Tn Corn DOSESCU Ph D Dre Cner Chrsn Unversy Buchres Consnn RAISCHI PhD Depren of Mhecs The Buchres Acdey of Econoc Sudes ANOTHER CATEGORY OF THE STOCHASTIC DEPENDENCE FOR ECONOMETRIC MODELING OF TIME SERIES DATA Absrc In hs pper we propose odly for concevng dgenerng process fro whch clsscl econoerc odelng srs by showng β-dependence beween rndo vrbles defned on probbly spces dfferen n generl whch cree rndo phenoenon S h es plce n severl seps nd wh evoluon n dscree e Key words: esurble spces nd ppngs probbly spces produc of probbly spces sochsc rnson funcon sochsc processes e seres econoc vrbles econoerc odel JEL CLASSIFICATION: C5 AMS000: 60G99 Inroducon In clsscl econoerc odels lner regresson odels or nonlner regresson odels sulneous equon odels syses of regresson equons odels dync regresson odels e seres odels nd ohers econoc vrbles of he rel world econoy re econoerclly odeled by rndo vrbles defned on he se probbly spce [] [] [5] [7] The supposon h ll rndo vrbles whch nerfere whn econoerc odels re defned on he se probbly spce llows us o hghly elbore preceps whch hve led o he developen of econoerc echnques Here re few exples of hese preceps: - correlon precep essenl n lner or nonlner regresson [] [7]; - uocorrelon precep essenl n he uoregressve odels [] [6]; - co- negred syses nd ulvre co negron [] [6]
Tn Corn Dosescu Consnn Rsch Rer 0 The bove-enoned supposon llows us o clcule he correlon coeffcen beween ny wo rndo vrbles lhough here s no econoc ground In he pper we e no consderon noher pproch of econoerc odelng whch ens: o replce he bove-enoned hypohess wh rndo vrbles hypoheses whch nerfere wh he econoerc odels re defned on dfferen probbly spces; b he econoc vrbles econoerclly odeled re pr of vrbles syses whch evolve n dscree e re pr of n nerdependen chn or crculr ype For exple we y consder he syse de of he followng hree croeconoc ndexes vrbles): GDP consupon nd nvesens Ths syse represens subsyse of he whole syse of ndexes whch re used for he enre econoy h re pr of he Nonl Accouns Syse Mng use of he ndex syse evoluon durng 990-995 where he ndexes re: Index Mesureen un 990 99 99 99 99 995 GDP Bll RON 8579 09 609 0057 97676 75597 ConsuponC) Bll RON 6795 675 65 558 775 565 InvesenI) Bll RON 680 0 888566 889 8006 9959 Source: The Ronn Sscs Yerboo 996 Accordng o b he ndces I GDP) I consupon) nd I nvesens) re coponens of rndo syse S wh evoluon n dscree e Ech ndex s econoerc odeled by sochsc process defned on proper probbly spce We consder I N rndo vrble oen whch odels ndex I = Then I ) depends on I due o he crculr dependency nd lso on I j or on I j ) due o he chn dependency The eleens bove enoned hve suggesed defnon of noher ype of dependency clled β-dependence beween rndo vrbles defned on spces of dfferen probbly The econoerc odels consruced on he grounds of he hypohess usng β-dependence re dfferen fro he clsscl econoerc odels The pper consss of he fundens of econoerc odelng of rndo phenoenon S h es plce n severl seps nd wh evoluon n dscree e In concluson β-dependence proposes ens of concevng dgenerng process Ths process for e seres d s suded n econoerc lerure nd proposes dfferen ehods o derve synhec e seres [5]
Anoher Cegory of he Sochsc Dependence for Econoerc Modelng of In hs pper we use noces nd resuls fro [] Probblsc odelng o genere sscl seres Le N For = nd N we denoe by V he se of he rel rndo vrbles whch re defned on probbly spce K P ) where s fne or denuerbly nfne Le us e he funcons : N V = nd le us e -uples = ) K ) N For fxed -uples = ) K )) le us consder for { } x = he rndo vrble x ) ω) wh ω ) where ens he vlue of N e We consder condons bvlen C C C h refer o he vlues x N = ; ore ccure for fxed he condon C refers o x nd j N s no bou x j Wh hose condons defne dependence beween Defnon Clled C C C dependence or β dependence beween he dependence defned by he followng wo procedures ) depends on = so: A N \{} e he spce of probbly h he rel rndo vrble I) s defned depends on he vlue s: x f x ee he condon posed by he C hen: ) K = P ) P P ) nd P s pr of he se clss of probbly lws s well s P ) ) : R here s he followng = ) resrcon ) f x does no ee he condon posed by he C hen: = ) K =K -) P = P ) nd = ) Noe h f = he ) procedure does no e sense b) depends on so: A N e he probbly spce on whch ) rndo vrble s defned depends on he vlue of s follows: x
Tn Corn Dosescu Consnn Rsch f x ees he condon posed by he C hen: ) K )= P )) P ) P P ) beng pr of he se clss of probbly dsrbuons such s P ) ) : ) R here s he followng ) = resrcon ) f x does no ee he condon posed by he C hen: ) = K )= K P ) = P nd ) = We use β dependence for he probblsc odelled of rndo phenoenon S wh n evoluon n seps on whch bvlen condons C C C nd C respecvely cs Ech sep s econoerc odeled whch s defned on he probbly spce e by he rndo vrble ) K P ) where s fne or denuerbly nfne = N Accordng o he β dependence es sense relon S ~ K )) N whch ens h he se of he phenoenon S e s chrcerzed by -uples ) K ) In hs cse he vlues of rndo vrbles e e seres d whose generon s descrbed by he evoluon of he phenoenon S Noe h β dependence nduces orderng of he rndo vrbles Rer A e he phenoenon S hs he se S ) ~ ) ) K )) whch depends on he se S wh N becuse ) depends on ccordng o b) procedure of he β- dependence nd ) depends on = ccordng o ) procedure Rer A e N for = he effecs of x s vlues upon he rndo vrbles re nsnneous In hs wy he sulney relons beween he rndo vrbles s posule Rer The b) procedure cn be regrded s vrn of feedbc beween he se of phenoenon S Rer I cn be ssued h 0 N so h = 0 e whever = here s he rndo vrble 0 ) defned on probbly spce
Anoher Cegory of he Sochsc Dependence for Econoerc Modelng of 0 ) K 0 ) P 0 )) K 0 )= P 0 )) he sr of he procedures ) nd b) Rer 5 The condons refer o he vlues of he rndo vrbles ensurng he copbly wh he phenoenon S Ech condon s bvlen e yes ssfed) / no oed) Rer 6 The probbly spces K P ) = nd N re sndrd Borel spces nd we cn pply he dsnegron heore Rer 7 x = x x ) K x ) N s he rndo vecor h nuerclly chrcerzes he se S Theore Le us consder N s he e whch he S) se of he phenoenon S s deerned by he S se ccordng o β dependence Then: ) here s sochsc rnson funcon P ) relve o K )) nd whever = here s sochsc rnson funcon Q ) relve o j ) K )) b) N here s sochsc rnson funcon P ) relve o j) K )) where = K ) Proof ) As x ees or does no ee he condon posed by he C re wo possble cses Cse x ees he condon posed by he C Accordng o b) procedure s defned P ) s: P ) : K ) [0] P ) P bu P ) s pr of he se clss of probbly lws s well s P ) where ) K )= P )) ) : ) R nd ) ) = Suppose = nd we defne sochsc rnson funcon Q ) fro ) K )) n Borel spce ) K )) f x ) ees he condon posed by he C Then ccordng o ) procedure we hve: ) ) K )= P ))
Tn Corn Dosescu Consnn Rsch In hs cse we defne wo probbles denoed by P ) nd P ) respecvely P ) s pr of he se clss of probbly lws s well s P ) nd P ) P P ) s n exenson of P ) o ) K )) where: P ω) ω P ) ω ) = 0 ω ) \ Noe h ) : ) R so we hve ) = Le us ) = { ω ) ) ω) = x ) ees he condon posed by he C } nd Q ): ) K )) [0] where: P ) ω ) Q ) ω ) = P ) ω ) \ ) Becuse A K ) Q ) A) s K )-esurble follows h Q ) s sochsc rnson funcon relve o ) K )) In ddon P ) Q ) s produc probbly on he Borel spce ) ) K ) K )) nd ) ) K ) K ) P ) Q ) ) s probbly spce If x ) does no ee he condon posed by he C hen ccordng o ) procedure we hve: ) = K )= K nd )= I follows h here s defned on he Borel spce ) K )) only he probbly P ) nd P ) = P ) In hs cse le us e Q ): ) K )) [0] Q ) ω ) = P ) ω ) Iedely follows h Q ) s sochsc rnson funcon relve o ) K )) nd ) ) K ) K ) P ) Q ) ) s probbly spce Suppose h he ) seen of he Theore s rue for ny N nd we wll deonsre for phenoenon S wh n evoluon n seps nd bvlen condons I follows h Q ) = s sochsc rnson funcon relve o j ) K )) nd
Anoher Cegory of he Sochsc Dependence for Econoerc Modelng of prculrly here s he probbly spce = where R )= P ) Q ) K Q ) ) K )R )) Le us e he funcon Q ): ) K )) [0] whch s defned by x ) s follows: If x ) ees he condon posed by he C hen ccordng o he ) procedure exended o he coponens we hve: = ) ) K )= P )) In hs cse we defne wo probbles on ) K )) denoed by P ) nd P ) respecvely: P ) : K ) [0] = P ) P P ) s pr of he se clss of probbly lws s well s P ) ; P ) : K ) [0] where: P ω) ω P ) ω ) = 0 ω ) \ Noe h we cn defne les one rndo vrble ) : ) R so h ) = Q where Le us e Q ): ) K )) [0] where: = P ) pr ) ω) ) ) ω ) = P ) pr ) ω) ) \ ) ) = { ω ) ) pr ees = he condon posed by he C } Becuse P ) ) A) ω = x ) )) ) nd P ) re probbles nd Q s K ) esurble A K ) follows h ) Q s sochsc rnson funcon relve o ) =
Tn Corn Dosescu Consnn Rsch K )) R ) Q ) K ) R ) s produc probbly nd ) = ) s probbly spce where R )= ) Q ) on whch he rndo vrble ) R s defned In concluson he seen ) s rue for ny N If x ) does no ee he condon posed by he C hen ccordng o he ) procedure exended o he seps nd ccordng o he defnon of β-dependence we hve: ) = ) K )= K P ) = P nd ) = In hs cse le us e Q ): ) K )) [0] = where Q ) ω ) = P ) ω ) Iedely follows h Q ) s sochsc rnson funcon relve o ) K )) R ) Q ) = s produc probbly ) K ) R ) ) s probbly spce where R )= R ) Q ) on whch he rndo vrble ) s defned In concluson he ) seen s rue for ny N n hs cse Cse x does no ee he condon posed by he C Accordng o b) procedure we hve : ) = ) K )= K P ) = P ): ) R nd )= Suppose h = Q s sochsc rnson funcon fro he ) K )) o he Borel spce ) K )) s n cse nd obn h ) ) K ) K ) nd le us e ) P ) Q ) ) s probbly spce Assung he seen rue for ny N follows s n Cse cn be shown h for rndo phenoenon S wh seps nd bvlen condons In hs cse h seen s rue for ny b) An nference h s slr o ) deonsron =
Anoher Cegory of he Sochsc Dependence for Econoerc Modelng of = Consequence In ers of Theore here s he probbly spce ) K )R )) where: Cses = R )= P ) Q ) K Q ) Deonsron follows fro ) seen of Theore Nex we consder wo neresng cses Cse = In hs cse he phenoenon S s reduced o sngle sep nd o bvlen condon C The se S of he phenoenon e s econoerc odeled by he rel rndo vrble whch s defned on he probbly spce K P ) N where s fne or denuerbly nfne For ny N here s dependence beween nd ) deerned by he b) procedure of he β dependence nd expressed by ens of he bvlen condon C whch refers o he vlues x = ω) ω Rer 8 If x does no ee he condon posed by he C no follows x=x) becuse ω e y be dfferen fro ω e Rer 9 C-dependence odels dependence of he rndo vrble self dfferen es whch we cll uodependence Auodependence no he se uocorrelon becuse generlly dfferen probbly spces occur dfferen es Theore Le N be he e whch he se S) of he phenoenon S s deerned by he se S under C-dependence Then: ) here s Q) sochsc rnson funcon relve o K)) nd for ny N here s Q) sochsc rnson funcon relve o ) K)) = b) Le us e N nd = ) K )) Then s esurble funcon defned on probbly spce ) ) K ) K K) K) R ) wh vlues n R where R = P Q ) K Q ) Proof ) As x ees he condon posed by he C or no wo cses re possble:
Tn Corn Dosescu Consnn Rsch Cse x ees he condon posed by he C Then ccordng o he Defnon cn be defned ) K)=P )) P ) P P ) beng pr of he se clss of probbly lws such s P ) : ) R here s he followng resrcon ) = Le us e he funcon P ) : K) [0] so P ω) ω P ) ω ) = 0 ω ) \ I follows h P ) s probbly on ) K)) nd le us e he funcon P ) ω Q ) ω ) = P ) ω ) \ where = { ω ω) = x ees he condon posed by he C} Iedely follows h Q) s sochsc rnson funcon relve o K)) nd ) ) K K) R ) s probbly spce where R = P Q ) s he produc of probbles P nd Q) Cse x does no ee he condon posed by he C Then ccordng o he Defnon we hve ) = K)= K P ) = P nd ) = I follows h on he Borel spce ) K)) s defned only he probbly P ) nd P ) = P ) = P Le us e Q): K)) [0] Q ) ω ) = P ) ω ) Iedely follows h Q) s sochsc rnson funcon relve o K)) nd ) ) K K) R ) s probbly spce where = P Q ) = P P ) R The exsence of he sochsc rnson funcon Q) N s n nference h s slr o ) deonsron of Theore b) Follows edely ccordng o ) Rer 0 In generl s no rndo vecor becuse s coponens re no defned on he se probbly spce
Anoher Cegory of he Sochsc Dependence for Econoerc Modelng of Cse = In hs cse he phenoenon S consss of wo seps nd wo bvlen condons C nd C nd se S of he syse e s econoerc odeled by he -uples ) N where s rndo vrble whch s defned on probbly spce K P ) wh fne or denuerbly nfne = A ech e le us consder nd hen Consequence If s 0 = he e whch whever N > 0 he se S) of he phenoenon S s deerned by he se S ccordng β-dependence hen n n N K R ) = = = n n = s probbly spce where K = K K R = P Q P n) n s sochsc rnson funcon relve o = j j) K n)) where n nd Q ) s sochsc rnson funcon relve o ) K )) nd Q n) n s sochsc rnson funcon relve o j) ) n An exple Follows edely ccordng o Theore n K n)) where Le us consder phenoenon S whch es plce n four seps whch c upon bvlen four condons C C C nd C respecvely nd whose se e s S N wh evoluon n dscree e Ech sep s econoerc odelled e by rel rndo vrble whch s defned on he probbly spce K P ) where = { K n } n N N K = P ) P )= Cn p q wh p q = = nd n = N λ K = P N ) nd P λ )= e λ > 0 N N! Condons C = us be copble wh he phenoenon S nd he res cn be defned nywy For exple: > p p N p fxed = N nd ) C : ) [ )] ) { } x C : { x s even nuber} b) C : x M )) N ; { > N fxed} = N 0 0 0
Tn Corn Dosescu Consnn Rsch Beween rndo vrbles = s defned β - dependence s follows: ) depends on = A he N \{} e he probbly spce on whch he rel rndo vrble s defned depends on he vlue x = ω) ω = If x does no ee he condon posed by he C hen = ) K = K -) P = P -) nd ) = ) If x ees he condon posed by he C hen here re wo suons: ) for = = { n ) } K = P )) P P -) P n ) )= p q ) R = ) Cn ) ) ) : here s he followng resrcon ) ; [ )] ) ) = N λ K = P N ) P P -) P λ )= e! λ λ N N R = where ) ) here s he followng resrcon = ) ) : ) ) b) depends on A he N e he probbly spce on whch he rndo s defned depends on he vlue of x If x does no ee he condon posed by he C hen ) = = If x ees he condon posed by he C hen ) = n } K )= P )) P ) P P vrble ) K )= K P )= P nd ) ) { ) ) ): ) R ))= n Cn p q here s he followng resrcon h ) = ) We consder he followng suon Le us e N s he e whch he se S) of phenoenon S s deerned by he se S ccordng by he β-dependence defned bove
Anoher Cegory of he Sochsc Dependence for Econoerc Modelng of Suppose h condons C C nd C re ssfed he e nd he e s ssfed condon C The sochsc rnson funcon P ) relve o K )) s defned s: P ) ω P ) ω ) = P ω \ where = { ω ω) = x ) ees he condon posed by he C } nd C P ))= 0 n p n ) The sochsc rnson funcons Q ) j K )) where = re defned s: q = n relve o P ) pr ) ω) ) Q ) ω ) = P ) pr ) ω) ) \ ) where ) = { ω ) ω condon posed by he C } P ))= C P P ) ) = ) 0 = n The sochsc rnson funcon Q ) j j ) K )) s defned s: ) pr ) )) = x n p n ) relve o P ) pr ) ) ) ω Q ) ω ) = P ) pr ) ) ) \ ) ω where ) = { j ) pr ) )) = ) ω x q ees he nd ω ees he condon posed by he C } P ))= λ ) λ N nd P ) = P [ λ )] λ )! e where
Tn Corn Dosescu Consnn Rsch Accordng o Theore follows h: = Q q f ) K )R )) s probbly spce where = R )= ) Q ) Q ) Q ) P For e f g h) ) nd ω = follows: R )){} e { f} { g} { h} )= P ) ω {} e ) Q ) e { f} e n e e f ) e f){ g} ) Q e f g){ h} ) = Cn p q Cn h g n g g [ λ ) ] λ ) ) C n For p q = h! A K ) nd e ) le us e A e = { f ) e f) pr ) A pr ) A } A e f = { g ) e f g) pr ) A pr ) A pr ) A } nd A e f g = { h ) e f g) A } Ou of bove resuls: R A = P ω e Q e A ) ) ) {}) ) e e f A g A h A p n ) ) e e f e f g Q e f) Ae f) Q e f g) Ae f g) Le us e ) = ) ) ) ) ) N he rndo funcon whch deernes he se S ) where ) : ) R In generl becuse no ll probbly spces correspond here s no rndo vecor = e f 5 Conclusons ) Accordng o defnon β - dependence llows he odfcon of he Borel spces on whch he rndo vrbles re defned condons h y be consderng nd ulvlen Thus he evoluon of rndo phenoenon h es plce n severl "seps" y be nfluenced by exernl condons bvlen or ulvlen b) Also fne se consss of severl econoc vrbles h re nerreled cn be consdered rndo phenoenon h es plce
Anoher Cegory of he Sochsc Dependence for Econoerc Modelng of n "seps" In hs cse "sep" s n econoc vrble beween "seps" cn defne convenen β - dependence nd vlues of econoc vrbles re obned s resul of econoerc odelng of he generon process c) A gol of fuure reserch would be o consruc rndo lgorh see [8]) o esblsh forecss on fuure vlues of econoc vrbles REFERENCES [] Cucu G Tudor C 978) Probbles nd Sochsc Processes voli n Ronn) Acdey of RSR Publshng House; []EngleRF Grnger CWJ 987) Co-negron nd Error Correcon: Represenon Eson nd Tesng Econoerc vol55 No Mr; [] Fuller AW996) Inroducon o Sscl Te Seres John Wley nd Sons Inc New Yor; [] Percch F 00) Econoercs John Wley & Sons Chceser New Yor; [5] Peron F Auslos M Roundo G 007): Generng Synhec Te Seres fro B-Sneppen Co-evoluon Model Pysc A 8 59-67 vlble onlne wwwscencedrecco; [6] Ruxnd Gh Soenescu Cpoeru) S 009): Bvre nd Mulvre Conegron nd her Applcon n Soc Mres Econoc Copuon nd Econoc Cybernecs Sudes nd Reserch Vol nr ASE Publshng; [7] Green W008) Econoerc Anlyss Sxh Edon Prence Hll Publshng;