VECTOR AUTOREGRESSIVE MOVING AVERAGE MODEL

Transcription

1 VECOR AUOREGRESSIVE MOVING AVERAGE MODEL Ran Kumar Paul Indan Agrculural Sac Reearch Inue Lbrar Avenue, New Delh Inroducon In man area of Agrculure relaed feld forecang roducon ver much eenal for roer lannng. Value oberved over a erod of me can be reaed a a me ere can be uded o ee he rend, ner-relaon, erodc ec. Aar from h, me ere model can be ued o foreca fuure value wh more recon. A model ung whch we are able o calculae he robabl ha a fuure value of he ere lng beween wo ecfed lm ermed a a ochac model. In h cone, a me ere condered a a ochac roce a amle or conecuve value of he me ere condered a a realaon of he ochac roce ha generaed he me ere. An moran cla of ochac model ha are wdel ued o rereen me ere he aonar model whch aume ha he ochac roce ha generae he me ere reman n eulbrum abou a conan mean level. A non-aonar roce doe no have an naural mean. A ochac roce ad o be rcl aonar f roere are unaffeced b change of me orgn. A euence of rom varable {a } called a whe noe ere f hee are rom drawng from a fxed drbuon wh ero mean conan varance. he nd of ochac model condered for me ere anal are baed on he conce ha a roce n whch ucceve value are hghl deenden can be regarded a generaed from a ere of ndeenden hoc, a, whch are whe noe. If a whe noe roce {a } ranformed no a ochac roce { } b a weghed um of a value o ha a a a, hen called a lnear flerng. Ung he bac hf oeraor B noaon, B oerae on he me ere { a } uch ha B a a, for value of,, we can wre he above euaon a a Ba B a ( B B a ( B a ( B B B whch a olnomal n he bac hf oeraor B nown a he ranfer funcon of he fler. If he euence of wegh n he ranfer funcon are fne or nfne convergen hen he fler ad o be able he generaed ochac roce { } ad o be aonar. When he euence of wegh { } nfne no convergng hen he roce { } non-aonar. For a aonar roce he arameer wll be he mean abou whch he roce vare for a non-aonar roce onl a reference on for he level of he roce. he mo oular cla of aonar e of ochac model ued for me ere modellng he Auoregreve Inegraed Movng Average (ARIMA model nroduced

2 M III: : Vecor Auoregreve Movng Average Model b Box e al (7. h cla nclude, auoregreve model, movng average model, rom wal model, auoregreve movng average model, negraed model eaonal model. In auoregreve model, he curren value of a roce exreed a a lnear aggregae of a value of he ere along wh a rom hoc. If b { }, we mean a aonar roce o rereen a me ere euence { } he roce whch he devaon from a cenral value, whch he mean of he roce, hen an auoregreve model of order, denoed b AR(, ae he form. Here { } a euence of rom hoc whch are aumed o be ndeendenl dencall drbued wh execaon ero conan varance. Parameer of h model are (,,,,,. Ung he bac hf oeraor noaon we can wre he AR( model a B B B ( B B B ha ( B ( B B B B a olnomal of degree n he bac hf oeraor B. B conderng he lnear fler rereenaon of he above AR( model he condon for aonar of he roce can be derved n erm of he roo of he characerc euaon ( B. h condon ha all he roo of he h degree olnomal euaon ( B are more han one n abolue erm. Anoher oular rereenaon of a me ere ung movng average model. In h model he devaon of are rereened b a lnear combnaon of a fne number of revou rom hoc. he exreon for a movng average model of order, MA( model, a a a a ( B a ( B B B B a h degree olnomal n he bachf oeraor B. Snce he lnear fler forma of h model he ame conan onl fne number of erm, a movng average model alwa aonar. Ung he nvere funcon ( B ( B, he lnear roce ( B a can have he nfne auoregreve rereenaon ( B a ( B B. h roce nverble when he euence of wegh converge. he reured condon for h ha he roo of he characerc olnomal euaon ( B le ou de he un crcle he condon for nverbl of a MA( model. he model ha combne he above wo nd of model he mxed auoregreve movng average (ARMA model. he auoregreve movng average model wh auoregreve erm movng average erm rereened b ARMA(, mahemacal exreon, a a a a whch can be wren a ( B ( B a M III-3

3 M III: : Vecor Auoregreve Movng Average Model ( B B B B a olnomal n he bac hf oeraor B of degree ( B B B B a olnomal n B of degree. he condon for aonar of he model ame a ha of an AR( model he condon for nverbl of he model he ame of an MA( model. Hence hee condon are ha roo of he wo olnomal ( B ( B le ou de he un crcle. In mo of he raccal uaon, he oberved me ere how non-aonar behavour do no var abou a conan mean. In uch uaon ma be oble o rereen he ere b an ARMA model wh a generaled AR oeraor olnomal, a ( B for whch one or more roo are un o ha can be facored a ( B ( B( B d ( B a olnomal n B wh all roo ou de he un crcle. Box e al (7 nroduced a cla of non-aonar model b negrang ARMA model wh rovon for he rereenaon for un roo non-aonar. h model erm volaon of he aonar condon b allowng ome roo o le on he un crcle. hee model are ermed a auoregreve negraed movng average (ARIMA model. If here are d roo ha fall on he un crcle, hen he model rereenaon d ( B ( B ( B a h euvalen o ranformng he orgnal ere no anoher ere b uccevel dfferencng d me hen rereenng he ulmae dfferenced ere b an ARMA(, model. he arameer d ermed a he order of dfferencng are reecvel he order of auoregreve movng average erm n he model. he general eaonal ARIMA model wh AR order, MA order, order of dfferencng d, eaonal, eaonal AR order P, eaonal MA order Q order of eaonal dfferencng D rereened b ARIMA(,d,(P,D,Q exreon d D ( B ( B ( B ( B a ( B, ( B, ( B B B P ( B B B, ( B B B ( B B B P Q Q whch are olnomal n B. he condon for aonar nverbl of he dfferenced ere d D ha he roo of all hee olnomal le ou de he un crcle. Mulvarae me ere model he maor wo reaon for analng modellng of more han one me ere euence ogeher, nown a mulle me ere or vecor me ere, are ( o under he dnamc relaonh among he dfferen me ere comonen ( o mrove he accurac of foreca of one ere b ulng he nformaon abou ha ere conaned n all oher me ere. Suoe here are me ere comonen { Z },{ Z},,{ Z} for,,, a euall aced me nerval. We can rereen hee comonen b a vecor Z Z, Z,, Z ( M III-33

4 M III: : Vecor Auoregreve Movng Average Model whch we call a a vecor of me ere. A vecor me ere roce Z rcl aonar f he robabl drbuon of he rom vecor Z, Z,, Z Z, Z,, Z n l l nl,,, n are he ame for arbrar me, all lag l,,, all n. ha he robabl drbuon of obervaon from a aonar roce nvaran wh reec o hf n me. For all uch ere E( Z for all (,,, he mean vecor for he ere E Z Z ( l l nown a he cro covarance marx of lag l for onl on he lag l when he ere aonar. ( ( ( ( ( ( ( ( ( ( If V dag (, (,, ( ( he varance of he h comonen ere {Z }, hen l,, ( wll deend / ( l V ( V ( ( ( ( he cro correlaon marx a lag l. For a aonar vecor roce, he rucure of he cro-covarance cro correlaon marce rovde a ueful ummar of nformaon on aec of dnamc ner relaon among he comonen ere of he roce. Samle emae of elemen of he lag l cro correlaon marx baed on a amle of e comued a ˆ ( l l ( Z Z l Z Z Z Z ( Z ( Z for,,, ; l,,,, he amle mean of he h comonen ere. he amle cro correlaon can be ued o denf low order vecor movng average model. For large amle e, under whe noe aumon, ( l are execed o be drbued a normal devae wh mean ero aroxmae varance h roer ued o e he gnfcance of ndvdual amle cro correlaon. Under M III-34

5 M III: : Vecor Auoregreve Movng Average Model whe noe aumon o e he combned gnfcance of he elemen of amle crocorrelaon marx ( for dfferen lag l, he ac defned a l r - - ( ( ( ( ( wh degree of freedom. Vecor AuoRegreve Model A unvarae auoregreon a ngle-euaon, ngle-varable lnear model n whch he curren value of a varable exlaned b own lagged value. Mulvarae model loo le he unvarae model wh he leer re-nerreed a vecor marce. One uch model he vecor auoregreon model. he vecor auoregreon (VAR model one of he mo ucceful, flexble, ea o ue model for he anal of mulvarae me ere. I a naural exenon of he unvarae auoregreve model o dnamc mulvarae me ere. hu hee model caure he lnear nerdeendence among mulle me ere. he VAR model ha roven o be eecall ueful for decrbng he dnamc behavor of economc fnancal me ere for forecang. I ofen rovde ueror foreca o hoe from unvarae me ere model elaborae heor-baed mulaneou euaon model. Foreca from VAR model are ue flexble becaue he can be made condonal on he oenal fuure ah of ecfed varable n he model. In addon o daa decron forecang, he VAR model alo ued for rucural nference olc anal. In rucural anal, ceran aumon abou he caual rucure of he daa under nvegaon are moed, he reulng caual mac of unexeced hoc or nnovaon o ecfed varable on he varable n he model are ummared. hee caual mac are uuall ummared wh mule reone funcon foreca error varance decomoon. A dadvanage here ha a large number of arameer mgh be needed for adeuae decron of daa. A good accoun of leraure on VAR model are avalable, o ce a few, Hamlon (994, Hacer Haem (8, Helmu (5, Runle (987, Soc Waon (, a ( Waler (3. A aonar vecor me ere { Z } wh comonen can be modeled b a vecor auoregreve model of order denoed b VAR(, exreon h can be wren a Where ( B. ( B I B B a marx olnomal n he bac hf oeraor B, vecor of he ere,,,, (,, are Z arameer marce,, he mean are ndeendenl dencall drbued rom nnovaon vecor havng ero mean conan covarance marx. M III-35

6 M III: : Vecor Auoregreve Movng Average Model Examle: he VAR( model wh hree me ere euence ( ( ( ( ( 3, ( ( ( ( ( 3, ( ( ( ( ( , 3 3 he correondng ndvdual unvarae model are ( ( ( ( ( ( 3 ( 3 ( 33,, 3, 3 (,, 3 3,,, 3 3, ( ( ( ( ( (,, 3 3,,, 3 3, ( ( ( ( ( ( 3 3 3, 3, 33 3, 3, 3, 33 3, 3 hu n each model, lagged erm, u o lag, of all he hree me ere euence are ncluded. he condon reured for aonar of a VAR( model can be brough ou b conderng an euvalen VAR( rereenaon. B reeaed ubuon for,,, n he VAR( model, he model can be rerucured no he form ( I Hence he roce vecor Z, Z,, Z are unuel deermned b Z he nal value of he roce he euence of nnovaon vecor. If all egen value of have modulu le han un, hen he roce Z a well defned ochac roce Z can be exreed a Z for,,, (. h condon euvalen o de( x for. Hence he condon for abl/aonar of a VAR( roce ha he roo of he deermnenal euaon de( x have all roo ou de he un crcle or euvalenl all he egen value of are le han one n abolue value. he general VAR( model Z Z Z can be brough o an euvalen VAR( model of a dmenonal roce a Y Y Y ( Z, Z,, Z, (,,,, (,,, whch are column vecor of lengh a uare marx gven b M III-36

7 M III: : Vecor Auoregreve Movng Average Model Baed on h rereenaon of VAR( model he condon for aonar of he model ha all egen value of he above marx have module value le han un. Euvalenl de( x for. Snce de( x de( x x he condon for aonar of he VAR( model ha he deermnanal olnomal de( x x have all roo ou de he un crcle. In erm of he bac hf oeraor B, he VAR( model can be wren a ( B Z ( B B B. Now conder a funcon ( B ( B o ha ( B ( B ( B B Oerang h funcon on he VAR( model gve, ( B ( B Z ( B ( B whch reduce o Z ( B ( B. Bu. ( B B ( becaue B If we wre Z. hen we can wre he VAR( model a whch can ex onl when he euence converge whch he condon for nverbl of he roce. he relaon beween hee coeffcen marce arameer M III-37

8 M III: : Vecor Auoregreve Movng Average Model marce can be obaned b exng ( B ( B hen euang coeffcen of ower of B, a wh I for. Emaon he arameer marce of VAR( model are emaed b generaled lea uare mehod. he VAR( model ( Z ( Z can be exreed a = ( Z X ( X (( Z,,( Z (,,. ( An euvalen form of he model Z Z = wh = ( he model can be rewren a Z X X (, Z,, Z (,,,. If he amle e n = -, hen defne n marce Z ( Z,, Z (,,. Le X be a marx of order n (+ wh h row a Z (, Z,, Z for,,., hen we have he relaon Y XB whch n he general form of a mulvarae lnear model can be olved for B a ˆ B ( X X X Y. M III-38

9 he emae of nnovaon deron marx obaned a Sm ( n S m ˆ Z ˆ ˆ Z. M III: : Vecor Auoregreve Movng Average Model For large amle e under aonar Gauan aumon he aroxmae large amle drbuon of whch he emae of vec( ( N, ˆ X X h roer ued o comue he ard error of he emae for furher eng. For he elecon of he order arameer of vecor auoregreve model dfferen order elecon crera are ued. If ( he maxmum lelhood emaor of he nnovaon deron marx obaned b fng a VAR( model o he daa, hen he Fnal Predcon Error creron ( FPE creron gven b FPE de ( (. ( he nformaon creron nroduced b Ae defned b log( maxmed lelhood AIC r r he amle e r he number of arameer emaed for he model aroxmaed a r AICr log( r c r he maxmum lelhood emae of he nnovaon deron marx c a conan. Snce here are ( arameer n he cae of a general VAR( model he mnmum AIC creron calculaed b AIC( ln (. ( Oher wo crera ued are he Baean nformaon creron BIC uggeed b Schwar (alo nown a SC creron he HQ creron rooed b Hannan Qunn. hee are gven defned a r log( BICr log( r r log(log( HQr log( r. In he cae of a general VAR( model hee creron are calculaed ung he formula ln ln ( HQ( ln ( M III-39

10 ln ( SC( ln. ( M III: : Vecor Auoregreve Movng Average Model he order ha eld mnmum value for hee creron are eleced a he uable order for he model. A lelhood rao e for eng he null hohe H of he VAR( model M III-4 : agan H : gven ha he e ac ( ln ln LR ( denoe he maxmum lelhood emae of when a VAR( model wa fed o he vecor me ere of lengh. h e ac ha an amoc drbuon wh degree of freedom. Examle A vecor auoregreve model of order wh he followng arameer marce gven below. Examne wheher he model aonar or no. Wre he ndvdual model. Model: Z Z Z ,, he arameer marx correondng o VAR( rereenaon of a VAR( model o ha we ge for a VAR( model I Egen value of he above marx are he module value of he egen value are Snce he module value of all he egen value are le han un he model aonar he ndvdual model of he VAR( model are Z.5Z,.6Z,.5Z,. 3Z,.3Z.Z.7Z. Z Z,,,, Examle: For he amle bvarae me ere (Samle daa - on US fxed nvemen change n bune nvenore he VAR( model can be emaed a

11 M III: : Vecor Auoregreve Movng Average Model Paral Cro Correlaon For a vecor me ere euence baed on elemen, he aral cro correlaon beween he vecor m gven he n beween vecor,, m he cro correlaon marx beween he elemen of vecor m, afer adumen of boh for her deendence on he elemen of nervenng vecor,, m denoed b P m. For a aonar vecor roce, he aral cro correlaon marx P m can be aroxmaed a follow. Conder he error vecor U m reulng from aroxmang he roce b a VAR model of order (m-, gven b m Um ( m m Y m ( (,( ( m ( ( m, ( m,, ( m( m Y( m.( (,, m. Smlarl b conderng a bacward VAR model of order (m- a m U ( m m m, m we can oban he bacward AR coeffcen marce from he e of euaon m ( ( ( m for l,,( m he bacward error vecor U m, m can be obaned a m m, m m ( m m U m ( m Ym, Paral cro correlaon marx a lag m hen defned a P Corr U, U V ( m E( U U V( m m ( m, m m, m, m m, V ( m dag (, V m dag (, m ( m ( m ( m( m,, a a oluon m m ( U m, ( ( Cov, ( m m m ( U m, m ( ( m Cov ( m ( m E ( Um, mu m, ( m ( m ( m ( m ( m ( m ( ( (,, ( m m ( ( m,, ( ( m. M III-4

12 M III: : Vecor Auoregreve Movng Average Model Under VAR( model aumon, he elemen of amle aral cro correlaon marx Pˆ m are aroxmael normall drbued wh ero mean varance for m h roer can be ued o e her gnfcance. Examle: For he amle daa - he followng are he lag 3 amle cro correlaon marx aral cro correlaon marx. Cro correlaon marx of lag Paral cro correlaon marx of lag VARMA Model he rereenaon of a vecor me ere b a Vecor Auoregreve model of order are ndeendenl dencall drbued rom vecor wh E( = E( = for all, whch are oherwe nown a whe noe ere. Inead of he whe noe ere, f we aume ha he error erm are auocorrelaed u o a ceran lag a, hen he whe noe erm can be relaced b an auo correlaed erm a a a he a are ndeendenl dencall drbued whe noe ere,, are marce of order. B mang h ubuon for, we ge he correondng model a a a a h model nown a Vecor Auoregreve Movng Average model wh order (, denoed b VARMA(,. h can be rewren a ( B ( B a B I B B (, B B I B B (, B I an den marx of order B he bac hf oeraor uch ha B. A vecor me ere can be modelled b a VARMA model whch a mulvarae analog of he unvarae ARMA model. he condon reured for aonar nverbl of a M III-4

13 M III: : Vecor Auoregreve Movng Average Model VARMA(, roce ha he ero of he deermnenal olnomal (B (B n B le oude he un crcle. A general VARMA(, model can have a VAR( rereenaon ung whch oble o e he aonar condon n erm of egen value of a characerc marx he form of he AR coeffcen marx for uch a rereenaon n erm of he arameer marce a marx of order ( ( he comonen marce are gven b I I a marx of order I a marx of order a marx of order wh all elemen ero I I I a marx of order,, If all he egen value of marx are le han one n abolue value hen he VARMA(, model aonar. In arcular for a VARMA(, model wh he exreon he condon for aonar ha all egen value of he coeffcen marx are le han one n abolue value he condon for nverbl of he model ha all egen value of he coeffcen marx are le han one n abolue value. Selecon of he order arameer are made b ung order elecon crera le AIC, BIC HQ. he Aae nformaon creron aroxmaed b r AICr log( r c, he Baean nformaon creron gven b Schwar (978 M III-43

14 M III: : Vecor Auoregreve Movng Average Model r log( BICr log( r, he creron rooed b Hannan Qunn (979 r log(log( HQr log( r r he maxmum lelhood emae of he nnovaon deron marx, r he number of arameer emaed, he amle e c a conan. he order ha eld mnmum value for hee creron eleced a he reured order for he model. For a VARMA(, model he number of arameer o be emaed ( o ha we ge hee creron for elecon of order of a VARMA(, model a AIC ( (, log(, BIC ( log( (, log( HQ ( log(log( (, log( Emaon of Model Parameer An emaon algorhm for vecor auoregreve movng average model arameer baed on he mehod gven b Sld (983 a followed. Conder he model ( B ( B whch aonar nverble Le Here ( B I B ( B I B n B B,, be a amle of vecor realaon of he ere. Defne marce ( for,, n;,, whch an n marx of obervaon, n ( for,, n;,, whch an n marx of redual, n Y ( B, B,, B whch a lagged daa marx of order n, A ( B, B,, B whch a lagged redual marx of order n, U ( A, Y whch a marx of order n (. B he marx wh order n. Now defne he marx,,,,, ( of arameer whch of order h (, elemen, M III-44 for,, n;,, of (. Ung hee marce he VARMA(, model can be wren for he amle daa marx a U. Fr mae he nal emae of redual vecor b fng a hgher order vecor auoregreve model. For h conruc a marx W ( B, B,, B

15 M III: : Vecor Auoregreve Movng Average Model he hgher order choen for he auoregreon. hen he nal redual marx ˆ ( obaned a ˆ ( W( WW W. Ung h redual marx conruc marce Â( Uˆ ( comue he fr emae of a ˆ( from lnear regreon a, ˆ ( [ U ( U(] U ( h nal value ˆ( aumed a ero. For he eraon o emae ˆ (, we oban he emae from he regreon U ( U( ˆ( U( a ˆ ( [ U( U( ] U ( we comue he new e of redual ung he recuron ˆ ( ˆ ( ˆ ( ˆ (. Ung he new emae of redual arameer marce he eraon connued ll convergence acheved o a afacor level of accurac. Reference Box, G. E. P., Jenn, G. M. Renel, G. C. (7. me-sere Anal: Forecang Conrol. 3 rd edon. Pearon educaon, Inda. Hamlon, Jame D. (994. me Sere Anal. Prnceon Unver Pre. Hacer, R. S. Haem-J, A. (8. Omal lag-lengh choce n able unable VAR model under uaon of homocedac ARCH, "Journal of Aled Sac", 35(6, Helmu L. (5. New Inroducon o Mulle me Sere Anal. Srnger. Runle, D. E. (987. Vecor Auoregreon Real. Journal of Bune Economc Sac, 5 (4, Soc, J.H. M.W. Waon (. Vecor Auoregreon. Journal of Economc Perecve, 5, -5. a, R. (. Anal of Fnancal me Sere. John Wle & Son. New Yor. Waler E. (3. Aled Economerc me Sere, nd Edon, John Wle & Son M III-45