CENTRE FOR CENTRAL BANKING STUDIES

Transcription

1 CENTRE FOR CENTRAL BANKING STUDIES ECONOMIC MODELLING AND FORECASTING Recen developmens in srcrl VAR modelling y Hroon Mmz nd Ole Rmmel Cenre for Cenrl Bnking Sdies Bnk of Englnd Ferry 25 ole.rmmel@nkofenglnd.co.k Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

2 Conens Inrodcion 3 2 Preprion 3 3 Closed-economy esimion of monery policy shocks 4 4 Seing p monery policy VAR model 5 5 Esiming monery policy VAR model 5 6 VAR idenificion 3 6. Imposing shor-rn resricions 4 6 Genering implse response fncions nd forecs error vrince decomposiions Non-recrsive idenificion schemes Imposing long-rn resricions Imposing hisoricl decomposiion Imposing sign resricions 37 References nd frher reding 42 ole.rmmel@nkofenglnd.co.k 2 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

3 Inrodcion The im of his exercise is o esime smll- nd medim-sized vecor oregressions (VAR) for he US; idenify srcrl shocks (sch s monery policy shocks) y imposing pproprie shor-rn, long-rn nd sign resricions sing EViews; nd ssess he resls sing implse response fncions (IRFs), forecs error vrince decomposiions (FEVDs) nd hisoricl decomposiions. Or d re conined in he EViews workfile svr_s.wf, which incldes he following for d series: he nemploymen re (nre); he nominl M2 money spply mesre (m2); he consmer price index (cpi); nd he federl fnds rge re (ffr) Bsed on his d se, we consrced wo oher series: he (nnl) inflion re (inf, clcled s (log(cpi) log(cpi(-2)))*)) nd he yer-on-yer growh re of he nominl M2 money spply mesre (dm2, clcled s (log(m2) log(m2(-2))*)). The d re monhly nd spn he period from Jnry 947 (947M in EViews noion) o Decemer 28 (28M2), lhogh no ll for series re ville for sch long period. Unless we re confiden in ssming h he nderlying monery policy shocks re ros o differen monery policy regimes (money-spply rgeing, exchnge-re rgeing, inflion rgeing, ec.) nd chnges in regimes over ime, i is imporn o esime prmeers in srcrl vecor oregressions (SVARs) on single policy regime. Any regime shif herefore reqires differen prmeerision of he SVAR model. This imporn cve my explin some of he coneriniive resls we will enconer in he following exercises, in which VARs re esimed nd SVARs idenified over long ime periods. 2 Preprions To open he EViews workfile from wihin EViews, choose File, Open, EViews Workfile, selec svr_s.wf nd click on Open. Alernively, yo cn dole-click on he workfile icon oside of EViews, which will open EViews omiclly. Whenever we egin working wih new d se, i is lwys good ide o ke some ime o simply exmine he d, so he firs hing we will do is o plo he d o mke sre h i looks fine. This will help ensre h here were no miskes in he d iself or in he process of reding in he d. I lso provides s wih chnce o oserve he generl (ime-series) ehvior of he series we will e working wih. A plo of or d is shown in Figre. For exmple, mny srcrl VAR sdies of US monery policy leve o he disinflionry period from 979 o 984, which consies differen monery policy regime. ole.rmmel@nkofenglnd.co.k 3 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

4 Figre : US nemploymen re (nre), nominl M2 money spply (m2), consmer price index (cpi) nd federl fnds rge re (ff), Jnry 947 Decemer 28 2 UNRATE, M2 8, 8 6, 6 4, 4 2, CPI 2 FFR Closed-economy esimion of monery policy shocks Monery economics focses on he ehvior of prices, monery ggreges, nominl nd rel ineres res nd op. VARs hve served s primry ool in mch of he empiricl nlysis of he inerrelionship eween hose vriles nd for ncovering he impc of monery phenomen on he rel economy. A smmry of he empiricl findings in his lierre cn e fond in, iner li, Leeper e l. (996) nd Chrisino e l. (999). On he sis of exensive empiricl VAR nlysis, Chrisino e l. (999) derived sylised fcs o he effecs of conrcionry monery policy shocks. They conclded h plsile models of he monery rnsmission mechnism shold e consisen wih les he following evidence on prices, op nd ineres res. Following conrcionry monery shock (mening wh?): (i) he ggrege price level iniilly responds very lile; (ii) ineres res iniilly rise; nd (iii) ggrege op iniilly flls, wih J-shped response, nd zero long-rn effec of he monery policy shock (long-rn monery policy nerliy). In empiricl work, monery policy shocks re defined s deviions from he monery policy rle h re oined y considering n exogenos shock which does no ler he response of he monery policy-mker o mcroeconomic condiions. In oher words, he sndrd VAR pproch ddresses only he effecs of nniciped chnges in monery policy, no he rgly more imporn effecs of he sysemic porion of monery policy or he choice of monery policy rle. ole.rmmel@nkofenglnd.co.k 4 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

5 VAR models of he effecs of monery policy shocks hve herefore exclsively concenred on simling shocks, leving he sysemic componen of monery policy s formlised in he monery policy recion fncion or feedck rle nlered. In oher words, we re in essence crrying o hogh experimen involving n nniciped monery policy shock (defined s he residl of he monery policy recion fncion) wihin n exising monery policy rle. This exercise is disinc from h of chnging he monery policy rle iself. 4 Seing p monery policy VAR model A key considerion efore ny esimion cn e emped is he form he vriles ms hve when hey ener he VAR: shold hey ener in levels, gps or firs differences? The nswer is simply i depends. I depends on wh he VAR is going o e sed for. For forecsing prposes, we ms void poenil sprios regressions h my resl in sprios forecss; for he prpose of idenifying shocks (sch s monery policy shocks) we hve o e crefl o he siliy of he VAR (wheher i cn e invered o yield corresponding vecor moving-verge represenion), he reliiliy of or implse response fncions nd he sisicl properies of he residls. Q. Before we esime or model, wh shold we do o ensre nised esimes? Answer: We cn se OLS o esime he VAR, so if we re ineresed in sing he esimed model for forecsing, sy, we need o ensre h ll vriles re eiher sionry or coinegred o void he sprios regression prolem ssocied wih ni roos. For srcrl idenificion, on he oher hnd, we re ineresed in consisen coefficien esimes s well s he inerrelionships eween he vriles, so we follow Cnov (27, p. 25) nd To minimise pre-esing prolems, we recommend sring y ssming covrince sionriy nd devie from i only if he d overwhelmingly sgges he opposie. As demonsred y Sims e l. (99), consisen esimes of VAR coefficiens re oined even when ni roos re presen. Moreover, s shown y Tod nd Ymmoo (995) nd Doldo nd Lükepohl (996), if ll vriles in he VAR re eiher I() or I() nd if nll hypohesis is considered h does no resric elemens in ech of he prmeer mrices A i s (i =, 2,, p), he sl ess hve heir sndrd sympoic norml disriions. Moreover, if he VAR order p 2, he -rios hve heir sl sympoic sndrd norml disriions, which mens h hey remin sile sisics for esing he nll hypohesis h (single) coefficien in one of he prmeer mrices is zero (while leving he oher prmeer mrices nresriced). This llevies he sprios regression prolem on he se of sndrd sympoic norml disriions. In ligh of he resls in Sims e l. (99), poenil non-sionriy in he VAR nder invesigion shold no ffec he model selecion process. Moreover, mximm likelihood esimion procedres my e pplied o VAR fied o he levels even if he vriles hve ni roos; hence, possile coinegrion resricions re ignored. This is freqenly done in (S)VAR modelling o void imposing oo mny resricions, nd we follow his pproch here. 5 Esiming monery policy VAR model Hving hogh o seing p VAR model for he nlysis of monery policy, we cn now proceed o he esimion sge. Q2. Esime n nresriced VAR. We sr or esimion wih nre, inf, dm2 nd ffr, inclding lso consn. ole.rmmel@nkofenglnd.co.k 5 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

6 Answer: Use Qick, Esime VAR, ener nre, inf, dm2 nd ffr in h order ino he Endogenos Vriles ox nd leve EView s defl seing of 2 for he lg inervl. The smple period for esimion shold e 97M o 28M2. An eqivlen wy of geing EViews o do he esimion is o ype he following commnd in he commnd window: vr vr.ls 2 nre inf dm2 ffr This commnd specifies VAR wih he nme vr wih n iniilly rirry lg lengh of wo. Esiming VAR generes lo of op, so Tle shows he firs nd he ls wo enries of he fll EViews VAR(2) op only. Tle : Aridged esimion resls for he VAR(2) model in nre, inf, dm2 nd ffr, Jnry 97 Decemer 28 Vecor Aoregression Esimes Smple: 97M 28M2 Inclded oservions: 468 Sndrd errors in ( ) & -sisics in [ ] UNRATE INF DM2 FFR UNRATE(-) (.483) (.986) (.756) (.549) [ ] [ ] [.] [ ] FFR(-) (.34) (72) (985) (.4276) [ ] [ 2.766] [ ] [ 3392] FFR(-2) (.342) (725) (989) (.4282) [ ] [-29554] [ ] [ ] C (.3678) (.7467) (.89) (.734) [ ] [ ] [.36647] [.737] R-sqred Adj. R-sqred Sm sq. resids S.E. eqion F-sisic Log likelihood Akike AIC Schwrz SC Men dependen S.D. dependen Deerminn resid covrince (dof dj.). Deerminn resid covrince.3 Log likelihood Akike informion crierion Schwrz crierion Q3. We hve seleced n rirry lg lengh of 2, is h pproprie? ole.rmmel@nkofenglnd.co.k 6 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

7 Answer: Too shor lg lengh my resl in inconsisen esimes nd n iniliy o cpre imporn dynmics in he d, while oo mny lgs cn resl in imprecise esimes in smll nd modere smples. As sch, dding more lgs improves he fi redces he degrees of freedom nd increses he dnger of over-fiing. An ojecive wy o decide eween hese compeing ojecives is o mximise some weighed mesre of hese wo prmeers. This is how he Akike informion crierion (AIC), he Schwrz or Byesin crierion (SC) nd he Hnnn-Qinn crierion (HQ) work. These hree sisics re mesres of he rde-off of improved fi gins loss of degrees of freedom, so h he es lg lengh shold minimise ll of hese hree sisics. 2 An lernive o he informion crierion is o sysemiclly es for he significnce of ech lg sing likelihood-rio es (discssed in Lükepohl (25), secion 4.3). This is he pproch fvored y Sims (98), who lso sggesed modificion o he likelihood-rio es o ke ino ccon smll-smple is. We shold follow his recommendion in prcice EViews does s well! Since n nresriced VAR of lg lengh p ness he sme resriced VAR of lg lengh (p ), he log-likelihood difference mliplied y he nmer of oservions less he nmer of regressors in he VAR shold e disried s χ 2 -disriion wih degrees of freedom eql o he nmer of resricions in he sysem, s, i.e., χ 2 (s). In oher words: LR = (T m) {log p- log p } ~ χ 2 (s) where T is he nmer of oservions, m is he nmer of prmeers esimed per eqion (nder he lernive) nd log j is he logrihm of he deerminn of he vrince-covrince mrix of he VAR wih j lgs, where j =,, 2,, p. The djsed es hs he sme sympoic disriion s he sndrd likelihood-rio es h does no inclde he djsmen for m, is less likely o rejec he nll hypohesis in smll smples. For ech lg lengh, if here is no improvemen in he fi from he inclsion of he ls lg hen he difference in errors shold no e significnly differen from whie noise. Mke sre h yo se he sme smple period for he resriced nd he nresriced model, i.e., do no se he exr oservion h ecomes ville when yo shoren he lg lengh. 3 For or exmple, we se generl-o-specific mehodology (some hors specify specific-o-generl insed, which I wold srongly discorge): 4 (i) we sr wih high lg lengh (sy 8 lgs for monhly d); 5 (ii) for ech lg, sy p, noe he deerminn of he residl vrince-covrince mrix given in he EViews op nd hen noe he deerminn of he residl vrince-covrince mrix for VAR of lg (p ); (iii) ke he difference of he logs of he deerminns; nd (iv) his difference mliplied y (T m) shold e disried s χ 2 -disriion wih s degrees of freedom. 6 2 Some progrms mximise he negive of hese mesres. 3 EViews will do his omiclly. 4 An informive reference in his regrd is Lükepohl (27). 5 Oviosly, his will depend on he freqency of yor d, so h if yo hve nnl d yo cold sr wih 2-3 lgs, if yo hve qrerly d yo cold sr wih, sy, -2 lgs, nd if yo hve monhly d wih 8-24 lgs. 6 The degrees of freedom will depend on he nmer of vriles s well s he nmer of lgs in he VAR. The ol nmer of vriles in VAR is given y n( + np) = n + n 2 p, where n is he nmer of vriles nd p is he nmer of lgs. In or cse of hree-vrile VAR(5), esiming he model wih for lgs rher hn five mens h we hve hree less prmeers o esime per eqion. In ol, we will hve nine less prmeers in he sysem in generl, n 2. The nmer of resricions, s, in he sysem wold herefore e 3*3 = 9. ole.rmmel@nkofenglnd.co.k 7 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

8 I is imporn o noe h he residls from he esimed VAR shold e well ehved, h is, here shold e no prolems wih ocorrelion nd non-normliy. Ths, whils he AIC, SC or HQ my e good sring poins for deermining he lg-lengh of he VAR, i is imporn o check for ocorrelion nd non-normliy. If we find h here is ocorrelion for he chosen lg-lengh, one ogh o increse he lg-lengh of he VAR nil he prolem disppers. Similrly, if here re prolems wih non-normliy, sefl rick is o dd exogenos vriles o he VAR (hey my correc he prolem), inclding he se of dmmy vriles nd ime rends. A he sme ime, we shold noe h he specificion of he monery policy VAR nd is sisicl deqcy is n isse h hs no received mch explici enion in he lierre. In mos of he pplied ppers, he lg lengh is eiher decided niformly on n d hoc sis, severl differen lg lenghs re esimed for rosness or he lg lengh is simply se on he sis of informion crieri. Virlly none of he cdemic ppers cied in his exercise nderke ny rigoros ssessmen of he sisicl deqcy of he esimed models. Q4. Tes for he pproprie nmer of lgs sing he AIC, SC, HQ nd he LR informion crierion. Answer: In or VAR window, selec View, Lg Srcre, Lg Lengh Crieri, hen ener 8 in he mximm lg specificion. The redo shold e s in Tle 2. VAR Lg Order Selecion Crieri Endogenos vriles: UNRATE INF DM2 FFR Exogenos vriles: C Smple: 97M 28M2 Inclded oservions: 468 Tle 2: VAR lg order selecion crieri Lg LogL LR FPE AIC SC HQ NA * 2.572* e e-5*.7933* e * 7e e e * indices lg order seleced y he crierion LR: seqenil modified LR es sisic (ech es 5% level) FPE: Finl predicion error AIC: Akike informion crierion SC: Schwrz informion crierion HQ: Hnnn-Qinn informion crierion ole.rmmel@nkofenglnd.co.k 8 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

9 The LR, AIC, SC nd HQ informion crieri sgges 6, 4, 2 nd 2 lgs respecively hese re given y he serisks nex o vle of he selecion crieri ssocied wih ech lg lengh. The correc lg lengh will depend on he crieri or mesre we se. This is ypicl of hese ess nd reserchers ofen se he crierion mos convenien for heir needs. Noe h he AIC is inconsisen nd overesimes he re lg order wih posiive proiliy; h oh SC nd HQ re consisen. The SC crierion is generlly more conservive in erms of lg lengh hn he AIC crierion, i.e., i selecs shorer lg hn he oher crieri. Ivnov nd Kilin (25) show h while he choice of informion crierion depends on he freqency of he d nd ype of model, HQ is ypiclly more pproprie for qrerly nd monhly d. Q5. Does he chosen VAR hve pproprie properies? Are he residls sionry, norml nd no ocorreled? Is he VAR sle? A sefl ip is o sr wih he VAR wih he minimm nmer of lgs ccording o he informion crieri (in his cse 2 lgs) nd check wheher here re prolems wih ocorrelion nd normliy. Answer (ocorrelion): As we re lredy sing VAR wih wo lgs, here is no need for s o reesime he model. For ocorrelion, which is y fr he mos imporn prolem o recify, click on View, choose Residl Tess nd pick he Aocorrelion LM es. The op for he VAR(2) model wih welve lgs is given in Tle 3. Tle 3: VAR(2) residl ocorrelion LM ess p o lg order welve VAR Residl Seril Correlion LM Tess Nll Hypohesis: no seril correlion lg order h Smple: 97M 28M2 Inclded oservions: 468 Lgs LM-S Pro Pros from chi-sqre wih 6 df. There pper o e prolems wih significn ocorrelion ll wo lgs (lgs 5 nd 6) convenionl significnce levels. To ddress he prolem of ocorrelion, ry dding frher lgs o he VAR (hving 6 lgs in ol seems o ge rid of he prolem of residl ocorrelion lgs nd 2, which re he mos pressing). This is illsred in Tle 4. ole.rmmel@nkofenglnd.co.k 9 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

10 Tle 4: VAR(6) residl seril correlion LM ess p o lg order welve VAR Residl Seril Correlion LM Tess Nll Hypohesis: no seril correlion lg order h Smple: 97M 28M2 Inclded oservions: 468 Lgs LM-S Pro Pros from chi-sqre wih 6 df. Adding frher lgs mkes very lile discernle difference o he ove op. All in ll, i is exremely difficl o find VAR lengh h elimines seril correlion ll lg lenghs. For h reson, I rein he VAR(6) model, s i is he mos prsimonios model h shows no seril correlion shor lgs. Answer (normliy): Click on View, choose Residl Tess nd pick Normliy Tes. For he normliy es we ge he resls in Tle 5. Tle 5: VAR(6) residl normliy ess VAR Residl Normliy Tess Orhogonlizion: Cholesky (Lkepohl) Nll Hypohesis: residls re mlivrie norml Smple: 97M 28M2 Inclded oservions: 468 Componen Skewness Chi-sq df Pro Join Componen Krosis Chi-sq df Pro Join Componen Jrqe-Ber df Pro ole.rmmel@nkofenglnd.co.k Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

11 Join Ths his VAR does hve prolems wih non-normliy (why?). The reson for his is ovios: we noe h ll one of he for residl series hve prolems wih skewness, while ll for residl series hve prolems wih (excess) krosis, leding o he rejecion of join normliy for ll for residl series. The non-normliy of he residls resls from nmer of very lrge oliers, s cn e seen in Figre 2. Very ovios is he disinflionry policy period from 979 o 984 in he ffr residls. In ddiion, inflion (inf) shows wo very lrge (negive) residls owrds he end of he smple period. Indeed, looking he residl dignosics more closely, we cn see he inflence of lrge oliers in he componens, s well s he overll Jrqe-Ber ess, for ll for residl series. This is, nfornely, prolem. Alhogh normliy is no necessry condiion for he vlidiy of mny of he sisicl procedres reled o VAR nd SVAR models, deviions from he normliy ssmpion my neverheless indice h improvemens o he model re possile. 7 Figre 2: Esimed residl series from he VAR(6) model wih nre, inf, dm2 nd ffr.6 UNRATE Residls 2 INF Residls DM2 Residls FFR Residls Rher hn EViews defl seing of Cholesky of covrince (Lükepohl) s he Orhogonlision mehod, some hors prefer o se Sqre roo of correlion (Doornik- Hendry). This is ecse we ms choose fcorision of he residls for he mlivrie normliy es, sch h residls re orhogonl o ech oher. The pproch de o Doornik nd Hnsen (28) 7 One wy of deling wih his prolem wold e o inrodce dmmies o ccon for some of he lrger oliers. ole.rmmel@nkofenglnd.co.k Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

12 hs wo dvnges over he one in Lükepohl (99, p ). Firs, Lükepohl s es ses he inverse of he lower ringlr Cholesky fcor of he residl covrince mrix, resling in es which is no invrin o re-ordering of he dependen vriles. Second, Doornik nd Hnsen perform smll-smple correcion o he rnsformed residls efore comping heir sisics. We shold noe h he finding of non-normliy is ros o he orhogonlision mehod, hogh. Answer (siliy): If he VAR is no sle, cerin resls (sch s implse response sndrd errors) re no vlid. Mos impornly, if he VAR is no sle, we will no e le o genere he vecor moving-verge (VMA) represenion from he VAR. In doing his procedre, here will e (n p ) = 24 roos overll, where n is he nmer of endogenos vriles (for) nd p is he priclr lg lengh (six). I is esy o check for siliy in EViews. Go o View, Lg Srcre nd click on AR Roos Tle. Yo shold ge he resls in Tle 6. Tle 6: Roos of he chrcerisic polynomil of he VAR(6) Roos of Chrcerisic Polynomil Endogenos vriles: UNRATE INF DM2 FFR Exogenos vriles: C Lg specificion: 6 Roo Modls i i i i i i i i i i i i i i i i i i i i.4799 No roo lies oside he ni circle. VAR sisfies he siliy condiion. On he pls side, we re finlly geing some posiive resls. The VAR is sle s none of he roos lie oside he ni circle: ll he modli of he roos of he chrcerisic polynomil re less hn one in mgnide. In cse one or more roos fll oside he ni circle, dding ime rend in EViews) s n exogenos vrile cn help he siliy of he VAR. Noe h dding ime rend will no lwys correc insiliy. We lso noe h few of he roos of he chrcerisic polynomil of he VAR exceed.9 in mgnide. We will rern o his poin ler on, he min rgmen is h if he chrcerisic roos ole.rmmel@nkofenglnd.co.k 2 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

13 re close o one, i will e dofl if he nlyic (sympoic) confidence inervls h EViews prodces will sill e ccre. If one or more of he chrcerisic roos exceed.9 in mgnide, we my wn o consider oosrpping he confidence inervls. Finlly, i is worh spending i of ime invesiging he individl coefficien esimes of he VAR(6) we hve js esimed. Or smll- o medim-sized for-vrile VAR wih six lgs nd one deerminisic regressor (he consn), which hs een esimed on 468 d poins, my lredy e overprmeerised. Using he empiricl resls on he viiliy of he sl sympoic disriion of es sisics y Tod nd Ymmoo (995) nd Doldo nd Lükepohl (996), we noe h js 8 of he esimed prmeers re sisiclly significn he 5 per cen level (sing he enchmrk of he -sisic eing lrger hn.96 in sole mgnide). 6 VAR idenificion When Sims (98) firs dvoced he se of VAR in economics, i ws in response o he previling orhodoxy he ime h ll economic models shold e srcrl models, i.e., h hey shold inclde idenifying resricions. Insed, he rged for he se of n nresriced VAR wih no disincion eing mde in he model eween endogenos nd exogenos vriles. The im ws o free-p economeric modelling from he consrins pplied y economic heory nd, in effec, o le he d spek. 8 We re now redy o emp idenificion of he srcrl VAR (SVAR) nd, y so doing, emp o idenify monery policy shocks. In priclr, EViews ssmes n nderlying srcrl eqion of he form: Ay = C(L)y + B () where he srcrl shocks re normlly disried, i.e., N(, Σ), where Σ is generlly ssmed o e digonl mrix, slly he ideniy mrix, sch h ~ N(, I n ). 9 Unfornely, we cnno esime his eqion direcly de o idenificion isses. Insed, we esime n nresriced VAR of he form: y = A - C(L)y + A - B = H(L)y + ε (2) For resons olined in he presenion, he mrices A, B nd he C i s (i =, 2,, p) re no seprely oservle from he esimed H i s nd he vrince-covrince mrix, E(ε ε ) = Ω, of he redced-form shocks, ε. So how cn we recover eqion () from eqion (2)? The solion is o impose resricions on or VAR o idenify n nderlying srcre wh kind of resricions re hese? Economic heory cn someimes ell s somehing o he srcre of he sysem we wish o esime. As economiss, we ms conver hese srcrl or heoreicl ssmpions ino fesile resricions on he VAR. Sch resricions cn inclde, for exmple: csl (recrsive) ordering of shock propgion, e.g., he Cholesky decomposiion; he fc h nominl vriles hve no long-rn effec on rel vriles; 8 B we hve lredy come cross les wo necessry resricions: (i) we need o choose he vriles h go ino he model nd (ii) we hve o choose finie lg lengh for he VAR. 9 For resons hving o do wih he so-clled AB model elow, I se differen noion from he presenion. ole.rmmel@nkofenglnd.co.k 3 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

14 he long-rn ehvior of vriles, e.g., he rel exchnge re is consn in he long rn; nd he fc h we cn derive heoreicl resricions on he signs of he implse responses resling from priclr srcrl shocks There re mny ypes of resricions h cn e sed o idenify srcrl VAR. EViews llows yo o impose differen ypes of resricions. One ype imposes resricions on he shor-rn ehvior of he sysem, wheres noher ype imposes resricions on he long-rn. Using he svrperns dd-in for js-idenified SVAR models, EViews llow oh ypes of resricions o e imposed he sme ime, s hs een done in Bjørnlnd nd Leiemo (29), for exmple. The dd-in llows yo o impose oh shor- nd long-rn resricions o oin non-recrsive orhogonlision of he error erms (s opposed o he recrsive Cholesky decomposiion) for implse response nlysis h wold mke more sense from mcroeconomic/srcrl poin of view. In order o se he dd-in, yo shold firs esime reglr VAR model. Afer h, yo cn eiher spply he nme of yor model or he covrince mrix. The op will e fcor mrix, which cn e sed frher in genering implse responses (i.e., s ser-specified implse definiion). Sring wih EViews 7, we cn lso impose sign resricions o idenify srcrl VARs, s we will demonsre frher elow. 6. Imposing shor-rn resricions To impose shor rn resricions in EViews, we se eqion (2): y = A - C(L)y + A - B We esime he rndom sochsic residl, A - B, from he residl, ε, of he esimed VAR. Compring he residls from eqions () nd (2), we find h: or, eqivlenly, h: ε = A - B (3) Aε = B (3 ) In reqiring h resricion or idenifying schemes ms e of he form given y (3 ) ove, EViews follows wh is known s he AB model, which is exensively descried in Amisno nd Ginnini (997). By imposing srcre on he mrices A nd B, we impose resricions on he srcrl VAR in eqion (). Reformling eqion (3), we hve ε ε = A - B B (A - ), nd, since E( ) = I n (he ideniy mrix) y ssmpion, we hve: E(ε ε ) = E(A - B B (A - ) ) = A - B E( ) B (A - ) = A - BB (A - ) = Ω (4) B cn we idenify ll he elemens in A nd B from Ω? Eqion (4) sys h for he n vriles in y, he symmery propery of he vrince-covrince mrix E(ε ε ) = Ω imposes n(n + )/2 (ideniy) resricions on he 2n 2 nknown elemens in A nd B. Ths, n ddiionl 2n 2 n(n + )/2 = (3n 2 n)/2 resricions ms e imposed. In or cse, nd sing he ove AB model, we hve VAR wih for endogenos vriles, reqiring (3*4*4 4)/2 = resricions. An exmple of his specificion sing he Cholesky decomposiion idenificion scheme is: ole.rmmel@nkofenglnd.co.k 4 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

15 A = , B = (5) where he A mrix y din of eing lower ringlr hs recrsive srcre. Coning resricions in he A nd B mrices ove, we hve 8 zero resricions (six in mrix A nd welve in mrix B) s well s noher for normlision resricions on he digonl of mrix A, giving s he reqired ol of resricions. In EViews, hese resricions cn e imposed in eiher mrix form or in ex form. Since imposing resricions in mrix form is relively srighforwrd, we will sr off y illsring he ex form. Q6. Impose he Cholesky decomposiion, which ssmes h shocks or innovions re propged in he order of nre, inf, dm2 nd ffr. As we will discss elow, he Cholesky decomposiion cn e inerpreed s recrsive conemporneos srcrl model. Answer: To impose he resricion ove in ex form, we selec Proc nd Esime Srcrl Fcorizion from he VAR window men. In he SVAR opions dilog, selec Tex (or Mrix s pproprie). Ech endogenos vrile hs n ssocied vrile nmer, in or exmple his for he nre for he inf for he dm2 residls; for he ffr residls The idenifying resricions re imposed in erms of he ε s, which re he residls from he redced-form VAR esimes, nd he s, which re he srcrl, fndmenl or primiive rndom (sochsic) errors in he srcrl sysem. Ener he following in he ex ox (i is esies o simply copy he sggesed shor-rn fcorision exmple from he op of he SVAR opions ox ino he whie Idenifying Resricions ox he = = C(2)*@e + = C(4)*@e + C(5)*@e2 + = C(7)*@e + C(8)*@e2 + C(9)*@e3 + C()*@4 The wy o inerpre hese resricions is h hey represen he enries in he A - B mrix linking ε nd vi eqion (5), i.e., ε = A - B. We mke se of he fc h he inverse of lower (pper) ringlr mrix is lso lower (pper) ringlr mrix. We cn ke closer look he nderlying mrix lger h resls in he se of EViews resricions. Wriing o eqion (5) nd he idenificion resricions (6) in fll mrix form, we hve: Aε = B Resricions sing ex form re lso more flexile, s we cn resric vles o e eql. We shold noe, however, h mos economic heories do no imply recrsive conemporneos sysems. ole.rmmel@nkofenglnd.co.k 5 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

16 6 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided ) ( ) ( ) ( ) ( ) ( ) ( We cn see h ε =, which we cn ssie ino he remining hree expressions for ε 2, ε 3 nd ε 4. This resls in n expression for ε 2 (= 2 ε + 2 ), which we cn solve for 2 nd ssie ino he eqions for ε 3 nd ε 4. Afer lile lger, we oin he following for eqions in he for nknowns ε, ε 2, ε 3 nd ε 4 : ε = ε 2 = 2 ε + 2 ε 3 = 3 ε ε ε 4 = 4 ε ε 2 34 ε which yo cn compre wih he for EViews resricions ove. The correspondence eween he esimed residls, ε (denoed in EViews), nd he srcrl shocks, (denoed in EViews) shold e ovios, s shold he correspondence eween c() (= ), c(2) (= 2 ), c(3) (= ), c(4) (= 3 ), c(5) (= ), c(6) (= 33 ), c(7) (= - 4 ), c(8) (= - ), c(9) (= - 34 ) nd c() (= 44 ). The op fer imposing he resricions on he VAR wih six lgs nd consn sing he ex form is given in Tle 7.

17 Tle 7: Js-idenified srcrl VAR(6) esimes, Jnry 97 Decemer 28 (ex form) Srcrl VAR Esimes Smple: 97M 28M2 Inclded oservions: 468 Esimion mehod: mehod of scoring (nlyic derivives) Convergence chieved fer 8 ierions Srcrl VAR is js-idenified Model: Ae = B where E[']=I Resricion Type: shor-rn ex = = C(2)*@e + = C(4)*@e + C(5)*@e2 + = C(7)*@e + C(8)*@e2 + C(9)*@e3 + C()*@4 represens UNRATE represens INF represens DM2 represens FFR residls Coefficien Sd. Error z-sisic Pro. C(2) C(4) C(5) C(7) C(8) C(9) C() C(3) C(6) C() Log likelihood Esimed A mrix: Esimed B mrix: Srcrl VAR proponens ry o void overidenifying he VAR srcre nd propose js enogh resricions o idenify he prmeers niqely, which is wh we hve js done. Noe h he EViews op explicily menions fc h he srcrl VAR is js idenified. Accordingly, mos SVAR models re js idenified. I is lwys good ide o consider recrsive solion firs, which cn serve s enchmrk for ler nlysis. We shold hen sk if here is nyhing nresonle o he recrsive solion, which cn e done y looking he implse response fncions, sy. A h sge, we cn hink o how he sysem shold e modified. The lernive pproch o inping idenifying resricions wold e o se he mrix form resricions. Under his pproch, yo wold cree wo mrices wih he following enries: ole.rmmel@nkofenglnd.co.k 7 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

18 A = NA NA NA NA NA NA NA, B = NA NA NA (6) Mrices re creed y going o Ojec, New Ojec in he workfile window nd selecing Mrix-Vecor-Coef from he lis of possiiliies. Mke sre o give i n pproprie nme I hve clled hem mrix_ nd mrix_. Dimension he mrix s reqired (in or cse, we hve for colmns nd for rows). Once he mrix comes p (ll enries will e zeros), selec Edi +/- o ccess he individl cells in he mrix yo hve creed. We cn hen ener he individl elemens of he firs mrix s shown ove, consising of ones, zeros nd NA s. Nmericl vles, sch s nd, se he respecive elemens of he mrix excly eql o h vle, while NA s ell EViews h hese re elemns of he mrix o e esimed. When yo re done, click on Edi +/- gin, nd hen close he window. Repe he exercise for he second mrix. Once we hve creed he wo mrices hey will pper in he lis of vriles in he workfile. Rern o he esimed VAR model, selec he Mrix specificion in he Srcrl Fcorision descried ove, selec Shor-rn pern nd ener he nmes of he mrices for A nd B s pproprie. Excep for he iniil chnge of sign of he esimed coefficiens c() o c() (which is no crried over o he finl represenion of he A nd B mrices in he oom pr of Tle 8), he resling op in Tle 8 shold e idenicl o he one oined sing he ex version in Tle 7 ove. Tle 8: Js-idenified srcrl VAR(6) esimes, Jnry 97 Decemer 28 (mrix form) Srcrl VAR Esimes Smple: 97M 28M2 Inclded oservions: 468 Esimion mehod: mehod of scoring (nlyic derivives) Convergence chieved fer 7 ierions Srcrl VAR is js-idenified Model: Ae = B where E[']=I Resricion Type: shor-rn pern mrix A = C() C(2) C(4) C(3) C(5) C(6) B = C(7) C(8) C(9) C() Coefficien Sd. Error z-sisic Pro. C() C(2) C(3) C(4) C(5) C(6) C(7) C(8) ole.rmmel@nkofenglnd.co.k 8 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

19 C(9) C() Log likelihood Esimed A mrix: Esimed B mrix: Genering implse response fncions nd forecs error vrince decomposiions Two sefl ops from VARs re he implse response fncion (IRF) nd he forecs error vrince decomposiion (FEVD). Implse responses show how he differen vriles in he sysem respond o (idenified) shocks, i.e., hey show he dynmic inercions eween he endogenos vriles in he VAR(p) process. Since we hve idenified he srcrl VAR, he implse responses will e depicing he responses o he srcrl shocks h hve n economic inerpreion. In oher words, once he srcrl model hs een idenified nd esimed, he effecs of he srcrl shocks,, cn e invesiged sing n implse response nlysis. The ler provides informion on he dynmics of he VAR sysem of eqions nd how ech vrile responds nd inercs o shocks in he oher vriles in he sysem. We do his ecse he resls of he implse response nlysis re ofen more informive hn he prmeer esimes of he (S)VAR coefficiens hemselves. The sme is re for forecs error vrince decomposiions, which re lso poplr ools for inerpreing VAR models. While implse response fncions rce he effec of shock o one endogenos vrile ono he oher vriles in he VAR, forecs error vrince decomposiions (or vrince decomposiions in shor) sepre he vriion in n endogenos vrile ino he conriions explined y he componen shocks in he VAR. In oher words, he vrince decomposiion ells s he proporion of he movemens in vrile de o is own shock verss shocks o he oher vriles. Ths, he vrince decomposiion provides informion o he relive impornce of ech (srcrl) shock in ffecing he vriles in he VAR. In mch empiricl work, i is ypicl for vrile o explin lmos ll of is own forecs error vrince shor horizons nd smller proporions longer horizons. Sch delyed effec of he oher endogenos vriles is no nexpeced, s he effecs from he oher vriles re propged hrogh he redced-form VAR wih lgs. Q7. Genere implse response fncions sing he Cholesky decomposiion following shock o ffr. Answer: Selec View nd Implse Response, which opens he Implse Responses men. 2 On he Disply, selec 3 periods, Mliple Grphs nd Anlyic (sympoic) for Response Sndrd Errors. Yo shold ener he vriles for which yo wish o genere innovions (Implses) nd he vriles for which yo wish o oserve he responses (Responses). Yo my eiher ener he nme of he endogenos vriles or he nmers corresponding o he ordering of he vriles. For exmple, for he for vriles in or VAR (nre, inf, dm2 nd ffr), yo my eiher ype: 2 Alernively, click on he Implse on he op of he VAR ox. ole.rmmel@nkofenglnd.co.k 9 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

20 nre inf dm2 ffr if yo wned o evle h priclr ordering of he vriles or even simply: Noe h he for nmers s enered ove chnge he ordering of he vriles, s wold correspond o nre dm2 ffr inf. B he order in which yo ener hese vriles only ffecs he disply of he resls nd nohing else. Type ffr (or 4) in he Implses ox nd leve he for vriles s hey re in he Responses ox (which shold correspond o heir originl order in he esimed VAR). This opion will show he implse response of ech vrile o srcrl shock o ffr (or 4 ). Rememer wh his ordering of he for vriles implies: he monery policy shock does no ffec he oher hree vriles conemporneosly. The wo sndrd error nds of he implse response fncions re sed on nlyicl (or sympoic, i.e., lrge-smple) resls. In smll smples, i migh e es o oosrp he sndrd error nds, which cn e esily done in EViews. We will do so in mine. By clicking on he Implse Definiion, yo will find h he ox Cholesky dof djsed is lredy chosen for yo his is EViews defl opion. Chnge his opion o Srcrl Decomposiion. We need o do his s we hve js chieved idenificion y sing eiher he ex or he mrix form. Oher implse definiions cn e chosen y selecing ny of he oher opions (s we will do ler on). Then click on OK, which shold ring p Figre 3, consising of he following for chrs of implse response fncions. 3 Figre 3: Implse response fncions of nre, inf, dm2 nd ffr o one sndrd-deviion shock in ffr (srcrl decomposiion) Response o Srcrl One S.D. Innovions ± 2 S.E. Response of UNRATE o Shock4 Response of INF o Shock Response of DM2 o Shock4 Response of FFR o Shock Noe h I hve djsed he xes of ech chr o mke hem conform wih he ler resls wih oosrpped sndrd error nds. ole.rmmel@nkofenglnd.co.k 2 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

21 In response o posiive one-sndrd deviion srcrl shock o ffr, nemploymen firs flls for for periods efore incresing herefer, inflion increses nd is posiive for some 2 periods (his is mnifesion of he so-clled inflion or price pzzle we wold no expec inflion o increse when we increse he monery policy insrmen) nd he response of he money-spply growh re is negive over he enire period. Once we dd he pls nd mins wo sndrd error nds, we cn see how significn hese effecs re. The posiive response of he nemploymen re occrs fer some 5 monhs, inflion shows no significn response o ffr over ll 3 periods, he negive response of dm2 lss for some five periods nd he shock of ffr o iself persiss from some 2 periods. 4 If we limi he nlysis o he convenionl Cholesky idenificion scheme, EViews llows yo o plo he implse responses of he Cholesky decomposiion wiho hving done h decomposiion in he firs plce. Selec View nd Implse Response, which opens he Implse Responses men. 5 On he Disply, selec 3 periods nd Mliple Grphs. In order o illsre he oosrpping pproch o clcling sndrd error nds, selec Mone Crlo wih, repeiions for Response Sndrd Errors. This opion shows he implse response of ech vrile o shocks o he nderlying fndmenl shocks ( s). The wo sndrd error nds of he implse response fncions re sed on, Mone Crlo simlions. All he enries for he vriles for which yo wish o genere innovions (Implses) nd he vriles for which yo wish o oserve he responses (Responses) shold e pproprie, so yo do no hve o chnge nyhing. This opion will show he implse response of ech vrile o srcrl shock o ffr (or 4 ). By clicking on he Implse Definiion, yo will find h he ox Cholesky dof djsed my lredy e chosen for yo his is EViews defl opion. If no, plese selec i. Resls of his srcrl fcorision re given in Figre 4 elow. As we hve crried o he sme srcrl idenificion scheme in wo differen wys, he men implse responses in Figre 3 shold e excly he sme s he men implse responses in Figre 3. 4 Some of he resls my e de o he choice s well s nmer of vriles in he VAR. Tking inspirion from Sims (992), we ogh o inclde forwrd-looking vrile, sch s he exchnge re or commodiy price index, ino he VAR. Sch forwrd-looking sse price cn is hogh o conin inflionry expecions. This shold enlrge he informion se of he monery-policy mker nd llevie he price pzzle. In fc, mos indsril-conry enchmrk VAR models of monery policy nowdys conin six or seven vriles. While some hors hve improved heir resls y dding vriles o he VAR, nless hose vriles re pr of he heoreicl model he resercher hs in mind, i is no cler on wh gronds hey re seleced, oher hn he fc h hey work. 5 Alernively, click on he Implse on he op of he VAR ox. ole.rmmel@nkofenglnd.co.k 2 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

22 Figre 4: Implse response fncions of nre, inf, dm2 nd ffr o one sndrd-deviion shock in ffr (Cholesky decomposiion) Response o Cholesky One S.D. Innovions ± 2 S.E. Response of UNRATE o FFR Response of INF o FFR Response of DM2 o FFR Response of FFR o FFR In ligh of Chrisino e l. s (999) hree sylised fcs o he effecs of conrcionry monery policy shocks, le s nlyse he impc of (posiive) shock o he shor-erm ineres re on he vriles in he VAR. The shor-erm ineres re oviosly increses s resl of one-ime posiive shock o iself (he increse is eql o.5, vle we hve come cross efore s c()!), he effec of he monery shock dies down over ime. Afer some 2 periods, he (oneime) increse in he shor-erm ineres re is no longer sisiclly significn. Unemploymen shows J-shped response, nd he iniil negive effec is no sisiclly significn. Afer some 5 periods, nemploymen shows sisiclly significn posiive response, i.e., nemploymen rises wih lg in response o (one-ime) increse of he monery policy re. 6 The growh re of M2 is negive for for periods. Wheher his corresponds o Chrisino e l. s (999) firs sylised fc of he price level iniilly responding very lile is open o dee. Q8. Genere forecs error vrince decomposiions for he idenificion scheme in or exmple. Answer: Selec View nd Vrince Decomposiion s well s he Tle opion, no sndrd errors nd 2 periods. The following le gives he vrince decomposiion of he hree vriles in he VAR o he idenified srcrl shocks ( for-qrer inervls o conserve spce). Agin, EViews llows yo o clcle he forecs error vrince decomposiion sing he Cholesky decomposiion wiho hving done h decomposiion in he firs plce. Vrince decomposiions in Tle 9 re given 6 Rnning he model for longer shows h his posiive response is emporry. The nemploymen re ecomes insignificnly differen from zero fer 37 periods. ole.rmmel@nkofenglnd.co.k Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

23 wiho sndrd errors, lhogh yo cn esily se sympoic sndrd errors or oosrp hem sing he Mone Crlo opion. Tle 9: Forecs error vrince decomposiions Vrince Decomposiion of UNRATE: Period S.E. UNRATE INF DM2 FFR Vrince Decomposiion of INF: Period S.E. UNRATE INF DM2 FFR Vrince Decomposiion of DM2: Period S.E. UNRATE INF DM2 FFR Vrince Decomposiion of FFR: Period S.E. UNRATE INF DM2 FFR Cholesky Ordering: UNRATE INF DM2 FFR Of some ineres is he effec of nominl on rel vriles, s sndrd resl of he SVAR lierre is h he monery policy shock explins relively smll frcion of he forecs error of rel civiy mesres or inflion. As sch, inflion, money-spply growh nd he shor-erm ineres re ogeher predic only smll percenge of he vrince of nemploymen, eql o some 2 per cen fer 2 periods. Noe how he vrince decomposiion of nre de o shock o iself is sill eql o some 8 per cen he end of he period. The picre is qie differen for inflion. The percenge of vrince of inflion explined y ffr never goes ove per cen. The sme is re for dm2. On he oher hnd, he percenge of vrince explined y nre increses qie qickly from slighly less hn per cen in he firs period o some 4 per cen in period 2. Finlly, even fer 2 periods, inflion sill explins he mjoriy (58 per cen) of is own vriion. ole.rmmel@nkofenglnd.co.k 23 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

24 The oher wo vrince decomposiions for dm2 nd ffr shold e ssessed long similr lines. 6.3 Non-recrsive idenificion schemes Recll h he min reqiremen of idenificion is o ensre h we cn niqely recover ll he prmeers in he A nd B mrices from he vrince-covrince mrix Ω of he esimed residls. We do his y imposing he necessry (3n 2 n)/2 ddiionl resricions. Recll from he presenion h i is only he overll nmer of idenifying resricions h mers; here is herefore nohing h necessrily reqires idenificion resricions o follow he Wold csl chin, i.e., recrsive srcre. This my i oo cvlier, hogh, s he following exmples show. We cn herefore impose idenifying srcres differen from he recrsive one. In he cse of he ove for-vrile VAR (nre, inf, dm2, ffr), we cold posle he following non-recrsive sysem for illsrive prposes only. In oher words, I hve spen no ime ll rying o come p wih n economic rionle for his priclr non-recrsive idenificion scheme. The shor-rn mrix for he srcrl fcorision is given y mrix nr: mrix nr = NA NA NA NA NA NA NA, mrix_2 = NA NA NA (7) Resls for his non-recrsive idenificion scheme re given in Tle elow. Tle : Non-recrsively idenified srcrl VAR(6) esimes, Jnry 97 Decemer 28 (mrix form) Srcrl VAR Esimes Smple: 97M 28M2 Inclded oservions: 468 Esimion mehod: mehod of scoring (nlyic derivives) Convergence chieved fer 7 ierions Srcrl VAR is js-idenified Model: Ae = B where E[']=I Resricion Type: shor-rn pern mrix A = C() C(5) C(2) C(3) C(4) C(6) B = C(7) C(8) C(9) C() Coefficien Sd. Error z-sisic Pro. C() C(2) C(3) C(4) C(5) C(6) C(7) ole.rmmel@nkofenglnd.co.k 24 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

25 C(8) C(9) C() Log likelihood Esimed A mrix: Esimed B mrix: While Tle 8 shows he resls of srcrl idenificion sing recrsive order nd he Cholesky decomposiion, Tle shows he resls of srcrl orhogonlision sing nonrecrsive idenificion scheme. How do we know which one is eer? The simple nswer is h we cnno choose eween hem. We noe h he mximised vle of he log-likelihood for oh models is , i.e., hey re oservionlly eqivlen. In oher words, we cnno choose eween hem sed on he fi of he d lone, nd some oher crierion is reqired. In mny cses, his will e he resonleness of he implse response fncions. (Compring he implse response fncions eween he wo idenificion schemes is lef s n opionl exercise.) On he oher hnd, he non-recrsive idenificion scheme p forwrd elow does hve some gronding in economic heory. Assme h d on he price level nd he nemploymen re emerge only wih lg nd h he monery policymker does no respond o chnges in inflion nd nemploymen wihin he period. In ddiion, inflion nd nemploymen do no respond wihin he period o chnges in money spply growh nd he monery policy re. In he cse of he ove forvrile VAR (nre, inf, dm2, ffr), we cold posle he following non-recrsive sysem: (8) The firs relionship in mrix (8) shows h he srcrl shocks o nemploymen re eqivlen o he esimed redced-form shocks, or h srcrl shocks re onomos of he oher vriles in he sysem. If we regrd nemploymen s rogh monhly proxy of (rel) op, he hird relionship in sysem (8) conins conemporneosly ll for of he rdiionl rgmens of liqidiy preference in n IS-LM model. For h reson, i cold e lelled money demnd eqion. Similrly, he finl relionship in mrix (8), which incldes he growh re of he money spply nd he shor-erm ineres re, cold e regrded s money spply eqion. Q9. Idenify SVAR wih his idenificion sysem. Aε = B ole.rmmel@nkofenglnd.co.k 25 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

26 3 2 ( 2 ( 34 ( 3 2 ( 34 ( 2 2 ( 34) ( 3 ( ) ( ) ) ) 2 ) 2 3 ) ) 34 ( 2 3 ( 34 ( ) ) ( 34 ( ) 2 ) ) 34 ( 34) ( ) 2 34 ) ( ( 34) 33 ( ) ( ) ) Answer: Algericlly, he mrix conins one more hn he reqired n(n )/2 = (4)(3)/2 = 6 zero resricions reqired for idenificion. We hs hve one over-idenifying resricion. These resricions re mos esily imposed sing he mrix form. We herefore need o cree he wo mrices A nd B y hnd s descried ove. 7 Once yo hve creed he wo new mrices, go ck o he VAR window, selec Proc, Esime Srcrl Fcorision..., selec Mrix, Shor-rn pern nd ener he wo mrices where pproprie. Click on OK o esime he elemens of he wo mrices. While only hree of he five elemens of he A mrix re sisiclly significn he 5 per cen level of significnce, he single over-idenifying resricion cnno e rejeced, s shown in Tle I hve clled hem mrix nr2 nd mrix_2 in he workfile. ole.rmmel@nkofenglnd.co.k 26 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

27 Tle : Non-recrsively idenified srcrl VAR(6) esimes, Jnry 97 Decemer 28 (mrix form) Srcrl VAR Esimes Smple: 97M 28M2 Inclded oservions: 468 Esimion mehod: mehod of scoring (nlyic derivives) Convergence chieved fer 7 ierions Srcrl VAR is over-idenified ( degrees of freedom) Model: Ae = B where E[']=I Resricion Type: shor-rn pern mrix A = C() C(2) C(3) C(5) C(4) B = C(6) C(7) C(8) C(9) Coefficien Sd. Error z-sisic Pro. C() C(2) C(3) C(4) C(5) C(6) C(7) C(8) C(9) Log likelihood LR es for over-idenificion: Chi-sqre() Proiliy.47 Esimed A mrix: Esimed B mrix: Q. How do he resls for he implse responses compre o Cholesky decomposiion? Answer: Click on Implse o ring p he Implse Responses window. On he Disply, selec 3 periods, Mliple Grphs nd Anlyic (sympoic) for Response Sndrd Errors. Yo shold ener he vriles for which yo wish o genere innovions (Implses) nd he vriles for which yo wish o oserve he responses (Responses). Type ffr (or 4) in he Implses ox nd leve he for vriles s hey re in he Responses ox. By clicking on he Implse Definiion, yo will find h he ox Cholesky dof djsed is lredy chosen for yo his is EViews defl opion. Chnge his opion o Srcrl Decomposiion. We need o do his s we hve js chieved ole.rmmel@nkofenglnd.co.k 27 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

28 idenificion y sing eiher he ex or he mrix form sed on non-recrsive, i.e., non-cholesky decomposiion. Oher implse definiions cn e chosen y selecing ny of he oher opions (s we will do ler on). Then click on OK, which shold ring p Figre 5, consising of he following for chrs of implse response fncions. Figre 5: Implse response fncions of nre, inf, dm2 nd ffr o one sndrd-deviion shock in ffr (non-recrsive decomposiion) Response o Srcrl One S.D. Innovions ± 2 S.E. Response of UNRATE o Shock4 Response of INF o Shock Response of DM2 o Shock4 Response of FFR o Shock Very rodly speking, he shpes of he new implse responses re comprle o hose in Figres 3 nd 4. B some differences emerge. To egin wih, nre responds wih lg o he shock in ffr, he response is very shor-lived (eween periods nd 2). Inflion gin increses, wih shor-lived significn increse p o period 4. The money-spply growh re shows mch lrger (more negive) nd significn response over he enire period nder oservion. Finlly, he shock of ffr o iself is now mch shorer-lived: he lower sndrd error nd crosses he zero line fer wo periods, mening he implse response fncion is no sisiclly differen from zero fer h poin. Le me rern o he erlier poin h i is only he overll nmer of idenifying resricions h mers; which migh e rher creless or excessively loose. Consider he following non-recrsive idenificion scheme: (9) ole.rmmel@nkofenglnd.co.k 28 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

29 Algericlly, sysem (9) gin conins he reqired nmer (6) of zero resricions reqired for idenificion. This sysem implies h for he firs wo eqions: ε = ε 2 = - 23 ε 3 24 ε whence i follows h he covrince eween ε nd ε 2 is eql o zero (cov(ε, ε 2 ) = ). The recrsive sysem discssed ove essenilly deermines he six nknown prmeers sing he six elemens in he vrince-covrince mrix of ε. B in he lernive A mrix in eqion (9), one elemen of he vrince-covrince mrix is eql o zero. i.e., i conins no informion o he prmeers. This leves s wih js five eqions he non-zero elemens in cov(ε ) o find six prmeers. As will e ppren when yo sk EViews o crry o he srcrl fcorision sing he shor-rn mrices mrix nr2 (which needs o e creed from eqion (9) ove) nd mrix_, his cnno e done nd EViews will e nle o provide n esime. In smmry, while here re some dvnges o ndoning he recrsive ssmpion, here is lso ssnil cos: roder se of economic relions ms e idenified. Moreover, even sensile economic models specified y non-recrsive shor-rn mrix A my no e esimle. The ler fc is no de o ny economic heory resons, o he violion of simple compionl consrins. 6.4 Imposing long-rn resricions A second ype of resricion h cn e imposed in EViews is long-rn resricion. This ype of resricion ws inrodced y Blnchrd nd Qh (989). Their resricion scheme is o consider he vecor moving-verge represenion nd hs is implse responses in deil, concenring on he impc h shocks hve on he long-rn of vriles. Since he long-rn is considered, he vriles h ener he VAR hve o e sionry. Noe h if some of he vriles re I(), hen i is possile, if oher vriles re I(), o decompose he I() vrile ino wo componens: permnen nd rnsiory componen. Ths, he Blnchrd nd Qh decomposiion is n lernive form of condcing Beveridge nd Nelson (98) decomposiions. Long-rn resricions re no ofen sed o idenify monery policy shocks wih he ype of vriles h we hve. Consider or srcrl VAR from eqion (): Rerrngemen of his eqion yields: y = A - C(L)y + A - B y = (I A - C(L)) - A - B () Eqion () shows how he rndom (sochsic) shocks ffec he long-rn levels of he vriles. If we define mrix M = (I A - C(L)) - A - B, he ggrege effec of shock is given y mrix M. Hence, if we ssme h he (long-rn) cmlive effec of s-shock i on vrile y j is zero, hen he colmn i nd row j elemen of mrix M shold e zero. For exmple, sppose yo hve wo-vrile VAR where yo wn o resric he long-rn response of he firs endogenos vrile, y, o he second srcrl shock, 2, o e zero, hen m 2 = : y y 2 m m 2 m 2 () ole.rmmel@nkofenglnd.co.k 29 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

30 In rn, knowing he vles of he mrix M ells s somehing o mrices A nd B. De o he nmer of overll resricions reqired, EViews lso imposes he (necessry) resricion h mrix A is he ideniy mrix. 8 I hen ses mrix M o esime mrix B. Long-rn resricions in EViews cn e specified eiher in mrix form (where he mrix M is enered) or in ex form. We will consider ivrie VAR sing he re of chnge of nemploymen nd he money growh re. We hen impose he long-rn resricion h only demnd shocks hve permnen effecs on nemploymen nd esime he implse response fncion ssocied wih his idenificion scheme. Q. Esime redced-form VAR(6) wih d(nre) nd dlog(m2) over he period from 99 M o 28 M2, impose he long-rn resricion h only demnd shocks hve permnen effecs on nemploymen nd esime he implse response fncions of he idenificion scheme.. Answer: Recll h vriles hve o e in sionry form in order o impose he long-rn idenificion scheme. Ths we need o rnsform he wo vriles o sionry form. The vriles we shll se re he firs difference of he nemploymen re (d(nre)) nd he monhly money growh vrile (dlog(m2)). A VAR wih hree lgs sing he firs difference of he nemploymen re nd monhly money growh s well s consn ppers o ge rid of he prolem of ocorrelion. The VAR is sle, he residls re non-norml, principly on ccon of he second series (his is lef s n opionl exercise). The new esimion resls re given in Tle 2. Tle 2: VAR(3) esimion resls for d(nre) nd dlog(m2), 99 M 28 M2 Vecor Aoregression Esimes Smple: 99M 28M2 Inclded oservions: 8 Sndrd errors in ( ) & -sisics in [ ] D(UNRATE) DLOG(M2) D(UNRATE(-)) (.6579) (.45) [-.555] [ ] D(UNRATE(-2)) (.6555) (.44) [ 3.365] [ ] D(UNRATE(-3)) (.684) (.5) [ 3.844] [-.4848] DLOG(M2(-)) (3.583) (.6926) [.67287] [ ] DLOG(M2(-2)) (37) (.727) [-.3665] [ 263] DLOG(M2(-3)) (3.73) (.6973) [.72739] [ 2.562] 8 The expression for he long-rn response in eqion () involves he inverse of A. Since EViews reqires ll resricions o e liner in he elemens of A nd B, he A mrix ms e he ideniy mrix when yo specify long-rn resricion. ole.rmmel@nkofenglnd.co.k 3 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

31 C (.825) (.4) [-.9358] [ ] R-sqred Adj. R-sqred Sm sq. resids S.E. eqion F-sisic Log likelihood Akike AIC Schwrz SC Men dependen S.D. dependen Deerminn resid covrince (dof dj.).65e-7 Deerminn resid covrince.55e-7 Log likelihood 45 Akike informion crierion Schwrz crierion We now hve o cree he long-rn impc mrix. d( nre ) m d log( m2 ) m 2 m demnd_ shock spply _ shock Once gin, imposing his (long-rn) resricion cn e done eiher in ex or mrix form. To impose his resricion sing he ex form To impose he resricion sing he mrix form, cree he long-rn mrix, which I hve clled mrix_lr in he workfile: m M m 2 m NA NA NA In oher words, he long-rn response of he firs vrile (rel op) o he second srcrl shock ( spply shock) is zero. The op cn e fond in Tle 3. Tle 3: Srcrl fcorision of SVAR(3) model for d(nre) nd dlog(m2), 99 M 28 M2 (long-rn resricions) Srcrl VAR Esimes Smple: 99M 28M2 Inclded oservions: 8 Esimion mehod: mehod of scoring (nlyic derivives) Convergence chieved fer 8 ierions Srcrl VAR is js-idenified Model: Ae = B where E[']=I Resricion Type: long-rn pern mrix Long-rn response pern: ole.rmmel@nkofenglnd.co.k 3 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

32 C() C(2) C(3) Coefficien Sd. Error z-sisic Pro. C() C(2) C(3) Log likelihood Esimed A mrix:.... Esimed B mrix: In erms of implse response fncions, we re now no longer considering Cholesky decomposiion. In order o ell EViews his, we hve o go o View/Implse Responses..., check he ox for Accmled Responses nd selec he Srcrl Decomposiion opion on he Implse Definiion. Noe h EViews does no give nme o he shocks lels hem seqenilly. In or cse, he shocks re referred o s Shock nd Shock2. In ddiion, i is no possile o ge EViews o clcle sndrd error nds rond he implse response fncions omiclly. The (ccmled why?) implse responses ssocied wih his idenificion re given in Figre 6. Figre 6: Accmled implse response fncions Accmled Response o Srcrl One S.D. Innovions Accmled Response of D(UNRATE) o Shock Accmled Response of D(UNRATE) o Shock Accmled Response of DLOG(M2) o Shock Accmled Response of DLOG(M2) o Shock ole.rmmel@nkofenglnd.co.k Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

33 Shock is he demnd shock, wheres Shock2 is he spply shock. Ths, demnd shock increses nemploymen (s expeced?) nd monhly money growh. A spply shock leves op nffeced in he long-rn (s i shold!) nd increses monhly money growh. From he esimed srcrl VAR, how cn we genere he fndmenl shocks, i.e., he 's in eqion (), sing EViews? To genere he fndmenl shocks, we se he eqion Aε = B, where ε is he error or residl from he VAR regression which hs een genered, nd mrices A nd B come from he esimed srcrl VAR. The fndmenl shocks re hen simply: ˆ B A. Q2. Genere he fndmenl shocks from he long-rn SVAR. Answer: Selec Proc nd Mke Residls. EViews will omiclly genere series nmed resid?? in he sme ordering s he VAR esime. In his cse we ssme h hey re resid nd resid2. Chnge he nmes s pproprie I hve clled hem lr_shock nd lr_shock2. Cree wo mrices, clling hem m_lr_ nd m_lr_ y yping in he commnd window: mrix(2,2) m_lr_ mrix(2,2) m_lr_ nd ener he esimed coefficien vles from he esimed long-rn SVAR, i.e.: for he enries of m_lr_ (rememer h EViews ssmes he A mrix o e he ideniy mrix, cf. Foonoe 8) nd: for m_lr_. These nmer come from Tle 3 ove. Type in he commnd window (or cree progrmme nd rn i): grop resgrop lr_shock lr_shock2 mrix resmrix mrix resfnd * m_lr_ show resfnd In he resfnd mrix ojec h will open omiclly, selec View, Grph, Line o see plo of he fndmenl srcrl shocks (Figre 7). ole.rmmel@nkofenglnd.co.k 33 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

34 C Figre 7: Fndmenl shocks C Imposing hisoricl decomposiion The forecs error vrince decomposiion is widely employed s sefl nlyicl ool for he ex pos nlysis of (S)VARs. As illsred in Tle 9, we fond, sy, h he frcion of he vrince of inflion welve-periods-hed explined y: he demnd shock (nre) is 34 per cen; he cos-psh shock (inf) is 688 per cen; nd he monery policy shock (ffr) is negligile.93 per cen Yo my sk he no nresonle qesion of why we migh e ineresed in heoreicl decomposiion of forecs errors. The ler convey no informion o he implicions of he srcrl shocks in he hisoricl smple nd hve nohing o do wih rying o idenify he drivers of he siness cycle. Using he concep of hisoricl decomposiion, we cn esime he individl conriions of ech srcrl shock o he movemens in he op gp, inflion nd he shor-erm ineres re over he smple period. I ws firs developed y Sims (98), lhogh he firs pper sed pon i wold seem o e Bridge nd Hrrison (985). The hisoricl decomposiions of ech vrile ino he esimed srcrl shocks re clcled s follows: he SVAR(3) model is wrien in compnion form (i.e., s VAR() model): y = c + Ay + (2) sing ckwrd ssiion nd he Wold decomposiion, he model vriles ech poin in ime (y T ) cn e represened s fncion of iniil vles (y ) pls he ccmled sm of ll he srcrl shocks of he model: y T T A y T T k A k (3) k The hisoricl decomposiion is mch more sefl decomposiion hn he FEVD, since i shows wh is driving he vriles in y over ime. ole.rmmel@nkofenglnd.co.k 34 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

35 To crry o hisoricl decomposiion in EViews on he wo-vrile VAR(6) we hve js esimed, we cn se he Hisoricl Decomposiion dd-in for EViews 7, which cn e downloded for free from: hp:// To clcle he hisoricl decomposiion of redced-form VAR or n idenified SVAR in EViews, go o Proc, Add-ins in he VAR window, i.e., we ms hve esimed VAR firs. 9 We cn se he ivrie VAR(3) on d(nre) nd dlog(m2) h we hve js esimed for his prpose. Upon selecing he Hisoricl Decomposiion opion, EViews will sr o crry o he clclions. 2 Once i hs finished, i will open new grph, which shows he hisoricl decomposiions, one for ech of he vriles in he VAR (reprodced in Figre 8). Q3. Clcle he hisoricl decomposiion of he chnge in he nemploymen re s well s he monhly re of money growh in erms of he wo shocks. 9 While h he VAR cn hve consn, oher exogenos vriles re he momen no sppored. A hisoricl decomposiion is lso no ville for vecor error-correcion models. 2 EViews clly clls some Ml code o do he clclions. This reqires no only h Ml needs o e inslled on he comper, lso h EViews cn commnice wih Ml. ole.rmmel@nkofenglnd.co.k 35 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

36 Figre 8: Hisoricl decomposiion D on D(UNRATE) Mins Trend de o D(UNRATE)shock de o DLOG(M2)shock D on DLOG(M2) Mins Trend de o D(UNRATE)shock de o DLOG(M2)shock Ech grph plos he de-rended d series nd he conriion o h series de o ech shock. For exmple, he op grph plos he de-rended chnge in he nemploymen re (he le line), he chnge in he nemploymen re de o is own shock (he red line) nd he chnge in he nemploymen re de o money-spply growh (he green line). Rememer h he firs shock ws lelled demnd shock nd he second shock spply shock. The nderlying nmers re sved in differen mrices: ole.rmmel@nkofenglnd.co.k 36 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

37 he mrix seo conins he esimed rend for ech endogenos vrile in he order hey ener he VAR(6); nd he mrix hiso conins he conriion of ech shock For ech series, here is se forecs pls he ol impc of srcrl shocks. Hence, he mrix dimension is eql o n(n + ). In or cse, here re wo vriles (d(nre)) nd dlog(m2)), so hism will e (2 3) mrix. Since d(nre) is he firs vrile, colmns, 2 nd 3 will correspond o he se forecs for d(nre), hisoricl impc of he firs shock nd he hisoricl impc of he second shock, respecively. As ove, he decomposiions re rrnged y he shock. The firs wo colmns re he conriions of he demnd shock o he wo endogenos vriles, while he nex wo colmns re he conriions of he spply shock o he wo endogenos vriles. Noe h hese mrices re overwrien every ime new VAR model wih sign resricions is esimed. 6.6 Imposing sign resricions Scepicism owrd rdiionl idenifying ssmpions sed on eiher shor- or long-rn resricions hs mde n lernive clss of SVAR models more poplr. Sring wih Fs (998), srcrl shocks in hese SVAR models re idenified y resricing he sign of he responses of seleced model vriles o srcrl shocks. In pplied work, Uhlig (25) hs shown h signidenified models my prodce ssnilly differen resls from convenionlly idenified SVAR models. To operionlise he mehod, idenificion in sign-idenified models reqires h ech idenified shock is ssocied wih niqe sign pern. Q4. For he originl VAR(6) model wih he for vriles nemploymen (nre), he yer-onyer growh re in he nominl M2 money spply (dm2), he nnl inflion re (inf) nd he federl fnds rge re (ffr), compe he implse responses sing he sign resricion scheme in Tle 4. The smple period is 99M o 28M2. Tle 4: Sign resricion schemes for for-vrile VAR(6) Unemploymen CPI Money Ineres re NI X X X X NI X X X X MD MS + + In he ove Tle, X mens no resricion, + denoes resricion h he conemporneos response is resriced o e posiive nd denoes negive resricion. Frhermore, MS denoes money-spply shock, MD denoes money-demnd shock nd NI mens no idenified. To esime VAR where he shocks re idenified sing conemporneos sign resricions, we cn se he VAR wih sign resricions dd-in for EViews Noe h his dd-in reqires Ml o e inslled on yor comper. I lso reqires h EViews cn commnice wih Ml. ole.rmmel@nkofenglnd.co.k 37 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.

38 Answer. To impose he resricions, we need o cree nd nme mrix specifying he sign resricions. For his exmple, he EViews mrix looks s follows: (4) In oher words, posiive resricion is specified s, negive resricion s - nd no resricion s. I hve sved his mrix s mrix_sign in svr_s.wf. To esime VAR where he shocks re idenified sing conemporneos sign resricions, we cn se he VAR wih sign resricions dd-in for EViews 7. Go o Add-ins nd choose VAR wih sign resricions. Upon opening he EViews VAR wih sign resricions dd-in from he Add-ins men, dilog ox will pper: Under Endogenos Vriles, we need o ype he nmes of he endogenos VAR vriles in he order hey ener he model, i.e., in wy consisen wih he sign resricions o e imposed. For exmple, he mrix of sign resricions creed ove implies h he vriles need o e enered in he order: nre inf dm2 ffr. Under A Pern mrix, we need o ener he nme of he mrix conining he sign resricions, i.e., mrix_sign. The remining hree oxes (Ener n esimion smple, Ener nmer of lgs nd Ener nmer of periods for IRFs, FEVD) re prey selfexplnory. Noe h he esimion smple is sill 99M 28M2, h we hve model wih six lgs nd h we wn o plo IRF s over 3 periods. Noe h his dd-in lso reqires Ml o e inslled on yor comper. I lso reqires h EViews cn commnice wih Ml. ole.rmmel@nkofenglnd.co.k 38 Bnk of Englnd The Bnk of Englnd does no ccep ny liiliy for misleding or inccre informion or omissions in he informion provided.