Pamela Labadie

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Research Papers in Economics

FEDERAL RESERVE BANK OF ST. LOUIS REVIEW JULY/AUGUST 2007 327
Commentary
Pamela A. Labadie
OVERVIEW OF THE MODEL
There is a single, exogenous, stochastic, and
perishable endowment good. The endowment
grows at the rate xt, with stochastic volatility vt:
Hence, the endowment process is characterized
by a parameter vector ￿ˆx,ˆv,ʸx,ʸv,˃v￿.
There is a representative agent with Epstein-
Zin preferences. The advantages of this preference
structure, with its property of separating relative
risk aversion from the elasticity of intertemporal
substitution, are well known. The preference
parameter vector is ￿β,ˁ,α￿. The pricing kernel
is the intertemporal marginal rate of substitution
in consumption, denoted log￿mt+1￿. The price of
default-free discount bonds can be determined
recursively through an arbitrage-free restriction
of the form
With this structure for the endowment process
and the preferences, the model can be mapped
into the Duffie-Kan affine term-structure model
with two factors. In particular, the authors guess
a form of the value function and then verify that
this guess is a solution. The value function then
implies the form of the real pricing kernel. The
process by which the authors relate the deeper
preference and endowment parameters to the
bE m b t
n
tt t
n ( )
++
− ( ) = 11
1 .
xx v
v
t xx x t t t
x
tv v
+ +
+
=− ( ) ++
=− ( ) +
1
1
2
1
1
1
1
ˆʸ ˆ ʵ
ˆʸ ˆ v vt vt
v v + + σʵ 1.
T
he questions addressed in the Gallmeyer
et al. (2007) paper are important ones:
How does monetary policy affect long-
term interest rates? How can we explain
the volatility of the long end of the yield curve
and its relationship with monetary policy? A
successful model is one that answers these ques-
tions and is useful to policymakers. To be useful
to policymakers, the model should yield quanti-
tative answers to questions such as, What will
happen to long-term rates if the central bank
raises the federal funds rate by 25 basis points?
Since money is not introduced in a way that is
essential,1 this model is not designed to answer
questions about the mechanism by which mone-
tary policy affects the yield curve. This is not
necessarily a deficiency, because models where
money is essential have yet to prove useful in
policy discussions and empirical results.2 The
strength of this model is that it provides quanti-
tative answers to the questions. Hence, it is rea-
sonable to set a standard of success where the
model-generated time series must, in some sense,
“look like” actual time series.
As in any paper where hard modeling deci-
sions are made, there are both strengths and weak-
nesses in the choices made by the authors. To
describe these choices concisely, I’ll provide a
brief overview of the model.
1 Wallace (2001) says that money is “essential” in an economy if it
permits allocations that would otherwise not be achieved.
2 See Kocherlakota (2002).
Pamela A. Labadie is a professor of economics at The George Washington University.
Federal Reserve Bank of St. Louis Review, July/August 2007, 89(4), pp. 327-29.
© 2007, The Federal Reserve Bank of St. Louis. Articles may be reprinted, reproduced, published, distributed, displayed, and transmitted in
their entirety if copyright notice, author name(s), and full citation are included. Abstracts, synopses, and other derivative works may be made
only with prior written permission of the Federal Reserve Bank of St. Louis.discrete-time affine term-structure model is inter-
esting and elegant.
To introduce money and inflation into the
model, the authors add a stochastic process to
the real pricing kernel:
where pt+1 is inflation. Hence, money, prices, and
inflation in the model are merely noise, creating
a wedge between the real and nominal pricing
kernels. Two specifications of the inflation process
are studied: exogenous and endogenous.
Exogenous Inflation
Inflation is conjectured to take the form
This specification does not link inflation to the
endowment process or to any money-growth
process. The state space is expanded to xt,vt,pt.
They calibrate the model to data and set the
parameters values at
The preference parameters are set at ˁ = 0 and
α = –2.91, where ˁ is the elasticity of intertem-
poral substitution and α is relative risk aversion,3
and there is little discussion of this choice. With
this set of parameter values, the model is used
to derive a yield curve. The result is a model-
generated yield curve that matches the shape of
the historical yield curve, but exhibits less volatil-
ity in long rates. Since an explicit goal of the
paper is to explain volatility at the long end of
the yield curve, this answer is not satisfactory.
Three comments arise: The first is why are these
values for the preference parameters chosen?
How does varying the preference parameters
change the results? Should we just aimlessly
search the parameter space for a better fit? This
ˆ
ʸ
σˆ
p
p
pp
=
=
=− ( )
08 4 7 1
00 0 9 3
00 0 6 31
2
1
2
.
.
..
pp tp p p t p t
p
++ =− ( ) ++ 11 1 ˆʸ ˆ σ ʵ.
log log , m mp t tt + ++ ( ) = ( )− 1 11
$
leads to my second comment. The model, as
specified, can be estimated using formal econo-
metric techniques. The endowment process and
inflation processes, along with the real and nomi-
nal pricing kernels, form a system of equations.
Using data on nominal interest rates, one can
estimate inflation and consumption growth, the
parameters of the model, and in particular the
preference parameters. The preference-parameter
estimates could then be used to generate a real
pricing kernel. What are the estimated preference
parameters and does economic intuition suggest
that they are sensible? How does the real pricing
kernel behave? It seems a missed opportunity.
Finally, the model, as posed, severely restricts
the price of inflation risk by fixing the price of
inflation risk at unity. A very simple cash-in-
advance model with fixed velocity has the prop-
erty that inflation is a function of both money
growth and output growth, so there is a state-
varying inflation risk premium. Even with a
zero mean, the inflation-risk premium may be
an additional source of variability and may help
to remedy the lack of volatility at the long end of
the yield curve in the model.
Endogenous Inflation
To make inflation endogenous, the authors
assume that monetary policy follows a nominal
interest rate rule (Taylor rule) of the form
This rule raises short-term rates aggressively in
response to inflation. There are many other
specifications that could be considered, and it
would be helpful to have a discussion on why
this rule is chosen over other specifications. Is
this the type of rule the authors believe monetary
authorities are using? Is this the rule that gives
the best results in the sense that the model-
generated yield curve matches the data? Is it
chosen for tractability?
This process must be consistent with the
other equations in the model, which requires the
derivation of an inflation process consistent with
the interest rate rule and other equations. To link
the rule to the nominal pricing kernel and bond-
ix p s tx t p t t =+ + + ττ τ .
328 JULY/AUGUST 2007 FEDERAL RESERVE BANK OF ST. LOUIS REVIEW
3 These are the values chosen in the version of the paper presented
at the conference.
Labadiemarket equilibrium, they use a guess-and-verify
method to derive a consistent inflation process
of the form
The state space is now zt = ￿xt,vt,st￿, and there are
additional restrictions on the means and condi-
tional variances. Once again they calibrate the
model, choosing parameter values taken from the
data. The calibrated model fits the average yield
curve and has a volatility pattern closer to the
data, especially at the long end.
This is a major goal of the paper and, in that
sense, it is successful; but the question arises as
to whether the endogenous inflation process in
any way resembles the inflation process in the
data. My conjecture is that it does not—and in
some important ways—and the differences need
to be made explicit. How do the endogenous and
exogenous inflation processes compare? If the
calibrated model fits the average yield curve and
closely matches the volatility, but is based on an
inflation process that differs significantly from
the actual inflation process, how useful is this to
policymakers? How useful is a model that fits
the yield curve and its volatility but is greatly at
odds with the actual inflation process? Common
sense would suggest it is of limited usefulness.
CONCLUSION
The authors are to be commended for devising
a model with such a rich potential for explaining
yield curves and their volatility. Linking the
Epstein-Zin preferences to a discrete-time affine
px v s tx t v t s t =+ + + ˀˀ ˀ ˀ.
term-structure model is no easy task, although
they seem to do so effortlessly. The model is
devised to be estimated with standard econometric
methods, using data on bond prices, consumption
growth, and inflation. Such an exercise would
provide useful insights into the real pricing kernel
and the parameters of the Epstein-Zin preferences.
The exogenous inflation specification is too
restrictive and should permit a variable inflation-
risk premium. Finally, alternative interest rate
rules should be examined with the explicit goal
of generating an inflation process using the model
that matches the actual inflation process, accord-
ing to some explicit criterion. While much of this
may sound negative, I want to emphasize that
this model has the potential to be very useful to
policymakers and the steps needed to make it so
are straightforward ones to take.
REFERENCES
Gallmeyer, Michael F.; and Hollifield, Burton;
Palomino, Francisco and Zin, Stanley. “Aritrage-
Free Bond Pricing with Dynamic Macroeconomic
Models.” Federal Reserve Bank of St. Louis Review,
July/August 2007, 89(4), pp. 305-26.
Kocherlakota, Narayana. “Money: What’s the Question
and Why Should We Care about the Answer?”
American Economic Review, May 2002, 92(2),
pp. 58-61.
Wallace, Neil. “Lawrence R. Klein Lecture 2000:
Whither Monetary Economics?” International
Economic Review, November 2001, 42(4), pp. 847-69.
Labadie
FEDERAL RESERVE BANK OF ST. LOUIS REVIEW JULY/AUGUST 2007 329330 JULY/AUGUST 2007 FEDERAL RESERVE BANK OF ST. LOUIS REVIEW

AritrageFree Bond Pricing with Dynamic Macroeconomic Models.” Federal Reserve Bank of St. Louis Review,

Klein Lecture 2000: Whither Monetary Economics?” International Economic Review,

Money: What’s the Question and Why Should We Care about the Answer?” American Economic Review,

Commentary on "Arbitrage-free bond pricing with dynamic macroeconomic models"

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Commentary on "Arbitrage-free bond pricing with dynamic macroeconomic models"

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