INTERNATIONALECONOMICREVIEW
Vol.45, No. 2, May2004
PRIORS FROM GENERAL EQUILIBRIUM MODELS FOR VARS*
BY MARCO DEL NEGRO AND FRANK SCHORFHEIDE1
Research Department, Federal Reserve Bank of Atlanta, US.A.;
Department of Economics, University of Pennsylvania, U.S.A.
Thisarticleuses a simpleNew Keynesiandynamicstochasticgeneralequilibriummodel as a priorfor a vectorautoregression,and showsthat the resulting
model is competitivewithstandardbenchmarksin termsof forecasting,andcan
be used for policyanalysis.
1.
INTRODUCTION
Dynamic stochastic general equilibrium (DSGE) models are popular nowadays
in macroeconomics. They are taught in virtually every economics Ph.D. program,
and represent a predominant share of publications in the field. Yet, when it comes
to policy making, these models are scarcely used-at least from a quantitative
point of view. The main quantitative workhorse for policy making at the Federal
Reserve System is FRB-US, a macroeconometric model built in the Cowles foundation tradition-a style of macroeconomics that is no longer taught in many Ph.D.
programs.2In their decision process, Fed policymakers rely heavily on forecasting.
They want to know the expected path of inflation in the next few quarters, and by
how much a 50 basis point increase in the federal funds rate would affect that path.
FRB-US offers answers to these questions-answers that many macroeconomists
would regard with suspicion given both the Lucas' (1976) critique and the fact that
in general the restrictions imposed by Cowles foundation models are at odds with
*
ManuscriptreceivedAugust2002;revisedMay2003.
1Partof thisresearchwasconductedwhilethe secondauthorwas
visitingtheResearchDepartment
of the FederalReserveBankof Atlanta,for whosehospitalityhe is grateful.Thesecondauthorgratefully acknowledgesfinancialsupportfrom the UniversityResearchFoundationof the Universityof
Pennsylvania.Wethanktwoanonymousreferees,SeanCampbell,FabioCanova,FrankDiebold,John
Geweke,Dave Gordon,ThomasLubik,JimNason, ChrisSims,Dan Waggoner,CharlesWhiteman,
Tao Zha, and seminarparticipantsat the 2003ASSA Meetings,Bank of Canada,BrownUniversity,
2002 CIRANO ForecastingConference,Duke University,EuropeanCentralBank,2002 European
EconometricSociety Meetings,FederalReserve Bank of Atlanta,2002 LACEA Meetings,Spring
2002MacroFederalReserveSystemMeetings,IndianaUniversity,2002MidwestMacroConference,
2002 SED Meetings,Universityof Pennsylvania,and UCSD for helpfulcommentsand discussions.
We would like to thankIskanderKaribzhanovfor translatingsome of our Matlabcode in C. The
viewsexpressedin this articledo not necessarilyreflectthose of the FederalReserveBankof Atlanta
or the FederalReserve System.Please addresscorrespondenceto: FrankSchorfheide,Department
of Economics,Universityof Pennsylvania,3718 LocustWalk,Philadelphia,PA 19104-6297.E-mail:
[email protected].
2 Some luckyfew, amongthem the authorsof this article,have had the privilegeof encountering
proponentsof this approachduringtheirgraduatestudies.
643
644
DEL NEGRO AND SCHORFHEIDE
dynamic general equilibrium macroeconomics (Sims, 1980). General equilibrium
models on the other hand have a hard time providing alternative answers. The
fact that these models are perceived to do badly in terms of forecasting, as they
are scarcely parameterized, is perhaps one of the reasons why they are not at the
forefront of policy making.
Although progress is being made in the development of DSGE models that
deliver acceptable forecasts, e.g., Smets and Wouters (2003), this article proposes
an approach that combines a stylized general equilibrium model with a vector
autoregression (VAR) to obtain a specification that both forecasts well and is
usable for policy analysis. Specifically, the approach involves using prior information coming from a DSGE model in the estimation of a VAR. Loosely speaking,
this prior can be thought of as the result of the following exercise: (i) simulate
time-series data from the DSGE model, (ii) fit a VAR to these data. In practice
we replace the sample moments of the simulated data by population moments
computed from the DSGE model solution. Since the DSGE model depends on
unknown structuralparameters, we use a hierarchical prior in our analysis by placing a distribution on the DSGE model parameters. A tightness parameter controls
the weight of the DSGE model prior relative to the weight of the actual sample.
Markov Chain Monte Carlo methods are used to generate draws from the joint
posterior distribution of the VAR and DSGE model parameters.
The article shows that the approach makes even a fairly stylized New Keynesian
DSGE model competitive with standard benchmarks in terms of forecasting real
output growth, inflation, and the nominal interest rate-the three variables that
are of most interest to policymakers.3 Up to this point our procedure borrows
from the work of DeJong et al. (1993) and Ingram and Whiteman (1994), who
were the first to use priors from DSGE models for VARs. Ingram and Whiteman
showed that prior information from the bare-bones stochastic growth model of
King et al. (1988) is helpful in forecasting real economic activity, such as output,
consumption, investment, and hours worked.
In addition to documenting the forecasting performance of a trivariate VAR
with a prior derived from a monetary DSGE model, this article makes two contributions that significantly extend the earlier work. First, we show formally how
posterior inference for the VAR parameters can be translated into posterior inference for DSGE model parameters. Second, we propose procedures to conduct
two types of policy experiments within our framework. The first policy analysis is
based on identified VAR impulse responses to monetary policy shocks. To obtain
identification we construct an orthonormal matrix from the VAR approximation
3 Ireland
(2004) pursues a similar goal by augmenting a linearized DSGE model with unobservable
errors, following Sargent (1989) and Altug (1989). Unlike in these earlier articles, Ireland does not
restrict the errors to be uncorrelated across variables ("measurement errors"). Instead he allows them
to follow a stationary VAR. Although the resulting hybrid model is suitable for forecasting, Ireland
provides no general discussion about the extent to which the latent errors are distinguishable from the
structural shocks (identification) and no framework for policy analysis. Robertson et al. (2002) also
provide an alternative approach, based on relative entropy, to incorporate moment restrictions from
economic theory into a model's forecasts.
PRIORSFROMDSGE MODELSFOR VARS
645
of the DSGE model to map the reduced-forminnovationsinto structuralshocks.
Thisprocedureinducesa DSGE model-basedpriordistributionfor the VAR impulse responses,whichis updatedwith the sampleinformation.
The second policy experimentis more ambitious.We want to forecastthe effects of a change in the policy rule. Suppose a policymakerhas a DSGE model
that forecastspoorly and a VAR model that is silent about the effects of a policy
regime change.In order to assess the effects of a policy interventionaccurately,
one has to correct the DSGE model predictionsto bring its baseline forecasts
underthe existingpolicyregimeon track.Based on the VAR with DSGE model
prior (DSGE-VAR) we propose to implementsuch a correction.Since we lack
knowledgeabout the relationshipbetween policyparametersand DSGE model
misspecification,we assumethat the pre-interventioncorrectionremainsappropriateunderthe new policy regime.To illustrateour approachwe try to forecast
the impactof the changefrom the Martin-Burns-Millerregimeto the VolckerGreenspanregime on the volatilityof inflationand compareit to the variability
of actualinflationin the Volcker-Greenspanperiod.
The articleis organizedas follows.Section2 containsa briefdescriptionof the
DSGE model that we are usingto constructthe priordistribution.Section3 discussesthe specificationof the DSGE model priorandexploresthe joint posterior
distributionof VAR andDSGE modelparametersfroma theoreticalperspective.
Empiricalresults for a VAR in output growth,inflation,and interest rates are
presentedin Section4. Section5 concludes.Proofsand computationaldetailsare
providedin the Appendix.
2.
A SIMPLE MONETARY DSGE MODEL
Oureconometricprocedureis appliedto a trivariateVAR for output,inflation,
and interest rates. The prior distributionfor the VAR is derived from a New
KeynesianDSGE model. To make this articleself-contained,we brieflyreview
the model specification,which is adopted from Lubik and Schorfheide(2004).
Detailed derivationscanbe found,for instance,in KingandWolman(1999),King
(2000),and Woodford(2003).
The model economy consists of a representativehousehold, a continuumof
monopolisticallycompetitivefirms,and a monetarypolicy authoritythat adjusts
the nominalinterestrate in response to deviationsof inflationand output from
theirtargets.TherepresentativehouseholdderivesutilityfromrealbalancesM/ P
and consumptionsC relativeto a habit stock. We assumethat the habit stock is
given by the level of technologyA. This assumptionassuresthat the economy
evolvesalonga balancedgrowthpath.Thehouseholdderivesdisutilityfromhours
worked,h, and maximizes
(i)
s
*E^t
sE,+tX
i -((CAs
-
1
log
-hs
Ms
~s
646
DEL NEGRO AND SCHORFHEIDE
where / is the discountfactor, r is the risk aversionparameter,and X is a scale
factor. P is the economy-widenominal price level that the household takes as
given. The (gross) inflation rate is defined as Tt = Pt/Pt-i.
The householdsuppliesperfectlyelastic labor servicesto the firmsperiod by
period and receives the real wage W. The household has access to a domestic
capitalmarketwhere nominalgovernmentbonds B are tradedthat pay (gross)
interestR. Furthermore,it receives aggregateresidualprofitsD from the firms
and has to pay lump-sumtaxes T. Thus,the households'budget constraintis of
the form
(2)
C, +
Bt
Pt
+
Mt
Pt
__t
W
Pt
+
Mt -
+ Rt
Pt
-_
Pt
+ Dt
The usualtransversalityconditionon assetaccumulationrulesout Ponzischemes.
The productionsector is describedby a continuumof monopolisticallycomdemandcurvefor its differentiated
petitivefirmseach facinga downward-sloping
product
(3)
)
(Ptt=0
=
-1/ Pt
( Xt(i)
xtu}X
This demandfunctioncan be derivedin the usual way from Dixit-Stiglitzpreferences,wherebyPt(j) is the profit-maximizing
price consistentwith production
level Xt(j). The parameter v is the elasticity of substitution between two differ-
entiatedgoods. The aggregateprice level and aggregatedemandXt are beyond
the controlof the individualfirm.Nominalrigidityis introducedby assumingthat
firmsface quadraticadjustmentcosts in nominal prices.When a firm wants to
changeits price beyond the economy-wideinflationrate Ir*, it incursmenu costs
in the form of lost output
T(hPt(()t(j) -vr
t
The parameter (p> 0 governs the degree of stickiness in the economy.
Productionis assumedto be linear in labor ht(j), which each firmhires from
the household
(5)
Xt(j) = Atht(j)
TotalfactorproductivityAt is an exogenouslygiven unit root processof the form
(6)
In At = In y + InAt1 +
where
(7)
Zt = PzZt-1 + Cz,t
Zt
647
PRIORSFROMDSGE MODELSFOR VARS
and E(,t can be broadlyinterpretedas a technologyshock that affects all firms
in the same way.The specificationof the technologyprocessinducesa stochastic
trendinto the model.4
Firmj chooses its labor input ht(j) and price Pt(j) to maximizethe present
value of futureprofits
00
Et EQsDs(i)
(8)
s=t
subjectto (5) and (6), where the time s profitis given by
- Whs(j))
)
P (J)s(j
PS
Ds(j) =
2Ps-l(j)
Xs)(j--)
Here Q is the time-dependentdiscountfactor that firmsuse to evaluate future
profit streams.Although firmsare heterogenousex ante, we only consider the
symmetricequilibriumin which all firmsbehave identicallyand can be aggregated into a single representativemonopolisticallycompetitivefirm.Under the
assumptionthat households have access to a complete set of state-contingent
claims Qt+l/Qt
in equilibrium. Since the household is the recipi-
= P(Ct/Ct+l)T
ent of the firms'residualpaymentsit directsfirmsto make decisionsbased on the
household'sintertemporalrate of substitution.
The centralbankfollowsa nominalinterestrateruleby adjustingits instrument
in responseto deviationsof inflationandoutputfromtheirrespectivetargetlevels
(9)
(9)
gt
(gt-1)P
R
R-
R
R*
\f)R-(
)
't)~l(m_~t,2-pR)
(7)
*)
( Xt)
Xt*
e(Rt
e
where R* is the steady-statenominalrate and Xt is potentialoutput,whichwe
define as X* = At after normalizingthe steady-statehours worked to one. The
parameter0 < PR < 1 determinesthe degreeof interestratesmoothing.Thecentral
bank supplies the money demandedby the household to support the desired
nominalinterest rate. The monetarypolicy shock ER,t can be interpretedas an
unanticipateddeviationfromthe policyrule.
The governmentconsumesa fraction(t of each individualgood j. We define
gt = 1/(1 - (t) and assumethatgt = ln(gt/g*) followsa stationaryAR(1) process
(10)
gt = Pggt-1 + Eg,t
where (g,t can be broadlyinterpretedas governmentspendingshock. The governmentlevies a lump-sumtax (or subsidy) Tt/Pt to financeany shortfallin governmentrevenues(or to rebate any surplus)
4 Since our simple DSGE model lacks an internal propagation mechanism that can generate serially
correlated output growth rates we assume that technology growth follows a stationary AR(1) process.
648
("1)
DEL NEGRO AND SCHORFHEIDE
'
tXt + Rt-l
- =
+
+ Bt
Pt
Pt
Pt
Pt
t +
Pt
The fiscal authority accommodates the monetary policy of the central bank and
endogenously adjusts the primary surplus to changes in the government's outstanding liabilities.
To solve the model, optimality conditions are derived for the maximization
problems. Consumption, output, wages, and the marginal utility of consumption
are detrended by the total factor productivity At. The model has a deterministic
steady state in terms of the detrended variables. Define the percentage deviation
of a variable yt from its trend Yt*as 5t = In Y - In t*. The log-linearized system
can be reduced to three equations in output, inflation, and nominal interest rates
-
Et[t+1i]) + (1 - pg)gt + Pz-Zt
(12)
Xt = Et[Xt+1]-
(13)
fit = -1Et [tt+l] + K[Xt- gt]
(14)
Rt = pRtt-1 + (1 - PR) (4lft
Tr-1(R
+
22Xt) + ER,t
where r* = y/B is the steady-state real interest rate and K is a function of the
price adjustment costs and the demand elasticity. The parameter measures the
overall degree of distortion in the economy. Equation (12) is an intertemporal
consumption Euler equation, whereas Equation (13) is derived from the firms'
optimal price-setting problem and governs inflation dynamics. Equation (14) is
the log-linearized monetary policy rule.5 The linear rational expectations system
given by Equations (7), (10), and (12)-(14) can be solved, for instance, with the
algorithm described in Sims (2002).
The relationship between the steady-state deviations and observable output
growth, inflation, and interest rates is given by the following measurement equations
(15)
A lnxt = In y + Ait + 5t
A In P = Inr* +
it
In R' =4[(lnr* + ln *) + Rt]
In the subsequent empirical analysis a period t corresponds to one quarter. Output
growth and inflation are quarter-to-quarterchanges, whereas the interest rate, Rt,
is annualized. The evolution of Xt, ft, , ,and Ztis given by the solution to the linear
rational expectations system given above. The DSGE model has three structural
5 In this article we restrict the
parameter space to values that lead to a unique stable solution of the
linear rational expectations system. Lubik and Schorfheide (2004) discuss the econometric analysis
of linear rational expectations models when the parameter space is not restricted to the determinacy
region.
649
PRIORSFROMDSGE MODELSFOR VARS
shocks that we collect in the vector Et = [ER,t,Eg,t, E,t]'. We assume that the
shocks are normallydistributedand independentof each other and over time.
Their standard deviations are denoted by a R, gg,and a z, respectively. The DSGE
model parametersare stackedinto the vector
0 = [ n y, In r*, Inr*, K, T,Og, 1l, *2, PR, Pg,
(16)
Pz,
R,
Oz]/
The DSGE model imposes tight restrictions across the parameters of the movingaverage (MA) representation for output growth, inflation, and interest rates.
3.
USING THE DSGE MODEL TO OBTAIN A PRIOR FOR A VAR
Let Yt = [Aln xt, Aln Pt, In Rt]. A less restrictive MA representation for the
n x 1 vector Ytthan the one implied by the DSGE model of the previous section
can be obtainedfrom a VAR model
(17)
Yt = Oo +
lYt-i +
* ( pYt-p + Ut
where ut is a vector of one-step-ahead forecast errors. VARs have a long tradition
in applied macroeconomics as a tool for forecasting, policy analysis, and business
cycle analysis. One drawback of VARs is that they are not very parsimonious:
In many applications, data availability poses a serious constraint on the number
of endogenous variables and the number of lags that can effectively be included
in a VAR without overfitting the data. A solution to this problem of too many
parameters is to constrain the parameter estimates by "shrinking"them toward a
specific point in the parameter space.6 For instance, Doan et al. (1984) proposed to
shrink the estimates of a VAR that included the level of output, and prices, toward
univariaterandomwalk representations.The justificationfor this shrinkagewas
statistical rather than economic: No-change forecasts implied by the random walk
model were known to be fairly accurate. Our method shrinks the VAR estimates
toward restrictions that are derived from a DSGE model. The spirit of our ap-
proachis to take the DSGE model restrictionsseriouslywithoutimposingthem
dogmatically.
Shrinkage estimators can be interpreted as Bayes estimators that are derived
from prior distributions that concentrate much of their probability mass near the
desired parameter restrictions. Priors are a way of systematically adding observations. In fact, many prior distributions can be incorporated into the estimation
by augmenting the actual data set with dummy observations.7 Loosely speaking,
we implement the DSGE model prior by generating dummy observations from
the DSGE model, and adding them to the actual data. The ratio of dummy over
actual observations-which we call X in the remainder of the article-measures
the weight of the prior relative to the sample.
6
Shrinkage estimators have a long tradition in statistics, see, e.g., Lehmann (1983).
7 This so-called mixed estimation is
due to Theil and
originally
instance, by Sims and Zha (1998).
Goldberger (1961), and used, for
650
DEL NEGRO AND SCHORFHEIDE
The DSGE modelis indexedby a vector0 of so-calleddeep parameters.Instead
of conditioningour analysison a specificvalue of 0 we place a priordistribution
on the DSGE model parameters.In the econometricparlance,we constructa
hierarchicalprior consistingof a marginaldistributionfor 0 and a conditional
distributionfor the VAR parametersgiven 0, which is generated throughthe
dummyobservations.Bayes theoremthen leadsus to a jointposteriordistribution
for the DSGE model and VAR parameters.Our implementationof the DSGE
modelpriorhasanimportantadvantageoverpreviousapproaches.Welearnabout
the DSGE model parametersby implicitlysearchingfor valuesof 0 for whichthe
distancebetweenthe VAR estimateandthe vectorautoregressiverepresentation
of the DSGE model is small.This informationabout 0 will be exploited for the
policy analysisin Section4.
We have just given an informal descriptionof our procedure.Much of the
remainderof this section-for the econometricallyorientedreader-provides an
accurateanddetailedexposition.In Section3.1we discussthe likelihoodfunction.
Section3.2 describesthe priordistribution.The posteriordistributionis analyzed
in Section3.3.
3.1. LikelihoodFunction. In order to constructa likelihoodfunctionwe assume that the innovationsut in Equation(17) have a multivariatenormaldistributionJA(0,Eu)conditionalon past observationsof Yt.Let Y be the T x n matrix
with rows y'. Let k = 1 + np, Xbe the T x k matrix with rows xt = [1, y-,...,
p]'. The VAR
y_tp], U be the T x n matrix with rows u', and E = [Vo, 1i,...,
can be expressedas Y = X >+ U with likelihoodfunction
(18)
p(Y
y),Eu)
o I-
T/2exp
l -tr[Eul1(Y'Y-
Y'X'
Y- Y
'X
+
'X'X)]
conditionalon observationsy1-p,..., yo. Although the DSGE model presented
in Section 2 does not have a finite-ordervector autoregressiverepresentationin
terms of Yt,the VAR can be interpretedas an approximationof the MA representation of the DSGE model. The magnitudeof the discrepancybecomes
smalleras more lags are includedin the VAR. Since0 is of muchlowerdimension
than the VAR parametervector, the DSGE model imposes restrictionson the
(approximate)vector autoregressiverepresentationof yt.
3.2. PriorDistribution. Supposethe actualobservationsare augmentedwith
T* = T artificialobservations(Y*, X*) generatedfromthe DSGE model based
on the parametervector 0. The likelihoodfunctionfor the combinedsample of
artificialand actualobservationsis obtainedby premultiplying(18) with
(19) p(Y*(O) I D, E)
IEu-T/2 exp j-tr[u
('* (Y*Y
2~ u ~
-~ Y
Y
*
~X- ~
* + ^>XX'
~ ~X*X~] )]}
651
PRIORSFROMDSGE MODELSFOR VARS
The factorization
p(Y*(o), Yj ,C,)
(20)
= p(Y*(0) jI , E,)p(YI *,
U,)
suggeststhat the term p(Y*(O)1 p, Su) can be interpretedas a priordensityfor
'P and Eu. It summarizesthe informationabout the VAR parameterscontained
in the sampleof artificialobservations.
If we in fact were to constructour priorby generatingrandomdrawsfromthe
DSGE model, a repeated applicationof our procedurewould lead to stochastic
variationin the prior distribution,which is undesirable.In order to remove the
stochasticvariationfrom p(Y*(O)Ia, Su) we replacethe nonstandardizedsample moments
*'*,
Y*' X*, and X"X* by their expected values. According to
our DSGE model, the vector Ytis covariancestationaryand the expectedvalues
of the samplemomentsare given by the (scaled)populationmoments Tr*y(0),
ATrr(0), andATr f(0), where,for instance,rfyy() = Ee[ytyj].Sincethese populationmomentscanbe computedanalytically,ourprocedureis veryefficientfrom
the computationalstandpoint.Formally,the use of populationmomentsimplies
that we replaceEquation(19) with
(21)
p(P,
u
) = c-1 (0)lE
TE-(ry()
exp-tr[
-r;
- '
(0)
(0)0 + cr*x(0)4)]
where we also addedan initialimproperpriorp(QI,Su) x I I-(n+1)/2.
T > k + n and rxx(0)is invertible,the priordensityis properand
Providedthat T
In
nondegenerate. this case the normalizationfactorc(0) can be chosento ensure
that the densityintegratesto one (the formulafor c(0) is givenin the Appendix).
Define the functions
(22)
*(0) = rFx
(23)
*(0)= r*(f) - r*
y()r*-(o)r()
(0)r()
Conditionalon 0 the prior distributionof the VAR parameters(21) is of the
Inverted-Wishart
(ZW) - Normal(.A) form8
(24)
(25)
8 If
- k,
, 10 - ZW(AT.TE*(), ,T
n)
|I u 0
, 7(*(0), Eu? (ATrx(0)) 1)
r'x(0) is of reduced rank, then the prior concentrates its mass in a lower dimensional subspace of
the domain of a. In our application the number of structural shocks equals the number of endogenous
variables n to which the model is fitted and Frx() is full rank. For DSGE models with less than n
structural shocks we recommend to ensure that rFx(0) is full rank by introducing additional shocks or
measurement errors.
652
DEL NEGRO AND SCHORFHEIDE
The specification of the prior is completed with a distribution of the DSGE model
parameters, details of which we discuss in Section 4. Overall our prior has the
hierarchical structure
p(1, E,, 0) = p(<, E, IO)p(0)
(26)
The functions V*(0) and 12(0) trace out a subspace of the VAR parameter
space and can be interpreted as follows. Suppose that data are generated from a
DSGE model with parameters 0. Among the pth-order VARs the one with the
coefficient matrix 1I*(0) minimizes the one-step-ahead quadratic forecast error
loss. The corresponding forecast error covariance matrix is given by XS(0).
Our prior is designed to assign probability mass outside of the subspace traced
out by (*(0) and S*(0).9 We use the covariance matrix ES(0) 0 (XTr7x(0))-1 to
distribute probability mass around I*(0) and average over 0 with respect to a prior
p(O). The orientation of the prior contours is such that the prior is fairly diffuse in
the directions of the VAR parameter space that we expect to estimate imprecisely
according to the DSGE model. Our prior differs from the one used by DeJong
et al. (1993), who used a simulation procedure to approximate (in our notation)
the marginal prior for the VAR coefficients p(I, E) = fp((,
,, IO)p(O)dOby a
conjugate 1W-1A/ prior.
3.3. Posterior Distribution. In order to study the posterior distribution we factorize it into the posterior density of the VAR parameters given the DSGE model
parameters and the marginal posterior density of the DSGE model parameters
p(<, E,, 0 1Y) = p((, E, IY, )p(O l Y)
(27)
Let (>(0) and E,(0) be the maximum-likelihood estimates of ? and Eu, respectively, based on artificial sample and actual sample
(28)
t(0)
l()
(29)
= (xTr;x(0) + X'X)- (xTry + X'Y)
= (X + )T[( Ty(0)
+ Y'Y)
- (iTr
(O) + Y'X)(XTF* (e)
+
'1( TVr(0) +
X'Y)]
'x)
Since conditional on 0 the DSGE model prior and the likelihood function are
conjugate, it is straightforward to show, e.g., like Zellner (1971), that the posterior
distribution of 1 and E is also of the Inverted Wishart-Normal form
9
Ingram and Whiteman (1994) used a Gaussian prior for the DSGE model parameters 0 - A(0, V )
and approximated the function I*(0) equation by equation with a first-order Taylor series around the
prior mean 0 to induce a prior distribution for the VAR parameters.
PRIORSFROMDSGE MODELSFOR VARS
(30)
(31)
653
E IY,0 - ZW((X+ 1)TX((0), (1 + X)T- k, n)
i IY,
xu, 0 - A;((0),
SuX (xTrxx(0) + X'x)-1)
The formula for the marginal posterior density of 0 and the description of a
Markov Chain Monte Carlo algorithm that generates draws from the joint posterior of 0, S,, and 0 are provided in the Appendix. The ability to compute the population moments r* (0), Fry(0), and rlx(0) analytically from the log-linearized
solution to the DSGE model and the use of conjugate priors for the VAR parameters makes the approach very efficient from a computational point of view:
25,000 draws from the posterior distribution of all the items of interest-including
forecast paths and impulse responses-can be obtained in less than 10 minutes
using a 1.2 GHz PC. In the remainder of this section we discuss the choice of X
and the source of information about the DSGE model parameters.
3.3.1. Choice of L. The hyperparameter X determines the effective sample
size for the artificial observations, which is AT. If A.is small the prior is diffuse,
and the actual observations dominate the artificial observations in the posterior.
According to Equation (28), the posterior mean of ? conditional on 0 equals the
OLS estimate of I if . = 0.
For large values of X the prior concentrates along the restriction functions 0* (0)
and XS(0). We will show that as the length of the artificial sample increases the
VAR parameter estimates will stay closer to the restrictions implied by the DSGE
model. In the limit our procedure is equivalent to estimating the VAR subject to
the DSGE model restrictions. Formally,as A -> oo, conditional on 0 the posterior
mean 4(0) approaches c*(0) and the variance Su(0) goes to zero. However, this
result does not imply that the actual observations have no influence on the overall
posterior distribution. An examination of the marginal posterior of the DSGE
model parameters can clarify this point.
The posterior p(0 I Y) can be obtained by combining the marginal-likelihood
function
(32)
p(YI 0)= f p(YI c, Eu)p(),
,uI0)d(), Eu)
with the prior p(O) (see Appendix). If T is fixed and A tends to infinity the
marginal-likelihoodfunctionp(YI 0) approachesthe (quasi)-likelihoodfunction
of the DSGE model10
(33) p*(y0)
Eu*(0)
l /exp -2tr[ Eu*(0)(Y- X*())'(Y-
X*(O))J
10 Since the DSGE model
typically does not have a finite-order VAR specification p*(YI 0) is a
quasi-likelihood function from the perspective of the structural model.
654
DEL NEGRO AND SCHORFHEIDE
The function p*(YI 0) is obtained by replacing the unrestricted parameters 0 and
Eu in Equation (18) with the restriction functions 1*(0) and S,(0). The result is
summarized in the following proposition that is proved in the Appendix.
1.
PROPOSITION
Let 0 be the mode of p(Y 0). For a fixed set of observations Y,
ln
P(Y 0)
p(YI )
> ln P*(Y I0)
p*(YI0)
uniformlyfor in compact subsets of
as
oo
X
for which S (0) and r* (0) are nonsingular.
Not surprisingly,the empirical performance of a VAR with DSGE model prior
will crucially depend on the choice of X.We use a data-driven procedure to determine an appropriate value X of the hyperparameter. We maximize the marginal
data density
(34)
px(Y)=
J
P(YI 0)p(0) dO
with respect to X over some grid A = {l1,...,lq}. Rather than averaging our
conclusions about all possible values of ., we condition on the value X. with the
highest posterior probability.This marginal data density can also be used to choose
an appropriate lag length for the VAR.
3.3.2. Learning about the DSGE model parameters. A major improvement
of our procedure over earlier approaches is that it enables posterior inference
with respect to the DSGE model parameters, whereas previous approaches only
delivered a posterior for the VAR parameters. Although the likelihood function
(18) itself does not directly depend on the DSGE model parameters, the marginal
distribution of 0 will nevertheless be updated through the sample information.
Similar to Smith (1993), an estimate of 0 is implicitly obtained by projecting the
VAR estimates onto the restrictions implied by the DSGE model.
The joint posterior density can be written as
(35)
p(0, Eu, 0 I Y) = p(DP,Eu I Y)p(0 I D, Su)
Learning about 0 takes place indirectly through learning about the VAR
parameters.
We expect that the best fit of the VAR model is achieved for values of X that
allow moderate deviations from the DSGE model restrictions. Define the function
(36) q(IY)=
expf-
ln ~u
(0)l -tr[E^
-1 tr[umle
2 m[Eem:*(O)
--
()]
-r
e)
Xme
()()(*(O)
-
4)mle)]}
655
PRIORSFROMDSGE MODELSFOR VARS
where
"mle and Su,mlemaximizethe likelihood
function (18). Using a second-order
Taylorexpansion,it can be shownthat the logarithmof q(OIY) is approximately
a quadraticfunctionof the discrepancybetween the VAR estimatesand the restrictionfunctionsgeneratedfromthe DSGE model.Forlong samplesand under
moderatelytightpriorsthe marginallog-likelihoodfunctioncan be approximated
as follows:
2. Let 0 be the mode of p(YI 0). Suppose T
PROPOSITION
T --- oo. Then
1
-
XT
In
q(YJO)
p(YI O)
) + Op(max[( T)-',
p
=l In q(y
p(Y\0)
> oo, X -.
0, and
])
q(Y(0)
The approximation holds uniformly for 0 in compact subsets of ( for which S2 (0)
and rFx(0) are nonsingular.
The intuitionfor this resultis the following.The weightof the priorrelativeto
the likelihoodfunctionis small (X -> 0), so that for all values of 0 the posterior
distributionof the VAR parametersconcentratesaround mle. The conditional
densityof 0 given (c and Euprojects >mleonto the subspace >*(0).The amountof
informationaccumulatedin the marginallikelihoodp(YI 0) relativeto the prior
depends on the rate at which XT diverges.The more weight is placed on the
artificialobservationsfrom the DSGE model (X convergesto zero slowly), the
more curvatureand informationthere is in p(YI 0). Accordingto Proposition2
the posteriorestimate of 0 can be interpretedas a minimumdistanceestimate
(e.g., Chamberlain,1984 and Moon and Schorfheide,2002) that is obtained by
minimizingthe weighted discrepancybetween the unrestrictedVAR estimates
4)mle and the restrictionfunction I*(0).
4.
EMPIRICAL APPLICATION
Thissectiondescribesthe resultsobtainedwhenwe applythe priorfromthe New
Keynesianmodeldescribedin Section2 on a trivariateVAR in realoutputgrowth,
inflation,andinterestrates.Weuse quarterlydata,andthe lag lengthin the VAR is
four quarters.11
Section4 consistsof fourparts.Section4.1 discussesthe priorand
DSGE model parameters.Section 4.2 describesthe forecasting
for
the
posterior
results.We show that the DSGE model priorleads to a substantialimprovement
relativeto an unrestrictedVAR andin severaldimensionsdominatesa VAR with
Minnesotaprior(Minn-VAR).
11The data for real output growth come from the Bureau of Economic Analysis (Gross Domestic
Product-SAAR, Billions Chained 1996$). The data for inflation come from the Bureau of Labor Statistics (CPI-U: All Items, seasonally adjusted, 1982-1984 = 100). The interest rate series are constructed
as in Clarida et al. (2000): For each quarter the interest rate is computed as the average federal funds
rate ( source: Haver Analytics) during the first month of the quarter, including business days only. Our
sample ranges from 1955:111to 2001:111.
656
DEL NEGRO AND SCHORFHEIDE
Policy analysis with a DSGE-VAR is discussed in Sections 4.3 and 4.4. First,
we construct impulse response functions to study the effects of modest interventions (Leeper and Zha, 2003) in terms of unanticipated deviations from the
monetary policy reaction function. The DSGE model is used to obtain an identification scheme for the VAR. Second, we use the DSGE-VAR to predict the
effects of a policy rule change. Whereas in the context of VARs the analysis
of regime changes is generally subject to the Lucas' critique, our approach can
be seen as a weighted average of two extremes: (i) using the DSGE model to
forecast the effects of the policy change (X = oo), and (ii) using the VAR to
make forecasts (X = 0), thereby ignoring the effects of the policy intervention.
In our framework, the choice of the prior weight X reflects the degree of misspecification of the structural model. We try to predict the impact of the change
from the Martin-Burns-Miller regime to the Volcker-Greenspan regime using the
DSGE-VAR. The results suggest that the approach is promising, at least in some
dimensions.
4.1. Prior and Posterior of 0. All empirical results are generated with the
prior distribution reported in Table 1. The model parameters In y, In r*, In r*, o R,
ag, and Oa are scaled by 100 to convert their units into percentages. The priors
for the quarterly steady-state growth rate, inflation rate, and real interest rate are
fairly diffuse and have means of 0.5%, 1.0%, and 0.5%, respectively. With 90%
prior probability the risk aversion parameter r is between 1.2 and 2.8, whereas
the slope of the Phillips curve K is between 0.06 and 0.51. The latter interval
TABLE1
PRIOR DISTRIBUTIONS FOR DSGE MODEL PARAMETERS
Name
Range
In y
In 7r*
In r*
JR
JR
IR+
/R+
JR+
IR+
/R+
[0,1)
[0.1)
[0,1)
JR+
R+
R+
K
r
lrJ
f2
PR
Pg
Pz
CrR
ag
aoz
Density
Normal
Normal
Gamma
Gamma
Gamma
Gamma
Gamma
Beta
Beta
Beta
Inv. Gamma
Inv. Gamma
Inv. Gamma
Mean
SD
0.500
1.000
0.500
0.300
2.000
1.500
0.125
0.500
0.800
0.300
0.251
0.630
0.875
0.250
0.500
0.250
0.150
0.500
0.250
0.100
0.200
0.100
0.100
0.139
0.323
0.430
NOTES:
The model parameters In y, In 7r*,In r*, rR, og, and orzare scaled by
100 to convert them into percentages. The Inverse Gamma priors are of the
form p(a Iv, s) oc c-v-le-s
/2a,2 where v = 4 and s equals 0.2, 0.5, and 0.7,
respectively. Approximately 1.5% of the prior mass lies in the indeterminacy
region of the parameter space. The prior is truncated in order to restrict it to
the determinacy region of the DSGE model (SD is standard deviation).
657
PRIORSFROMDSGE MODELSFOR VARS
TABLE 2
POSTERIOR
OF DSGEMODELPARAMETERS:
1959:III-1979:II
Posterior, . = 1
Prior
Posterior,
Name
CI (Low)
CI (High)
CI (Low)
CI (High)
CI (Low)
In y
In r*
In r*
K
0.101
0.219
0.132
0.063
1.197
1.121
0.001
0.157
0.922
1.863
0.880
0.513
2.788
1.910
0.260
0.812
0.473
0.433
0.113
0.101
1.336
1.011
0.120
0.530
1.021
1.613
0.463
0.516
2.816
1.559
0.497
0.756
0.616
0.553
0.126
0.081
1.684
1.009
0.150
0.550
T
Vf
'2
PR
=- 10
CI (High)
1.045
1.678
0.384
0.416
3.225
1.512
0.545
0.747
NOTES:We report 90% confidence intervals (CI) based on the output of the Metropolis-Hastings
Algorithm. The model parameters In y, In zr*, and In r* are scaled by 100 to convert them into
percentages.
is consistent with the values that have been used in calibrationexercises,e.g.,
Claridaet al. (2000).The priorsfor the policyparametersV1and 1V2are centered
at Taylor's(1999) values.12The prioris truncatedat the indeterminacyregion of
the parameterspace.
As stressedin Section 3, our procedurealso generatesposteriorestimatesfor
the DSGE model parameters.Such estimates are presented in Table 2 for the
To illustratethat the extent of learningabout 0
sample period 1959:III11-1979:II11.
X
of
on
the
the
DSGE
modelprior,Table2 reports90%posterior
weight
depends
confidencesets for X = 1 and X = 10. A comparisonof priorand posteriorintervals indicatesthat for X = 1 the data lead to a modest updating.The confidence
intervalsfor most parametersshrink.For instance,the priorconfidenceinterval
for the interestrate smoothingparameterPR rangesfrom 0.15 to 0.81, whereas
the posteriorintervalnarrowsto [0.21, 0.54]. The parameterthat characterizes
the responsivenessof the centralbank to inflationis reviseddownward,whereas
the confidenceintervalfor the output coefficientshifted upwards.The posterior
meansfor Vf1and1V2are1.3and0.3,respectively.Theupdatingis slightlymorepronouncedfor X= 10,when the artificialsamplesize is 10 timesas long as the actual
sample.
4.2. ForecastingResults. Theobjectiveof thissubsectionis to showthatVARs
with DSGE model priorsproduceforecaststhatimproveon those obtainedusing
unrestrictedVARs, and are competitivewith those obtained using the popular
Minnesotaprior.The Minnesotaprior shrinksthe VAR coefficientsto univariate unit root representations.Although it has been empiricallysuccessful,e.g.,
12Since the inflation rate and the interest rate in the DSGE model are
quarter to quarter, the value
of Vr2corresponds to one-fourth of the value obtained in univariate Taylor rule regressions that use
annualized interest rate and inflation data.
658
DEL NEGRO AND SCHORFHEIDE
Litterman (1986), Todd (1984), it lacks economic justification and ignores information with respect to co-movements of the endogenous variables.13
In a particular instance, this point has already been made by Ingram and
Whiteman (1994). However, we provide two extensions of their results. First,
we show that DSGE model priors can be helpful in forecasting not only real but
also nominal variables. Second, whereas Ingram and Whiteman document the ex
post forecasting performance of their DSGE-VAR as a function of the relative
weight of the prior, we also study the forecasting performance under the requirement that a hyperparameter X is chosen ex ante for each forecast based on the
marginal posterior data density p (Y). This is an important extension because the
forecasting performance of the VAR is sensitive to X and it has to be guaranteed
that a good X can be chosen before the actual forecast errors become available.
Most of the remainder of the section will present results from a forecasting
exercise using a rolling sample from 1975:1 to 1997:111(90 periods). For each
date in the forecasting interval we used 80 observations in order to estimate the
VAR, that is, a ratio of data to parameters of about 6 to 1 for each equation. This
choice is motivated by the fact that the data-parameter ratio in larger models that
are being used for actual forecasting, such as the Atlanta Fed VAR, is of the same
magnitude. It is important to remark that the results presented in this section have
no pretense of being general: They are specific to the particular DSGE model, and
the particular VAR being estimated.
We begin by investigating how the forecasting performance of the VAR changes
as a function of X and the forecast horizon. Figure 1 plots the percentage improvement (or loss, if negative) in root mean square forecast errors (rmse) of
the DSGE-VAR relative to the unrestricted VAR for cumulative real output
growth, cumulative inflation, and the federal funds rate.14 The fourth panel of
Figure 1 depicts the improvements in the multivariate forecasting performance
statistic proposed by Doan et al. (1984).15The grid of values of Xranges from 0 to
13In this article the Minnesota
prior is implemented as
= (In ? (X,XT) + tHml)) (vec(XYrT) + tHm1m)
where the parameter t denotes the weight of the Minnesota prior, ?m is the prior mean, and Hm is the
prior tightness. The values of )mand Hm are the same as in Doan et al. (1984), with the exception of
the prior mean for the first lag of output growth and inflation. Since these two variables enter the VAR
in growth rates, as opposed to log levels, to be consistent with the random walk hypothesis the prior
mean for the first lag of the "own" regressor in the output growth and inflation equations is 0 and not
1. The Minnesota prior is augmented by a proper ZW prior for Su. The weight of the Minnesota prior
is controlled by the hyperparameter l. The hyperparameter is selected ex ante using a modification of
Equation (34). This value hovers around 0.5, depending on the sample. The value used in Doan et al.
(1984) is t = 1.
14Neither the
output growth rates nor the inflation rates are annualized.
15 The
"ln-det" statistic is defined as the converse of the natural logarithm of the determinant of the
error covariance matrix of the forecasts, divided by two (to convert from variance to standard error)
times the number of variables that are forecasted (to obtain an average figure). The improvement
in the multivariate forecasting performance statistics is computed by taking the difference between
the multivariate statistics multiplied by 100 to obtain percentage figures. This number can be seen as
the average in the improvements for the individual variables, adjusted to take into account the joint
forecasting performance, i.e., the correlation in forecast errors.
659
PRIORSFROM DSGE MODELSFOR VARS
inflation
real GDP growth
20
..
o 10 .-
o0
0
1qua
quarters
multivariatestatistic
Fed Fund
20 -
10
o
0o 10-
5
0
quartersahead
.
4qua2
quartersahead
lnflO1
lnf 1.81
nf10
NOTES: The plot shows the percentage gain (loss) in RMSEs relative to an unrestricted VAR. The
rolling sample is 1975:III-1997:III (90 periods). At each date in the sample, 80 observations are used
in order to estimate the VAR.
FIGURE1
FORECASTING PERFORMANCE AS A FUNCTION OF THE WEIGHT OF THE PRIOR
oo, where X = oo means forecasting with the VAR approximation of the DSGE
model.
By definition the gain for X= 0 is zero. As the weight of the prior is increased we
observe a substantial gain in the forecast performance. The surface for the multivariate statistic is fairly flat for values of Xbetween 0.5 and 5, but then deteriorates
sharply as . approaches infinity. Overall our performance measures have an inverse U-shape as a function of the hyperparameter X. This indicates that there is
a benefit from shrinking the VAR estimates toward the DSGE model restrictions
without dogmatically imposing them. The ex post optimal X for long-run forecasts tends to be larger than for short-run forecasts. In order to obtain accurate
forecasts over long horizons one has to estimate powers of the autoregressive
coefficients (. The large sampling variance of these estimates can be reduced by
increasing the weight of the prior. However, once the length of the artificial saniple
relative to the actual sample exceeds 2, the variance reduction is dominated by an
increased bias and the forecasting accuracy generally deteriorates. Interestingly,
the deterioration is not sharp at all: In particular, for inflation and the interest rate
660
DEL NEGRO AND SCHORFHEIDE
TABLE3
PERCENTAGE GAIN (LOSS) IN RMSES: DSGE PRIOR VERSUS UNRESTRICTED VAR (V-UNR)
AND MINNESOTA PRIOR (V-MINN)
rGDP Growth
Inflation
Fed Funds
Multivariate
Horizon
V-unr
V-Minn
V-unr
V-Minn
V-unr
V-Minn
V-unr
V-Minn
1
2
4
6
8
10
12
14
16
17.335
16.977
15.057
14.116
12.387
14.418
15.078
16.236
19.122
1.072
6.965
5.803
3.452
4.230
7.986
12.512
17.233
21.575
8.389
7.247
8.761
10.460
11.481
12.261
12.626
12.995
13.238
1.653
1.339
4.767
7.240
7.794
8.351
9.011
9.634
10.116
7.250
5.024
5.008
6.648
8.420
8.242
6.404
6.059
5.823
-7.593
-4.895
-1.878
-0.713
-0.204
-0.639
0.726
1.146
2.389
12.842
10.993
9.630
10.388
11.023
12.864
12.419
12.611
13.428
0.943
2.884
3.959
4.290
5.187
6.463
7.537
8.481
9.512
NOTES:The rolling sample is 1975:111-1997:111
(90 periods). At each date in the sample, 80 observations
are used in order to estimate the VAR. The forecasts are computed based on the values X and l that
have the highest posterior probability based on the estimation sample.
the long-run forecasts from DSGE-VAR are still accurate even when the prior
weight is infinity.
Although Figure 1 documents that the DSGE model prior leads to an improved
forecasting performance, two questions remain. First, can an appropriate value of
A be selected ex ante? Second, how does the DSGE-VAR compare to a more
competitive forecasting model such as a VAR with Minnesota prior?
We argued in Section 3.3.1 that X can be chosen over a grid A to maximize the
marginal data density px(Y), given in Equation (34). This leads to the hyperparameter estimate
(37)
= arg max p (Y)
XeA
Depending on the sample, this value generally hovers around 0.6, which corresponds to 48 artificial observations from the DSGE model. However, the shape
of the marginal data density as a function of X is flat for values of A between 0.5
and 2, suggesting that the fit of the model is approximately the same within that
range. These estimates are broadly consistent with the value that leads to the best
ex post one-step-ahead forecast performance.16
Table 3 documents the rmse improvements of the DSGE-VAR based on A
relative to the unrestricted VAR and the Minn-VAR. The DSGE-VAR clearly
outperforms the unrestricted VAR, even if the weight of the prior is chosen ex
ante. The DSGE-VAR also dominates the Minn-VAR in terms of output growth
and inflation forecasts and according to the multivariate performance measure.
16A full
Bayesian procedure would average over . rather than condition on the highest posterior
probability X.However, in our experience the values of X that have nonnegligible posterior probability
produce very similar predictions so that the gain from averaging instead of conditioning is minimal.
PRIORSFROMDSGE MODELSFOR VARS
661
Although the improvement is small, around 1-1.5%, for short horizons, it is substantial for forecast horizons beyond two years. The DSGE-VAR forecasts of the
federal funds rate are unfortunately slightly worse than the Minn-VAR forecasts
for most of the horizons that we considered. Overall, these results suggest that the
DSGE-VAR is competitive with, and to some extent improves upon, the MinnVAR.17 When interpreting these results one has to bear in mind a key difference
between Minnesota and DSGE model prior. The Minnesota prior has only a statistical but not an economic justification. We will show subsequently that the DSGE
model prior can be exploited in the analysis of policy interventions.
4.3. Impulse Response Functions. In order to compute dynamic responses of
output, inflation, and interest rates to unanticipated changes in monetary policy
and to other structural shocks it is necessary to determine the mapping between
the structural shocks et and the one-step-ahead forecast errors ut. Let Str be
the Cholesky decomposition of Eu. It is well known that in any exactly identified structural VAR the relationship between ut and Et can be characterized as
follows:
(38)
Ut = EtrQEt
where Q is an orthonormal matrix and the structural shocks are from now on
standardized to have unit variance, that is E[te/]J = Z. According to Equation (17)
the initial impact of Et on the endogenous variables Ytin the VAR is given by
(39)
ayt) =V
\cEt/
VAR
The identification problem arises from the fact that the data are silent about the
choice of the rotation matrix 2. More prosaically, since Etr2Q2Q'tr = EtrEtr = Eu
the likelihood function is invariant to 2.
Macroeconomists generally require Q to have some ex ante justification and to
produce ex post impulse response functions that are "reasonable," i.e., conform
in one or more dimensions with the predictions of theoretical models. Since there
is no agreement on what these dimensions should be, a multitude of identification
strategies have been proposed (see Christiano et al., 1999, for a recent survey).
In this article we have used the DSGE model to derive a prior distribution for
the reduced-form VAR parameters. Hence, it is quite natural in our framework
to use the structural model also to identify the VAR. Thus, we will now construct
a rotation matrix Q2based on the dynamic equilibrium model.
The DSGE model itself is identified in the sense that for each value of 0 there is a
unique matrix Ao(O),obtained from the state space representation (15), that deter17We do not
report formal significance tests for superior forecast performance, such as the Diebold
and Mariano (1995) test, since the assumptions underlying those tests do not match the setup in our
article. Thus, the results should be interpreted as ex post accuracy comparisons, not as hypothesis tests.
Although not pursued here, Bayesian posterior odds could be used to choose among DSGE-VAR and
Minn-VAR ex ante.
662
DEL NEGRO AND SCHORFHEIDE
mines the contemporaneous effect of Et on Yt.Using a QR factorization of Ao(O),
the initial response of Ytto the structural shocks can be uniquely decomposed into
(40)
ay)
t
DSGE=
DSGE
Ao(O) =
(0)Q2*(O)
where ZEr(0) is lower triangular and Q*(0) is orthonormal. To identify the VAR,
we maintain the triangularization of its covariance matrix E" and replace the
rotation Q2in Equation (39) with the function Q*(0) that appears in (40).18
The implementation of this identification procedure is straightforward in our
framework. Since we are able to generate draws from the joint posterior distribution of D, EU, and 0, we can for each draw (i) use 1 to construct a MA representation of yt in terms of the reduced-form shocks ut, (ii) compute a Cholesky
decomposition of Eu, and (iii) calculate Q2= Q*(0) to obtain a MA representation
in terms of the structural shocks et.
A few remarks about the procedure are due. First, since the likelihood of the
reduced-form VAR is invariant with respect to Q, the rotation matrix that is used
to achieve identification is the same a posteriori as it is a priori for any fixed 0.
However, as explained in Section 3.3.2, the distribution of 0 is updated with the
sample information. Hence, we learn from the data which rotation to choose,
indirectly, via learning about the DSGE model parameters.
Second, the extent to which the posterior impulse responses are forced to look
like the DSGE model's responses will depend on the tightness of the prior. The
larger i, the more similar the responses will be. If the researcher chooses the
tightness of the prior endogenously based on the marginal data density (34) the
data-and not the researcher-will determine in which dimensions the posterior
impulse responses will conform to the model's responses, and in which dimensions
they will not.
Figure 2 depicts the impulse response functions with respect to monetary policy
shocks of cumulative real output growth, inflation, and the interest rate, normalized so that the initial impact of a monetary shock on the interest rate is 25 basis
points. Each plot shows the VAR impulse responses (dashed-and-dotted line),
the corresponding 90% error bands (dotted lines), and the DSGE model impulse
responses (solid lines). The estimates are based on a sample of 80 observations
ending in 2001:111.The impulse responses are computed for different values of
the tightness parameter A, namely A E {0.5, 1, 5}. As expected, the VAR impulse
responses become closer to the model's as the weight of the prior increases. Specifically, the distance between the posterior means of the VAR and the model's impulse responses decreases. In addition, the bands for the VAR impulse responses
narrow considerably.
It is interesting to observe that in some dimensions the VAR impulse responses
conform to the model's even for small values of the tightness parameter (X = 0.5).
18Since the VAR
representation of the DSGE model, as characterized by I*(0) and EZ(0), is only
an approximation, the Cholesky decomposition of E*(0) is not exactly equal to ETr(0).However, in
our experience the difference is for practical purposes negligible.
663
PRIORSFROMDSGE MODELSFOR VARS
real GDP growth-cumulative
40
Lo
inflation
Fed Fund
10
30
5
20
-
20
10 \
II
.
0
0
.
.
..
4
8
12
16
4
8
12
16
-15
*
o
-10 :
4
0
8
12
16
-10
0
..
4
..
.
...
8
12
16
30
20
T1
11
f<1
-
10
0'
0
4
8
12
-10
16
0
.,
4
8
12
16
Lr
10
0
4
8
12
16
15
'
-
-5
..
..
0
.
4
8
12
16
-._,_
.
0..0-
-10
0
4
8
,_ .. -
-
12
16
The dashed-dotted lines represent the posterior means of the VAR impulse response functions.
NOTES:
The dotted lines are 90% confidence bands. The solid lines represent the mean impulse responses from
the DSGE model. The impulse responses are based on the sample 1981:IV-2001:III.
2
FIGURE
IDENTIFIED IMPULSE RESPONSE FUNCTIONS
The sign and the magnitude of the VAR impulse responses on impact agree with
the model and are very precisely estimated. This suggests that Etr and ZE*(0) in
Equations (39) and (40) are very similar. The response of inflation to a money
shock is short-lived both in the DSGE model as well as in the VAR. In other
dimensions there is less agreement: Where the model predicts long-run money
neutrality,the VAR impulse responses indicate that there is substantial uncertainty
about the long-run effects of money shocks on output. Although these findings are
specific to this DSGE model, they seem to favor identification strategies based on
impulse responses on impact (as in Faust, 1998;Uhlig, 2001; Canova and DeNicol6,
2002) relative to strategies that rely on long-run neutrality.
Identification schemes based on zero-restrictions on the contemporaneous impact of the structural shocks often produce a price-puzzle in three- or four-variable
VARs. Although the price-puzzle can be avoided by including producer prices in
addition to consumer prices, it can also be avoided by using our identification
scheme that is not based on zero-restrictions.
664
DEL NEGRO AND SCHORFHEIDE
4.4. Regime Shifts. The analysis of welfare implications of different monetary
policy rules has become an active area of research (see, for instance, the articles
in Taylor, 1999). It is important for policymakers to have a set of tools that allows
them to predict the effects of switching from one policy rule to another. Suppose
a policymaker only has a DSGE model and an unrestricted VAR available. Moreover, suppose the forecast performance of the DSGE model is worse than that
of the VAR. However, because of the Lucas critique, the policymaker does not
fully trust the VAR to correctly predict the impact of a permanent change in the
policy rule. Based on the DSGE-VAR, we propose an approach by which the
policymaker uses the DGSE model to forecast the effect of the policy change, but
corrects the DSGE model predictions. The correction is the one that brings the
DSGEs forecasts under the existing policy regime on track. The tacit assumption
underlying this procedure is that the correction is itself policy invariant.
We begin by decomposing the joint posterior of VAR and DSGE model
parameters into
p(C>,Eu, 0 I Y) = p(t,
(41)
I0, Y)p(0 I Y)
To the extent that p(1>, Eu I0, Y) assigns mass away from the restriction functions
I*(0) and ES(0) it can be interpreted as a correction of the VAR representation
of the DSGE model given 0. This correction has been constructed from past
observations to optimize forecasting performance.
Partition 0 = [0[s), 0'p)]', where 0(p) corresponds to the parameters that the
central bank can affect with its conduct of monetary policy. Suppose that our
beliefs about the post-intervention policy parameters can be characterized by
the density p(O(p)),which possibly concentrates all its mass on a single value of
0(p). We combine p((p)) with the information that we have about the nonpolicy
parameters based on the available data. This leads to a modified posterior of 0
P(0 I Y) = p(0s) I Y)p(P))
(42)
where p(0(s) I Y) is the marginal posterior of the nonpolicy parameters.
Our inference with respect to the effect of the regime shift will be drawn from
px(c,>,S,
(43)
Y)=
p
Px(D,
E, I Y,
0)x(
Y)d0
We use the subscript Xto indicate that the conclusions depend on the weight given
to the DSGE model. It is instructive to take another look at the posterior mean
of the VAR parameters 4 conditional on the DSGE model parameters 0
(O) = (
(0)
+x1XX)T-1X'
(1
)+F*
T'X'Y)
The parameters 0 are now drawn from the modified distribution p(0 I Y) that reflects the change in the distribution of the policy parameters due to the intervention. The sample moments X'X, X'Y, and Y'Y, on the other hand, are functions of
PRIORS FROM DSGE MODELS FOR VARS
665
pre-intervention observations only. They remain part of our analysis because they
incorporate information about how to "correct" the DSGE model predictions to
achieve a good forecast performance.
If X = 0, the VAR posterior does not depend on 0 at all: Hence the researcher is
ignoring the DSGE model (and the regime shift itself) in computing her forecasts.
If A = oo, then the procedure is equivalent to analyzing the policy directly with
the (VAR approximation of the) DSGE model. It is clear that the Lucas critique
is fully observed only in the X = oo extreme.
Most of the current literature on monetary policy rules focuses on the effect of
these rules on the magnitude of economic fluctuations and the households' utility
over the business cycle. A popular measure of welfare besides agents' utility is the
volatility of the output gap and inflation. To illustrate our prediction approach we
are considering the effect of a change in the response of the federal funds rate to
deviations of inflation from its target rate (the parameter Ifl in the Taylor rule
(4)) on the standard deviation of real output growth and inflation.
A widely shared belief, e.g., Clarida et al. (2000), is that under the chairmanship
of Paul Volcker and Alan Greenspan the U.S. central bank responded more aggressively to rising inflation than under their predecessors William Martin, Arthur
Burns, and William Miller. Based on the empirical results in the Taylor rule literature we compare two policies. Under the first policy scenario f1 = 1.1, whereas
under the second policy scenario *1 = 1.8. The former can be loosely interpreted
as a continuation of the inactive Martin, Burns, and Miller policy,19whereas the
latter corresponds to a switch to a more active Volcker, Greenspan policy. To assess the two policies we generated draws from the modified posterior (42) and
simulate trajectories of 80 observations conditional on the parameter draws. For
each trajectory we discard the first eight quarters (hence we consider only the
paths post-1982:III) and then compute the standard deviation of output growth,
inflation, and the federal funds rate.
The results for various choices of X are summarized in the density plots of
Figure 3. The dashed densities corresponds to f1 = 1.1 and the solid densities to
f1 = 1.8. The vertical lines in the plots show the standard deviation of the actual
sample (post-1982:III) for the variables of interest. The 1982:1 threshold is taken
from Clarida et al. (2000). The transition period from high inflation to low inflation
between 1979 and 1982 implies that the actual standard deviation of inflation for
the whole sample is high, and in our view does not reflect the "steady-state"
variability of inflation under the Volcker-Greenspan policy. Hence we choose to
discard the first eight quarters.20
The DSGE model predicts that an increase in the Taylor rule parameter fr1 induces a lower equilibrium variability of inflation and therefore a lower variability
of the federal funds rate. This is indeed what can be observed in Figure 3 for X > 0.
19
Although some authors report estimates of if 1 < 1 we restrict ourselves to the determinacy region
of the DSGE model.
20 A
possible explanation is that the linear model fails to capture the transition period from high
inflation to low inflation. In the model, agents change their expectations instantaneously when the
policy change is announced whereas in reality there may be a learning process in which agents slowly
realize that the policy change is permanent (regime shift) rather than temporary (deviation from policy
rule).
666
DEL NEGRO AND SCHORFHEIDE
Fed Fund
inflation
realGDPgrowth
0.5
2
0
0
Lo
1
2
0
0
1
2
2
0
5
10
5
10
0
5
10
0
5
0
10
5
10
0.5
0
1
2
0
1
0
2
0
1
2
10
1
0
2
0
0
0.5
'f:A2
0
0
0.5
m2
0
0
0
1
2
2
0
a 0
2
2
0
1
2
0
-
0,
1
2
0.5
0
to the samplestandard
linescorrespond
NOTES:Thedottedhorizontal
deviationof the actualdata
and the solid lines are posterior predictie distributions of sample
from 1982:V to 1999:11.The dashed
SHIFT
OFA POLICY
REGIME
EFFECTS
Thedashedline
for the sametimeperiod,obtainedusingdataup to 1979:11.
standard
deviations
toii = l1.1,thesolidlinecorresponds
to hI =t1.8.
corresponds
FIGURE
3
EFFECTS OF A POLICY REGIME SHIFT
from 1.1 to
increases
the forecasted variability of inflation and
Whenever
1.8
tend to.8,
rigt
th
distribution
e is sk
t
interest rate decreases. It roeassuringts
= of
1.8 (solid line) concentrates around
of the predicted standard deviation for
under e = 1.1
the actual standard deviation. The s predictns
ti (dashed line) tend
interest
In
of
overestimate
rate, the predictions
to
inflation variability. terms the
under
a change 1.8 tend to underestimate Fed Fund variability, whereas the predicthebecomes largerdashed
tions under that= 1.1predicted
the
effect ofof policy change
distribution appears to be roughly on spot, the distribution is skewed to the right).
The predictions for the standard deviation of output do not change significantly
as *1 increases. The DSGE-VAR does not predict the reduction of the variability
in real output growth that took place after 1979. This reduction may be the outcome
of a change in the exogenous technology process rather than the effect of monetary
policy. Note that the predicted effect of the policy change becomes larger as the
weight of the prior increases. As more weight is given to the DSGE model, the
differences in the predictions becomes sharper, as one would expect.
Although the experiment just described is not pure out-of-sample prediction, since the policy experiment /1 = 1.8 was motivated by an analysis of the
PRIORSFROMDSGE MODELSFOR VARS
667
Volcker-Greenspansample,it illustratesthe potentialof our approach.We view
the procedureas a tool that lets the policymakerassess the effects of the policy
change as a functionof the confidenceplaced in the structuralmodel measured
by S.
5.
CONCLUSIONS
The articletakes the idea of Ingramand Whiteman(1994)-imposing priors
from general equilibriummodels on VARs- and develops it into a full-blown,
computationallyefficientstrategythatis usablefor policyanalysis.Ourapproach
involves the followingsteps: (i) Choose a DSGE model and a priordistribution
for its parameters.(ii) Solve the DGSE model and map the prior distribution
of its parametersinto a prior distributionfor the VAR parameters.Although a
log-linearapproximationof the DSGE model simplifiesthe computationof the
VAR approximationgivenby c*(0) and E,(O) considerably,it is not crucialto our
approach.(iii) Obtainvia Monte Carlomethods the joint posteriordistribution
of DSGE and VAR parameters,which can then be used to computepredictive
densities.
We applythis procedureto a VAR in real outputgrowth,inflation,andinterest
rates,and show that it is broadlysuccessfulin termsof forecastingperformance.
TheDSGE-VARclearlyoutperformsan unrestrictedVAR andthe vectorautoregressiveapproximationof the DSGE model itself at all horizons.Its forecasting
performanceis comparableto and in some dimensionsbetter than a VAR with
Minnesotaprior.
Unlike the Minnesotaprior,our DSGE model prior also offers help in terms
of policy analysis.We constructa VAR identificationscheme for the structural
shocksbased on a comparisonof the contemporaneousVAR responseswith the
DSGE model responses.The identificationof policyshocksenablesan analysisof
modestinterventions.Our approachprovidesan attractivealternativeto existing
identificationschemesbecauseit closelyties VAR identificationto a fullyspecified
generalequilibriummodel. Beyond the specificationof a DSGE model and the
possibly data-drivenselection of the prior weight, our identificationprocedure
does not requirethe researcherto make any additionalchoices.
We alsoillustratehow a VAR withDSGE modelpriorcanbe usedto predictthe
effectsof a permanentchangein the policyrule-a taskthatis generallyconsidered
infeasiblefor identifiedVARs. We use the approachto predictthe impactof the
changefrom the Martin-Burns-Millerregimeto the Volcker-Greenspanregime
on the volatilityof the variablesof interest.Wefindthatat leastin somedimensions
the approachfaresreasonablywell in termsof predictingthe effect of the change,
althoughfurtherresearchis needed to investigatethis issue more deeply.
More researchlies down the road. First,if the VAR is specifiedin terms of
output and prices ratherthan output growthand inflation,then the calculation
of expected samplemomentsof data generatedfrom the DSGE model becomes
more complicateddue to the nonstationarity.Second,our methodcould be used
to generatepriorsfor state-spacemodelsthatformallynest the DSGE model.This
extensionis conceptuallystraightforwardbut challengingfrom a computational
perspective.Third,it is worthwhileto apply the method to largerscale DSGE
668
DEL NEGRO AND SCHORFHEIDE
models, e.g., the Smets-Wouters (2003) model, and to make comparisons among
priors that are derived from different models, such as a New Keynesian model
versus a flexible price cash-in-advance model.
As envisioned in Diebold (1998), the combination of DSGE models and VARs
shows promise for macroeconomic forecasting and policy analysis.
APPENDIX
A.1. Derivations and Proofs. The normalization factor c(O) for the conditional prior density of the VAR parameters p(I, E~ I0) in Equation (21) is given
by
nT-k
(A.1)
c(O) = (2jr)'k 1TF(O)
XTE(0) 2|
)|
T- k + -i)/2]
2n2 k)r[(r
i=l
where F[.] denotes the gamma function.
The marginal likelihood function of 0 in Equation (32) is given by
(A.2)
P(YI 0) = p(YI d), E)p(I', E I0)/p(,
TX
X
Trx (0) + X'Al
E I Y)
(i+ 1)
Tru(
(0)
2
)~T-k
X7Tr (0)1 I XT* (0)1
X
(2r)- /22
2
n2
+ 1)T- k + 1 - i)/2]
rF[(XT- k +1 - i)/2]
Ii=ln r[((
n(XT-k)
2
2
i=
The second equality can be obtained from the normalization constants of the
Inverted Wishart-Normal distributions.
Define the sample moments Fxx = X'X/ T, fry =
=
and
T.
Let
Y'Y/
X'Y/ T,
0 = 1/k and 0 be the mode of the marginal logPyy
likelihood function given in Equation (A.2). Consider the log-likelihood ratio
PROOF OF PROPOSITION 1.
(A.3)
(A.3)
p(Y O),) =ln p(Y
T ln
n (IO)+ FXx
- -lI,AF*l10\f
In
/'/ 11
(2lnlc:(e)l
(1/0 + 1)T- k n 1/ +
2
l/q5
T
)
+ 1/ In
2
-+ n n
ln
rxx
*-
1 + (px'x (5)F(xx
1/?
669
PRIORSFROMDSGE MODELSFOR VARS
We derive an approximation of the log-likelihood ratio that is valid as ( ->
first-order Taylor approximation of the second term around ( = 0 yields
0. A
r =In IIZl
+ ptr[rxx1
o(r2)+
InI?+ Pr1xx
r'x
(A.4)
Notice that
(A.5)
1/4 + 1
1/?
=_ [r* - r* x- r- ]-1
yy
y
u
+
x[rF;y + rfyy
(r;x qryx)(rx
+ crpx)-
(rF* + orxy)]
The log-determinant of this term has the following first-order Taylor expansion
around 0 = 0:
In l/+
(A.6)
1/?0
^-
= In IIZ+ btr[E- (yy
-
r - *-
*'
+ *'rx,,*)]
+ 0(02
Combining these results yields
(A.7)
ln p(YI 0)
-
T
In
+
2T Iu*(0)
T
ln |E*(0)I
- 2-tr[-1(0o)(y
- ry
T
+ 2tr[E*- (0)(fr
- rfyxl*(0) - c4*'(0)rfX +
= n P(
*() -
*'(0)rxy + c$*'(0)fxx*
(0))]
*'(0)r,,xxD*(0))]+ 0(0)
0) + O(Q)
p*(YI0)
Thus, as 0 -> 0 the log-likelihood ratio converges to the log-likelihood ratio of
the quasi-likelihood functions. The convergence is uniform on compact subsets of
I
0 for which ES(0)
and 1XX\VIU~IV11UU
rX(0) are nonsingular.
\ IIU
670
DEL NEGRO AND SCHORFHEIDE
2. We rewrite the marginal log-likelihood ratio given
PROOFOFPROPOSITION
in Equation (A.3) in terms of X
T
u
-jl2 IIn |E:,(0)|-
I
Ay
P(YO)0)
ln^^^
P
(A.8)
n
2 In |Af:+ ?xx (0
XX
2
\(0)r^l
+ 1)T- k
ln I(X+ 1)*-
-(
T+n s^
+ 2
5:(0)1
2
(0)fu()
,,
++ rx (0)rx
jn^
n
In
2
( + 1)T - k In (X +
+
1)*-)u(
+
2
A Taylor series expansion of the second term around X = 0 yields
ln 1XI+ rF' rx
(A.9)
-
In Irx rxx + Xtr[rXl rX] + O(X2)
Notice that
(A.10)
(X + 1)E*-' Eu = [ryy - ryxrxx rry]-
x [xr;y+ yy - (xr'; + fryx)(xr*x+ rxx)-1(xr + rfX)]
The log-determinant of this term has the following first-order Taylor expansion
around X = 0
(A.11)
In |(X + 1)
=
=
'
S
I--tr[ieu,mle(y
Xtr[u,mleeu]
u- In | Eu,mle
- rmle
^*m
r+
yxmie
+ Xtr[ ,mle(*
*
leDi lerxmle)]
x )]D
ney
- (mle)
'rx((I
q 0(X2)
O(X2)
+
4(mle)]
+ 0(x2)
Combining the three terms leads to the following approximation of the loglikelihood ratio
(A.12)
p(Y.I) X- T In I
M
2
2U
p(YI )
In
]
+2
(0
U
- XT r-1
] +T
q----tr[Efmj,E:*'
U,mle 2u
2 tr[Eu'mleE(O)]
20)]
*
671
PRIORSFROMDSGE MODELSFOR VARS
XT
-
? -tr
2 utr[u'mle ( VO:e)
)mle)
xx((0 ]m
e)]
)-
+ Op(max[2 T, 1])
=nq(YI
0) +Op(max[X2T, 1])
q(yIl)
A.2. PracticalImplementation. We assumethat the parameterspace of X is
finite A = {/1,... ,lq}. In orderto select X,and to generatedrawsfrom the joint
posteriordistributionof DSGE model parametersand VAR parameterswe use
the followingscheme:
1. For each X E A use the Metropolisalgorithmdescribedin Schorfheide
(2000) to generate draws from px(OIY) oc px(Y 0) p(0). The steps
needed to evaluate px(YI 0) based on Equation (A.2) are as follows. For
each 0:
(i) Solve the DSGE model given by Equations(7), (10), and (12)-(14),
for instance,with the algorithmdescribedin Sims (2002). This leads
to a transitionequationof the form
St = T(O)st_l + R(0)Et
(A.13)
The measurementequation(15) can be writtenin stackedform as
(A.14)
Yt = Z(O)st + D(0) + vt
(In our implementationwe choose st such that vt = 0). Define the
variance-covariancematricesof the shocksas
E[Vt Vt] = Xvv(0)
IE[EtEt] = XE(0)
E[t Vt] = 5UV(0)
(ii) Compute the population moments ry(0), Frx(0), and F* (0) from the
state-space representation (A.13, A.14). Notice that
(A.15)
E[ytyt] = Zss Z' + ZREV + (ZREv)'
+ E,
+ DD'
E[YtYt-_h]= ZTh(QZssZ' + RZev) + DD'
where Qss= E[sts'] and can be obtained by solving the following
Lyapunov equation: Qss = T
ss T' + R2E, R'.
672
DEL NEGRO AND SCHORFHEIDE
2. Based on these draws apply Geweke's (1999) modified harmonic mean
estimator to obtain numerical approximations of the data densities px (Y).
3. Find the presample size Xthat has the highest data density.
4. Select the draws of {0(s)} that correspond to Xand use standard methods
to generate draws from p((, Su I Y, 0(s)) (see Equations (30) and (31))
for each 0(s).
Notice that this scheme can also be used to select among competing DSGE
models. Moreover, the whole procedure can be easily generalized to the case in
which we have a prior distribution over the hyperparameter X.
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