CHAPTER 5 Vecor Auoregression and Vecor Error-Correcion Models Vecor auoregression (VAR) was inroduced b Sims (198) as a echnique ha could be used b macroeconomiss o characerize he join dnamic behavior of a collecion of variables wihou requiring srong resricions of he kind needed o idenif underling srucural parameers. I has become a prevalen mehod of ime-series modeling. Alhough esimaing he equaions of a VAR does no require srong idenificaion assumpions, some of he mos useful applicaions of he esimaes, such as calculaing impulseresponse funcions (IRFs) or variance decomposiions do require idenifing resricions. A pical resricion akes he form of an assumpion abou he dnamic relaionship beween a pair of variables, for eample, ha affecs onl wih a lag, or ha does no affec in he long run. A VAR ssem conains a se of m variables, each of which is epressed as a linear funcion of p lags of iself and of all of he oher m 1 variables, plus an error erm. (I is possible o include eogenous variables such as seasonal dummies or ime rends in a VAR, bu we shall focus on he simple case.) Wih wo variables, and, an order-p VAR would be he wo equaions =β +β + +β +β + +β + v 1 1 p p 1 1 p p =β +β + +β +β + +β + v 1 1 p p 1 1 p p. (5.1) We adop he subscrip convenion ha β p represens he coefficien of in he equaion for a lag p. If we were o add anoher variable z o he ssem, here would be a hird equaion for z and erms involving p lagged values of z, for eample, β zp, would be added o he righhand side of each of he hree equaions. A ke feaure of equaions (5.1) is ha no curren variables appear on he righ-hand side of an of he equaions. This makes i plausible, hough no alwas cerain, ha he regressors of (5.1) are weakl eogenous and ha, if all of he variables are saionar and ergodic,
7 Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models OLS can produce asmpoicall desirable esimaors. Variables ha are known o be eogenous a common eample is seasonal dumm variables ma be added o he righ-hand side of he VAR equaions wihou difficul, and obviousl wihou including addiional equaions o model hem. Our eamples will no include such eogenous variables. The error erms in (5.1) represen he pars of and ha are no relaed o pas values of he wo variables: he unpredicable innovaion in each variable. These innovaions will, in general, be correlaed wih one anoher because here will usuall be some endenc for movemens in and o be correlaed, perhaps because of a conemporaneous causal relaionship (or because of he common influence of oher variables). A ke disincion in undersanding and appling VARs is beween he innovaion erms v in he VAR and underling eogenous, orhogonal shocks o each variable, which we shall call ε. The innovaion in is he par of ha canno be prediced b pas values of and. Some of his unpredicable variaion in ha we measure b v is surel due o ε, an eogenous shock o ha is has no relaionship o wha is happening wih or an oher variable ha migh be included in he ssem. However, if has a conemporaneous effec on, hen some par of v will be due o he indirec effec of he curren shock o, ε, which eners he equaion in (5.1) hrough he error erm because curren is no allowed o be on he righ-hand side. We will sud in he ne secion how, b making idenifing assumpions, we can idenif he eogenous shocks ε from our esimaes of he VAR coefficiens and residuals. Correlaion beween he error erms of wo equaions, such as ha presen in (5.1), usuall means ha we can gain efficienc b using he seemingl unrelaed regressions (SUR) ssem esimaor raher han esimaing he equaions individuall b OLS. However, he VAR ssem conforms o he one ecepion o ha rule: he regressors of all of he equaions are idenical, meaning ha SUR and OLS lead o idenical esimaors. The onl siuaion in which we gain b esimaing he VAR as a ssem of seemingl unrelaed regressions is when we impose resricions on he coefficiens of he VAR, a case ha we shall ignore here. When he variables of a VAR are coinegraed, we use a vecor error-correcion (VEC) model. A VEC for wo variables migh look like =β +β + +β +γ + +γ 1 1 p p 1 1 p p λ α α + v 1 1 1 =β +β + +β +γ + +γ 1 1 p p 1 1 p p λ α α + 1 1 1 v, (5.2) where =α +α 1 is he long-run coinegraing relaionship beween he wo variables and λ and λ are he error-correcion parameers ha measure how and reac o deviaions from long-run equilibrium.
Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models 71 When we appl he VEC model o more han wo variables, we mus consider he possibili ha more han one coinegraing relaionship eiss among he variables. For eample, if,, and z all end o be equal in he long run, hen = and = z (or, equivalenl, = z ) would be wo coinegraing relaionships. To deal wih his siuaion we need o generalize he procedure for esing for coinegraing relaionships o allow more han one coinegraing equaion, and we need a model ha allows muliple error-correcion erms in each equaion. 5.1 Forecasing and Granger Causali in a VAR In order o idenif srucural shocks and heir dnamic effecs we mus make addiional idenificaion assumpions. However, a simple VAR ssem such as (5.1) can be used for wo imporan economeric asks wihou making an addiional assumpions. We can use (5.1) as a convenien mehod o generae forecass for and, and we can aemp o infer informaion abou he direcion or direcions of causali beween and using he echnique of Granger causali analsis. 5.1.1 Forecasing wih a VAR The srucure of equaions (5.1) is designed o model how he values of he variables in period are relaed o pas values. This makes he VAR a naural for he ask of forecasing he fuure pahs of and condiional on heir pas hisories. Suppose ha we have a sample of observaions on and ha ends in period T, and ha we wish o forecas heir values in T + 1, T + 2, ec. To keep he algebra simple, suppose ha p = 1, so here is onl one lagged value on he righ-hand side. For period T + 1, our VAR is =β +β +β + v T + 1 1 T 1 T T + 1 =β +β +β + v. T + 1 1 T 1 T T + 1 (5.3) Taking he epecaion condiional on he relevan informaion from he sample ( T and T ) gives ( T + 1 T, T ) =β +β 1 T +β 1 T + ( T + 1 T, T ) ( T + 1 T T ) =β +β 1 T +β 1 T + ( T + 1 T T ) E E v E, E v,. (5.4) The condiional epecaion of he VAR error erms on he righ-hand side mus be zero in order for OLS o esimae he coefficiens consisenl. Wheher or no his assumpion is valid will depend on he serial correlaion properies of he v erms we have seen ha seriall correlaed errors and lagged dependen variables of he kind presen in he VAR can be a oic combinaion.
72 Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models j Thus, we wan o make sure ha E( v v 1 v 1), =. As we saw in an earlier chaper, adding lagged values of and can ofen eliminae serial correlaion of he error, and his mehod is now more common han using GLS procedures o correc for possible auocorrelaion. We assume ha our VAR ssem has sufficien lag lengh ha he error erm is no seriall correlaed, so ha he condiional epecaion of he error erm for all periods afer T is zero. This means ha he final erm on he righ-hand side of each equaion in (5.4) is zero, so ( +, ) E =β +β +β T 1 T T 1 T 1 T E, =β +β +β. T + 1 T T 1 T 1 T (5.5) If we knew he β coefficiens, we could use (5.5) o calculae a forecas for period T + 1. Naurall, we use our esimaed VAR coefficiens in place of he rue values o calculae our predicions (, ) P ˆ =β ˆ +β ˆ +βˆ T + 1 T T T + 1 T 1 T 1 T P, ˆ =β ˆ +β ˆ +βˆ. T + 1 T T T + 1 T 1 T 1 T (5.6) The forecas error in he predicions in (5.6) will come from wo sources: he unpredicable period T + 1 error erm and he errors we make in esimaing he β coefficiens. Formall, ( ˆ ) ( ˆ ) ( ˆ ) ( ˆ ) ( ˆ ) ( ˆ ) ˆ = β β + β β + β β + v T + 1 T + 1 T 1 1 T 1 1 T T + 1 ˆ = β β + β β + β β + v. T + 1 T + 1 T 1 1 T 1 1 T T + 1 If our esimaes of he β coefficiens are consisen and here is no serial correlaion in v, hen he epecaion of he forecas error is asmpoicall zero. The variance of he forecas error is 2 2 ( ˆ ) ( ˆ ) ( ˆ ) ( ˆ T + 1 T + 1 T = β + β 1 T + β1 ) T + 2cov ( βˆ, β ˆ 1) + 2cov ( βˆ, β ˆ 1) + 2cov ( βˆ ˆ 1, β 1) + var ( vt + 1 ) 2 2 ( ˆ 1 1 ) ( ˆ ) ( ˆ 1 ) ( ˆ T + T + T = β + β T + β1 ) T + 2cov ( βˆ ˆ, 1) 2cov ( ˆ ˆ, 1) 2cov ˆ β T + β β T + ( β1, βˆ 1) + var ( v T + 1 ). var var var var var var var var T T T T T T As our consisen esimaes of he β coefficiens converge o he rue values (as T ges large), all of he erms in his epression converge o zero ecep he las one. Thus, in calculaing
Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models 73 he variance of he forecas error, he error in esimaing he coefficiens is ofen negleced, giving var ( ˆ + + ) var ( v + ) ( ˆ + + ) ( v + ) σ 2 T 1 T 1 T T 1 v, var var σ. 2 T 1 T 1 T T 1 v, (5.7) One of he mos useful aribues of he VAR is ha i can be used recursivel o eend forecass ino he fuure. For period T + 2, so b recursive epecaions (, ) E =β +β +β T + 2 T + 1 T + 1 1 T + 1 1 T + 1 E, =β +β +β, T + 2 T + 1 T + 1 1 T + 1 1 T + 1 (, ) =β +β (, ) +β (, ) E E E T + 2 T T 1 T + 1 T T 1 T + 1 T T ( ) ( ) =β +β β +β +β +β β +β +β 1 1 T 1 T 1 1 T 1 T (, ) =β +β (, ) +β (, ) E E E T + 2 T T 1 T + 1 T T 1 T + 1 T T ( ) ( ) =β +β β +β +β +β β +β +β 1 1 T 1 T 1 1 T 1 T The corresponding forecass are again obained b subsiuing coefficien esimaes o ge. (, ) P ˆ =β ˆ +β ˆ ˆ +βˆ ˆ T + 2 T T T + 2 T 1 T + 1 T 1 T + 1 T P, ˆ =β ˆ +β ˆ ˆ +βˆ ˆ. T + 2 T T T + 2 T 1 T + 1 T 1 T + 1 T (5.8) If we once again ignore error in esimaing he coefficiens, hen he wo-period-ahead forecas error in (5.8) is ˆ β +β + v T + 2 T + 2 T 1 T + 1 T + 1 T 1 T + 1 T + 1 T T + 2 β v +β v + v 1 T + 1 1 T + 1 T + 2 ˆ β +β + v T + 2 T + 2 T 1 T + 1 T + 1 T 1 T + 1 T + 1 T T + 2 β v +β v + v 1 T + 1 1 T + 1 T + 2 In general, he error erms for period T + 1 will be correlaed across equaions, so he variance of he wo-period-ahead forecas is approimael,. ( ˆ + + ) var β σ +β σ + 2β β σ +σ 2 2 2 2 2 T 2 T 2 T 1 v, 1 v, 1 1 v, v, ( ˆ + + ) = 1+β σ +β σ + 2 β β σ, 2 2 2 2 1 v, 1 v, 1 1 v, var β σ +β σ + 2β β σ +σ 2 2 2 2 2 T 2 T 2 T 1 v, 1 v, 1 1 v, v, =β σ + 1+β σ + 2 β β σ. 2 2 2 2 1 v, 1 v, 1 1 v, (5.9)
74 Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models The wo-period-error forecas error has larger variance han he one-period-ahead error because he errors ha we make in forecasing period T + 1 propagae ino errors in he forecas for T + 2. As our forecas horizon increases, he variance ges larger and larger, reflecing our inabili o forecas a grea disance ino he fuure even if (as we have opimisicall assumed here) we have accurae esimaes of he coefficiens. The calculaions in equaion (5.9) become increasingl comple as one considers longer forecas horizons. Including more han wo variables in he VAR or more han one lag on he righ-hand side also increases he number of erms in boh (5.8) and (5.9) rapidl. We are forunae ha modern saisical sofware, including Saa, has auomaed hese asks for us. We now discuss he basics of esimaing a VAR in Saa. 5.1.2 Esimaing and forecasing a simple VAR in Saa A one level, esimaing a VAR is a simple ask because i is esimaed wih OLS, he Saa regression command will handle he esimaion. However, for everhing we do wih a VAR beond esimaion, we need o consider he ssem as a whole, so Saa provides a famil of procedures ha are ailored o he VAR applicaion. The wo essenial VAR commands are var and varbasic. The laer is eas o use (poeniall as eas as lising he variables ou wan in he ssem), bu lacks he fleibili of he former o deal wih asmmeric lag paerns across equaions, addiional eogenous variables ha have no equaions of heir own, and coefficien consrains across equaions. We discuss var firs; laer in he chaper we will go back and show he use of he simpler varbasic command. (As alwas, onl simple eamples of Saa commands are shown here. The curren Saa manual available hrough he Saa Help menu conains full documenaion of all opions and variaions, along wih addiional eamples.) To run a simple VAR for variables and wih wo lags or each variable in each equaion and no consrains or eogenous variables, we can simpl pe var, lags(1/2) Noice ha we need 1/2 raher han jus 2 in he lag specificaion because we wan lags 1 hrough 2, no jus he second lag. The oupu from his command will give he β coefficiens from OLS esimaion of he wo regressions, plus some ssem and individual-equaion goodness-of-fi saisics. Once we have esimaed a VAR model, here are a varie of ess ha can be used o help us deermine wheher we have a good model. In erms of model validaion, one imporan proper for our esimaes o have desirable asmpoic properies is ha he model j mus be sable in he sense ha he esimaed coefficiens impl ha + s/ v and j + s/ v (j =, ) become small as s ges large. If hese condiions do no hold, hen he VAR implies ha and are no joinl ergodic: he effecs of shocks do no die ou.
Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models 75 The Saa command varsable (which usuall needs no argumens or opions) calculaes he eigenvalues of a companion mari for he ssem. If all of he calculaed eigenvalues (which can be comple) are less han one (in modulus, if he have imaginar pars), hen he model is sable. This condiion is he vecor eension o he saionari condiion ha he roos of an auoregressive polnomial of a single variable lie ouside he uni circle. If he varsable command repors an eigenvalue wih modulus greaer han one, hen he VAR is unsable and forecass will eplode. This can arise when he variables in he model are non-saionar or when he model is misspecified. Differencing (and perhaps, afer checking for coinegraion, using a VEC) ma ield a sable ssem. If he VAR is sable, hen he main issue in specificaion is lag lengh. We discussed lag lengh issues above in he cone of single-variable disribued lag models. The issues and mehods in a VAR are similar, bu appl simulaneousl o all of he equaions of he model and all of he variables, since we convenionall choose a common lengh for all lags. Forecasing wih a VAR assumes ha here is no serial correlaion in he error erm. The Saa command varlmar implemens a VAR version of he Lagrange muliplier es for se- j j cov v, v = wih rial correlaion in he residual. This command ess he null hpoheses ( s) j indeing he variables of he model. The main opion in he varlmar command allows ou o specif he highes order of auocorrelaion (he defaul is 2) ha ou wan o es in he residual. For eample varlmar, mlag (4) would perform he above es individuall for s = 1, s = 2, s = 3, and s = 4. If he Lagrange muliplier es rejecs he null hpohesis of no serial correlaion, hen ou ma wan o include addiional lags in he equaions and perform he es again. The Akaike Informaion Crierion (AIC) and Schwarz-Baesian Informaion Crierion (SBIC) are ofen used o choose he opimal lag lengh in single-variable disribued-lag models. These and oher crieria have been eended o he VAR case and are repored b he varsoc command. Tping varsoc, malag(4) ells Saa o esimae VARs for lag lengh, (jus consans), 1, 2, 3, and 4, and compue he log-likelihood funcion and various informaion crieria for each choice. The oupu of he varsoc command includes likelihood-raio es saisics for he null hpohesis ha he ne lag is zero. The opimal lag lengh b each crierion is indicaed b an aserisk in he able of resuls. In general, he various crieria will no agree, so ou will need o eercise some degree of judgmen in choosing among he recommendaions. Anoher wa of deciding on lag lengh is o use sandard (Wald) es saisics o es wheher all of he coefficiens a each lag are zero. Saa auomaes his in he varwle command, which requires no opions. Once ou have seled on a VAR model ha includes an appropriae number of lags, is sable, and has seriall uncorrelaed errors, ou can proceed o use he model o generae forecass. There are wo commands for creaing and graphing forecass. The fcas compue command calculaes he predicions of he VAR and sores hem in a se of new vari-
76 Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models ables. If ou wan our forecass o sar in he period immediael following he las period of he esimaing sample, hen he onl opion ou need in he fcas compue command is sep(#), wih which ou specif he forecas horizon (how man periods ahead ou wan o forecas). The forecas variables are sored in variables ha aach a prefi ha ou specif o he names of he VAR variables being forecased. For eample, o forecas our VAR model for 1 periods beginning afer he esimaing sample and sore prediced values of in pred_ and in pred_, ou could pe fcas compue pred_, sep(1) The fcas compue command also generaes sandard errors of he forecass and uses hem o calculae upper and lower confidence bounds. Afer compuing he forecass, ou can graph hem along wih he confidence b ping fcas graph pred_*. If ou have acual observed values for he variables for he forecas period, he can be added o he graph wih he observed opion (separaed from he command b a comma, as alwas wih Saa opions). 5.1.3 Granger causali One of he firs, and undeniable, maims ha ever economerician or saisician is augh is ha correlaion does no impl causali. Correlaion or covariance is a smmeric, bivariae relaionship; cov (, ) = cov (, ). We canno, in general, infer anhing abou he eisence or direcion of causali beween and b observing non-zero covariance. Even if our saisical analsis is successful in esablishing ha he covariance is highl unlikel o have occurred b chance, such a relaionship could occur because causes, because causes, because each causes he oher, or because and are responding o some hird variable wihou an causal relaionship beween hem. However, Clive Granger defined he concep of Granger causali, which, under some conroversial assumpions, can be used o shed ligh on he direcion of possible causali beween pairs of variables. The formal definiion of Granger causali asks wheher pas values of aid in he predicion of, condiional on having alread accouned for he effecs on of pas values of (and perhaps of pas values of oher variables). If he do, he is said o Granger cause. The VAR is a naural framework for eamining Granger causali. Consider he wovariable ssem in equaions (5.1). The firs equaion models as a linear funcion of is own pas values, plus pas values of. If Granger causes (which we wrie as ), hen some or all of he lagged values have non-zero effecs: lagged affecs condiional on he effecs of lagged. Tesing for Granger causali in (5.1) amouns o esing he join blocks of coefficiens β s and β s o see if he are zero. The null hpohesis ( does no Granger cause ) in his VAR is H : β =β =... =β =, 1 2 p
Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models 77 which can be esed using a sandard Wald F or χ 2 es. Similarl, he null hpohesis can be epressed in he VAR as H : β =β =... =β =. 1 2 p Running boh of hese ess can ield four possible oucomes, as shown in Table 5-1: no Granger causali, one-wa Granger causali in eiher direcion, or feedback, wih Granger causali running boh was. Table 5-1. Granger causali es oucomes Fail o rejec: β 1 =β 2 =... =β s = Rejec: β 1 =β 2 =... =β s = Fail o rejec: β 1 =β 2 =... =β s = (no Granger causali) ( Granger causes ) Rejec: β 1 =β 2 =... =β s = ( Granger causes ) (bi-direcional Granger causali, or feedback) There are muliple was o perform Granger causali ess beween a pair of variables, so no resul is unique or definiive. Wihin he wo-variable VAR, one ma obain differen resuls wih differen lag lenghs p. Moreover, including addiional variables in he VAR ssem ma change he oucome of he Wald ess ha underpin Granger causali. In a hreevariable VAR, here are hree pairs of variables, (, ), (, z), and (, z) ha can be esed for Granger causali in boh direcions: si ess wih 36 possible combinaions of oucomes. The effec of lagged on can disappear when lagged values of a hird variable z are added o he regression. For eample, if z and z, hen omiing z from he VAR ssem could lead us o conclude ha even if here is no direc Granger causali in he larger ssem. Is Granger causali reall causali? Obviousl, if he maim abou correlaion and causali is rue, hen here mus be somehing rick happening, and indeed here is. Granger causali ess wheher lagged values of one variable condiionall help predic anoher variable. Under wha condiions can we inerpre his as causali? Two assumpions are sufficien. Firs, we use emporal priori in an imporan wa in Granger causali. We inerpre correlaion beween lagged and he par of curren ha is orhogonal o is own lagged values as. Could his insead reflec curren causing lagged? To rule his ou, inerpreing Granger causali as more general causali requires ha we assume ha he fuure canno cause he presen. While his ma ofen be a reasonable assumpion, modern economic heor has shown us ha epecaions of fuure variables (which are likel o be corre-
78 Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models laed wih he fuure variables hemselves) can change agens curren choices, which migh resul in causali ha would appear o violae his assumpion. Second, an causal relaionship ha is sricl immediae in he sense ha a change in leads o a change in bu no change in an fuure values of would fl under he radar of a Granger causali es, which onl measures and ess lagged effecs. Mos causal economic relaionships are dnamic in ha effecs are no full realized wihin a single ime period, so his difficul ma no presen a pracical problem in man cases. To summarize, we mus be ver careful in inerpreing he resul of Granger causali ess o reflec rue causali in an non-economeric sense. Onl if we can rule ou he possibili of he fuure causing he presen and sricl immediae causal effecs can we confidenl hink of Granger causali as causali. Saa implemens Granger causali ess auomaicall wih vargranger, which ess all of he pairs of variables in a VAR for Granger causali. In ssems wih more han wo variables, i also ess he join hpohesis ha all of he oher variables fail o Granger cause each variable in urn. This join es amouns o esing wheher all of he lagged erms oher han hose of he dependen variable have zero effecs. 5.2 Idenificaion of Srucural Shocks in a VAR Ssem Two variables ha have a dnamic relaionship in a VAR ssem are also likel o have some degree of conemporaneous associaion. This will be refleced in correlaion in he innovaion erms v in (5.1) because here is no oher place in he equaions for his associaion o be manifesed. I is naural o hink of he VAR ssem as he reduced form of a srucural model in which conemporaneous effecs among he variables have been solved ou. We now consider a simple wo-equaion srucural model in which affecs conemporaneousl bu has no immediae effec on. Suppose ha wo variables, and, evolve over ime according o he srucural model =α +α +θ +ε 1 1 1 1 =φ +φ +δ +δ +ε 1 1 1 1, (5.1) where he ε erms are eogenous whie-noise shocks o and ha are orhogonal (uncorrelaed) o one anoher: var ( ε ) =σ, var, cov ε, ε =. The ε shocks 2 2 are ε =σ and ( ) changes in he variables ha come from ouside he VAR ssem. Because he are (assumed o be) eogenous, we can measure he effec of an eogenous change in on he pah of and b looking a he dnamic marginal effecs of ε, for eample, + s/ ε. This is he ke disincion beween he VAR error erms v and he eogenous srucural shocks ε depending on he idenifing assumpions we make, we canno generall inerpre a change in v as an eogenous shock o one variable.
Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models 79 We assume ha and are saionar and ergodic, which imposes resricions on he auoregressive coefficiens of he model. 1 The firs equaion of (5.1) is alread in he form of a VAR equaion: i epresses he curren value of as a funcion of lagged values of and. If we solve (5.1) b subsiuing for in he second equaion using he firs equaion, we ge 1 1 ( 1 1 1 1 ) 1 1 ( ) =φ +φ +δ α +α +θ +ε +δ +ε = φ +δ α + φ +δ θ + δ +δ α + ε +δ ε 1 1 1 1 1 1, (5.11) which also has he VAR form. Thus, we can wrie he reduced-form ssem of (5.1) as =β +β +β + v 1 1 1 1 =β +β +β + v 1 1 1 1, (5.12) wih β =φ +δ α β =α v β =φ +δ θ β =θ 1 1 1 1 1 β =δ +δ α β =α 1 1 1 1 1 =ε +δ ε v =ε. (5.13) Given our assumpions abou he disribuions of he eogenous shocks ε, we can deermine he variances and covariance of he VAR error erms v as var var 2 ( v ) = σ 2 2 2 ( v ) =σ +δσ ( v v ) E( v v ) E 2 cov, = = ε ε +δε =δσ. (5.14) Le s now consider wha can be esimaed using he VAR ssem (5.12) and o wha een hese esimaes allow us o infer value of he parameers in he srucural ssem (5.1). In erms of coefficiens, here are si β coefficiens ha can be esimaed in he VAR and seven srucural coefficiens in (5.1). This seems a pessimisic sar o he ask of idenificaion. However, we can also esimae he variances and covariance of he v erms using he VAR residuals: var ( v ), var ( v ), and cov (, ) v v. Condiions (5.14) allow us o esimae hree parameers σ 2, 2 σ, δ from he hree esimaed variances and covariance: 1 In a single-variable auoregressive model, we would require ha he coefficien φ1 for 1 be in he range ( 1, 1). The corresponding condiions in he vecor seing are more involved, bu similar in naure.
8 Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models ( v ) cov ( v, v ) var ( v ) ˆ v δ ( v ) 2 σ ˆ = var, δ ˆ =, 2 2 σ ˆ = var var. (5.15) Armed wih an esimae of δ from he covariance erm, we can now use he si β coefficiens o esimae he remaining si srucural coefficiens using (5.13). The ssem is jus idenified. So how did we manage o achieve idenificaion hrough his back-door covariance mehod? Le s consider wh v and v, which are he innovaions in and ha canno be prediced from pas values, migh be correlaed. In a general model he could be correlaed because (1) has an effec on, (2) has an effec on, or (3) he eogenous srucural shocks o and, ε and ε, are correlaed wih one anoher. Our VAR esimaes give us no mehod of discriminaing among hese hree possible sources of correlaion, so wihou ruling ou wo of hem we canno achieve idenificaion. In he model of (5.1), we have ruled ou, b assumpion, (2) and (3): does no affec and he eogenous shocks o and are orhogonal. Thus, we are inerpreing he covariance beween v and v as reflecing he conemporaneous effec of on. This allows us o idenif he coefficien measuring cov v, v as we do in he second equaion of (5.15). his effec δ in (5.1) based on he ( ) This idenifing assumpion also allows us o reconsruc esimaes of he eogenous srucural shocks ε from he residuals of he VAR: ε ˆ = vˆ ε ˆ = ˆ ˆ. ˆ v δv This makes i clear ha we are inerpreing he VAR residual for o be an eogenous, srucural shock o. In order o erac he srucural shock o, we subrac he par of v ˆ, δ ˆ ˆ ˆ ˆ v =δε, ha is due o he effec of he shock o on. From an economeric sandpoin, we could equall well make he opposie assumpion, assuming ha affecs raher han vice versa, which would inerpre v ˆ as ε ˆ and calculae ε ˆ as he par of v ˆ ha is no eplained b v ˆ. Choosing which inerpreaion o use mus be done on he basis of heor: which variable is more plausibl eogenous wihin he immediae period. We ma ge differen resuls depending on which idenificaion assumpion we choose, so if here is no clear choice i ma be useful o eamine wheher resuls are robus across differen choices. Idenificaion of he underling srucural shocks is necessar if we are o esimae he effecs of an eogenous shock o a single variable on he dnamic pahs of all of he variables
Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models 81 of he ssem, which we call impulse-response funcions (IRFs). We discuss he compuaion and inerpreaion of IRFs in he ne secion. In our eample, we idenified shocks b limiing he conemporaneous effecs among he variables. Wih onl wo variables, here are wo possible choices: (1) he assumpion we made, ha affecs immediael bu does no have an immediae effec on or (2) he opposie assumpion, ha affecs immediael bu does no affec ecep wih a lag. We can hink of he choice beween hese alernaives as an ordering of he variables, wih he variables ling higher in he order having insananeous effecs on hose lower in he order, bu he lower variables onl affecing hose above hem wih a lag. This ordering or orhogonalizaion sraeg of idenificaion eends direcl o VAR ssems wih more han wo variables. Sims s seminal VAR ssem had si variables, which he ordered as he mone suppl, real oupu, unemplomen, wages, prices, and impor prices. B adoping his ordering, Sims was imposing an arra of idenifing resricions abou he conemporaneous effecs of shocks on he variables of he ssem. The mone shock, because i was a he op of he lis, could affec all of he variables in he ssem wihin he curren period. The shock o oupu affecs all variables immediael ecep mone, because mone lies above i on he lis. The variable a he boom of he lis, impor prices, is assumed o have no conemporaneous effec on an of he oher variables of he ssem. Alhough idenificaion b ordering is sill common, subsequen research has shown ha oher kinds of resricions can be used. For eample, in some macroeconomic models we can assume ha changes in a variable such as he mone suppl would have no long-run effec on anoher variable such as real oupu. In a simple ssem such as (5.1), his migh show up as he assumpion ha δ + δ 1 =, for eample. Imposing his condiion would allow he seven srucural coefficiens of (5.1) o be idenified from he si β coefficiens of he VAR wihou using resricions on he covariances. 5.3 Inerpreing he Resuls of Idenified VARs When we can idenif he srucural shocks o each variable in a VAR, we can perform wo kinds of analsis o eplain how each shock affecs he dnamic pah of all of he variables of he ssem. Impulse-response funcions (IRFs) measure he dnamic marginal effecs of each shock on all of he variables over ime. Variance decomposiions eamine how imporan each of he shocks is as a componen of he overall (unpredicable) variance of each of he variables over ime. I is imporan o sress ha, unlike forecass and Granger causali ess, boh IRFs and variance decomposiions can onl be calculaed based on a se of idenifing assumpions and ha a differen se of idenificaion assumpions ma lead o differen conclusions. Suppose ha we have an n-variable VAR wih lags up o order p. If he variables of he ssem are 1, 2,, n, hen we can wrie he n equaions of he VAR as
82 Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models n p i j i =β i + β ijs s + v, i = 1, 2,..., n. j= 1 s= 1 (5.16) We assume ha we have a se of idenifing resricions on he model eiher an ordering of assumed conemporaneous causali or anoher se of assumpions so ha we can idenif he n orhogonal srucural shocks i ε from he n VAR error erms The impulse-response funcions are he n n se of dnamic marginal effecs of a oneime shock o variable j on iself or anoher variable i: i v. ε i + s j, s =, 1, 2,... (5.17) Noe ha here is in principle no limi on how far ino he fuure hese dnamic impulse responses can eend. If he VAR is sable, hen he IRFs should converge o zero as he ime from he shock s ges large one-ime shocks should no have permanen effecs. As noed above, non-convergen IRFs and unsable VARs are indicaions of non-saionari in he variables of he model, which ma be correced b differencing. IRFs are usuall presened graphicall wih he ime lag s running from zero up o some user-se limi S on he horizonal ais and he impac a he s-order lag on he verical. The can also be epressed in abular form if he numbers hemselves are imporan. One common forma for he enire collecion of IRFs corresponding o a VAR is as an n n mari of graphs, wih he impulse variable (he shock) on one dimension and he response variable on he oher. Each of he n 2 IRF graphs ells us how a shock o one variable affecs anoher (or he same) variable. There are wo common convenions for deermining he size of he shock o he impulse variable. One is o use a shock of magniude one. Since we can hink of he impulse shock as he ε in he denominaor of (5.17), seing he shock o one means ha he values repored are he dnamic marginal effecs as in (5.17). However, a shock of size one does no alwas make economic sense: Suppose ha he shock variable is banks raio of reserves o deposis, epressed as a fracion. An increase of one in his variable, sa from.1 o 1.1, would be implausible. To aid in inerpreaion, some sofware packages normalize he size of he shocks o be one sandard deviaion of he variable raher han one uni. Under his convenion, he values ploed are ε i + s j σ ˆ, s =, 1, 2,... j and are inerpreed as he change in each response variable resuling from a one-sandarddeviaion increase in he impulse variable. This makes he magniude of he changes in he response variables more realisic, bu does no allow he IRF values o be inerpreed direcl as dnamic marginal effecs.
Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models 83 Because he VAR model is linear, he marginal effecs in (5.17) are consan, so which normalizaion o choose for he shocks one uni or one sandard deviaion is arbirar and should be done o faciliae inerpreaion. Saa uses he convenion of he one-uni impulse in is simple IRFs and one sandard deviaion in is orhogonalized IRFs. If he impulse variable is he same as he response variable, hen he IRF ells us how persisen shocks o ha variable end o be. B definiion, ε i i = 1, so he zero-order own impulse response is alwas one. If he VAR is sable, reflecing he saionari and ergodici of he underling variables, hen he own impulse responses deca o zero as he ime horizon increases: i s lim + =. s i ε If he impulse responses deca o zero onl slowl hen shocks o he variable end o change is value for man periods, whereas a shor impulse response paern indicaes ha shocks are more ransior. For cross-variable effecs, where he impulse and response variables are differen, general paerns of posiive or negaive responses are possible. Depending on he idenificaion assumpion (he ordering ), he zero-period response ma be zero or non-zero. B assumpion, shocks o variables near he boom of he ordering have no curren-period effec on variables higher in he order, so he zero-lag impulse response in such cases is eacl zero. 5.4 A VAR Eample: GDP Growh in US and Canada To illusrae he various applicaions of VAR analsis, we eamine he join behavior of US and Canadian real GDP growh using a quarerl sample from 1975q1 hrough 211q4. Each of he series is an annual coninuousl-compounded growh rae. For eample, USGR = 4 ( lnusgdp lnusgdp 1 ), wih he 4 included o epress he growh rae as an annual raher han quarerl rae and he 1 o pu he rae in percenage erms. 5.4.1 Geing he specificaion righ As a preliminar check, we verif ha boh growh series are saionar. To be conservaive, we include four lagged differences o eliminae serial correlaion in he error erm of he Dicke-Fuller regression.. dfuller usgr, lags(4) Augmened Dicke-Fuller es for uni roo Number of obs = 143
84 Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models ---------- Inerpolaed Dicke-Fuller --------- Tes 1% Criical 5% Criical 1% Criical Saisic Value Value Value ------------------------------------------------------------------------------ Z() -4.565-3.496-2.887-2.577 ------------------------------------------------------------------------------ MacKinnon approimae p-value for Z() =.1. dfuller cgr, lags(4) Augmened Dicke-Fuller es for uni roo Number of obs = 143 ---------- Inerpolaed Dicke-Fuller --------- Tes 1% Criical 5% Criical 1% Criical Saisic Value Value Value ------------------------------------------------------------------------------ Z() -4.769-3.496-2.887-2.577 ------------------------------------------------------------------------------ MacKinnon approimae p-value for Z() =.1 In boh cases, we comforabl rejec he presence of a uni roo in he growh series because he es saisic is more negaive han he criical value, even a a 1% level of significance. Phillips-Perron ess lead o similar conclusions. Therefore, we conclude ha VAR analsis can be performed on he wo growh series wihou differencing. To assess he opimal lag lengh, we use he Saa varsoc command wih a maimum lag lengh of four:. varsoc usgr cgr, malag(4) Selecion-order crieria Sample: 1976q1-211q4 Number of obs = 144 +---------------------------------------------------------------------------+ lag LL LR df p FPE AIC HQIC SBIC ----+---------------------------------------------------------------------- -79.83 67.486 9.88653 9.9329 9.92777 1-671.726 76.28* 4. 41.9769* 9.41286* 9.46314* 9.5366* 2-67.318 2.8151 4.589 43.5178 9.44887 9.53267 9.6551 3-667.543 5.55 4.235 44.2688 9.46588 9.58321 9.75461 4-664.14 6.867 4.146 44.6449 9.47417 9.6251 9.84539 +---------------------------------------------------------------------------+ Endogenous: usgr cgr Eogenous: _cons Noe ha all of he regressions leading o he numbers in he able are run for a sample beginning in 1976q1, which is he earlies dae for which 4 lags are available, even hough he regressions wih fewer han 4 lags could use a longer sample. In his VAR, all of he crieria suppor a lag of lengh one, so ha is wha we choose.
Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models 85 5.4.2 Analsis wihou making idenificaion assumpions Alhough we could accomplish he asks we desire using varbasic, we will use he more general commands o demonsrae heir use. To run he VAR regressions, we use var:. var usgr cgr, lags(1) Vecor auoregression Sample: 1975q2-211q4 No. of obs = 147 Log likelihood = -686.263 AIC = 9.415324 FPE = 42.837 HQIC = 9.464918 De(Sigma_ml) = 38.78127 SBIC = 9.537383 Equaion Parms RMSE R-sq chi2 P>chi2 ---------------------------------------------------------------- usgr 3 2.95659.1789 32.277. cgr 3 2.3983.3861 92.43759. ---------------------------------------------------------------- ------------------------------------------------------------------------------ Coef. Sd. Err. z P> z [95% Conf. Inerval] -------------+---------------------------------------------------------------- usgr usgr L1..2512771.89854 2.8.5.751735.427387 cgr L1..234161.975328 2.4.16.429453.4252668 _cons 1.494612.33545 4.46..8372394 2.151985 -------------+---------------------------------------------------------------- cgr usgr L1..3759117.726571 5.17..233565.5183169 cgr L1..2859551.788694 3.63..1313739.445362 _cons.8755421.2712199 3.23.1.343969 1.47123 ------------------------------------------------------------------------------ We have no e aemped an shock idenificaion, so a his poin he ordering of he variables in he command is arbirar. The VAR regressions are run saring he sample a he earlies possible dae wih one lag, which is 1975q2 because our firs available observaion is 1975q1. Because i uses hree addiional observaions, he repored AIC and SBIC values from he VAR oupu do no mach hose from he varsoc able above. To assess he validi of our VAR, we es for sabili and for auocorrelaion of he residuals. The varsable command eamines he dnamic sabili of he ssem. None of he eigenvalues is even close o one, so our ssem is sable.
86 Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models. varsable Eigenvalue sabili condiion +----------------------------------------+ Eigenvalue Modulus --------------------------+-------------.5657757.565776 -.2854353.28544 +----------------------------------------+ All he eigenvalues lie inside he uni circle. VAR saisfies sabili condiion. The varlmar performs a Lagrange muliplier es for he join null hpohesis of no auocorrelaion of he residuals of he wo equaions:. varlmar, mlag(4) Lagrange-muliplier es +--------------------------------------+ lag chi2 df Prob > chi2 ------+------------------------------- 1 4.688 4.3284 2 5.5347 4.2367 3 3.5253 4.4744 4 7.1422 4.12856 +--------------------------------------+ H: no auocorrelaion a lag order We canno rejec he null of no residual auocorrelaion a orders 1 hrough 4 a an convenional significance level, so we have no evidence o conradic he validi of our VAR. To deermine if he growh raes of he US and Canada affec one anoher over ime, we can perform Granger causali ess using our VAR.. vargranger Granger causali Wald ess +------------------------------------------------------------------+ Equaion Ecluded chi2 df Prob > chi2 --------------------------------------+--------------------------- usgr cgr 5.7613 1.16 usgr ALL 5.7613 1.16 --------------------------------------+--------------------------- cgr usgr 26.768 1. cgr ALL 26.768 1. +------------------------------------------------------------------+ We see srong evidence ha lagged Canadian growh helps predic US growh (he p-value is.16) and overwhelming evidence ha lagged US growh helps predic Canadian growh (pvalue less han.1). I is no surprising, given he relaive sizes of he economies, ha he US migh have a sronger effec on Canada han vice versa. Noe ha because we have onl one lag in our VAR, he Granger causali ess have onl one degree of freedom and are
Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models 87 equivalen o he single-coefficien ess in he VAR regression ables. The coefficien of Canadian growh in he US equaion has a z value of 2.4, which is he square roo of he 5.7613 value repored in he vargranger able; boh have idenical p-values. If we had more han one lag in he regressions, hen he z values in he VAR able would be ess of individual lag coefficiens and he χ 2 values in he Granger causali able would be join ess of he blocks of lag coefficiens associaed wih each variable. We ne eplore he implicaions of our VARs for he behavior of GDP in he wo counries in 212 and 213. Wha does our model forecas? To find ou, we issue ha command: fcas compue p_, sep(8). This command produces no oupu, bu changes our daase in wo was. Firs, i eends he daase b eigh observaions o 213q4, filling in appropriae values for he dae variable. Second, i adds eigh variables o he daase wih values for hose eigh quarers. The new variables p_usgr and p_cgr conain he forecass, and he variables p_usgr_se, p_usgr_lb, and p_usgr_ub (and corresponding variables for Canada) conain he sandard error, 95% lower bound, and 95% upper bound for he forecass. We can eamine hese forecas values in he daa browser or wih an saisical commands in he Saa arsenal, bu i is ofen mos informaive o graph hem. The command fcas graph p_usgr p_cgr generaes he graph shown in Figure 5-1. The graph shows ha he confidence bands on our forecass are ver large: our VARs do no forecas ver confidenl. The poin forecass predic lile change in growh raes. The US growh rae is prediced o decline ver slighl and hen hold sead near is mean; he Canadian growh rae o increase a bi and hen converge o is mean. If he goal of our VAR eercise was o obain insighful and reliable forecass, we have no succeeded! Esimaion, Granger causali, and forecasing can all be accomplished wihou an idenificaion assumpions. Bu his is as far as we can go wih our VAR analsis wihou making some assumpions o allow us o idenif he srucural shocks o US and Canadian GDP growh.
88 Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models Forecas for usgr Forecas for cgr -5 5 1-5 5 1 211q3 212q1 212q3 213q1 213q3 211q3 212q1 212q3 213q1 213q3 95% CI forecas Figure 5-1. Graph of forecass 5.4.3 Impulse-response funcions and variance decomposiions Onl if he residuals of he wo equaions are conemporaneousl uncorrelaed can we inerpre hem as srucural shocks. We would epec ha innovaions o US and Canadian growh in an period would end o be posiivel correlaed, and indeed he cross-equaion correlaion coefficien for he residuals in our VAR regressions is.44. In order o idenif he effecs of shocks o US or Canadian growh on he subsequen ime pahs of boh, we mus make an assumpion abou wheher he correlaion is due o curren US growh affecing curren Canadian growh or due o Canadian growh affecing US growh. Given he relaive sizes of he wo economies, heor suggess ha US growh would have a sronger effec on Canada han vice versa, and his is suppored somewha b he evidence for lagged effecs from our Granger causali ess (alhough boh have srong effecs on he oher). Thus, we choose as our preferred idenificaion paern inerpreing he conemporaneous correlaion as he effec of US growh on Canadian growh: US growh is firs in our ordering and Canadian growh is second. We begin b creaing a.irf file called uscan.irf o conain our impulseresponse funcions:. irf se "uscan" (file uscan.irf creaed) (file uscan.irf now acive)
Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models 89 We hen creae an IRF enr in he file called L1 o hold he resuls of our one-lag VARs, running he IRF effec horizon ou 2 quarers (five ears):. irf creae L1, sep(2) order(usgr cgr) (file uscan.irf updaed) We specified he ordering eplicil in creaing he IRF. However, because he ordering is he same as he order in which he variables were lised in he var command iself, Saa would have chosen his ordering b defaul. We can now use he irf graph command o produce impulse-response funcion and variance decomposiion graphs. To ge he (orhogonalized) IRFs, we pe. irf graph oirf, irf(l1) usep(8) I is imporan o specif oirf raher han irf because he laer gives impulse responses assuming (counerfacuall) ha he VAR residuals are uncorrelaed. The resuling IRFs are shown in Figure 5-2. 3 L1, cgr, cgr L1, cgr, usgr 2 1 3 L1, usgr, cgr L1, usgr, usgr 2 1 2 4 6 8 2 4 6 8 sep 95% CI orhogonalized irf Graphs b irfname, impulse variable, and response variable Figure 5-2. IRFs for preferred ordering
9 Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models The diagonal panels in Figure 5-2 show he effecs of shocks o each counr s GDP growh on fuure values of is own growh. In boh cases, he shock dies ou quickl, reflecing he saionari of he variables. A one-sandard deviaion shock o Canadian GDP growh in he op-lef panel is jus over 2 percen; a corresponding shock o U.S. growh is close o 3 percen. The off-diagonal panels (boom-lef and op-righ) show he effecs of a growh shock in one counr on he pah of growh in he oher. In he boom-lef panel, we see ha a onesandard-deviaion (abou 3 percenage poins) shock in U.S. growh raises Canadian growh b abou 1 percenage poin in he curren quarer, hen b a bi more in he ne quarer as he lagged effec kicks in. From he second lag on, he effec decas rapidl o zero, wih he saisical significance of he effec vanishing a abou one ear. In he op-righ panel, we see he esimaed effecs of a shock o Canadian growh on growh in he Unied Saes. The firs hing o noice is ha he effec is zero in he curren period (a zero lag). This is a direc resul of our idenificaion assumpion: we imposed he condiion ha Canadian growh has no immediae effec on U.S. growh in order o idenif he shocks. The second noeworh resul is ha he dnamic effec ha occurs in he second period is much smaller han he effec of he U.S. on Canada. This is as we epeced. Bu how much of his greaer dependence of Canada on he Unied Saes is reall he daa speaking and how much is our assumpion ha conemporaneous correlaion in shocks runs onl from he U.S. o Canada? Recall ha our idenificaion assumpion imposes he condiion ha an common shocks ha affec boh counries are assumed o be U.S. shocks, wih Canada shocks being he par of he Canadian VAR innovaion ha is no eplained b he common shock. This migh cause he Canadian shocks o have smaller variance (which i does in Figure 5-2) and migh also overesimae he effec of he U.S. shocks on Canada. To assess he sensiivi of our conclusions o he ordering assumpion, we eamine he IRFs making he opposie assumpion: ha conemporaneous correlaion in he innovaions is due o Canada shocks affecing he U.S. Figure 5-3 shows he graphs of he reverseordering IRFs. As epeced, he effec of he U.S. on Canada (lower lef) now begins a zero and he effec of Canada on he U.S. (upper righ) does no. Beond his, here are a couple of ineresing changes when we reverse he order. Firs, noe ha boh shocks now have a sandard deviaion of abou 2.5 raher han he U.S. shock having a much larger sandard deviaion. This occurs because we now aribue he common par of he innovaion o he Canadian shock raher han he U.S. shock. Second, afer he iniial period in which he U.S.-o-Canada effec is consrained o be zero, he wo effecs are of similar magniudes and die ou in a similar wa. This eample shows he difficul of idenifing impulse responses in VARs. The implicaions can depend on he idenificaion assumpion we make, so if we are no sure which assumpion is beer we ma be lef wih considerable uncerain in inerpreing our resuls.
Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models 91 3 l1cus, cgr, cgr l1cus, cgr, usgr 2 1 3 l1cus, usgr, cgr l1cus, usgr, usgr 2 1 2 4 6 8 2 4 6 8 sep 95% CI orhogonalized irf Graphs b irfname, impulse variable, and response variable Figure 5-3. IRFs wih reversed ordering We can also use Saa s irf graph command o plo he cumulaive effec of a permanen shock o one of he variables. For he preferred ordering his looks like Figure 5-4. Using he op-lef panel, a permanen posiive shock of one sandard deviaion o Canada s growh an eogenous increase of abou 2 percenage poins of growh ha is susained over ime would evenuall cause Canadian growh o be abou 3.5 percenage poins higher. This magnificaion comes from wo effecs. Firs, shocks o Canadian growh end o persis for a period or wo afer he shock, so growh increases more as a resul. Second, a posiive shock o Canadian growh increases U.S. growh (even wih no eogenous shock in he U.S.), which feeds back posiivel on Canadian growh. The same muliplier effec happens in he oher panels.
92 Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models 6 L1, cgr, cgr L1, cgr, usgr 4 2 6 L1, usgr, cgr L1, usgr, usgr 4 2 2 4 6 8 2 4 6 8 sep 95% CI cumulaive orhogonalized irf Graphs b irfname, impulse variable, and response variable Figure 5-4. Cumulaive IRFs wih preferred ordering Anoher ool ha is available for analsis of idenified VARs is he forecas-error variance decomposiion, which measures he een o which each shock conribues o uneplained movemens (forecas errors) in each variable. Figure 5-5 resuls from he Saa command: irf graph fevd, irf(l1) usep(8)and shows how each shock conribues o he variaion in each variable. All variance decomposiions sar a zero because here is no forecas error a a zero lag. The lef-column panels show ha (wih he preferred idenificaion assumpion) he Canadian shock conribues abou 8% of he variance in he one-period-ahead forecas error for Canadian growh, wih he U.S. shock conribuing he oher 2%. As our forecas horizon moves furher ino he fuure, he effec of he U.S. shock on Canadian growh increases and he shares converge o less han 6% of variaion in Canadian growh being due o he Canadian shock and more han 4% due o he U.S. shock. The righ-column panels indicae ha ver lile (less han 5%) of he variaion in U.S. growh is aribuable o Canadian growh shocks in he shor run or long run.
Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models 93 1 L1, cgr, cgr L1, cgr, usgr.5 1 L1, usgr, cgr L1, usgr, usgr.5 2 4 6 8 2 4 6 8 sep 95% CI fracion of mse due o impulse Graphs b irfname, impulse variable, and response variable Figure 5-5. Variance decomposiions wih preferred ordering Like IRFs, variance decomposiions can be sensiive o he idenificaion assumpions we make. If we compue IRFs and variance decomposiions for alernaive orderings and find ha he resuls are similar, hen we gain confidence ha our conclusions are no sensiive o he (perhaps arbirar) assumpions we make abou conemporaneous causali. If alernaive assumpions lead o differen conclusions, we mus be more careful abou drawing conclusions. Figure 5-6 shows he ver differen resuls ha we ge when we reverse he conemporaneous causal ordering. Now he Canadian shock (which includes he shock ha is common o boh counries under his assumpion) eplains mos (8%) of he variaion in Canadian growh and much (3%) of he variaion in growh in he Unied Saes. I ma seem frusraing o reach quie differen conclusions depending on a poeniall arbirar assumpion abou he direcion of immediae causaion. In his case, hough, he differences beween he resuls sugges some possible inerpreaions. Firs, he Unied Saes has a sronger effec on Canada han vice versa. Inerpreing he VAR resuls in favor of Canada s effec (b puing hem firs in he order) gives Canada a subsanial effec on he U.S. bu he U.S. shocks are clearl sill imporan for boh counries, bu inerpreing hem in favor of he U.S. effec viruall wipes ou he effec of Canada on he Unied Saes.
94 Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models Second, because of he wa he resuls var beween orderings, i is clear ha much of he variaion in growh in boh counries is due o a common shock. Whichever counr is (perhaps arbiraril) assigned ownership of his shock seems o have a large effec relaive wih he oher. While his doesn resolve he causali quesion, i is ver useful informaion abou he co-movemen of U.S. and Canadian growh. 1 l1cus, cgr, cgr l1cus, cgr, usgr.5 1 l1cus, usgr, cgr l1cus, usgr, usgr.5 2 4 6 8 2 4 6 8 sep 95% CI fracion of mse due o impulse Graphs b irfname, impulse variable, and response variable Figure 5-6. Variance decomposiion wih reversed ordering 5.5 Coinegraion in a VAR: Vecor Error-Correcion Models In our analsis of vecor auoregressions, we have assumed ha he variables of he model are saionar and ergodic. We saw in he previous chaper ha variables ha are individuall non-saionar ma be coinegraed: wo (or more) variables ma have common underling sochasic rends along which he move ogeher on a non-saionar pah. For he simple case of wo variables and one coinegraing relaionship, we saw ha an errorcorrecion model is he appropriae economeric specificaion. In his model, he equaion is differenced and an error-correcion erm measuring he previous period s deviaion from long-run equilibrium is included. We now consider how coinegraed variables can be used in a VAR using a vecor errorcorrecion (VEC) model. Firs we eamine he wo-variable case, which eends he simple
Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models 95 single-equaion error-correcion model o wo equaions in a sraighforward wa. We hen generalize he model o more han wo variables and equaions, which allows for he possibili of more han one coinegraing relaionship. This requires a new es for coinegraion and a generalizaion of he error-correcion model o include muliple error-correcion erms. 5.5.1 A wo-variable VEC model If wo I(1) series and are coinegraed, hen here is eis unique α and α 1 such ha u α α 1 is I(). In he single-equaion model of coinegraion where we hough of as he dependen variable and as an eogenous regressor, we saw ha he error-correcion model =β +β +λ u +ε =β +β +λ α α +ε (5.18) 1 1 1 1 1 1 was an appropriae specificaion. All erms in equaion (5.18) are I() as long as he α coefficiens (he coinegraing vecor ) are known or a leas consisenl esimaed. The u 1 erm is he magniude b which was above or below is long-run equilibrium value in he previous period. The coefficien λ (which we epec o be negaive) represens he amoun of correcion of his period-( 1) disequilibrium ha happens in period. For eample, if λ is.25, hen one quarer of he gap beween 1 and is equilibrium value would end (all else equal) o be reversed (because he sign is negaive) in period. The VEC model eends his single-equaion error-correcion model o allow and o evolve joinl over ime as in a VAR ssem. In he wo-variable case, here can be onl one coinegraing relaionship and he equaion of he VEC ssem is similar o (5.18), ecep ha we mirror he VAR specificaion b puing lagged differences of and on he righhand side. Wih onl one lagged difference (here can be more) he bivariae VEC can be wrien =β +β +β +λ α α + v 1 1 1 1 1 1 1 =β +β +β +λ α α + v 1 1 1 1 1 1 1,. (5.19) As in (5.18), all of he erms in boh equaions of (5.19) are I() if he variables are coinegraed wih coinegraing vecor (1, α, α 1 ), in oher words, if α α 1 is saionar. The λ coefficiens are again he error-correcion coefficiens, measuring he response of each variable o he degree of deviaion from long-run equilibrium in he previous period. We epec λ < for he same reason as above: if 1 is above is long-run value in relaion o 1 hen he error-correcion erm in parenheses is posiive and his should lead, oher hings consan, o downward movemen in in period. The epeced sign of λ depends on he sign of α 1. We epec / 1 = λ α 1 < for he same reason ha we epec / 1 = λ < : if 1 is above is long-run relaion o, hen we epec o be negaive, oher hings consan.
96 Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models A simple, concree eample ma help clarif he role of he error-correcion erms in a VEC model. Le he long-run coinegraing relaionship be =, so ha α = and α 1 = 1. The parenheical error-correcion erm in each equaion of (5.19) is now 1 1, he difference beween and in he previous period. Suppose ha because of previous shocks, = + so ha is above is long-run equilibrium relaionship o b one uni (or, 1 1 1 equivalenl, is below is long-run equilibrium relaionship o b one uni). To move oward long-run equilibrium in period, we epec (if here are no oher changes) < and >. Using equaion (5.19), changes in response o his equilibrium b ( ) 1 1 λ =λ, so for sable adjusmen o occur λ < ; is oo high so i mus decrease in response o he disequilibrium. The corresponding change in from equaion (5.19) is ( ) 1 1 λ =λ. Since is oo low, sable adjusmen requires ha he response in be posiive, so we need λ >. Noe ha if he long-run relaionship beween and were inverse (α 1 < ), hen would need o decrease in order o move oward equilibrium and we would need λ <. The epeced sign on λ depends on he sign of α 1. If heor ells us he coefficiens α and α 1 of he coinegraing relaionship, as in he case of purchasing-power pari, hen we can calculae he error-correcion erm in (5.19) and esimae i as a sandard VAR. However, we usuall do no know hese coefficiens, so he mus be esimaed. Single-equaion coinegraed models can be esimaed eiher direcl or in wo seps. We can use OLS o esimae he coinegraing relaionship he coinegraing vecor (1, α, α 1 ) and impose hese esimaes on he error-correcion model, or we can esimae he α coefficiens joinl wih he β coefficiens on he differences. In a VAR/VEC ssem, separae esimaion of he coinegraing relaionship is likel o be problemaic because i ma be implausible o assume ha eiher or is even weakl eogenous. (However, noe ha he idenifing resricions we impose o calculae IRFs require eacl his assumpion.) Thus, i is common o esimae boh ses of coefficiens simulaneousl in a VEC model. We shall see an eample of VEC esimaion and inerpreaion shorl, bu firs we consider models wih more han wo variables and equaions because hese models raise imporan addiional consideraions. 5.5.2 A hree-variable VEC wih (pariall) known coinegraing relaionships We now consider a vecor error-correcion model wih hree variables,, and z. This siuaion is more comple because he number of linear combinaions of he hree variables ha are saionar could be, 1, or 2. In oher words, here could be zero, one, or wo common rends among he hree variables. If here are no coinegraing relaionships, hen he series are no coinegraed and a VAR in differences is he appropriae specificaion. There is no long-run relaionship o which he levels of he variables end o reurn, so here is no basis for an error-correcion erm in an equaion.
Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models 97 There would be one coinegraing relaionship among he hree variables if here is one long-run equilibrium condiion ing he levels of he variables ogeher. An eample would be he purchasing-power-pari (PPP) condiion beween wo counries under floaing echange raes. Suppose ha P 1 is he price of a baske of goods in counr one, P 2 is he price of he same baske in counr 2, and X is he echange rae: he number of unis of counr one s currenc ha bus one uni of counr wo s. If goods are o cos he same in boh 1 2 counries purchasing-power pari hen X = P / P. An increase in prices in counr one should be refleced, in long-run equilibrium, b an increase of equal proporion in he amoun of counr-one currenc needed o bu a uni of counr-wo s currenc. In pracice, economiss have o rel on price indees whose marke baskes differ across counries, so he PPP equaion would need a consan of proporionali o reflec his difference: X= AP P. Denoing logs of he variables b small leers, his implies 1 2 ha / =α + p p 1 2 is a long-run equilibrium condiion oward which he variables should end. We could esimae a VEC ssem (wih one lag, for simplici) for he evoluion of he hree variables, p 1, and p 2 wih one coinegraing relaionship (wih some known coefficiens): =β +β +β p +β p +λ α p + p + v 1 2 1 2 1 1 11 1 21 1 1 1 1 p =β +β +β p +β p +λ α p + p + v 1 1 2 1 2 1 1 11 1 111 1 121 1 1 1 1 1 p =β +β +β p +β p +λ α p + p + v 2 1 2 2 2 2 2 21 1 211 1 221 1 2 1 1 1. (5.2) If he echange rae is ou of equilibrium, sa, oo high, hen we epec some combinaion of adjusmens in, p 1, and p 2 o move back oward long-run equilibrium. The error-correcion coefficiens λ, λ 1, and λ 2 measure hese responses. Using he logic described above, we would epec λ and λ 2 o be negaive and λ 1 o be posiive. This does no ehaus he possible degree of coinegraion among hese variables, however. Suppose ha counr one is on a gold sandard, so ha he price level in ha counr ends o be consan in he long run. 2 This would impose a second long-run equilibrium condiion a second coinegraing relaionship on he variables: p = α 1 1. The VEC ssem incorporaing boh coinegraing relaionships would look like 2 Anoher eample we could use would be a fied-echange-rae ssem in which one counr keeps near a consan level in he long run.
98 Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models 1 2 1 2 1 1 1 11 1 21 1 ( 1 1 1 ) ( 1 1 ) =β 1 +β1 1 1 +β111 1 +β121 1 +λ1 ( 1 α 1 + 1 ) +µ 1 ( 1 α 1 ) + 2 1 2 p =β 2 +β2 1 1 +β211 p 1 +β221 p 1 +λ 2 ( 1 ) ( ) =β +β +β p +β p +λ α p + p +µ p α + v p p p p p p v 1 1 2 1 2 1 1 α + +µ α + 2 2 1 2 p 1 p 1 2 p 1 1 v. In equaions (5.2) and (5.21) we sared wih a ssem in which we knew he naure of he coinegraing relaionship(s) among he variables. I is more common ha we mus es for he possibili of coinegraion (and deermine how man coinegraing relaionships eis) and esimae a full se of α parameers for hem. We now urn o ha process, hen o an eample of an esimaed VEC model. 5.5.3 Tesing for and esimaing coinegraing relaionships in a VEC The mos common ess o deermine he number of coinegraing relaionships among he series in a VAR/VEC are due o Johansen (1995). Alhough he mahemaics of he ess involve mehods ha are beond our reach, he inuiion is ver similar o esing for uni roos in he polnomial represening an AR process. If we have n I (1) variables ha are modeled joinl in a dnamic ssem, here can be up o n 1 coinegraing relaionships linking hem. Sock and Wason (ref) hink of each coinegraing relaionship as a common rend linking some or all of he series in he ssem. we shall hink of coinegraing relaionship and common rend as snonmous. The coinegraing rank of he ssem is he number of such common rends, or he number of coinegraing relaionships. 3 To deermine he coinegraing rank r, we perform a sequence of ess. Firs we es he null hpohesis of r = agains r 1 o deermine if here is a leas one coinegraing relaionship. If we fail o rejec r =, hen we conclude ha here are no coinegraing relaionships or common rends among he series. In his case, we do no need a VEC model and can simpl use a VAR in he differences of he series. If we rejec r = a he iniial sage hen a leas some of he series are coinegraed and we wan o deermine he number of coinegraing relaionships. We proceed o a second sep o es he null hpohesis ha r 1 agains r 2. If we canno rejec he hpohesis of no more han one common rend, hen we esimae a VEC ssem wih one coinegraing relaionship, such as (5.2). If we rejec he hpohesis ha r 1, hen we proceed furher o es r 2 agains r 3, and so on. We choose r o be he smalles value a which we fail o rejec he null hpohesis ha here are no addiional coinegraing relaionships. (5.21) 3 For hose familiar wih linear algebra, he erm rank refers o he rank of a mari characerizing he dnamic ssem. If a dnamic ssem of n variables has r coinegraing relaionships, hen he rank of he mari is n r. This means ha he mari has r eigenvalues ha are zero and n r ha are no. The Johansen ess are based on deermining he number of nonzero eigenvalues.
Chaper 4: Vecor Auoregression and Vecor Error-Correcion Models 99 Johansen proposed several relaed ess ha can be used a each sage. The mos common (and he defaul in Saa) is he race saisic. The Saa command vecrank prins he race saisic or, alernaivel, he maimum-eigenvalue saisic (wih he ma opion) or various informaion crieria (wih he ic opion). The Johansen procedure invoked in Saa b he vec command esimaes boh he parameers of he adjusmen process (he β coefficiens on he lagged changes in all variables) and he long-run coinegraing relaionships hemselves (he α coefficiens on he long-run relaionships) b maimum likelihood. We mus ell Saa wheher o include consan erms in he differenced VEC regressions remember ha a consan erm in a differenced equaion corresponds o a rend erm in he levels or perhaps rend erms (which would be a quadraic rend in he levels). I is also possible o include seasonal variables where appropriae or o impose consrains on he coefficiens of eiher he coinegraing relaionships or he adjusmen equaions. Once he VEC ssem has been esimaed, we can proceed o calculae IRFs and variance decomposiions, or o generae forecass as we would wih a VAR. References Johansen, Soren. 1995. Likelihood-Based Inference in Coinegraed Vecor Auoregressive Models. Oford: Clarendon Press. Sims, Chrisopher A. 198. Macroeconomics and Reali. Economerica 48 (1):1-48.