Download Structural Estimates of the U.S. Sacrifice Ratio

Structural Estimates of the U.S. Sacrifice Ratio
Author(s): Stephen G. Cecchetti and Robert W. Rich
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Source: Journal of Business & Economic Statistics, Vol. 19, No. 4 (Oct., 2001), pp. 416-427
Published by: American Statistical Association
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Structural
Estimates
of
U.S.
the
Sacrifice
Ratio
StephenG. CECCHETTI
Departmentof Economics, The Ohio State University,Columbus, OH 43210-1172
([email protected])
Robert W. RICH
Domestic Research Function,Federal Reserve Bank of New York,New York,NY 10045-0001
([email protected])
This article investigatesthe statisticalpropertiesof the U.S. sacrifice ratio--the cumulativeoutput loss
arising from a permanentreduction in inflation. We derive estimates of the sacrifice ratio from three
structuralvector autoregressionmodels and then conducta series of simulationexercises to analyze their
sampling distribution.We obtain point estimates of the sacrifice ratio that are consistent with results
reportedin earlier studies. However, the estimates are very imprecise, which we suggest reflects the
poor quality of instrumentsused in estimation.We conclude that the estimatesprovide a very unreliable
guide for assessing the output cost of disinflationpolicy.
KEY WORDS: Disinflation;Identification;Structuralshocks; Vector autoregression.
The successful conduct of monetary policy requires policy makers to both specify a set of objectives for the performance of the economy and understandthe effects of policies
designed to attain these goals. Stabilizing prices, one of the
dual goals of U.S. monetarypolicy, is no different.It is generally agreed, and documentedby Barro (1996) and the papers
in Feldstein (1999), that permanentlylow levels of inflation
create long-run benefits for society, increasing the level and
possibly the trendgrowthrate of output.Thereis also a strong
belief that engineeringinflationreductionsinvolves short-term
costs associated with losses in output. Policy makers' decisions on the timing and extent of inflation reduction depend
on a balancing of the benefits and costs of moving to a new,
lower level of inflation,which in the end requiresestimates of
the size of each.
This study focuses on the size of one of these aspects of
disinflationpolicy for the U.S. over the postwarperiod. Specifically, we investigate the output cost of disinflation policy,
usually referredto as the sacrifice ratio. The sacrifice ratio is
the cumulative loss in output, measured as a percent of one
year's gross domestic product (GDP), resulting from a onepercentage-pointpermanentreductionin inflation.
Although the sacrifice ratio is a key considerationfor policy makers, estimating its size is a difficult exercise because
it requiresthe identificationof changes in the stance of monetary policy and an evaluationof their impact on the path of
output and inflation. Neither one of these determinationsis
straightforwardbecause it is difficult to gauge the timing and
extent of shifts in policy as well as to separatethe movements
in output and inflation into those that were caused by policy
and those thatwere not. Finally,even if an estimateof the sacrifice ratio can be constructed,it is critical for policy makers
to know something about the precision of the estimate.
To investigatethe U.S. sacrificeratio and its statisticalproperties, we employ structuralvector autoregression(VAR) estimation methods. Within this framework,we study models of
increasingcomplexity,beginningwith Cecchetti's (1994) twovariablesystem, then consideringShapiroand Watson's(1988)
three-variablesystem, and finally examining Gali's (1992)
four-variablesystem. In each of these models, we derive estimates of the sacrifice ratio under a differentset of identifying
restrictionsfor the structuralshocks. To assess the reliability
of the sacrifice-ratioestimates, we undertakea series of simulation exercises, based on parametricbootstraps,that allow us
to constructconfidence intervals.
Our modeling strategyfollows the conventionalpractice in
the structuralVAR literatureof decomposing monetary policy into a systematic and a random component to identify
changes in the stance of policy. The systematic component
can be thoughtof as a reactionfunction and describesthe historical response of the monetaryauthorityto movements in a
set of key economic variables, while the random component
is labeled as "monetary-policyshocks" and signifies actions
of the monetary authority that cannot be explained by the
policy rule. Using the estimated structuralVAR system, we
can trace the dynamic responses of variables to a monetarypolicy shock and thereby assess the quantitativeimpact of a
shift in policy on outputand inflation.Thus, our evaluationof
the sacrifice ratio focuses on unanticipatedpolicy shocks, in
which the effects of a contractionarypolicy correspondto a
"pure"(exogenous) monetarytighteningratherthan a systematic (endogenous) response to other shocks.
As an alternativeto the use of the structuralVAR framework and its focus on the dynamic responses to innovations
in the system, one might consider studying the effects of the
systematiccomponentof monetarypolicy. Attemptingto measure the sacrificeratio, and its precision, by looking at periods
in which the monetary authorityadopted a stronger (or easier) anti-inflationarypolicy would requirethat experimentsin
which there are statisticallymeasurableshifts in policy regime
exist in the data. It is our belief that such episodes are not
available. This is confirmedby a series of tests for parameter
stability, in which we find that the models display little evidence of structuralbreaks over the sample period we examine. But beyond the practical issue of whether we have an
appropriatehistory in the sample period, there is the problem of finding an econometric technique that would permit
416
? 2001 American Statistical Association
Journal of Business & Economic Statistics
October 2001, Vol. 19, No. 4
Cecchettiand Rich: StructuralEstimatesof the U.S. SacrificeRatio
a straightforwardevaluation of the effect of changes in the
policy rule on the economy. Because of their reduced-form
nature, structuralVAR techniques do not allow us to distinguish between the direct effects of shocks and the indirect
effects that arise from their eliciting systematic responses by
the monetaryauthority.
We examine quarterlyU.S. data over the period 1959-1997
and consider the behavior of the sacrifice ratio over a one- to
five-yearhorizon. The analysis generatesestimates of the sacrifice ratio that vary substantiallyacross the three structural
VAR models. Specifically,the estimatesrange from about 1 to
nearly 10, suggesting that somewherebetween 1%and 10%of
one year's GDP must be sacrificedfor inflationto fall one percentage point. Although the point estimates are broadly consistent with the results of previous studies, the analysis also
suggests that the sacrifice ratio is very imprecisely estimated.
For example, a 90% confidence intervalcovers 0 for the point
estimates in each model. Further,the sacrifice-ratioestimates
display even greaterimprecisionas the models allow for additional structuralshocks. Although the simplest two-variable
system indicates that the true value of the sacrifice ratio may
lie somewherebetween -0.5 and +3.8, the four-variablesystem suggests a possible range that extends from -43 to +68.
We do not take these latter estimates too literally because the
values seem extremely implausible given our reading of the
recent history. Nevertheless, they serve as a caution to policy
makers in that the point estimates do not seem to provide a
very reliable guide for gauging the output cost of disinflation
policy.
After presenting the basic results, we look into the possible sources of the imprecision in the estimates. Our findings do not suggest that the imprecisionarises from structural
instability of the reduced-formVAR's or from a deterioration
in the forecasting performance of higher-dimensionalmodels. Rather,the evidence points to weak instrumentsimplicitly
used to estimate the models. This latter finding is consistent with the conclusions of Pagan and Robertson (1998),
who reported substantialrandomnessin structuralVAR estimates of the liquidity effect and documentedthat poor instrument quality is an important source of the imprecision.
Because the models examined in this study and by Pagan and
Robertsonshare similaridentificationschemes, futureresearch
employing the structuralVAR methodology may need to be
particularlyconscious of the issue of instrumentquality and
its implications for the reliability of the estimates.
1. THE SACRIFICERATIO
Most people believe that attemptson the partof a monetary
authorityto lower the inflation rate will lead to a period of
increasedunemploymentand reduced output.The reason disinflationaryepisodes have this effect on real economic activity
is that inflation displays a great deal of persistence or inertia. That is, price inflation [measuredby indexes such as the
consumerprice index (CPI)] tends to move slowly over time,
exhibiting very different behavior from things like stock or
commodity prices. Thus, the adjustmentprocess during a disinflation requires the monetary authority to slow aggregate
417
demand growth, creating a period of temporaryslack in the
economy that will lower the inflation rate only eventually.
There are a number of explanations for inflation's
slow adjustment and the absence of costless disinflation.
Fuhrer (1995) provided a review of recent discussions and
focused on three possibilities. First, inflation persistence may
arise from the overlap and nonsynchronizationof wage and
price contracts in the economy. Because wages and prices
adjust at different times, as well as to each other, slowing
the process takes time. Second, people's inflationexpectations
may adjustslowly over time, being based on a sort of adaptive
mechanism.Because decisions aboutwages and prices depend
on expectations of future changes, slow adaptationis selffulfilling, creating inertia. And third, if people do not believe
that the monetary authority is truly committed to reducing
inflation, then inflation will not fall as rapidly. That is, the
credibilityof the policy makeris importantin determiningthe
dynamics of inflation, with less credibility leading to more
persistence.
The view that reductions in inflation are accompaniedby
a period of decreased output (relative to trend) has generated considerabledebate among economists on how to lessen
the costs of disinflation. Some discussions, including those
of Okun (1978), Gordon and King (1982), Taylor (1983),
Sargent (1983), Schelde-Andersen (1992), and Ball (1994),
have focused on the speed of disinflation and whether the
monetaryauthorityshould adopt a gradualistapproachor subject the economy to a "cold turkey"remedy.Otherdiscussions
have focused on identifying the sources of inflation persistence and analyzing the implications of the level of inflation
(Ball, Mankiw,and Romer 1988), the degree of nominal wage
rigidity (Grubb, Jackman, and Layard 1983), and the extent
of centralbank independence(Jordan1997) for the pursuitof
cost-reducingstrategies.
Although the question of how to lessen the costs of disinflation raises a numberof importantissues, the present analysis does not focus on this debate. Rather, we contend that
discussions about the design and implementationof disinflation policy cannot proceed without some understandingof the
quantitativeimpact of monetary policy on output and inflation. Thus, it is our view that the measurementof the sacrifice
ratio is a prerequisiteto any study attemptingto evaluate its
key determinantsor the impact of alternativepolicies on the
costs of disinflation.
With this practicalissue in mind, a numberof authorshave
estimated sacrifice ratios for the United States using a variety of techniques. Okun (1978) examined a family of Phillips
curve models and derived estimates that range from 6% to
18% of a year's gross nationalproduct,with a mean of 10%.
Gordon and King (1982) used traditionaland VAR models to
obtain estimates of the sacrifice ratio that range from 0 to 8.
Mankiw (1991) examined the 1982-1985 Volckerdisinflation
estiand used Okun'slaw to arriveat a "back-of-the-envelope"
movements
Ball
examined
mate of 2.8. More recently,
(1994)
in trend output and trend inflation over various disinflation
episodes and obtainedestimates that vary from 1.8 to 3.3.
Although the estimates calculated by Ball (1994) and
Mankiw (1991) are of roughly the same order of magnitude,
suggesting that a consensus may exist about the size of the
418
Journalof Business & Economic Statistics,October2001
4
3
2-
l
01
-1 _
-2-358
68
78
88
98
Figure 1. QuarterlyGrowthRate of Output:Real GDP: 1959:Q11997:Q4.
sacrifice ratio, there are several issues remaining.First, prior
studies do not, in our view, adequatelycontrol for the impact
of nonmonetaryfactors on the behavior of output and inflation. Consider the plots of the quarterlygrowth rate of real
GDP and the CPI displayed in Figures 1 and 2. Although
some of the movements in output and inflation over the postWorldWarII period are surely attributableto monetary-policy
actions, it is unreasonable on either theoretical grounds or
from visual inspectionto believe they all are. Thus, computing
a meaningfulestimate of the sacrificeratio requiresmore than
simply calculating a measure of the association between output and inflationduringarbitrarilyselected episodes. Rather,it
S
depends
critically on isolating which movements result from
monetaryinfluences.
Second, previous studies such as that of Gordon and King
(1982) have failed to account for the policy process. Some of
the actions undertakenby a monetary authorityare intended
to accommodate or offset shocks to the economy. However,
the analysis of Gordonand King did not allow the movements
in a policy variableto be separatedinto those associated with
a shift in policy and those reflecting a systematic response to
the state of the economy. This type of decomposition, which
is necessary to assess the effects of monetary policy on the
economy, requiresthe specificationand estimationof a structural economic model.
Another importantissue concerns the periods selected for
the empiricalanalysis. Studies such as thatof Ball (1994) have
focused solely on specific disinflationaryepisodes-periods
when contractionarymonetary policies are thought to have
resultedin the reductionin both inflationand output.However,
it is not obvious that estimates of the sacrifice ratio should
exclude a priori episodes in which inflation and output are
both increasing. Such an approachwould only be justified if
there were an accepted asymmetryin the impact of monetary
policy on output and prices. In the absence of such evidence,
4.0
3.5
3.0
C
2.5
2.0
.
1.5s
1.0
0.5
0.0
economic expansionswould contain episodes in which output
and inflation increased as a result of expansionarymonetary
policy. Such episodes would then be as informativeabout the
sacrifice ratio as a disinflation.
Filardo (1998) was an exception because he presentedevidence thatthe sacrificeratiofor the United States varies across
threeregimes correspondingto periodsof weak, moderate,and
strong output growth. Using a measure of the sacrifice ratio
similar to that used in this study, Filardo derived sacrificeratio estimates of 5.0 in the weak-growthregime and 2.1 in
the strong-growthregime.
Although the findings of Filardo offer new and interesting
insights into the outputcost of disinflation,his study, as well
as previous work, was silent on the accuracyof the estimates.
Specifically, there has been no serious attemptto characterize
the statisticalprecision of sacrifice-ratiomeasures.Estimation
of economic relationshipsand magnitudesinherentlyinvolves
some uncertainty,and it is extremely importantto quantify
their reliability.For example, policy makersmay be reluctant
to undertakecertain policy actions unless they can attach a
high degree of confidence to the predictedoutcomes. Characterizing the precision of sacrifice-ratioestimates is a primary
goal of our analysis.
We now turnto a discussion of the structuralVAR methodology and a descriptionof the models that we use to construct
sacrifice-ratioestimates.
2. STRUCTURAL
VECTORAUTOREGRESSIONS
2.1 IdentifyingMonetary-PolicyShocks and
DerivingEstimates of the Sacrifice Ratio
The structuralVAR methodology remains a popular technique for analyzing the effects of monetary policy on the
economy. A structuralVAR can be viewed as a dynamic
simultaneous-equationsmodel with identifying restrictions
based on economic theory. In particular,the structuralVAR
relates the observed movementsin a variableto a set of structural shocks-innovations that are fundamentalin the sense
that they have an economic interpretation.In the formulation of their identificationassumptions,the models we employ
appealto economic theories that then allow us to interpretone
of the structuralinnovations as a monetary-policyshock. For
this reason, we find the structuralVAR methodology attractive in evaluating the impact of monetary policy on output
and inflation and giving us a measure of the sacrifice ratio.
Admittedly, the structuralVAR methodology is not without
its limitations. The estimated effects of shocks can vary considerablyas a result of slight modificationsto the identifying
restrictions.Thus, it may be quite importantto examine a set
of models when drawing inferences based on this approach.
Our strategy for deriving an estimate of the sacrifice ratio
can be illustratedwithin a relatively simple system that only
includes output and inflation.Following Cecchetti (1994), we
consider the following structuralVAR model:
,
Ayt =
58
Figure 2.
1997:Q4.
68
78
88
98
Quarterly Growth Rate of Prices: CPI Inflation: 1959:Q1-
b lAy,_i
+ b 2A
t+
i=1
A1rr,= b , Ay, +
-b2
ti+
i=l
i=1
bi , Ayti
+"
i=1
b227
ti
Et
(1)
Cecchettiand Rich: StructuralEstimatesof the U.S. SacrificeRatio
419
that can be rewrittenmore conveniently as
Because the structuralshocks are not observable,we obtain
estimates of the structuralimpulse responses by using the
B(L) A•,
(2) reduced-formVAR representationof (1) in conjunction with
et
identifying restrictions.Formally,the unrestrictedVAR reprewhere yt is the log of output at time t, 7rtis the inflation rate sentation is given by
between time t - 1 and t, A denotes the difference operator
=
=
DX
- D2Xt-2
(5)
DkXt-k
..D(L)Xt
-t,
Xt -D(1 - L), B(L) - [Bij(L)] for i, j = 1, 2 is a (2 x 2) matrix
of polynomial lags, and E, = [E45,
Et]' is a vector innovation where Xt is an (n x 1) vector of endogenous variables,D(L)
to
that
contains
the
shocks
aggregatesupply (Ey) and is a kth-orderlag polynomial matrix, and t, is the (n x 1)
process
It
assumed
that Et has zero mean vector of innovationsto the system. It is assumed that has
demand
is
aggregate
(E7).
,t
and is serially uncorrelatedwith covariance matrix
E[Et,,] = zero mean and is serially uncorrelatedwith covariancematrix
fl for all t. The model includes the change in the inflation
E[/xjt/] = I for all t.
rate to allow shocks to have a permanenteffect on its level.
Equation(5) can be estimatedand invertedto yield its unreOurprimaryinterestis in the impactof the structuralshocks strictedVMA representation:
on output and inflation over time. To evaluate these magnitudes, we can look at the vector moving average (VMA) rep(6)
X, = L,t + ClL-1 + C2.t-2 +'-' = C(L)jt.
of
the
of
which
resentation (1),
impulse responses
provides
From (2) and (3), the VAR and VMA representationsof the
the system to the structuralshocks. This is written as
structuralsystem can be written,respectively,as
aIylEt-i+
i=o
Ay,]
Lata
I
-i=O
i Y _+
a2et-i
BoXt = B X_1 +B2Xt-2
a2rt-i
i=O
X, = AoEt+ AI •E,_ -'
i=O
[A11(L)A12(L) rY
(7)
t+
and
a 22 t-i
SA21(L) A22(L) LEt
+- - +BkXt-k
(3)
If we initially use aggregatedemand shocks to identify shifts
in monetarypolicy, then (3) providesa particularlyconvenient
representationto assess the dynamicimpactof a monetarypolicy shock on outputand inflation.An estimate of the sacrifice
ratio can then be computed based on the structuralimpulse
response functions from (3).
For inflation, the sum of the first r coefficients in A22(L)
measures the effect of a monetary-policyshock on its level 7
periods forward.In the case of output, however, the sacrifice
ratio requiresus to consider the cumulativeeffect on its level
resulting from a monetarypolicy shock. This quantitycan be
expressed as a function of the coefficients in A12(L). Taken
together,the relative impact of monetarypolicy on outputand
inflation, and hence the sacrifice ratio, over the time horizon
7 is just the ratio of these effects and can be calculated as
= A(L)Et.
(8)
Equations(6) and (8) imply that
E[(AoEt)(Ao0t)']
=
AoIAo=
=
E[(,t)(,
tt)]
(9)
and
A(L) = C(L)A0.
(10)
The ability to link the unrestrictedVAR to a structuralVAR
model hinges crucially on the estimation of the matrix A0.
Because A0 has (n x n) unique elements, complete identification requiresa total of n2 restrictions.This is a necessary but
not sufficient condition for identificationbecause sufficiency
requiresthe matrix A0 to be invertible.
One set of identifying restrictionsis based on the assumption that the structuralshocks are uncorrelatedand have unit
variance. That is, fl = I, where I, is the (n x n) identity
matrix. In addition to assuming that fl is the identity matrix,
there are three other sets of identifying restrictions that are
employed in the estimation of A0. Two of these sets focus
on the effects of structural shocks on particular variables
df
lfl
S(7)=
(4) and involve short-runrestrictions(A0 restrictions)or long-run
Lj=oa)
restrictions[A(1) restrictions].The third involves restrictions
on the coefficients of contemporaneousvariablesin the struct
re
u
For
aimt2
-aioaa12
diinf2m
oi=n
/io=0
ii=0
tural equations(B0 restrictions).
Following Blanchardand Quah (1989), our additionalidentifying restriction for the Cecchetti model is that aggregate
i=O
demand shocks have no permanent effect on the level of
ai
output. This is equivalent to the condition that A12(1) =
L ai=a2 = 0.
(4)
(
The Cecchetti model only identifies two shocks and associates shifts in monetary policy with the aggregate demand
For a disinflationarymonetary strategy undertakenat time t, disturbance.Although this identificationscheme is useful for
the numeratormeasuresthe cumulativeoutputloss throughthe illustrating the structuralVAR methodology and may profirst 7 periods (ignoring discounting), while the denominator vide a good approximationfor analyzing the relative importance of nominal and real shocks, the frameworkcould yield
is the difference in the level of inflation 7 periods later.
(
Journalof Business & Economic Statistics,October2001
420
misleading estimates of the sacrifice ratio. Specifically, the
restrictions in the Cecchetti model fail to identify separate
components of the aggregate demand disturbance.Thus, the
estimated monetary-policyshock would not only encompass
policy shifts but also other shocks related to government
spending or shifts in consumptionor investmentfunctions.
To provide a more detailed analysis, we also derive estimates of the sacrificeratio from models developedby Shapiro
and Watson (1988) and Gali (1992). These models allow us
to decompose the aggregatedemanddisturbanceinto separate
componentsand thereforecan be used to judge the sensitivity
of the results to alternativemeasures of the monetary-policy
shock.
Drawing on the work of Shapiro and Watson (1988), we
consider a three-variablesystem given by
E
SAt
[ E
Ar"tj= A(L)
(it - 7Tt)
(11)
L
L
where it is a short-termnominal interestrate, (it - rt) is an ex
post real interestrate, and A(L) is a (3 x 3) matrixof polynomial lags. Although we refer to this as the "Shapiro-Watson
model," our specification actually differs from theirs in two
small ways. First, Shapiroand Watsondecomposedthe aggregate supply disturbanceinto a technology shock and a laborsupply shock. Second, they also included oil prices as an
exogenous regressorin their structuralVAR system.
Based on this three-variablesystem, we are able to identify three structuralshocks, where eY continues to denote an
aggregatesupply disturbanceand where the aggregatedemand
disturbanceis now decomposed into an LM shock and an IS
and E1s. We identify the
shock, denoted, respectively,by "LM
structuralshocks using both short-runand long-run restrictions. The Blanchard-Quahrestrictionallows us to identify the
aggregatesupply shock. In addition,we discriminatebetween
IS and LM shocks by assuming that monetarypolicy has no
contemporaneouseffect on output (a?2= 0). We estimate the
sacrifice ratio by identifying monetary-policyshifts with LM
shocks.
The Gali (1992) model, which allows for the identification
of a fourth structuralshock, can be written as
Ayt
L
i
- 'lt)
(Amt(
=iEMS
A(L)
(12)
M,,D
L sI
where mt is the log of the money supply, (Amt --r)
is the
growth of real money balances, and A(L) is a (4 x 4) matrix
of polynomial lags.
The structural demand shocks els, eMD, and
EtS
denote
(a2 = a3 = 0). The second assumptionis that contemporaneous prices do not enter the money-supplyrule (b?3+ b4 = 0).
Our estimate of the sacrifice ratio uses money-supply shocks
to identify changes in the stance of monetarypolicy.
Although the inflationrate does not appearas an individual
variablein (12), we can recoverits impulse responsefunctions
from those estimated for Ait and (i, - rt). Specifically, we
can use the following relationship
7rt = it[= (1 - L)-'Ait] - (it - 7rt)
(13)
to obtainan estimateof the impactof a monetary-policyshock
on the level of inflation 7 periods forward:
T
+'r
aTrt
t
)
i
r
a22
--a32.
i=O
(14)
2.2 ComputingConfidenceIntervals
The sacrifice-ratioestimates we compute are not the true
values but are randomvariableswith distributions.It is natural
to compute and reportsome measure of the precision for our
estimates. Thereare severalpossible proceduresfor computing
confidence intervals for S, (r). One possibility would be to
note that the sacrifice ratio is a function of the parametersof
the structuralVMA and use the delta method.This is computationally demandingand infeasible for the more complex models we study. Insteadwe employ simulation-basedmethods to
constructexact small-sampledistributionsof the estimatorof
the sacrifice ratio. The procedurecan be briefly described as
follows.
Let 0 denote the vector of parametersthat constitute the
reduced-formVAR model, and let 6 and I denote, respectively, the estimatedparametervector and estimatedvariancecovariance matrix of the residuals of the reduced-formVAR
using the observed set of data. Our simulation approachis
based on specific distributionalassumptionsabout (0, 1) and
follows the procedure described by Doan (1992, chap. 10).
Specifically, we construct a sequence of random draws of
I from an inverted Wishart distribution,each of which is
then used to generate a draw for 0. For each draw of
(0i, Vi), we can impose the relevant identification restrictions and estimate an artificial structuralVAR model, compute its impulse response functions, and then construct an
estimate of the sacrifice ratio denoted by Si(Oi, iW). In this
manner, a series of N simulations can be undertakenand
used to construct N estimates of the sacrifice ratio denoted
by Sl(O', E'), S2(02, 2) ...,
SN(ON, CN). We can then construct confidence intervalsfor S, (7) based on the range that
includes the specified percent of the values for Si(Oi,Ei)
Simulationmethods can also provide insights into the presence of bias in the point estimates. Specifically, we can
use informationfrom the simulations to report median biascorrected point estimates of the sacrifice ratio using the following formula:
a money-supply shock, a money-demandshock, and an IS
shock, respectively.The identificationof the structuralshocks
are again based on short-runand long-run restrictions. We
retain the Blanchard-Quahrestriction and assume that the
(15)
s, (7) = Sr (7) -[S;5 (r7)- SE (7)],
three aggregate demand shocks have no permanent effect
on the level of output. Further,we follow Gali and adopt where S•, 7) is the bias-adjustedpoint estimate of the sacritwo additional assumptions. The first is that neither money fice ratio, S,,(7) is the sacrifice-ratioestimate from the strucdemand nor money supply affect output contemporaneously tural VAR model, and S?;5(7) is the median sacrifice-ratio
Cecchettiand Rich: StructuralEstimatesof the U.S. SacrificeRatio
estimate from the simulations.The term in bracketsmeasures
the estimated bias. A similar procedurecan be used to construct bias-adjustedestimates of the impulse response functions and confidence intervals.
As a final note, one could arguethatour resultsmay actually
understatethe imprecision associated with the sacrifice-ratio
estimates. The analysis abstractsfrom real-time data considerations and its implicationsfor the conduct of monetarypolicy. For example, variables such as output (or unemployment
rate) gaps are typically constructedand used by the monetary
authorityfor forecasting purposes or in Taylor-typerules to
gauge policy actions. Because these variables are unobserved
and very imprecisely measured (Orphanides2000a,b), their
inclusion in the models we study would only lead to greater
unreliabilityin the sacrifice-ratioestimates.
The analysis now turns to a presentationof the results and
a discussion of the point estimates of the sacrifice ratio and
their associated confidence intervals.
RESULTS
3. EMPIRICAL
We construct sacrifice-ratioestimates from the three structuralVAR models using quarterlydata over the sample period
1959:Q1-1997:Q4. Outputis measuredby real GDP and inflation is measuredby the CPI. The sacrifice-ratioestimates are
based on a differenttransformationof the data for output and
prices. Outputgrowth is measured at a quarterlyrate in percentage terms, while inflationis measuredat an annualrate in
percentage terms. The short-terminterest rate representsthe
yield on three-monthTreasurybills and the monetary aggregate is measuredby Ml. The appendix describes the data in
furtherdetail.
It is worth noting that preliminary analysis of the data
supports the specification of the models. In particular,we
examined the stationarityproperties of the various series to
determinetheir degree of integrationand the presence of cointegrating relationships. The results from the application of
Dickey-Fuller (1981) tests providedevidence that y, ir, i, and
Am contain a unit root, but that (i - 7) and (Am - Ir) are stationary variables. The findings that both output and inflation
contain a unit root are particularlyimportantfor the identification schemes and the concept of a sacrifice ratio. The evidence of a unit root in the outputprocess allows the long-run
restriction on the effects of aggregate demand shocks to be
well defined and meaningful,while the evidence of a unit root
in the inflationprocess allows for permanentshifts in its level.
The lag length of the reduced-formVAR was set equal to
8 for the Cecchetti model and equal to 4 for the ShapiroWatsonand Gali models. Estimationof the sacrificeratio also
requiresthe selection of a horizon for the long-runrestriction
on aggregatedemand shocks. Following Cecchetti (1994), we
assume that aggregatedemandshocks completely die out after
20 years and truncatethe structuralVMA representationsat
80 quarters.
Table 1 presents the point estimates of the sacrifice ratio
at horizons of one to five years, and Figure 3 displays the
(median) bias-adjusted estimated responses of output and
inflation to a one-unit monetarypolicy shock across the three
421
Table1. Sacrifice-RatioEstimatesforthe UnitedStates, 1959:Q11997:Q4;CumulativeOutputLoss as a Percentage of Real GDP
Model
7= 4
1.3219
Cecchetti
Shapiro-Watson 0.1854
0.8016
Gall
r=8
r = 12
r = 16
r = 20
1.3204
0.6211
3.7092
1.5700
1.0014
6.1791
1.5219
1.1548
8.1434
1.3763
1.2768
9.8709
models, together with 90% confidence bands. The point estimates should be interpretedas the cumulativeoutputloss correspondingto a permanentone-percentage-pointdecline in the
rate of inflation measuredon an annualbasis.
The Cecchetti model yields sacrifice-ratioestimates that are
relatively constant as the horizon grows, while the other two
models show a clear upward trend, with higher output loss
at longer horizons. Interestingly,the 20-quarterhorizon estimates from the Cecchetti and Shapiro-Watsonmodels, 1.38
and 1.28, respectively,are very similarto the values of 1.8 and
1.4 calculated by Ball (1994) and Schelde-Andersen(1992)
for the Volckerdisinflation.In contrast,the Gali model generates a markedlyhigher estimate of 9.87, which is much closer
to the mean value obtainedby Okun (1978).
An examinationof the bias-adjustedimpulse responsefunctions in Figure 3 reveals a pattern that is qualitatively similar across models and accords with the predicted effects
of a monetary tightening. Output declines in response to a
monetary-policyshock before eventually returningto its initial level. (The nature of the identificationrestrictionsin the
Shapiro-Watsonand Gali models precludes a contemporaneous responseof outputto the monetary-policyshock.) Inflation
also decreases and displays a permanentdecline in response
to the monetary-policyshock.
As shown, there are some differencesin the extent and size
of the impact of the monetary-policyshock across models. For
example, most of the response of output and inflation in the
Cecchetti model occurs within the first few quartersafter the
shock. Relative to the Cecchetti and Shapiro-Watsonmodels,
the Gali model suggests a deeper and more protractedoutput
decline along with a smaller decrease in inflation. Both of
these effects lead to the higher point estimates of the sacrifice
ratio that emerge from this model.
Figure 3 also indicates that the impulse response functions
are not estimatedvery precisely. Because of the natureof the
long-run identifying restrictions,one would expect the confidence intervals for output to narrow and converge around 0
as the horizon steadily increases. However, the results indicate that the confidence intervalsat shorterhorizons typically
cover 0 and can be fairly wide. In the case of inflation, the
confidence intervalsfor the Shapiro-Watsonand Gali models
seem particularlywide and either cover or lie very close to
0. An initial reading of this evidence reveals the imprecision
associated with the quantities of interest and clearly hints at
the potential difficulty of obtaining reliable estimates of the
sacrifice ratio.
Table 2 presents the results for the simulated distributions
of the sacrifice ratio based on the proceduresoutlined in the
previous section, with 10,000 replications.We report(median)
422
Journal of Business & Economic Statistics, October 2001
A: DynamicResponse to a MonetaryPolicy Shock
B: DynamicResponse to a MonetaryPolicy Shock
Real GDP- Cecchetti
0.75
0.4
0.50
0.2
"" "* -0.25
...
.
0.0
0
- Cecchetti
Inflation
0.6
-
.......
..
0.00
-0.2-
-0.25
-0.4
-0.50.
.10
-1.25-
-1.2
-1.50
-1.4
-1.75
0
5
10
15
20
0
C: DynamicResponse to a MonetaryPolicy Shock
5
15
20
D: DynamicResponse to a MonetaryPolicy Shock
Real GDP- Shapiro-Watson
- Shapiro-Watson
Inflation
0.6
0.4
10
0.75
"
0.2
.
...
-
0.50
0.25
...
-0.2 -e
-0.25
-1.0
-1.25
-1.2
-1.50
-1.4
-
-1.75
0
5
10
15
20
0
E: DynamicResponse to a MonetaryPolicy Shock
5
Real GDP- Gali
0.75
0-50-1.50
..
a
...1.25
-1.4
0
0
5
10
5
101
Figur 3.
-0.75
?..
aMonetary
Bias-Adjusted
-1.75F: Dynamic
PolicyIpulse
ShockResponseFunctions
wons RepfidenceIntervals.Policy
E:
Dynamic
Respne3.to
Meia
20
- Gali
Inflation
0.4
-1.0
15
F: DynamicResponse to a MonetaryPolicy Shock
0.6
-1.2
0.2 -.0.25
-0.6
10
15
20
0
5
0051
Bias-Ajste
Imus
10
15
Shock
20
52
Repos
bias-adjustedestimates of the sacrifice ratio and 90% confidence intervalsfor each horizon. The empirical density functions for SE (7 = 20) are plotted in Figure 4.
There are several strikingresults that emerge from Table 2
and Figure 4. The first is that a 90% confidence interval
for the sacrifice ratio includes 0 for all three models. That
Fucioswth9%Cofdncneras
is, we cannot with any reasonable degree of certainty rule
out the possibility that SE(7)r)= at each horizon. Second, the
Shapiro-Watsonand Gali models display a marked increase
in the imprecision of the sacrifice-ratioestimates as the horizon lengthens. Last, the results are sensitive to the measure of monetary-policyshocks. In particular,expanding the
Cecchettiand Rich: StructuralEstimatesof the U.S. SacrificeRatio
423
Table2. MedianBias-AdjustedEstimatesand ConfidenceIntervalsfor
SacrificeRatio;SimulationExperimentBased on 10,000 Replications
S,7(r)
Medianbias-adjusted
90%confidence interval
Medianbiasadjustedestimate
Horizon
(quarters)
Cecchettimodel
r= 4
7==8
1.3072
1.3203
1.5806
1.5482
1.3591
T=12
S=16
7 = 20
(0.0110,2.4686)
(-0.3782, 2.6474)
(-0.5048,3.8141)
(-0.3341,3.9162)
(-0.4200, 3.2170)
Shapiro-Watsonmodel
7= 4
7= 8
7= 12
0.4131
2.0049
3.4605
3.9607
4.3495
T=16
r = 20
(-2.4175, 4.5281)
(-11.5053, 18.6673)
(-22.7007, 30.0720)
(-27.6783,36.4351)
(-30.2247,43.0066)
Galimodel
7=4
1.0911
4.9281
8.2839
11.2572
14.0334
'"=8
S=12
7= 16
7 = 20
(-2.4804, 5.2475)
(-13.4865,25.2386)
(-34.1267, 48.7030)
(-41.9125, 55.3507)
(-43.4513, 67.9279)
two-variable system to identify separate components of the
aggregate demand disturbanceyields ranges for the sacrifice
ratio that are highly variableand implausible.Takentogether,
these findings speak directly to the unreliabilityof the point
estimates and suggest that we have little understandingabout
CecchettiModel
0.35
-
0.3
0.25
0.2
0.15
-
0.1
-
0.05
0
-60
-40
-20
0
20
40
60
Model
Shapiro-Watson
0.1
0.09-
0.08
0.07
0.060.05-
the quantitative impact of monetary policy on output and
inflation.
Looking at Figure 4, we see that the simulated distribution of the sacrifice ratio for the Cecchetti model is nonnormal, although it is reasonably symmetric. In addition, the
bias-correctedbootstrappedestimates of the sacrificeratio are
nearly identical with the original estimates, providing little
evidence of bias. For the Shapiro-Watson and Gali models, however, the simulated distributionsof the sacrifice ratio
reveal a markedlydifferentpicture.The distributionsare nonnormal, asymmetric,and extremely long tailed. Further,there
is now increasedevidence of bias in the point estimates from
both models.
To explore the behavior of the sacrifice-ratioestimates in
furtherdetail, we examined the simulated distributionof the
estimatorsof the impact responses of outputand inflationto a
monetary-policyshock. Because the featuresof the estimators
for each model were broadly similar across horizons, we only
display and discuss the resultsfor a horizonof r = 20 quarters.
Figure 5 plots the correspondingempirical density functions.
The graphs relate to the relevant quantities in the numerator
and denominatorof the sacrifice ratio and therefore provide
estimates of the cumulative response of the level of output
and the response of the level of inflationto a monetary-policy
shock.
The impact response estimatorshave very nonormaldistributionsthatbecome distinctlybimodal as the numberof structural shocks increases. In addition,the simulateddistributions
of (Iaii'12)do not restrictthe outputresponseto be negative and display greater variability as the complexity of the
models increases. In the case of the response of inflationto a
monetary-policyshock, the estimatedsampling distributionof
Ca2 lies exclusively below 0 for the Cecchetti model, while
the corresponding sampling distributions for the ShapiroWatsonand Gali models display a pronouncedrightwardshift
and are centered near 0. This latter result would seem to
account for the extremely wide confidence bands associated
with the sacrifice-ratioestimatesfrom the Shapiro-Watsonand
Gali models. Because the estimatedinflationresponse appears
in the denominatorof the expressionfor the sacrificeratio, values of this magnitudeclose to 0 will lead, for a given output
response, to larger (absolute) estimates of the sacrifice ratio.
The Shapiro-Watsonand Gali models also seem capable of
generatingpositive estimates of the sacrifice ratio in the presence of perverse output and inflationresponses.
0.04
0.03
0.02
4.
0.01O
-60
-40
-20
-60
-40
-20
0
20
40
60
0
20
40
60
EXPLORINGTHE SOURCES OF THE IMPRECISION
There are a numberof possible sources for the imprecision
find in the estimates of the sacrifice ratio. In this section
we
GallModel
we examine three. First, we look at the possibility that the
0.07
0.06
estimatedVAR's contain structuralbreaks.If they do, then the
wide confidence bands we obtain could reflect specification
error arising from the estimation of a time-invariantmodel.
We examine this possibility with a set of structuralbreak
tests based on the work of Andrews (1993) and Andrews and
Ploberger(1994). Much to our surprise,we are able to conclude that all three of the models we analyze are stable over
Estimates.
Sacrifice-Ratio
for
Functions
4.
EmpiricalDensity
the sample period we study.
Figure
0.05
0.04
0.03
0.02
0.01
424
Journal of Business & Economic Statistics, October 2001
Output:CecchettiModel
CecchettiModel
Inflation:
0.6
a
0.6 2.5
0.5
- 0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
- 0.1
2.5
22
-2
1.5
0
0
-20
-16
-12
-8
-4
0
4
8
12
16
0
0.3
- 0.25
0.25
0
-5
20
2
0.5
0.5
Model
Output:Shapiro-Watson
0.3
-1.5
-4
-3
-2
-1
1
0
2
3
4
5
Model
Inflation:
Shapiro-Watson
0.8
0.8
0.7.0.7
0.6 -.0.6
0.2
0.2
0.15
0.15
- 0.1
0.1
- 0.05
0.05
0
0
-20
-16
-12
-8
-4
0
4
8
12
16
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.
0.1
0.
0.1
0
20
0
-5
-4
Output:GaliModel
0.14
0.12
- 0.12
0.1
- 0.1
0.08
-0.08
0.06
0.06
0.04
0.04
0.02
-0.02
0
-16
-12
-8
-4
0
4
8
12
-2
-1
0
1
2
3
4
5
GaliModel
Inflation:
0.14
0
-20
-3
16
20
1.2
]221.2
1
0.8
- 0.8
0.6
- 0.6
- 0.2
0.2
0
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
Figure5. EmpiricalDensityFunctionsfor Responses of Outputand Inflation.
Second, we ask whetherthe results reflect the fact that the
higher-dimensionalmodels provide worse forecasts of output
and inflation. If they do, then it may not be the addition of
more structuralshocks that leads to less precise estimates of
the sacrificeratio. Instead,it may be that the Shapiro-Watson
and Gali models are just poorer characterizationsof the data.
To address this issue, we compare the out-of-sample forecasting performanceof each of these models for output and
Cecchettiand Rich: StructuralEstimatesof the U.S. SacrificeRatio
inflation. The findings indicate that the more complex models
provide better,not worse, forecasts, so this is not the explanation for our results.
Finally, we look at the issue of instrumentquality. Estimation of structural VAR models can be viewed in an
instrumental-variablessetting. The use of weak instruments
could lead to imprecision in the calculation of the elements
that go into the computationof the sacrifice ratio. We provide
some evidence here that this is a likely source of at least part
of the imprecisionin the sacrifice-ratioestimates.
Before we proceed to examine each of these three possible sources of imprecision in detail, we would like to comment on one avenue we do not pursue. Specifically, we do
not attempt to provide a detailed investigation into the relative contributionsof A0, the matrix of the contemporaneous
effects of the structuraldisturbanceson the endogenous variables, and C(L), the reduced-formVMA representation,to
the variability of the sacrifice-ratioestimates. The most natural way to attempt to parse the relative impact of these on
the imprecision of the sacrifice ratio would be to hold one of
these two quantitiesfixed and then use simulateddata to estimate the variabilityinduced by the other. Unfortunately,this
approachis not feasible because the value of AOdepends on
C(L) throughthe identifying restrictions,so it cannot be held
constant while new data are generatedto reestimateC(L).
425
or implicit monetary-policyreaction function and, if the policy regime were to change, then one would expect the coefficients in these equations to change as well. In other words,
we have every reason to believe that the models we estimate
may contain structuralbreaks.
To investigate this issue, we use tests proposed by
Andrews (1993) and Andrews and Ploberger(1994) to study
the possibility of a break in the specification of the models. Table 3 reports the results of the three commonly used
tests for structuralchange, all of which are Lagrange multiplier (LM) tests and allow for an endogenous determination
of the break date. We report the value of the LM test statistics, togetherwith their p values, for each equationin each of
the three models. The results are quite striking. It is only the
Andrews-Plobergeraverage LM test that provides evidence
of a structuralbreak for one of the equations in the StockWatson model and two of the equations in the Gali model.
Even for these cases, however,the other two tests fail to reject
structuralstability. From this we conclude that instability of
the VAR's is unlikely to be the source of our results.
4.2 ForecastingAccuracy
If the two-variablesystem is just a better model than the
three- or four-variablesystems, then the explanation for our
results would be straightforward.The obvious dimension in
4.1 Structural
Breaks
which to comparemodels of this type is to examine their foreThe results in Section 3 are estimatedover a nearly 40-year casting ability. We do this by calculatingthe accuracyof outperiod beginning in 1959. Many things may have changed in of-sample forecasts for outputand inflationusing the relevant
the United States over this time, not the least of which is Fed- reduced-formequations (or their equivalents)for each of the
eral Reserve policy. Each of our models includes an explicit three models.
Table3. StructuralBreakTests:1959:Q1-1997:Q4
Modell
equation
Andrews/Quandt
supremumLMtest
AndrewslPloberger
exponentialLMtest
AndrewslPloberger
average LMtest
Cecchettimodel
Outputequation
Inflationequation
Outputequation
Inflationequation
Ex post real interestrate equation
23.180
8.981
(0.9199)
(0.8295)
27.999
10.626
(0.6211)
(0.5555)
Stock-Watsonmodel
16.612
(0.5084)
18.851
(0.2631)
15.455
(0.9936)
22.166
(0.6616)
24.579
(0.4600)
6.276
(0.8999)
8.841
(0.4274)
10.233
(0.2225)
11.876
(0.6187)
15.521
(0.1790)
18.722
(0.0372)
12.206
(0.3081)
12.135
(0.3176)
12.123
(0.3192)
13.860
(0.1404)
20.649
(0.1337)
19.530
(0.2067)
22.524
(0.0586)
23.910
(0.0298)
Galimodel
Outputequation
Nominalinterestrate equation
Ex post real interestrate equation
Real money balances
equation
30.524
(0.4320)
29.371
(0.5166)
27.114
(0.6877)
34.415
(0.2041)
NOTE: Structural
stabilitytests correspondto the jointtest forconstancyof all regressionparametersin a particular
equation.Values
of the test statisticare reportedin columns2-4, withasymptoticp values reportedbelow in parentheses.
Journalof Business & EconomicStatistics,October2001
426
Table4. RollingOne-Quarter-Ahead
Out-of-SampleForecasts; 1959:
Q1-1975:Q1 (1997:Q4)Root Mean SquaredError
Model
Cecchetti
Shapiro-Watson
Gali
Outputgrowth
(Levelof) Inflation
0.7348
0.6956
0.7072
1.3943
1.3666
1.2313
Table 4 reportsthe results of an exercise in which we estimate each model, generateone-quarter-ahead
forecasts,add an
additionalquarter'sworthof data,and repeatthe exercise. This
expanding-sampleestimation begins with data from 1959:Q1
through 1975:Q1 and rolls forward through 1997:Q4. We
reportthe root mean squarederrorof these one-step-aheadoutof-sample forecasts. The conclusion is clear. As we increase
the numberof variablesin the system, the models improve at
forecasting output growth. The accuracyof the inflationforecasts is roughly equivalent,with the two-variablesystem performing worst. The last result is not a surprisebecause adding
information rarely improves inflation forecasts. But overall,
we do not find evidence indicatingthat the forecastingperformance deterioratesas the complexity of the model increases.
4.3 Instrument
Quality
Pagan and Robertson(1998) recently showed that structural
VAR models can be cast in a generalizedmethod of moments
frameworkand that the restrictionsused to identify the structural shocks generate instrumentsfor the estimation of A0. In
addition, they found that structuralVAR estimates of the liquidity effect are extremely imprecise due in large part to the
poor quality of instrumentsused in estimation.The results of
Pagan and Robertson suggest that instrumentquality may be
a relevantconsiderationfor our results.
Following the approachof Pagan and Robertson,we adopt
the testing procedureoutlined by Shea (1997) to evaluate the
quality of instruments used in the estimation of the three
structuralVAR models. An advantageof this methodology is
that it can be applied to equations with multiple endogenous
explanatoryvariables.The testingprocedureyields a partialR2
measure indicatingthe extent to which the instrumentset has
componentsimportantto an endogenousvariablethatare independent of those importantto the other endogenous variables.
Table 5 reports the partial R2 measure for the instruments
used in estimating the structuralequations. Although we are
able to examine the quality of instrumentsfor each equation,
the results for the output equation and inflation equation in
the Cecchetti and Shapiro-Watsonmodels are of particular
interest. In the case of the Gali model, we will focus our
attentionon the outputequationas well as on the nominaland
real interest-rateequations.
Across all three models, the long-run restrictions yield
instrumentsof relatively low quality for estimating the coefficients of the output (aggregate supply) equation. This is
especially true in the case of the Gali model, in which the
instruments seem to be particularlypoor. For the inflation
equation, or the nominal and real interest-rateequations in
the Gali model, there is evidence of some improvementin
the quality of the instruments.However, some caution may
be needed in interpretingthe high observedvalues of the partial R2 measure.As Pagan and Robertsonnoted, the short-run
restrictions used for identification in these latter equations
generate instrumentsthat are (structuraland reduced-form)
residualsfrom previouslyestimatedequations.Although these
variablesare assumedto be orthogonalto the structuralerrors,
the high partial R2 measure indicates that they may not be
valid instruments and suggests the presence of additional
bias in the instrumental-variableestimates. Taken together,
the evidence of weak or invalid instrumentsspeaks directly
to the tenuous natureof the identifying assumptionsand the
limited ability of current empirical methodologies to allow
researchersto draw reliable inferences about structuraleconomic relationships.
Table5. PartialR2Measurefor Instrumentsin the StructuralVARModels
Cecchettimodel
Endogenous variables
Outputequation
Inflationequation
Ayt
0.858085
Art
0.121203
Shapiro-Watsonmodel
Endogenous variables
Outputequation
Inflationequation
Real interestrate equation
Ayt
-
At
0.932317
0.239066
-
0.772303
0.477639
(it - t)
0.071403
0.096139
Galimodel
Endogenous variables
Outputequation
Nominalinterestrate equation
Real interestrate equation
Moneygrowth
equation
AyAi
-
0.091705
(i (it)
0.048001
(Amt - ITt)
0.130205
0.058115
0.832205
0.627003
0.215293
0.266718
0.209473
0.206824
0.184180
Cecchettiand Rich: StructuralEstimatesof the U.S. SacrificeRatio
5. CONCLUSION
This study examines the output cost of disinflation policy
for the United States over the postwar period. Across various
models, the estimates of the sacrificeratio imply that a permanent one-percentage-pointreductionin inflation entails a loss
of approximately1-10% of a year's real GDP. The confidence
intervals around the point estimates, however, indicate that
none of the point estimatesdiffer from 0 at conventionallevels
of statisticalsignificance.Further,the high degree of imprecision associatedwith the estimates suggests that our knowledge
about the actual impact of monetarypolicy on the behaviorof
the economy is quite limited.
The evidence from this study supports the view that the
identifying restrictions used in estimation are tenuous and
generate weak or invalid instruments.Alternative identification schemes are unlikely to change this outcome. Unlike standard situationsthat allow for the applicationof instrumentalvariables procedures, the instrument set available for the
estimation of structuralVAR models is very restricted.This
lack of valid instrumentsimposes severe restrictions on the
nature of the structuralshocks that can be identified. Thus,
although a better understandingof the true costs of disinflation would be of particularinterest and importanceto policy
makers, we are skeptical that current data and econometric
techniquescan provide a meaningful set of estimates.
ACKNOWLEDGMENTS
We have benefited from the suggestions of attendees
of the System Committee on Macroeconomics Conference
in Chicago and seminar participants at the University of
Arkansas,the Universityof Connecticut,and the Universityof
Californiaat Davis. We thank Paul Bennett, Jean Boivin, Tim
Cogley, Jeff Fuhrer,Jordi Gali, EdwardGreen, Jim Hamilton,
Jim Kahn, Ken Kuttner, Rick Mishkin, Adrian Pagan,
Simon Potter,the associate editor, and an anonymousreferee
for helpful comments on an earlier version of this article. We
also thankJordiGali and Ken Kuttnerfor their assistancewith
programs used in performing the computations. The views
expressed in this article are ours and do not necessarilyreflect
the position of the Federal Reserve Bank of New York or the
Federal Reserve System. We are solely responsible for any
remainingerrors.
APPENDIX:DATA
This appendix discusses the data used in the estimation.
All data are quarterly.The estimates are conducted over the
sample period 1959:Ql-1997:Q4.
Output. The outputseries GDPH is measuredas real GDP
in chain-weighted 1992 dollars.
Prices. The price data are a quarterly average of the
monthly consumerprice series PCU for all urbanconsumers.
InterestRates. The interest-ratedata are a quarterlyaverage of the monthly series FTBS3, which representsthe yield
on three-monthTreasurybills.
427
Money Stock. The data on the money stock are for Ml and
are a quarterlyaverage of the series FM1. The measures of
M1 priorto 1959 are taken from the Federal Reserve Bulletin.
[Received March 1999. Revised February2001.]
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