American Economic Association - NYU Stern

American Economic Association
Were There Regime Switches in U.S. Monetary Policy?
Author(s): Christopher A. Sims and Tao Zha
Reviewed work(s):
Source: The American Economic Review, Vol. 96, No. 1 (Mar., 2006), pp. 54-81
Published by: American Economic Association
Stable URL: http://www.jstor.org/stable/30034354 .
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WereThereRegimeSwitches in U.S. MonetaryPolicy?

By CHRISTOPHERA. SIMS AND TAO ZHA*

A multivariateregime-switchingmodelfor monetarypolicy is confrontedwith U.S.
data. The bestfit allows time variation in disturbancevariances only. Withcoeffi-
cients allowed to change, the bestfit is withchange only in the monetarypolicy rule
and thereare threeestimatedregimescorrespondingroughlyto periods whenmost
observersbelieve thatmonetarypolicy actually differed.But the differencesamong
regimesare not large enoughto accountfor the rise, thendecline, in inflationof the
1970s and 1980s. Our estimates implymonetarytargetingwas central in the early
1980s, but also importantsporadically in the 1970s. (JELE52, E47, C53)

It is widely thoughtthatU.S. monetarypolicy severaleconometrictests and do not find strong
changeda greatdeal, andfor the better,between evidenceagainststabilityof coefficients.An ear-
the 1970s and the 1980s. RichardClaridaet al. lier version of this paper (entitled "Macroeco-
nomic Switching")and subsequentstudies(Fabio
(2000) (CGG) andThomasA. Lubikand Frank CanovaandLucaGambetti2, 004;Chang-JinKim
Schorfheide (2004) find that the policy rule and CharlesR. Nelson, 2004; Timothy Cogley
apparentlyfollowed in the 1970s was one that, and Sargent,2005; Primiceri,2005a) show little
when embeddedin a stochasticgeneralequilib- evidence in favor of the view thatthe monetary
riummodel, would imply nonuniquenessof the policy rulehas changeddrastically.
equilibriumandhence vulnerabilityof the econ-
omy to "sunspot" fluctuations of arbitrarily This paperfollows the structuralVAR liter-
large size. Their estimated policy rule for the ature in making explicit identifying assump-
laterperiod,on the otherhand,implied no such tions to isolate estimates of monetary policy
indeterminacy.These resultsapparentlyprovide behaviorand its effects on the economy, while
an explanationof the volatile and rising infla- keeping the model free of the many additional
tion of the 1970s and of its subsequentdecline. restrictive assumptions needed to give every
parameterand equationa behavioralinterpreta-
Thereareotherinterpretationosf the evidence, tion or to allow structuralinterpretationof a
however.GiorgioPrimiceri(2005b) andThomas single-equation model. We use a model that
J. Sargentet al. (forthcoming)estimatemodels allows explicitly for changes in policy regime,
thatfindonly modestchangesin policyin thepast including as special cases both short-livedos-
four decades.Ben S. Bernankeand Ilian Mihov cillatingpolicy changes andunidirectional,per-
(1998),EricM. LeeperandZha(2003),andJames sistent shifts toward improved policy. We
H. Stock and MarkW. Watson(2003) perform compare versions of the model with Bayesian
posteriorodds ratios, a method that automati-
* Sims: Departmentof Economics,PrincetonUniversity, cally penalizes models with unneeded free
PrincetonN, J08544-1021(e-mail:[email protected]);ha:
ResearchDepartmentF, ederalReserveBank of Atlanta,At- parameters.
lanta, GA 30309-4470 (e-mail:[email protected]). e Our most importantempiricalfinding is that
thankthe refereesfor helpfulcommentsand Dan Waggoner
for many valuablediscussions,as well as for his help on C the versionof ourmodel thatfits best is one that
programmingE. ric Wang providedexcellent researchassis-
tance in computationon the Linux operatingsystem. The shows no change at all in coefficients, eitherof
technicalsupportof parallelcomputingfrom the College of the policy rule or of the privatesector block of
ComputerScience at the GeorgiaInstituteof Technologyis the model. What changes across "regimes"is
greatlyacknowledged.This researchis supportedin partby only the variances of structuraldisturbances.
CEPSandby NSF GrantSES 0350686. The views expressed That is, this version of the model explains dif-
hereindo not necessarilyreflectthose of the FederalReserve ferences in the behavior of the economy be-
Bankof Atlantaor the FederalReserveSystem. tween periods as reflecting variation in the
sourcesof economic disturbances,not as varia-

54

VOL.96 NO. 1 SIMSAND ZHA: WERETHEREREGIMESWITCHESIN U.S. MONETARYPOLICY? 55

tion in the dynamics of the effects of a given show as strong a monetary-aggregate-targeting
disturbance on the economy. The Volcker flavor as the "Volcker regime," it does tend
reserves-targetingperiodemergesas a periodof much more strongly in that direction than the
high variancein disturbancesof the policy rule. "Greenspanregime."We call this fourthregime
This findinglends empiricalsupportto the com- the "Burnsregime,"even thoughthe Greenspan
mon practicein the literatureof combiningthe regime was in place throughapproximatelythe
samples before and after the reserve-targeting same proportionof the Burns chairmanshipas
period to estimate the model, as long as het- was the Burns regime. (For the rest of this
eroskedasticityis properlytaken into account. paper,we dropthe quoteson the regimenames,
hoping the reader can bear in mind that the
We also considermodels in whichparameters correspondenceof the regimes to chairmanship
do change.We have looked at models whereall terms is rough.)
parametersin all equationscan change, where
only nonmonetary-policycoefficients change, We displaycounterfactualsimulationsof his-
and where only monetary-policy coefficients tory with alternatemonetarypolicy regimes. If
can change. In these cases, we allow structural we simulate history with the estimated time
variancesto shift size at the same time coeffi- series of shocks, but with the coefficients of the

cients change, andwe have also triedmodels in policy rule set at the estimatedGreenspanpol-
which the times of coefficient changes are sto- icy throughoutthe period 1961-1987, the rise
chasticallyindependentof the times of variance and fall of inflation follows the historicalpath
changes. We have allowed the number of re- extremely closely. This is not because the
gimes to vary, including the case of a single model is incapable of showing an effect of
regime, and we have considered specifications monetary policy. If we, instead, use a policy
in which regime change is constrained to be rulethatuses the Greenspancoefficients,except
monotonic, so that old regimes are constrained that it doubles the coefficients on inflation,the
never to recur.None of these models fits nearly counterfactual historical simulation shows
as well as the best-fittingmodel in which only
residual varianceschange across regimes. Par- much lower inflationthroughoutthe 1970s and
ticularlyill-fitting are the models with a single early 1980s-at the cost of considerablylower
regime and the model that constrains regime output growth through that period. A similar
changes to be monotonic. lower inflationpathemergesif we fix the policy
rule at the point estimate for the Volcker
The best-fittingmodel among those that do
allow coefficients to change is one that con- reserve-targetingregime.
strainsthe changes to occur only in the mone- Although the estimateddifferences in policy
tary policy equation, while coefficients in the
other equations remain constant. Like Cogley behavior and their effects on the economy in
and Sargent (2005) and Primiceri(2005a), we this four-statemodel are substantivelyinterest-
find that the point estimates of the changes are ing and consistent with the results from the
not trivial, even though the data leave their recent literature(Primiceri, 2005a; Sargent et
magnitudesuncertain.The model finds the best al., forthcoming),they arenot as drasticas what
fit with four regimes. One occurs in only a few is impliedby the sunspot-equilibriummodel. In
briefspansof months,oneof whichis September- particular,for all three main regimes, our esti-
October2001, and has very high residualvari- mates imply that, with high probability,mone-
ance in money demand.Anothercorrespondsto tary policy responses to inflation were strong
the Volcker reserve-targetingperiod and shows enough to guaranteea determinateequilibrium
clearly the targeting of monetary aggregates, price level.
ratherthan interest rates, in that regime. An-
other regime has been in place throughnearly Therearea numberof likely explanationsfor
the contrastbetween our finding here and the
all of the years of the GreenspanFederal Re- findings in some other empirical papers. Per-
haps the most importantis thatratherthanaim-
serve chairmanship-but was also in place
ing at findingsome model we can interpretthat
through most of the 1960s. A fourth regime
is not rejected by the data, we aim at fully
occurred in several multiyear episodes in the
characterizingthe uncertaintyaboutourresults.
late 1960s and early 1970s. Though it does not When we runour counterfactualhistoricalsim-

ulationsby drawingfromthe posteriordistribu-

56 THEAMERICANECONOMICREVIEW MARCH2006

tion of the coefficientsof the policy rule instead moneyin ourframeworkimprovesthe relativefit
of fixing the coefficientsat particularvalues, we of modelsthatallow variationin the policy rule.
can see thatthe shapeof uncertaintyaboutthese
policy rules differs more than do their most We think our results have implications for
likely values. When we simulate history with futureresearchon theoreticalmodels with more
the Greenspan,Bums, and Volcker rule distri- detailed behavioralstructure:
butions,the medianpathsfor inflationand out-
put show visible differences, with the Volcker (a) The Taylor rule formalism, valuable as it
and Greenspanmedianpaths similarand lower may be as a way to characterizepolicy over
than the Burns median path. The Volcker and the last 20 years, can be seriously mislead-
Greenspandistributionsshow a risk of defla- ing if we try to use it to interpretother
tion, while the Burnsdistributiondoes not, and historical periods, where monetary aggre-
gate growth was an importantfactor in the
the Volcker andGreenspanpathsshow a riskof thinkingof policymakers.
periods of outputgrowth below -5 percent at
an annualrate, while the Burns path does not. (b) It is time to abandonthe idea that policy
The outputgrowthratealong the medianBurns change is best modelled as a once-and-for-
pathis slightly above the historicalgrowthrate, all, nonstochastic regime switch.' Policy
while it is notably below (1/2to 1 percent at changes, if they have occurred, have not
annualrate)the historicalratealong the Green- been monotonic, and they have been diffi-
span and Volcker medians.The Burnsdistribu- cult to detect.Both the rationalpublicin our
tion shows a risk of inflationnot coming down models and econometriciansmust treatthe
at all in the 1980s, while neitherthe Volckernor changes in policy probabilistically,with a
model of how and when the policy shifts
the Greenspanpath shows such a risk. In other occur and with recognition of the uncer-
words, even thoughthe data are best explained tainty abouttheir natureand timing.
by a model with no change at all in policy rule
coefficients,if one looks for changes,andone is I. The DebateoverMonetaryPolicyChange
willing to considerpolicy rulesthatareunlikely
but not impossible, one can tell a story consis- The literaturein this areais largeenoughthat
tent with the view thatthe Burnspolicy, had it we will not try to discuss papers in it one by
persisted(insteadof endingaround1977, as the one. Ratherwe lay out what seems to us a few
model estimates it did), would have failed to of the most importantreasons why our results
end inflation. differ from much of the previous empirical
work in the area:
There are also substantive differences be-
(a) As we pointed out above, our specification
tween our model and the rest of the literature includes a monetaryaggregatein the reac-
tion function. Most of the previous litera-
which maycontributeto ourfindingthatthereis ture does not. We think this is a possibly
little evidence of policy change. Of particular importantsource of bias in estimatesof the
note is the fact that,unlikemuchpreviouswork, reactionfunction.
which fits a "Taylorrule"to the whole period,
we include a monetaryaggregatein our policy (b) Much of the previous literature either
reaction function. The Federal Reserve is by makes no allowance for heteroskedasticity
law required to provide the target paths for or allows only implausiblyrestrictedforms
various monetary aggregates, and during the of heteroskedasticity.Particularlycommon
1970s the behavior of these aggregates was have been specificationsin which thereis a
central to discussions of monetarypolicy. We single change in residual variance in the
show that constrainingthe monetaryaggregate sample, and specifications that generate
not to appearin our monetarypolicy equation "robust standarderrors"by allowing for
greatlyworsens the model's fit to the historical
1 Recent work by Troy Davig and Leeper (2005) repre-
data,andwe arguethatit is likely thatexcluding sents an attemptin this direction.

the aggregatefromthe equationwas a sourceof

bias in earlierwork. However, while excluding

money mighthave led to a spuriousfindingof a

violation of the "Taylor principle," including

VOL.96 NO. 1 SIMSAND ZHA: WERETHEREREGIMESWITCHESIN U.S. MONETARYPOLICY? 57

heteroskedasticity that is a function of shocks. Single equation approaches obvi-
right-hand-sidevariables.It is clear to the ously do not. It seems to us that empirical
eye, and apparentin our estimationresults, work that has been based on multivariate
thatresidualvariancesin the reactionfunc- models and has includedchecks for plausi-
bility of responses to monetary policy
tion rose sharply at the end of 1979, then shocks has tended to find less evidence of
dropped back a few years later. A single changing monetarypolicy.
shift in variance cannot capture this fact. (d) It is interestingto considerchangesin mon-
And the persistent shifts in variances that etary policy and to connect estimated
we find could not be well modeled as func- changes to historicalevents. Indeed,we do
some of thatin this paper,with a model we
tions of right-hand-sidevariables. As we do not think is our best. As a result, ab-
have already noted, failure to allow prop- stracts,introductions,and conclusions often
erly for heteroskedasticitycan stronglybias seem to support the idea that there have
statistical tests in favor of finding signifi- been changes in monetary policy even
cant shifts in coefficients. This is apparent when a look at plotted confidence or prob-
from the contrast between the results of ability bands aroundtime paths of coeffi-
cients or functions of them can be seen to
Cogley and Sargent (2002) and the later include constant paths. So in some cases
version of Cogley and Sargent (2005), thereis morecontrastbetween the abstracts
which does allow for a fairly generalform of papersin the literatureand our abstract
of heteroskedasticity. than there is in the detailed results.
(c) Identification in these models is fragile.
This is particularlytrue for the forward- II. Classof Models
looking Taylor rule specification of CGG,
for two reasons. The generalframeworkis describedby non-
linear stochastic dynamic simultaneous equa-
One is thatestimatingthis single equation tions of the form:
is basedon claimingthata list of instrumental
variablesis availablethatcanbe usedto con- (1) yA0o(st)= xA+(s,) + t = 1, ... , T

trol for the endogeneityof expected future (2) Pr(st= ilst=k)- = Pik ii,,k 1...
inflationandoutput.Buttheseinstrumentasre
availableonly because of a claim that we where s is an unobserved state, y is an n X 1
know a priorithatthey do not enterdirectly vector of endogenous variables, x is an m X I
into the reactionfunction-they can affect vector of exogenous and lagged endogenous
monetarypolicy only throughtheireffectson variables, Ao is an n x n matrix of parame-
expected futurevariables.We find it inher- ters, A, is an m x n matrix of parameters,T
ently implausiblethat,for example,the mon- is a sample size, and h is the total numberof
etary authorityreactsto an expected future states.
3-percentinflationrate in exactly the same
way, whethertherecentpastlevel of inflation Denote the longest lag lengthin the system of
has been 1.5 percentor 6 percent. equation (1) by v. The vector of right-hand
variables,xt, is orderedfrom the n endogenous
The otherproblemwith this specification variablesfor the firstlag downto the n variables
is that the Fisherrelationis always lurking for the last (vth)lag with the last element of x,
in the background.The Fisherrelationcon- being the constantterm.
nects currentnominalrates to expected fu-
tureinflationratesandto real interestrates, For t = 1,...., T, denote
which are in turn plausibly determinedby
expected outputgrowthrates.So one might Yt = {yl, ... ,Yt}.
easily find an equationthathad the form of
the forward-lookingTaylor rule, satisfied

the identifying restrictions,but was some-

thing otherthan a policy reactionfunction.
Multivariatemodels allow a check on the

identifying assumptionsvia examinationof

the impulse responses to monetarypolicy

58 THEAMERICANECONOMICREVIEW MARCH2006

We treat as given the initial lagged values of simplify the model by restrictingthe degree of
time variationin the model's parameters.2
endogenous variables Yo a=ssu(m7eYd1tvo.-..h. aveYtoh}e.
Structuraldisturbancesare We rewriteA , as

distribution:

(7) A+(s,) = D(s,) + S Ao(s,)

3"(EYtt-_1) = .N( 0, I,) mXn mXn mXn nXn

where

where oM(a,b) refers to the normal pdf with S= mn
mean a and covariance matrix b, and I, is an
(m - n)x n
n X n identity matrix. Following James D.
Hamilton (1989) and SiddharthaChib (1996), If we place a priordistributionon D(s,) thathas
mean zero, our prior is centeredon the same
we impose no restrictionson the transitionma- reduced-formrandomwalkmodelthatis theprior
trix P = [Pik]. mean in existing Bayesian VAR models (Sims
andZha, 1998).As canbe seen from(4)-(7), this
The reduced-formsystem of equations im- formof priorimpliesthatsmallerA-' values,and
plied by (1) is: thus smallerreduced-formresidualvariances,are
associatedwith tighterconcentrationof the prior
(3) = xtB(s,) + u"(st) t = 1, .... T abouttherandomwalkformof thereducedform.
y;
On the other hand, small values of D are also
where associatedwith tighterconcentrationof the prior
abouttherandomwalkreducedform,withoutany
(4) B(st) = A+(s,)Ao '(s,), correspondingeffect on reduced-formresidual
variances.
(5) u,(s,)= A'-'(s)Et,
Note thatthis setupcentersthe prioron mod-
(6) E[ut(st)u,(s,)'] = (AO(s,)A'(s,))-'. els in which the moving averagerepresentation3
has the form
In the reduced-form(4)-(6), B(s,) and u,(s) in-
volve the structurapl arametersandshocksacross o0S -s
equations,makingit impossibleto distinguishre- E-
gime shifts from one structurael quationto an-
other.In contrast,the structuraflorm (1) allows s=O
one to identifyeach structurael quation,such as
the policy rule,for regimeswitches. This ties our beliefs about lagged effects of
structuralinnovationi on variablej to our be-
If we let all parametersvaryacrossstates,it is liefs aboutcontemporaneouseffects of innova-
relatively straightforwardto apply the existing tion i on variablej. Any priorthat centerson a
methods of Chib (1996) and Sims and Zha
(1998) to the model estimationbecause Ao(s,) 2 In all the models studied here, we incorporatethe
andA+(s,) in each given statecan be estimated
independentlyof the parametersin otherstates. RobertB. Litterman(1986) lag-decay priorthateffectively
But with such an unrestrictedform for the time dampensthe unreasonableinfluenceof long lags. Thus the
overparameterizationproblems associated with typical
variation,if the system of equationsis large or VARs do not apply here. In addition,the marginallikeli-
the lag length is long, the numberof free pa- hood or the Schwarz criterion used in this paper as a
rameters in the model becomes impractically
large.For a typical monthlymodel with 13 lags measure of fit, by design, would penalize an excessive
and six endogenousvariables,for example, the
numberof parametersin A+(s,) is of order468 numberof parametersthat overfit the data.
for each state. Given the post-war macroeco- 3 Of course the expressionwe give here for the MAR is
nomic data, however, it is not uncommon to
have some states lasting for only a few years, valid only if the innovationsare not stationaryinfinitelyfar
and thus the numberof associatedobservations back into the past, but instead are, e.g., zero before some
startupdate.Orthe expressioncan be thoughof as the limit
is far less than 468. It is thereforeessential to as p -- 1 of stationaryMARs with coefficients of the form
((1 - pL)Ao)-1

VOL.96 NO. 1 SIMSAND ZHA: WERETHEREREGIMESWITCHESIN U.S. MONETARYPOLICY? 59

randomwalk reduced form, while leaving be- We have considered models with Case II
liefs about reduced form residual covariances specificationsfor all equations,with Case II for
the policy equationand Case III for all others,
independentof beliefs about reduced form co- with Case IIIfor the policy equationandCase II
efficients, will have the same effect. For exam- for all others, and with Case III for all equa-
ple, the standard "Minnesota prior" on the tions. That is, we have examined models with
reduced form, combined with any identifica- time variationin coefficients in all equations,
tion scheme based on restrictions on contem- with time variationin coefficients in policy or
privatesector equationsonly, and with no time
poraneous coefficients, will center on MARs variationin coefficients.In all of these cases, we
of this form. If one thinks of the model as a allow time variation in structuraldisturbance
variancesof all equations.The model with time
discrete approximationto an underlying con- variationin coefficients in all equations might
tinuous-time system, this type of prior is rea- be expected to fit best if there were policy
sonable. It is implausible that the effects of regime changes, and the nonlinear effects of
structuralinnovationsshow sharpdiscontinuities these changes on private sector dynamics, via
acrosslags. changes in private sector forecastingbehavior,
were important.That this is possible was the
We consider the following three cases of main point of RobertE. Lucas (1972).
restrictedtime variationfor Ao(s,) and D(st):
However, as Sims (1987) has explained at
(8) aoj(s,), dij,f(St),Cj(S,) more length, once we recognize that changes
in policy must in principle themselves be
= aoj d,9, Case I modeled as stochastic, Lucas's argumentcan
dij be seen as a claim that a certain sort of
j(st) Case II nonlinearity is important. Even if the public
aoojj(js(st)t,)d,diijj,eeAj(sijt()s,,c),j(s,) Case III believes that policy is time-varying and tries
to adjust its expectation-formation accord-
where j(s,) is a scale factorfor thejthstructural ingly, its behavior could be well approxi-
mated as linear and non-time-varying. As
equation, a0,(s,) is the jth column of Ao(s,), with any use of a linear approximation, it is
di(s,) is the jth column of D(s,), di,e(s,) is the an empirical matter whether the linear ap-
elementof di(s,)for the ith variableatthe fth lag, proximation is adequate for a particularsam-
and the last element of dj(s,), cj(s,), is the con- ple or counterfactual analysis.4
stant term for equationj. The parameterAii(s,)
changes with variablesbut does not varyacross We considerthe model with Case III for all
lags. This allows long-run responses to vary equationsbecause we are interestedin whether
over time, while constrainingthe dynamicform it fits betterthanthe othermodels, as would be
of the responsesto varyonly throughAii,which true if policy had changed within the sample
can be thought of as indexing the degree of and Lucas-critiquenonlinearitieswere impor-
inertia in the variable interpretedas the "left- tant. We consider the other combinations be-
hand side." Of course, in this simultaneous cause it is possible thatcoefficientsin the policy
equationssetup,theremay notbe a variablethat have not changed enough for the changes to
is uniquely appropriateas "left-handside" in emerge clearly from the data, or enough to
equation i. The specification insures, though, generate detectable correspondingchanges in
that whichever variable we think of as on the privatesector behavior.

left-handside, the time variationin dynamicsis 4 Anotherearlypaperemphasizingthe need for stochas-
one-dimensional, in that it affects all "right- tic modeling of policy change is Thomas F. Cooley et al.
hand-side"variablesin the same way. The bar (1984). More recently Leeper and Zha (2003) have drawn
symbol over ao,, dije, and cj means that these out the implicationsof this way of thinkingfor the practice
parametersare state-independent(i.e., constant of monetarypolicy.
across time).

Case I is a constant-coefficient structural

equation. Case II is an equation with time-
varying disturbancevariancesonly. Case III is
an equation with time-varying coefficients, as
well as time-varyingdisturbancevariances.

60 THEAMERICANECONOMICREVIEW MARCH2006

TABLE1-IDENTIFYINRGESTRICTIOONNSAO(s,) shown in Table 1, we introducestochasticprior
informationfavoring a negative contemporane-
Variable Sector ous response of money demandto the interest
(below) (right) Inf Fed MD Prod Prod Prod rate and a positive contemporaneousresponse
of the interest rate to money (see Appendix).
Pcom X More precisely, we use a priorthat makes the
M coefficients on R and M in the money demand
R XX X column of A0 positively correlatedand in the
monetarypolicy column of A0 negatively cor-
y XX X related.This liquidity effect priorhas little in-
P fluenceon the correlationof posteriorestimates
X XX X X of the coefficients in the policy and the money
U demandequations,but it makes point estimates
XX X X XX of coefficients and impulse responsesmore sta-
ble across different sample periods. The insta-
III. Data,Identificationa, nd ModelFit bility we eliminate here arises from the
difficultyof separatingmoney demandand sup-
We use monthlyU.S. datafrom1959:1-2003:3. ply in some subperiods,and for this reason is
Each model has 13 lags and includes the con- associated with imprecise estimates in both
stantterm and six commonly used endogenous equations. Since a finding of change in mone-
variables: a commodity price index (Pcom), tary policy across periods requiressome preci-
M2 divisia (M), the federal funds rate (R), in- sion in the estimates of policy rule coefficients
terpolatedmonthly real GDP (y), the core per- in those periods, the liquidity-effectpriors are
sonal consumption expenditure (PCE) price as likely to strengthenas to weakenevidence for
index (P), and the unemploymentrate (U). All changesin the policy rule.We takeup this issue
variables are expressed in naturallogs except again in discussion of the results, below.
for the federalfundsrateandthe unemployment
rate, which are expressed in percent.5 We modelandcomparethe five specifications:

The identification of monetary policy, fol- Constant:a constant-parameteBr VAR (i.e., all
lowing Leeper and Zha (2003), is describedin equationsare Case I);
Table 1. The X's in Table 1 indicate the unre-
stricted parameters in Ao(s,), and the blank Variancesonly: all equationsare Case II;
spaces indicatethe parametersthatarerestricted Monetarypolicy: all equationsexcept the mon-
to be zero. The "Fed" column representsthe
FederalReserve contemporaneousbehavior;the etarypolicy rule areCase II, while the policy
"Inf" column describes the informationsector rule is Case III;
(the commoditymarket);the "MD"columnrep- Private sector: equations in the private sector
resents the money demand equation; and the are Case III and monetarypolicy is Case II;
block consisting of the last three columns rep- All change: all equationsare Case III.
resents the productionsector, whose variables
are arbitrarilyordered in an upper triangular There are two major factors that make the
form.6 estimationand inference of our models a diffi-
cult task. One factor is simultaneousrelation-
In addition to the exact zero restrictions ships in the structuralcoefficient matrixAo(s,).
The other factor is the types of restrictedtime
5 As robustness checks, we also used the M2 stock variations specified in (8). Without these ele-
ments, the shape of the posteriordensity would
instead of M2 divisia and the CPI (as well as the GDP be much more regular, and more straightfor-
ward Gibbs sampling methods would apply.
deflator) instead of the core PCE price index, and the The Appendixoutlines the methods and briefly
paper's main conclusions remainedunchanged.
variables,differentsample periods, and differentdeveloped
6 While we provideno discussion here of why delays in economies.
reactionof the privatesectorto financialvariablesmightbe
plausible, explanations of inertia, and examination of its
effects, are common in the recent literature(Sims, 1998;

Rochelle M. Edge, 2000; Sims, 2003; LawrenceChristiano
et al., 2005). The economic and theoreticaljustificationof
the identificationpresentedin Table 1 can also be found in
Leeperet al. (1996) and Sims and Zha (forthcoming).This
identificationhas provento be stableacrossdifferentsets of

VOL.96 NO. 1 SIMSAND ZHA: WERETHEREREGIMESWITCHESIN U.S. MONETARYPOLICY? 61

TABLE2-COMPREHENSIVEMEASURESOF FIT imply extremeodds ratiosin favor of the high-
er-marginal-data-densitymodel. For the upper
Log marginaldatadensities rows in the table,the MonteCarlo(MC) errorin
Constant 12,998.20 these numbers(based on two million MCMC
draws)is from o2 to o4. Forthe lower rows in
Variances Monetary Private All each column, the error is larger (from o+3to
change o5). These estimates of MC errorare conser-
only policy sector vative, based on our own experience with mul-
tiple startingpoints for the chain. Conventional
2 states 13,345.71 13,383.36 13,280.74 13,308.80 measures of accuracy based on serial covari-
13,434.25 13,446.13 13,380.77 13,426.78 ances of the draws,for example, would suggest
3 states 13,466.86 13,480.18 much smallererrorbands.When the whole pri-
4 states 13,455.26 13,400.10 * * vate sector, or the whole model, is allowed to
5 states 13,510.31 * * change accordingto Case III, the marginaldata
6 states 13,530.71 * * * density is distinctly lower thanthat of the best
7 states 13,540.32 * * models for a given row of thetableandforthose
8 states 13,544.07 * * * versionsof the model for whichwe could obtain
9 states 13,538.03 * * * convergence. The best fit is for the nine-state
10 states * * * variances-onlymodel, though any of the seven
throughten state versions of that model have
* similar fit. The marginaldata density for these
variances-onlymodels is higher by at least 50
discusses both analytical and computational on a log scale thanthatfor any othermodel.The
difficulties. best of the models allowing time variation in
coefficients is the monetarypolicy model with
The firstset of resultsto consideris measures four states, whose marginal data density is
of model fit, with the comparison based on higher by at least 50 than that of any other
posterior marginal data densities. The results model that allows change in coefficients.8
are displayed in Table 2. For the models with
largernumbersof free parameters,the Markov IV. Best-FitModel
Chain Monte Carlo (MCMC) sample averages
that are the basis of the numbersin the table There are a numberof best-fit models, all of
behave erratically,and we display "*"for these them variances-only models with from seven
cases ratherthana specific number.Thoughthe to ten states. Since the results from these
estimated marginaldata densities (MDDs) for models are quite similar, we reportthe results
these cases are erratic,they remain far below from only the nine-statevariances-onlymodel.
the levels of MDDs shown in the same column The transition matrix for the nine states is
above them.In otherwords,thoughdisplayinga shown in Table 3. The states appearto behave
single numberfor their MDD values might in- similarly, and they have a fairly evenly spread
dicate misleading precision, it is clear that the set of steady-stateprobabilities,ranging from
MDDs for these cases arevery muchlowerthan 0.078 to 0.19.
those of the cases for which we do display
numbers.7 The firststateis used as a benchmarkwith its
variances being normalizedto one. As can be
Note thatthis is a log-likelihoodscale, so that seen fromFigure 1, this stateprevailsin most of
differences of one or two in absolute value the Greenspan regime and includes several
mean little, while differences of ten or more
8 Note, though,thatthe "privatesector"and"allchange"
7The main reason for the slow convergence of our models may be doing less well because of parametercount.
estimatedposteriorprobabilitiesof models is thatthe simul- It could be that more tightly parameterizedmodels of co-
taneity in our model creates zeroes in the likelihood at efficient change in the privatesectorwould look betterin a
points in the parameterspace whereAois less thanfull rank. table like this.
Because our application of the modified harmonic mean
method for estimating the posterior probability did not
allow for these zeroes, ourestimatesarebased on averaging
draws from a distributionwith first, but not second, mo-
ments. The estimatesconverge, but do so very slowly; and
standardconvergencediagnosticsbasedon secondmoments
are useless. We have ideas for how to do this betterif we
were to approachthe problemagain.

62 THEAMERICANECONOMICREVIEW MARCH2006

TABLE3-TRANSITION MATRIXFORNINE-STATE Estimatesof the model's dynamic responses
VARIANCES-ONLMY ODEL
arevery similarto those producedby previously

0.9643 0.0063 0.0117 0.0064 0.0108 identifiedVAR models, so we will not presenta
0.0030 0.9394 0.0047 0.0070 0.0210 full set of impulseresponses.The resultsare as
0.0104 0.0159 0.9455 0.0064 0.0046 sensible as for previous models, yet we have a
0.0026 0.0043 0.0042 0.9476 0.0040 more accuratepictureof uncertaintybecause of
0.0058 0.0155 0.0044 0.0068 0.9425 its stochasticallyevolving shock variances.The
0.0027 0.0056 0.0058 0.0064 0.0051 responses to a monetary policy shock for the
0.0052 0.0042 0.0081 0.0068 0.0040 first state, togetherwith errorbands, are shown
0.0033 0.0041 0.0069 0.0062 0.0038 in Figure 2.9 Note that, though commodity
0.0026 0.0046 0.0087 0.0065 0.0042

0.0057 0.0107 0.0095 0.0049 prices and the money stock decline following a
0.0062 0.0061 0.0069 0.0112 shock that tightens monetarypolicy, the point
0.0063 0.0064 0.0096 0.0057 estimatesshow P decliningonly aftera delay of
0.0058 0.0056 0.0062 0.0051 several years, and this decline is small and
0.0185 0.0058 0.0064 0.0057
0.9406 0.0120 0.0062 0.0050 uncertain.
0.0057 0.9423 0.0062 0.0053
0.0056 0.0054 0.9429 0.0049 Table 6 reportsartificiallong-runresponses
0.0056 0.0056 0.0062 0.9522 of the policy rate to other macro variables, as
often presentedin the literature.By "artificial,"
we mean that these are neither an equilibrium

outcome nor multivariate impulse responses,
butarecalculatedfromthe policy reactionfunc-

yearsin the 1960s. The variancesin otherstates tion alone, askingwhatwouldbe the permanent
do not simply scale up and down across all response in R to a permanentincrease in the
structuralequations.Some states affect a group level or rate of change of the variablein ques-
of structuralshocksjointly, as can be seen from tion, if all other variables remained constant.
Table 4. The ninthstateprevailsin the Volcker
reserve-targetingperiod and primarilyinflates The long-runresponse to the level of the vari-
the varianceof the policy shock (Figure 1 and be.,where ae
Table 4.) The eighth state inflatesthe variances able is calculatedas i = althel,l.a=go of the "right-
of several private-sectorequations, and it pre- is the the
vails only for the two months of September coefficient on
and October 2001. This is clearly a "9/11"
state. The other states exist sporadically over hand-side"variableand 5e is the coefficient on
the 1970s, as well as over the period from the thlag of the "left-hand-side"variablein the
1983 to 1987 and some years in the 1960s. policy rule.The long-runresponseto the change
Among these states, the shock variances of the variableis calculated as 1'= o
change irregularlyfrom state to state. For the ali
1970s, short-lived states with changing shock 5stu.cIhn Table 6, the differenced(logie)=voari-
variances reflect several economic disruptions 1e=o as Ay and AP are annualized to
(e.g., two big oil shocks) and the ambivalent ables
way monetary policy was conducted in re-
sponse to those disturbances. match the annual rate of interest R. Absence

For this variances-onlymodel, the structural of sunspots in the price level will be associ-
parametersand impulse responses vary across ated with the sum of these long-run responses
states only up to scales. Table 5 reports the to nominal variables (here APCom, AM, and
estimateof contemporaneouscoefficient matrix
for the first state. As can be seen from the "M AP) exceeding one. For this model, the sum is
1.76, well above one, though the errorbands
Policy" column, the policy rule shows a much on individual coefficient leave room for some
larger contemporaneouscoefficient on R than
on M, implyingthe FederalReserve pays much uncertainty.
more attentionwithin the month to the interest
V. PolicyRegimeSwitches
rate than the money stock.
In this section, we present the key results
from the four-state model with time-varying

coefficients in the policy rule. There are two

9 The shape of the impulse responses as seen on scaled
plots is the same across states.

VOL.96 NO. I SIMSAND ZHA: WERETHEREREGIMESWITCHESIN U.S. MONETARYPOLICY? 63

15 0.8

FFR10 p5
0.4
5
0.0
0.8 0.8

pl p6
0.4 0.4

0.0 0.0
0.8 0.8

p2 p7
0.4 0.4

0.0 0.0
0.8 0.8

p3 p8
0.4 0.4

0.0 0.0
0.8
p4 0.04
p9
0.4

0.00 1970 1980 1990 2000 0.0 1970 1980 1990 2000
1960 1960

FIGURE 1. NINE-STATE VARIANCES-ONLY PROBABILITIES
Note: The Fed Funds Rate is in the upperleft.

TABLE 4-RELATIVE SHOCK STANDARD DEVIATIONS ACROSS STATES FOR NINE-STATE VARIANCES-ONLY MODEL

First state Financial M policy M demand Privatey PrivateP PrivateU
Second state
Third state 1.00 1.00 1.00 1.00 1.00 1.00
0.95 1.47 1.03 2.07 1.19 1.69
Fourthstate 1.28 1.65 1.84 1.11 1.12 0.91
Fifth state 2.01 2.65 1.93 1.59 1.29 1.37
1.38 2.95 1.24 1.01 0.96 1.17
Sixth state 2.67 2.99 2.32 2.52 0.95 2.13
Seventh state 2.40 4.43 1.21 1.59 2.58 1.05
Eighth state 2.55 4.49 11.44 4.10 10.48 2.67
Ninth state 1.49 12.57 1.53 1.44 1.48 1.44

64 THEAMERICANECONOMICREVIEW MARCH2006

Pcom TABLE5-CONTEMPORANEOUCS OEFFICIENMTATRIXFORNINE-STATEVARIANCES-ONLYMODEL
M
R Financial M policy M demand Privatey PrivateP PrivateU
y
P 70.64 0.00 0.00 0.00 0.00 0.00
U 9.21 - 130.24 -669.91 0.00 0.00 0.00
-27.30 -70.10 0.00 0.00 0.00
- 14.21 688.52 308.75 -20.77 51.94
-5.54 0.00 19.85 0.00 -1061.30 32.38
82.37 216.07 0.00 0.00 766.38
-0.00
0.00 0.00

PCOM M2

resp -0.005 10 20 30 40 0.000 10 20 30 40
Months Months
015 resp -0.002
-0 FFR y
-0.004
0 0

resp 0.0010 -0.0005
resp

-0.00005 10 20 30 40 -0.0020 10 20 30 40
Months 0 Months

P U

resp Oe+00 resp 4e-04

-le-03 10 20 30 40 Oe+00 10 20 30 40
0 Months 0 Months

FIGURE2. RESPONSESTOA MONETARYPOLICYSHOCK
(Nine-state, variances-onlymodel)

Note: Eachgraphshows, over48 months,the modal's estimatedresponse(blackest),the medianresponse,and68-percentand
90-percent probabilitybands.

VOL.96 NO. 1 SIMSAND ZHA: WERETHEREREGIMESWITCHESIN U.S. MONETARYPOLICY? 65

TABLE6--LONG-RUN POLICYRESPONSESIN NINE-STATE 0.143 for the Volcker state, and 0.116 for the
VARIANCES-ONLYMODEL
fourthstate.FromTable 7 one can also see that
Responses Posteriorpeak 0.68 probability
of R to estimate interval the probabilityof switchingfromthe Greenspan
andBurnsstatesto the Volckerandfourthstates
A Pcom 0.21 (0.17, 0.73)
AM 0.16 (-0.48, 0.44) is reducedby one-half as comparedto the prob-
Ay 0.71 ability of switching the other way.
AP 1.39 (0.69, 3.36)
U -1.01 (0.45, 2.21) Differences in the contemporaneous coef-
(-2.80, -0.42) ficient matrixshow up across states as well. In
Table 8 we can see that the Greenspan re-
reasons why this model may be of interest, gime's contemporaneouscoefficient matrix is
despite the fact thatit is dominatedin fit by the broadly similar to that estimated for the full
model with only disturbancevariances chang- sample with the variances-only model (Table
ing. First,this model's fit is substantiallybetter 5). In particular, both policy rules show a
than all other models that allow change in co- much larger contemporaneous coefficient on
efficients(Table2). Second,the model reflectsa R than on M. On the other hand, we see from
prevailingview thatthe endogenouscomponent Tables 9 and 10 that the Burns and Volcker
of U.S. monetarypolicy has changed substan-
tially since 1960 and its simulatedresults cap- states both have much larger contemporane-
ture some importantaspects of conventional ous coefficients on M, with the M coefficient
wisdom about policy changes from the 1970s
throughthe 1980s to 1990s. being relatively larger for the Volcker state.
These results are consistent with the observa-
Figure3 shows the impliedstate-probabilities
over time produced by this four-state model. tion that Burns seemed to pay a lot of atten-
We can see thatstate 1 has prevailedfor most of tion to money growth in the early 1970s and
our full sample periodand for the entireperiod less (more) attention to money growth (the
from the late 1980s onward.We call this state interest rate) in the last few years of his tenure
(Arthur F. Burns, 1987; Henry W. Chappell,
the Greenspanstateof policy, but of course one Rob Roy McGregor, and Todd Vermilyea
needs to bear in mind that this policy regime (CMGV), 2005) and that Greenspanmade the
was dominantin most of the 1960s and in the interest rate the explicit policy instrument.

latter half of the 1970s as well. State 2 is the The long-runpolicy responsesto macrovari-
ables show a similar pattern, as reported in
next most common, occurringmost frequently Table 11. The Greenspanregime shows slightly
from the early 1960s throughthe early 1970s strongerpoint estimates of the responses of the
(the firstoil shock period),thoughwith no sus- funds rate to money growth and inflationthan
tainedperiodsof prevalencethatmatchthose of those impliedby the variances-onlymodel (Ta-
state 1. We call this the Burns regime, even ble 6), but with greateruncertaintybecause of
though it matches up with Burns's chairman- the smaller effective sample period. For the
ship even less well than the Greenspanregime VolckerandBurnsregimes,the responsesof the
matcheswith Greenspan's.State 3 prevailsdur- federal funds rate are, variableby variable, so
ing the Volcker reserve targeting period and ill-determined that we instead present re-
nowhere else, except one very brief period sponses of money growth, which seems closer
around 1970. State 4 occurs only for a few to the short-runpolicy target in those regimes.
isolated months, including 9/11, and seems We see that the Volcker regime makes money
clearly to be picking up outliersratherthanany unresponsive to all variables (measured by
systematicchange of coefficients. both point estimates and error bands). The
Burns regime shows a disturbingly high re-
The estimateof the transitionmatrixis shown sponsiveness of money growth to inflation,
though the point estimate is still below one,
in Table 7. The four states behave quite differ-
ently. Nearlyhalf of the steady-stateprobability which is only partially offset by a negative
(0.49) goes to the Greenspanstate.Forthe other
half, the probabilityis 0.25 for the Burns state, response to the rate of change in commodity

prices.

Because the Burnsregimelooks like the most

likely candidatefor a potentialsunspot incuba-

66 THEAMERICANECONOMICREVIEW MARCH2006
2000
pl 0.6 1970 1980 1990
0.0 Year
1960

p2 0.6 1970 1980 1990 2000
Year
0.0
1960

p3 0.6 1970 1980 1990 2000
Year
0.0
1960

p4 0.6
0.0

1960 1970 1980 1990 2000
Year

FIGUR3E. STATEPROBABILITIES
(Four-statemonetarypolicy changing)
Note: The time path of the Fed Funds Rate is in the backgroundof each figure.

TABLE7-TRANSITIONMATRIXFORFOUR-STATE on all nominalvariables-the rate of change in
POLICY-ONLMYODEL commodityprices,money growth,andinflation.
This response is surprisinglywell-determined,
0.9627 0.0460 0.0203 0.0334 probablybecause of collinearityin the sample
0.0214 0.9388 0.0195 0.0174 among the nominalvariables.10The 68-percent
0.0077 0.0073 0.9414 0.0238
0.0082 0.0079 0.0188 0.9254

tor, we triednormalizingthatregime's reaction 10Note that if we calculated long-runresponses of the
function on the interestrate and calculatingits
long-runresponseto the sum of the coefficients interestratefor this regime, variableby variable,we would
get very large, opposite-signed numbersthat would have
high uncertaintyand be difficult to interpret.

VOL.96 NO. 1 SIMSAND ZHA: WERETHEREREGIMESWITCHESIN U.S. MONETARYPOLICY? 67

TABLE 8-CONTEMPORANEOUS COEFFICIENT MATRIX FOR FIRST STATE IN FOUR-STATE POLICY-ONLY MODEL

Pcom Financial M policy M demand Privatey PrivateP Private U
M
R 68.03 0.00 0.00 0.00 0.00 0.00
34.19 - 208.60 -559.30 0.00 0.00 0.00
y -32.62 -172.64 0.00 -0.00 0.00
P -4.49 559.48 272.37 -17.51 51.94
U 0.00 11.87 0.00 -1029.19 25.45
8.65 0.00 -54.58 0.00 0.00 705.57
84.70 0.00
0.00

TABLE 9-CONTEMPORANEOUS COEFFICIENT MATRIX FOR SECOND STATE IN FOUR-STATE POLICY-ONLY MODEL

Pcom Financial M policy M demand Privatey PrivateP PrivateU
M
R 38.20 0.00 0.00 0.00 0.00 0.00
19.20 -221.50 -401.63 0.00 0.00 0.00
y -18.32 -123.97 0.00 -0.00 0.00
P -2.52 188.29 206.87 -13.72 42.40
U 4.86 0.00 8.52 0.00 - 806.18 20.77
47.56 0.00 -39.19 0.00 0.00 576.00
0.00
0.00

TABLE 10-CONTEMPORANEOUS COEFFICIENTMATRIX FOR THIRD STATE IN FOUR-STATE POLICY-ONLY MODEL

Pcom Financial M policy M demand Privatey PrivateP PrivateU
M
R 50.43 0.00 0.00 0.00 0.00 0.00
25.35 - 393.51 - 241.46 0.00 0.00 0.00
y -24.18 0.00 -0.00 0.00
P -3.33 136.05 -74.53 235.35 -12.82 41.12
U 0.00 5.12 0.00 -753.62 20.15
6.41 0.00 0.00 0.00 558.70
62.78 0.00 -23.56
0.00

probabilitybandis (0.94, 3.50), which makesit the debate on the effects of monetary policy
very likely that the regime was not a sunspot changes.
incubator.
A. SuppressingPolicy Shocks
VI. HistoricalCounterfactuals
The first and simplest of our counterfactual
As a way to quantify the importance of simulations sets the disturbancesin the policy
policy change over time, the four-state time- equation to zero in the nine-state model. Dis-
varying model makes it an internally coherent turbancesand coefficients are otherwise set at
exercise to calculate what would have hap- high-likelihood values, so that if the policy
pened if regime changes had not occurred, or rule disturbances had been left in place, the
had occurred when they otherwise didn't, at simulations would have shown a perfect fit.
particularhistorical dates. We have run quite As can be seen from Figure 4, the model
a few of these experiments, but the main leaves the time path of inflation almost un-
conclusion is that the estimated policych- changed. Policy shocks play a crucialroleonly
anges do make a noticeable difference, but in attributintghefluctuationosf thefundsratein the
not a drastic difference. In the following, we late 1970sandtheearly1980s.Thehistoryof infla-
display examples that seem most relevant to tion is attributedalmost entirely to nonpolicy

68 THEAMERICANECONOMICREVIEW MARCH2006

TABLEI l-LONG-RUN POLICYRESPONSESIN FOUR-STATE TABLE12-ANNUAL AVERAGEOUTPUTGROWTHRATES
POLICY-ONLYMODEL OVER1961-1986, ACTUALANDCOUNTERFACTUAL

First state (Greenspan) Actual Burns Greenspan Volcker

Responses Posteriorpeak 0.68 probability 3.7206 4.0560 3.2454 2.8956
of R to estimate interval

A Pcom 0.09 (-0.19, 0.24) also reproduceshistory very closely, matching
AM 0.23 (-0.46, 2.08) the rise andthe subsequentfall in inflation.This
Ay (-1.28, 0.64) policy keeps inflation slightly lower in the
AP 0.43 (-0.09, 2.48) 1960s and 1970s, butthenin the mid-1980s lets
U 1.99 (-0.91, 0.46)
-1.29 the inflation level out at a somewhat higher
Responses value.
of A M to Second state (Burns)
The modest differencesacross these policies
A Pcom Posteriorpeak 0.68 probability do not mean the model implies thatno changes
R estimate interval in monetarypolicy could have preventeda rise
Ay in inflationto near-double-digitlevels. Though
AP -0.24 (-0.50, 0.01) the Volcker reaction function is estimated im-
U 0.09 (-0.02, 0.49)
0.18 (-0.43, 0.35) preciselybecauseof the shortperiodin which it
Responses 0.92 (-0.17, 1.74) prevailed, if we repeat our exercise with the
of A M to 0.05 (-0.025, 0.09) point estimateof the Volckerpolicy functionin
place, we obtain the results in Figure 7. This
A Pcom Thirdstate (Volcker) policy would have kept money growth much
R lower, would have kept inflation lower by
Ay Posteriorpeak 0.68 probability aroundtwo percentagepoints at its peak, and
AP estimate interval would have loweredaverageoutputgrowth.Al-
U thoughthe outputeffect may be difficultto see
-0.12 (-0.06, 0.05) from Figures 5 to 7, Table 12 shows the sub-
0.01 (-0.02, 0.20) stantialimplieddifferencesin outputgrowthfor
0.13 (-0.70, 0.64) the threeregimepoint estimatesthroughoutthis
0.23 (-0.51, 0.28) entire period.
0.02 (-0.04, 0.06)
These results are not reflective simply of the
sources,thoughof course feeding systematically Volcker policy's focus on growth of monetary
througha fixedmonetarypolicyrule. aggregates.If we simply doublethe coefficients
on inflationin the Greenspanmonetarypolicy
B. Keepinga Fixed Greenspanor Volcker rule, while again leaving disturbancesin other
Rule in Place Throughout equations at historical values and suppressing
monetary policy shocks, we arrive at Fig-
If we run a similar simulationbut with the ure8. Peakinflationis cut nearlyin half, andthe
four-statemonetarypolicy model by placingthe inflationratehoversaroundzerofor muchof the
estimated Greenspan rule through the pre-
Greenspan period 1961-1987, we obtain the 1961-1987 period.
results shown in Figure 5. This simulation Without any a priori imposed structureon
tracks history almost as well as the previous
one. Thus, the model attributesthe rise and fall private sector behavior,the model nonetheless
in inflation neither to monetarypolicy shocks shows a type of neutralityresult. By the 1980s,
nor to changes in policy regime. In particular, even thoughinflationis running4 or 5 percent-
the model reproducesthe high peak inflation age points below the actual historical values,
rates of the early 1980s, even though the with this "inflationhawk Greenspan"policy,
Greenspan reaction function is in place output is trackingthe historical values almost
throughout. perfectly. The model thus appearsto allow for
the public's learningthat a new, lower level of
With the Burns policy in place throughout inflationprevails.On the otherhand,the tighter
this history, instead, we obtain the counterfac- monetarypolicy cuts outputgrowth startingin
tual history shown in Figure6. This simulation

VOL.96 NO. 1 SIMSAND ZHA: WERETHEREREGIMESWITCHESIN U.S. MONETARYPOLICY? 69

20
Actual
Counterfactual

15

cr 10

0

0 1965 1970 1975 1980 1985 1990 1995 2000 2005
1960 Year

10

Actual
8 Counterfactual

6

CL

4

2

0 1965 1970 1975 1980 1985 1990 1995 2000 2005
1960 Year

FIGURE4. COUNTERFACTUAPALTHSWITHNo COEFFICIENCTHANGESANDNo POLICYSHOCKS
(Nine-state, variances-onlymodel)

the early 1960s, andkeeps it well below histor- monetarypolicy regime of the late 80s and 90s
ical values for most of the 60s, 70s, and 80s. was not enough differentfrom the policy actu-
ally in place in 60s and 70s to have made any
Both of these policy rules which lower the in- substantial difference to the time path of
flationratealso lower the outputgrowthrate,as inflation.
can be seen from Figures 7 and 8.
C. Distributionsof Policy Functions
The counterfactual simulations that imply
lower inflation create a markedchange in the Although the policy rules in place before the
stochasticprocessfollowed by outputandinfla- end of 1979 and after 1982 are estimated to
tion. It is, therefore,quite possible thatthe out- have similarconsequencesfor the rise and fall
put costs of the strongeranti-inflationarypolicy in inflation, the estimates leave uncertainty
stance would not have been so persistent, as about those policies. Point estimates for both
shown in the graphs. Our point is not that regimes show, as we noted above, cumulative
stricteranti-inflationarypolicy would have had responses of the funds rate to inflation that
outputcosts as great as shown in these graphs. imply a unique price level. Nonetheless, the
Ourpoint is only thatif the Greenspanrule had Burnsregimepoint estimatesarelower, andthe
been different enough to prevent the rise in uncertainty about the estimates leaves more
inflation in the 1960s and 1970s, our model probabilityin the region arounda unit response

would have shownthatthe regimechangemade
a difference.In fact, ourbest estimateis thatthe

70 THEAMERICANECONOMICREVIEW MARCH2006
20
15 1970 1980 1990 14 1970 1980 1990
10 12
10
LL 8

5 C"
0
1960 6
4
2
0
1960

Year Year

10 10

8 8

6 6
CL
4
4 =h

2 2

0 0
1960
-2

1970 1980 1990 -4 1970 1980 1990

1960

Year Year

FIGURE 5. FIXED GREENSPAN POLICY THROUGHOUT 1961-1987

(Four-statemonetarypolicy model)
Note: Each graphshows the actualpath (thick line) and the counterfactualpath (thin line).

than with the Greenspanregime. As might be cients in the reactionfunction,are being drawn
expected, the model's simulatedtime paths re- from the distributioncorrespondingto a single
spond nonlinearlyas the region with less than
unit cumulative response of the funds rate to regime.
inflationis approached.As a result, if we con- The Greenspanregime results are shown in
duct our counterfactualsimulationsby drawing
from the distributionof policy rule coefficients Figure 9, where we see that the median simu-
for the Burns and Greenspanregimes, rather lated pathdisplays substantiallylower inflation
than simply imposing the most likely values, than what was historically observed. It is im-
differences between the coefficients become portantto bearin mindthatthis is not the actual
pathfor any one policy. This is clear when we
more apparent.In the simulationswe describe look at the medianpathfor interestrates,which
below, the historical shocks are kept on their is almost uniformly lower than the historical
historical path, with variances changing with path.If these medianpathswere actualpathsfor
regimeaccordingto ourestimatedposteriordis- any given policy, it wouldbe a mysteryhow the
tribution,but the policy regime distributionis policy could lower inflationwithoutever raising
kept fixed in one regime for all coefficients in interest rates. But as can be seen from the
the policy equation.This meansthatthe scale of
monetarypolicy shocks, as well as the coeffi- graphsfor point-estimatepolicies, policies that
lower inflationraise interestrates in some cru-

cial periods, and this is followed by long peri-

VOL.96 NO. 1 SIMSAND ZHA: WERETHEREREGIMESWITCHESIN U.S. MONETARYPOLICY? 71
20 20

15 15

a 10 CM10
5 5

0. 1970 1980 1990 0 1970 1980 1990
1960
1960

Year Year

10 10

8 8

6 6

.. 4
Year
4
2
2
0
0
1960 -2

1970 1980 1990 -4 1970 1980 1990

1960

Year Year

FIGURE6. FIXEDBURNSPOLICYTHROUGHOU1T961-1987
(Four-statemonetarypolicy model)

Note: Each graphshows the actualpath (thick line) and the counterfactualpath (thin line).

ods of lower inflation, and hence of lower VII. RobustnessAnalysis
nominal interest rates. When we display the
median path across many policies that imply In this section, we study a numberof other
periods of tighter policy, but imply different relevantmodels to check the robustnessof our
timing for the periods of tighterpolicy, we see results. The insights from these exercises rein-
a uniformlylower path of interestrates. force the points made in the previous sections.

Note that these simulated draws from the A. The Economywith Policy Changes

Greenspanpolicy distributionimply a substan- We consider an economy with two mone-
tial risk of deflationin the 1980s, as well as a tary policy rules estimated in our four-state,
risk of outputgrowth below -5 percent. policy-only model: one is the rule associated
with the Burns regime and the other rule is the
A similarexercisewith the Bums regimedis- Greenspan interest-smoothing policy. This
tributionproducesthe resultsin Figure10. There economy consists of the same six variables as
is littleriskof outputloss;moneygrowthtendsto our actual data and starts with the Burns pol-
be higher than the historicalpath. The risk of icy, which lasts for 236 months (correspond-
deflationis lower, but now thereis a substantial ing to September 1979 in our sample) and

riskof no declineat all in inflationin the 1980s,
consistentwith the conventionalview aboutthe

effects of the Burnspolicy.

72 THEAMERICANECONOMICREVIEW MARCH2006
15
20

15 10
M2 5
10
FFR 0

5

0.

-5- 1960 1970 1980 1990 -5 1970 1980 1990

1960

Year Year

10 10

8 8

6 6
P
4
4 y

2 2

0 0
1960
-2

1970 1980 1990 -4 1970 1980 1990
1960

Year Year

FIGUR7E. FIXEDVOLCKEPROLICTYHROUGHO1U96T1-1987
(Four-statemonetarypolicy model)

Note: Each graphshows the actualpath (thick line) and the counterfactualpath (thin line).

then monetarypolicy switches, once for all, to In eight out of the ten datasets,the estimated
the estimated Greenspan policy rule. At the transitionmatrixfor the two-statemonetarypol-
time of the switch in policy rules, the scale of icy model has one absorbingstate, which is of
nonpolicy shocks also changes as in our esti- coursecorrectin the simulateddata.12Thus,the
mated four-state model. We simulated ten methodwe have used would have been likely to
samples, each with the same sample length as
our actual data and each with initial values set College of ComputerScience at the Georgia Institute of
at the actual data from 1959:01 to 1960:01. Technology, which designed a Linux-basedprogramcalled
For each simulated dataset, we consider four "STAMPEDE"specifically for this project. This program
models: monetary policy models for two and allows us to runourjobs efficientlyon a clusterof comput-
three states and variances-only models for ers simultaneously.
two and three states.11
12 To obtainan absorbingstate,our originalprioron the
" Computationsfor these simulateddataarequiteinten- transitionmatrixis modifiedso thatthe Dirichletweight akk
sive. For each model, it takes about a week on a single on the diagonalelement of the transitionmatrixis 1.0. The
processorcomputerto get the marginaldatadensity. There original prior gives the weight value of 5.0, which effec-
is a total of 40 models (which would be a ten-month tively puts an upperboundon the estimateof Pkk away from
computation).We acknowledgetechnical supportfrom the 1.0. In this case, we obtainthe posteriorprobabilitiesof this
state being nearone for almost the entireperiod for which
the state actually prevails.

VOL.96 NO. 1 SIMSAND ZHA: WERETHEREREGIMESWITCHESIN U.S. MONETARYPOLICY? 73
1990
20

Actual
15 Counterfactual

FFR10

0 1965 1970 1975 1980 1985
Year
C
1960

10

5
Y

0 Actual

Counterfactual

-5 1965 1970 1975 1980 1985 1990
1960

Year

10
Actual

Counterfactual

5

P

0

-5 1965 1970 1975 1980 1985 1990
1960 Year

FIGURE8. COUNTERFACTUAPLATHSIF POLICYRESPONSETOINFLATIONHAD BEENTWICEAS STRONGAS IN THEESTIMATED
GREENSPANPOLICYTHROUGHOU1T961-1987
(Four-state monetarypolicy model)

detect a permanentregime shift if that is what states. The variances-only model with three
had occurred. stateshadposteriorprobabilityless than10-6 in
all ten simulations.The posteriorprobabilityon
Figure 11 shows the cdf, acrossthe ten Monte the three-statemodel with policy change(which
Carlo samples, of the posteriorprobabilitythat of courseis overparameterizedb, ut containsthe
there was a change in policy coefficients. In truemodel) reacheda maximumof around0.04
seven of ten cases the posteriorprobabilityof a in one simulation, and otherwise was even
changewas over 0.99. In one it was around0.2, smaller than the posterior probability on the
and in two it was 0.02 to 0.03. The log odds three-state,variances-onlymodel.
ratio correspondingto the most extreme odds
against the policy change (i.e., in favor of a These experiments give our methods a stiff
variance-change-onlymodel) was 3.78. The log test. The estimated Greenspanand Burns pol-
odds ratio in favor of variances only in our icy rules that we use imply very similar qual-
analysis of the historicaldatais about60, many itative behavior in our counterfactual simula-
times strongerthanthe most extremefindingin
these Monte Carlo simulations. tions with point estimators. Yet even with
these two similar policy rules, our method is
It is also worthnotingthatthe resultsshowed able to detect the switch for a majority of
no tendency to favor spuriousvariance-change
samples.

74 THEAMERICANECONOMICREVIEW MARCH2006

25 1970 1980 1990 20 1970 1980 1990
20 15
15 10
10 5
FFR 5 M2
0
0 -5
-5 -10
-10 -15
1960 1960

Year Year

15 15

10 10

5 5
P Y

0- 0

-5 -5

-10 1970 1980 1990 -10 1970 1980 1990
1960 1960

Year Year

FIGURE9. THE GREENSPANPOLICYRULEDISTRIBUTIOTNHROUGHOUTTHEPRE-GREENSPAPNERIOD

(Four-statemonetarypolicy model)

Note: Each graphshows the actualpath(thick line), the mediancounterfactualpath (thin darkline), and the 68-percentand
90-percentprobabilitybands (thin light lines).

B. OtherRelevantModels ces for the two types of stateareQ, andQ2, we
get the desired independentevolution by treat-
IndependenCt oefficienat nd VarianceStates.-- ing each pair of values for the two states as a
The results so far assume that coefficients and sinEgsletimstaatteinagnadsseettotifnmg Pod=elsQw1it0h
variancesswitch at the same time. Forthe mon- Q2.
etary policy model, the potentialproblemwith independent
this approachis thatthe numberof statesfor the
coefficients on the policy equation must in- mean and variance states at the same scale of
crease with those for the variance state. In a
single equationmodel, Sims (2001) found that parameterizationas our main models would be
making the transitionsof variance and coeffi- a majorcomputationaltask, which we have not
cient statesindependentdeliveredthe best fit. In undertaken.We have instead calculated maxi-
our framework, this can be done by giving
special structureto the transitionmatrix P. If mumlog posteriordensity (LPD) values (rather
therearetwo independentlyevolving statevari- thanlog likelihood (LLH)values) for a number
ables, one indexing variancesand one indexing of somewhatsmallerscale models of this type,
equation coefficients, and the transitionmatri- which we can label 2v, 2v2p, 3v, 3v2p, and
4v. The "nv"models aremodels with n variance

states and no policy coefficient changes. The
"nv2p" models are models with n variance
states and two policy rule coefficient states,

VOL.96 NO. 1 SIMSAND ZHA: WERETHEREREGIMESWITCHESIN U.S. MONETARYPOLICY? 75

30 40
25
20 30
FFR15
10 20
5 M2
0
1960 10

20 1970 1980 1990 -10 1970 1980 1990
-10

1960

Year Year

15

15 10
P 10 Y5

50

0 1970 1980 1990 -5 1970 1980 1990
1960 1960

Year Year

FIGURE10. THEBURNSPOLICRYULEDISTRIBUTITOHNROUGHOTUHTEPRE-GREENSPPAENRIOD
(Four-statemonetarypolicy model)

Note: Each graphshows the actualpath(thick line), the mediancounterfactualpath(thin darkline), and the 68-percentand
90-percentprobabilitybands (thin light lines).

evolving independently.Because we have only 2v 2v2p 3v 3v2p 4v
LPDs, we can't computeposteriorodds, but we 0.0 11.1 91.7 78.7 127.9
can (as Sims did in his single-equationpaper)
comparethe models by the Schwarzcriterion.13 From this pattern of results it appears that a
The best of the modelsby this criterionis the 4v model with just two coefficient policy re-
model. With the 2v model as base (therefore gimes is not competitive with variance-only
with the zero value), the Schwarzcriteriaare: models, even if the variance changes are al-
lowed to evolve independently of the coeffi-
13 The Schwarz, or Bayesian Information,Criterion,is cient regimes.
usuallydescribedas log likelihoodminusnumberof param-
eters times log of sample size divided by two. Under stan- Note thatthese resultsmay explain why pre-
dardregularityconditionsit is guaranteedto be maximalat vious researchers (Lubik and Schorfheide,
the model with highestposteriorodds, if the sampleis large 2004; Claridaet al., 2000, e.g.) who allow only
enough. Though we use LPD in place of LLH, the same a single change in residual variances find evi-
asymptotic reasoning that justifies the criterion based on dence of coefficient change. Those studies are
likelihood applies here. making a comparisonlike our 2v versus 2v2p
comparisonin the table,which favors2v2p. It is

76 THEAMERICANECONOMICREVIEW MARCH2006

1.0 TABLE 13-LOG MARGINAL DATA DENSITIES FOR OTHER
MODELS

0.8 Permanent
regime
Excluding money change
in policy rule
Monetary
draw0s.6 Variances Monetary policy
of only policy
13,154.08
Prop0o.r4tion 2 states 13,330.89 13,347.46 13,414.53
3 states 13,432.88 13,419.88 13,412.85
0.2 4 states 13,462.40 13,296.58

0.0 0.2 0.4 0.6 0.8 1.0 cluding a monetary aggregate to describe the
0.0 Probabiliotyfa change policy rule under the Burns and Volcker re-
gimes. Here we exclude this variablefrom the
FIGURE 11. SIMULATED DATA: CUMULATIVE DENSITY policy reaction function to see if this worsens
FUNCTION OF POSTERIOR PROBABILITIES THAT POLICY the fit. The thirdcolumnof Table 13 reportsthe
measures of fit for a model with four states,
CHANGES
allowing the monetary policy rule to change
only when we allow at least three variance only coefficients, and with, as usual, variances
states that the addition of a coefficient state allowed to change with the state in all other
ceases to improve fit. equations.The fit is considerablyworse thanthe
correspondingcases when money is included
Permanent Policy Shifts.-Our experiments (see the thirdcolumnof Table2), by about60 in
log odds units.
with artificialdatasuggest thatour specification
could identify a permanentpolicy shift if it The fit is also worse when we exclude
occurred.Because it is a widespreadview that
therewas a single permanentshift in U.S. mon- money from the reaction function in the vari-
ances-only model, but the odds ratio is much
etary policy around 1979, however, it may less extreme. The log odds difference be-
nonethelessbe of interestto see whatemergesif tween the four-state, variances-only model
we economize on parametersby imposing on and the version of that model with money
our model the requirementthat there is an ab- excluded from the reaction function is 4.46.
sorbingstate-that is, thereis a state that,once
entered, remains in place for the rest of the This implies an odds ratio in favor of the
sample.This is equivalentto requiringthatone model, including money of over 80 in un-
column of the transitionmatrix, which repre- logged units, but this ratio is much less ex-
sents the probabilityof enteringeach state con- treme than the result for the model that allows
ditional on being in this state, is a unit vector
coefficient variation in the monetary policy
with a one at the diagonalposition. rule. This is not surprising, since the most
The fourth column of Table 13 displays the salient difference among the three main esti-
mated policy reaction functions is in the de-
marginal data densities of the monetary pol- gree to which they give weight to a monetary
icy models with permanent changes on the aggregate. If we shut down this type of dif-
coefficients of the policy equation. Compar- ference among policies, the model with coef-
ing to the thirdcolumn of Table 2, we see that ficient variationin the policy rule is penalized
the log posterior weight on these models is much more than the model that fits a single
lower by at least 60 more than the log poste- rule to the whole sample. As we have already
rior weights on the models that do not impose pointed out, it seems possible that a model
the absorbing state restriction. whose prior focused the search for policy
variation in particular economically reason-
Excludingthe MonetaryAggregate.-In Sec- able directions might be more competitive
tion V, we have shown the importanceof in- with the variances-only model. But the results
here suggest that such a model, if it is possi-

VOL.96 NO. 1 SIMSAND ZHA: WERETHEREREGIMESWITCHESIN U.S. MONETARYPOLICY? 77

ble at all, is not likely to succeed if it excludes ourpointestimatesimply thatthe impacton the
money from the reaction function. economy of changes in the systematic part of
monetarypolicy was not as big as commonly
VIII. Conclusion thought. Nonetheless, our estimates do imply
thata permanenrteserve-targetinpgolicy like that
Monetarypolicy andits historyarecomplex, of 1979-1982, or a policy thatgreatlyamplified
and abstracttheoreticalmodels that we use to the reactionof interestrates to inflation,could
organizethoughtaboutthem can hide what was
really going on. Explorationsof datawith rela- have keptinflationsubstantiallylower,while ex-
tively few preconceptions,like this exploration, actinga cost in loweroutputgrowth.
may bring out regularitiesthat have been slip-
ping through abstractdiscussion. In this case, In our estimatesthat enforce changes in pol-
we think this has happened. icy rule, the strongest evidence for monetary
policy change is that for shifting emphasis on
Our best-fit model suggests that neither ad- monetaryaggregatesin the policy reactionfunc-
ditive disturbances to a linear monetary pol- tion. This accordswith the prominentrole mon-
icy reaction function nor changes in the etarism played in policy discussions of the
coefficients of that function have been a pri- 1970s. If furtherresearch succeeds in finding
mary source of the rise and fall of inflation clear evidence of changes in monetarypolicy
over our sample period. Instead, stable mon- behavior in this period, it will most likely be
etary policy reactions to a changing array of throughfocusing attentionon the changingim-
major disturbances generated the historical pact of monetarismon policy behavior.
pattern. Oil price shocks and the Vietnam
War and its financing produced disturbances Policy actionswere difficultto predict,andif
in the 1960s and 1970s which have not recurred therewere shiftsin the systematiccomponentof
on such a scale since. With such a large role policy, they were of a sortthatis difficultfor us
assignedto "privatesector shocks,"it would be to track precisely, even with hindsight. While
useful to consider a model that allows more our results leave room for those with strong
detailed interpretationof these shocks. Recent beliefs that monetarypolicy changed substan-
work by Gambettiet al. (2005) is an attemptin tially to maintainthose beliefs, it is nonetheless
this direction. clear that whatever the changes, they were of
uncertaintiming, not permanent,and not easily
Even if one gives all the priorweight to the understood,even today.Models thattreatpolicy
four-statepolicy model, which assumes the ex- changes as permanent,nonstochastic,transpar-
istence of regime changes in monetarypolicy, ent regimechangesarenotuseful in understand-
ing this history.

APPENDIX:ESTIMATIONAND INFERENCE

A. The Prior.-The identification specified in Table 1 is a special case of standardlinear
restrictionsimposed on Ao and D as

a, = Uj b j= ... n,

nhX 1 nh X oj o X 1

mdhXj 1 = mVh xjr gj j=1 ... n,

rjx 1

aj [ aoj(h)1 d;=[ djh(1)
aoj(h) dj (h)

whereb-andgj arethefreeparameter"ssqueezedo"utof aj anddjby thelinearrestrictionso,jand

78 THEAMERICANECONOMICREVIEW MARCH2006

rj are the numbersof the correspondingfree parameters,columns of Uj are orthonormalvectors in

the Euclideanspace Fnh,and columns of Vj are orthonormalvectors in Rmh

The priordistributionsfor the free parametersbj andgj have the following Gaussianforms:

IT(b=j)N(0,Ho0),

SIT(=g1.)oN(H0,f+).

For all the models studiedin this paper,we set HojandHIj the same way as Sims andZha (1998)
but scale themby the numberof states(h) so thatthe Case I model in (8) coincides with the standard
Bayesian VAR with constantparameters.The liquidityeffect prioris implementedby adjustingthe
off-diagonal elements of Ho;that correspondto the coefficients of M and R forj = 2, 3 such that
the correlationfor the policy equation (the second equation) is -0.8 and the correlationfor the
money demandequation(the thirdequation)is 0.8. Because we use monthlydata,the tightnessof
the referenceprioris set as, in the notationof Sims and Zha (1998), A0 = 0.6, Al = 0.1, A2= 1.0,
C. Robertsonand Ellis W. Tallman,2001).
AO3(Tk=h) ea1np.2dr,iFoAr(4Od=i)sdte0rn.i1bo,ute/tLis5otnh=feor5s.to0aj,n(kad)naidsrdt/Lgaakme=nm5aa.s0p-r(ds(f(ejew(kJit)oh)h=Pn
F(ag, P ) for k E {1,..., h}, where oj(k)

being a scale factor (not an inverse scale
factoras in the notationof some textbooks).The priorpdf for Aij(k)is rN(0,o-2)for k E {1, ... , h}.
The priorof the transitionmatrixP takes a Dirichletform as suggested by Chib (1996). For the

kthcolumn of P, Pk,the priordensity is

7r(pk)= 7r(Plk, ... Phk) = D(alk, ... , ahk) p ... Ph

where aik > 0 for i = I,..., h.
The hyperparametersa , f3, and ax are newly introducedand have no reference values in the

literature.We set ag = P = 1 andax = 50 as the benchmarkandthen performa sensitivitycheck
by varying these values. The priorsetting ax = 50 is reasonablebecause the posteriorestimateof
Aij(k)can be as large as 40 or 50 even with a much smallervalue of o,.14
Therearetwo steps in settingup a priorfor Pk.First,the priormode of Pikis chosen to be viksuch
that Vkk= 0.95 and vik= 0.05/(h - 1) for i 0 k. Note that hi= vk = 1. In the second step, given
0.025), we solve for akkthrougha thirdpolynomialandthen for
vikand eVleamre(pnktsko)(fwthhiecvheicstsoertatko througha system of h - 1 linearequations.This priorexpresses
all other

the belief that the average durationof each state is about 20 months. We also experiencedwith
differentpriorvalues for P, includinga very diffuse priorfor P by letting vikbe evenly distributed
across i for given k andby lettingthe priorstandarddeviationof Pikbe much largerthan0.025. The
results seem insensitive to these priorvalues.

B. Posterior Estimate.-We gather different groups of free parametersas follows, with the
understandingthatwe sometimesinterchangethe use of free parametersandoriginal(butrestricted)
parameters.

p={pk, k= 1,..., h};

= {(j(k),j = 1, ..., n, k = 1, ..., h}, for Case II;
A= i,j = 1 k= for Case III;
{ii(k), 1, h},
..... n, ....,

g9={gjj= 1, ...., n};

14 Indeed, a tighterprioron Aij(k)tends to lower the marginallikelihood for the same model.

VOL.96 NO. 1 SIMSAND ZHA: WERETHEREREGIMESWITCHESIN U.S. MONETARYPOLICY? 79

b ={bj,j= 1,..., n};

0 =V{p,1, g b}.

The overalllikelihoodfunction 0) can be obtainedby integratingover unobservedstatesthe
conditional likelihood at each timter(tYaTnId by recursively multiplying these conditional likelihood

functions forward(Kim and Nelson, 1999).

From the Bayes rule, the posteriordistributionof 0 conditionalon the data is

7r(OlIY)oc'r(O)T(YTlo)

where the prior Tr(0)is specified in Section A of the Appendix above.
In orderto avoid very long startupperiods for the MCMCsampler,it is importantto begin with

at least an approximateestimate of the peak of the posteriordensity T(0IOYTM).oreover, such an
estimateis used as a referencepointin normalizationto obtainlikelihood-basedstatisticalinferences.
Because the numberof parametersis quite large for our models (over 500), we used an eclectic
approach,combining the stochastic expectation-maximizingalgorithmwith various optimization
routines.For some models, the convergencetook about 15 hourson an IntelPentium4 2.0 GHz PC;
for others, it took as long as a week.15

C. Inference.-Our objective is to obtain the posterior distributionof functions of 0 such as

impulseresponses,forecasts,historicaldecompositions,andlong-runresponsesof policy. It involves
integratingover large dimensionsmanyhighly nonlinearfunctions.We follow Sim and Zha (2004)
and uGsiebtbhsesGamibpblsersainmvpollverestosaombtpaliinntghaeljtoeirnntadtiivsetrlyibfruotmionth-re(0fo, lSloTwIYinTgw)choenrdeitSioTn=alp{osost,esr1i,o..r.,dissTtr}i-.
The

butions:

Pr(S, Yr,p, y, g, b),

Tr(IpYT,ST, Y,g, b),

7r( Yr, ST,P, g, b),

'r(g p, y, b),
Yr, S,t,

7r(bIY, ST,p, y, g).

It has been shownin the literaturethatsuch a Gibbssamplingprocedureproducesthe uniquelimiting
distributionthatis the posteriordistributionof S, and 0 (e.g., JohnGeweke, 1999). The probability
density functions of these conditional distributionsare quite complicated but can be nonetheless
simulated.

D. Normalization.-To obtainaccurateposteriordistributionsof functionsof 0 (such as long-run
responsesandhistoricaldecompositions)w, e mustnormalizeboththe signs of structuraelquationsand
thelabelsof states;otherwise,theposteriordistributionws ill be symmetricwithmultiplemodes,making
statisticalinferencesof interestmeaningless.Such normalizationis also necessaryto achieveefficiency

15 We are still improvingour algorithm.Once it is finished,it is possible that the computingtime could be considerably
reduced.

80 THEAMERICANECONOMICREVIEW MARCH2006

in evaluatingthemarginallikelihoodformodelcomparison.1F6orbothpurposes,we normalizethesigns
of structuraelquationsthe sameway. Specifically,we use thenormalizationruleof DanielF. Waggoner

andZha (2003) to determinethe columnsigns of Ao(k)andA+(k) for any given k E 11,..., h}.
Two additional normalizationsare (a) scale normalizationon oj(k) and Xj(k) and (b) label
normalizationon the states. We simulateMCMC posteriordraws of 0 with (j(k) = 1 andXj(k) =
'hX for all j E {1, ... , n}, and k E
h}, where the notation1hxl denotes the h x 1 vector
of l's. For each posteriordraw, we la{b1,e.l..t.,he states so that the posteriorprobabilitiesof each state

for all t Ef{ TJ matchclosest to the posteriorestimates of those probabilities.17

To estimat1e..t.h.,e marginaldatadensity7r(YTf)or each model, we applyboththe modifiedharmonic
meanmethod(MHM)of AlanE. GelfandandDepakK. Dey (1994) andthe methodof ChibandIvan
Jeliazkov(2001).TheMHMmethodis quiteefficientformostmodelsconsideredin thispaper,butit may
give unreliableestimatesfor some modelswhose posteriordistributionhs avemultiplemodes.In sucha
situation,we also use the ChibandJeliazkovmethodto checkthe robustnessof the estimate.

16 Note that the marginal data density is invariantto the way parametersare normalized, as long as the Jacobian
transformationsof the parametersare taken into accountexplicitly.

17 This label normalizationis a computationallyefficient way to approximateWald normalizationdiscussed by Hamilton

et al. (2004).

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