Simulation of Non-normal Autocorrelated Variables

Transcription

1 Jounal of Moden Appled Sascal Mehods Volume 5 Issue Acle Smulaon of Non-nomal Auocoelaed Vaables HT Holgesson Jönöpng Inenaonal Busness School Sweden homasholgesson@bshse Follow hs and addonal wos a: hp://dgalcommonswayneedu/masm Pa of he Appled Sascs Commons Socal and Behavoal Scences Commons and he Sascal Theoy Commons Recommended Caon Holgesson HT (005 "Smulaon of Non-nomal Auocoelaed Vaables" Jounal of Moden Appled Sascal Mehods: Vol 5 : Iss Acle 5 DOI: 07/masm/ Avalable a: hp://dgalcommonswayneedu/masm/vol5/ss/5 Ths Regula Acle s bough o you fo fee and open access by he Open Access Jounals a DgalCommons@WayneSae I has been acceped fo ncluson n Jounal of Moden Appled Sascal Mehods by an auhozed edo of DgalCommons@WayneSae

2 Jounal of Moden Appled Sascal Mehods Copygh 006 JMASM Inc Novembe 006 Vol 5 No /06/$9500 Smulaon of Non-nomal Auocoelaed Vaables HT Holgesson Depamen of conomcs and Sascs Jönöpng Inenaonal Busness School All sascal mehods ely on assumpons o some exen Two assumpons fequenly me n sascal analyses ae hose of nomal dsbuon and ndependence When examnng obusness popees of such assumpons by Mone Calo smulaons s heefoe cucal ha he possble effecs of auocoelaon and non-nomaly ae no confounded so ha he sepaae effecs may be nvesgaed Ths acle pesens a numbe of non-nomal vaables wh non-confounded auocoelaon hus allowng he analys o specfy auocoelaon o shape popees whle eepng he ohe effec fxed Key wods: Auocoelaon non-nomaly confoundng obusness Inoducon All sascal mehods ely on assumpons o some exen These assumpons fo example may be ha some momens ae fne o ha he vaance s homogenous a all daa pons Ohe assumpons nvolve nomal dsbuon o ndependence If some of he assumpons ae volaed hen he expeced popees of he mehod may no longe hold Fo example a sascal hypohess es ha eques ndependence of he daa may seously ove eec unde he null hypohess f he daa possess auocoelaon I s heefoe mpoan o nvesgae he obusness popees of sascal mehods befoe hey ae appled o eal daa Alhough moden compues ae developng a a apd pace has become nceasngly popula o H T Holgesson s Assocae Pofesso a Jönöpng Inenaonal Busness School wh eseach and eachng Hs eseach has manly been focused on assessng dsbuonal popees Reseach also ncludes mulvaae analyss and fnancal sascs pefom obusness sudes of such assumpons by Mone Calo smulaons Howeve when examnng obusness o auocoelaon and non-nomaly some echncal poblems ase Because auocoelaon usually s geneaed by a ecusve sequence of andom numbes he cenal lm heoem wll foce he auocoelaed vaable o be moe nomal when compaed o he vaable used o geneae he sequence Fo example magne he poblem of nvesgang he obusness of a non-nomaly es o auocoelaon If a sewed vaable s used o ecusvely geneae a sewed and auocoelaed vaable hen hs new vaable wll be moe symmec han he ognal one and wll be moe symmec he lage he auocoelaon s Thus he smulaon sudy wll no eveal he sepaae effecs of auocoelaon and non-nomaly as was nended Seveal such examples ae o be found n he leaue Fo example Shuu (000 examned he obusness of an auocoelaon es o non-nomaly by geneang a fs ode auoegessve pocess wh non-nomal dsubances and Ba and Ng (005 appled he fs ode auoegessve pocess wh nonnomal dsubances o nvesgae he obusness of a non-nomaly es o auocoelaon Such effecs of confoundng can be avoded by usng alenave mehods of geneang he vaables 408

3 HT HOLGRSSON 409 A numbe of vaables ae poposed ha ae non-nomal bu auocoelaed and allow fo sepaae conol of he shape popees (sewness/uoss and auocoelaon All poposed vaables ae easy o geneae n sandad sofwae pacages Non-Nomal And Auocoelaed Vaables The queson of how o smulae andom vaables wh gven dsbuonal popees have been gven a gea deal of aenon (see Johnson (987 fo a geneal descpon One of he mos fequenly used mehods s he socalled nvese mehod Assume he poblem consss of geneang a vaae X whose dsbuon s specfed by F and ha F s scly nceasng wh nvese funcon F Then he nvese mehod consss of fs geneang a unfomly dsbued vaae U and hen calculang he vaable of nees by X F ( U Ths mehod s fas and smple because all sascal sofwae povdes facles fo geneang unfomly dsbued vaaes Unfounaely he ssue becomes much moe complcaed when geneang auocoelaed vaables wh gven dsbuon because he mappng U X usually wll change he auocoelaon paen of U dascally Fuhemoe s no an easy as o geneae auocoelaed vaaes whch ae unfomly dsbued Ths suggess ha he nvese mehod s no vey useful fo he poblem of geneang auocoelaed vaaes wh gven magnal dsbuon and so ohe mehods ae usually appled fo ha pupose In pacula such vaaes ae fequenly geneaed by fne ode ARMA models ofen he AR( pocess wh gven dsbuon of he dsubances Unfounaely hs mehod wll no esul n he dsbuon amed a Ths can be seen fom he followng: Consde he lnea pocess defned by 0 ψ δ ( whee δ ae some zeo mean ndependenly dencally dsbued vaables and he ψ s ae consans such ha ψ 0 < Whou loss of genealy assume ha 0 and Va ( δ 0 < σδ < The sequence ( s nown as he nfne ode movng aveage pocess also efeed o as a lnea pocess and encompasses all saonay vaables accodng o Wold s decomposon heoem I s seen decly fom ( ha because he gh hand s he sum of a (possble nfne numbe of vaables The lef hand sde wll n geneal be a leas as nomally dsbued as he δ due o he cenal lm heoem Fo example he AR( pocess defned by φ + δ ( s a specal case of ( whee ψ φ Then follows ha he dsbuon of wll be moe nomal when compaed o δ fo φ 0 Now s no geneally ue ha ψ > 0 An example s he fne ode MA(q model Howeve s eadly seen ha aleady he sewness of an MA( pocess s close o ha of a nomal dsbuon when compaed o he sewness of δ In fac may be shown ha fo a pocess defned by δ + θδ he sewness of s gven by ( + θ ( + θ δ [ ] [ ] : ( whee δ s he sewness coeffcen of δ (see Appendx Hence f fo example δ 5 and θ 07 he sewness becomes

4 40 SIMULATION OF NON-NORMAL AUTOCORRLATD VARIABLS whch s less han half ha of δ Thus f one wshes o nvesgae he powe popees of an auocoelaon es when appled o non-nomal daa and apples model ( fo dffeen values of θ hen he effec of non-nomaly wll be confounded wh he powe popees because he sewness s a dec funcon of he auocoelaon In lgh of he above dscusson one may wonde how auocoelaed and non-nomal vaables should be geneaed hen The fac ha auocoelaon n geneal wll smooh ou nonnomaly suggess ha auocoelaed nomally dsbued vaables should be geneaed a a fs sep and he non-nomaly should be mposed n he second sep Fuhemoe he ansfomaon o non-nomaly should esul n a smple elaon beween he auocoelaon of he ognal vaable and ha of he ansfomed hus allowng fo oal conol of he auocoelaon paen In he followng ha pncple wll be appled n a sees of heoems ha descbe he geneaon of he vaables and s popees Theoem Le φ δ δ ~ N 0 φ < and defne : σ whee σ V ( φ Then ~ χ( ndependenly of he value of φ and he auocoelaons of s gven by ρ φ d + whee Poof of Theoem : The vaance of he AR( pocess s well nown o be gven by σ ( φ and hence ~ N( 0 and he ch-squae dsbuon of follows decly The γ of ae gven by: auocovaances γ σ σ ( φ + ε σ ( + + φ φ ε ε φσ σ + φσ ε σ + σ ε σ φ + { φ } + φ + 0+ σ ( φ + φ + σ φγ + φ + ( ( φγ Fuhemoe ( 0 ( ( φ γ By usng ecuson follows ha ha γ ( φ The auocoelaon funcon of s hus deemned by ρ ( 0 γ γ φ and Theoem follows In ohe wods he auocoelaon behaves le ha of an AR( pocess wh auoegessve paamee φ whle he dsbuon s χ ( Also noe ha he shape popey s ndependen of he auoegessve paamee Now hs vaable s hghly sewed and may no be appopae n suaons whee neanomal dsbuons ae equed The nex

5 HT HOLGRSSON 4 heoem poposes a geneal χ ( dsbued vaable (whch lms a nomal dsbuon as nceases wh non-confounded auocoelaon: Theoem Le Z : whee ae muually ndependen vaables defned as n Theoem wh common auocoelaon paamee φ Then he sewness and uoss of Z ae gven by Z 8 and Z + especvely ndependenly of φ and he auocoelaon of Z s gven by ρ φ ndependenly of Z Poof of Theoem : The ch-squae dsbuon follows decly fom he fac ha a sum of ndependen χ ( vaaes s dsbued as χ ( The sewness and uoss of such vaaes ae well nown and can be found n Johnson e al (994 The auocovaance s obaned by usng he popey γ ( (gven n he devaon of Theoem : γ Z Z Z Z : [ ] [ ] [ ] + ( Hence he sewness and uoss of Z gven by: + + γ ( ( φ 0 In pacula Z ρ ( φ γ 0 φ and so Z as was o be shown In ohe wods he shape-pa of he dsbuon of Z s χ ( and he auocoelaon-pa of he dsbuon s ρz ( φ and none of he effecs s confounded o he ohe Thus he sewness and uoss can be deemned ove an abay (hough dscee ange of values ndependenly of he auocoelaon The nex heoem poposes a symmec non-nomal vaable wh non-confounded auocoelaon: Theoem Le Z be defned as n Theoem Also le c φc + ε whee ε ~ N ( 0 and defne c : c ( φ d and W : c Z Then he sewness and uoss of W ae gven by W 0 and W ( + ndependenly of φ whle he auocoelaons ae gven by φ ndependenly of Poof of Theoem : Fsly noe ha c and ( Z ae muually ndependen and ha W [ ] 0 and [ ] W [ ] [ ] W 0 W c Z W 4 ( W ( ( W W 4 4 c Z c Z W ae 4 4 { c }{ ( c Z Z } { } +

6 4 SIMULATION OF NON-NORMAL AUTOCORRLATD VARIABLS Hence he W vaable s symmec wh uoss deemned by wh ange 9< W 45 Fuhemoe by usng he esuls of he poofs of Theoem and Theoem he auocovaances may be obaned: ( [ WW ] 0 [ Z] [ ] ( ( ( ZZ Z Z { ( ZZ } ( γ czc W cc Z Z φ + { } φ { } { } φ + φ φ γ φ ( φ Hence γ ( W φ and n pacula γ W ( 0 Thus he auocoelaons ae gven by ρ ( W φ φ ndependenly of as was o be shown Thus Theoem povdes a symmec bu nonnomal vaable whee he auocoelaons can be dencal o hose of an AR( pocess by pung he auocoelaon paamee of he ognal vaable equal o φ In geneal he vaables pesened n Theoems - shae he popey ha hey all have auocoelaons ha decay slowly n he sense ha hey ae non zeo a all lags f φ 0 In many nsances s of nees o geneae auocoelaons ha ae zeo above a cean lag Theefoe some sho memoy pocesses of MA( ype wll also be poposed These ae pesened below: Theoem 4 Le δ θδ δ ~ N 0 and defne : σ whee σ + θ d whee Then ~ χ( ndependenly of φ and he auocoelaons ae gven by ( ρ θ + θ and ρ 0 > ndependenly of Poof of Theoem 4 The ch-squae dsbuon follows vally as s a sandad nomally dsbued vaae The second ode momens ae gven by γ 0 V γ ( + ( ( θ ( ( θ ( δ θδ ( δ θδ + δδ + δθδ ( ( + θ θδ + θδ δ 4 ( θ θ ( θ θ ( +θ An analogous poof eveals ha γ ( 0 f > Hence he auocoelaons of ae gven by ( ρ θ + θ θ + θ ρ 0 > as was o be shown Thus he auocoelaons of ae hose of an MA( pocess whee he auocoelaon a lag and θ + θ whch n un s bounded beween 0 and 05 (he maxmum beng eached a θ Ths vaable may also be exended o an abay χ ( dsbuon: equals he oo of

7 HT HOLGRSSON 4 Theoem 5 Le Z : whee ae muually ndependen vaables defned as n Theoem 4 wh common auoegessve paamee θ Then he sewness and uoss of Z ae gven by Z 8 and Z + ndependenly of θ and he auocoelaons of Z ae gven by ( ρ θ + θ and ρ 0 > ndependenly of Poof of Theoem 5 The sewness and uoss ae movaed n Theoem Analogous o he poof of Theoem γ Z ( ( and as he vaance s gven by γ Z ( 0 Va [ Z ] follows ha he auocoelaons ae gven by and { } ( ( + ρ θ θ ( θ θ + ρ 0 > as was o be shown The Z vaable may also be symmezed accodng o he followng: Theoem 6 Le Also le c ε θε Z be defned as n Theoem 5 ε ~ N 0 whee and and defne c : c ( + θ W : c Z Then he sewness and uoss of W ae gven by W 0 and W ( + ndependenly of θ and he auocoelaons of W ae gven by ( ρ W W 0 ndependenly of Poof of Theoem 6 The sewness and uoss ae gven n he poof of Theoem The auocovaance of W s Hence W γ ( WW [ ] [ ] ( ( θ ( Z Z θ Cov[ Z Z ] W cc Z Z (( + ( θ θγ ( { } ( 0 W ( ( γ γ θ θ + θ 0 γ W > and he heoem follows Hence he auocoelaons behave le hose of an MA( pocess wh MA paamee θ + θ whch deemned by he oo of s bounded n he neval ( 00 wh maxmum a θ ± 7 The vaables poposed above ae all unvaae I wll somemes be of nees o geneae mulvaae vaables wh coss coelaon beween pawse magnal vaables When mposng such coss coelaon one wsh o do ha n a manne ha does no ale he magnal dsbuons Ths can be acheved by leng one o seveal vaables used o fom he magnal vaables be dencal (fxed n all magnal vaables The nex heoem descbes such vaables and s man popees: Theoem 7 Le P be muually ndependen vaables defned as n Theoem o Theoem 4 dependng on whehe he AR( o MA( pocess have been used o geneae he auocoelaon wh common auocoelaon paamee φ (o θ Then

8 44 SIMULATION OF NON-NORMAL AUTOCORRLATD VARIABLS defne Z : h + h+ P and le Z Z ZP be a andom veco of he magnal vaables Z Then he coss coelaons beween wo magnal vaables of Z ae gven by ( Co Z Z h Poof of Theoem 7 Assume fo he momen ha h Then on obsevng ha each Z s a ch squae vaae he covaance becomes: Cov Z ( Z Z Z Z Z ( ( ( ( ( ( Because he magnal Z vaables ae dsbued as χ ( follows ha he coelaon s gven by Co ( Z Z By applyng an analogous poof fo a geneal 0 h s seen ha Cov( Z Z h and hence he coss coelaon s gven by Co ( Z Z h / h whch complees he poof In ohe wods Theoem 7 poposes a andom veco wh magnal vaables of he nd descbed n Theoem (o Theoem 5 hough wh coss coelaons gven by h I s also possble o geneae symmec mulvaae vaables as shown n he nex heoem: Theoem 8 Le le W : c ( Z Z be defned as n Theoem 7 and whee c s defned as n Theoem (o Theoem 6 Then he coss coelaon beween W and W s gven by ( Co W W h ndependenly of φ (o θ Poof of Theoem 8 The coss covaance s gven by Cov W W Fnally Va W ( W ( W 0 ( c ( Z c( Z c ( Z ( Z ( Z Z ( h h + ( ( ( ( W W c Z 0 ( ( Va ( Z c Z Ths complees he poof The vaables poposed above allow fo sepaae conol of shape popees and auocoelaons In pacula he vaables fomed by ansfomaons of AR( pocesses

9 HT HOLGRSSON 45 (hose of Theoems - behave le AR( pocesses and he doman of he auoegessve paamee emans < φ < In ohe wods hese vaables povde a ool fo geneang AR( pocesses wh non-nomal dsbuons On he ohe hand he domans of he auoegessve paamee (whch s equal o he whole eal lne hough usually ep n he span < θ < of he MA( ype vaables (hose of Theoems 4-6 do no eman when ansfomed o non-nomaly Ths mgh be a dawbac n some nsances hough hey do povde a ool fo nvesgang he effec of non-nomal sho memoy pocesses In geneal he poposed vaables should cove mos obusness poblems me n pacce Fuhe eseach n he mae could nvolve he developmen of geneang non-nomal pocesses wh long memoy of ARFIMA ype wh auocoelaons nonconfounded wh shape popees Ohe elevan ssues nvolve non- saonay pocesses wh gven shape popees Ths howeve s beyond he scope of hs acle and s lef fo fuue sudes Concluson In hs acle s agued ha cae mus be aen when smulang auocoelaed and nonnomal vaables so ha he auocoelaon s no a funcon of he shape popey and vce vesa Fuhemoe a numbe of andom vaables specally desgned fo smulaon sudes concenng shape popees and auocoelaon ae poposed The vaables nvolve unvaae o mulvaae dsbuons whch ae symmec o sewed and have sho memoy o long memoy Thus hey cove a faly wde ange of applcaons Fuhemoe all vaables ae easly geneaed n any sandad sascal sofwae pacages ha have facles o geneae AR o MA pocesses Refeences Ba J & Ng S (005 Tess fo sewness uoss and nomaly fo me sees daa Jounal of Busness & conomc Sascs ( Johnson M (987 Mulvaae sascal smulaon Wley Johnson N L Koz S & Balashnan N (994 Connous unvaae dsbuons Vol Wley Shuu G (000 The obusness of he sysemwse Beauch-Godfey auo-coelaon es fo non-nomal eo ems Communcaons n Sascs Smulaon & Compuaon 9( Appendx h h Le : X : ε fo Xh ε h h Then f ε s a sequence of d (zeo mean unfomly negable vaables he followng holds: ( θε X 0 + θε 0 θε θε θ [ ] 0 ε ε θ 0 ( θε X 0 θε + θ εθε 0 + θε θε θε θ 0 ε ε θ 0

10 46 SIMULATION OF NON-NORMAL AUTOCORRLATD VARIABLS ( θε 4 X θε θ ε θε + θ ε θ ε + 6 θ ε θ ε θ ε + 6 θε θεθεθε l l l 4 4 θ 0 ε + θ ε θ ε 4 ε4 0 ε θ + θ θ Hence f an MA( pocess s deemned by X ε + θε follows ha he sewness s gven by X X X ε 0 ( ε θ 0 ε θ ( + θ ( + θ ( + θ ( + θ ε ε (A (A Because ( ( + θ ( + θ s bounded n he span follows ha X ε Fuhemoe he uoss of he MA( pocess s gven by X 4 X X 4 ε4 θ + 0 εθ θ ε 4 0 ( ε θ 0 θ + ( ε θ 0 ε θ ( + ( + ε ( + ( ε ( + 4 ( + ε ( + ε ( + 4 ( + θ θ + ( + θ ( + θ ( ε 4 ε ε θ θ θ + θ θ ε 4 θ θ + θ θ ε ε θ θ (A Noe ha (A s f ε Fuhemoe may be shown ha he uoss of X s always close o he uoss of he nomal dsbuon when compaed o ε e X ε