Stability Analysis for VAR systems. )', a VAR model of order p (VAR(p)) can be written as:

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Sbl Anlss for VAR ssems For se of n me seres vrbles (,,, n ', VAR model of order p (VAR(p n be wren s: ( A + A + + Ap p + u where he A s re (nxn oeffen mres nd u ( u, u,, un ' s n unobservble d zero men error erm I Sbl of he Sonr VAR ssem: (Glser, Mheml Mehods for Eonomss The sbl of VAR n be exmned b lulng he roos of: ( I A L A L A( L n The hrers polnoml s defned s: Π( ( I A z A z z n The roos of Π (z wll gve he neessr nformon bou he sonr or nonsonr of he proess The neessr nd suffen ondon for sbl s h ll hrers roos le ousde he un rle Then Π s of full rnk nd ll vrbles re sonr In hs seon, we ssume hs s he se Ler we llow for less hn full rnk mres (Johnsen mehodolog Clulon of he egenvlues nd egenveors Gven n (nxn squre mrx A, we re lookng for slr nd veor suh h A hen s n egenvlues (or hrers vlue or len roo of A Then here wll be up o n egenvlues, whh wll gve up o n lnerl ndependen ssoed egenveors suh h or A I [ A I] For here o be nonrvl soluon, he mrx [ A I ] mus be sngulr Then mus be suh h A I Ex: A

Expndng he deermnn of hs mrx gves he hrers equon: 7,, 7 ( ( + Noe: n egenveor s onl deermned up o slr mulple: If s n egenveor, hen µ s lso n egenveor where µ s slr: ( ( ( ( μ μ μ μ µ µ µ A A The ssoed egenveors re hose h ssf he equons for he hree dsn vlues of he egenvlues The egenveor ssoed wh, whh ssfes he equon for hs mrx s found s Noe h onl olumns nd re lnerl ndependen (rnk so we n hoose he frs elemen of he mrx rbrrl Se nd he oher wo elemens re Smlrl, he egenveor ssoed wh 7, whh ssfes he equon for hs mrx s found s Noe h rk(a gn beuse hs me he ls wo rows re lnerl dependen Thus onl he x mrx on he LHS s nonsngulr We n delee he ls row nd move mulpled b he ls olumn o he RHS Now he frs wo elemens wll be Remnder: For hrers equon of he pe + + b b b β α ± ±, Rel roos: β α ±, Imgnr roos: β α ±, where Modulus β α +

expressed n erms of he ls elemen We n fx rbrrl nd solve for he wo ohers: ssume Then (9,, ' s n egenveor orrespondng o he egenvlue 7 We n fnd smlrl he ls egenveor o be (7,,' Jordn Cnonl Form: Form new mrx C whose olumns re he hree egenveors 9 7 C You n lule o fnd h he mrx produ Q AQ Λ C AC 7 Thus, for n squre mrx A, here s nonsngulr mrx C suh h ( C AC s dgonl wh he egenvlues on he dgonl ( The egenveors orrespondng o dsn egenvlues of smmer mrx re orhogonl (lnerl ndependen II Sbl Condons for Sonr nd Nonsonr VAR Ssems (Johnson nd DNrdo, Ch 9+Appdx To dsuss hese ondons we sr wh smple models nd generlze We wll see: VAR( wh vrbles: VAR( wh k vrbles (ex: VAR( wh vrbles VAR(p wh k vrbles VAR( wh wo vrbles (p, k ( b +, +, + ε ( b + + + or: ε ( b + A + ε, whh n be wren wh he lg operor ( ( I AL b + ε Eh vrble s expressed s lner ombnon of self nd ll oher vrbles (plus nereps, dummes, me rends The dnms of he ssem wll depend on he properes of he A mrx

Ω s The error erm s veor whe nose proess wh E( ε nd E( ε ε ' s s where he ovrne mrx Ω s ssumed o be posve defne he errors re serll unorreled bu n be onemporneousl orreled Soluon o : ( Homogenous equon: Om he error erm b + A smples soluon: Then, (5 Π b f Π s nonsngulr ( Π I A As soluon r d Subsung n he homogenous (rvl soluon equon (5: ( I A d ---egenvlues The nonrvl soluon requres he deermnn o be zero: I A Ge he egenvlues ( ' s ( Subsue he egenvlues no he homogenous ssem, o ge he orrespondng egenveors ( C' s ( Afer lulng he nonhomogenous soluon nd ddng o he homogenous equon, we obn he omplee soluon (n mrx form: (6 + + (LR vlue s rses f he wo egenvlues hve he modulus< We n rewre ( I AL b n ( s polnoml o see he sbl ondons n erms of he egenvlues: B( L b where B(LI-AL, B L ( L( ( L The sbl ondon: ( Modulus s < Π nonsngulr, he deermnn no, he ssem s sonr In (6 onverges o ( Modulus > Π nonsngulr, bu he ssem s explosve, no onvergene Ths s beuse one or more of he ' s grows whou bound s nreses, so does from (6 No pl proess observed n he mro/fnne seres, herefore we do no onsder hs se ( Modulus un roo, Π s sngulr, he deermnn s s nonsonr, we need o look no he VECM spefon A lo of he mro/fnne models fll no hs egor (v Modulus I( vrbles, VAR s I( In generl A s no smmer Look for onegrng veors

Relon beween VAR vrbles nd egenvlues Defne he egenvlues nd he orrespondng egenveors of he mrx A s: Λ nd C If he egenvlues re dsn hen he egenveors re lnerl ndependen, nd C s nonsngulr C AC Λ nd A CΛC Theorem: ( for n squre mrx A, here s nonsngulr mrx C suh h C AC s dgonl wh he egenvlues on he dgonl ( The egenveors orrespondng o dsn egenvlues of smmer mrx re orhogonl (lnerl ndependen Defne new veor of vrbles w suh h Cw or w C eh s lner ombnon of w s (or eh w s lner ombnon of s Mulpl ( b + + ε b C : C C A b + AC + C ε w b * +Λw + e, b* C b, Λ C AC, e C ε or: (7 w b * + w, + e w b * + w + e, ( < for, Boh egenvlues hve modulus < Eh w s herefore I(, nd sne s re lner ombnons of w s, eh s I( You n herefore ppl he sndrd nferene proedures nd esme eh equon seprel As we sw bove, Π I A s nonsngulr, s full rnk ( here, nd unque s equlbrum exss: ( I A b Π b The vlues of re suh h n shok de ou qukl nd devons from equlbrum re rnsor ( > for or One of he egenvlues hs modulus > Sne eh s lner ombnon of boh w s, s unbounded nd he proess s explosve ( nd < Now w s rndom wlk wh drf, or I(, w s I( Eh s I( sne eh s lner ombnon of boh w s, herefore VAR s nonsonr Is here lner ombnon of nd h removes he sohs rend nd mkes I(, e boh vrbles re onegred?

Consder gn w he C * *,, + + * *,, where * represen he oeffens n C mrx We know h w s I(, hus [ * *] s onegrng veor Look for Relon beween he CI veor * *] nd he Π mrx suh h * * [ ] Π [] [ Reprmeerze equon ( o gve: ( b Π + ε where Π I A The egenvlues of Π re he omplemens of he egenvlues of A: µ Sne he egenvlues of Π re nd Thus, s sngulr mrx wh rnk Le us deompose Π Sne Π I A nd A CΛC, we n wre Π I C AC ( CI CΛ C C( I Λ C Thus: (9 Π C C * * * * ( ( [ * *] αβ ' So Π, whh hs rnk, s forzed no he produ of row veor β nd olumn veor α, lled n ouer produ: The row veor β he onegrng veor The olumn veor α he lodng mrx he weghs wh whh he CI veor eners no eh equon of he VAR ----------- Noe: ompre (9 o he se where Π s full rnk wh : Π C C sd o be of redued rnk ------------ ( ( ( * * * * ( You n see wh αβ ' s Combnng ( nd (9 we ge he veor error orreon model of he VAR: ( b b ( ( ( ( * *,, + + * *,, + ε + ε b b ( w ( w,, + ε + ε All vrbles here re I(: s n frs dfferenes nd w s

The w (EC erm mesures he exen o whh s deve from her equlbrum LR vlues Alhough ll he vrbles re I(, he sndrd nferene proedures re no vld (smlr o he unvre se where n order o es wheher seres s I(, we hve o use n ADF es nd no he sss on he AR oeffen --See exmple below (v Repeed unr egenvlues: We n no longer hve dgonl egenvlue mrx s before Bu s possble o fnd nonsngulr mrx P suh h P AP J nd A PJP where J (he Jordn mrx The problem wh hs se s h lhough Π s sll rnk, he rnsformon of s no w s leds o I( vrbles, he onegron veor gves lner ombnon of I( vrbles nd s hus I( nd no I( Thus s CI(,, he vrbles n he VAR re ll I( bu he nferene proedures re nonsndrd Exmple of se wh nd < Fnd he mres α nd β from VAR( wh k: (,, + e, ( 6 + + e,,, Reprmerzng he VAR no VECM gves us: + e,,, 6, 6, + e, n mrx form: e ( + 6 6 e or: Y ΠY + u Bu we nno nfer he lodng mrx nd he onegrng mrx seprel from hs To fnd α nd β seprel, we need o lule he egenveor mrx: Ge he egenvlues from he soluon o A I A I Egenveors orrespondng o :, 6 6

6 6 So se [ ]' 6 here s lner dependen Egenvlues orrespondng o 6 6 here s lner dependen So se 6 [ ]' The egenveor mrx nd s nverse re: 5 5 C C 5 5 Now we n wre he VAR n VECM b deomposng Π : C C + u Y ΠY + u Y +,, [ 5 5] u Ths s he sme expresson s n ( bu now we hve boh he lodng nd he onegrng mres: α nd β ' [ 5 5] VAR( wh k vrbles: ( b + A + A + ε Noe: ou n lso dd n deermns erms suh s rend, breks b spefng he model s: A + A + ΦD + ε Se he error erm nd exmne he properes of he ssem We sll hve he LR soluon (or he prulr soluon s n (5 Π b bu now Π I A A exss f Π s nonzero To see hs, look he egenvlues We gn r he sme soluon for he homogenous equon nd subsue n o ge he hrers equon

I A A The number of roos pk where porder of he VAR nd k#vrbles Here we wll hve k roos If ll egenvlues hve modulus< hen Π s non sngulr nd he soluon k + wll onverge o s grows The nlss wr he modulus of he roos (<,, > s he sme s n he VAR( se If he proess s sonr hen we n nver he VAR model nd express s funon of presen nd ps shoks, nd he exogenous (deermns omponensimpulse Responses: Ex: Clule he roos of -dmensonl VAR(: np nd fnd he effe of shok on dependen vrble: (Juselus Ch The hrers funon of Π ( z I A z A z where z s,,,, Π( z I z z,,,, I z z z z,,,,,z,,z,z z ( (,, z z z,,z ( (,, z z,, z z Therefore Π ( z (,z,z (,z,z (,z +,z (,z +,z Regroupng smlr erms: Π ( z z z z z ( z( z ( z ( z The deermnn s h order polnoml n z gvng hrers roos: z / ; z / ; z / ; z / Effes of shok (or sruurl hnge dumm on dependen vrble: If Π s nverble (ll roos n he un rle, we n wre Π ε We n hen lule he effe of shok on :

Π ( L ε j Π( z Π ( L ε j for, T ( z( z( z ( z We re ssumng h ll roos hve modulus less hn The hrers roos gve nformon bou he dnm behvor of he proess To see how he shok s propged, expnd he ls omponen: ( ε ( + L + L ε j + j You wll hve o do he sme hng wh eh roo Thus, eh shok wll ffe urren nd fuure vlues of The perssene of he shok depends on he mgnude of he roos The lrger he re he more perssen wll be he shoks -If he roos re rel nd <, he shok wll exponenll de ou -If one or more roo s mgnr hen shok wll be ll bu exponenll delnng -If one or more roos les on he un rle, he shok wll be permnen nd nd wll show nonsonr behvor VAR s no nverble, hen we need o look no VECM We n lso lule he roos b reformulng he VAR(p no he ompnon mrx VAR( form nd solve for he wo egenvlues: Alernve pproh: ompnon mrx A VAR(p n be rnsformed no VAR( Consder he equon (6 gn We n rewre s: + u A + A In mrx form: A A I p u + Clule he egenvlues from he oeffen mrx: I A A A A I A A A I I I ( ( Now we ge he roos drel nsed of he z s, whh were he nverse of he roos, obned b solvng he hrers polnoml Johnsen nd Juselus refer o he s s egenvlues roos nd o z s hrers roos

In he se of he ompnon mrx, here re wo roos If he roos o he hrers polnoml re ousde he un rle, hen he egenvlues of he ompnon mrx re nsde he un rle nd he ssem s sble To rep: -The soluon o I za gves he sonr roos (hrers roos ousde he un rle -The soluon o I A gves he sonr roos (egenvlues nsde he un rle -If he roos of Π (z re ll ousde he un rle or he egenvlues of he ompnon mrx re nsde he un rle, he proess s sonr -If one or more of he roos of Π (z or hose of he ompnon mrx re on he un rle hen he proess s nonsonr -If one or more roos of Π (z s nsde he un rle or he egenvlues of he ompnon mrx re ousde he un rle, he proess s explosve

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