Are Forecasting Models Usable for Policy Analysis - Christopher a. Sims

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Are Forecasting Models Usablefor Policy Analysis?*

A. v

one o he early papers descbg he applicatos o vecor autoreession (VAR) mels o economics, Thoms Sargent ( 1979 emphase tat wle such models were usel or orecasg, tey coul not be used

o icy anayss. Recenly ths int has been vigorousy easseed, by agent mse ( 184 and byEdwad Lee ( 185 among ohes

The t has eured vigoous easseon becauseVAR mels are use widey, and ew who use em saypue-n the sense o never g

au er

mations o icy-or vey ong here re seer reoabe ways to generate conditona orecas om VAR mes Once one has seen how easy it is ob aorecast condiona on a ceai coraon o licy

variables, it is tempg me such orecs fcult to be sure ht hey d no u one'sopnons on icy choices be Lie a leaer

demonsating the vaue o VA mels orecasnhas also develope meh use hem o me orecassconional on poc choice een use em computng op les (See, or exple, Literm 182, 1984

A smila paox conces he so o e lgeommecia economec orecsig mes I waspesent at a ecent oun tabe dscussion n whch a distnguished heoreica economst oadey asseed a, snce ese commercia mel are n or orectin o o s m sous oomsThe ratona expeco cique o he use o such models in icy aysis is no more h ten years o

an in a gd p o te economics proession has thesiion o the esablishe othodoxy Yet hen acuallicy choices are beng made at all eves o the publicand private sectors, oecass m these lage modesth cononl ucondionl remain peiveyinuenal he ationa expectatons schl beg wth apam for povidn ateative to such mes o uanttave icy anaysis, but the po has had ittepracca mpac

Why

s s?Are e people who auly make poicy

uabe to under he tii o using VAR modes or conventona enomec mels as an ai to poicyhoice? Or e he absact argumens against using such mels hs way missing somehing mant

Why Making Plicy With a Frcting dlIs Sup to Be a MikeThere are o relate versions o he gument aganst usig oecsng mel or i analysis One is that such mels nohing more ha sar descrptionso he hsoical da usually based on sple corelatons We such a descripion can b exaatd ito a

usel orecas, supg at it ca e basis o projeg e eec o icy choice amouns o takng coelaon o indcae causaon whch we udestan

b allaous

"Rsech for pap up by National Sciece FoudonG SE-8309329.



Te gment in ts om applies dectly to VR

odels ch e oightly descptive statisticlodes that do nohing more th suze coeations a coneniet way Te argumet also applies directlyto he use of arge commercial mes sce mayecoomists (ysel included) regrd the ecoomic terpretations those models gve their own equations astraied d ragile Despite the ragilit o those interpretations the mels ae clerly usel forecastingEcoomist who accept this form of the gmet againstorecating mels as policy tls may be ready to rejecthe use o comercil models or policy anysis despiteadmitng thei value or forecastg

Sget 1984 puts foad a second version o thegme ags usg forecasg models or polcy

aaysis He obses hat VAR moe usualy ico rae plcy vables ito th model symmercally whoher vrabls teag hem l s rndom viabes eaees that, n prncile, plicy choices ae o

vrabes, wih uce abut them imtuece o ac behavior. However, when we chosea picy we do o usully rol he dice; sead eordarly have to make a unqu choice ic to us ha adterisc chracter We cano reasonab au such a choic wi coes of a oel inwhch plicy is ded by soe radom mechs

We properly inprd, bo versios o theument are crrec, evn uoloc. It s issbe tus a stasc mdel anlyze cy whout gin

bnd e coelais mak econoic teeton f . Geerag such a erpretation s hatecomcias ca identcation of a mel An onecat aalyze e choc of licy varables withoutcug trou salss web o a model whch a

licy vaabls ae den sid e me In aes agm ev al use of a l r forca

Bu s cc d t cosue betio us of frec ls to gd choice. Thy on u a when e n a ay tuse such ml ude chc e e •or explici, sule i wi n dent m m·prea a us such me de

licy chic w hav pre ther dnt

intrpreo aavs real coe •� nme c aysis bd on spe 11 wch sle, we iden ssu� JUt �uf cien b a frc el o n issu ised vus aives bae ( srong den assuns wch lea w 1th

r recastg prps.

dd r, rr, d dTese thee tes-reduced fom scre d identicationre used i deren ways by economists and eused ouide of economics i sill other as n onecommo pae of usage a reduced fon s a modlthat descrbes how some historcal daa wch we ccall was geeraed by some rdom mechnismhen we estmate a reduced orm el we consctsome statiscs tat summarze the ll data set Thereduced f mel can be thought o a rationale or a paicula d f daa suma

Asttur or scural ml is a mel we ca usei decisioak geerates predicons of result Z of vio k o acosA e ight take. The notions of scre ad reduced o e implicit n most of

use of da d dcisionmkg but when a econmerci uss hs oos he generally elicitlydisplays a rduc f rbab dsbutoX /or da a a nco of reduced fom paameter d dslays a scur as condona dsrbutonq(Z a) o rsul ven acons deenng on now su aer deig a mode sen assrg a cocto bween the reduced form ade scure so a esmaes o the reduced omper ca b used to detee the scraamr 1

Reduc slest s identtion is the terpretaon f srcaly obsed varatio i data a ay ta alows e vaaon be used to predct h

osequences o acon no ye deken Mngis ccn be da and consequences of dcsions is vr via, d it always is more dcult theore r is acn w cemplae takg m aysc bseabl v If we ar lucy ther may much src radm vation n actons at ae si e aco w now conemplae Then weay b ab u rar drecy to decd teel f u as. Bu somemes w mu ca acs v dr hos obsedstorc wch c identcao bcoms ordcul ad cs

n eccs, scurl mels e en oulae o ta v ar i vecor hs an coomic

ter a is, e lmn of

will be hg lk

I In engeeg onl th, is used f o wa t wo ca f paraeer. n dt the queso o wheher or no ame' m, >·ne bby tbo o h B sage e reated o eome usedecbi he ext hee s not mu nt n exnng



"teret-elatc of demand for money or elai

of uttuton between electc pwer and labor n mlkpuon In ontrst the eleents of the f veor reusually hder to teret becaue they reect combeuence of behavor y ectors of the econoyThere no unque tandard however, for when a par �eter ha n econoic intepretaton The paameters a oel ay have ntepretation, yet the odelay not b stur, the sese at we ay be uable

t ue t predict te coequences of he aions whhere u. Converely, we may be able to ue a mel ake u predicons acuraely even hough ome or allof it paete do not have neat tepreation

The foal mahematis of aeig a coection een a rued fo el nd a nonstura el

that has a aing interetaion is no derent o thefoal maematic of aseg a coneion ween e reduced fo and a cre May economi avecome t of cra mel as els wit aisig ereaions for a paamete The uaguable asseion that preding he ee of iyrequies ideniaion of a ca el us cmes, via a ei cnsion a oure of seriousmiundeig2

amples of Identification in DecisionmakingTo brig s absa disssion down o eah present a eies of exaples of e apliion of saisiaels deciioakig

oThe Spreadsheet

Perhaps the iple ad os widely us fo of quiave melng a a aid iy oie ipreeet nacia meig Here a iden

ies and a fe iple relaio og ag agoes haeig average isia avior uss a aeork for pjeing e ceuences of busines deiions. n ye of e ee i noexpli! t probabii me, oug e e of aerage pahavr to exalae e ipies soe iefsau he probabiiy se of e e reuefo is ipicit he pai aegoies e re d e the aog daa i e of sss reu on q pui rai nd sofoh . Th ure generay i fo by s at ce soaly obse average aos e acig daa ae liely o de vaious ges buine ae d ence c je eee of vaiaions n ae Eveone dea ha suh me ae subje to y iaio; non teles, beaue he els ie soe cniseny on

pection of lon ls of ieelaed numbe ey ae

helpl he mel' dentg umption ae eldom the focu of uch explt attntion though ues understand that the assumption e crude apprxatons and

therefore judenlly scount the el reults accordinglyo Clinical Trials in MedicineAother exaple, n ome way even spler e n clinca tal in medine We mgt have reuls om a udy of 20 paten, 1 0 given e old treatment d10 gven the new eamen Te reul mght bered as crosclassied no 16 categoe by agerace d sex of he paien Te coplete crostabula

tio c be regrded as a et of reduced fo paeter etmates. Say hat 80 of hoe ven the new treament

were cured while ony 30 of toe ven e old teament were cured A dctr usig ths udy igt take t toen hat he new eaent s beer nd proeed toprefer t over he old for al his piens. He would be takig the overa pion of te paien in the tudy who beneed om ea eaen be a g icaor of te probabiiy a a pien of is il benet m the teaten is ura mel woud piity count he vaon aross age-rae-sex ategoes s ur

tnt rando vaiaio.Bu upe e ds paie for wom the

tudy is relevan is a 24yea-old blak mae The dtor happns o rea a ere were ony blak eeen 15 and 25 yea old e udy an, ter that

tose ven te old eaent sived and oe ve� henew eaen di Given is ifoaio, te dcr probably wil cse e e eaen bu thechoice is no auma ad wi deend on deai of thedaa d on e ds iefs au ow he eaents

workat is, wheer ere is ay raon upse theeaen would w iereny genera on young blak e

Tis kid of esioi fa aises i y pae I tis pai paie e e paien e sple o

2Te sense whih tctu s he, t somewhat n ni we mmon out n e xt weU nded n eer eaen ofse ssu-r exmple th Hz 196) d Kpm d Buc

( 1959). Enm

shl sae a

tha

tctuis awayset lm usen

he pr o he ee aons In

nas ducd r e sat sme nal nebe mel a th se wa m ven b ls n One mqe a e orpr n& e ees o s a edu o a a m earae s o haora neo Sb e d b stal d n a sense d�ed I seno h i usng ud i hs wa tgh hs pa I wl k me sg whh me ake nee stur.



that results rm the sample are relevant? C the resulbe extraplated t age r etc ups nt represented ithe saple at al Shuld a veteriran give ay weigh tts study hums in decidig hw t treat hrses Thisis the identicatin prblem.

o c ecnmcs, he use a recas ipes at smedecisn is bein made whch may be uenced by beesabut the ure pat he ecnmy. The reduced rmmde descs te daa at each hisca date t as drawnm a prbaby dsrbutn cndtined n daa up hrugh -1, usuy w the rm tha dstributn tesae a daes t In a pe recasn san, hesrce ase a he sbun net r's daa

is unfeced by ay acn we may take, bu tha sreated daa up ugh e curent pe in the sae way a da a ha een eaed daa up ug 1a a hsc daest. B hee s ways a quesn as whee n s pcl dae s een m wha we have seen scay: Wl s que b the ne in whch ur recasg me ay breaks dw And hee s as aways a quesn a t whethe r n tedecsn we e man ds i ac have n impac nne pers da I e decsn s whee r n aeu a hme mtgage then ee is pbably n pac;bu s wheter n t lay 10,wrkes, teaswe may n s cea

o ccc PcyWhen macrcnmc mes e used pjec theeect cy, the ms cmmn prcede bes w a mei whch cy vaes appe epcty d wch hse vabes e en a predeemned Tha s he me s a whee nncy vabes a a ven dae e deeed y cy vrabes a ta dae gven Pcy ns e ten deed a te

·pa the cy vaes d e equa temde ae se a e vabescespn e vs cy pn

We e ea ts pcede s sahte epren dencan a reducem s n se e me s ler ad cmple n

tha t c sve

vabes cnan ecep elcy vaaes Lg e cy varales and

he he vaes a sysem equains al evarabes ha e

(l)

(2)

Y(t)A = X(t)B + Z(tl)C e(t)

Xt Zt-lD + ut

where Z(t-1 s the vect al lagged values

andYtha ener the system, where the rst blck equatins

is he mdel, ad whee the secnd blck equatins aepredctn equans r e pcy vables e ndpath r ue Y cesndng t pa X by n he X pas ad slving r usg the rst blk equatins When we d ts we are settig the disturbaces e(t) te rs blck euatins t zer whileadjusng the resduas u(t) n the sec blck t achevehe hyphec pa X whch epesent lcychces Vran i licy s bei represened enrelythrugh vaan i u(t), wt n varatn e(

The equat te mdel have been tted tsca daa F be easnabe t prec heeec cy by se e t er d vg u it musts be e a scy he s and ()s haveeen ureaed d secnd be reanable t suppse thatprectn er r plcy vables can be dentied

wh cy chces I is nt necess that the ysce pecn e picy varables be icy

· chnes b a suces predcn er r plicyvrabes mus have he se s eect n heecnmy a pcy chages r exple in a perid when te mneta autt was maitag stablenees ras mney sc varas wuldat east i e sh ndrecy eec sh in ney demadTrea mney stck a a cy varae accrdi t e cnventn mehl lcy pectin wud hen e seus msakes sce it wud pl a

sen dencan e sica eec mneydemd s w he eec cygeneated shis inmney s

o A Exc TA sc mde need n cna plic arabes irde be use n pjecn e eects picy Theses o ee ecnmerc identicatn a me spply d demand say

Demad

3 p(t) = 8 a q(t) fy(t) u

upply

(4 p(t) = ¢ + q(t) + w(t) '

3 If it no mple i c comple by , <"uans ory y pedeeed ve. rde k pn • t�ee h h plicit in e, e all m ewdle e



where p s prce, q s quantty y s ncome d w s

weather. Though ths model does not contan an excse vaable t can e used o predct the eecs of an excse ax Le e excse ax rate a bf hen tere

was no i the pst we c s pedt he ees of a paid by rucer by ircig in e supplyequation 4), making it

(5) [1 - )]p) = + q) + 1w) + v)

Sovng or p and q we an obtai

6) p) = [o + y) + u)]

- a[ + w) + v)J}

+ { a[ )]

7) ) = {[1 )][o + y) + u)]

+ [ + w) + v)J}

+ { a[ l )]}

wch c used o redct he eet of the rate) for ven vaues o weaher d nome wh re eaeas deened ousde s market hi exple he hsa esnses o pres d quanttes to movemen in weather and nme e eng

sed to predi he resnses

a revously unedexcse tx Ts an inelealy appeang k, somuh so ha eonomis somemes orge ha wouldrely e eer o have some soa daa m a per in whh ther w aal exse a ae heystem 6)7) could e esmaed dey When wehave o deive 6)7) om(3)4) we rely on somestong assupon ong hee te ampon hathe dynas whh have en hsoay ua omel marke pre luuaon w aso sue melresnses o a more pesen hge pe due aton Another s the ason ha i supplyhavor s perfectly com ve4

o Rational Extaton Pl AnyThe deal lcy evauaon prue o e raonexctton schl-wh e (1984) h nda an al ea ve me losely e VAmels-es wh a dyna probab mel or heeonomy. We an label s whole mel There s aloa pbab law or y vaables whh we lal hou he y vables have ved hsay

he ratonal expectaton schl makes a sha dstncton

ween changes i lcy varables and changes i theprobablty law goveng polcy vaables Only the latter are taken o be "real changes i polcy. In the dealework, h not changed at all torcaly. Weonemplae makng a sne, eanent hne in Getng historicl daa in whih X h not changed to te us somethng aut what wl hapen to Y when we dochange is not imssibe is quaitativey smar te problem of recng te eet of a neverbeforetredexce , a discuss prevously, houh the raonaexaons idenaon pblem s even more ineresng and elleually hallenng

s Hasen and Sagent(1980) outlned t he ratonalexctaons rescrpo s that etes ofobjective

nios of agen" be eated ese paeters oen descrbed as haaezng es and technol o, are ken o e nvaian he goveent decon eX and also o e sucent o deteme the reacon of the economy o peane onee change n the

decon le Boh n clag o able o estmate alhe relevan aspes o es ad enolo and n lang o be able use hem o pred reacton ohges in we wil have o make asumpons ha ean oer of maude more elarat and questonable

h hoe requi he eise xampleJut a n he exce exae we ae relly bete

o he hge n prbab law or ly vables ha we onemplae mplemenng ha preeden that

ae h Y and X need ime argumen d we have conder ho o con a dyc mel connecng X) o Y). I thi mel the vaue of lcy vables w e deer o tepstX () s determned hen he vaues o vables are genera by X().( ) e s no hen he probab law generag ly vaable bu a pee or sge o ha pbab lawNoneheless X) hat e e the e of ch what we nee he dbuo of the uepa o ) ven our derae hoe o a pah or X)he a ha t) d () now repsen pbablaws no values o re varables ds no hge hebac au of the deco ble

By ndeg d evuaton edue

appled a saon where y hans aaly s e e sure o e enoy by hg eaon oaon les e raon expeaon rique

4Ifnd suly mtve uaon(6)-7 wll not a dp ds der e t, ee & e sc haio f , qu ime ehe d my �igshbem geee y me y .





concreely hee may be a derence Some may feel he

qesons abo gen's deniing assmpions appg e penaion sdes re more or less elng ose a deng predicion eo erest raes wih icy The is a n y emprcl sudy

here wil debaable quesons aut idencaion qesons ha will eave s more or ess ncomfoabeat applyig e concsions The raionl ec ions mework raises sch isses om a erent nge,b it cot avoid hem.

Rational Expeations Models VeusPoli Analysis With Forcasting ModelsObviosly, in pnciple, a raionl eions qli�m mel c s g a forecsg mel s a VAR

mel I fac a world whc people were ratonland markes were comttve, one migt exect that thest forecastg mels would be ratonal expectationsibriu mels

Possibly n so obviosly, ere is no locl consstency beeen a world of compettve mkes andrato ole nd a world whc te est forecstg mes are VARmels d otl cy c maeby sg VA mels uder te slest of idening assptons This later it reues some eendemathemaicl rmet nd is develod ly in a recent

par of me (Ss 1985).Thus, in the absact, a mel culd clude lner

VARmels d nonle ratol exectatons eul�um mels s specil cses of a more gener work Bt in acal probles of licy choice ndforecsing, ere re s. Raonl etonsmels have not been sed successly for lrgscalemacroecooc forecstg Desite te aet navete, nostcstc sreasheet aroaces melng e more only sed nvennl enmec mels, VA mels or rao exttomels. nc comute nd e med, a mels comlexi mst o

te oe h, oe eve icmechaniss of rato eto e t d

hat we kow eou at te rver tem om the dat, e it myb wo accet te burden of unbelevable sl ssuo wc re nevible n a sable raon exo mel Oneacceps at he resulg mel w t a forectng mel ecause of ef tt vg raonalexctatos echsms exlcity te mel maymake it more acurate i some ks of ly nlyssc e mel is worse s a forecstg mel, it s

obvous tat decrese accuray some of licy

lysis is beg raded o agans a ope for ncresed

accac some oeOn e other hnd, one ght believe hat we know tltle aut the natre of dnmc economc havior forcare melng of rato exctatons dyamcs o be very helpl In ha cse, usng a mel whch ms a ndg condonl dstribion of economic ocomes ven lc acion, without diling ll e dnc optaton ndelyng at condonal sbonmay lk more atactve

VAR meths have oe dee conceptual advanage over cuen plemettons of ratonl excionseilibrim mels: ey mke the conecion of hemel t a eced fo forecstng me ompeeelici Ts mens that one c vldae he pobabii

ssmpons in e reduced fo by vng it a dry n ough he hsorcl dt, hvg it generate forecss wth daa avaable at each date the pas The mes own pobabi disrion for forect eo cn bcompredt e obseed sple sribon of forec eos In cos, sdd econometc mels d,even moe o rational exttos eulbrm melscno do h A major of te uce au forecs from pes of mels is ce a

he agie deniing ssuptions built nto emTog e mels e probablstc e dsrbions he generae for her ow forecst eo are of med pracical e nce they iore er risg om nees he o sccato. Ecnomss gen

erally end o ford cy coclusions nd fore ey ere surer of tem e og t bb o obetve evidence. T tendenc isrenfoc b he se of arenty scientc mahe macal me hat iore re surce of nceia e coclsions But VARmes, or y ohe

a c gp een mel stic spccaiod e oe forecst er stibuon hold e pro of ndng sussion muc more rmly on obcive fac.

ctg e ee of cton identg o wil r me se ofVAR melshee uons wb of u vl. The of Vb icy evon s to splay exciy e m of ide ssutos o a celld uc fo fot meL mes i t pate y of uce aut ident a ys of ce aut he pbabi o e dat

t ems tat tere e wo ve ieren paches macoc cy lsishe a heavily emc stle, wc tes alysis



cosely o forecasng models; he second s he rational

econs sle, whch s more complcaed, lesscosely ed o d more respecble n he eyes of most mcroeconomiss, nd less sed From wh hs beensad so f, prhaps the reader w aee hat he sse is

whehe, a pticu apcaon, one apach o heohe res lusible ideniing assumtionsThere s no generl rgmen ht one proach voidsdcles whch he oher pprach does no

Identiing VAR ModelsThs secon of he ape descbes a class of nveient idenng rescons on VAR mels Beneng mit reques some previous esure economec (Less cicay incned edes cn st pe

has gain some insigt om te examle esend n thenext secion wihout eadng his one)The splest wy o acheve an identng nteet

on of VA mdel for licy choice is ssume ht cy viation can be idened wih he resdl one(or more) of e mels euaions. I tat cse, lcypojecos cn b made n the stad way-by pg he cy equaons m he moel nd eaing heohers exc Ths wy of achevg dencon s, s we ntd ou exacy wha ndeles he usl e of ecnometc mels to oet e ees of cy,

whehe hey re excilyVAR mels o not is alsoeacly he idenng son ndelyg raionaexctos monetars mels, n whch upredcable

chngesi

money stk e laed be generte eniely by eac lcy choices5 It is, howeve, arescive aoach t iteetg a mel.

Recenly a numb of enomis have exlod ways o epret VAmels whc presee a g de of he splc and conveence of deng lcy whpdcion eo i licy vables, wt beng quiso esctve6 e se a foows We supse hat hee e havoy dt souces o vaton lleced n vec et) at hese soues dive vron n he economy Some elemen of et) arernm ucaton icy A mel connecs heobseable ve of da Y t), cuent d pt vauesof he dvng disrbaces acdig

(8) - As)Yt-s) = - Bs e-s). •

The VAR mel fo Y wl have he fo

(9 ) Yt) C(sY(t-s + u(

whee u is onspa prdicion rror

Euation 9 ca b soved o yed he mpse esose

mes o G(, wich satis( 10) Y( = lm G(u( + Hm(mr- whee H(m) depens on m b he G seqence oes oThe ow,colum ofG(),G() es he esseofY(+) to a it disrbace u,(. Agpaof epl ofVA modes as descpive devces s a he u veco, or les some elemens of i is oen de plusble deng sspo ey e sme ssome elemens of he e vecor. In cse pf he Gmatix can be ieeed direcly s esoses o ocyacon.

However, he genea siaon is hat e avio

disrbces of equon (8) are no he sme s he 's. f e Ys e a ch enogh lst of dt so we co f weew e A nd B maes, cover e( ech fnowedge of cuent d ast vaues of Y t ad eprocess s serly dependen, hen 8 pes efollowg:7

(ll ( O)() BOetThen u) = (0) BO)e and we can sbse ion (1 0) ob reses of Y o e y epcg G() wi G(() BO), for

e C and G coecient seqences i eqons (9)and ( 0) ca aways b recoveed om he da by

rlatvely sple estmaon prcues. ad B nequaon (8) cnot b revered om e aa wiodeng smpo t esc her o f hedentng asumpon resrct oy ( 0),B( 0

va ( e(t)) hen they may mply resrcons on

va(()) but tey w not ly any esctions on or G Sce ecient esimaion of s neced byscon on we cn n hs cse proceed sepwse estma C by he usul smple VAR mehod dfo n u estmte of om he eseVR esiduas second, use he resricons o ercesats of AO), BO) ad n om he uesrced

SFor expl of sch aon on mons m. 1977 o t d Wlc 976

o xape, Le ( 84) Ba on !8 ) Bc 985 omng an Mon 8)

d H 8 v ed ou e umio }' ach o&h st o a o owee eng mo ev u d th oe w oe oabe t c i e i onc h e h a whch o dy n ns o cyo It e hatO qee ht e nme s n e e b bi i



matx The esponses of the system to policy disu bances hen ae available om he fomulaG()A(0)1 B(O), n which we ca use he G om heogin unesrcted V R estimate togeer with he(O) and B(O) eates om the econd step of heestaion

The most sghtfoard example of identig sricions onA(O), B(O), and n is the Wold causl chain.

ccodng to his idea ! should be diagona, B(O) and A (0) should e iangular and noaz to haveones dow the main diagonal when variables ae orderedaccodig t causal prioity Using the fact that with B( 0)= =A()A(), the triaulriy of A() plie

hat, once e have put the variables the prope order, we c recover A(O) d n om : a :'s unique LDL decomsitio. That is, it is nown that tere is a unque ay to express a pitve dente ma n e fom' here L i lower iangur wih ones dow i diagonl, ndD is diagonal. pplying a standd LDLalgort : ves us A(O) L d a D This igul ohogonaizaon ha come a sdrd practice as pa of he itereaton of economeic V Rmels.

ne c also apply he Wold cha idea only to oblck of variale i te sytem, hch ruiresA(O) t lk riulr d n t b cooly block diagon This blk Wold chai ework uderes a e stan economec eo of simultaus equ ios mels, hat it is equivalent t dividing Y t tocomnen, one endogenou and the oter preter

ed. If A(O) i loer block rilar, then the upper p of the Y vector meet the usu cteria for pretered vaiales, nely tat ter current d pat vauesare ucoelated with curent vaues of the disturance i he equations cesndg t te loer lk. e lck Wold chi sture e not y iel aceve identca iontere rem more ee ccen (0) d n i e ma In e d siultu equatio, additon ident re

ions are imps on eA suence [The sence is en as B(O) [ B() = 0, for noe] teadditiona estiction volve A() for iive then

hey will generaly imply ro on C d ivdate he simple twostage identcaton eork ut te

stadard ework ds aly te ce here etri ions e imps oy onA (0). e uu rk d order ondtions for idenication apply ad the usu aay of simultaeus equatios estiat c api s equation( 1 1 ith) = were the hole mel ad theestiat of : obted om the urestrctd V mel

wee the moment ma of te dat The std

aymptoic distibution heo applies ithout modica ion to estimatos consructed om ( 1 1 ) and he est mated ma (n example of empical eseach exploitg s convenent amework appes Mon985

Thee i� no need, hough, to emain tied to theconvention that estictio on n are appied only nconjunction with block iaulaity esictons onA(O).The eason it came conventiona to pose such esicons is at econometcias do accept idea atoen it il be easonale t assume that behavioaly disct sources f schatc vaiation should be ndeenden T idea c uel even hen not coupled toa block iangularity supton onA(O). We might, for exple, assume n to e dagon d acieve idenica ion y impsng addtiona restictions onA(O), ereysisg at al ehavioray ds souces of d

urbance hould e uncelated.

m f g a V e exame a sple svaiable querly psta VR mel of he US. economy over the period1948:119793 The r tcated at 19793 avoid te lkely need llo for a in money supplyhavio aound 19794. The variales e ea GNP

( Y, ea buses ed invesent /} GNP pce dear(P), e M meure of money(), uemployment ( U), ad Treubill ate (R ). list of variable is choen t allo a interestig discussion whie stil bi magealy sho The list is sho

howeve, for u to b sure t it ill alow us to· disguh among havioray disi souces of dsace. paticular, sce it des not clude anymeta et clely cnrled by licy authoi

ties, it force u t l er e money suply eectsarii m the ay te ytm uses reees andeec aig diry m actos y e Federasee Systm o, sice te ist icludes no sca vaals, chges i exctio aut tue adspending icieich culd it can show

up i the mel oy rly.

A Model With Weak Restrictions

When a V A mel is t tee dat, usg four lagso each variale plu a cnstt t, d it a weak ayei pror it idendent rdom the

8e ren wrk t nvenenty is h h ikeh fr he da blc dg m. Th h Dub ( l 970) ne gt hat twse esimin, eng -sepper em xy yl h m ymc r l jnmmm io



prior ean, the resulting esiated syste has the

pulse responses plotted in Cha 19Each of the sall

aphs represents the resnse, over 32 subsequentquters of a vaiable n a gven row to a onestandaddeviation innovation (predicion eror) a viable n aven oum. In the notaton of the previous section theA matr has been taken to be lower trangula, wih thevariables ordered s shown in the chart The disturbancescn be ntereted as behavorall distic one belevesthat a behavor veson of the syste woud have thefo of a Wod causa chain The vetc scae s thesame for a te pot i a gven row, so e reaive size ofeec of vaous innovaions on a gven vaiabe cn becoped by ig at e reave szes of pottedresponses acros at aabe's ow

Mny of he resnse pae n h 1 do apa hae staghod behaor tereons The uemploymen shk (column 5 for exaple, lke alabor supply disbance. Unemploymen ses tema-

Ch

ly reu to no whn wo ye or so nd output

rises steadly over aut wo ye hen reins at agher level vesent lso ollows he pate of outputterest rates decline slighy and teraily Moneyepd shly wihout a corespondngly lage epson e pce level ndicaing accoodation ofreal bces to a hgher level of re actviy t isremakable that the uneployent dsurbnce eectemeges as the lagest single nuence on the ong-unbehavor of output

Toug seerl oher of the esponse pates endeeves e telng of ntereve "sries e theone we just deveoped for unepoyent et us focusnow on a pbe: Do we see oney suppy nd denddsurbces n hs ch? The raona epectations

oneast teretaton

hat one inovatons ae

9T pror u ad mplemntatn sipcatin f ue b D Liten, d S 8

Respnse Functions of Variables n a Weakly Resticted VAR Mel*

e OupRea GN

vetmts xPcP Oeat

Mo

Umpymett

Inre ReT·B)

Invato o

i l (2) 5 yl ntet Peak

O nvesmet Pri Mey Rt at . e

� 0096

l 08

� . 00

.� 0073

,-.3t

L 4 Ye Ae O

"Eh l gph 1 e of t vbl lbl t ow to o-ddto b< ot ty m t e top o t cu ah S potte ov 8y (32qt� p th o 0 gv ow l t h l l a a ve wed to the dtac t e ta getg ovet t ow' vbl ll vbl n log\hms xcp o e uploym ae ee . w1 are ke a pere vetl cal a el o a at eva1o o t zot x S t e dt o tgph m h ow d t ze of pea po note t the g of eah owCa t the epoe

\o r\y rOnz nvn



money supply disubances. The money coln (4) of

Cha l sows that ices esd stroy desistently although with a delay, money novaions, wlere vies esnd ve weky e weess of tereal esnses oes not t rtionl excttions monetaist theo we Also thee is a negible ntest ateesnse Since thee is no mmeiate esse thelevel of ices o output it is icult to eln how, with xed money demand schedule he ubic is led toccept jump n te money stk

Many economists would the, in y cse, tetsho-tem nteest tes as the monety licy vabeThe inteest ate coln ( 6) of Ch l sows substtiltem eatve eects o outut of a se teestates It also shows a lge peitent negave eect of

te nteest te se on te money stok Howeve, teei tte istent eect of te monet conton onpices, and indeed an tial sitive eect o ices.Fheoe, with a xed moey demd scedule it had to see why the quickly dssiatd ie nomanteest tes les to a ve eistent dene n temoney stk

Two Dierent Identifications of the Model

Cs 2 nd 3 eesent two dent identctons of VAR system In each case, idencaton is lted toseat demd d suly of moey om eac otedom a blockofemg equatos I eac ce thebevioa distubces e e tken o be muyuncoelted

oThe Fist Idetfiti

Te mel geneatng C 2 slates that e moneysuply equation connec money iovaon wh teest ate novions and that o othe viables ete ests on he idea that he monet auoi dhebanks c see nteest ates d dicato of movemein monetay aegates mmeatel but c o ea tte eminng vales n te enomy a ayecuse dt on tese vbes e eleed lae. Temoney demand equaon in mel aows moenovatios to deed o ovatio i itt atsouut d te ic level Iveset demd is lated o make invesent ovios uivet tonvesment demnd ovios-hat ientdemnd eacts ote vale ony wh a dela Theree ohe equtions e st o detee outpce and unemloyment ovaios m moedeedet ovatos d te iovaio i ivestmet and inteest ates Tese uaos enoaled o ave a blk gul s of ceon pce, ouut, d unemoyment Note ee

mt idet estctio: te moe stock ova

to

ot llowed to te equatios othe t moesuly d demand; moneta seto vaiales feed acknto e emnde of e model ony vi nteest tes

Te equtions fo system d thei estatcoecient e st below:

Money Suply

( 12) r= 712m + e(460)

Moey Demd

( 13) = 283 + 22p - 8 1 + e

(103 (128 (3)

Ouut

(1) y= 135 + . 132i + e(308 _(02

Pice

( 15 ) = -lO + 045 + 364i + es(12 ( 05 (015)

Uemlomet

( 6) u = - 16 2 - 48i 898p + e(05) (25) (72) ( )

Iveset Demd

(7 i=e •

Te data o l viables e loed excet fo U dwhch meued cent A stddeo e ven n enes low e cien 10

Te moe demd d sul uaios hassurii reble ve tat te e idetied oyb exclusio esictio o iovatios e esse toa moey supl sck sow i quite ieet

m eihe te moey sk o teest shock C , beg cle o a deen of ose o esnse

OTe stdd e b the axime derivative mxcumua durin the seh f the likeommum BFGSuatet al ( Gl Muay d Wt 19 8 . 9) pximae Hei n nt ve acre, even wen t ch algrithmh wor wel mm



Ch Respnse Futions of VA Mel Vaiabes for Two Differet Idetificatios*

Ch Ree te Fit etiiti

Rsps (

MnySupp

(2)Mony

Oa

Ivaio o3] (4)

Ou s 5( (6) Rte m

akRspup �- 0086Real GNP ,. me - _ 03(Ral 8USS F Pice

(GNP Oea

Moy)_ 0

0

95.00

UR -,-Lr 27

es ReT·BI .36� � � w0 4 8 0 4 8 0 4 8 0 4 8 0 4 8 0 4 8Y A

Ch e te eiti

2)3 ()

5 6)oSu

D PkR R le ese

O _.- 008! -a_.� 032 F _ - oiMoy (1)

UtR

�._ 0097

-. _

, .L

0086

27es Re· _- 4� L� w.

0 4 8 0 4 8 0 4 8 0� 8 0 4 8�e A

·e e fr Ch 1 . The deg es1s ed ee re desed e ber eses res \� \ £ t iden'l assumpos scrb 10 h x.



paes ow a mony supply shock (C ha olumn )

cosists of smultaneous jumps upward n interest ratesnd donrd in money stock he money stock moement ncrees o�r he next fe urers nd reminse peristent The interest rate movement remanstemporay but now th delayed yet persstent movementn prices s real blances so that the persistence ne nomnl money stock despite the eanescence of thenterest rate rse, is not puzzlng Output nestment ndunemployment ll tke plausible busness-cyclelke excursions n respone to the money supply shock

The estated resonse to a money demand disturbance (Cha 2, column 2) is harder to rationzeThat a money demand shock should rise interest ratesnd the money stock nitlly nd result n a persistentdeclne in ouput is not unresonable. But that it shouldhave no mmedate eect on pces and n the long nshould rase pces is nt so reasonable. There ae severalssible explnations n tes of eesses n idenng sumptions. One is he prblem that by pung Ml nthe money supply euaon, we forego separating bnkbehaior m Federl Resee behavor d therebyssibly also forego poperly sepaatg demd omsupply behavior Another explaton is hat withoutscl vriables a change in he cuent debt or nexcted ue rel fer suluses might generatewhat is idened here a a money demnd shock. A rsen the debt toug a cuent decit, unatched by a sen expected tre rel suluses would geerate thefollowing eecs: an crease desed money holdn(through an ncreae n the scae of the public's nomnanesment folio), a rise in the nterest rate, ndupwrd presure on pces Finlly, he mel es notcontan exchange rate or mmd pe vable Afal he vaue of he dol o a se coy could create ncreed tansactions demd fo moneyicipat GNP-dear ation, a rise in iterest raesd a declne in output

S Befo vg u o d oy dm ovo or ee more e m e ofidening resction we o moneynoaton wa alowed t e equaon

dectly.T

specicato m ao t nestated money demnd s no mo reable butit mde the pce uaton de tv lea money demd sh Ltd lo ovioequation for ths interetton of the system renozed so that the pice equaton is noized on money,nd vice ve

Money Supply

(8) r= 820m+ ( 30)

Money Demand

( 19) m = 29y - 0088i + 90p - .0082r + e2( 08) (01 64) (.22) (.0022)

Output

(20) = -21r + 3i + 3(21) (02)

ce

(21) p = -58 + 22y - 50m + (25) (06 (1 1)

Unemployment

(22) = -2 20ly 149i 87p + 6(.04) ( 1 1 ( 36 (1 . 17

Invesent emnd

(2 i=e•

Ch 3 itelf no provide a more beievable resnse

o the money demnd shck column 2) Interest ratesrse money s ies, nd pice fl at t Ouut fllsnd stys low he money stc remn hgh nd pcesevenaly se t ela te er re blances isnot ey to undd y ouut sould rmn depress for so long e iteret ras d re bncesreu n but he sne is sm enough that itmay not siicly sic Note hat he eec ofthe money demd sh on othe vables e llrelatvely e but tt te moey demd sckp o moy of t orde of mad th t at of the moneysupply sh m ha hou money supplyshk n mel a e iuence on th

pces d re bnes ycle havior, the ertuer moveme te money stk cnt a lgecomnent of moy demd he mel suesthat stble ony suly avo wod help stbilzehe enomy but it o sues hat stble moneysupply avior i quite deret m stble money sckn a quey e e





References ,

B, J. 19 Unancipd money wh d nemploymen eUn Ses. American Economic Review 6 (March): 101-15.

Beae Ben. Fohmn Alatve exploraons ofhe mneie crlon CeRheer Confeen Ses on bic Pocymd: NohHollad.

Blcha, Olvier. 1985. empicl sl ivego of he havior ofwaes, p, d o. Pe nd nmple ion prNovemr

Bchard Oivier d Wan, Mk 1984. all siess cycles ai?n pper pa he Natonl B of Economc RechConfen Buis Cycl, Mch

hom; tte, Ro; ad Sms Ctophe 1984 Fog dndionl pojion ig eaic p io. Economtc

Rviws 3 ( l: 1-1.

D J 1970. Testg f serial coel i etsqus sion hme of he e e l dependent vribles. Econometrica 38(May) 4021 .

Gil Philip E. Muay Walr; ad Wgh M H. 981. Pccalmetho of optimatio New Y Aic Ps.

Hn, Pr d Snt om J. 1980. Foulaing d dynmic e rn eo e Joual Economi Dnamic and Contl 2 (Feb) 46.

Hu, 962. the crl f tnt Logic and methodolo in th l cin . 23239 SfodSaford Univity

Kpms llg, aBach AusF 1959. Sletpiieinvolvng mhec onig. SIM Rviw 1 3848.

Lemer, Edwrd E. 19 85 . Vr aeeio for cs fer Unddi mon ims, Kl Ber d AllaH.Meer.Ceeh Cferen Se Pbic Pol 22 (S255304 A: NohHola

Lie Ro B. 1982 Opa conr

e money suly.Fdl Rs

Bank ofMinneapo Qaerly Review 6 Fl): 1-9 .

. 984. ectof ine g Depentg Par 254. Fer Rse B of Miel

Mon Cn 1985. e mp of vet c on U.S sl adiidu. PhD dse U oM

Sarent m J. 1979. Estag vt tss usg mn

b on expcih.edel Ree n ofMinnpolQua Review 3 (Su) 815

-=· 192 e g ! dqet R E. H 4-97. Nao of Eceh Proj R. Chic U Chc

-;

-. 194. Aute EconomiRviw 4 (May) IS

Sen om J, d Hn, P. 194. To ivectr auteo. e e Wo P 227 Fee Bk of Minne

Sen m J. and Wl, Ne 1 976 ed heh of enoc cy. o Mo oms 2 (April16983

Sms Chriopher A. 1 985. A e k fo poicyalyss. Mit Cene foEc Dss P No.223. Ao, fohmi& i e T Siu Econo hed at A ex My

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