The Importance of Common-Cyclical Features in VAR Analysis: A Monte-Carlo Study (Preliminary Version)

Ensaios Econômicos

The Importance of Common-Cyclical

Fea-tures in VAR Analysis: A Monte-Carlo Study

(Preliminary Version)

Farshid Vahid, João Victor Issler

Os artigos publicados são de inteira responsabilidade de seus autores. As

opiniões neles emitidas não exprimem, necessariamente, o ponto de vista da

Fundação Getulio Vargas.

ESCOLA DE PÓS-GRADUAÇÃO EM ECONOMIA Diretor Geral: Renato Fragelli Cardoso

Diretor de Ensino: Luis Henrique Bertolino Braido Diretor de Pesquisa: João Victor Issler

Diretor de Publicações Cientícas: Ricardo de Oliveira Cavalcanti

Vahid, Farshid

The Importance of Common-Cyclical Features in VAR Analysis: A Monte-Carlo Study (Preliminary Version)/

Farshid Vahid, João Victor Issler Rio de Janeiro : FGV,EPGE, 2010

(Ensaios Econômicos; 352) Inclui bibliografia.

T heImportanceofCommon-CyclicalFeatures inVA R

A nalysis: A M onte-CarloStudy

¤

FarshidV ahid

D epartmentofEconometrics andB usiness Statistics

M onashU niversity

Clayton, V ictoria3168

A ustralia

Farshid.V ahid@ B usEco.monash.edu.au

Joã oV ictorIssler

G raduateSchoolofEconomics – EP G E

G etulioV argas Foundation

P raiadeB otafogo19 0 s. 1125-8

R iodeJaneiro, R J 22253-9 00

B razil

jissler@ fgv.br

A ugust19 9 9 , preliminaryversion, comments arewelcome.

¤_{T his paper was written while Joã o V ictorIsslerwas visiting M onash U niversity, which hospitalty is}

A bstract

D espite the belief, supported by recentapplied research, thataggregate data dis-play short-run comovement, there has been little discussion about the econometric consequences of these data “features.” W e use exhaustive M onte-Carlo simulations toinvestigate the importance ofrestrictions implied by common-cyclicalfeatures for estimates and forecasts based on vectorautoregressive and errorcorrection models. First, weshowthatthe“best” empiricalmodeldeveloped withoutcommon cycles re-strictions neednotnestthe“best” modeldevelopedwiththoserere-strictions, duetothe use ofinformation criteria forchoosing the lag orderofthe two alternative models. Second, weshowthatthecosts ofignoringcommon-cyclicalfeatures in V A R analysis may be high in terms offorecasting accuracy and e¢ciency ofestimates ofvariance decomposition coe¢cients. A lthough these costs are more pronounced when the lag orderofV A R models areknown, theyarealsonon-trivialwhenitis selectedusingthe conventionaltools availabletoappliedresearchers. T hird, we…ndthatifthedatahave common-cyclicalfeatures and theresearcherwants tousean information criterium to selectthe laglength, the H annan-Q uinn criterium is the mostappropriate, since the A kaike and the Schwarz criteria have a tendency to over- and under-predictthe lag length respectivelyin oursimulations.

1. Introduction

Common-cyclicalmovements in detrended economic variables have been soprevalentthat theyhaveacquiredthestatusof“stylizedfacts.” L ucas(19 7 7 )statesthatthemainregularities observed in cyclical‡uctuations ofeconomictimeseries arein theircomovement, which he itemizes as follows:

(i)O utputmovementsacrossbroadlyde…nedsectorsmovetogether. (InM itchell’s

M artins-Filho, A man U llah, and participants ofthe L atin A merican and European M eetings ofthe Econo-metric Society of19 9 9 , whoare notresponsible forany remainingerrors in this paper. Joã oV ictorIssler acknowledges thesupportofCN P q-B raziland P R O N EX .

terminology, theyexhibithighconformity;in modern timeseries language, they havehighcoherence.) (ii) P roductionofproducerandconsumerdurables exhibit much greateramplitudethan does theproduction ofnondurables. (iii) P roduc-tion and prices ofagriculturalgoods and naturalresources have lowerthan av-erage conformity. (iv) B usiness pro…ts showhigh conformity and much higher amplitude than otherseries. (v) P rices generally are procyclical. (vi) Short-term interestrates are procyclical; long-Short-term rates slightly so. (vii) M onetary aggregates andvelocitymeasures areprocyclical.

From anempiricalstandpointcommoncycles havebeenshowntobea“feature” ofava-rietyofmacroeconomicdatasets. Forexample, CampbellandM ankiw(19 89 ) …ndacommon cyclebetweenconsumptionandincomeformostG -7 countries. EngleandKozicki(19 9 3) …nd common internationalcycles usingG N P dataforO ECD countries. ForU S data, Isslerand V ahid(19 9 8) …nd common cycles formacroeconomicaggregates and Engleand Issler(19 9 5) and Carlinoand Sill(19 9 8) …nd common cycles forsectoraland regionalG N P s respectively. Similartomostapplied macroeconomicresearch done in the last…fteen years, these stud-ies investigatedcommon-cyclicalfeatures usingvector-autoregressive(V A R ) orvectorerror-correction (V EC) models.

A lthoughV A R andV EC models havebecomethe“workinghorse” ofmacroeconometric studies, one of theirshortcomings is the excessive number of parameters relative to the average sample size one is usually forced to work with. Forexample, when dealing with post-warquarterly data, and a V A R with three variables and eightlags, there are seventy …vemeanparameters tobeestimatedfrom abouttwohundreddatapoints oneachvariable. Cointegration places some restrictions on V A R coe¢cients, especially when cointegrating vectors arebroughtin from economictheory;seeEngleand G ranger(19 87 ). In thesecases, the reduction in the numberoffree coe¢cients is notoverwhelming. Ifthe three-variable system has one known cointegrating vector, the numberof free parameters reduces from seventy…vetosixtyninebyestimatingaV EC model. Common-cyclicalfeatures canfurther

reducethenumberofconditional-meanparameters estimatedinV EC models byconsidering the restrictions implied by them; see V ahid and Engle(19 9 3). Ifthe three variables in the V EC modelshare one common cycle, then the numberofmean parameters reduces from sixtyninetotwentyseven.

T he objective ofthis paperis to investigate the importance ofrestrictions implied by common-cyclicalfeaturesforforecasts, impulse-responsefunctions, andvariance-decomposition offorecasterrorsofeconomictime-seriesbasedonV A R andV EC models. A sfarasweknow, noworkhas studied thee¤ects oftheserestrictions. H owever, considerablee¤ orthas been putinexaminingtheimportanceoflong-runcomovementconstraints inV A R models, espe-ciallyforforecasting;see, amongothers, EngleandY oo(19 87 ), Clements andH endry(19 9 5), and L inandT say(19 9 6).

A s shownbyEngleandY oo, theforecastinggains ofimposinglong-run constraints hap-pens as thehorizongets large. Infact, intheirsimulations, theunconstrainedV A R forecasts betterthan theV EC modelforshorthorizons. B ecauseforlonghorizons forecastinguncer-taintygetshelplesslylargetime-seriesmodelsaremostusefulforforecastinginshorthorizons. H ence, thepayo¤ ofinvestigatingtheseshort-run constraints arebigrelativetothoseofin-vestigatinglong-run constraints, sincethey may beaway ofimprovingthee¤ectiveness of time-series models forhorizons wheretheyaremostuseful.

W eassess thee¤ectsofcommon-cyclicalfeatures onV A R andV EC modelsusingM onte-Carlo simulations. T he focus here is on the small-sample properties of the estimates of impulse-responsefunctions andvariance-decomposition offorecasterrors, as wellas on out-of-sampleforecastingaccuracymeasures. W edesign thesimulations in such awaythatthe results wouldberelevantforappliedmacroeconomists dealingwithalimitednumberofdata points and trying to estimate a relatively large numberof parameters. T o that end, we considera variety ofD ata G eneratingP rocesses (D G P s) and sample sizes, which are kept closetothe“typical” dataapplied researchers oftenencounterin practice.

V A R and V EC models with common cycles fallinto the generalcategory ofreduced-rank multivariate models, because common serialcorrelation implies ofreduced-rank restrictions on

theirparametermatrices1_{. R esearchers may be reluctantto incorporate these parameter}

restrictions fora purely statisticalreason related tothe asymmetry oftheconsequences of over- versus under-parametrization ofeconometricmodels. O nemightthinkthatfailingto incorporatecommon-cyclerestrictions when theyaretruewillonlycauseine¢ciency, while imposingthem whentheyarefalsewillcauseinconsistency. H ence, itmayseem wisertolive withapossiblyine¢cientunconstrainedmodelratherthanamisspeci…edinconsistentmodel. T he factthatallmodels are mostprobably misspeci…ed, and the large body ofempirical evidenceonthesuperiorforecastperformanceofparsimonious models notwithstanding, this reasoningwould be correctonly ifthe empiricalmodelthatdoes nothave common cycles builtintoitnests theempiricalmodelwith common cycles. H owever, weshowthat, using theaveragetoolsofanappliedresearcher, andthesamedataset, the“best” empiricalmodel developedwithoutcommon-cyclerestrictionsneednotnestthe“best” modeldevelopedwith thoserestrictions.

T heunderlyingreasonforthisratherparadoxicalresulthastodowiththeselectionoflag orderforthetwoalternativemodels. T hecommonpracticeinV A R analysisistouseamodel-selection criterium tochoosethelaglength. Standard model-analysisistouseamodel-selection criteriamay…nd too smallalaglengthofreduced-rankV A R ssimplybecausethisistheonlypossiblewayavailable toachieveparsimony. H owever, ifthelaglengthandtheV A R rankarechosensimultaneously, as suggested by L ütkepohl(19 9 3, page202), thelaglength selected forreduced-rankV A R s can be potentially biggerthan thatselected by the standard criterium. Forexample, the Schwarz criterium mightchoose aV A R (1) as thebestunconstrained V A R , whilethesame criterium mightchooseaV A R (4) withonecommoncycleforthesamedataset. O bviously, a V A R (1) cannotnestaV A R (4) withcommoncycles. H ence, theconsiderationofcomovement inthemodelselection stagemaydrasticallyalterthe…nalmodelchosen.

O ursimulations revealthat, when thetrueD G P is areduced-rankV A R orV EC model, the laglength chosen by the standard model-selection criteria and thatchosen when rank

1_{Classic references on reduced-rank V A R ’s include V elu, R einsel and W ickern(19 8 6), A hn and R}

and orderareselected simultaneously can bequitedi¤ erent. Standard information criteria whichplaceastrongpenaltyonoverparameterization, suchastheSchwarzorH annan-Q uinn criteria, maychoosetoosmallalag-lengthwhenthetruemodelhascommoncycles. H owever, theycanimproveconsiderablyiftherankorderisselectedsimultaneouslywiththelaglength. W e…ndstrongevidenceinfavoroftheabilityoftheH annan-Q uinncriterium inchoosingthe correctlagand rankorderoverall. R egardingtheA kaikeinformation criterium, weobserve thatits tendency to choose an overparameterized modelwhen the lag orderand rank are selected simultaneouslyworsens compared tothecasewhen onlythelaglengthis selected.

Forhorizons up tosixteenperiods ahead, usingseveralmeasures offorecastingaccuracy, we…ndthatforecasts produced bythe“best” reduced-rankmodelaregenerallysuperiorto thoseproduced bythe“best” modelwhen onlylag-orderis selected. T hesameconclusions areobtainedwhencomparingthevariancedecompositions ofthe“best” reducedrankmodel withthatofthe“best” fullrankmodelwhen thesamplesizeis 2002_{. Indeed, on average, if}

theH annan-Q uinncriterium isusedtoselectlagorderandrank, forecastingaccuracycanbe improvedbyup to20% , andmean-squared-errorofpredictingthetruevariancecontribution canbecutup tohalfinshorthorizons. T his is asizablee¤ectwhichillustrates thepotential gains associatedtoconsideringcommon-cyclicalfeatures whenevertheyexist.

T heoutlineofthepaperis as follows. Section2 states thereduced-rankrestrictions that common-cyclical‡uctuations imposeonparameters ofV A R andV EC models, anddiscusses therelativemerits ofdeterminingtherankorderby statisticaltests versus information cri-teria. Section 3 explains thedesign oftheM onte-Carlodesign used throughoutthe paper. Section 4 presents thesimulation results forasmallsystem ofthreevariables and section 5 presents thesameresults foralargersystem ofsix variables. Finally, section 6summarizes themainconclusions ofthepaper.

2_{N oticethatin thetextbookexample L ütkepohl(19 9 3, pp. 202-3) lagselection was identicalwhetheror}

nottherankwas alsochosen, and in thatcase, heobserved thattheforecasts and variancedecompositions werequitesimilarforthereduced rankand fullrankmodels.

2. Commoncycles inV A R andV EC models

T omatch thestylized facts ofmostmacroeconomicvariables, weassumethattheobjective ofresearch is to build a time series modelforthe growth rate ofa vectorofn economic

variables. W edenotethelevelofthesevariables attimetbyYt, theirlogarithms byyt, and

theirgrowth rates (i.e. the …rstdi¤erence ofthe logarithm ofYt) by¢yt. W e make the

reasonableassumptionthat¢ ytis stationary, addtheassumptionthat¢ ythasmeanzeroto

simplifynotation, withoutloss ofgenerality, andstartwith theW oldrepresentation of¢ yt:

¢ yt= C (L)"t; (2.1)

whereC (L) = P1_{j= 0} CjLjis amatrix polynomialin the lagoperatorL, Lkz t= z t¡k, with

C0 = In. From theworkofB everidgeand N elson(19 8 1) and Stockand W atson(19 88 ) itis

possibletodecomposethelog-levelseriesytintocommon trends and cycles (whichwerefer

toas theB everidge-N elson-Stock-W atson — orB N SW forshort— decomposition). U sing theidentityC (L) = C (1) + ¢C¤_{(L), disconsideringtheinitialvalues iny}

0, andintegrating

bothsides of(2.1) weget3_:

yt = C (1) t X j= 1 "j+ C¤(L)"t = Tt+ Ct; (2.2)

whereTt= C (1)Ptj= 1"jandCt= C¤(L)"tstackrespectively thetrend and

cyclicalcom-ponents ofyt. In the B N SW decomposition then variables inytare decomposed inton

random-walk components (stochastic trends) andn stationary components (stochastic

cy-cles). IfC (1) has rankn ¡q (q > 0 ), the stochastic trends inytcan be characterized as

linearcombinations ofonlyn_¡qcommonrandom walks, inwhichcaseytis

saidtobecoin-tegratedorhavecommonstochastictrends, withqlinearlyindependentcointegratingvectors

stackedinthematrix®0_{;seeEngleandG ranger(19 8 7 ). IfC}¤_{(L)hasrankr(n ¡r > 0 ), then}

thestochasticcyclesinytcanbecharacterizedaslinearcombinationsofrcommonstochastic

cycles, withn_¡r cofeaturevectors stackedinthematrix®e0;seeV ahidandEngle(19 9 3). In this paper, weinvestigatethecosts ofignoringthis singularityinthestochasticcyclesCt.

Ifthere aren _¡q common trends in the system, then a vectorerror-correction (V EC) model, ¢ yt = A1¢ yt¡1+ :::+ Ap¢ yt¡p + °®0yt¡1+ "t = · A1 ::: Ap °®0 ¸ 2 6 6 6 6 6 6 6 6 4 ¢ yt¡1 ... ¢ yt¡p yt¡1 3 7 7 7 7 7 7 7 7 5 + "t; (2.3)

would bea parsimonious representation forytrelativetothe V A R in log-levels, where the

columnsofthen_£qmatrix® arethecointegratingvectorsand°istheadjustment-coe¢cient matrix. Ifthere is nocointegration, then the term°®0_y

t¡1 on the right-hand side of(2.3)

vanishesandthemodelreducestoaV A R in…rstdi¤ erences. H ence, inwhatfollows, without loss ofgenerality, wefocus ourattention on theV EC model.

Ifthere arer common stochastic cycles inyt, the matrix ·

A1 ::: Ap °®0 ¸

, which includes allthe parameters in the conditionalmean of¢ yt, musthave rankr. T his is a

consequenceofthefactthatC¤(L)in(2.2) has rankr, whichimplies thatthereareexactly n¡r non-colinearlinearcombinations ofytwhicharerandom walks anddonotexhibitany

cycles. Since the …rstdi¤erences ofthesen ¡r linearcombinations are white noise, they

are unpredictable using the past. H ence, ·A1 ::: Ap °®0 ¸

musthave rankr, with a

common null-space. T his implies thattheV EC modelhas itselfthefollowingparsimonious representation: 2 6 4 In¡r ®e ¤0 0 Ir 3 7 5¢ yt= 2 6 4 0 ¢¢¢ 0 0 A¤₁ ¢¢¢A¤_p (°®0)¤ 3 7 5 2 6 6 6 6 66 6 6 4 ¢ yt¡1 ... ¢ yt¡p yt¡1 3 7 7 7 7 77 7 7 5 + vt; (2.4)

wherethecofeaturematrix®e0=

·

In¡r ®e¤0 ¸

, whichstacks thelinearly-independentcombi-nations of¢ ytthateliminates thecommonserialcorrelationinthesystem’s components, is

rotatedinordertoyieldann¡r identitysub-matrixinits …rstn¡r rows andcolumns,A¤_i

and(°®0₎¤_{representrespectivelythepartitionsofA}

iand°®0correspondingtotheremaining

r equations in thesystem, andvt= 2 6 4 In¡r ®~ ¤0 0 Ir 3 7 5 "t. Since 2 6 4 In¡r ®~ ¤0 0 Ir 3 7 5 is invertible, itis possibletorecover(2.3) from (2.4).

Common-cycle constraints imply importantrestrictions forthe impulse-response func-tions, variance-decomposition offorecasterrors, and multi-step ahead forecasts. T he exis-tence ofr common cycles implies thatthere aren ¡r non-collinearlinearcombinations of

¢ ytwhicharewhitenoise. T hus, from (2.1), allmatricesCi, i= 1;2;¢¢¢, musthaverankr.

T hesematricesCi, usuallynormalizedtobeconsistentwithorthogonalerrors, form thebasis

ofimpulse-responsefunctions andthecomponents offorecast-errorvariancedecompositions. Forexample, whentheyarepost-multipliedbythecholeskyfactorofthevariance-covariance matrix of"t, they yield the so-called orthogonalized impulse responses. H ence, itbecomes

clearthatcommoncycles implythattheimpulseresponses ofdi¤ erentvariables tothesame shocks arelinearlydependent. T herefore, iftheobjects ofinterestaretheimpulseresponses (orvariancedecompositionsofforecasterrors) of¢ yt, common-cyclerestrictionscanbeused

toachieveparsimonyand theire¢cientestimationinthis multivariatecontext.

A similarargumentapplies to forecasts of¢ ytathorizonh. T hese can be recursively

calculatedfrom:

¢ yf_t+h = A1¢ ytf+h¡1+ :::+ Ap¢ yft+h¡p + ¦y f

t+h¡1 (2.5)

where the superscriptf stands forforecasts using information up to periodt, ¦ = °®0,

and actualvariables are used instead offorecasts on the right-hand side where available. Sincecommon cycles implythatthematrix·A1 ::: Ap ¦

¸

has reduced rank, equation (2.5) clearlyshows thattheywillalsoimplythattheforecasts of¢ ytatanyhorizonwillbe

linearlydependent. A gain, ifforecastingis theobjectiveofthemultivariatemodelbuilding exercise, common-cyclerestrictions canalsobeusedtoachieveparsimonyandtheire¢cient

estimation.

In aV A R contextabove, therearetwoways in which parsimony can beachieved. T he …rstis byimposinglong-runconstraints (cointegration), i.e., equation (2.3), andthesecond is by imposing short-run constraints (common cycles), i.e., equation (2.4). T he literature on forecastinghas focused on long-run constraints;see, forexample, Engleand Y oo(19 87 ), Clements and H endry(19 9 5), and L in and T say(19 9 6). A s argued in the Introduction, the payo¤ ofinvestigatingshort-runconstraints arelargerelativetothoseofinvestigatinglong-runconstraints, whichmotivates ourpresentresearch e¤ ort.

2.1. Informationcriteriaforreduced-rankmodels

O urmotivation is to build V A R -based models for¢ ytwhich can be used forforecasting,

impulse-response orvariance-decomposition analysis. A criticalstep in constructingthese models is howtoselectthelaglengthoftheV A R (V EC). M odel-selection criteriaareoften used in practicetoselectlaglength. In principle, theyareusefulbecausetheydonotfavor anyspeci…cmodelagainstothers (nullversus alternativeinhypothesis testing, forexample); see the discussion in G ranger, King and W hite(19 9 5). Such criteria may choose di¤ erent lag orders depending on whetherornotwe allowforreduced-rank parametermatrices in the V EC model4_{. G iven the potentiale¢ciency gains ofusing common-cycle restrictions,}

we investigate empirically the performance ofwidely-used selection criteria when only lag lengthis selectedandwhenlaglengthandrankorderareselected. Finalresults canthenbe comparedtohelp buildingastrategyforempiricalwork.

FollowingL ütkepohl(19 9 3), we focus on the A kaike, H annan-Q uinn and Schwarz infor-mationcriteriaforthesimultaneous selectionoflagandrankordersinV A R s(V EC models). FortheV EC modelinequation(2.3), weassumethatcointegratingvectorsareeitherknown

4_{V ahid and Engle(19 9 3) focused on testing theories thatimplied common cycles within a V EC-model}

framework. T hey derived a statistical test forthe hypothesis ofrcommon cycles, and recommended a sequentialtestingprocedurewhich coulddeterminer:T heirprocedurerequiredthatthenumberoflags, i.e.

from theory, orarecorrectlyestimatedinadvance. T his is donehereforsimplicity, avoiding thewellknown problem ofdealingsimultaneouslywithintegratedandstationaryregressors inV EC models;seeT odaandP hillips(19 9 3), forexample. U nderthis assumption, theV EC modelin (2.3) can bewritten as:

¢ yt= · A1 ::: Ap ° ¸ 2 6 6 6 6 6 6 6 6 4 ¢ yt¡1 ... ¢ yt¡p ®0_y t¡1 3 7 7 7 7 7 7 7 7 5 + "t: (2.6)

T helagorderpandthenumberofcommoncyclesr, whichistherankof ·

A1 ::: Ap ° ¸

, canbesimultaneouslychosen tominimizeoneofthefollowingmodelselection criteria5_:

AI C(p;r) = ln¯¯¯§^"(p;r) ¯ ¯ ¯+ 2 T £r£(n p+ n¡r+ q) (2.7 ) H Q (p;r) = ln¯¯¯§^"(p;r) ¯ ¯ ¯+ 2 lnlnT T £r£(np + n ¡r+ q) (2.8) SC (p;r) = ln¯¯¯§^"(p;r) ¯ ¯ ¯+ lnT T £r£(np+ n¡r+ q) (2.9 )

whereqisthenumberofcointegratingvectors,n isthedimensionofthesystem,r istherank

ofV EC model,p is thenumberoflagged di¤ erences inthemodel,_§^_{"(p;r)is theestimated}

variance-covariancematrixoftheerrors oftheV EC modelwithp lags and rankr;andT is

thenumberofobservations. V ahid and Engle(19 9 3) showed thatr _¸q;which implies that,

givenq, models ofranksmallerthanqneednotbeconsidered.

Calculatingtheinformationcriteriain(2.7 )-(2.9 ) forfull-rankmodels(r= n) isstraight-forward, sincetheycanbeestimated, equationbyequation, usingO L S. O ntheotherhand, forreduced-rank models, computingthesecriteriafordi¤erentp andr may seem di¢cult,

sinceonemaythinkthattheirestimation is required. H owever, thefollowingL emmastates awellknown resultthatrelatesln¯¯¯§^"(p;r)

¯ ¯

¯tothe squared canonicalcorrelations between

5_{W hen the variables are notcointegratedq= 0, and these criteria are the same as those suggested in}

¢ ytand ·

¢y_t0_¡1 ¢¢¢¢ y0_t¡p (®0yt¡1)0 ¸0

makingcomputation ofthesecriteriainthis case alsostraightforward. L emma2.1.T heminimum ofln¯¯¯_T1 PTt= 1"t"0t ¯ ¯ ¯undertheassumptionthat · A1 ::: Ap ° ¸ has rankr is ln ¯ ¯ ¯ ¯ ¯ 1 T T X t= 1 ¢yt¢ y0t ¯ ¯ ¯ ¯ ¯+ n X i=n¡r+ 1 ln(1_¡¸i);

where¸1 < ¸2 < ¢¢¢< ¸n are the sample squared canonicalcorrelations between¢ ytand the setofregressorsx0 t´ · ¢y0 t¡1 ¢¢¢¢ yt0¡p (®0yt¡1)0 ¸

:T he sample squared canonical correlations aretheeigenvalues of 0 @ XT t=p+ 1 ¢yt¢ yt0 1 A ¡10 @ XT t=p+ 1 ¢ytx0t 1 A 0 @ XT t=p+ 1 xtx0t 1 A ¡10 @ XT t=p+ 1 xt¢ yt0 1 A :

P roof.SeeT so(19 81).

T his lemmashows that, afterdroppingthecommonconstanttermln¯¯¯_T1 PtT= 1¢yt¢ yt0 ¯ ¯ ¯in

(2.7 )-(2.9 ), these model-selection criteriacan be expressed in terms ofthe eigenvalues(¸i)

as: AI C(p;r) = n X i=n¡r+ 1 ln(1¡¸i(p)) + 2 T £r£(np+ n¡r+ q) (2.10) H Q (p;r) = n X i=n¡r+ 1 ln(1_¡¸i(p)) + 2 lnlnT T £r£(n p+ n¡r+ q) (2.11) SC (p;r) = n X i=n¡r+ 1 ln(1¡¸i(p)) + lnT T £r£(n p + n ¡r+ q): (2.12)

H ence, for…xedp; the model-selection criteria forany rank can be easily calculated after

theseeigenvalues arecomputed. T heeigenvalues can beeasily calculated byanystatistical program whichhas acanonicalcorrelationprocedure6_{. N oticethat, for…xedT,n andq, the}

model-selectioncriteriain(2.10)-(2.12) dependonlyonthelaglengthpandontherankofthe parametermatricesintheV EC modelr, i.e., V EC modelswherethenumberofcointegrating

vectors is known.

6_{Forexample, SA S orST A T A , oranymatrixprogram suchas G A U SS, orbyslightlymodifyinganyofthe}

plethoraofcomputerprograms which usethis lemmatocalculatetheJohansen cointegration teststatistics (seechapter20 ofH amilton(19 9 4)).

3. M onte-Carlodesign

Ifsamples are “large” and the variables have common cycles, ourintuition tells us that ignoringthem willnotbeveryharmful. T his is based on theexpectation thatwith “large” samples, lag-orderselection is likely to be unambiguous, and parameterestimates willbe precise, sothatthereduced rankconstraints willbe(approximately) truefortheestimated parameters, even when theyarenotimposed attheestimation stage. H ence, theestimated models with orwithout common-cycle restrictions willbe so close, that theirresults for forecasting, impulse-response, andvariance-decomposition analysis willbeverysimilar.

T his intuition should not, however, becarried overtothecaseof“small” samples. A s a matteroffact, e¢ciencygains arepotentiallyrelevantwhen samplesizes aresmalland de-greesoffreedom arescarce. Inthiscontext, usinginformationcriteriatoselectlagordermay notbe unambiguous. T hestandard practiceis todisregard thepossibility ofreduced-rank models intheformulaofwidely-usedmodel-selectioncriteria, i.e., setr = n in(2.10)-(2.12).

T his creates thepotentialproblem ofmodelmisspeci…cationinselectinglaglength. Indeed, selectinglagorderimposingthatthemodelisfull-rankcanyieldacompletelydi¤ erentresult thanselectinglagorderandranksimultaneously. W einvestigatethis issueusing1000 simu-lationsof100 reduced-rankV A R switheither100 or200 observationseach, tabulatingresults whenlaglengthaloneis chosenandwhenrankand laglengtharechosen simultaneously.

T o make presentation manageable, we chose to work with three- and six-dimensional V A R s. In theapplied macroeconomics literature, models thatonlyconsidertherealsideof the economy areusually three-dimensional. Forexample, in testingthereal-business-cycle modelinKing, P losserandR ebelo(19 9 8), Kingetal.(19 9 1) estimateaV A R includingoutput, consumption, and investment. Isslerand Ferreira(19 9 8) useaV A R includingoutput, labor, and capitalinputs, to estimate long-run elasticities ofthe aggregate production function. Six-dimensional V A R s usually include a tri-variate real-variable sub-system, as well as a monetary sub-system, including the realmoney supply, realinterestrate, and the levelof in‡ation;seeKingetal. forexample.

T he…rstparameterwesetin theM onte-Carlodesignis thelaglengthp. Itis chosen in ordertoalloweitherthe possibility ofunder- orover-parameterization ofthe V A R (V EC) model. T his does nothappen in eitherL ütkepohl(19 8 5) orin somesimulations in N ickels-burg(19 8 2). T he …rstuses a D G P with a true lag orderof1 in his simulations, making under-parameterization virtually impossible. T his favors information criteriawhich heavily penalize over-parameterization, e.g., the Schwarz criterium7 _{. T he second sets the true lag}

orderto four, butthe maximum lag offouras wellin the estimation stage. T his makes over-parameterization impossible, favoring liberalcriteria such as the A IC. T o avoid both problems, forthe three-dimensionalsystem, we chose the true lag orderof4 allowing for models ofup tolag8. Forthesix-dimensionalsystem, in ordertosavedegrees offreedom, weuseas theD G P aV A R withtwolags andallowestimation up tolagsix.

N ext, we discuss the choice ofvariance-covariance matrix forthe V A R error"tin the

M onte-Carlo design. T he properties ofestimated V A R s are only invariantto scaling the variance-covariance matrix ofthe errors by a scalar. H owever, the following lemma shows thatinordertocovertheentirespaceofreduced-rankV A R processesoforder(p), onecan…x thevariance-covariancematrixof"ttobetheidentitymatrixwithoutanyloss ofgenerality.

L emma3.1.A nyarbitraryfullranklineartransformationofareduced-rankV A R , generates anotherV A R with thesamerank.

P roof.ConsideraV AR(p),

yt= A1yt¡1+ ¢¢¢+ Apyt¡p + "t:

A ssume thatPis a fullrankn _£n matrix which orthogonalizes the variance-covariance matrix of"t. D e…nez t´Pyt, B1 ´PA1P¡1 , ¢¢¢, Bp ´PApP¡1, and´t´P"t, where

E(´_t´0

t) = In. W ehave,

z t ´ Pyt= PA1yt¡1+ ¢¢¢+ PApyt¡p + P"t

7 _{N otsurprisingly, this is exactly the criterium thatdoes best in choosing the correct lag orderin his}

= PA1P¡1Pyt¡1 + ¢¢¢+ PApP¡1Pyt¡p + P"t

= B1z t¡1+ ¢¢¢+ Bpz t¡p + ´t: (3.1)

SincePis offullrank, theBi’shavethesamerankas oftheirAi’s counterpart. Sinceallthe

Ai’s havethesameleftnull-space, sowillallthePAi’s, and thereforesowilltheBi’s.

W e now turn to choice ofthe coe¢cients in the conditionalmean ofthe V A R (V EC) model. A n exhaustive M onte-Carlo study overthe entire modelspace is unfeasible. Itis customary, as in L ütkepohl(19 85), tochoose severalsets ofeigenvalues forthe companion matrix8 _{ofthe V A R , and to choose arbitrary parametermatrices which give rise to those}

eigenvalues, averagingtheresults overalltheseD G P s. A lthough theresults generatedfrom such adesign strategy mightbeusefulforgeneraltime-series analysts, they areunsuitable foreconomists whoworkwith aggregatemacroeconomics data. T his is becausethecyclical structure (i.e. signaltonoise ratio) ofmodels includingmacroeconomicaggregates can be quiteweak, especiallyforsystems whichdonotcontainamonetarysector. Forexample, the systemR2 _{(ameasuresimilartoR}2 _{forunivariatemodelswhichisdiscussedintheA ppendix)}

forKingetal.’s(19 9 1) V EC modelofU S per-capitaincome, consumption, andinvestmentis 0.44, whereasthesystemR2 _{for160 outofthe200 D G P swhichL ütkepohl(19 85)averagesover}

areabove0.5, and9 6ofthesearegreaterthan0.8 . Sincethispaperisintendedprimarilyfor appliedmacroeconomists, adesignwhichgives toomuchweighttomodels withhighsystem

R 2 _{would beinappropriate.}

H ere, westartwith a“typical” macroeconometricstudyin ordertoselecttheD G P and the system R2 _{associated with it. T he data setused forchoosingourparametervalues is}

thesameas inKingetal.(19 9 1)9_{. Forthethree-variablesystem, weconsidered100 di¤ erent}

D G P s. T heirsetofparametervalues are drawn randomly from the estimated asymptotic 9 5% con…dence region ofthe parameters ofreduced-rank V A R s oforderfour. T he latter

8_{T hecompanion matrixofaV A R(p)is thecoe¢cientmatrixofits V A R (1) representation. T hecondition}

forV A R(p)tobestationaryis thatalloftheeigenvalues ofits companion matrixareinsidetheunitcircle.

9_{Kingetal.(19 9 1) chosealaglengthofeightfortheirthreevariablemodelandalaglengthofsixfortheir}

arebased on estimates offourth-orderV A R s ofthe…rst-di¤erences ofthelogarithms ofU S per-capita private income, consumption, and investmentoverthe period 19 47 .1 to19 88.4, in quarterly frequency. Forthesix-variable model, wealsoconsidered 100 D G P s, drawned usingthesamemethod, after…ttingreduced-rankV A R s ofordertwotothe…rstdi¤ erences oflogarithms ofper-capitaprivate income, consumption, investment, realmoney balances, interestrate and in‡ation overthe period 19 47 .1 to 19 88 .4. Forallcases, we have been carefultoverify thatalloftheserandomly drawn D G P s satisfy thestationarity conditions forV ectorA utoregressions10_.

B y choosingourD G P s from this “plausible” subsetofthe parameterspace, we believe thatourresults would be directly relevantforapplied macroeconomists. Forcomparison withL ütkepohl(19 8 5), themedianofthesystemR 2 _{measureforourgeneratedthree-variable}

D G P s is between0.5 and 0.6,withless than 5% largerthan0.7 andnonegreaterthan0.8. T heM onte-Carlosimulationconsists ofthefollowing:

1. U sing each ofthese 100 D G P s, we generate 1000 samples (ofeither100 or200 ob-servations), record the laglength chosen by traditional(full-rank) A IC, H Q , and SC measures, i.e., theinformation criteriastatedin(2.10)-(2.12) whenr = n, andthelag

lengthand rankorderchosen bytheinformationcriteriastatedin (2.10)-(2.12). 1. Inallcases, toreducetheimpactofinitialvaluesonsimulatedseries, wegenerated

500 observations. O nly the last 116or216observations were selected forthe forecastingexerciseand onlythelast100 and 200 wereselected fortheimpulse-responseandvariance-decompositionexercises.

2. W ethencomparetheabilityofeachofthesestrategies ofmodelselectioninestimating theD G P ’s truelaglength and rank. T his comparison allows measuringthechanceof misspeci…cationarisingfrom ignoringcommon-cycles atthelag-lengthselectionstage.

10_{T he range of the absolute value of the maximum eigenvalue of the tri-variate rank-one D G P s is}

(0:49;0:87). Forthe tri-variate rank-twoD G P s this range is(0:63;0:92), and forthe six-variate rank-four D G P s itis(0:56;0:96).

3. B ased on theinformation-criteriaresults, a“best” modelforeach criterium is chosen when traditional(full-rank) criteriaareused and a“best” modelforeach criterium is alsochosen when the criteria in (2.10)-(2.12) are used. Foreach type ofinformation criteriathesetwo“best” models arecompared regarding:

1. out-of-sampleforecastingaccuracyup to16periods ahead, and,

2. mean-squared-errorin estimatingvariancedecompositions offorecasterrors and impulse-responsefunctions.

W eturn nowtothespeci…cs oftheexercises on forecastingevaluation and on variance-decompositionandimpulse-responsefunctionestimates.

3.1. M easuringforecastingaccuracy

A ppropriateevaluationofforecasts depends onthespeci…cusethattheforecasts areneeded for, i.e. the“lossfunction” oftheuser. T hefactthatwehaveappliedeconomistsasourtarget audiencedoesnotpointtoanyspeci…cwaythatweshouldevaluatetheforecastsofalternative models. First, itis notreasonabletoimposeany typeofasymmetry in evaluatingforecast errors. Second, amacroeconomistwhomodels thegrowth rateofincome, consumption and investment, mightinfactbeinterestedinthegrowthratesofincome, savingsandinvestment, orshe mightbe interested in forecastingthe levels based on the growth rates. T herefore, itis importanttoevaluate the forecasting performance ofdi¤erentmodels on the basis of measures thatare invariantto lineartransformation offorecasts atone horizon, oracross di¤erent horizons. O ne measure that satis…es this invariance property is thegeneralized forecasterrorsecond moment(G F ESM )introduced by Clements and H endry(19 9 4). Itis thedeterminantoftheexpected valueoftheouterproductofthevectorofstacked forecast errors ofallfuture times up to the horizon ofinterest. Forexample, ifforecasts up toh

quarters aheadareofinterest, this measurewillbe: G F ESM = ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ E 0 B B B BB B B B @ ~ "t+ 1 ~ "t+ 2 ... ~ "t+h 1 C C C CC C C C A 0 B B B BB B B B @ ~ "t+ 1 ~ "t+ 2 ... ~ "t+h 1 C C C CC C C C A 0¯_¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯

where~"t+h is then-dimensionalforecasterrorathorizonh ofourn-variablemodel. Itis

ob-vious thatthis measureis invarianttoelementaryoperations thatinvolvedi¤ erentvariables, and alsotoelementary operations thatinvolve the same variable in di¤erenthorizons. In theM onte-Carlo, theaboveexpectation is evaluated foreverymodel, byaveragingoverthe simulations.

W e also considered here two additionalmeasures offorecasting accuracy. T he …rstis thedeterminantofthemeansquaredforecasterrormatrixatdi¤erenthorizons(jM SF Ehj),

and the second is the trace ofthe mean squared forecasterrormatrix(TM SF E). T he

determinantoftheM SF E is invariantto elementary operations on forecasts ofdi¤ erent

variables ata single horizon, butitis notinvariantto elementary operations on forecasts acrossdi¤ erenthorizons. T hetraceofthemeansquaredforecasterrormatrixisnotinvariant toeitherofthesetransformations.

T hereisonecomplicationarisingfrom thefactthatwearesimulating100 di¤ erentD G P s. Inthiscase, thesimpleaveragingofthesemeasuresacrossdi¤erentD G P sisnotappropriate, sincetheforecasterrorsofdi¤erentD G P sdonothaveidenticalvariance-covariancematrices. L ütkepohl(19 85) normalizes the forecasterrors by theirtrue variance-covariance matrix in eachcasetogeti.i.d. observations. U nfortunately, thiswouldbeaverytimeconsumingpro-cedureforameasurelikeG F ESM whichinvolves stackederrors ofmanyhorizons. Instead, wecalculatethepercentagechangein thesemeasures in the“best” full-rankmodeland the “best” reduced-rankmodelchosenbyeachcriterium foreveryD G P, andthenaveragethese changes overallD G P s.

3.2. P recisionofimpulse-responseandvariance-decompositionestimates

A lthoughinmanycasestheobjectsofinterestinappliedstudiesthatuseV A R orV ECmodels areimpulse-responsefunctions andvariance-decompositions offorecasterrors, allsimulation studies we aware offocus on forecastcomparisons alone. T hus, studying the precision of estimates ofimpulse-responsefunctions andvariance-decompositioncoe¢cients fordi¤ erent V A R and V EC models has alsoahigh payo¤.

Impulse-responsefunctionsandvariance-decompositionofforecasterrorsdi¤erfrom multi-step forecasts ofV A R ofV EC models inwhichtheyarenotonlynon-linearfunctions ofthe estimatesofmean-parametermatricesbutofthevariance-covariancematrixofsystem errors aswell. G iventhisaddeddimensiontotheproblem, onecannotaprioriexpecttogetsimilar results totheforecastingexercise.

M oreover, thereafewissuesthatarespeci…ctotheanalysisofimpulse-responsefunctions andvariancedecompositionofforecasterrors. First, errors havetobeorthogonalforresults to be meaningful11_{. A s is wellknown, there are severaltechniques thatyield orthogonal}

errors. H ere, wechosetousetheCholeski decompositionforthevariance-covariancematrix to orthogonalize shocks, since this is by far the most popular method used in practice. Second, forathree-variablesystem therearenineimpulse-responseandvariance-component coe¢cients ineachhorizon. Forasix-variablesystem therearethirtysix. Inordertoreport results inacompactway, themean-squarederrors ofeachofthem is computedfortherank-restricted and theuncomputedfortherank-restricted V A R model. T hen, thepercentageimprovementin M SE of the restricted modelis computed foreach ofthese coe¢cients. Finally, foreach horizon, themean percentageimprovementacross allcoe¢cients is computed. Itshould benoticed thatthis methodensures that…nalresults donotdependontheunitofmeasurementofthe variables in thesystem. T hird, in ordertokeep thesizeofourtables down toaminimum, onlyvariance-decompositionresultsarereported, sinceresultsforimpulseresponsesfollowed

11_{Inthis case, theresults inourL emma2 arenotapplicable, sinceM onte-Carloresultsarenotindependent}

averysimilarpatterntothosefoundinthevariance-decompositionexercise.

3.3. A benchmarkforfuture reference

A s in any simulation study is usefulto generate a benchmark case to be used forfuture reference. H ere, wechoseas anaturalbenchmarksimulation thecasewheretheresearcher knowsthetruelaglengthoftheV A R , therebyrulingoutanychanceofmodelmisspeci…cation. A nydi¤ erencesbetweenreduced-rankandfull-rankV A R modelsre‡ectsolelye¢ciencygains oftheformer.

T osave spacewe presenttheresults ofthis exercise only forthe three-variable system. T able1 shows thepercentageimprovementofdi¤erentmeasures offorecastaccuracyandof mean-squared error(M SE) ofvariance-decomposition coe¢cients when reduced-rank V EC models areallowedfor. IfG F ESM is considered, forecastingaccuracycanimproveas much

as 7 3% , with amedian improvementacross allhorizons and sample sizes ofabout30% . If

jM SF EhjandT M SF E are considered the gains are smaller(22% and 2% respectively).

Italso becomes clearthatthey happen mostly atshorthorizons, which does nothappen whenG F ESM is considered, sincethelatteraccumulates forecasting-accuracygains across

horizons. T hegainsinM SE ofvariance-decompositioncoe¢cients arealsosizable- theycan beas highas 66% withamedian gainofabout45% , althoughatthe…rsthorizonthereis a loss in M SE thatcan reachup to43% .

T hese results showthepotentiale¢ciency gains when thelag-length ofV A R and V EC models are known. A lthough they serve as a benchmark, these gains are unrealistic for empiricalstudies, sincethere laglengths mustbe estimated beforehand. W enextconsider reduced-rank models lag-length selection using information criteria undertwodistinctsit-uations. First, when the information criteria in (2.10)-(2.12) are used settingr = n, and

second, when they are used allowingforthe possibility ofreduced rank in V A R and V EC models.

criteriachooseadi¤erentlaglengthwhen, asisthenorm inpractice, onlyfullrankmodelsare considered?T heseresults arethenconfrontedwiththoseobtainedwhenthesamecriterium is used to pick the lag orderand rank atthe same time; (ii) do di¤ erences in the chosen modelsbythesetwoclassesofmodelselectioncriterialeadtomajordi¤erencesinforecasting accuracy?A nd(iii) dodi¤ erencesinthechosenmodelsbythesetwoclassesofmodelselection criterialead tomajordi¤erences ofaccuracyofestimates ofimpulse-responseand variance-decomposition coe¢cients. A secondary result, which can be ofinterestto practitioners, is the relative performance ofdi¤erent model-selection criteria forreduced-rank V A R s in choosingtheircorrectlagandrankorder.

4. M onte-Carlosimulationresults forthe three-variable model

4.1. Selectionoflagandrankorder

T able 2.a shows the frequency oflag-orderselection in 1000 simulations of100 trivariate V A R s(4) with rank1 by A IC, H Q and SC when only fullrankmodels areconsidered, and when rank and lagorders are determined simultaneously. T hetop halfofthis tablecorre-sponds toasamplesizeof100, and thebottom halfcorretablecorre-sponds tosamples of200 observa-tions. T able2.b shows theanalogous frequencies whenthetrueD G P is atrivariateV A R (4) ofrank2.

T heseT ablescon…rm thatselectingthelagandrankorderjointly, canleadtochoiceofa modelwhichis ofhigherlag-orderthanthelag-lengthchosenwhenonlyfullrankmodels are considered. Forexample, thetophalfofT able2.ashowsthatforsamplesof100 observations, themodalchoiceofallthree criteriais V A R (1), withAI C choosingthetrue lagof4 only 14 percentofthe time. T he othertwo criteria choose a V A R (4) with a frequency ofless than 1 percent. H owever, when thelagand rankarechosen simultaneously, thereis alarge reductioninthenumberoftimesthattheV A R (1) ischosen, regardlessofthecriterium used. Furthermore, thefrequencyofthecorrectlagchosen increases signi…cantly. In both T ables 2.aand 2.b, AI C chooses thecorrectlagand rankmore often than theothertwocriteria,

withH Q aclosesecond. T hemodalchoiceoftheSchwarzcriterium stays ataV A R (1) even with200 observations.

T wo points are worth noting. First, even in those cases in which the criteria choose thewronglag-length, theyarelikely tochoosethecorrectrank. T heonly exception isSC

when the true rank is 2 and there are only 100 observations. T his suggests thatcommon cycles can be detected even ifthe wrong lag-length is chosen. T his is plausible, because the property thata linearcombination ofvariables has nodependence with the past(the necessaryand su¢cientcondition forcommon cycles), is unrelated towhatthosecycles are and whetherthey arecorrectly speci…ed. T hesecond pointis thatAI C has atendency to over-predictthetruelaglength, even when samplesizeis 200, onceonechooses lag-length and ranksimultaneously. G iven theevidenceon theadversee¤ects ofoverparameterization on forecastingin timeseries models in theliterature(seeL ütkepohl, 19 8 5), this cautions us thatthecosts ofusingAI C tochooselagand rankorderjointlymayoutweigh its bene…ts.

T heanalysis oftheforecasts in thenextsubsection con…rms thatthis is indeedthecase.

4.2. Forecasts

T ables 3.aand 3.b showthepercentageimprovementin thesemeasures when lagand rank arechosen simultaneously, overwhen lag-length is chosen aloneimposingr = n. A general

conclusionis thatreduced-rankmodels havenoforecastabilitybeyond8 periods, and most ofthe advantageoflookingforcommon cycles is in forecastingone tofourperiods ahead. D espitethat, therearestillnon-trivialgains forconsideringthepossibility ofreduced-rank models: G F ESM canbereducedup to32% ,_jM SF Ehjup to11% andTM SF Eup to8 % ,

whenthetruerankis one. Forranktwothesepotentialreductions arerespectively31% , 9 % , and 5% . T hesenumbers areabouthalfas largeas ourbenchmarkcasewhenG F ESM and

jM SF Ehjareconsidered, butarelargerwhenTM SF Eis considered.

T heresults in T ables 3.aand 3.b allowalsocomparingthethreeinformation criteriain termsoftheirrelativeperformance. R egardlessoftheforecasting-accuracymeasureused,H Q

providesthebestforecastingperformanceacrossinformationcriteria, whileAI C providesthe worst. O urresults forH Q andSC showthatthere are sizable bene…ts from choosinglag

and rankjointly. In thesecases, thesecriteriagiverisetomodels thatareclosertothetrue D G P, withoutincreasingthechanceofovershootingthecorrectlag.

O n the other hand, considering the forecast performance ofAI C, and the increased possibility ofover-prediction ofthelag-orderwhen lagand rankarechosen simultaneously, especiallywhensamplesizeis 100, weconcludethatthejointdeterminationoflagandorder byAI C inmodelswithsmallR2 _{isnotappropriate. Indeed, ifonewantstouseAI C, andalso}

wants toallowforpossibilityofcommon cycles, itwould bebettertoemploythefollowing strategy. First, useAI C tochoosethelaglengthtestingforcommoncycles usingthetestin

V ahid and Engle(19 9 3). T hen, imposecommon cycles ifthatis notrejected bythetesting procedure. In this way, the possibility ofovershootingthe correctlag length is somewhat controlled.

4.3. V ariance-decompositionresults

T he percentimprovementofestimates offorecast-errorvariancedecomposition coe¢cients are presented in T able 4. Firstnotice thatthe smallestgains (largestlosses) are obtained forhorizonone, whereresults depend exclusivelyontheestimateofthevariance-covariance matrixoftheerrors12_{;asimilarresultis obtainedin ourbenchmarkcase. T his maybedue}

tothefactthatthevariancecontributions arearatioofelements ofthevariance-covariance matrix. H ence, althoughtherestrictedmodelestimatesthelattermoreprecisely, itperforms worsein estimatingthevarianceratios compared totheunrestricted model. Second, when sample size is 100 observations, there is no consistentpattern ofresults favoringreduced-rankcriteria(i.e., whenlagandrankarejointlyselected). T hird, increasingthesamplesize to200 observations improves the gains ofreduced-rank models regardless ofthe criterium considered. T hehighestrealized gain happens when theH Q criterium is used in this case,

12_{N oticethat, forhorizon one, when theCholeski decomposition is employed, thereareonly…vevariance}

although this says nothingaboutthe relative performance across criteria forestimation of variance-decompositionestimates.

5. M onte-Carlosimulationresults forthe six-variable model

T he D G P s ofthe six-variable M onte-Carlosimulations are drawn uniformly from thecon-…denceregion ofan estimated six-variableV A R (2) ofrank4, based on U S macroeconomic aggregates. T he six variables are the growth rates ofper-capita income, consumption, in-vestmentandrealmoneybalances, andthe…rstdi¤erenceofinterestrates andthein‡ation rate, from 19 47 :1 to19 8 8:4. T his is the data setused in King, etal(19 9 1). T he addition ofthemonetaryvariables tothesystem, increases thesystemR2 _{from 0.44 to0.81. T his is}

possibly duetothefactthatmonetary variables aremoredependenton thepastthan real variables, andthatthereisastrongcross correlationbetweenthegrowthratesofrealmoney balances andincome.

T hemaximum lag-length considered is 6, which is thelaglength thatKingetal.(19 9 1) used in theiranalysis. In ordertoreduce thecomputationalcosts, we have drawn only 20 D G P s and performed 500 simulations in each case. In contrasttothethree-variablemodel, theD G P s inthis exerciseallhaveR2 _{between0:8 and0:9;whichimplies that…nitesamples}

are more informative aboutthe structure ofthe D G P than in the previous three variable case, andthatinformationcriteriamustbemoresuccessfulinselectingthecorrectlag. T his is clearfrom thelagandlag-rankorders chosenbythedi¤erentcriteriareportedinT able5. T able5 showsthefrequencyofmakingthecorrectchoice, usingeachofthethreecriteria. T he problem withAI C over-shooting the correctlag, when lag and rank orderis chosen

simultaneously, is much more pronounced here, especially in samples of100 observations. In this case, theregularAI C only overshoots thetruelag-orderin less than 2 percent, but

the lag-rank version oftheAI C overshoots thetrue lagin 9 .4 percentofallofthe10,000

simulations. Ifwe add the 8.3 percentage oftimes thatAI C overpredicts the rank even

modelin morethan17 percentin allsimulations.

T he interestinginformation in T able 5, however, is the relative success ofthe H annan-Q uinn criterium in choosing the correct lag and rank order, without the risk of overpa-rameterizingthe model. T his repeats whatwe observed in ourthree-variable simulations. Even with a sample size of100, the H annan-Q uinn criterium chooses the correctlag 9 4.2 percentofthetimes, and leads tooverparameterized models in less than 0.5 percentofall simulations. W ith a sample size of200, the H annan-Q uinn criterium is almostperfectin choosingthecorrectlag-order(9 9 .9 8 percent), and is thebestamongthe threecriteriafor choosingthecorrectlag-rankcombination9 0.7 percentinallsimulations. A lthoughtheper-centageofthetimes thattheSchwarz criterium chooses thetruelagimproves when thelag and rankorderarechosen simultaneously, this criterium is stillsigni…cantlybiased towards under-parameterizingthemodel, evenwhenthesignalis strongand thesamplesizeis 200.

T heconfusioncausedbyAI C whenitisusedtochooselagandrankordersimultaneously,

is re‡ected in its forecastperformance. T able 6shows the improvementin the forecasting performanceofmodelschosenbythedi¤ erentcriteriawhenrankorderischoseninthemodel selection stageoverwhen itis ignored. W hen thesamplesizeis 100, usingAI C tochoose

thelag-lengthand rankordersimultaneouslyleads toverypoorforecastperformanceofthe selected model, forlongerthan one step ahead horizons. G iven the relative success ofthe H annan-Q uinn criterium in choosingthecorrectlag-rankorder, itleads tomodels with the bestoverallforecastperformance, withAI C beingaclosesecondonlywhenthesamplesize

is 200. U nlikethecasewherethesignaltonoiseratiowas small, theunderparameterization of the true model by the Schwarz criterium is re‡ected by a very poorrelative forecast performanceofmodels chosen bythis criterium.

6.Conclusions

T his paperargues thatthestylized factthat“macroeconomicvariables movetogetherover the business cycle” should be taken seriously in econometric models using them, usually

a V A R ora V EC model. T he M onte-Carlo analysis in this papersuggests the following messages topractitioners whoanalyzedatawhichexhibitcomovement:

1. T herearenon-trivialgainsinforecastingaccuracyandinM SEofestimatesofvariance-decompositioncoe¢cients whenweallowV A R andV EC modelstobeofreducedrank. T heseresultsarestrongerinourbenchmarkcase(lagorderisknown), butarecon…rmed as wellforthemorerealisticcasewherelagorderis selectedusingpopularinformation criteria.

2. Informationcriteriathatallowforreduced-rankV A R andV EC models perform better in choosing the correctlag length compared to these same information criteria that disregardreduced-rankmodels(n = r).

3. IfH Q andSC areused in acontextwhich allows them topickreduced rankmodels,

thentheproblem thattheyunderpredictthetruelag-length is signi…cantlyremedied. 4. D onotuseAI C tochooselagand rankorderatthesame time, especially when the

samplesizeis small.

5. T he H annan-Q uinn criterium seems to be the bestcriterium forchoosing lag-length and rankorderatthesametime.

T heanalysis alsocon…rmsthatAI C canleadtoover-parameterizedmodels withadverse

consequencesforforecastperformance. H owever, itrevealsthatL ütkepohl’s(19 8 5) conclusion thatmodels selected bySchwarz criterium leadtobestforecasts is anartifactofhis M onte-Carlodesign.

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A . System

R

andsignal-to-noise ratio

Inamultipleregressionwithstochasticregressors andi.i.d. errors,y= X ¯ + ";thelimiting signal-to-noiseratio(sn r)can bede…nedas:

sn r= ¯ 0_lim T! 1 E ³_X0X T ´ ¯ ¾2 " ; (A .1) whereE(""0_{) = ¾}2

"¢I, and theproportion ofthevariation ofdependentvariableexplained

bythemodel, i.e. thepopulationR 2_{, is:}

R 2 = ¯ 0_lim T! 1 E ³ X0X T ´ ¯ ¾2 "+ ¯0limT! 1 E ³ X 0_X T ´ ¯ = sn r (1 + snr):

SincetheasymptoticvarianceofpT³¯b_¡¯´isAV AR(¯b) = ¾2 " ³ limT! 1 E ³ X0X T ´´¡1 , we canwrite(A .1) as: sn r= ¯0³AV AR(¯b)´¡1¯ : (A .2) ConsidernowaV AR (p): yt= A1yt¡1+ ¢¢¢+ Apyt¡p + "t: (A .3)

T heanalogous measureofsn r foritis:

sn r= ¯0³§ -- ¡1´¯ (A .4) where¯ = vec(A),A= · A1 ::: Ap ¸ , E³"t"0t¡j ´ = - , and: § = 0 B B B B B B BB @ ¡0 ¡1 ¢¢¢¡p¡1 ¡0 1 ¡0 ¢¢¢¡p¡2 ¢¢¢ ¢¢¢ ¢¢¢ ¢¢¢ ¡0 p¡1 ¡0p¡2 ¢¢¢ ¡0 1 C C C C C C CC A ; where¡j= E ³ ytyt0¡j ´

:N oticethat§ iscompletelydeterminedby(A;- )viatheY ule-W alker equations13_{. A ftersomealgebra, itcan beshown that(A .4) is equalto:}

sn r= ¯0³§ _-- ¡1´¯ = trace³¡0- ¡1 ´

¡n:

U singthis lastresult, onecan then de…nethesystemR 2 _tobe:

R2 = trace(¡0- ¡

1_)¡n

1 + trace(¡0- ¡1)¡n :

Table 1: P ercentage improvementinM SE offorecast-errorvariance decompositions, anddi¤ erentforecastmeasures whenthelaglength is known

horizon T ruerankis one T ruerankis two

(h) G FESM jM SFEhj T M SFE V ar. D ec. G FESM jM SFEhj T M SFE V ar. D ec.

Samplesize100 1 22.22 22.22 1.9 7 -14.10 12.08 12.08 0.9 8 -39 .60 4 60.41 8 .66 1.7 2 45.55 34.56 5.02 0.9 6 45.23 8 7 0.54 1.7 0 1.39 46.9 1 42.57 1.32 0.7 7 43.84 12 7 2.34 0.46 1.03 46.25 44.66 0.45 0.59 46.9 1 16 7 2.8 6 0.21 0.8 1 45.9 5 45.26 0.25 0.47 47 .7 3 Samplesize200 1 11.22 11.22 1.14 -7 .9 2 6.55 6.55 0.60 -42.53 4 29 .53 3.80 0.8 8 64.50 17 .8 0 2.32 0.52 9 .32 8 32.9 7 0.52 0.64 66.20 20.7 2 0.46 0.37 13.18 12 33.37 0.11 0.45 64.07 21.19 0.07 0.27 14.51 16 33.47 0.05 0.34 63.52 21.25 0.06 0.20 14.9 0

T able2.a: Frequencyoflag(p)and lag-rank(p;r)choice bydi¤ erentcriteriawhenthe truemodels are(4;1 ) Selected lag 1 2 3 4 5 6 7 8 Selected rank 1 2 3 1 2 3 1 2 3 1T _{2 3 1 2 3 1 2 3 1 2 3 1 2 3} N umberofobservations= 100 A IC (p) ¡ ¡ 57 .0 _{¡ ¡} 13.1 _{¡ ¡} 12.6 _{¡ ¡} 14.0 _{¡ ¡} 2.0 _{¡ ¡} 0.7 _{¡ ¡} 0.3 _{¡ ¡} 0.3 A IC(p;r) 10.8 2.5 0.4 7 .4 2.0 0.1 14.4 2.4 0.1 32.7 3.4 * 8.3 1.1 * 5.0 0.6 * 3.8 0.4 * 4.0 0.5 * HQ (p) ¡ ¡ 9 2.9 _{¡ ¡} 4.7 _{¡ ¡} 1.7 _{¡ ¡} 0.7 _{¡ ¡} * _{¡ ¡} * _{¡ ¡} 0 _{¡ ¡} 0 HQ (p;r) 39 .2 1.9 0.2 13.3 0.3 * 17 .0 0.1 * 24.3 0.1 * 2.4 * 0 0.7 * 0 0.3 0 0 0.1 0 0 S C(p) ¡ ¡ 9 9 .6 ¡ ¡ 0.4 ¡ ¡ * ¡ ¡ * ¡ ¡ 0 ¡ ¡ 0 ¡ ¡ 0 ¡ ¡ 0 S C(p;r) 7 3.8 0.4 * 10.7 * 0 8 .4 0 0 6.6 0 0 0.1 0 0 * 0 0 0 0 0 0 0 0 N umberofobservations= 200 A IC (p) _{¡ ¡} 25.9 _{¡ ¡} 10.7 _{¡ ¡} 20.0 _{¡ ¡} 40.0 _{¡ ¡} 2.7 _{¡ ¡} 0.5 _{¡ ¡} 0.2 _{¡ ¡} * A IC(p;r) 2.2 0.7 0.1 3.3 0.8 * 12.1 1.8 * 56.4 4.1 0.1 9 .1 0.8 * 4.1 0.3 0 2.3 0.1 0 1.6 0.1 0 HQ (p) _{¡ ¡} 8 0.1 _{¡ ¡} 7 .8 _{¡ ¡} 7 .2 _{¡ ¡} 4.9 _{¡ ¡} * _{¡ ¡} 0 _{¡ ¡} 0 _{¡ ¡} 0 HQ (p;r) 16.1 0.6 0.1 8 .9 0.1 * 20.7 0.1 0 51.3 * 0 1.9 0 0 0.2 0 0 * 0 0 * 0 0 S C(p) _{¡ ¡} 9 8 .6 _{¡ ¡} 1.0 _{¡ ¡} 0.3 _{¡ ¡} 0.1 _{¡ ¡} 0 _{¡ ¡} 0 _{¡ ¡} 0 _{¡ ¡} 0 S C(p;r) 49 .4 0.1 * 11.1 * 0 17 .1 0 0 22.3 0 0 0.1 0 0 * 0 0 0 0 0 0 0 0 N umbers in each cellrepresentpercentagetimes thatthemodelselectioncriterion correspondingtothatrowchosethelag-rankordercorrespondingtothat

column in 100,000 simulations (1000 simulations of100 di¤ erentD G P s). T hetruelag-orderis identi…ed withsuperscriptT. A * corresponds toanon-zero valueless than 0.05 percent. N umbers in arowmaynotaddup to100.0 exactlybecauseofrounding.

Table 2.b: Frequencyoflag(p)andlag-rank(p;r)choice bydi¤ erentcriteriawhenthe true models are(4;2) Selected lag 1 2 3 4 5 6 7 8 Selected rank 1 2 3 1 2 3 1 2 3 1 2T _{3 1 2 3 1 2 3 1 2 3 1 2 3} N umberofobservations= 100 A IC (p) ¡ ¡ 19 .9 _{¡ ¡} 10.2 _{¡ ¡} 21.3 _{¡ ¡} 41.3 _{¡ ¡} 4.6 _{¡ ¡} 1.5 _{¡ ¡} 0.7 _{¡ ¡} 0.5 A IC(p;r) 1.1 4.9 1.0 1.0 4.7 0.6 2.5 15.5 1.2 4.3 43.7 1.8 1.2 7 .0 0.3 0.8 3.1 0.1 0.7 1.8 * 0.9 1.6 * HQ (p) ¡ ¡ 64.1 _{¡ ¡} 13.1 _{¡ ¡} 12.7 _{¡ ¡} 9 .9 _{¡ ¡} 0.1 _{¡ ¡} * _{¡ ¡} 0 _{¡ ¡} 0 HQ (p;r) 8 .6 19 .6 1.9 5.0 8 .1 0.2 8.1 14.8 0.1 10.5 20.8 * 1.1 0.6 0 0.4 0.1 0 0.1 * 0 0.1 * 0 S C(p) ¡ ¡ 9 3.2 ¡ ¡ 5.1 ¡ ¡ 1.5 ¡ ¡ 0.2 ¡ ¡ 0 ¡ ¡ 0 ¡ ¡ 0 ¡ ¡ 0 S C(p;r) 30.3 30.6 1.2 9 .5 4.8 * 9 .4 4.3 * 7 .9 1.9 0 0.2 * 0 * 0 0 * 0 0 0 0 0 N umberofobservations= 200 A IC (p) _{¡ ¡} 3.3 _{¡ ¡} 2.7 _{¡ ¡} 16.3 _{¡ ¡} 7 2.2 _{¡ ¡} 4.3 _{¡ ¡} 0.8 _{¡ ¡} 0.2 _{¡ ¡} 0.1 A IC(p;r) 0.1 0.6 0.2 0.1 0.9 0.1 0.3 10.2 0.7 0.9 7 2.3 2.5 0.2 7 .1 0.2 0.1 2.0 * 0.1 0.8 * 0.1 0.4 * HQ (p) _{¡ ¡} 27 .9 _{¡ ¡} 9 .6 _{¡ ¡} 23.3 _{¡ ¡} 39 .2 _{¡ ¡} * _{¡ ¡} 0 _{¡ ¡} 0 _{¡ ¡} 0 HQ (p;r) 1.3 7 .5 0.6 0.9 4.7 * 3.4 20.0 * 4.7 56.2 * 0.2 0.4 0 * * 0 * 0 0 * 0 0 S C(p) _{¡ ¡} 7 4.4 _{¡ ¡} 10.4 _{¡ ¡} 10.3 _{¡ ¡} 5.0 _{¡ ¡} 0 _{¡ ¡} 0 _{¡ ¡} 0 _{¡ ¡} 0 S C(p;r) 9 .2 26.9 0.7 4.3 6.8 * 8.2 15.1 0 9 .1 19 .8 0 * * 0 * 0 0 0 0 0 0 0 0 N umbers in each cellrepresentpercentagetimes thatthemodelselectioncriterion correspondingtothatrowchosethelag-rankordercorrespondingtothat

column in 100,000 simulations (1000 simulations of100 di¤ erentD G P s). T hetruelag-orderis identi…ed withsuperscriptT. A * corresponds toanon-zero valueless than 0.05 percent. N umbers in arowmaynotaddup to100.0 exactlybecauseofrounding.

Table 3a: P ercentage improvementindi¤ erentmeasures ofaccuracyinforecasts generatedbythe bestpossiblyreduced rankV A R overthe bestfullrankV A R chosenbythe same modelselectioncriterionwhenthetrue models are trivariate(4,1)

horizon A IC H Q SC

(h) G FESM jM SFEhj T M SFE G FESM jM SFEhj T M SFE G FESM jM SFEhj T M SFE

Samplesize100 1 6.6w _6.6w _0.0w _6.8b _6.8b _2.8b _{5.3 5.3 1.6} 4 10.8w _{2.3 1.1 16.1}b _6.1b _4.8b _{10.9 4.1}w _3.0w 8 4.0 -1.0 0.0 15.1 -0.3 2.7 11.0 -0.1 1.7 12 2.0 -0.6 -0.2 14.2 -0.2 1.7 10.7 -0.1 1.1 16 1.0 -0.3 -0.2 13.7 -0.2 1.2 10.5 -0.1 0.8 Samplesize200 1 9 .1 9 .1 2.0 11.0 11.0 6.7 8 .3 8 .3 5.3 4 22.2 3.2 2.0 30.8 8 .2 7 .7 22.5 7 .1 6.8 8 22.1 0.1 1.0 31.8 0.5 4.4 23.4 0.4 3.9 12 22.1 0.0 0.7 31.7 0.0 2.8 23.4 0.0 2.6 16 22.0 0.0 0.5 31.7 0.0 2.1 23.3 0.0 1.9

G FESM is Clements and H endry’s generalizedforecasterrorsecond momentmeasure,_jM SFEhjis thedeterminantofthemean squared forecasterrormatrix

Table 3b: P ercentage improvementindi¤ erentmeasures ofaccuracyinforecasts generatedbythe bestpossiblyreducedrankV A R overthe bestfullrankV A R chosenbythe same modelselectioncriterionwhenthetrue models are trivariate(4,2)

horizon A IC H Q SC

(h) G FESM jM SFEhj T M SFE G FESM jM SFEhj T M SFE G FESM jM SFEhj T M SFE

Samplesize100 1 7 .6 7 .6 0.1 5.9 5.9 2.2 1.6 1.6 1.4 4 19 .2 2.9 0.5 19 .2 6.1 3.9 10.1 6.1 4.3 8 20.4 0.1 0.1 19 .7 -0.0 2.2 10.0 -0.0 2.5 12 20.5 0.0 0.0 19 .6 -0.1 1.4 9 .4 -0.0 1.6 16 20.5 0.1 0.0 19 .4 -0.1 1.0 9 .1 -0.1 1.2 Samplesize200 1 5.9 5.9 0.7 6.8 6.8 2.3 8 .8 8 .8 5.4 4 15.3 2.0 0.5 20.5 4.3 2.6 28.7 8 .9 6.5 8 17 .1 0.2 0.3 21.7 0.3 1.5 31.1 0.6 3.7 12 17 .3 0.0 0.2 21.8 0.0 1.0 31.3 0.1 2.5 16 17 .3 0.0 0.1 21.7 0.0 1.0 31.2 -0.0 1.8

G FESM is Clements and H endry’s generalizedforecasterrorsecond momentmeasure,_jM SFEhjis thedeterminantofthemean squared forecasterrormatrix

T able4: P ercentage improvementinM SE offorecast-errorvariance decompositiongenerated bythe bestpossiblyreducedrankV A R overthe bestfullrankV A R chosenbythesame modelselectioncriterion

horizon T ruerankis one T ruerankis two

(h) A IC H Q SC A IC H Q SC Samplesize100 1 -20.9 9 -5.51 1.47 -14.39 -8 .9 8 -9 .26 4 -5.68 1.17 7 .20 13.04 5.41 -8 .61 8 -8.00 -0.8 5 4.87 8.8 7 2.08 -10.7 7 12 -9 .26 -2.00 4.20 7 .9 7 1.42 -11.14 16 -9 .7 9 -2.44 3.9 6 7 .56 1.19 -11.28 Samplesize200 1 -8.10 12.37 20.25 -4.45 0.52 9 .68 4 29 .15 50.65 40.9 1 27 .15 32.84 21.20 8 27 .40 57 .51 38.60 24.7 5 31.37 20.17 12 25.7 9 56.30 37 .41 24.06 31.03 19 .7 3 16 25.35 55.81 37 .05 23.8 4 30.9 0 19 .56

Table 5: Frequencyoflag(p)andlag-rank(p;r)choice bydi¤ erentcriteriawhenthetrue modelis asixvariable V A R (2,4)

Selected lag 1 2 3-6

Selected rank rank< 4 rank= 4 rank>4 rank<4 rank= 4T _{rank>4 rank< 4 rank= 4 rank>4}

N umberofobservations= 100 A IC(p) _{¡ ¡} 1.2 _{¡ ¡} 9 7 .1 _{¡ ¡} 1.7 A IC(p;r) 0.1 0.1 * 9 .3 7 2.8 8 .3 2.7 6.3 0.5 HQ (p) _{¡ ¡} 25.2 _{¡ ¡} 7 4.8 _{¡ ¡} 0 HQ (p;r) 4.3 1.4 0.1 40.6 53.2 0.4 0.1 * 0 S C(p) _{¡ ¡} 7 7 .5 _{¡ ¡} 22.5 _{¡ ¡} 0 S C(p;r) 32.0 2.8 * 52.5 12.8 0 0 0 0 N umberofobservations= 200 A IC(p) _{¡ ¡} 0 _{¡ ¡} 9 9 .8 _{¡ ¡} 0.2 A IC(p;r) 0 0 0 0.6 8 9 .5 8 .5 * 1.2 0.2 HQ (p) ¡ ¡ 0.8 _{¡ ¡} 9 9 .2 _{¡ ¡} 0 HQ (p;r) * 0 0 9 .0 9 0.7 0.2 0 0 0 S C(p) ¡ ¡ 28 .6 _{¡ ¡} 7 1.4 _{¡ ¡} 0 S C(p;r) 3.2 1.2 0 40.4 55.2 0 0 0 0

N umbers in each cellrepresentpercentagetimes thatthemodelselectioncriterion correspondingtothatrowchosethelag-rankordercorrespondingtothat column in 10,000 simulations (500 simulations of20 di¤ erentD G P s). T hetruelag-orderis identi…ed with superscriptT. A * corresponds toanon-zerovalue

Table 6: P ercentageimprovementindi¤ erentmeasures ofaccuracyinforecasts whenthe truemodelis asixvariable V A R (2,4)

horizon A IC H Q SC

(h) G FESM _jM SFEhj T M SFE G FESM jM SFEhj T M SFE G FESM jM SFEhj T M SFE

Samplesize100 1 1.7 1.7 -1.2 14.1 14.1 1.1 15.4 15.4 4.4 4 -6.6 -2.2 -0.3 25.0 3.2 1.3 30.2 6.0 2.4 8 -15.4 -1.0 -0.3 25.1 0.8 0.7 24.6 2.4 1.2 12 -18.2 -0.7 -0.2 25.4 0.7 0.5 21.7 0.7 0.8 16 -18.3 -0.8 -0.2 25.5 0.1 0.4 20.8 -0.4 0.6 Samplesize200 1 5.8 5.8 0.1 6.8 6.8 0.3 13.7 13.7 2.5 4 11.6 1.2 0.4 12.9 1.4 0.4 25.8 3.8 1.4 8 12.2 0.3 0.2 13.7 0.3 0.3 26.2 1.5 0.7 12 12.1 0.1 0.2 13.6 0.2 0.2 25.7 0.6 0.5 16 12.2 0.0 0.1 13.8 0.0 0.1 25.2 0.2 0.4

G FESM is Clements and H endry’s generalizedforecasterrorsecond momentmeasure,_jM SFEhjis thedeterminantofthemean squared forecasterrormatrix