Forec astingand T urningP oint P red ic tionsina B ayesianP anel
VAR M od el
Fab io Canova
¤U niversitat P omp eu Fab ra,U niversity ofSoutham ptonand CE P R
and
M atteo Cic c arelli
yU niversitat d 'Alac ant
T hisDraf
t: O c tob er 19 99
A bstract
W eprovidemethodsforforecastingvariablesandpredictingturningpointsinpanelB ayesian V A R s.W especifya° exiblemodelwhichaccountsforbothinterdependenciesinthecrosssection and timevariations in theparameters.P osteriordistributions fortheparameters areobtained fora particulartype ofdi®use, forM innesota-type and forhierarchicalpriors. Formulas for multistep, multiunitpointand average forecasts are provided.A n application tothe problem offorecastingthegrowth rateofoutputand ofpredictingturningpoints in theG -7illustrates theapproach.A comparison with alternativeforecastingmethods is alsoprovided.
Key W ords: Forecasting, T urningP oints, B ayesian M ethods, P anelV A R , M arkov Chains M onte CarloM ethods
JelClassi¯cation nos:C1 1 ,C1 5,E32,E37
¤W e w ould like to thank Alb ert M arc et and the partic ipantsofthe 19 9 9 E uropeanM eetingsofthe E c onom etric
Soc iety for c om m entsand suggestions.
yCanova,Departam ent d 'E c onom ia iE m presa,U P F ,R am onT riasFargas2 5-2 7,0 80 0 5 B arc elona,Spain.E -m ail:
c [email protected]; Cic c arelli,Fund am entosd elAn¶alisisE c on¶om ic o,U A,Ap.Correos9 9 ,0 30 80 Alic ante,Spain.
1
In
trod uc tion
P anelV A R models have become increasingly popularin macroeconomics to study the transmis-sionofshocks across countries (B allabriga,SebastianandV alles (1 995)),thepropagatione®ects of monetary policy in the European U nion (G erlach and Smets (1 996))and the average di®erential responseofdeveloped and underdeveloped countries todomesticand externaldisturbances (H o®-maisterand R old¶os (1 997),R ebucci (1 998)).A tthe same time,recentdevelopments in computer technology have permitted the estimation ofincreasingly complex multicountry V A R models in reasonabletime,makingthem potentiallyusableforavarietyofforecastingand policypurposes. D espitethis interest,the theory forpanelV A R is somewhatunderdeveloped.A ftertheworks ofChamberlain (1 982, 1 984)and H oltz{ Eakin etal. (1 988), who specify panelV A R models for micro data, to the best ofour knowledge only P esaran and Smith (1 996), Canova and M arcet (1 997)and H siao et al. (1 998)have considered problems connected with the speci¯cation and the estimation of(univariate)dynamic macro panels. G arcia Ferreret al. (1 987), Z ellnerand H ong(1 989),Z ellner,H ongand M in (1 991 ),on theotherhand,haveprovided B ayesian shrinkage estimators and predictors forsimilarmodels.In general,aresearcherfocuses on thespeci¯cation
yit=A(L )yit¡1+ "it
whereyitis aG{ dimensionalvector,i = 1;:::;N ;A(L )is a matrix in the lag operator;"it=
®i+ ±t+ uit, where±tis a time e®ect;®iis a unitspeci¯c e®ectanduita disturbance term.In
somecases (seee.g.H oltz{ Eakin etal.(1 988))aspeci¯cation with timevaryingslopecoe±cients and a ¯xed e®ectis used.T womain restrictions characterize this speci¯cation.First, itassumes commonslopecoe±cients.Second,itdoes notallowforinterdependencies across units.W iththese restrictions,theinterestis typically in estimatingtheaveragedynamics ofthesystem in response toshocks (thematrixA(L )).
G arcia Ferrer et al., Canova and M arcet and P esaran and Smith, instead, use a univariate dynamicmodeloftheform
yit=®i+ ½iyit¡1+ x0it¯i+ v0t±i+ "it
whereyitis ascalar,xitis asetofkexogenous unitspeci¯cregressors,vtis asetofhexogenous
regressors common toallunits while½i,¯iand±iareunitspeci¯cvectors ofcoe±cients.In some
speci¯cations thesevectors ofcoe±cients areassumed tohaveanexchangeableprior.T worestric-tions are implicitalsoin this speci¯cation.First, notime variation is allowed in the parameters. Second,therearenointerdependencies eitheramongdi®erentvariables within units oramongthe samevariableacross units.
T he task ofthis paperis torelax these restrictions and study the issues ofspeci¯cation, esti-mation and forecastingin amacro-panelV A R modelwith interdependencies.O urpointofviewis B ayesian.Such an approach has been widelyused in theV A R literaturesincetheworks ofD oan, L ittermanandSims (1 984),L itterman(1 986),andSims andZ ha(1 998)andprovides aconvenient frameworkwhere onecan allowforboth interdependencies and meaningfultime variations in the coe±cients.T hespeci¯cation weconsiderhas thegeneralform
1 IN T R O D U CT IO N 3 whereYs(s < t)is avectorofG N elements (thereG variables foreach uniti= 1;:::N ).B ecause
coe±cients vary across units and alongtime, estimation ofthe parameters is impossible without imposing restrictions. H owever, instead of constraining the coe±cients to be the same across units, we assume that they are random and a prior distribution onAit(L )is introduced. W e
decompose the parametervectorinto two components, one which is unitspeci¯c and the other which is time speci¯c.W especify a° exible prioron these twocomponents which parsimoniously takes into accountpossible interdependencies in the cross section and allows fortime variations in theevolution oftheparameters overtime.T he priorshares features with thoseofL indlay and Smith(1 972),D oan,L ittermanandSims (1 984)andH siaoetal.(1 998)anditis speci¯edtohavea hierarchicalstructure,whichallows forvarious degrees ofignorancein theresearcher's information abouttheparameters.
B esides important considerations concerning the speci¯cation of the model, B ayesian V A R s are known produce betterforecasts than unrestricted V A R and, in many situations, A R IM A or structuralmodels (Canova(1 995)forreferences).B yallowinginterdependencies and somedegree ofinformationpoolingacross units weintroduceanadditionallevelof° exibilitywhichmayimprove theforecastingability ofthesemodels.
W e analyze severalspecialcases ofourspeci¯cation and computeB ayesian estimators forthe individualcoe±cients and fortheirmean values overthe cross section.In some cases analytical formulas forthe posteriormean are availableusingstandard formulas.W heneverthe parameters ofthe priorare unknown, we employ the predictive density ofthe modelto estimate them and plug-in ourestimates in therelevantformulas in an empiricalB ayes fashion.
Inthecaseoffullyhierarchicalpriors,aM arkovChainM onteCarlomethod(theG ibbssampler) isemployedtocalculateposteriordistributions.Suchanapproachisparticularlyusefulinoursetup sinceitexploits therecursivefeatures oftheposteriordistribution.W eproviderecursiveformulas formultistep, multiunitforecasts,consistentwith the information available ateach pointin time usingtheposterioroftheparametersorthepredictivedensityoffutureobservations.T hepredictive densityoffutureobservation is alsoused tocomputeturningpointprobabilities.
T oillustratetheforecastingabilityoftheproposedapproach,weapplythemethodologytothe problem ofpredictingoutputgrowth,offorecastingturningpoints inoutputgrowthandcomputing the probability ofa recession in the G -7 usingthree variables (outputgrowth, realstock returns and realmoney growth)foreach country in thepanel.T oevaluatetheperformanceofthemodel wealsoprovideaforecastingcomparison with otherspeci¯cations suggested in theliterature.W e showthatourpanelV A R approach improves overexisting univariate and simple B V A R models when we measure the forecasting performance using the T heil-U and the M A D criteria, both at the one step and atthe foursteps horizons. T he improvements are ofthe orderof5-1 0 % with the T heil-U and about2-4% with the M A D .T he forecasting performance ofourspeci¯cation is alsoslightlybetterthentheoneofaB V A R modelwhichmechanicallyextends theL ittermanprior tothe panelcase.In terms ofturning pointpredictions, the twoversions ofourpanelapproach are able to recognize about 80 % of turning points in the sample and they turn out to be the bestforthis task, along with Z ellner's g-priorshrinkage approach. T he simple extension ofthe L itterman's priortothepanelcasedoes poorlyalongthis dimensionand,amongalltheprocedures employed, is the second worst. Finally, we show thatthe proposed method is competitive with
the best speci¯cations in predicting the peak in U S economic activity occurred in 1 990 :3 when usingtheinformationavailablein1 988:4,apeakwhichwas missedbymanyofthecommercialand government forecasting procedures. D epending on the speci¯cation, ourapproach ¯nds 20 -55% probabilityofadownward turn atthatdate.
T herestofthepaperis organized as follows.T he nextsection gives the generalmodelspeci-¯cation and the assumptions we make.Section 3 provides thegeneralities ofB ayesian estimation ofthe model.Section 4 speci¯es the priorand discusses the computationalissues involved.Sec-tion 5 describes formulas formulti-step,multi-units forecasting.Secinvolved.Sec-tion6 contains theforecasting application toapanelV A R modelfortheG -7.Section 7concludes.
2
T he generalspec i¯c ation
T hestatisticalreduced form modelweuseis oftheform: yit= N X j= 1 p X l= 1 bjit;lyjt¡l+ ditvt+ uit (1 )
wherei= 1;:::;N ;t= 1;:::;T;yitis aG{ dimensionalvectorforeachi,bjit;lareG£G matrices,dit
isG£q;vtis aq£1 vectorofexogenous variables common toallunits anduitis aG{ dimensional
vectorofrandom disturbances.H erepis thenumberoflags,G thenumberofendogenous variables andqthenumberofexogenous variables.
T he generality of(1 )comes from atleasttwo features. First, the coe±cients are allowed to vary both across units and across time. Second, there are interdependencies among units, since bjit;l6= 0 forj6=iand foranyl.B oth features constitute the main di®erence with the literature (H oltz-Eakin atal.(1 988),R ebucci (1 998))thatconsiders panelV A R models.Itis easy toverify thatifwe setditvt=at; bit=bt 8i; uit=Ãtfi+ »itbjit;l= 0; j6=i; 8l;ourspeci¯cation
collapses totheoneused byH oltz-Eakin etal.(1 988). W erewrite(1 )in astacked regression manner
Yt=W t°t+ Ut (2) whereW t=IN G-Xt0;Xt= ³ y0t¡1; y0t¡2;¢¢¢yt0¡p; v0t ´0 ;°t= (°01t;:::;°N t0 )0and°it= (¯it10;:::;¯itG0)0.
H ereys (s < t)is aN G{ dimensionalvector,¯itg arek{ dimensionalvectors, withk=N G p+ q,
containing,stacked,thegrows ofthecoe±cientmatricesbitanddit,whileYtandUtareN G£1
matrices containingtheendogenous variables and therandom disturbances ofthemodel.
Ifthe°itaredi®erentforeach cross{ sectionalunitin di®erenttimeperiods,thereis nowayto
obtain meaningfulestimates ofthem.O nepossibility is tovieweach coe±cientvectoras random with agiven probability distribution.W emakethefollowingassumptions:
1 .Foreachi,theG k£1 vector°ithas atimeinvariantand atimevaryingcomponent,thatis
2 T H E G EN ER A L SP ECIFICA T IO N 5 2.Foreachi,theG k£1 vectoroftimeinvariantcomponents®ifollows anormaldistribution
®i»N (Ri®;¹¢ i) (4)
whereRi=IG -Ei,¢ i=V -Ei-1Ei,andtheG£G matrixV andthek£kmatrix-1 are
symmetricandpositivede¯nite.H ereEiis ak£kmatrixthatcommutes thekcoe±cients of
unitiforeachoftheG equations withthoseofunitone.W ealsoassumethatcov(®i;®j)= 0
fori6=j.
3.T hemean vector¹® is common toallunits and is assumed tohaveanormaldistribution ¹
® »N (¹;ª) (5)
4.Foreachiwe write the vectorofthe time varying components as¸it=Ri¸t, where¸tis
independentof®iforanyi.T heG k£1 vector¸tevolves accordingto
¸t=B ¸t¡1+ et; (6)
whereB =½¤IG kand,conditionalonUtandW t,et»N (0;§"),with §"=V --2,and -2
is apositivede¯nite,symmetricmatrix.T heinitialcondition is such that¸0 »N
³
~ ¸0;-0
´
. 5.ConditionalonW t,thevectorofrandom disturbancesUthas anormaldistribution
Ut»N (0;§u): (7)
W e assume that§u= §-H;where § is aN £N matrix andHis aG £G matrix, both
positivede¯niteand symmetric.
G iven the previous assumptions, the structure ofthe model(1 )can be summarized with the followinga{ priorihierarchicalscheme
Yt j Ft;®;¸t»N (W t®+ Z t¸t;§u) ® j Ft»N (SN ® ;¹¢ ) ¹ ® j Ft»N (¹;ª) ¸t j Ft»N ³ ^ ¸tjt¡1;-^tjt¡1 ´ (8) whereFtis theinformationsetatt(whichincludesY0,thepresampleinformation,andW t);SN =
eN -Ri;Z t=W tSN ;¢ =diag(¢ 1;::;¢ n), ^¸tjt¡1 =B ^¸t¡1jt¡1;^-tjt¡1 =B-^t¡1jt¡1B0+ §",eN
is a vectorofones ofdimensionN and the notationtjt¡1 indicates values attpredicted with information att¡1 .
A ssumptions 1 -4 decompose the parameters vectorin 2 components: one is unitspeci¯c and constantovertime;the otheris common across units butvaries with time.T he priorpossibility fortime{ variationincreases the° exibilityofthespeci¯cationand provides ageneralmechanism to accountforstructuralshifts withoutexplicitlymodellingthesourceoftheshift.T hefactthatthe
time{ varying parametervectoris common across units does notpreventunit{ speci¯c structural shifts,since°itcan bere-written as
°it= (1 ¡½)®i+ ½°it¡1+ eit (9)
whereunitspeci¯cvariations oftimeoccurthrough thecommon coe±cient½:
A ssumptions 2 and 3 can beused torecoverthevector® orthemean coe±cientvector¹®.In this sense,we can distinguish between "¯xed"and "random"e®ects,followingtheterminology of L indleyandSmith(1 972).B y¯xede®ects wemeantheestimationofthevector°it,whiletheterm
random e®ects refers totheestimationof¹°t= ¹®+¸t.Forexample,inthecontextofaV A R without
interdependencies, (i.e.bjit;l= 0 ; j6=i), we may be more interested in the relationships among thevariables ofthesystem fora"typical"unit,in which caseinterestcenters in theestimation of the random e®ect¹°t.If, instead, we are interested in the relationships across units, forexample,
wishingto¯nd thee®ectofashockin thegvariableofunitjon thevariables ofuniti;webetter estimate°itforeachuniti.Inthecontextofforecasting,wemaybeconcernedwithpointprediction
usingtheaveragecoe±cientvector¹°torinpredictingfuturevalues ofthevariables ofinterestusing
information availableforeach unit.
T he assumed Kroneker structure for the variance{ covariance matrices is convenient to nest interestinghypothesis.Forinstance,when -1 = 0 ,thereis noheterogeneityin thecross sectional
dimension ofthe panel.IfB =IG k, coe±cients evolve overtime as a random walk, while when
B =IG kand-2 = 0 ,themodelreduces toastandarddynamicpanelmodelwithnotime{ variation
in thecoe±cientvector.Finally,whenV = 0 neitherheterogeneitynortimevariation arepresent in themodel.
T he prior speci¯cation is fully symmetric in the sense that it is the same regardless of the variables and of the units we are considering. In some applications where it is interesting to considersomepriorasymmetries,this restriction may notbeneeded.In thatcasewesetEi=IN
sothatRi=IG -IN and(3)becomes°it=®i+¸twhere®i»N (¹®;¢ )andthepriordistributions
for¹® and¸tarethesameas before.
A s compared tostandard B V A R models,we allowforsomedegree ofa-priori poolingofcross sectionalinformation via the exchangeable prioron®.T his may be importantifthere are some similarities in thetimeseries characteristics ofthevectorofvariables considered across units since coe±cients ofotherunits maycontain usefulinformation forestimatingthecoe±cients oftheunit underconsideration.A singlecountryV A R with¯xedcoe±cients is nestedinourspeci¯cationand can beobtained bysettingbjit;l= 0;8j6=i;8land letting¤;ª;§"gotozero.
3 P osterior E stim ates
3.
1
F ixed e®ec tsm od el
G ivenpriorinformationon°t,andassumingthat~¸0,¹ andthecovariancematrices areknown,we
can obtain theposteriordistribution oftheparametervectorbycombiningthelikelihood function conditionalonFtwith thepriordistribution for°tin theusualway.From (8)thelikelihood is
3 P O ST ER IO R EST IM A T ES 7 and theprior,given information attot,is
p(°tjFt)=N ³ ^ °t¡1;H^t¡1 ´ (1 0 ) where ^°t¡1 =SN ³ ¹+ ^¸tjt¡1 ´ and ^Ht¡1 = (SN ªSN0 + ¢ )+ SN -^tjt¡1SN0:
Standard calculations give us thatthe posterior¼(°tjFt; Yt)is normalwith mean°¤tand
varianceHt¤where: °¤t = Ht¤³W t0§u¡1Yt+ ^Ht¡1¡1^°t¡1 ´ Ht¤ = h ^ Ht¡1¡1+ W t0§¡1uW t i¡1 (1 1 ) H ence°¤
tis astandard weighted average ofpriorand sampleinformation.W ith aknown §uand
startingfrom initialconditions ^°0 and ^H0 we can alsoobtain posteriormoments for°tusingthe
followingrecursiveformulas: °¤t = ^°t¡1+ ^Ht¡1W t0 h W tH^t¡1W t0+ §u i¡1 (Yt¡W t^°t¡1) Ht¤ = H^t¡1¡H^t¡1W t0 h W tH^t¡1W t0+ §u i¡1 W tH^t¡1 (1 2)
H ereinformation about°t¤andHt¤is updated in aKalman ¯lterfashion.
In somecases attention may becentered in obtainingposteriordistributions of® and¸t
sepa-rately.Itis straightforward toshowthat:
à ® YtjFt ! »N " à SN ¹ Z t ³ ¹+ ^¸tjt¡1 ´ ! ; à Á11 Á12 Á2 1 Á2 2 ! # whereÁ11= (SN ªSN0 + ¢ );Á12 =Á11W t0;Á2 1=W tÁ11;Á2 2 =W tÁ11W t0+ Z t-^tjt¡1Z t0+ §u. U singthepropertiesofmultivariatenormaldistributions,theconditionalmarginal¼1(® jFt;Yt) is normalwithmean®¤=SN ¹+Á12Á¡1 2 2 h Yt¡Z t ³ ¹+ ^¸tjt¡1 ´i andvarianceV¤ ® =Á11¡Á12Á¡2 21Á2 1.
R epeatingthe same argumentwe obtain thatthe conditionalmarginal¼2(¸tjYt; Ft)is
nor-mal with mean¸¤t = ^¸tjt¡1 + ^-tjt¡1Z t0Á¡2 21
h Yt¡Z t ³ ¹+ ^¸tjt¡1 ´i and variance -¤t = ^-tjt¡1 ¡ ^
-tjt¡1Z t0Á2 2¡1Z t-^tjt¡1:A s usual,themean oftheposteriordistribution is used as apointestimate
fortheparametervectorwhilethevarianceprovides ameasureofdispersion.
Fortheformulas tobeoperationalweneedattimet= 1 aspeci¯cationfor§uandfortheprior
distributions of® and¸t,which in turn requires thespeci¯cation ofthematricesB,§",¢ ,ª,-0
and ofthevectors¹ and ~¸.W ewillreturn on this issuein thenextsection.
3.
2
R and om e®ec tsm od el
W hen interestcenters on theestimation ofthemean vector¹° = ¹® + ¸t,werewritethemodelas
where ¹°t= ¹®+ ¸tand´t=ut+ W tv.
T he posteriordistributions of ¹® and¸tcan be obtained by combining the priors and the
re-spectivelikelihoods.T hesum oftheposteriormeans of¹® and¸tthen gives us apointestimateof
themean coe±cientvectorateacht.
Standardmanipulations giveus thattheposterior¼3(¹® jYt; Ft)»N (¹®¤;ª¤)andtheposterior
¼2(¸tjYt; Ft)»N (¸¤t;-¤t)where ¹ ®¤=¹¡ªZ t0hZ t ³ ª+ ^-tjt¡1 ´ Z t0+ §u+ W t¢ W t0 i¡1h Yt¡Z t ³ ¹+ ^¸tjt¡1 ´i (1 4) ª¤= ª¡ªZ t0hZ t ³ ª+ ^-tjt¡1 ´ Z t0+ §u+ W t¢ W t0 i¡1 Z tª (1 5)
whiletheexpressions for¸¤tand -¤tarethesameas before.T his implies thattheposterior¼4(¹°tj
Yt; Ft)»N (¹°t¤;Ht¤)where ¹ °t¤ = ³ ¹+ ^¸tjt¡1 ´ + ³ª+ ^-tjt¡1 ´ Z t0 h Z t ³ ª+ ^-tjt¡1 ´ Z t0+ §u+ W t¢ W t0 i¡1 £hYt¡Z t ³ ¹+ ^¸tjt¡1 ´i (1 6) Ht¤ = ³ª+ ^-tjt¡1 ´ ¡³ª+ ^-tjt¡1 ´ Z t0hZ t ³ ª+ ^-tjt¡1 ´ Z t0+ §u+ W t¢ W t0 i¡1 £Z t ³ ª+ ^-tjt¡1 ´ (1 7)
4
Settingup the priors
Fortheformulas described intheprevious section tobeoperational,weneed tospecifythevector ³ = (¹;¸~o;-o;§u;§";B ;ª;¢ ). T he results of section 3 were obtained under the assumption
thatthis vectorofparameters was known. In practice, this is hardly the case: to getposterior distributions for the parameters we need to make assumptions on the³ vector and to obtain marginalposteriors we need to integrate nuisance parameters outofthe jointposteriordensity. T his integration,in general, is di±cult, even with brute force numericalmethods,given the large numberofparameters typicallycontained in³.
T hereareseveralways toproceed.O neis toassumeadi®useprioronsomeofthecomponents of the parameter vector, while still assuming that others are known. A nother is to specify a L itterman-typepriorwheretheunknownelementsof³dependonasmallvectorofhyperparameters tobe estimated from the data in EmpiricalB ayes fashion.T he third is to assume explicitprior distributions for the parameter vector and proceed directly to the numerical integration using M arkov Chains-M onteCarlomethods.W eexaminetheseapproaches in turn.
4 .
1
Di®use P riors
Imposingdi®usepriorsisinterestinginourcontextasawaytodescribetheignoranceofaresearcher on someaspects ofthepriordistribution.Itis wellknown (seeZ ellner(1 971 ))thatajointdi®use
4 SET T IN G U P T H E P R IO R S 9 priorforalltheelements of³leads toposteriors whichcontainthesampleinformationsummarized in a leastsquare fashion.A lso, as shown by Kadiyala and Karlsson (1 997), such priorproduces posteriordependence amongthe coe±cients ofdi®erentequations, i.e.the jointposteriorforthe N G k£1 vectorofcoe±cients does notfactorintotheproductoftheposteriorforthekcoe±cients ofeach ofthe N G equations.H ere we concentrate attention on twospecialcases ofinterest:one wherethereis noinformation on thelocationofthemean oftheunitspeci¯ce®ect¡ª¡1 = 0¢and onewherethereis noinformation on thetimevaryingcomponentofthecoe±cients eitherattime zero³-¡0 1= 0´orataparticularpointin time³-^¡tj1t¡1 = 0´.A llothercomponents ofthevector ofparameters areassumed tobeknown.
4.1 .1 Case 1 :Ignorance about®¹
W hen thepriordistribution ofthesecond stageofthehierarchyis proportionaltoaconstant,the posteriordistribution changes accordingtothefollowingproposition:
P roposition 4.1 G iven the prior(8),ifª¡1 = 0;conditionalonYtandFt,
(i) T heposteriordistribution¼3(¹® jYt; Ft)is normalwith mean ¹®¤¤and varianceª¤¤where
¹ ®¤¤= ª¤¤Z t0 h Z t-^tjt¡1Z t0+ §u+ W t¢ W t0 i¡1³ Yt¡Z t¸^tjt¡1 ´ ª¤¤¡1 =Z t0hZ t-^tjt¡1Z t0+ §u+ W t¢ W t0 i¡1 Z t
(ii) T heposteriordistribution¼1(® jYt; Ft)is normalwith mean®¤¤and varianceV®¤¤where
®¤¤=V®¤¤W t0³§u+ Z t-^tjt¡1Z t0 ´¡1³ Yt¡Z t¸^tjt¡1 ´ ; V®¤¤¡1=W t0 ³ §u+ Z t-^tjt¡1Z t0 ´¡1 W t+ F (1 8) withF = ¢ ¡1¡¢ ¡1S N ¡SN0¢ ¡1SN ¢¡1SN0¢ ¡1.
(iii) T heposteriordistribution of¸tis equaltotheprior,i.e.,
¼2 (¸tjYt;X t)=p(¸tjXt):
(T heproofofallpropositions is in theappendix).
N otice thatthe di®use prioron ¹® does notallowtoupdate the priorinformation we have on ¸t.In fact, in this case, the posteriordistribution of°tdoes notdepend on the priorfor¸t.T o
see this note that, with ª¡1 = 0;we have that ^Ht¡1SN
³
ª+ ^-tjt¡1
´
SN0 + ¢ =F ¡1 and using
the factthat ^Ht¡1¡1°^t¡1 =F SN ¡¹+ ¹¸t¡1¢= 0 we have°¤t=
£
F + W t0§¡u1W t¤¡1¡W t0§¡u1Yt¢and
Ht¤=
£
4.1 .2 Case 2:Ignorance about¸t
T here are two simple ways of attaching a di®use prior to the time varying component of the coe±cientvector.O nepossibilityis toconsiderlackofinformation attimezero(-¡01 = 0 ).W hen thepriordistributionfor¸0 is proportionaltoaconstant,giventheautoregressivestructurefor¸t,
andprovided½ < 1 ,theprocess tends to"forget"theinitialcondition.Inotherwords,subsequent realizations of¸tmake less and less uncertain ourinformation on thetimevaryingcomponentof
the coe±cients so that-¡01 = 0 does notimply ^-¡tj1t¡1 = 0 atallpoints in time and, forlarge enough T ,theposteriorfor® and¸tis theonepresented in section 3.
A notherpossibility is to set§¡1" = 0 .T o implementthis di®use prior, we assume -¡2 1 = 0 . N oticethat^-tjt¡1 =B -^t¡1jt¡1B 0+ §".T hereforeif§¡"1 = 0 , ^-¡tj1t¡1 = 0 .Inthis caseitis possible
toprovethefollowingresult
P roposition 4.2 G iven the prior(8),if§¡"1 = 0;then
(i) T heposteriordistribution of¹® is equaltotheprior,i.e., ¼3(¹® jYt;X t)=p(¹® jX t)
(ii) T heposteriordistribution¼1(® jYt; Ft)is normalwith mean®¤¤and varianceV®¤¤where
®¤¤ = V®¤¤ h W t0S ³ Yt¡Z t^¸tjt¡1 ´ + ¡SN ªSN0 + ¢ ¢¡1 SN ¹ i ; V®¤¤¡1 = W t0T W t+ ¡SN ªSN0 + ¢ ¢¡1 andT = §¡u1¡§¡u1Z t¡Z t0§¡u1Z t¢¡1Z t0§¡u1
(iii) T heposteriordistribution¼2(¸tjYt; Ft)is normalwith mean¸¤¤t and variance-¤¤t where
¸¤¤t = -¤¤t nZ t0£W t¡SN ªSN0 + ¢ ¢ W t0+ §u¤¡1(Yt¡Z t¹) o -¤¤¡t 1 =Z t0 £ W t¡SN ªSN0 + ¢ ¢ W t0+ §u¤¡1Z t
T he assumption §¡1e = 0 implies that ^-¡tj1t¡1 = 0 , atallpoints in time. T his implication is unreasonable or, atleast, excessively myopic, because itprevents researchers to learn from past realizations of¸tandtobeless uncertainonits meanas times goes by.T heassumption ^-¡tj1t¡1= 0
can be more realistic ifwe attach this in¯nite uncertainty tothe coe±cients only ata particular pointin time(let's say,t=to),perhaps totakecareofastructuralbreak,afterwhich theprocess
restarts and behaves as itdid beforethebreak.
Itis worth notingthatin both cases 1 and 2, the posteriormean and variance for°tare the
sameas thoseobtainedwhenonlypriorinformationon® is used.T his is notsurprisingifwewrite (8)as athreestagehierarchy Yt j Ft; °t»N (W t°t;§u) °t j Ft;® ; ¸¹ t»N [EN (¹®+ ¸t);¢ ] (¹®+ ¸t) j Ft; ¹;¸^tjt¡1 »N h³ ¹+ ^¸tjt¡1 ´ ;ª+ ^-tjt¡1 i :
4 SET T IN G U P T H E P R IO R S 1 1 A ssuming ª¡1 = 0 or§¡1" = 0 is equivalentto assume a di®use prioron the third stage ofthe hierarchy.
4 .
2
Litterm an-type prior
N ext, we modify the so-called M innesota priorto accountforthe presence ofmultiple units in theV A R .T heM innesotaprior,described in L itterman (1 986),D oan,L itterman and Sims (1 984), Ingram and W hiteman (1 995),B allabriga,etal.(1 998)amongothers is away toaccountforthe nearnonstationarity ofmany macroeconomictime series and,atthesametime,toweakly reduce the dimensionality ofa V A R model.G iven thatthe intertemporaldependence ofthe variables is believedtobestrong,thepriormeanoftheV A R coe±cients onthe¯rstownlagis setequaltoone andthemeanofremainingcoe±cients is equaltozero.T hecovariancematrixofthecoe±cients is diagonal(sowe haveprior| and posterior| independencebetween equations)and theelements arespeci¯ed in away thatcoe±cients ofhigherorderlags arelikely tobeclosetozero(theprior variance decreases when the lag length increases).M oreover, since mostofthe variations in the V A R variables is accounted forbyown lags,coe±cients ofvariables otherthan thedependentone areassigned asmallerrelative variance.T he prioron the constantterm, otherdeterministicand exogenous variables is di®use.Finally,thevariance-covariancematrixoftheerrorterm is assumed tobe¯xed and known.
For a panel V A R setup we introduce the following modi¯cations. T he covariance{ matrices -o;ª;¢ ,areassumedtohavethesamea-priori structure.T ake,forexample,¢ =diag(¢ 1;:::;¢ n),
where¢ i=V -Ei-1Ei.
T hematrix-1 is assumed tobediagonaland its elements havethefollowingstructure:
¾2gijs= Ã µ1®µ±(g3 i;js) lµ2 1 ¾js !2 g;j= 1;:::;G i;s= 1;:::;N l= 1;:::p where±(gi;js)= 0 ifi=sand 1 otherwiseand
¾2gm = (µ1®µ4)2 m= 1;:::;q
H ere,girepresentsequationgofuniti,jstheendogenousvariablejofunits,lthelag,mexogenous
ordeterministicvariables.
T hehyperparameterµ1® controls thetightness ofbeliefs forthevector®;µ2 therateatwhich
the prior variance decays with the lag;µ3 the degree of uncertainty for the coe±cients of the
variables ofunitsin the equations ofuniti;µ4 the degree ofuncertainty ofthe coe±cients of
the exogenous variables and¾js are the diagonalelements ofthe matrix §uused as scale factors toaccountfordi®erences in units ofmeasurement.A lso, assume thatV =H(see equation (7)). N otice that we don't have priorindependence between equations. H ence ourpriorinformation speci¯es that, for example, the coe±cient on lag 1 ofthe G N P equation for the U S may have somerelationship with thesamecoe±cientin theP R ICE equation forU S.M oreover,wehavenot speci¯edahyperparameterwhichcontrols theoveralltightness ofbeliefs becausetherandomness of thecoe±cientsdependson®iand¸tandweparametrizetheuncertaintyineachofthem separately.
Finally,thereis nodistinctionbetweenownversus othercountries variables.B ecauseofthisV and -1 arecommon toallunits and thepriorhas asymmetricstructure(seeSims and Z ha(1 998)).
T hestructures forª and -o aresimilarwithµ1® beingreplaced byµ1 ¹® andµ1¸,respectively.
T ocompletethespeci¯cation weneed tohaveameasuretheelements ofthematrixHand of the¾'s.FollowingL itterman,these parameters areestimated from the datatotune up the prior tothespeci¯capplication.
T hepriortime{ varyingfeatures ofthemodelaredetermined byspecifyingthematricesB,§".
W eassumethatB isdiagonalandthateachofthek£kdiagonalblocksBgsatis¯es:Bg =diag(µ5).
Furthermore,weassume§"=µ6-o.H ereµ5controls theevolution ofthelawofmotion of¸tand
µ6theheteroskecasticityinthecoe±cients.N otethatatime{ invariantmodelis obtainedbysetting
µ5= 1 andµ6= 0 .H omoschedastictimevariations areobtained by settingµ6= 0 .
Finally,weassumethatthek£1 vectors¹g and ~¸og havethefollowingstructures:
¹g = 2 6 6 6 6 6 66 6 6 4 0 . . . µ7 0 . . . 0 3 7 7 7 7 7 77 7 7 5 ; ¸~og = 2 6 6 6 6 6 66 6 6 4 0 . . . 1 ¡µ7 0 . . . 0 3 7 7 7 7 7 77 7 7 5
where¹g and ~¸og arethegth{ elements ofthemeanvectors¹ and~¸o andµ7controls thepriormean
on the¯rstown lagcoe±cientofthedependentvariablein equationgforuniti.
Summingup,ourpriorinformation is afunction ofa9{ dimensionalvectorofhyperparameters £ = (µ1®;µ1¸;µ1¹®;µ2;µ3;µ4;µ5;µ6;µ7).Estimatesof£canbeobtainedbymaximizingthepredictive
density of the model as in D oan, L itterman and Sims (1 984). P osterior distributions for the parameters are then obtained by plugging-in the resulting estimates for¹;¸~o;-o;§u;§";B ;ª;¢
intheformulas wehavederived insection3 inanempiricalB ayes fashion(seee.g.B erger(1 985)). Compared with B allabriga etal.(1 998), whoused a M innesota prioron a panelV A R model for the Spanish, G erman and French economies, our speci¯cation allows for unit speci¯c time variations in the variance ofthe process (µ66= 0 );itseparates the priorinformation forthe time
and the individualcomponent(they have one parameterin place ofµ1®;µ1¸;µ1¹®)and introduces
a further level of uncertainty by specifying a prior for ¹®. Furthermore, our prior speci¯cation is symmetric and it allows fora-priori pooling ofthe information present in the cross sectional dimension ofthepanel.N oneofthesefeatures is presentin theirspeci¯cation.
4 .
3 Inf
orm ative priors
W hen the prior for the vector of parameters is informative, the posterior distribution for the parameter vector does not have an analytical closed form. N evertheless, we can implement a hierarchicalB ayes analysis usingasampling{ based approach,such as theG ibbs sampler,(seee.g. G eman and G eman (1 984),G elfand and Smith (1 990 ),G elfand and al.(1 990 )amongothers).
T hebasicideaoftheapproach is toconstructa(computable)M arkovchain on ageneralstate spacesuch thatthelimitingdistribution ofthechain is thejointposteriorofinterest.Supposewe
4 SET T IN G U P T H E P R IO R S 1 3 haveaparametervector# withkcomponents (#1;#2;:::;#k)and thattheposteriordistributions
¼(#jj#s; s6=j)areavailable.T henthealgorithm works asfollows.W estartfrom arbitraryvalues
for#(o)1 ;#(o)2 ;:::;#k(o).Settingi= 1;we cycle through the conditionaldistributions sampling#(1)1 from ¼³#1 j#(o)2 ;:::;#(o)k
´ #(i2) from ¼³#2 j#(1)1 ;:::;#(ko) ´ up to#(ik)from ¼³#kj#(1)1 ;:::;#(1)k¡1 ´ : N ext,weseti= 2 andrepeatthecycle.A fteriteratingonthis cycle,say,M times,thesamplevalue #(M )=³#(M ) 1 ;# (M ) 2 ;:::;# (M ) k ´
can be regarded as a drawingfrom the truejointposteriordensity. O ncethis simulated samplehas been obtained, any posteriormomentofinterestorany marginal densitycan beestimated,usingtheergodictheorem.Convergencetothedesired distribution can bechecked as suggested in G elfand and Smith (1 990 ).
InordertoapplytheG ibbs samplertoourpanelV A R modelweneedtospecifypriorinforma-tion sothatthe condimodelweneedtospecifypriorinforma-tionalposteriordistribumodelweneedtospecifypriorinforma-tion forcomponents ofthe parametervectorcan be obtained analytically.R ecallthatourhierarchicalmodelis given by:
Yt = W t® + Z t¸t+ ut; ®i = SN ®¹+ "i ¹ ® = ¹+ v ¸t = B ¸t¡1+ et whereut»N (0;§-H);"i»N (0;V -Ei-1Ei);v »N (0;ª);¸o »N (0;V --2)et»
N (0;V -´-2)and´ is the tightness on time variation: if´ = 0 andB =Ithen¸ is time
in-variant.W eassumethatthecovariancematrices areindependent,thatV ;ª,´,and¹ areknown and that §»iW N (¾o;M o),H »iW G (ho;Po);-1 »iW k(w1;W 1), and -2 »iW k(w2;W 2),
where the notation ©»iW p(v;Z )means thatthe symmetric positive de¯nite matrix © follows
ap{ dimensionalinverted W ishartdistribution withvdegrees offreedom and scale matrixZ :W e also assume thatforeach ofthese distributions the degrees offreedom and the scale matrix are known.T heseassumptions areinconsequentialandtheanalysis goes through,evenwhenconsistent estimates aresubstituted forthetrueones.
G iven this priorinformation,theposteriordensityoftheparametervector#= (®;§;H;® ;¹-1, f¸tgTt= 0;-2)is given by
¼(#jYT; FT)/ f(YT j#T; FT)p(#jFT) (1 9)
whereYT = (Y1;:::;YT)is thesampledataandp(#jFT)is thepriorinformation availableatT.
G iven the di±culty to obtain marginal posteriors directly from the integration of (1 9), we iterate on the conditionaldistributions ofthe parameters, which can easily be obtained from the conditionalposterior(1 9). T o dealwith the presence oftime varying parameters we adaptthe results ofCarterandKhon(1 994)andChib andG reenberg(1 996).Infact,conditionalonf¸tgTt= 0,
the distribution ofthe remaining parameters can be derived withoutdi±culty. L etáx be the
vector#containingallthe parameters butx.T hen the conditionaldistributions forparameters otherthanf¸tgare:
-1 j á-1; YT; FT »iW k
³
w1+ N G ;W^1
-2 j á-2; YT; FT »iW k ³ w2 + T G ;W^2 ´ § j á§; YT; FT »iW N ³ ¾o+ G T ;M^o ´ (20 ) H j áH ; YT; FT »iW G ³ ho+ N T ;P^o ´ ® j á®; YT; FT »N ³ ^ ® ;V^® ´ ¹ ® j á¹®; YT; FT »N ³ ®¤;V^¤´
wheretheexpressions for ^W 1;W^2;M^o;P^o;®;^ V^®; ®¤;V^¤aregiven in theappendix.
FollowingChib (1 996)the parametervector¸tcan be included in the G ibbs samplervia the
distribution ¼(¸o;:::;¸T jYT; FT;ÃT)whereÃt´#¡f¸tgt.W ecanre{ writesuchadistributionas
¼(¸T jYT; FT;ÃT)£¼(¸T¡1jYT; FT;ÃT¡1;¸T)£¢¢¢£¼(¸ojYT; FT;;Ã0;¸1;:::¸T) (21 )
A drawfrom thejointdistribution can beobtained bydrawing~¸T from¼(¸T jYT; FT;ÃT);then
~ ¸T¡1 from¼ ³ ¸T¡1 jYT; FT;;ÃT¡1;¸~T ´ and soon.L et¸s= (¸
s;:::;¸T)andYs= (Ys;:::;YT)for
s·T.T hedensityofthetypicalterm in (21 )is ¼³¸tjYT; FT;Ãt;¸t+ 1 ´ / ¼³¸tjYt; Ft;;Ãt ´ ¼(¸t+ 1 jYt; Ft;Ãt¡1; ¸t)f(Yt+ 1;¸t+ 1 jYt; Ft; ¸t; ¸t+ 1) / ¼³¸tjYt; Ft;Ãt ´ ¼(¸t+ 1jFt;Ãt¡1; ¸t) (22)
T he lastrow follows from the factthat, conditionalon¸t+ 1;the jointdensity of(Yt+ 1;¸t+ 1)is
independentof¸tand,conditionalon¸t,¸t+ 1 is independentofYt.
T he second density of(22)in G aussian with moments½¸tand §". T he ¯rstwas derived in
section 3, and it is G aussian with mean ^¸tjt= ^¸tjt¡1 + ^-tjt¡1Z t0Á¡12 2
³
Yt¡Z t¹¡Z t¸^tjt¡1
´
and variance ^-tjt= ^-tjt¡1¡-^tjt¡1Z t0Á¡2 21Z t-^tjt¡1. H ence,¼¡¸tjYT; Ft;Ãt;¸t+ 1¢»N (^¸t;-^t)where
^ ¸t= ^¸tjt+ ½¡1M t ³ ¸t+ 1¡½¸^tjt ´ ;^-t= ^-tjt¡M t-t+ 1j;tM t0andM t=½2-^tjt-^¡1t+ 1jt.
T obe concrete the followingalgorithm can be used tosamplef¸tg:¯rst, startingfrom given
initialconditions, we run the Kalman ¯lter to recursively get ^¸tand ^-t;then we simulate ~¸T
from a normalwith mean ^¸TjT and variance ^-TjT;~¸T¡1 fromN
³ ^ ¸T¡1;-^T¡1 ´ , and so on until ~ ¸o is simulated fromN ³ ^ ¸o;-^o ´ where, foreacht;¸^t= ^¸tjt+ ½¡1M t ³ ~ ¸t+ 1¡½^¸tjt ´ and ^-t= ^ -tjt¡M t-^t+ 1j;tM t0.
O ne specialcase ofthe setup described in this subsection deserve some attention. Suppose informativepriors on alltheparameters exceptthaton H ,whoseprioris nowdi®use,sothatthe priorfor§uis di®useas well.T hen thesetup resembles theN ormal-D i®usepriorofKadiyalaand
Karlsson(1 997)andimplies thatposteriordependenceamongthecoe±cients ofdi®erentequations obtains even when there is priorindependence.H ence, the majordi®erence ofourpriorwith the speci¯cation used by these authors is thatwe use a three stage hierarchy, sothatboth the mean and thevarianceof°tarerandom variables,whiletheytakethemeanand thevarianceof°ttobe
5 FO R ECA ST IN G 1 5 ¯xed.N ote alsothatourspeci¯cation does notrestrict§utobe diagonaland therefore permits
complicated interactions amongvariables within and across countries.
Finally, itis worth mentioning thatin allthe setups we have considered in this section, our priorspeci¯cation maintains a knonekerstructure forthe statisticalmodel. Such a speci¯cation is usefulsince,on onehand,itallows tohandlethecomputations forrelatively largesystems in a simplefashionand,ontheother,imposes symmetryrestrictions whichappeartobedesirableinan unrestrictedV A R system ofthetypeexaminedhere.Clearlytheserestrictionsmaybeinappropriate forstructuralorrestricted V A R systems andalternativespeci¯cations,alongthelines ofSims and Z ha(1 998),should beused.
5 Forec asting
O nce posteriorestimates are obtained, forecasts can be computed.In orderto obtain multistep forecastingformulas forapanelV A R and tocomputeturningpoints probabilities,itis convenient torewrite(1 )in acompanion V A R (1 )form
Yit= N
X
j= 1
BitjYjt¡1+ Ditzt+ Uit (23)
whereYitandUitareG p£1 vectors,Bitjis aG p£G pmatrixandDitis aG p£qmatrix.
Stackingfori,and repeatedlysubstitutingwehave Yt= "h¡1 Y r= 0 Bt¡r # Yt¡h + h¡1 X s= 0 "s¡1 Y r= 0 Bt¡r # Dt¡szt¡s+ hX¡1 s= 0 "s¡1 Y r= 0 Bt¡r # Ut¡s (24) or yt=J "h¡1 Y r= 0 Bt¡r # Yt¡h + hX¡1 s= 0 ©stDt¡szt¡s+ hX¡1 s= 0 ©stut¡s (25)
where ©st=Qs¡1r= 0Bt¡r, andJ=IN -J1,J1 = [IG 0 ]andJis a selection matrix such that
JYt=yt,JUt=utandJ0JUt=Ut.T heexpression in (25)can beused tocomputetheh{ steps
ahead forecastoftheN G{ dimensionalvectorYt.
First,wecomputea"point"forecastforyt+h.T heforecastfunction is given by
yt(h)=J "h¡1 Y r= 0 Bt+h¡r # Yt+ hX¡1 s= 0 ©st+hDt+h¡szt+h¡s (26) or,recursively yt(h)=JB~t+hYt(h¡1 )+ ~Dt+hzt+h
where ~Dt+h is theN G £qmatrix [d1td2t::::dN t]0and ~Bt+h =diag(B1t;B2t;:::;Bnt)withBit=
³
B1
it;B2it;:::;BitN
´
.O newaytoobtainah{ step aheadforecasts is tousetheposteriormeanof ~Bt+h
t.Estimates fortheposteriormeanofthecoe±cients can beobtainedfrom therecursiveformulas for¸t(and, consequently, for°t)usingexpressions like (9)orby drawingfrom distributions like
(20 )and (21 )in a recursive fashion.Callthis estimates ^Bt+hjtand ^Dt+hjt.T he forecasterroris
yt+h ¡y^t(h)=Phs= 0¡1©st+hut+h¡s+ [yt(h)¡y^t(h)]. T o measure the forecasting performance it
is usefulto compute the M ean Square Error(M SE)orthe M ean A bsolute Error(M A D )ofthe estimated forecastwhich aregiven by
M S E (^yt(h)) = hX¡1 s= 0 ©st+h§u©0st+h + M S E [yt(h)¡y^t(h)] M A D(^yt(h)) = hX¡1 s= 0 jut+h¡sj+ M A D[yt(h)¡y^t(h)]
T he ¯rstterm on the R H S ofeach equation can be obtained using posteriormean estimates of Bt+h¡r and ofUt,conditionalon the information attimet,whileforthesecond term an
approx-imation can be computed alongthelines ofL Äutkepohl(1 991 , p.86{ 89).Clearly, ifa researcheris interested in pointforecasts usingtheaveragevalueoftheparameters,then theprevious formulas applyusingfor ^Bt+hjtand ^Dt+hjttheposteriors derived in section 3.2.
In many situations, itmay be more appealing to compute "average" forecastsh{ step ahead usingthe predictive densityf(Yt+h jFt)=Rf(Yt+h jFt;#)p(#jFt)wheref(Yt+h jFt;#)is the
conditionaldensityofthefutureobservationvectorgiven#,andp(#jFt)is theposteriorpdfof#
attimet.T ocomputeforecasts forYt+h wecansamplefrom thepredictivedensitynumerically.For
eachi= 1;:::;M wedraw#(i)from theposteriordistribution and simulatethevectorY(i) t+h from
the densityf³Yt+h jFt;#(i)
´
.nYt(i+)hoM
i= 1 constitutes a sample, from which we can compute the
necessarymoments.T hevalueoftheforecastisthentheergodicaverage ^Yt+h =M ¡1PMi= 1Y (i) t+h and
its numericalvariancecanbeestimatedusingvar³Y^t+h
´ =M ¡1hQo+ Prs= 1 ³ 1 ¡ s r+ 1 ´ (Qs+ Q0s) i whereQs=M ¡1PMi=s+ 1 h Yt(i+)h ¡Y^t+h ih Yt(i+)h ¡Y^t+h i0 . N otethatsincethecomputationoftheimpulseresponsefunctionfororthogonalizedshocks is a simplecorollaryofthecalculationofforecasts,theapproachweprovideheretocalculatepointand averageforecasts canalsobeused tocomputeimpulseresponses.Infact,giventheinformationup totimet,computingimpulseresponseatt+ his equivalenttocalculatingthedi®erence between theconditionalforecasts att+ h,giventhatatt+ 1 therehas beenaoneunitimpulseinoneofthe orthogonalshocks,andtheunconditionalforecast,i.e.withthevalueofthevectorthatwouldhave occurredwithoutshocks (seeKoop (1 992)foranapplicationtostructuralV A R models).T his idea is exploited in arecentpaperbyW aggonerand Z ha(1 998).T heauthors,usingaversion of(25), develop twobayesianmethods forcomputingprobabilitydistributions ofconditionalforecasts.T he lastterm in(25)represents thedynamicimpactofstructuralshocks whicha®ectfuturerealizations ofvariables through the impulse responsematrix ©st.W ith conditions orconstraints imposed on
this lastterm wecan producewhattheycallconditionalforecasts.
In ordertocompute structuralimpulse responses and theirerrorbands we mustwork with a structuralV A R ,e.g.imposesomerestrictions on thecontemporaneous coe±cientmatrix.A prior
5 FO R ECA ST IN G 1 7 (° atorinformative)can then beassigned tothenon-zeroelements ofthis matrix,as suggested by Sims andZ ha(1 998).T heextensionoftheirapproachtopaneldateis howevernotstraightforward and wepostponethis issuetofuturework.
T urningpointpredictions can alsobecomputed from thepredictivedensity offutureobserva-tions (seein Z ellner,H ongand M in (1 991 )).L etus de¯neturningpoints as follows:
D e¯nition5.1 A downward turn foruniti attimet+ h+ 1 occurs ifSit+h the growth rate of the reference variable (typically, G N P )satis¯es forallh Sit+h¡2; Sit+h¡1 < Sit+h > Sit+h+ 1.A n upwardturn forunitiattimet+ h+ 1 occurs ifthe growth rate ofthe reference variable satis¯es
Sit+h¡2; Sit+h¡1 > Sit+h < Sit+h+ 1.
Similarly,wede¯neanon-downward turn and anon-upward turn:
D e¯nition5.2 A non-downwardturn forunitiattimet+ h+ 1 occurs ifSit+h satis¯es forallh
Sit+h¡2; Sit+h¡1 < Sit+h ·Sit+h+ 1.A non-upward turn forunitiattimet+ h+ 1 occurs ifthe growth rate ofthe reference variable satis¯esSit+h¡2; Sit+h¡1 > Sit+h ¸Sit+h+ 1.
A lthough thereareotherde¯nitions in theliterature(seee.g.L ahiri and M oore(1 991 ))this is themostusedoneanditsu±ces forourpurposes.L et ~f(Yi;t+h jFt)=RYp;t+hf(Yt+h jFt)dYp;t+h be themarginalpredictivedensityforthevariables ofunitiafterintegratingtheremainingp variables and letK(S1
it+h jFt)=R:::Rf(Sit1+h :::SitG+h jFt)dSit2+h :::dSitG+h be the marginalpredictive
density forthe growth rate ofthereference variable, which we ordertobe the ¯rstin thelist, in uniti.
T akenowthesimplestcaseofh= 0 .T ocomputetheprobabilityofaturningpointwehaveto calculateSit1+ 1.G iventhemarginalpredictivedensityK,theprobabilityofadownturninunitiis
PDt=P r(Sit1+ 1 < Sit1jSit1¡2;S1it¡1 < Sit1;Ft) = ZS1 it ¡1 K ³ Sit1+ 1jSit1¡2;Sit1¡1;Sit1;Ft ´ dSit1 (27)
and theprobabilityofan upturn is
PU t=P r(Sit1+ 1> Sit1jSit1¡2;Sit1¡1 > Sit1;Ft) = Z1 S1 it K ³Sit1+ 1 jSit1¡2;Sit1¡1;Sit1;Ft ´ dSit1 (28)
U sing a numericalsample from the predictive density satisfyingSit1¡2; Sit1¡1 < Sit1, we can
approximatetheseprobabilities usingthefrequencies ofrealizations which areless then orgreater thenSit.W ith a symmetric loss function, minimization ofthe expected loss leads to predictthe
occurrenceofturningpointatt+ 1 ifPDt> 0:5 orPU t> 0:5.
Forh6= 0 theprobabilityofaturningpointcan becomputedusingthejointpredictivedensity forallfutureobservations,i.e.in thecaseofadownturn,
PDt+h =P r(Sit1+h+ 1 < Sit1+h > Sit1+h¡2;Sit1+h¡1 jFt) = ZS1 it ¡1 Z1 S1 it Z1 S1 it K³Sit1+h+ 1 < Sit1+h > Sit1+h¡2;Sit1+h¡1 jFt ´ dSit1+hdSit1+h¡1dSit1+h¡2 (29)
G iventheavailablepaneldatastructurewemayalsobeinterestedincomputingtheprobability thataturningpointoccursjointlyform·N units ofpanel.Forexample,wewouldliketocompute theprobabilitythatatt+ 1 therewillbearecessioninEuropeancountries.L et ~K(St1+h jFt)bethe
jointpredictivedensity ofthe referencevariableforthemunits ofinterest.T hen the probability ofadownturn is: PDtm =P r(Sit1+ 1< Sit1 i= 1;:::mjSit1¡2;S1it¡1 < Sit1;Ft;) = ZS1 1t ¡1 ::: ZS1 m t ¡1 ~ K ³St1+ 1jSt1¡2;St1¡1 < St1;Ft ´ dS1t1 :::dSm t1 (30 )
6 Anappl
ic ation
Inthis sectionweapplythemethodologytotheproblem offorecastinggrowthrates andpredicting turningpoints in the G -7countries.Foreach country we considerthreenationalvariables (G N P, realstockreturns andrealmoneygrowth)andaworldone(themedianrealstockreturninO ECD countries)which is assumed tobeexogenous in each equation.H encethereare21 variables in the panelV A R .T hese variables are chosen aftera rough speci¯cation search overabout1 0 variables because they appeartohave the highestin-sample pairwise and multiple correlation with output growth.A mongthevariables wetriedarethenominalinterestrate,theslopeoftheterm structure and in° ation. D ata is sampled quarterly from 1 973,1 to 1 993,4 and taken from IM F statistics. D atafrom 1 973,1 to1 988,4 is used toestimatetheparameters and datafrom 1 989,1 to1 993,4 to evaluatetheforecastingperformanceand topredictturningpoints.
W e comparethe forecastingperformance ofourpanelV A R speci¯cations with thoseobtained with othermodels suggested in the literature.A s a benchmark we ¯rstrun twoversions ofa tri-variable V A R (2)modelforeach country separately.T he ¯rstone is an unrestricted (V A R ).T he second aweaklyrestricted V A R (B V A R )whereweuseastandardL itterman-priorwithameanof oneonthe¯rstlag,ageneraltightness of0 .1 5,nodecayinthelags and aweightof0 .5 onthelags ofothervariables.Since these twomodels donotexploitcross sectionalinformation nordothey allowfortimevariation,they can beused as abenchmarktomeasuretheimprovements obtained byspeci¯cations which allowanyofthesetwofeatures in themodel.
A lso forcomparison, we run a single equation A R (3)modelforG N P growth foreach single country, augmented with twolags ofrealstock returns,1 lagofrealmoney balances and one lag ofthemedian world realstockreturn.T his is thespeci¯cation used byG arciaFerreretal(1 987), Z ellnerandH ong(1 989)andZ ellner,H ongandM in(1 991 )toforecastannualgrowthratesofoutput in1 8 countries.W ith theextendedsampleandthehigherfrequencyofthedatawehaveavailable, we con¯rm theirresults forallofthe G -7 countries. T his modelrepresents a restricted version ofthe previous unrestricted V A R where insigni¯cantlags are purged from the speci¯cation.T he forecastingpowerofthis modelis measured when parameters areestimated with O L S (O L S)and with thethreeshrinkageprocedures:aridgeestimator(R ID G E),an estimatorobtained assuming anexchangableprioronthecoe±cients(asinG arciaFerreretal.(1 987))(EX CH A N G EA B L E)and an estimatorobtained usingag-prior(as in Z ellnerand H ong(1 989))(G -P R IO R ).T hetwolatter estimators attemptto improve upon O L S by combining the information coming from each unit
6 A N A P P L ICA T IO N 1 9 with theonefrom thepooled sample.T hey di®erin theway they combineindividualand pooled information.N oticethatnoneoftheseestimators allows fortimevariations in thecoe±cients.
Finally,as aterm ofcomparison,weuseaversion ofthepanelV A R speci¯cation suggested by B allabrigaetal(1 998)(P B V A R ).T hismodelspeci¯cationdoesnotusetheinformationcomingfrom thecross sectioneveryvariableis treatedinthesamewayregardless ofthecountrywhereis from -butallows fortimevariations inthecoe±cients ofthemodel.T hemodelhas thesamestructureas D oan,L itterman and Sims (1 984)and assumes thatthecoe±cientvector¯tfortheentiresystem
has an A R (1 )structure ofthe form¯t=M ¯t¡1 + utwhereut, conditionalon the information
available, is normalwith mean zero and variance §u. T he matrices¯0,M , and §udepend on
7 hyperparameters: ¯ve parameters controlling the structure of§u0 (a generaltightness (µ1), a tightness onvariables ofthesamecountry(µ3),atightness onthevariables ofothercountries (µ4),
ageometriclagdecay with parameter(µ2), and atightness on world variables (µ5));a parameter
describingthestructureofM (µ6);and aparametercontrollingthepriormean on the ¯rstlagof
¯0 (µ7).T able1 reports theoptimalvalues selectedbymaximizingthein-samplepredictivedensity
ofthemodelwith asimplexalgorithm.
W eproduceforecastsfrom twoversionsofourpanelV A R model:onewithamodi¯edM innesota-prior(PA N EL 1 ),andonewithafullyhierarchicalspeci¯cation(PA N EL 2).Intheformer,thenine priorparameters are selected to maximize the predictive density usinga simplex method.T heir optimalvalues arereportedintable2.ForbothP B V A R andPA N EL 1 forecastsarecomputedusing theposteriormeanofthecoe±cients,afterwehaveplugged-intheestimatesofthepriorparameters intheformula.ForP A N EL 2 posteriorestimates ofthecoe±cients arecomputednumericallyusing M CM C methods and forecasts aredirectly obtained from theseestimates.
In setting up the panelV A R models we assume thatH = V, whereV is known. For the PA N EL 1 speci¯cationwecomputethescalefactorsV andthematrix§uas follows.W eestimatea
trivariateV A R foreach countryand taketheaverageoftheestimated variance{ covariancematrix oftheresiduals across countries as ameasureofV.Furthermore,foreach ofthethreevariablewe estimatea7{ variableV A R (thesamevariableacross countries)and storethevariance{ covariance matrices oftheresiduals.A n estimateof§uis obtained as:
^ §u= 3 X j= 1 0 B B B B @ ¾1 0 ¢¢¢ 0 0 ¾2 ¢¢¢ 0 . . . ... ... ... 0 0 ¢¢¢¾7 1 C C C C A j -0 B @ 0 0 0 0 vj 0 0 0 0 1 C A
wherethe¯rstmatrixcontains onthediagonaltheestimatedstandarddeviations obtainedbyrun-ningthe three 7{ variate V A R s;while the second matrix contains justone elementdi®erentfrom zero, the (j;j)element, which is obtained from the diagonalofthe matrixV.Forthe PA N EL 2 speci¯cation weneed tochoosethescaleand thedegrees offreedom in thevarious W ishartdistri-bution.W e stillsetH=V withV estimated as before.FollowingKadiyalaand Karlsson (1 997) wesetthedegrees offreedom¾0 =N + 2 + (T ¡p)¤G ;!1 =k+ 2 + N + g;!2 =k+ 2 + (t¡p)¤G
whilethescalematricesM 0,W 1 andW 2 aresuchthat§u;¢ ;§²havethesamestructureas in the
W ecomparetheforecastingabilityofvarious models usingboth theT heil-U Statistics and the M ean A bsoluteD eviation (M A D )at1 and 4 periods ahead.T hesestatistics arereported in table 3.N otethatthevarious speci¯cations weusearein increasingorderofcomplexityand ° exibility. T herefore, ateach stage we can assess the forecasting improvements obtained adding one extra featuretothemodel.
T oexaminetheperformanceofvarious models as business cycleindicators wecomputeturning points predictions oneperiodahead.FollowingZ ellneretal.(1 991 ),wecomputethetotalnumber ofturning points, the numberofdownturns and no-downturns, and the numberofupturns and no-upturns in the sample (across allcountries)and foreach procedure we reportthe numberof correctcases in table4.
Finally,foreach model,wecomputetheprobabilitythattherewillbeadownward turn in the growthrateofU S outputin1 989:1-1 993:4,giventheinformationavailablein1 988:4.A ccordingto the o±cialN B ER classi¯cation the longexpansion ofthe 1 980 's terminated in 1 990 :3 and itwas followed by abriefand shallowrecession.T heprobabilities fortheninemodels foreach ofthe1 6 periods weconsiderarepresented in table5.
T he forecastingperformances ofunivariate O L S, ridge and exchangeable procedures are very similar.T heminimum and maximum values oftheT heil-U across countries atoneand foursteps forthelattertwoare slightly smaller,butthemean and themedian atboth steps are practically identical.O n the otherhand,aunivariatemodelwhere theparameters areshrunk with ag-prior is somewhatbetterthan O L S in allthe dimensions: the maximum, the median, the mean and theminimum valueacross countries oftheT heil-U atboth steps aresigni¯cantlylowerthan those obtained with O L S.
U nrestricted V A R models are notvery successfulin forecastinggrowth rates ofoutput, given thelargenumberofparameters tobeestimated.T his noticeableinparticularinthecaseofJapan, G ermanyandtheU K wheretheT heil-U aresigni¯cantlyworsethanthoseobtainedwithunivariate speci¯cations attheonestep horizon.H owever,unrestrictedV A R modeloutperform allunivariate speci¯cations atthe fourstep horizon.H ence, the presence ofinterdependencies across variables helps in predicting the evolution ofthe growth rate ofoutput in the medium run. B V A R are signi¯cantly betterthan V A R and univariate approaches atthe onestep horizon.In terms ofthe median value the gains are ofthe orderof5-6% overunivariate speci¯cations and ofmore than 1 0 % overthe unrestricted V A R .H owever, the performance atthe fourstep horizon turns outto be inferiorto the one of unrestricted V A R , and comparable to the one of univariate shrinkage procedures.T his is tobeexpectedsincetoimprovetheperformanceatshorthorizons B V A R tend toreduceboththememoryandtheinterdependencies ofthesystem,whichwehaveseenareuseful exactly when medium-longrun forecasts havetobemade.
A dding time variation in the coe±cients and interdependencies across countries substantially improves the forecasting performance both atshortand atmedium horizons. Forexample, the median T heil-U atonestep goes from 0 .85 with asimpleB V A R to0 .82 with thepanelversion of this modeland for5 countries theT heil-U is lowerbyas much as 1 0 %.Similarly,themean across countries drops byabout3% withtheP B V A R speci¯cation.T heimprovementis noticeablealsoat longerhorizons.T hedistributionoftheT heil-U across countries atthefourstep horizonis similar totheoneobtained with aunrestricted V A R ,which is thebestamongthebenchmarkmodels.
6 A N A P P L ICA T IO N 21 O urre¯nementofthe L itterman's prior, which allowforboth cross sectionaland time series a-priori restrictions, gives a performance which is essentially similar to the one ofthe P B V A R modelboth at the one and atthe fourstep horizons. Few features ofthe optimally estimated parameters are worth discussing.First, whileµ6, the time variation parameterin the variance of
¸ is di®erentfrom zero,itdoes notappeartoadd much totheperformanceofthemodel.H ence, atleastwith quarterlydata,allowingforheteroskedasticitydoes nothelp in improvingthequality oftheforecasts.Second,whilein the P B V A R ,thecoe±cientvectorevolves with apersistenceof 0 .95 butwithverysmallvariance,inourP A N EL 1 speci¯cation thetimevaryingcomponentofthe coe±cients is closetobeawhitenoise.N otethatthis di®erenceis inconsequentialforforecasting andcanbeexplainedbyexaminingtheroleoftheparametersregulatingthecrosssectionalprior(i.e. thetightness on® and ¹®).T heseparameters forceahighdegreeofcoherenceacross countriesinthe timeinvariantcomponentandleavethetimevaryingcomponenttorandomlyevolve.IntheP B V A R this distinctionis notpossibleandtoproducecoe±cients whicharealmostconstantovertimeitis necessarytohaveclosetoarandom walkdynamics coupledwith asmallvariance.U singequation (9),onecanseeinfactthatcoe±cients ofthePA N EL 1 modelareapproximatelyconstantovertime andaretightlylinked toeach otherbecauseoftherestrictions imposed on®i.T heomissionofthe
¯xed e®ectcomponent, which is precisely whatthe P B V A R does, biases upward estimates ofthe persistence parameterand this may explain why the twoestimated speci¯cations aresodi®erent. T hird,themaximizedvalueofpredictivedensityofthePA N EL 1 modelis signi¯cantlyhigherthen theoneoftheP B V A R model(-36.90 vs.-985.35)suggestingthattheourspeci¯cation¯ts thedata forthein-sampleperiodbetter.H owever,this superiorin-sample¯tappears tobeunimportantfor forecastingout-of-sample.T hatis, the (wrong)restrictions thatthe P B V A R imposes and which biases thepersistenceparameterofthetimevaryingcoe±cients donottranslatein poorforecasts atthe horizons we consider.W e conjecture thatthis may have todowith the peculiarity ofthe forecastingsamplemorethan with truesimilarities between thetwospeci¯cations.
T he performance of the PA N EL 2 speci¯cation is also comparable to the one obtained with P B V A R atthe one step horizon. H owever, while the ranking ofthe T heil-U across countries in P B V A R and PA N EL 1 were identical, there is some reshu²ing with the PA N EL 2 speci¯cation. T hatis,themodelis somewhatbetterforJapan and Franceand somewhatworsefortheU S and Italy.A tthefourstep horizon theperformanceofthemodelis signi¯cantly worsethan anyother model.W hile we have notbeen able to¯nd a reason forthis result, we conjecture thatthis has todowith the factthatthe presence ofalarge amountofrandomness in the speci¯cation ofthe modelcompounds atlonghorizons andworsens signi¯cantlyits performance.Infact,thedi®erence betweenPA N EL 1 andPA N EL 2 speci¯cations,apartfrom problem ofprecisionofestimates is only in thefactthatthereis an additionallayerofuncertaintyin thepriorofthemodel.
T herelativeperformanceofthevarious models with theM A D is somewhatsimilartotheone obtained with theT heil-U atboth horizons.H owever,fourfeatures deserveacomment.First,all univariateshrinkageprocedures appeartobebetterthanO L S attheonestep horizon.T hesameis trueatfoursteps horizons exceptforthecaseofg-prior,which is nowsigni¯cantlyworse.Second, unrestricted and simple B V A R display a somewhatmediocre performance both atone and four steps horizons.In general,thedistribution oftheM A D across countries is moreconcentrated but themean and the median areabovethoseobtained with univariate shrinkageapproaches.T hird,
the improvements obtained with panelV A R approaches are signi¯cantand ourre¯nementofthe L itterman'spriorproducesthebestdistributionofM A D attheonestephorizon.T heimprovements areprimarilyconcentrated forthosecountries which arein thecentralpartofthedistribution and this is re° ected in thelowermedian value we obtain.Fourth,the PA N EL 2 speci¯cation is better thananyotherwhenweusethemeanM A D acrosscountrytomeasuretheforecastingspeci¯cations at both horizons. T hat is, PA N EL 2 produces a distribution ofM A D across countries which is centered belowthe one obtained with othermodels and more concentrated.N otice alsothatthe signi¯cantforecastingdi®erences produced by PA N EL 2 fortheT heil-U and theM A D atthefour step horizon probablyhavetodowith thedi®erentway thetwocriteriatreatforecastingoutliers. In sum,usinginterdependencies,addingtime variation in the coe±cients and usingcross sec-tionalrestrictions in the priorforthe coe±cients helps in improving forecasts atshort-medium horizon.N evertheless,itshouldbepointedoutthatthedistribution offorecastingstatistics across countries is verywide,forexample,theM A D forItalyis 6 times theoneoftheU S.T his di®erences indicate thatthe process forthegrowth rate ofG D P in some countries does notsharemuch fea-tures withthegrowthrateofG D P ofotherG -7countries andthatsigni¯cantimprovements onthe results wepresentcanbeobtained byrestrictingattentiontothesubsetofthecountries whichare moresimilar.A lsonoticethattheforecastingperformanceforU S andCanadaG D P growthis very similaracross speci¯cations andjointlyimproves withthecomplexityofthemodel,con¯rmingthat there are forecastingexternalities which can be obtained by cross-sectionally linkingthe national models forthetwocountries.
H owgood arevarious approaches in predictingturningpoints? O utof96 totalactualturning points in the sample, univariate approaches recognize between 72 and 75. D i®erences primar-ily emerge when we try to predictupturns and non-upturns and forthis type ofturning points, Z ellner's-gapproach is betterthan theothers.U nrestricted V A R models arevery poorin this di-mension and recognizeabout1 0 % less turningpoints than Z ellner's-gapproach.T heperformance oftheB V A R modelis comparabletotheoneofunivariateR idgeandExchangeableapproaches but, contrarytothem,itpredicts upturns and non-upturns betterthandownturns and non-downturns. T he performance ofthe P B V A R modelis surprisingly poor:itis the second worstin recognizing the totalnumberofturning points and is comparable to unrestricted V A R s in predicting down-turns and non-downdown-turns. Finally, ourtwo P anelapproaches produce 73 and 74 turning point forecasts and recognize the same numberofupturns and non-upturns. Comparatively speaking, theysubstantiallyimproveoverP B V A R and arecompetitivewith thebestapproaches.
T hree furtherconclusions can be drawn from table 4. First, di®erent models are better in recognizingdi®erenttypes ofturningpoints.Ifpredictingdownturns (andnon-downturns)is more importantthan predicting upturns (and non-upturns)ourresults suggestthatV A R , B V A R and P B V A R shouldnotbeused.Second,whileinterms oflinearforecastingstatistics therewas aclear rankingofprocedures,with morecomplicated ones doingabetterjob,when welookatnonlinear forecasting statistics, simple univariate approaches, and O L S in particular, are as good as other morere¯ned approaches.T hird,P anelV A R models ofthetypewehaveproposed doabetterjob thananyotherprocedurewhenwejointlyuselinearandnonlinearstatistics tomeasureforecasting performance.
7 CO N CL U SIO N S 23 knowifthey are also good in predictinga speci¯cepisode ofinterest, i.e., the downward turn in realactivityoccurredintheU S in1 990 :3.T his is interestingbecausealternativeapproaches,which wereforecastingpretty wellin the sample 1 970 -1 980 ,failed to¯nd any relevantsigns in the data thatwould predictthata downturn and a shortrecession were forthcoming (see e.g. Stock and W atson (1 993)). Interestingly enough, and contrary to mostforecasting models, allprocedures predictthatthereis asigni¯cantprobabilitythatapeakineconomicactivitywilloccurat1 990 :3. Forunivariateprocedures this probabilityis muchlargerthanthethresholdof0 .5 whichweuseto considerthedateadownward turn.In factallfourunivariateapproaches predicttheexistenceof a peak with probability above 0 .64.Single country V A R , with and withouta B ayesian priorare worse than univariate procedures (probability 0 .32 and 0 .36 respectively)butthis may be due to the largernumberofparameters to be estimated with the information available at1 988:4. T he P B V A R speci¯cationis overwhelminglypredictingadownwardturnin1 990 :3 (probabilityis 0 .82) and does notproduce any false alarm in the neighborhood ofthis date.T he second P anelV A R approachimproves oversinglecountryV A R substantiallyandaproduceprobabilityofadownturn in 1 990 :3 which is comparable with those ofunivariate approaches.T he performance ofthe ¯rst P anelapproach is poorandfails toproduceaprobabilityin excess of0 .5 in 1 990 :3.N otealsothat whileunivariateapproaches havethetendencytoproduceafalsealarm in 1 989:4,probablydueto thestockmarketcrash ofthefallof1 989,theprobabilities produced byV A R and B V A R atdates otherthan1 990 :3 aresmallandneverexceed0 .5.T heP B V A R model,ontheotherhand,produces ahighprobabilityofadownturnin1 991 :3 adatewhereadownturnmaterialized.T hesecondpanel speci¯cation alsoproduces ahigh probabilityofadownward turn in 1 991 :3 whiletheprobabilities atotherdates are small.Finally notice thatthe peak in 1 989:2 is missed by allapproaches:the ones which give highestprobability tothis eventare the P B V A R (0 .42)and the ¯rstP anelV A R approach (0 .41 ).
In conclusion, ourproposed B ayesian P A N EL V A R approach is atleastas good as any other approach we have examined and in many cases improves the forecastingperformance ofexisting speci¯cation. T his is true when we compare procedures using linearand non-linearforecasting statistics and when welookatspeci¯chistoricalepisodes.
7 Concl
usions
T he task ofthis paperwas to describe the issues ofspeci¯cation, estimation and forecasting in a macro-panelV A R modelwith interdependencies.T he pointofviewused is B ayesian.Such an approachhas beenwidelyusedintheV A R literaturesincetheworks ofD oan,L ittermanandSims (1 984), L itterman (1 986), and Sims and Z ha (1 998)and provides a convenientframework where one can allowforboth interdependencies and meaningfultime variations in the coe±cients.W e decompose the parametervectorinto two components, one which is unitspeci¯c and the other which is time speci¯c.W especify a° exible prioron these twocomponents which parsimoniously takes into accountpossible interdependencies in the cross section and allows fortime variations in theevolution oftheparameters overtime.T he priorshares features with thoseofL indlay and Smith(1 972),D oan,L ittermanandSims (1 984)andH siaoetal.(1 998)anditis speci¯edtohavea hierarchicalstructure,whichallows forvarious degrees ofignorancein theresearcher's information
abouttheparameters.
B ayesian V A R s areknown producebetterforecasts than unrestricted V A R and,inmanysitua-tions,A R IM A orstructuralmodels (Canova(1 995)forreferences).B yallowinginterdependencies and some degree ofinformation pooling across units in the modelspeci¯cation we introduce an additionallevelof° exibilitywhich mayimprovetheforecastingabilityofthesemodels.
W e analyze severalspecialcases ofourspeci¯cation and computeB ayesian estimators forthe mean parameterin the cross section and forthe individualcoe±cients.In some cases analytical formulas forthe posteriormean are availableusingstandard formulas.W heneverthe parameters ofthe priorare unknown, we employ the predictive density ofthe modelto estimate them and plug-in ourestimates in therelevantformulas in an empiricalB ayes fashion.
Inthecaseoffullyhierarchicalpriors,aM arkovChainM onteCarlomethod(theG ibbssampler) is employed to calculate posteriordistributions. Such an approach is particularly usefulin our setup since itexploits the recursive features ofthe posteriordistribution.R ecursive formulas for multistep,multiunitforecasts,consistentwith theinformation availableateach pointin time,are providedusingtheposterioroftheparameters orthepredictivedensityoffutureobservations.T he predictivedensity offutureobservation is alsoused tocomputeturningpointprobabilities.
T o illustrate the performance of the proposed approach, we apply the methodology to the problem ofpredictingoutputgrowth,offorecastingturningpoints inoutputgrowthandcomputing theprobabilityofarecessionintheG -7usingathreevariables (outputgrowth,realstockreturns and realmoney growth)foreach country in the panel. T o evaluate the modelwe also provide a forecastingcomparison with otherspeci¯cations suggested in the literature.W e showthatour panelV A R approachimproves overexistingunivariateandsimpleB V A R models whenwemeasure the forecasting performance using the T heil-U and the M A D criteria both at the one step and at the foursteps horizons. T he improvements are ofthe orderof5-1 0 % with the T heil-U and about2-4% with theM A D .T heforecastingperformanceofourspeci¯cation is alsoslightlybetter then theoneofaB V A R modelwhich mechanicallyextends theL itterman priortothepanelcase. In terms ofturningpointprediction,thetwoversions ofourpanelapproach areabletorecognize about80 % ofturningpoints inthesampleandtheyturnouttobethebestforthis task,alongwith Z ellner's g-priorshrinkage approach.T he simple extension ofthe L itterman's priorto the panel casedoes poorlyalongthis dimensionand,amongalltheprocedures employedis thesecondworst. Finally,alltheprocedures produceahigh probability ofadownturn at90 :3,the date selected by theN B ER committeetoterminatethelongexpansion ofthe80 's.In this instance,ourapproach is competitive with the bestand avoids the false alarms thatotherapproaches produce atother dates.
W econsidertheworkpresentedinthis paperasthe¯rststep indevelopingacoherenttheoryfor B ayesian P anelV A R models which take intoconsideration both the nature ofinterdependencies, the similarities in the statistical model across units and the existence of time variation in the coe±cients.Extensions ofthetheoryoutlinedhereincludetheformulationofinterestinghypothesis on thenatureoftheinterdependencies,on thesimilarities across units and on timevariations and thedevelopmentoftools toundertakestructuralidenti¯cation in thesemodels.T heworkofSims and Z ha(1 998)is thestartingpointforextensions in this lattercase.
A P P EN D IX 25
Append ix
P roofofproposition1
(i)N oticethat(1 4)and (1 5)can bewritten as ¹ ®¤= ª¤ · Z t0³Z t-^tjt¡1Z t0+ §u+ W t¢ W t0 ´¡1³ Yt¡Z t¸^tjt¡1 ´ + ª¡1¹ ¸ (31 ) ª¤= · ª¡1+ Z t0 ³ Z t-^tjt¡1Z t0+ §u+ W t¢ W t0 ´¡1 Z t ¸¡1 (32) Settingª¡1 = 0 ,theresultfollows.
(ii)T heposteriordistributionof® is normalwithmean®¤=SN ¹+Á12Á¡12 2 [Yt¡Z t(¹+ ^¸tjt¡1)]
and varianceV®¤=Á11¡Á12Á2 2¡1Á2 1 whereÁ11= (SN ªSN0 + ¢ );Á12 = (SN ªSN0 + ¢ )W t0;Á2 1=
W tÁ11;Á2 2 =W tÁ11W t0+ Z t-^tjt¡1Z t0+ §u.In amorecompactwayV®¤can bewritten as
V®¤= · W t0³§u+ Z t-^tjt¡1Z t0 ´¡1 W t+ ¡SN ªSN0 + ¢ ¢¡1¸¡1 : (33) N oticethat(SN ªSN0 + ¢ )¡ 1 = ¢ ¡1¡¢ ¡1S N ¡SN0¢ ¡1SN + ª¡1¢¡1SN0¢ ¡1.H ence,forª¡1 = 0 ,
this expression is equaltoF .M oreovertheposteriormean can bewritten as ®¤=V®¤ · W t0³§u+ Z t-^tjt¡1Z t0 ´¡1³ Yt¡Z t¸^tjt¡1 ´ + ¡SN ªSN0 + ¢ ¢¡1 SN ¹ ¸ (34) U singtheprevious resultand thefactthatF SN = 0 ,theresultfollows.
(iii)T heposteriormean and varianceof¸tcan bewritten as
¸¤t= -¤tnZ t0£W t¡SN ªSN0 + ¢ ¢ W t0+ §u¤¡1(Yt¡Z t¹)+ ^-¡tj1t¡1¸^tjt¡1 o (35) -¤¡1t =Z t0£W t¡SN ªSN0 + ¢ ¢ W t0+ §u¤¡1Z t+ ^-¡1tjt¡1 (36)
T he matrixW t(SN ªSN0 + ¢ )W t0+ §ucan bewritten asZ tªZ t0+ (W t¢ W t0+ §u)and its inverse
is equaltoM ¡1¡M ¡1Z t¡Z t0M ¡1Z t+ ª¡1¢¡1Z t0M ¡1 whereM = (W t¢ W t0+ §u).Settingª¡1=
0;thelastexpressionreduces toM ¡1hI¡Z t¡Z t0M ¡1Z t¢¡1Z t0M ¡1
i
.P remultiplyingthis matrixby Z t0,wegetazeromatrix.H ence,from (35)and (36)-¤¡t 1 = ^-¡tj1t¡1 ¸¤t= ^¸tjt¡1 and theposterior
P roofofproposition2
R ecallthat§¡"1 = 0 implies ^-¡tj1t¡1 = 0:
(i)Considerthe posteriormoments (31 )and (32).[(W t¢ W t0+ §u)+ Z t-^tjt¡1Z t0]¡1 =M ¡1¡
M ¡1Z t
³
Z t0M ¡1Z t+ ^-¡tj1t¡1
´¡1
Z t0M ¡1 whereM was previously de¯ned. W hen ^-t¡j1t¡1 = 0;this
reduces toM ¡1 ¡M ¡1Z t¡Z t0M ¡1Z t¢¡1Z t0M ¡1 which gives a zeromatrix ifpremultiplied byZ t0.
Consequentlyª¤= ª ¹®¤=¹ and theposteriordistribution of¹® is justequaltoits prior.
(ii)Considerthe posteriormoments (33)and (34).W hen ^-¡tj1t¡1 = 0 ,³§u+ Z t-^tjt¡1Z t0
´¡1
= S = §¡1u ¡§¡1uZ t¡Z t0§¡1uZ t¢¡1Z t0§¡1u.Substitutinginto(33)and (34),gives theresult.
(iii)T heproofofthis statementcomes straightfrom (35)and (36).
D e¯nition ofthe matrices forthe G ibbs sampler
^ W 1 = W 1+ X i ¡ AiEi¡A¹¢0V¡1¡AiEi¡A¹¢; ^ W 2 = W 2 + X t (¤t¡½¤t¡1)0V1¡1(¤t¡½¤t¡1) ^ M o = M o+ X t ¡ Yt¡BtW 0t ¢ H¡1¡Y t¡BtW 0t ¢0 ^ Po = Po+ 0 X t ¡ Yt¡BtW 0t ¢0 §¡1¡Y t¡BtW 0t ¢ ^ ® = ^V® à X t W t0(§-H)¡1(Yt¡Z t¸t)+ ¢ ¡1SN ®¹ ! ^ V® = à X t W t0(§-H)¡1W t+ ¢ ¡1 !¡1 ®¤ = ^V¤ à (V --1)¡1 X i Ri®i+ ª¡1¹ ! ^ V¤ = ³N (V --1)¡1+ ª¡1 ´¡1
whereYtisN £G,BtisN £G kandW t= (IG -X t0):M odel(1 )is justarawvectorization of
Yt=BtW 0t+ U t,whereBt= [vecr(B1t);:::;vecr(BN t)]0:H erevecr()is therowvectorization of
a matrix;Bit=Ai+ ¤tEiis aG £kmatrix and the parametervectors®iand¸tin (4)and (6)
