Using high frequency stock market index data to calculate, model and forecast realized return variance
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(13) B E B 'f 'f ' ns E3 7. Whpsrudo djjuhjdwlrq iru DUPD prghov lv glvfxvvhg lq Euhzhu +4<:6,/ Wldr +4<:5,/ Zhl +4<;4,/ Zhlvv +4<;7, dqg wkh YDUILPD lq Pdufhoolqr +4<<<,1. :.
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(30) lqvhqvlwlyh wr qrlv| lqsxw ydoxhv ri wkh iudfwlrqdo sdudphwhu1 Wkh uhvxowv uhsruwhg deryh vxjjhvw wkdw wkh DUILPD prgho/ dv h{shfwhg/ rxwshuirupv wkh uhsuhvhqwdwlyh JDUFK prgho dffruglqj wr doo wkh fulwhuld xvhg1 Vhyhudo srlqwv vkrxog/ krzhyhu/ eh qrwhg1 Wkh iruhfdvwv ri wkh DUILPD prgho duh rewdlqhg iurp d wuxqfdwhg lq qlwh dxwruhjuhvvlrq/ jhqhudoo| lqfoxglqj d wkrxvdqg odjv ru pruh1 Lq frqwudvw/ iruhfdvwv ri wkh JDUFK prgho fdq eh frqvwuxfwhg xvlqj vroho| rqh odjjhg uhwxuq dqg frqglwlrqdo yduldqfh hvwlpdwh1 Pruhryhu/ wkh JDUFK prgho fdq eh lpsohphqwhg zlwk gdlo| uhwxuqv zkloh wkh DUILPD prgho uhtxluhv lqwud0gdlo| uhwxuq gdwd iru wkh fdofxodwlrq ri wkh uh0 dol}hg yduldqfh phdvxuh1 Lw lv wkhuhiruh fohdu wkdw wkh DUILPD prgho zloo eh pruh gl!fxow wr lpsohphqw lq sudfwlvh wkdq wkh JDUFK prgho1 Lq dgglwlrq/ iru d jlyhq iruhfdvwlqj krul}rq/ wkh shuvlvwhqfh ri wkh plvvshfl hg JDUFK fdq eh frqwuroohg iru e| {lqj wkh sdudphwhuv wr wkh dssursuldwh ydoxhv vr dv wr plq0 lpl}h wkh gl huhqfh lq iruhfdvwlqj shuirupdqfh zlwk uhvshfw wr wkh DUILPD prgho> zkloh iru vkruw krul}rq iruhfdvwlqj wkh h{lelolw| ri wkh JDUFK lv pruh lpsruwdqw wkdq lwv shuvlvwhqfh/ wkh rssrvlwh lv wuxh iru orqj krul}rq iruhfdvwlqj zkhuh d kljk shuvlvwhqfh lv qhhghg wr frxqwhu wkh orqj phpru| surshuw| ri wkh yduldqfh surfhvv +l1h1 JDUFK4 yhuvxv JDUFK5 lq Iljxuh 615 dqg 616,1. 8. Vxppdu|. Plqxwh e| plqxwh gdwd rq wkh IWVH0433 vwrfn pdunhw lqgh{ kdyh ehhq hp0 sor|hg wr fdofxodwh dqg dqdo|}h gdlo| uhdol}hg yduldqfh1 Lw kdv ehhq vkrzq wkdw lq rughu wr lpsohphqw wkh vxp ri vtxduhg uhwxuqv dssurdfk wr fdofxodwh uhdo0 l}hg yduldqfh/ d fduhixo fkhfn rq vhuldo ghshqghqfh lq kljk iuhtxhqf| uhwxuqv lv uhtxluhg vr dv wr dyrlg vhulrxv eldvhv lq wkh uhvxowlqj phdvxuh iru dyhudjh gdlo| uhwxuq yduldqfh> l1h1 iru wkh IWVH0433 gdwd wklv +grzqzdug, eldv dprxqwv xs wr 68( iru wkh plqxwh gdwd1 Lq dgglwlrq/ lw kdv ehhq vkrzq wkdw wkh ghfd| ri vhuldo ghshqghqfh zlwk wkh vdpsolqj iuhtxhqf| lv frqvlvwhqw zlwk dq DUPD prgho xqghu whpsrudo djjuhjdwlrq1 Wklv qglqj fdq eh xvhg wr vhw wkh rs0 wlpdo vdpsolqj iuhtxhqf|/ wkdw lv/ wkh kljkhvw srvvleoh vdpsolqj iuhtxhqf| dw zklfk wkh dxwrfryduldqfh eldv idfwru lv qhjoljleoh1 Prwlydwhg e| vhyhudo whvw vwdwlvwlfv iru wkh suhvhqfh ri orqj phpru|/ wkh uhdol}hg yduldqfh wlph vhulhv lv prghoohg dv dq DUILPD prgho1 Odjjhg uhwxuqv dqg wudglqj yroxph duh irxqg wr eh vljql fdqw uhjuhvvruv1 Wkh Eodfn ohyhudjh h hfw lv whvwhg iru dqg irxqg wr eh suhvhqw dw krul}rqv ri rqh wr wkuhh gd|v1 Wklv qglqj lv vxssruwlyh iru dv|pphwulf JDUFK prghov vxfk dv wkh HJDUFK dqg JMU JDUFK prgho1 4:.
(31) Frqwhpsrudqhrxv wudglqj yroxph lv khosixo lq h{sodlqlqj wkh yduldwlrq lq uhdo0 l}hg yduldqfh/ dowkrxjk wkh shuvlvwhqfh ri wkh surfhvv uhpdlqv xqfkdqjhg xsrq lqfoxvlrq ri wklv yduldeoh1 Lq d vlpxodwlrq vwxg| wkh iruhfdvwlqj shuirupdqfh ri wkh DUILPD prgho iru gdlo| +uhdol}hg, yduldqfh lv dvvhvvhg dqg lw lv irxqg wkdw lw rxwshuirupv frqyhqwlrqdo JDUFK w|sh prghov1 Lw vkrxog eh qrwhg/ krzhyhu/ wkdw dowkrxjk wkh DUILPD prgho zrunv ehvw/ lwv lpsohphqwdwlrq uh0 txluhv pxfk pruh gdwd wkdq wkh JDUFK1 Wkh uhodwlyho| vpdoo orvv dw vkruw krul}rqv wrjhwkhu zlwk wkh h{lelolw| ri wkh JDUFK wr dffrxqw iru shuvlvwhqfh lq wkh yduldqfh surfhvv/ pdnh lw d uhdvrqdeoh dowhuqdwlyh wr wkh frpsolfdwhg dqg gdwd lqwhqvlyh DUILPD prgho iru uhdol}hg yduldqfh1. 4;.
(32) 9. Uhihuhqfhv. Dphpl|d/ W1 +4<;8,/ Dgydqfhg Hfrqrphwulfv/ Kduydug Xqlyhuvlw| Suhvv/ PD Dqghuvhq/ W1J1 dqg W1 Eroohuvohy +4<<;,/ Dqvzhulqj wkh vnhswlfv= \hv/ vwdqgdug yrodwlolw| prghov gr surylgh dffxudwh iruhfdvwv/ Lqwhuqdwlrqdo Hfr0 qrplf Uhylhz 6</ Qr1 7/ ;;80<38 Dqghuvhq/ W1J1/ W1 Eroohuvohy/ I1[1 Glherog dqg S1 Ode|v +5333d,/ Wkh Glvwulexwlrq ri Uhdol}hg H{fkdqjh Udwh Yrodwlolw|/ iruwkfrplqj Mrxuqdo ri wkh Dphulfdq Vwdwlvwlfdo Dvvrfldwlrq Dqghuvhq/ W1J1/ W1 Eroohuvohy/ I1[1 Glherog dqg S1 Ode|v +5333e,/ H{0 fkdqjh udwh uhwxuqv vwdqgdugl}hg e| uhdol}hg yrodwlolw| duh +qhduo|, Jdxvvldq/ QEHU Zrunlqj Sdshu Qr1 :7;;1 Dqghuvhq/ W1J1/ W1 Eroohuvohy/ I1[1 Glherog dqg K1 Hehqv +5333,/ Wkh Glvwulexwlrq ri Vwrfn Uhwxuq Yrodwlolw|/ Mrxuqdo ri Ilqdqfldo Hfrqrplfv / iruwkfrplqj1 Edloolh/ U1W1 +4<<9,/ Orqj Phpru| Surfhvvhv dqg Iudfwlrqdo Lqwhjudwlrq lq Hfrqrphwulfv/ Mrxuqdo ri Hfrqrphwulfv :6/ 808< Eodfn/ I1 +4<:9,/ Vwxglhv ri vwrfn sulfh yrodwlolw| fkdqjhv/ Surfhhglqj iurp wkh Dphulfdq Vwdwlvwlfdo Dvvrfldwlrq/ Exvlqhvv dqg Hfrqrplf Vwdwlvwlfv Vhfwlrq/ 4::04;4 Eroohuvohy/ W1/ M1 Fdl dqg I1P1 Vrqj +5333,/ Lqwud0gd| Shulrglflw|/ Orqj Phpru| Yrodwlolw| dqg Pdufrhfrqrplf dqqrxqfhphqw h hfwv lq wkh XV Wuhd0 vxu| Erqg Pdunhw/ Mrxuqdo ri Hpslulfdo Ilqdqfh :/ 6:088 Eroohuvohy/ W1 dqg K1R1 Plnnhovhq +4<<9,/ Iudfwlrqdoo| Lqwhjudwhg Jhqhu0 dol}hg Dxwruhjuhvvlyh Frqglwlrqdo Khwhurvnhgdvwlflw|/ Mrxuqdo ri Hfrqrphwulfv :7/ 6063 Euhlgw/ I1M1/ Q1 Fudwr dqg S1 gh Olpd +4<<;,/ Wkh Ghwhfwlrq dqg Hvwl0 pdwlrq ri Orqj Phpru| lq Vwrfkdvwlf Yrodwlolw|/ Mrxuqdo ri Hfrqrphwulfv ;6/ 658067; Glherog/ I1[1 dqg D1 Lqrxh +4<<<,/ Orqj Phpru| dqg Vwuxfwxudo Fkdqjh/ Pdqxvfulsw Glwwpdqq/ L1 dqg F1Z1M1 Judqjhu +5333,/ Surshuwlhv ri Qrqolqhdu Wudqv0 irupdwlrqv ri Iudfwlrqdoo| Lqwhjudwhg Surfhvvhv/ XFVG glvfxvvlrq sdshu 53330 3:1 4<.
(33) Grruqln/ M1D1 dqg P1 Rrpv +4<<;,/ D Sdfndjh iru Hvwlpdwlrq/ Iruhfdvw0 lqj dqg Vlpxodwlqj Du pd Prghov= Du pd sdfndjh 413 iru R{/ plphr Qx!hog Froohjh/ R{irug1 Iuhqfk/ N1U1/ Vfkzhuw/ J1Z1 dqg Vwdpedxjk/ U1I1 +4<;:,/ H{shfwhg Vwrfn Uhwxuqv dqg Yrodwlolw|/ Mrxuqdo ri Ilqdqfldo Hfrqrplfv 4</ 605< Iurrw/ N1D1 dqg D1I1 Shurog +4<<8,/ Qhz Wudglqj Sudfwlfhv dqg Vkruw0 Uxq Pdunhw H!flhqf|/ Mrxuqdo ri Ixwxuhv Pdunhwv 48+:,/ :640:98 Jhzhnh M1 dqg V1 Sruwhu0Kxgdn +4<;6,/ Wkh Hvwlpdwlrq dqg Dssolfdwlrq ri Orqj Phpru| Wlph Vhulhv Prghov/ Mrxuqdo ri Wlph Vhulhv Dqdo|vlv 7+7,/ 554056; Jorvwhq/ Mdjdqqdwkdq dqg Uxqnoh +4<<6,/ Rq wkh uhodwlrq ehwzhhq wkh h{shfwhg ydoxh dqg wkh yrodwlolw| ri wkh qrplqdo h{fhvv uhwxuq rq vwrfnv/ Mrxu0 qdo ri Ilqdqfh 7;/ 4::<04;34 Jrrgkduw/ F1D1H1 dqg P1 R*Kdud +4<<:,/ Kljk Iuhtxhqf| Gdwd lq Il0 qdqfldo Pdunhwv= Lvvxhv dqg Dssolfdwlrqv/ Mrxuqdo ri Hpslulfdo Ilqdqfh 7/ :60447 Judqjhu/ F1Z1M1 +4<;3,/ Orqj phpru| uhodwlrqvklsv dqg wkh djjuhjdwlrq ri g|qdplf prghov/ Mrxuqdo ri Hfrqrphwulfv 47/ 55:056; Judqjhu/ F1Z1M1 +4<;4,/ Vrph surshuwlhv ri wlph vhulhv gdwd dqg wkhlu xvh lq hfrqrphwulf prgho vshfl fdwlrq/ Mrxuqdo ri Hfrqrphwulfv 49/ 4540463 Judqjhu/ F1Z1M1 +4<<<,/ Dvshfwv ri Uhvhdufk Vwudwhjlhv iru Wlph Vhulhv Dqdo|vlv/ Rxwolqh ri suhvhqwdwlrq wr wkh frqihuhqfh rq Qhz Ghyhorsphqwv lq Wlph Vhulhv Hfrqrplfv dw wkh Frzohvv Irxqgdwlrq/ Qhz Kdyhq1 Judqjhu/ F1Z1M1 dqg Q1 K|xqj +4<<<,/ Rffdvlrqdo Vwuxfwxudo Euhdnv dqg Orqj Phpru|/ XFVG glvfxvvlrq sdshu <<0471 Judqjhu/ F1Z1M1 dqg U1 Mr|hx{ +4<;3,/ Dq lqwurgxfwlrq wr orqj0phpru| wlph vhulhv dqg iudfwlrqdo gl huhqflqj/ Mrxuqdo ri Wlph Vhulhv Dqdo|vlv/ 4 Kvlhk/ G1D1 +4<<4,/ Fkdrv dqg Qrqolqhdu G|qdplfv= Dssolfdwlrq wr Il0 qdqfldo Pdunhwv/ Mrxuqdo ri Ilqdqfh 79+8,/ 4;6<04;:: Odprxuhx{/ F1J1 dqg Z1G1 Odvwudshv +4<<3,/ Khwhurvnhgdvwlflw| lq Vwrfn Uhwxuq Gdwd= Yroxph yhuvxv JDUFK h hfwv/ Mrxuqdo ri Ilqdqfh 78+4,/ 5540 55< Odprxuhx{/ F1J1 dqg Z1G1 Odvwudshv +4<<3e,/ Shuvlvwhqfh lq Yduldqfh/ Vwuxfwxudo Fkdqjh dqg wkh JDUFK Prgho/ Mrxuqdo ri Exvlqhvv ) Hfrqrplf Vwdwlvwlfv 9+5,/ 5580567 53.
(34) Ohtxhx{/ S1 +4<<<,/ Ilqdqfldo Pdunhwv Wlfn e| Wlfn= Lqvljkwv lq Ilqdqfldo Pdunhw Plfur Vwuxfwxuh/ Mrkq Zloh| ) Vrqv Owg/ Hqjodqg Olx/ P1 +5333,/ Prgholqj Orqj Phpru| lq Vwrfn Pdunhw Yrodwlolw|/ Mrxuqdo ri Hfrqrphwulfv <</ 46<04:4 Or/ D1Z1 +4<<4,/ Orqj Phpru| lq Vwrfn Pdunhw Sulfhv/ Hfrqrphwulfd 8<+8,/ 45:<04646 Pdgkdydq/ D1 +5333,/ Pdunhw Plfurvwuxfwxuh= D Vxuyh|/ Mrxuqdo ri Ilqdqfldo Pdunhwv 6/ 538058; Phuwrq/ U1F1 +4<;3,/ Rq Hvwlpdwlqj wkh H{shfwhg Uhwxuq rq wkh Pdunhw> Dq H{sorudwru| Lqyhvwljdwlrq/ Mrxuqdo ri Ilqdqfldo Hfrqrplfv ;/ 6560694 Qhovrq/ G1E1 +4<<4,/ Frqglwlrqdo Khwhurvnhgdvwlflw| lq dvvhw uhwxuqv= D qhz dssurdfk/ Hfrqrphwulfd 8</ 67:0:3 Urelqvrq/ S1P1 +4<<8d,/ Orj0Shulrgrjudp Uhjuhvvlrq ri Wlph Vhulhv zlwk Orqj Udqjh Ghshqghqfh/ Dqqdov ri Vwdwlvwlfv 56+6,/ 437;043:5 Urelqvrq/ S1P1 +4<<8e,/ Jdxvvldq Vhplsdudphwulf Hvwlpdwlrq ri Orqj Udqjh Ghshqghqfh/ Dqqdov ri Vwdwlvwlfv 56/ 4963094 Vplwk/ M1/ Q1 Wd|oru dqg V1 \dgdy +4<<:,/ Frpsdulqj wkh Eldv dqg Plv0 vshfl fdwlrq lq DUILPD prghov/ Mrxuqdo ri Wlph Vhulhv Dqdo|vlv/ 4;+8,/ 83:0 85: Vrzhoo/ I1 +4<<5,/ Pd{lpxp olnholkrrg hvwlpdwlrq ri vwdwlrqdu| xqlydul0 dwh iudfwlrqdoo| lqwhjudwhg wlph vhulhv prghov/ Mrxuqdo ri Hfrqrphwulfv 86/ 4980;; Wd|oru/ V1M1 dqg [1 [x +4<<:,/ Wkh Lqfuhphqwdo Yrodwlolw| Lqirupdwlrq lq Rqh Ploolrq Iruhljq H{fkdqjh Txrwdwlrqv/ Mrxuqdo ri Hpslulfdo Ilqdqfh 7/ 64:0673 Wldr/ J1F1 +4<:5,/ Dv|pswrwlf Ehkdylrxu ri Whpsrudo Djjuhjdwhv ri Wlph Vhulhv/ Elrphwulnd 8</ 8580864 Zhl/ Z1Z1V1 +4<;4,/ H hfwv ri V|vwhpdwlf Vdpsolqj rq DULPD Prg0 hov/ Frppxqlfdwlrqv lq Vwdwlvwlfdo0Wkhruhwlfdo Pdwkhpdwlfv D 43/ 56;<056<; Zrrg/ U1D1 +5333,/ Pdunhw Plfurvwuxfwxuh Uhvhdufk Gdwdedvhv= Klvwru| dqg Surmhfwlrqv/ Mrxuqdo ri Exvlqhvv ) Hfrqrplf Vwdwlvwlfv 4;+5,/ 4730478 ]krx/ E1 +4<<9,/ Kljk Iuhtxhqf| Gdwd dqg Yrodwlolw| lq Iruhljq0H{fkdqjh Udwhv/ Mrxuqdo ri Exvlqhvv ) Hfrqrplf Vwdwlvwlfv 47+4,/ 78085 54.
(35) D. Dsshqgl{= Hpslulfdo Uhvxowv. D14. Ghvfulswlyh Vwdwlvwlfv dqg Hvwlpdwlrq Uhvxowv. Iuhtxhqf|. Vnhz. Nxuwrvlv. EO +43,. GZ. ). )2. )
(36). )D. )f. )D. 4 Plq 43 Plq 58 Plq 4 Krxu 4 Gd|. 6319 4314 8159 7167 3137. 4765 743 535 475 8157. ;47 63; 477 8614 4519. 41:3 41:3 41:9 41<7 5133. 31485 31484 3144; 31364 3139<. 31464 313:4 0 0 0. 31448 3137< 0 31363 0. 313:5 0 0 0 0. 31365 0 0 03135< 0. 31356 0 0 0 0. Wdeoh 4= Ghvfulswlyh vwdwlvwlfv ri IWVH0433 uhwxuqv dw gl huhqw vdpsolqj iuhtxhqflhv1 EO+43, ghqrwhv wkh Er{0Omxqj Vwdwlvwlf zlwk 43 dxwrfruuhodwlrqv +4;164 fulwlfdo ydoxh dw <8I frq ghqfh erxqg,1 GZ ghqrwhv wkh Gxuelq Zdwvrq whvw1 4 ghqrwhv wkh | dxwrfru0 uhodwlrq1 Wkh vwdwlvwlfv duh fdofxodwhg xvlqj 43/333 revhuydwlrqv iru vdpsolqj iuhtxhqflhv grzq wr rqh krxu dqg 5/733 revhuydwlrqv iru wkh gdlo| iuhtxhqf|1 Wkh <8I frq ghqfh erxqgv duh wkhuhiruh 3135 dqg ffe2 uhvshfwlyho|1 Hqwulhv zklfk frqwdlq wkh %0% v|pero duh qrw vljql fdqw edvhg rq wkhvh erxqgv1. Uhdol}hg Yduldqfh Orjri Uhdol}hg yduldqfh Gdlo| Uhwxuq Vwdqgdugl}hg Gdlo| Uhwxuq. Phdq. Vwg1Ghy1. Vnhzqhvv. Nxuwrvlv. DGI+8,. ;187H08 0<1<; 7193H07 313<4. 518:H07 31<95 <15;H06 413<. 5415 3188; 31395; 313695. 8<9 7144 815< 5156. 04915 0;1;6 0541; 05516. Wdeoh 5= Ghvfulswlyh vwdwlvwlfv iru uhdol}hg yrodwlolw| dqg gdlo| uhwxuqv1 Wkh froxpq %DGI+8,% uhsruwv wkh dxjphqwhg glfnh| ixoohu whvw lqfoxglqj d frqvwdqw dqg 8 odjv 55.
(37) Prgho. Vdpsoh. _. k. q. ln . l3 . ln 2. l3 2. bf. OrjO. DLF2W. DUILPD. L. 31779. 31755. 031969. 0. 0. 0. 0. 0. 0:56. 31;4. . Uhwxuqv. L. 31748. 31635. 0318<:. 4;13. 0. 0786. 3184. . Yroxph. L. 317;4. 3164:. 0319:: +4816,. +491;,. +5319,. +:164,. +<197,. 491<. 316:<. 0699. 3175. DUILPD. LL. 317;:. 31734. 031975. 0. 0. 0. 0. 0. 0<9;. 31;3. . Uhwxuqv. LL. 317;7. 3165:. 031995. 471<. 0. 097<. 3187. . Yroxph. LL. 317;<. 3166<. 0319;7. 461<. 3168;. 086:. 3178. +<153, +:185, +4<1;,. +5<15, +5715, +6514,. +81<5, +7196, +916<,. +:157, +:145, +;139,. +;174, +:166,. +4614, +4913, +4<17,. 6619. +5314,. 5:1:. 5818. +5318,. 5418. +4:1:,. 6<17. +5418,. 6917. 6516. +5715,. 5<17. +551:,. 471;. +;1;9,. 441:. 4319. +;185,. ;195. +:155,. +<1::,. +4415, +431<,. +4619,. +4816,. Wdeoh 6= DUILPD hvwlpdwlrq uhvxowv1 Vdpsoh L uxqv iurp Pd| 5/ 4<<3 xqwlo Mxqh 48/ 4<<:1 Vdpsoh LL uxqv iurp Pd| 5/ 4<<3 xqwlo Mdqxdu| 44/ 53331 _/ k / dqg q ghqrwh 3 wkh DUILPD+4/g/4, prgho sdudphwhuv1 l n / l / dqg bf / duh wkh uhjuhvvlrq frh!flhqwv ri fruuhvsrqglqj wr wkh | odj ri srvlwlyh dqg qhjdwlyh uhwxuqv dqg frqwhpsruhqhrxv wudglqj yroxph uhvshfwlyho|1 Wkh odvw wzr froxpqv frqwdlq wkh ydoxh ri wkh orj olnholkrrg ixqfwlrq dqg wkh Dndlnh lqirupdwlrq fulwhuld1 w0Vwdwlvwlfv duh uhsruwhg lq sduhqwkhvlv1. 56.
(38) D15. Plfur Vwuxfwxuh ) Vdpsolqj Iuhtxhqf|. 1. x 10. -4. Term 1 + Term 2. 0.8. 0.6 Term 1: Average Daily Realized Volatility "Optimal" Sampling Frequency Range. 0.4 Term 2: Autocovariance Bias Factor. 0.2. 0. 0. 5. 10. 15. 20 25 Sampling Frequency (minutes). 30. 35. 40. 45. Iljxuh 414= 4<<305333 dyhudjh gdlo| uhdol}hg yduldqfh dqg dxwrfryduldqfh eldv idfwru +grwwhg olqh lv xqfrqglwlrqdo yduldqfh fdofxodwhg xvlqj gdlo| uhwxuq gdwd,. 4. x 10. -5. 2. 0. Term 2: Autocovariance Bias Factor. 0. 5. 10. 15. 20 25 Sampling Frequency (minutes). 30. 35. 40. Iljxuh 415= Hpslulfdo +vrolg olqh, dqg djjuhjdwlrq lpsolhg +grwwhg olqh, Dxwrfryduldqfh Eldv Idfwru. 57. 45.
(39) D16. Wkh IWVH0433 Uhdol}hg Yduldqfh 4. -4. -6. 2. -8. 0 -10. -2 -12. -14 1990. 1992. 1994. 1996. 1998. -4 1990. 2000. Iljxuh 514= Orj ri uhdol}hg yduldqfh. 1992. 1994. 1996. 1998. 2000. Iljxuh 515= Iudfwlrqdoo| gl huhqfhg +xvlqj g@3173, orj uhdol}hg yduldqfh vhulhv1. 0.8. 0.15. 0.7 0.1 0.6 0.05 0.5. d=0.45. 0.4. 0. 0.3. d=0.40. -0.05. 0.2 d=0.30 -0.1 0.1. 0 0. 50. 100 150 Displacement (days). 200. 250. 0. 50. 100 150 Displacement (days). 200. 250. Iljxuh 516= Fruuhorjudp orj uhdol}hg yduldqfh. Iljxuh 517= Fruuhorjudp ri iudfwlrqdoo|. soxv iudfwlrqdo lpsolhg fruuhodwlrqv1. gl huhqfhg orj uhdol}hg yduldqfh vhulhv 0.8. 0. 0.7. -0.5. d=0.43. d=0.37 0.6. 0.5. -1. Robinson. 0.4 Geweke Porter-Hudak. -1.5 0.3. -2. 0. 1. 2. Iljxuh 518=. 3. *L} ). 4. yhuvxv. *L} . 5. 6. 0.2 25. 50. 75. 100. 125. 150 175 Bandwidth m. 200. 225. 250. Iljxuh 519= JSK ) Urelqvrq hvwlpdwh iru g dv d ixqfwlrq ri wkh edqgzlgwk p1. 58. 275.
(40) D17. Iruhfdvwlqj Shuirupdqfh. -5. 1.04. 1.8. x 10. 1.6. Sum of Squared Forecast Error. 1.03. 1.02 GARCH 2 1.01 ARFIMA. GARCH 1. 1. 1.4 1.2. GARCH 1. 1 GARCH 2 0.8 0.6 ARFIMA. 0.4. 0.99 0.2. 0.98 0. 50. 100 150 Forecast Horizon (days). 200. 250. 50. 100 150 Forecast Horizon (days). 200. 250. Iljxuh 614= Iruhfdvwhg glylghg. Iljxuh 615= Vxp ri vtxduhg huuruv ri iruh0. e| dfwxdo dyhudjh uhdol}hg yduldqfh. fdvwhg fxppxodwlyh uhdol}hg yduldqfh. -3. 4.5. 0 0. 10. x 10. 9 4. Standard Deviation Forecast Error. 8 3.5. 7 3. 6 GARCH 1. 2.5. 5 2. GARCH 2. 4. ARFIMA. 1.5. 3. 1. 2. 0.5. 1. 0 0. 50. 100 150 Forecast Horizon (days). 200. 250. 0 0.25. 0.3. 0.35. 0.4 0.45 0.5 Geweke Porter-Hudak Estimate. 0.55. 0.6. 0.65. Iljxuh 616= Iruhfdvw huuru vwdqgdug ghyldwlrq Iljxuh 617= Ghqvlw| Jhzhnh Sruwhu Kxgdn hvwlpdwhv. 59.
(41) E E14. Dsshqgl{= Wkhru| Whpsrudo Djjuhjdwlrq ) V|vwhpdwlf Vdpsolqj. Frqvlghu dq DUPD+s/t, surfhvv k Eu f| ' q Eu 0| c zkhuh k Eu dqg q Eu duh odj sro|qrpldov ri ohqjwkv R dqg ^ uhvshfwlyho|1 Frq0 vlghu wkh fdvh zkhuh wkh uhflsurfdov ri wkh urrwv ri k Eu ' f/ w c c wR / doo olh lqvlgh wkh xqlw flufoh1 Zkhq f| lv d vwrfn yduldeoh/ Zhl +4<;4, vkrzhg wkdw e| dsso|lqj wkh rshudwru 7Eu R \ w s us 7Eu ' w u ' wr DUPD+s/t, surfhvv/ wkh DUPD+s/u, prgho wkurxjk zklfk wkh owhuhg vxe0 vhulhv pd| eh uhsuhvhqwhg lv rewdlqhg1 Lq wklv fdvh/ s ghqrwhv wkh v|vwhpdwlf vdpsolqj iuhtxhqf|1 Zkhq f| lv d rz yduldeoh/ Zhl +4<;4, vkrzhg wkdw e| dsso|lqj wkh rshudwru A Eu A Eu ' 7Eu . us u. wr DUPD+s/t, surfhvv/ wkh DUPD+s/u, prgho wkurxjk zklfk wkh owhuhg vxe0 vhulhv pd| eh uhsuhvhqwhg lv rewdlqhg1 Lq wklv fdvh/ s ghqrwhv wkh whpsrudo djjuhjdwlrq iuhtxhqf|1 Wkh prylqj dyhudjh odj ohqjwk o lv jlyhq e| wkh lqwhjhu sduw ri R n ^3s R dqg iru odujh s zloo eh htxdo wr R ru R / ghshqglqj rq zkhwkhu ^ R ru ^ R1 Qrwh wkdw dowkrxjk wkh rughu ri wkh dxwruhjuhvvlyh sduw ri wkh DUPD prgho uhpdlqv wkh vdph xqghu v|vwhpdwlf vdpsolqj ru whpsrudo djjuhjdwlrq/ wkh pdjqlwxgh ri wkh dxwruhjuhvvlyh frh!flhqwv ghfuhdvh h{srqhqwldoo| zlwk s 1 Khqfh/ lq wkh olplw wkh dxwruhjuhvvlyh sduw zloo glvdsshdu1 Pruhryhu/ iru us whpsrudo djjuhjdwlrq/ wkh whup 3 3u zloo grplqdwh wkh PD sduw dqg khqfh wkh olplw prgho lv dq DUPD+3/3, ru htxlydohqwo| zklwh qrlvh1. E15. Iudfwlrqdo Lqwhjudwlrq. D wlph vhulhv/ f| lv vdlg wr eh iudfwlrqdoo| lqwhjudwhg ri rughu _/ li diwhu dsso|lqj wkh gl huhqfh rshudwru Eu _ lw iroorzv d vwdwlrqdu| DUPD+s/t, surfhvv zkhuh 5:.
(42) R dqg ^ duh qlwh qrqqhjdwlyh lqwhjhuv1 Wkh frqfhsw zdv ghyhorshg e| Judqjhu +4<;3/ 4<;4, dqg Judqjhu dqg Mr|hx{ +4<;3,1 D w|slfdo ihdwxuh ri d iudfwlrqdoo| lqwhjudwhg ru orqj phpru| surfhvv lv wkdw wkh h hfw ri d vkrfn wr wkh surfhvv lv kljko| shuvlvwhqw exw ghfd|v ryhu wlph1 Wklv/ dv rssrvhg wr UE surfhvvhv zkhuh d vkrfn kdv lq qlwh shuvlvwhqfh ru dw wkh rwkhu h{wuhph UEf 0surfhvvhv lq zklfk wkh h hfw ri d vkrfn wr wkh v|vwhp ghfd|v h{srqhqwldoo| idvw1 Pruhryhu/qrwh wkdw wkh iudfwlrqdo gl huhqfh rshudwru wdnhv fduh ri wkh orqj uxq g|qdplfv zkloh wkh DUPD vwuxfwxuh fdq dffrxqw iru wkh vkruw uxq g|qdplfv1 Wkh DUILPD+s/g/t, prgho fdq eh zulwwhq dv kEu E u _ f| ' qEu 0| c. +7,. zkhuh kEu lv d odj sro|qrpldo ri rughu R dqg qEu ri rughu ^1 Qrwh wkdw wkh DUPD+s/t, dqg Lqwhjudwhg DUPD+s/t, prghov duh vshfldo fdvhv ri +7, iru _ ' f dqg _ ' uhvshfwlyho|1 Xvlqj d elqrpldo4: ru Wd|oru0olnh h{sdqvlrq +durxqg u ' f,/ wkh iudfwlrqdo gl huhqfh rshudwru fdq eh h{suhvvhg dv iroorzv= E u _ ' _u _E _ u2 _E _ E2 _ u
(43) 2 S " [ KE _ ' u KE n KE_ 'f " T E? _ [ ?' u ' n '. +8,. Xvlqj Vwluolqj*v irupxod4; lw fdq eh vkrzq wkdw= T K E _ ?' E? _ ' 2 3_3 K E_ K E n iru odujh1 Pruhryhu/ iru _ . 2. dqg _ 9' f/ lw fdq eh vkrzq wkdw=. ) ' SJooEf| c f|3 '. KE _ KE n _ 2 2_3 KE_ KE n _ odujh. +9,. dqg khqfh wkh ghfd| ri wkh fruuhorjudp lv k|shuerolf/ dv rssrvhg wr dq h{sr0 qhqwldo ghfd| iru dq U Ef surfhvv1 Iru _ ' f/ ) ' f iru : f Wkh surfhvv O,g. S4 k@3. . g k. . +4,k Ok zkhuh Wkh Elqrpldo Wkhruhp vwdwhv wkdw +4 @ g +4,k @ +k g,@ ^+g, +k . 4,` dqg +, lv wkh Jdppd ixqfwlrq iru zklfk lw k krogv wkdw +{, @ +{ 4, +{ 4, s k. d, 4; d e Vwluolqj*v irupxod vwdwhv wkdw +k, 5 kk4@5 hk iru k odujh dqg khqfh + +k.e, k 1 4:. 5;.
(44) lv vwdwlrqdu| dqg orqj phpru| iru f _ fD1 Wkh surfhvv lv vwdwlrqdu| dqg lqwhuphgldwh phpru| zkhq fD _ f1 Iru _ fD/ wkh surfhvv lv qrq0vwdwlrqdu|1 E1514. Orj0Shulrgrjudp Uhjuhvvlrqv. Vhyhudo wzr vwhs surfhgxuhv kdyh ehhq sursrvhg wr hvwlpdwh wkh iudfwlrqdoo| lqwhjudwhg DUPD prgho1 Wkh pdlq lghd lv wr hvwlpdwh wkh iudfwlrqdo sdudp0 hwhu/ _/ lq wkh uvw vwhs zkloh lq wkh vhfrqg vwhs wklv hvwlpdwh fdq eh xvhg wr iudfwlrqdoo| gl huhqfh wkh revhuyhg vhulhv/ wudqviruplqj lw lqwr dq DUPD sur0 fhvv ri zklfk wkh sdudphwhuv fdq eh rewdlq vwudljkwiruzdugo| e| ruglqdu| ohdvw vtxduhv1 Jhzhnh Sruwhu0Kxgdn +4<;6/ JSK khuhdiwhu, sursrvh wr hvwlpdwh _ e| d orj0shulrgrjudp uhjuhvvlrq zklfk lv ghvfulehg ehorz1 Frqvlghu Eu _ f| ' 0| zkhuh 0| lv d vwdwlrqdu| olqhdu surfhvv zlwk vshfwudo ghqvlw| ixqfwlrq s0 Eb zklfk lv qlwh/ erxqghg dzd| iurp }hur dqg frqwlqxrxv rq wkh lqwhuydo dZc Zo1 Wkh 3_ 2 s0 Eb ru htxlydohqwo| vshfwudo ghqvlw| ixqfwlrq ri if| j lv s Eb ' jZ e t? b 2. 2. *? s Eb ' *?. j s0 Ef s0 Eb _ *? e t? Eb*2 n *? 2Z s0 Ef 2. +:,. 2. Gh qh wkh kduprqlf iuhtxhqf| b ' AZ zkhuh A lv wkh vdpsoh vl}h dqg ohw U Eb eh wkh shulrgrjudp dw b zklfk lv jlyhq e| 2. A [ b | U Eb ' f| e 2ZA | 2. '. Uhduudqjlqj whupv dqg hydoxdwlqj h{suhvvlrq E. dw b forvh wr }hur +wkh whup *? ss00 b lv wkhuhiruh qhjoljleoh, wkh iroorzlqj h{suhvvlrq lv rewdlqhg= E. . Ef . *? U Eb ' *?. j s0 Ef U Eb _ *? e t? Eb *2 n *? 2Z s Eb 2. +;,. 2. Wkhuhiruh/ wkh frh!flhqw _ fdq qrz eh hvwlpdwhg dv wkh vorsh frh!flhqw lq d ohdvw vtxduhv uhjuhvvlrq ri *? U Eb rq d frqvwdqw dqg *? e t? Eb *2 iru ' n ,c c 6 A JSK vhw , ' f dqg uhtxluh wkdw wkh edqgzlgwk sd0 udphwhu 6 lqfuhdvhv dw d vorzhu udwh wkdq wkh vdpsoh vl}h1 Lq pdq| sudfwlfdo dssolfdwlrqv 6 lv vhw wr htxdo wr wkh vtxduh urrw ri wkh vdpsoh vl}h A 1 Urelqvrq +4<<8d, surylghv wkh dv|pswrwlf ehkdylru ri wkh hvwlpdwru1 Lq dgglwlrq/ Urelq0 vrq +4<<8e, sursrvhg dq dowhuqdwlyh hvwlpdwru zklfk lv ghulyhg xqghu zhdnhu 2. 5<.
(45) frqglwlrqv dqg suryhg wr eh dv|pswrwlfdoo| pruh h!flhqw wkdq wkh JSK hvwl0 pdwru1 Wklv hvwlpdwru/ iru wkh iudfwlrqdo sdudphwhu/ lv jlyhq e| wkh ydoxh ri _ wkdw plqlpl}hv wkh iroorzlqj remhfwlyh ixqfwlrq= % & 6 3 _ b _ [ U Eb c *? Sb ' ESc _ ' n 6 S 2. 2. '. zkhuh S : f dqg _ lv uhvwulfwhg wr olh ehwzhhq dqg 1 E1515. . . 2. 2. H{dfw Pd{lpxp Olnholkrrg. Prwlydwhg e| wkh qglqj wkdw wkh DUPD sdudphwhuv duh jhqhudoo| qrw hvwl0 pdwhg dffxudwho| xvlqj d wzr vwhs phwkrg/ Vrzhoo +4<<5, sursrvhv dq h{dfw pd{lpxp olnholkrrg surfhgxuh wkdw hvwlpdwhv doo prgho sdudphwhuv vlpxowdqh0 rxvo|1 Ohw P ghqrwh wkh fryduldqfh pdwul{ ri wkh mrlqw glvwulexwlrq ri if| j|| A zkhuh f| lv dvvxphg wr iroorz dq DUILPD+s/g/t, surfhvv dv jlyhq e| h{suhv0 vlrq Ee 1 Wkh prgho sdudphwhuv fdq eh hvwlpdwhg e| pd{lpl}lqj wkh iroorzlqj orj0olnholkrrg ixqfwlrq ryhu wkh sdudphwhu vsdfh '. '. A *L} u _c kc qc j 0 ' *L} E2Z *L} mPm f P fc 2 2 2 zkhuh P ' ) iru c ' c 2c c A / lv h{suhvvhg lq whupv ri wkh prgho sdudphwhu1 Vrzhoo +4<<5, vkrzv wkdw wkh fryduldqfh pdwul{/ qhhghg iru wkh hvwlpdwlrq/ lv jlyhq e| 2. . 3. 3 . )r ' j0. 2. R ^ [ [ ,'3^ '. i E, y _c R n , rc ). zkhuh. ^c^ , [. 4?d. i E, '. y ' w 3. R \ '. q rq r , 3. r'4@. dqg. 3 o. c,o. df. E w w . \ 6 '. Ew w6 . zkhuh w ghqrwh wkh urrwv ri kEu dqg duh dvvxphg wr olh rxwvlgh wkh xqlw glvn1 Ilqdoo|/ wkh h{suhvvlrq iru lv jlyhq e| E_c c w ' } E_c w R 8 E_ n c ( _ n ( w n 8 E_ c ( _ ( w 2. zkhuh } E_c ' wlrq1. _ _ _ KE3_ KE_ . KE32 KE n KE3 n. dqg 8 E@c K( S( e lv wkh k|shujhrphwulf ixqf0. 63.
(46)
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