{*},1) = 0$, and $\bld{z}^{*}$ is the solution to Eq.\
(15)
when the parameter value is $\bld{\lambda}^{*}$.  This homotopy
function
provides the economist with a
function $\bld{z}(\tau)$ that satisfies $H(\bld{z}(\tau),\tau) =
0$
for all $\tau \in [0,1]$.  Hence, the economist can gradually move
$\tau$ from zero to one and $\bld{z}(\tau)$ will map out the path
to
the solution $\bld{z}^{*}$ by providing the economist with the
Newton-like algorithm's starting points for the next value of
$\tau$.

In our optimization problem we not only apply homotopy theory in
solving Eq.\ (12), we also use the same concept in determining the
initial values of the coefficient vectors $\bld{v}_{i},$ for
$i=1,2,3,4$.
In the above discussion on homotopy theory we assumed that the
economist knows the solution when the parameter vector is equal to
$\bld{\lambda}_{0}$.  However, for our optimization problem we do
not
initially know the solution for any value of $\bld{\lambda}$.  To
provide an initial solution we instead utilize the PEA approach for
a
simple case of the optimization problem, i.e.  we choose a
$\bld{\lambda}$ for which the PEA converges.  We choose the
limiting case where $\sigma \rightarrow 0$ and $\alpha
\rightarrow 0$ causes the Euler equations of the optimization
problem
to be
\begin{eqnarray}
X_{s} & = & E_{s} \left[
\left( \rho \frac{p_{s}^{*}}{p_{s+1}^{*}} (1+R_{s}) \right)^{-1}
X_{s+1} \right]^{-1} \\
m_{1s} & = & E_{s} \left[ \left( \frac{1}{\delta_{1}}
\frac{\beta \rho}{1-\beta} \frac{p_{s}^{*} R_{s}}{p_{s+1}^{*}}
\right)^{-1} X_{s+1}
\right] \\
m_{2s} & = & E_{s} \left[ \left( \frac{1}{\delta_{2}}
\frac{\beta \rho}{1-\beta} \frac{p_{s}^{*}
(R_{s}-r_{2s})}{p_{s+1}^{*}}
\right)^{-1} X_{s+1} \right] \\
m_{3s} & = & E_{s} \left[ \left( \frac{1}{\delta_{3}}
\frac{\beta \rho}{1-\beta} \frac{p_{s}^{*}
(R_{s}-r_{3s})}{p_{s+1}^{*}}
\right)^{-1} X_{s+1}
\right]
\end{eqnarray}
and $\Phi$ of Eq.\ (12) to be the identity function.

The reader will notice that under this limiting parameter vector
solving for $\bld{z}_{s}$ in each iteration of the PEA approach no
longer requires a Newton-like numerical procedure. Rather, we are
able to explicitly solve for the vector $\bld{z}_{s}$ at each
iteration by setting the solutions equal to the parameterized value
of
the expectation operator,
$\exp[poly(\tilde{\bld{\sigma}}_{s},\bld{v}_{i}]$.

After convergence is reached with the PEA algorithm, homotopy theory
suggests that the economist use the calculated coefficient vectors,
$\bld{v}_{i},\;\mbox{for}\; i=1,2,3,4,$ as the starting coefficients
for the PEA approach and the associated PEA solutions $\{
\bld{z}_{s} \}$ as the initial guesses for the Newton-like
algorithm when the parameter values are either $\sigma \rightarrow
0$ and $\alpha = 0.1$ or $\sigma = -0.1$ and $\alpha \rightarrow
0$.  The coefficients found with the PEA approach for the new
parameter vector are then used as starting values for the PEA approach
when $\bld{\lambda}$'s values are again slightly changed.  This
iterative process of using the previous optimization problems
solutions as the starting values for the next optimization problem
continues until the economist reaches the specific combination of
parameters she desires.

For the special limiting case of $\sigma \rightarrow 0$ and $\alpha
\rightarrow 0$ we set all of the components of $\bld{v}_{i}^{0}$
equal to zero except for the coefficients of the constants.  These
coefficients are equal to the natural logarithm of the expected value
of $g_{i}(\bld{1}_{s+1}, \bld{\phi}_{s+1}),\;\mbox{for}\;
i=1,2,3,4,$.\footnote{This is the method that is recommend by Bansal
et. al. (1992).} The initial solutions $\{
\bld{z}_{s}^{0} \}$ are also all set equal to the expected
values of $g_{i}(\bld{1}_{s+1},\bld{\phi}_{s+1})$.\footnote{Another
equally valid method for setting the initial values of
$\bld{v}_{i}$'s
is to regress $g_{i,s}$ on
$\exp[poly(\tilde{\bld{\sigma}}_{s},\bld{v}_{i})]$ with $\{
\bld{z}_{s} \}$ equal to the steady state solution.}

\section{Simulation Results}
In this section we present some preliminary finding for the PEA
solutions to this paper's dynamic optimization problem.  All the
calculations were programmed in Fortan 77 and compiled and executed
on
a SPARC 10 workstation.\footnote{I am deeply indebted to Bob Hussey
for providing and explaining the initial coding from which I was
able
to change to fit the model found in this paper.} We experimented
with
different convergence levels and discovered that the strictest
convergence criterion for which the PEA approach would consistently
converge under 40,000 iterations was $\xi = 0.03$.  Convergence
of the solutions varied with each model but if the PEA algorithm
converged it usually did so within thirty minutes.  In addition,
by
applying homotopy theory once we obtained a PEA solution
convergence
for similar parameter values was much quicker and took less than
a
hundred iterations.

However, convergence of the PEA approach does not guarantee that
the
calculated solutions will be equal to the `true' solutions of the
optimization problem. Because the PEA approach is a numerical
algorithm it can only provide an approximation of the `true'
solution.
Hence, to determine if the approximation is close to the `true'
solution
we employ the den Haan, Marcet test statistic (DHM-stat) [den Haan
and
Marcet (1992) and Taylor and Uhlig (1990)].  The DHM-stat provides
a
test of the theoretical martingale property $E[ \bld{\nu}_{s}
\otimes
\bld{h}_{sij}] = 0$ where $\bld{\nu}_{s}$ is the $4 \times 1$
residual
vector from the Euler equations and $\bld{h}_{sij} =
\bld{\sigma}_{s-j}^{i}$. If Hansen's (1982) regularity conditions
hold and the approximation is an exact solution satisfying the
above martingale property then the DHM-stat
\begin{eqnarray}
T B A^{-1} B
\end{eqnarray}
where $B=1/T \sum_{s=t}^{t+T} [\bld{\nu}_{s} \otimes
\bld{h}_{sij}]$ and
$A = 1/T \sum_{s=t}^{t+T} [\bld{\nu}_{s} \otimes \bld{h}_{sij}]
[\bld{\nu}_{s} \otimes \bld{h}_{sij}]'$, will be distributed in law
$\chi^{2}$ with 20 degrees of freedom.  In the following results
we
have set the DHM-stat's $i$ and $j$ equal to one.

In each case the order of the approximating polynomial is equal to
one.  We attempted to increase the order of the polynomial to two,
both with and without the cross variable terms, but the PEA approach
became explosive after a few iterations.  The problem associated with
the higher order polynomial may be the result of the the polynomial's
large number of state arguments.  If this same problem exists
for other flexible functional forms is a topic for future research.

Table 1 shows all the parameter values, except for $\alpha$ and
$\sigma$, used in the PEA algorithm.  To insure that $\bld{\lambda}$
accurately reflects the empirical world we set the parameters
contained in the Euler equations (5)-(8) equal to the generalized
method of moment estimates found by Barnett and Yue (1991).  The
other
parameters associated with the two Markovian processes (9)-(10)
were
set equal to the estimated parameters for the two VAR(1) using
monthly
data from January 1960 to December 1990.  Lastly, in Table 1 the
parameter values $H_{i_{jk}}$ for $i=1,2,$ and $j,k=1,2,3$,
represent the Cholesky factorization of the random disturbances
covariance matrix, i.e.  $H_{i}H_{i}^{'} = \bld{\Omega}_{i}$.

To estimate the parameters of Eq.\ (10) we used the Consumer Price
Index for $p_{s}^{*}$ and nominal GNP for $I_{s}$.  The interest
rate
data for Eq.\ (9)'s regression was supplied to us by the St.\ Louis
Federal Reserve Bank.  Each investment's rate of return was
adjusted
to a common one-month maturity with Farr and Johnson's (1985) yield
curve adjustment to eliminate liquidity premiums associated with
longer maturity investments.  In addition, the investment's
interest
rates were adjusted to an annualized one-month yield on a bond
interest basis (365 day) as opposed to a bank basis (360 day).  For
the regression of Eq.\ (9) the benchmark assets rate of return was
measured by
\begin{eqnarray}
R_{s} = \max[r_{BAA}, (r_{is},\; i=1,2,3) ] \nonumber
\end{eqnarray}
where $r_{BAA}$ is the rate on Moody's BAA corporate bond, $r_{2s}$
was measured by the average of the following investment
returns
\begin{itemize}
\item RMMDAC: Rate paid on Money Market Deposit Accounts at
commercial banks
\item RMMDAT: Rate paid on Money Market Deposit Accounts at thrift
institutions
\item RSDCB: Rate on savings deposits less RMMDAC
\item RSDSL: Rate on savings deposits at FDIC-Insured savings banks
\item RSNOWC: Rate paid on Super NOWs at commercial banks
\item RSNOWT: Rate paid on Super NOWs at thrift institutions
\item RONRP: Rate paid on overnight dealer financing in the
repurchase
market
\item RONED: Rate paid on overnight eurodollars from London
\item RMMMF: Average yield on Money Market Mutual Funds
\end{itemize}
and $r_{3s}$ was the average of the returns
\begin{itemize}
\item RLTDCB: Rate on large time deposits at commercial banks
\item RSTDTH: Rate paid on small time deposits and retail
repurchase
agreements at thrifts.
\end{itemize}
In the above definition for interest rates we have implicitly
assumed
that $\bld{m}_{s}$ contains three aggregated elements.  In other
words, there exists a vector of investment vehicles $\bld{a}_{s}$
that
are weakly separable in the blocks described
above.\footnote{Barnett
and Yue (1990) used this same sub-nested blocking of $\bld{a}_{s}$
to
obtain the GMM estimates for this model.}

The size of our simulation is equal to 100, i.e. $T=100$.  We chose
a
small simulation size because of the explosive nature of the PEA
approach for large simulations if the solutions to the Euler
equations
(5)-(8) are not stationary processes.\footnote{Marcet and Marshall
(1992) provide a set of conditions and an alternative
PEA approach that overcomes the problems associated with
unstable solutions.  They advocate taking a large number of samples
of
random disturbances to generate the exogenous state variables and
use
the sample average of the polynomial's coefficient estimates,
$v_{i}$'s, as the coefficients for approximating the expectation
operators.}  This volatile behavior of the PEA approach results
from
the size of the set containing the endogenous state variables
becoming
bigger with larger $T$.  The increased size of the endogenous state
variable set presents the PEA approach with the difficulty of
trying to
determine which observations in the state space are relevant and
which
ones are not.  Because the PEA approach weights each observation
equally
as more and more irrelevant exogenous state observations occur the
PEA
method becomes explosive trying to fit the irrelevant observations
with the relevant states.  To overcome this problem of endogenous
oversampling for large simulations the economist can produce a
number
of small simulations.\footnote{See Marshall (1992) and Marcet and
Marimon (1991) for examples of this approach.}

\subsection{Limiting Case}
Table 2 shows the summary statistics for the limiting case.  The Den
Haan-Marcet statistic is also found in Table 2 with its $p$-value in
parenthesis. In the limiting case the DHM-stat statistically accepts
the hypothesis that the PEA algorithm has provided correct solutions.

In Fig.\ 1 the growth rates of the theoretical and Divisia monetary
aggregate for the limiting case are plotted. From these plots we found
that the Divisia monetary aggregate's growth rate identically tracks
the theoretical's growth rate up to two decimal places.  This result
favorably compares to the three decimal accuracy of the Divisia
aggregate when the representative agent has risk neutral preferences
[Barnett (1980)].  Furthermore, our discovery supports Barnett and
Yue's (1990) empirical findings that the Divisia aggregate tracked
reasonable well a GMM estimated theoretical monetary aggregate.
Hence, by increasing the measure of relative risk aversion from zero
to one the Divisia monetary aggregate looses its second-order
explanatory powers, but it continues to provide information about the
theoretical monetary aggregate up to the second decimal place.

Using the solutions from the limiting case we applied homotopy theory
to obtain the solutions for the cases where $\sigma
\rightarrow 0$ but $\alpha$ is allowed to move from zero to
$0.6$.\footnote{We applied homotopy theory for the cases $\alpha
= 0.1,\; 0.2,\; 0.3,\; 0.4$ by using the PEA solutions from $\sigma
\rightarrow 0 $ and $\alpha
\rightarrow 0$ as the initial starting values for the Newton-like
solutions of the system of equations.  For the other two cases $\alpha
= 0.5,\; 0.6$ we respectively used the polynomial coefficients from
the PEA solutions for $\alpha = 0.45,\; 0.5$ as the starting
coefficients for the PEA approach.} By keeping $\sigma$ the same and
allowing $\alpha$ to change we are able to keep the level of relative
risk aversion equal to one, but alter the theoretical monetary
aggregate from a Cobb-Douglas function to a CES aggregation function.
Furthermore, this change in parameter values moves us closer to the
empirically estimated value of $\alpha$, which Barnett and Yue (1991)
found to be equal to $0.8426$.  Unfortunately, we were unable to
obtain convergence with the PEA approach for values of $\alpha$ larger
than $0.7$.

The summary statistics for these cases are found in Table 3 and 4.
The reader will note that in each of these tables the DHM-stat rejects
the hypothesis that the approximate solutions are equal to the true
rational expectation equilibrium.  These results suggest that the PEA
approach loses its ability to find approximate solutions that are
close to the real Euler equation solutions when the Euler equations
can not be written in reduced form.  This shortcoming could be the
result of either the PEA approach or the inability of the Newton-like
algorithm we used to solve the system of equations.

Even with the rejection of the approximate solutions, in Fig.\
2 through 7 the Divisia monetary aggregate's growth rate
continues to closely track that of the theoretical aggregate.
Although the tracking by the Divisia monetary aggregate's growth
rate
is no longer identical to the second decimal place of the true
aggregate's growth, the closeness found in these figures suggests
that the Divisia monetary aggregate continues to produce reliable
nonparametric approximations of the theoretical aggregate.

\subsection{Higher Risk Case}
In this subsection we present the PEA solutions for our dynamic
optimization problem when the value of $\sigma$ becomes smaller,
i.e.
the level of relative risk aversion increases.  However,
unlike the previous solutions in this subsection we use the
convergence
criterion advocated by Marshall (1992) to determine the stopping
point
for the PEA algorithm.\footnote{We initially used the convergence
criterion found in Eq.\ (14) but the PEA algorithm failed to
converge
within 40,000 iterations.} Marshall argues that the PEA algorithm
converges to an approximate equilibrium if the coefficient vectors,
$\bld{v}_{i}$ for $i=1,2,3,4$, are the optimal prediction vectors.
The $\bld{v}_{i}$'s are considered optimal coefficient vectors if
they
minimize the sum of squared residuals associated with the PEA
approaches' least square regressions.  Hence, the convergence
criterion
used in this subsection is
\begin{eqnarray}
\max_{i} \left| \frac{SSR(\bld{v}_{i}^{k}) -
SSR(\bld{v}_{i}^{k-1})}{SSR(\bld{v}_{i}^{k-1})} \right| \leq \xi
\end{eqnarray}
where $SSR(\bld{v}_{i}^{k})=\sum_{s=t+1}^{t+T} \nu_{isk}^{2}$, and
$\nu_{isk}$ is the $s$ observation of the residuals associated with
the estimated coefficient vector from the $k$th iteration.  As
before all the results were calculated using $\xi = 0.03$.

The summary statistics for the PEA solutions are found in Table 5 and
6.\footnote{In each case we used homotopy theory to provide the
initial values for the polynomial's coefficients.  For those models
with $\sigma = -0.1,\, -0.2,\, -0.3$ we used the parameter values
found for the solution to the limiting case as initial values, while
for $\sigma = -0.4,\, -0.5,\, -0.6,\, -0.7$ we used the PEA's solution
parameters from the model with $\sigma = -0.3$ and $\alpha
\rightarrow 0.0$.}  Once again the DHM-stat with $p$-values
approximately equal to zero soundly rejects the hypothesis that the
approximated solutions are equal to the true rational expectation
solutions.  Furthermore, the Divisia and the theoretical monetary
aggregate's growth rates plotted in Fig.\ 8 to 15 reveal either the
poor performance of the Divisia monetary aggregate in tracking the
true aggregate, or the inability of the PEA approach to come close to
approximating the true solutions.  From the encouraging results found
when $\sigma \rightarrow 0$ and $\alpha
\rightarrow 0$ we suspect that it is the later.

The reader will note that as $\sigma$ becomes negative the Euler
equations (5)-(8) include both the current and future period's
theoretical monetary aggregate.  This, along with the complex
nature
of the implicit system of equations, seems to create a serious
problem
for the PEA approach.  We can only hypothesis that this is the
case,
but in the future we desire to determine if this hypothesis is
correct
by finding the approximate rational expectation solutions to our
dynamic
optimization problem with one of the other Euler equation
techniques
supplied by Coleman (1990, 1991), Baxter (1991), or Baxter,
Crucini, and
Rouwenhorst (1990).

\section{Conclusion}
In conclusion we have provided the PEA solutions to a dynamic
optimization problem where a representative agent allocates her
earnings between a consumption aggregate good and a monetary aggregate
comprised of three monetary assets.  It is important to find the
approximate rational expectation equilibrium to this optimization
problem in order to determine how strong the link between index and
aggregation theory is under risk.  The stronger this link the more
reliable the Divisia monetary aggregate is as a nonparametric
approximation to the `true' theoretical monetary aggregate.  In
addition, by incorporating the PEA approach in this task we have also
provided a dynamic programming problem that proves to be a formidable
test to the PEA's ability to produce reliable solutions.  We found
both success and failure in these objectives.

The results found in this paper show that the PEA method preformed
poorly in those cases where the parameter values caused the Euler
equations to become an implicit set of equations.  This occurred
whenever the dynamic programming problem's $\alpha$ or $\sigma$ did
not equal the limiting case values of $\alpha \rightarrow 0$ and
$\sigma \rightarrow 0$.  Hence, when either the level of relative risk
aversion increased above one, or if the theoretical monetary aggregate
was a CES function rather than the limiting case's Cobb-Douglas, the
PEA approach failed to provide approximate solutions that were close
to the `true' rational expectation equilibrium.

The poor tracking performance of the Divisia monetary aggregates also
occurred in these same cases.  Although the Divisia's tracking ability
was considerably worse for those cases where the level of relative
risk aversion was higher, the Divisia aggregate did provide helpful
information about the growth of the theoretical monetary aggregate
when a CES function was used as the theoretical aggregate.  However,
knowing how poorly the PEA approach preformed in providing correct
solution for these cases we reserve judgement on how good or bad the
Divisia aggregate tracks the theoretical aggregate when relative risk
increases above one or the theoretical monetary aggregate is a CES
function.  A better numerical method for calculating the rational
expectation solutions needs to be developed before any conclusion on
this matter can be made.

The one case where both the PEA approach and the Divisia aggregate
preformed at their best was the limiting case where $\sigma
\rightarrow 0$ and $\alpha \rightarrow 0$.   Under this special
limiting case the PEA approach obtained solutions that were
statistically close to the optimization problem's true solutions as
measured by the Den Haan, Marcet statistic.  The success of the PEA
approach also provided the Divisia aggregate with those solutions that
caused it to identically track the growth of the theoretical aggregate
up to its second decimal place.  Hence, our findings for the special
limiting case validates Barnett and Yue's (1990) empirical results that
the Divisia aggregate closely tracks the estimated theoretical
aggregate, but it also provides the area of economic aggregation with
the incentive to find a Diewert-like theorem that links aggregation
theory with index theory when risk aversion is present.


\newpage
\begin{thebibliography}{99}
\bibitem{bans: pea} Bansal, Ravi, A.\ Ronald Gallant, Robert Hussey
and George Tauchen (1992), ``Computational Aspects of Nonparametric
Simulation Estimation.'' Forthcoming in David Belsley (eds.), {\em
Computational Economics and Econometrics II,} (Academic Publishers).
\bibitem{barn: agg1} Barnett, William A.\ (1980), ``Economic
Monetary
Aggregates: An Application of Index Number and Aggregation
Theory,''
{\em Journal of Econometrics}, 14, pp.\ 11-48.
\bibitem{barn: rev} Barnett, William A., Douglas Fischer and
Apostolos Serletis (1992), ``Consumer Theory and the Demand for
Money,'' {\em Journal of Economic Literature,} 30, pp.\ 2086-2119.
\bibitem{barn: jona} Barnett, William A. and Andrew B.\ Jonas (1983),
``The M\"{u}ntz-Szatz Demand System: An Application of a Globally Well
Behaved Series Expansion,'' {\em Economics Letters,} 11, pp. 337-342.
\bibitem{barn: aim2} Barnett, William A.\, and Piyu Yue (1988),
``Semiparametric Estimation of
the Asymptotically Ideal Model (AIM): The AIM Demand System,'' in
G. Rhodes and T. Formby (eds.), {\em Nonparametric and Robust
Inference}, Advances in Econometrics, vol. 7, (Greenwich
Connecticut: JAI), 229-252.
\bibitem{barn: agg2} Barnett, William A.\ and Piyu Yue (1991),
``Exact Monetary
Aggregation Under Risk,'' Working Paper \#163, Washington
University.
\bibitem{bax: aap} Baxter, Marianne (1991), ``Approximating
Suboptimal
Dynamic Equilibria: An Euler Equation Approach,'' {\em Journal of
Monetary Economics,} 27, pp.\ 173-200.
\bibitem{bax: aap1} Baxter, Marianne, Mario J.\ Crucini, and K.\
Geert
Rouwenhorst (1990), ``Solving the Stochastic Growth Model by a
Discrete-State-Space, Euler-Equation Approach,'' {\em Journal of
Business and Economic Statistics}, 8, pp.\ 19-21.
\bibitem{col: eul1} Coleman, Wilbur John II (1990), ``Solving the
Stochastic Growth Model by Policy-Function Iteration,'' {\em
Journal
of Business and Economic Statistics,} 8, pp.\ 27-29.
\bibitem{col: eul2} Coleman, Wilbur John II (1991), ``Equilibrium
in a
Production Economy with an Income Tax,'' {\em Econometrica}, 59,
pp.\
1091-1104.
\bibitem{dhm: pea2} den Haan, Wouter J. (1990a),  ``The Optimal
Inflation Path in a Sidrauski-type Model with Uncertainty,'' {\em
Journal of Monetary Economics,} 25, pp.\ 389-409.
\bibitem{dhm: pea3} den Haan, Wouter J. (1990b),  ``The Term
Structure
of Interest Rates in Real and Monetary Production Economies,''
Working
Paper, Carnegie-Mellon University.
\bibitem{dhm: pea} den Haan, Wouter J.\ and Albert Marcet (1990),
``Solving the Stochastic Growth Model by Parameterizing
Expectations,'' {\em Journal of Business and Economic Statistics,}
8,
pp.\ 31-34.
\bibitem{dhm: pea} den Haan, Wouter J.\ and Albert Marcet (1992),
``Accuracy in Simulations,'' Working Paper \#92-30, University of
California, San Diego.
\bibitem{diew: agg} Diewert, W.\ Erwin (1976),  ``Exact and
Superlative Index Numbers,'' {\em Journal of Econometrics,} 4, pp.\
115-145.
\bibitem{farr: adj} Farr, H.\ T.\ and D.\ Johnson (1985),
``Revisions
in the Monetary Services (Divisia) Indexes of Monetary
Aggregates,''
mimographed (Board of Governors of the Federal Reserve System,
Washington, D.\ C.).
\bibitem{gall: four1} Gallant, A.\ Ronald (1982), ``Unbiased
Determination of Production Technologies,'' {\em Journal of
Econometrics}, 20, pp. 285-323.
\bibitem{garc: homo} Garcia, C.\ B.\ and W.\ I.\ Zangwill, (?) {\em
Pathways to Solutions, Fixed Points, and Equilibria,} Prentice
Hall.
\bibitem{han: gmm} Hansen, Lars P.\ (1982),  ``Large Sample
Properties of Generalized Method of Moments Estimators,'' {\em
Econometrica,} 50, pp.\ 1029-1054.
\bibitem{keet: pea} Ketterer, J.A. and Albert Marcet (1989),
``Introduction of Derivative Securities: A General Equilibrium
Approach,'' manuscript.
\bibitem{marmon: exp} Marcet, Albert and R.\ Marimon (1990),
``Communication, Commitment, and Growth,'' manuscript, Universitat
Pompeu Fabra, Barcelona.
\bibitem{marmar: exp} Marcet, Albert and David A.\ Marshall (1992),
``Convergence of Approximate Model Solutions to Rational
Expectations
Equilibria Using The Method of Parameterized Expectations,''
Working
Paper \#73, Northwestern University.
\bibitem{marsar: lear1} Marcet, Albert and Thomas J.\ Sargent
(1989a),
``Convergence of Least Squares Learning Mechanisms in
Self-Referential
Linear Stochastic Models,'' {\em Journal of Economic Theory,} 48,
pp.\
337-368.
\bibitem{marsar: lear1} Marcet, Albert and Thomas J.\ Sargent
(1989b),
``Convergence of Least-Squares Learning in Environments with Hidden
State Variables and Private Information,'' {\em Journal of
Political
Economy,} 97, pp.\ 1306-1322.
\bibitem{marsin: pea} Marcet, Albert and Kenneth Singleton (1990),
``Simulation Analysis of Dynamic Stochastic Models: Applications
to
Theory and Estimation,'' manuscript.
\bibitem{marsh: mon} Marshall, David A.\ (1992),  ``Inflation and
Asset
Returns in a Monetary Economy,'' {\em Journal of Finance,} 47, pp.\
1315-1342.
\bibitem{pot: mon} Poterba, James M. and Julio J. Rotemberg (1987),
``Money in the Utility Function: An Empirical Implementation,'' in
William A.\ Barnett and Kenneth Singleton (eds.), {\em New Approaches to
Monetary Economics,} (Cambridge University Press, Cambridge),
pp.\ 219-240.
\bibitem{tayuh: rea} Taylor, John B.\ and Harald Uhlig (1990),
``Solving Nonlinear Stochastic Growth Models: A Comparison of
Alternative Solution Methods,''  {\em Journal of Business and
Economic
Statistics,} 8, pp.\ 1-17.
\end{thebibliography}

\newpage
\begin{table}
\begin{center}
\begin{tabular}{cr}
$\rho$  & 0.89750 \\
$\beta$ & 0.95350 \\
$\delta_{1}$ & 0.46560 \\
$\delta_{2}$ & 0.33710 \\
$\delta_{3}$ & 0.19730 \\
$a_{11}$ & 0.00252 \\
$a_{12}$ & 0.00040 \\
$a_{13}$ & 0.00178 \\
$a_{21}$ & 0.00149 \\
$a_{22}$ & 0.00414 \\
$A_{1_{11}}$ & 0.90260 \\
$A_{1_{21}}$ & -0.00031 \\
$A_{1_{31}}$ & -0.01017 \\
$A_{1_{12}}$ & 0.00755 \\
$A_{1_{22}}$ & 0.99056 \\
$A_{1_{32}}$ & 0.01083 \\
$A_{1_{13}}$ & 0.09682 \\
$A_{1_{23}}$ & 0.00181 \\
$A_{1_{33}}$ & 0.98477 \\
$A_{2_{11}}$ & 0.65470 \\
$A_{2_{21}}$ & -0.32513 \\
$A_{2_{12}}$ & -0.02609 \\
$A_{2_{22}}$ & -0.15840 \\
$H_{1_{11}}$ & 0.00817 \\
$H_{1_{12}}$ & 0.00181 \\
$H_{1_{22}}$ & 0.00192 \\
$H_{1_{13}}$ & 0.00451 \\
$H_{1_{23}}$ & 0.00148 \\
$H_{1_{33}}$ & 0.00352 \\
$H_{2_{11}}$ & 0.00246 \\
$H_{2_{12}}$ & 2.0D-05 \\
$H_{2_{22}}$ & 0.00624
\end{tabular}
\caption{Parameter Values}
\end{center}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{||crr||}
\hline \hline
\multicolumn{3}{||c||}{$\sigma \rightarrow 0$ $\alpha \rightarrow
0$} \\
\hline
   &\multicolumn{1}{c}{$\mu$} & \multicolumn{1}{c||}{$\sigma^{2}$}
\\
\hline
$X$ & 24655.300 & 1.4D+07  \\
$m_{1}$ & 6954.800 & 1.8D+06 \\
$m_{2}$ & 10662.800 & 5.1D+06 \\
$m_{3}$ & 17586.900 & 3.9D+07 \\
$M$ & 9688.920 & 4.0D+06  \\
$Q$ & 77.120 & 3.2D+02 \\
$\dot{M}$ & -0.005 & 7.9D-03 \\
$\dot{Q}$ & -0.006 & 8.4D-03 \\
DMS & \multicolumn{2}{l||}{ 0.0018 (1.0) }  \\
\hline \hline
\end{tabular}
\end{center}
\caption{Limiting Case}
\end{table}


\begin{table}
\begin{center}
\begin{tabular}{||lcrr||}
\hline \hline
\multicolumn{1}{||c}{Case}&   &\multicolumn{1}{c}{$\mu$} &
\multicolumn{1}{c||}{$\sigma^{2}$} \\
\hline
$\sigma \rightarrow 0.0\; \alpha = 0.1$ &
$X$ & 2.4D+07 &   \\
 & $m_{1}$ & 6.5D+06 &  \\
 & $m_{2}$ & 1.0D+07 &  \\
 & $m_{3}$ & 1.8D+07 &  \\
 & $M$ & 9.6D+06 & 0.4D+13  \\
 & $Q$ & 74.6529 & 0.3D+03 \\
 & $\dot{M}$ & -0.004 & 0.9D-02 \\
 & $\dot{Q}$ & -0.005 & 0.1D-01 \\
 & DMS & \multicolumn{2}{l||}{ 98.99 (0.0) }  \\
\hline
$\sigma \rightarrow 0.0\; \alpha = 0.2$ &
$X$ & 6.7D+12 &   \\
 & $m_{1}$ & 1.7D+12 &  \\
 & $m_{2}$ & 2.9D+12 &  \\
 & $m_{3}$ & 5.5D+12 &  \\
 & $M$ & ****** & 3.2D+23  \\
 & $Q$ & 69.78200 & 3.2D+02 \\
 & $\dot{M}$ & -0.00371 & 1.1D-02 \\
 & $\dot{Q}$ & -0.00486 & 1.3D-02 \\
 & DMS & \multicolumn{2}{l||}{ 98.99 (0.0) }  \\
\hline
$\sigma \rightarrow 0.0\; \alpha = 0.3$ &
$X$ & 1.2D+05 &   \\
 & $m_{1}$ & 3.0D+04 &  \\
 & $m_{2}$ & 5.4D+04 &  \\
 & $m_{3}$ & 1.1D+05 &  \\
 & $M$ & 5.0D+04 & 1.2D+08  \\
 & $Q$ & 65.30593 & 3.3D+02 \\
 & $\dot{M}$ & -0.00369 & 0.00124 \\
 & $\dot{Q}$ & -0.00478 & 0.00146 \\
 & DMS & \multicolumn{2}{l||}{ 98.99 (0.0) }  \\
\hline \hline
\end{tabular}
\end{center}
\caption{}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{||lcrr||}
\hline \hline
\multicolumn{1}{||c}{Case}&   &\multicolumn{1}{c}{$\mu$} &
\multicolumn{1}{c||}{$\sigma^{2}$} \\
\hline
$\sigma \rightarrow 0.0 \; \alpha = 0.4$ &
$X$ & 1.2D+05 &   \\
 & $m_{1}$ & 3.0D+04 &  \\
 & $m_{2}$ & 5.5D+04 &  \\
 & $m_{3}$ & 1.3D+05 &  \\
 & $M$ & 5.0D+04 & 1.4D+08  \\
 & $Q$ & 60.65087 & 3.4D+02 \\
 & $\dot{M}$ & -0.00350 & 0.00141 \\
 & $\dot{Q}$ & -0.00439 & 0.00174 \\
 & DMS & \multicolumn{2}{l||}{ 98.99 (0.0) }  \\
\hline
$\sigma \rightarrow 0.0\; \alpha = 0.5$ &
$X$ & 3.1D+05 &   \\
 & $m_{1}$ & 5.9D+04 &  \\
 & $m_{2}$ & 1.4D+05 &  \\
 & $m_{3}$ & 3.9D+05 &  \\
 & $M$ & 1.3D+05 & 9.2D+08  \\
 & $Q$ & 55.79256 & 3.5D+02 \\
 & $\dot{M}$ & -0.00267 & 0.00172 \\
 & $\dot{Q}$ & -0.00313 & 0.00224 \\
 & DMS & \multicolumn{2}{l||}{ 98.99 (0.0) }  \\
\hline
$\sigma \rightarrow 0.0\; \alpha = 0.6$ &
$X$ & 1.5D+07 &   \\
 & $m_{1}$ & 2.3D+06 &  \\
 & $m_{2}$ & 6.7D+06 &  \\
 & $m_{3}$ & 2.4D+07 &  \\
 & $M$ & 6.8D+06 & 2.6D+12  \\
 & $Q$ & 51.51806 & 3.6D+02 \\
 & $\dot{M}$ & -0.00048 & 0.00231 \\
 & $\dot{Q}$ & -0.00055 & 0.00304 \\
 & DMS & \multicolumn{2}{l||}{ 98.99 (0.0) }  \\
\hline \hline
\end{tabular}
\end{center}
\caption{}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{||lcrr||}
\hline \hline
\multicolumn{1}{||c}{Case}&   &\multicolumn{1}{c}{$\mu$} &
\multicolumn{1}{c||}{$\sigma^{2}$} \\
\hline
$\sigma = -0.1\; \alpha \rightarrow 0.0$ &
$X$ & 11486.50 &   \\
 & $m_{1}$ & 3289.34 &  \\
 & $m_{2}$ & 5211.88 &  \\
 & $m_{3}$ & 7938.32 &  \\
 & $M$ & 67.58 & 30.61  \\
 & $Q$ & 87.76 & 378.42 \\
 & $\dot{M}$ & -0.00244 & 0.00064 \\
 & $\dot{Q}$ & -0.00472 & 0.00851 \\
 & DMS & \multicolumn{2}{l||}{ 98.99 (0.0) }  \\
\hline
$\sigma = -0.2\; \alpha \rightarrow 0.0$ &
$X$ & 7973.33 &   \\
 & $m_{1}$ & 2314.42 &  \\
 & $m_{2}$ & 3742.51 &  \\
 & $m_{3}$ & 5637.62 &  \\
 & $M$ & 58.70 & 22.89 \\
 & $Q$ & 89.22 & 392.83 \\
 & $\dot{M}$ & -0.00232 & 0.00069 \\
 & $\dot{Q}$ & -0.00403 & 0.00938 \\
 & DMS & \multicolumn{2}{l||}{ 98.99 (0.0) }  \\
\hline
$ \sigma = -0.3\; \alpha \rightarrow 0.0$ &
$X$ & 11230.4 &   \\
 & $m_{1}$ & 3179.4 &  \\
 & $m_{2}$ & 4875.3 &  \\
 & $m_{3}$ & 7877.2 &  \\
 & $M$ & 66.22 & 0.00291  \\
 & $Q$ & 68.48 & 0.00028 \\
 & $\dot{M}$ & -0.00253 & 0.00114 \\
 & $\dot{Q}$ & -0.00167 & 0.01704 \\
 & DMS & \multicolumn{2}{l||}{ 98.99 (0.0) }  \\
\hline \hline
\end{tabular}
\end{center}
\caption{}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{||lcrr||}
\hline \hline
\multicolumn{1}{||c}{Case}&   &\multicolumn{1}{c}{$\mu$} &
\multicolumn{1}{c||}{$\sigma^{2}$} \\
\hline
$ \sigma = -0.4\; \alpha \rightarrow 0.0$ &
$X$ & 6147.51 &   \\
 & $m_{1}$ & 1791.93 &  \\
 & $m_{2}$ & 2855.18 &  \\
 & $m_{3}$ & 4586.94 &  \\
 & $M$ & 52.81606 & 1.7D+01  \\
 & $Q$ & 78.64297 & 3.1D+02  \\
 & $\dot{M}$ & -0.00240 & 0.00062 \\
 & $\dot{Q}$ & -0.00462 & 0.00892 \\
 & DMS & \multicolumn{2}{l||}{ 98.97 (0.0) }  \\
\hline
$\sigma = -0.5\; \alpha \rightarrow 0.0$ &
$X$ & 3500.93 &   \\
 & $m_{1}$ & 1040.49 &  \\
 & $m_{2}$ & 1706.88 &  \\
 & $m_{3}$ & 2784.05 &  \\
 & $M$ & 42.68298 & 1.1D+01  \\
 & $Q$ & 83.70820 & 3.4D+02 \\
 & $\dot{M}$ & -0.00227 & 0.00058 \\
 & $\dot{Q}$ & -0.00448 & 0.00859 \\
 & DMS & \multicolumn{2}{l||}{ 98.97 (0.0) }  \\
\hline
$\sigma = -0.6\; \alpha \rightarrow 0.0$ &
$X$ & 2138.06 &   \\
 & $m_{1}$ & 644.63  &  \\
 & $m_{2}$ & 1081.45 &  \\
 & $m_{3}$ & 1780.79 &  \\
 & $M$ & 35.44056 & 0.7D+01  \\
 & $Q$ & 87.30851 & 3.7D+02 \\
 & $\dot{M}$ & -0.00217 & 0.00059 \\
 & $\dot{Q}$ & -0.00414 & 0.00894 \\
 & DMS & \multicolumn{2}{l||}{ 98.97 (0.0) }  \\
\hline
$ \sigma = -0.7\; \alpha \rightarrow 0.0$ &
$X$ & 1412.10 &   \\
 & $m_{1}$ & 424.82 &  \\
 & $m_{2}$ & 710.43 &  \\
 & $m_{3}$ & 1201.31 &  \\
 & $M$ & 30.10261 & 0.5D+01 \\
 & $Q$ & 86.53571 & 3.5D+03 \\
 & $\dot{M}$ & -0.00209 & 0.00060 \\
 & $\dot{Q}$ & -0.00392 & 0.00913 \\
 & DMS & \multicolumn{2}{l||}{ 98.99 (0.0) }  \\
\hline \hline
\end{tabular}
\end{center}
\caption{}
\end{table}

\end{document}



