THE ECONOMETRIC MODELING OF A SYSTEM OF THREE RANDOM VARIABLES WITH THE β DEPENDENCE

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1 Tin Corin DOSESCU hd Dimiri Cnmir Chrisin Univrsiy Buchrs Consnin RAISCHI hd Dprmn o Mhmics Th Buchrs Acdmy o Economic Sudis THE ECONOMETRIC MODELING OF A SYSTEM OF THREE RANDOM VARIABLES WITH THE β DEENDENCE Absrc Wihin clssicl conomric modlling bcus o h complxiy o d gnring procss w hv chosn i o b prsnd wih h hlp o s o ssumpions whr h rndom is s conrolld s possibl so h h civiy should b moniord nd vn dosd This ppr hs chosn sysmic nd cybrnic pproch o h disply o h d gnring procss Th irs sg o h d gnring procss dcoding h uhors sudy sysm md o hr rndom vribls dind on dirn probbiliy spc bwn which hr is spcil priori dpndnc clld β dpndnc Th rsrch includs sing probbilisic modl o such sysm nd mking rprsniv xmpl Th rsrch nds wih n nlysis o h sysmic nd cybrnic rprcussions corrsponding o h β dpndnc Ky words: probbiliy spcs produc o probbiliy spcs sochsic procsss im sris conomic vribls conomric modl JEL CLASSIFICATION: C5 AMS000: 60G99 INTRODUCTION In sysm o conomic vribls h clssicl rgrssion nlysis consiss minly in [] [] [7] [8] [9] [0] [] in considring on o h sysm vribls s dpndn on h rs o h vribls nd ll h conomic vribls vlus producd by n undrling d gnring procss on which w mk s o ssumpions For xmpl w my considr h sysm md o h ollowing hr mcroconomic indxs vribls: GD consumpion nd invsmns This sysm rprsns subsysm o h whol sysm o indxs which r usd or h nir conomy h r pr o h Nionl Accouns Sysm Mking us o h indx sysm voluion during whr h indxs r:

2 Tin Corin Dosscu Consnin Rischi Indx Msurmn uni GD Mild li Consumpio Mild li n C Invsmns I Mild li Sourc: Th Romnin Sisics Yrbook 996 W shll rsrch wh hppns whn w ry rgrssion nlysis bsd on h conomic links bwn h hr indxs In cs o h clssicl rgrssion nlysis w irs prsuppos h ch indx is non-rndom vribl Thn w prsuppos h on o h indxs or xmpl invsmns dpnds on h ohr indxs nd w mk n conomric modl o linr rgrssion such s I = GN b C ε Hr w hv som prmrs such s nd b which will b simd ccording o h d in h bl nd h rndom vribl ε clld disurbnc which cumuls h c o ohr unknown or nglcd cors on which w mk crin ssumpions Th nir oucom dpnds on h d xising in h bl I is obvious h h d r h rsul o ll conomic procsss wihin h nionl conomy which hv bn mixd wih ohr procsss involving h populion bhviour nionl lvl h inrnionl conomic rlions s wll s h world wid sus o h conomy Wihin h clssicl conomric modlling bcus o h complx d gnring procss w hv chosn h i will b prsnd ccording o s o ssumpions whr h rndom is s conrolld s possibl so h h civiy b moniord nd vn dosd Th prsn ppr chooss sysmic nd cybrnic pproch bsd on d gnring procss Such n pproch lds h sysm conomric modlling md o hr indxs o sysm o hr rndom vribls dind on dirn probbiliy spcs or which hr is no nd o clcul h covrinc mong h pirs o rndom vribls or o drmin h rgrssion uncion in rlion o h ohrs In conclusion hr is no us or h conomric modls o rgrssion wihin his pproch Morovr n imporn scion o h clssicl conomric modlling is no dqu or such n conomic indx sysm Th prsn ppr proposs h rsrch o sysm md o hr rndom vribls bwn which hr is spcil priori dpndnc clld β dpndnc s h irs sg o dcoding h d gnring procss Th ppr includs sing probbilisic modl o such sysm nd h consrucion

3 Th Economric Modling o Sysm o Thr Rndom Vribls wih β o rprsniv xmpl I nds wih n nlysis o h sysmic nd cybrnic rprcussions corrsponding o h β dpndnc THE RANDOM SYSTEM WITH CONSTITUENTS THAT GENERATES TIME SERIES DATA W considr rndom sysm S which dvlops in discr im nd is dind by h ordrd ripl N * Th ordr rrs o h c h ch im w irs considr hn nd hn i is rndom vribl which is dind on h probbiliy spc K i i hving i s h is h mos numrbl i = i L us considr xi = i ω wih ω i nd i = whr x i mns h vlu o h rndom vribl i im i = Th vlus o h hr rndom vribls mk hr im sris d h corrspond o h indicors mrkd I I nd I Thr is rciprocl dpndnc bwn h hr rndom vribls h is dind s ollows: Th β dpndnc s dpndnc on A N * \{} im h probbiliy spc on which rndom vribl is dind dpnds on h vlu o s ollows: I x ms h condiion imposd by C hn K =B bing pr o h sm clss o probbiliy lws or probbiliy disribuions such s nd : R whr ω ω = k ω \ k R \ I x dos no m h condiion imposd by C hn = K = K - = nd = b s dpndnc on A N * \{} im h probbiliy spc on which rndom vribl is dind dpnds on h vlu o s ollows: I x ms h condiion imposd by C hn

4 Tin Corin Dosscu Consnin Rischi K =B bing pr o h sm clss o probbiliy lws or probbiliy disribuions such s nd : R whr ω ω = k ω \ k R \ I x dos no m h condiion imposd by C hn = K = K - = nd = c s dpndnc on A im N * h probbiliy spc on which rndom vribl is dind dpnds on h vlu o s ollows: I x ms h condiion imposd by C hn K = B bing pr o h sm clss o probbiliy lws or probbiliy disribuions such s nd : R whr ω ω = k ω \ k R\ I x dos no m h condiion imposd by C hn = K = K = nd = Th rlion S ~ is mningul on h bsis o h β dpndnc in ordr o dsign h c h h s o h rndom sysm im is chrcrizd by h riniy Rmrk A N * im h sysm hs h s S ~ which dpnds on h s S dpnds on bcus ccording o h c procdur rom h β dpndnc On h ohr hnd dpnds on ccording o procdur nd dpnds on ccording o b procdur Rmrk A im h c o x s vlu upon h rndom vribl r nd h c o x s vlu upon h rndom vribl

5 Th Economric Modling o Sysm o Thr Rndom Vribls wih β insnnous In his wy h simulniy rlions bwn h hr rndom vribls r posul Rmrk Th procdur c is vriy o dbck bwn consiuns o sysm S Thorm T Th probbiliy modl o h s S dpnds on h vlus o x which ppr in h S s s ollows: I x ms h condiion imposd by C hn h probbiliy modl o S is rprsnd by h hr-dimnsionl rndom vribl = which is dind on h probbiliy = spc K K= B : K [0] { } { } { g} = { } { } { g} g = = { ω ω = x \ ms h condiion imposd by h C } : K [0] K = B \ = = { ω ω = x ms h condiion imposd by h C } : K [0 ] K = B Also hppns wih A = pr A pr A} nd A K { A = ={ g g } A = {} A A or A = A A d d I x dos no m h condiion imposd by C hn h probbiliy modl o S is rprsnd by h hr-dimnsionl rndom vribl = which is dind on h probbiliy

6 Tin Corin Dosscu Consnin Rischi spc K = K = B : K [0] { } { } { g} = { } { } g = = { ω \ ω = x ms h condiion imposd by h C } : K [0] K = B \ = = { ω ω = x ms h condiion imposd by h C } : K [0 ] K = B Also hppns A = pr A pr A} nd = A K ={ g g A {} = A { } A A or = A A d d A roo Suppos h x ms h C condiion Thn ccording o h c procdur rom h β dpndnc w hv K = B h is K is h Borlin gnrd ild o bu is pr o h sm clss o probbiliy lws s wll s Following h smn bov im h rndom vribl is dind on h probbiliy spc K According o procdur rom h β dpndnc w my only hv wo siuions:

7 Th Economric Modling o Sysm o Thr Rndom Vribls wih β x ms h C condiion In his cs K = B nd is pr o h sm clss o probbiliy lws s wll s W rm h probbiliy lw Following im h rndom vribl is dind on h probbiliy spc K x dos no m h C condiion In his cs = K =K = nd im h rndom vribl is dind on h probbiliy spc K Ou o h wo siuions w my dduc h hr r dind on h msurd spc K wo probbiliis: nd whr ω=0 or ω \ According o b procdur rom h β dpndnc w my only hv wo siuions: x ms h C condiion In his cs K = B nd is pr o h sm clss o probbiliy lws s wll s W rm h probbiliy lw Following im h rndom vribl is dind on h probbiliy spc K x dos no m h C condiion In his cs = K =K = nd im h rndom vribl is dind on h probbiliy spc K Ou o h wo siuions w my dduc h hr r dind on h msurd spc K wo probbiliis: nd whr ω=0 or ω \ In conclusion w hv h probbiliy spc K h msurbl spc K whr wo probbiliis nd r dind nd h msurbl spc K whr wo probbiliis nd r dind According o h proposiion rom h Annx [4] [4] i ollows h h probbiliy modl o h s S is givn by h hr-dimnsionl rndom vribl = which is dind on h probbiliy spc K whr:

8 Tin Corin Dosscu Consnin Rischi = K = = K K K nd : K [0] g { } { } { g} = { } { } { g} whr: = \ nd is h s dind in h horm conn nd \ = is h s dind in h horm conn Ou o bov rsuls nd A K wih A = pr A pr A} nd = { ={ g g = g A i ollows: A g A A {} = } nd by h rlion: A A A A or = A A d d Suppos h x dos no m h C condiion Thn ccording o c dpndnc w hv: = K =K rom h β = nd = According o rom h β dpndnc hr r wo siuions h w hv dscribd in In conclusion w hv h probbiliy spc K h msurbl spc K on which h wo probbiliis nd r dind nd prsnd in nd h msurbl spc K whr wo probbiliis nd r dind nd prsnd in According o h proposiion rom h Annx i ollows h h probbiliy modl o h s S is givn by h hr-dimnsionl rndom vribl = which is dind on h

9 Th Economric Modling o Sysm o Thr Rndom Vribls wih β probbiliy K = K nd : K whr: = K K K [0] g { } { } g = { } { } g whr { } { } nd probbiliis bing dind in Ou o bov rsuls A K wih A = pr A pr A} nd = { ={ g g = g A i ollows: A g A A {} = A } nd by h rlion: A A A or = A A d d EAMLE W considr sysm S md up o hr urns U U nd U Urn U conins n blls o h sm siz numbrd rom o n n 0 rom which r whi nd h rs blck 0< < n Urn U conins n blls o h sm siz numbrd rom o n nd urn U conins n blls o h sm siz numbrd rom o n Exrcions r don rom sysm S An xrcion rom sysm S consiss in rndom xrcion rom urn U ollowd by n xrcion o h sm yp rom urn U nd nx xrcion rom urn U W considr hr condiions CC nd C which rr o h rsuls o h xrcions I h xrcd bll rom urn U is whi i i ms h condiionc hn w inroduc sm yp bll ino urnu ssigning i h numbr n On conrry siuion h srucur o urn U dos no chng Thn w do h xrcion rom urn U I h rsul o h xrcion rom urn U ms h condiion C hn w U ssigning i h numbr n on conrry inroduc sm bll ino urn siuion h srucur o urn dos no chng Thn w do h xrcion rom urn U

10 Tin Corin Dosscu Consnin Rischi I h rsul o h xrcion rom urn U is n vn numbr i i ms h condiion C hn w inroduc whi bll ino urn U ssigning i h numbr n on conrry siuion h srucur o urn U dos no chng Thn w do h ollowing xrcion rom h sysm W will ch o ch urn probbiliy spc nd rndom vribl dind on his spc In ordr o do his w will considr h ollowing noions L us k s h numbr o xrcions rom sysm S N * n i h numbr o blls rom urn U bor h xrcion o h cgory i= W i noic h ni ni N * L us k s h numbr o whi blls rom h urn U bor h xrcion o h cgory W noic h N * Th probbiliy spc chd o urn U bor h xrcion o h cgory is = K whr ω n = { ω K ωn } = ω ω \ n = { ω ω h vn o obining whi bll} L us k : R is h rndom vribl h rprsns h rsul o h xrcion o h cgory rom urn U nd l us k x = ω Th probbiliy spc chd o urn U bor h xrcion o h cgory is K whr = { ω K ω n } K = ω = ω n : L us k R is h rndom vribl h rprsns h rsul o h xrcion o h cgory rom urn U nd l us k x = ω Th probbiliy spc chd o urn U bor h xrcion o h cgory is K whr = { ω K ω n } K = ω = ω n

11 Th Economric Modling o Sysm o Thr Rndom Vribls wih β L us k : R is h rndom vribl h rprsns h rsul o h xrcion o h cgory rom urn U nd l us k x = ω ω Following h signiicnc o h hr rndom vribls w cn S ~ wri According o h xrcion mod rom h sysm S dind bov hr is bwn h hr rndom vriblsw mus nlyz dpndnc β succssivly h hr dpndncis rom h diniion o β s dpndnc on In h xrcion o h cgory N * h probbiliy spc on which h rndom vribl is dind dpnds on h vlu o s ollows: - i x is whi bll i i ms h condiion imposd by C hn = { ω K ω } = { ω K ω } n n bcus n >n - K = ω = ω n I ollows h ω ω : R whr ω ω = n = n ω \ -i x is blck bll i i dos no m h condiion imposd by C hn = K =K - = nd = b s dpndnc on In h xrcion o h cgory N * h probbiliy spc on which h rndom vribl is dind dpnds on h vlu o s ollows: - i x ms h condiion imposd by C hn = { ω K ω n } = { ω K ω n } bcus n >n - K = ω = ω n I ollows h ω ω : R whr ω ω = n = n ω \ - i x dos no m h condiion imposd by C hn = K =K - = nd =

12 Tin Corin Dosscu Consnin Rischi c s dpndnc on Th xrcion o h cgory N * Hppnd h probbiliy spc on which h rndom vribl is dind dpnds on h vlu o s ollows: - i x is bll wih n vn numbr i i ms h condiion imposd by C hn = { ω K ω } bcus n <n ; n K = ω = ω < < n I ollows h ω ω : R whr ω ω = n ω \ - i x is bll wih n odd numbr i i dos no m h condiion imposd by C hn = K =K = nd = L us pply h Thorm T o h sysm I ollows h h probbiliy modl o h s S dpnds on h vlu o x which pprs wih h s S whn h xrcion hppns I x is bll wih n vn numbr hn h probbiliy modl is rprsnd by h hr-dimnsionl rndom vribl = dind on h probbiliy spc K whr = K =K K K : [0] g { } { } { g} = { } { } { g} K = \ = { ω x ms h condiion imposd by h C } : K [0] ω = ω n K = B

13 Th Economric Modling o Sysm o Thr Rndom Vribls wih β \ = = { ω ω = x ms h condiion imposd by h C } : K [0 ] ω = ω n K = B Also hppns wih A = pr A pr A} nd A K A = ={ g g A = {} { } A A I x dos no m h condiion imposd by C hn h probbiliy modl o S is rprsnd by h hr-dimnsionl rndom vribl dind on h probbiliy spc = K whr: = K K [0] nd : g { } { } g = { } { } = =K K K { } { g} = \ \ I is h sm cs or A K hving A nd A rom bov A = {} A A

14 Tin Corin Dosscu Consnin Rischi 4 CONCLUSIONS Ths r h conclusions on h β dpndnc Th condiions C i i= r o drminisic nur nd hy orm n inrc bwn h S sysm nd h our nvironmn This wy h sysm is opn nd hy llow cybrnic pproch o i bcus hr is h possibiliy o rgul hrough conomic or ohr msurs Th condiions C i i= r bivln: ccomplishd/nonccomplishd I is possibl o ormul mulivln condiions which would rin communicion wih h our nvironmn Th nur o h hr componns blonging o h β dpndnc is dirn Thus h nd b dpndncs rr o h inlunc o rndom vribl on h ollowing on whil h c dpndnc is dbck yp nd i inroducs circulion rlion bwn nd 4 I nd momns hos hr condiions r ulilld hn h rndom vribls i i= r dind on h vrious probbiliy spcs which dir rom hos nd momns * 5 I hr is im spn T N so h ll condiions r no ulilld ROOSITION L hn = T i= i i In his insnc bwn hos hr rndom vribls o S sysm h β dpndnc is no displyd nd h clssicl rgrssion sudy mks sns Also in h hypohsis h S sysm is subsysm o h sysm bwn hos hr rndom vribls hr my xis nohr dpndnc which is displyd h lvl o h sysm E F G ANNE b ss h r h mos numrbl Thn h probbiliis on h msurd spc E F G E F G whr M is h s o sids o h s M r in on-o-on nd ono corrspondnc wih h sysms Q Q E Q > 0 Q F Q { } > 0 whr Q is probbiliy on h msurd spc E E Q E r h probbiliis on h msurd spc F Q r h probbiliis on h msurd spc G o on probbiliy on h msurd spc E F G E F G probbiliis: F bu G Th corrspondnc is hus: chs

15 Th Economric Modling o Sysm o Thr Rndom Vribls wih β {} Q= o pre wih Q {} = pre {} = F G Q = pr { } o prf wih Q { } = { } prf { } = pre {} { } G {} F G = pr { } pr { } o prg wih Q { g} = E I F { } { } prg { g} pre I prf {} { } { g} = = { } { } G E F { g} whr Q nd Q {} { } G = Q condiiond probbiliis; o on sysm Q E Q > 0 Q F Q { } > 0 Q chs h probbiliy on h msurd spc E F G E F G wih {} { } { g} = Q {} Q { } Q { g} nd or A E F G wih A F pre A prf A A = { g G g A} w hv A Q{} Q A Q A = or A = Q A dq dq { } = nd REFERENCES [] Andri T 00 Sisică şi conomri Economic ublishing Hous Buchrs; [] Agpi Adrin Bădin L 009 Conidnc Inrvls or h Gussimion Algorihm: A Boosrp Approch Economic Compuion nd Economic Cybrnics Sudis nd Rsrch Vol 4 no ASE ublishing Hous Buchrs; [] Bourbonnis R 00 Economri - mnul xrciss corrigs Dunod ris; [4] Cuculscu I 988 Tori probbiliăńilor ALL ublishing Hous Buchrs ; [5] Dosscu T C nd Rischi C 008 Th Economric Modling o Rndom Sysm h Gnrs Tim Sris D Economic Compuion nd Economic Cybrnics Sudis nd Rsrch Vol 4 n0-4 ASE ublishing Hous Buchrs; [6] Fullr A W 996 Inroducion o Sisicl Tim Sris John Wily nd Sons Inc Nw York; [7] Grn W008 Economric Anlysis Sixh Ediion rnic Hll ublishing; r

16 Tin Corin Dosscu Consnin Rischi [8] Grobnr D F Shnnon W Fry C h nd Smih K D 005 Businss Sisics Updd sixh diion rson Educion Inrnionl; [9] Iosiscu M Mihoc Gh nd Thodorscu R 966: Tori probbiliăńilor şi sisică mmică Thnic ublishing Hous Buchrs; [0] Krry rson 000 An Inroducion o Applid Economrics: A Tim Sris Approch ublishd by ALGRAVE; [] Ruxnd Gh Soin Adr 008 Modlling o h Rlionship bwn Forign Dirc Invsmn nd Economic Growh Economic Compuion nd Economic Cybrnics Sudis nd Rsrch Vol 4 no -4 ASE ublishing Hous Buchrs; [] Spăru S 007 Modl şi mod conomric ASE ublishing Hous Buchrs; [] Sock H Jms 00 Inroducion o Economrics Addison Wsly Boson Sn Frncisco Nw York ris Monrl; [4] Tudor C 004: Tori robbilińílor Univrsiy o Buchrs ublishing Hous