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The Multivariate Split Normal Distribution and Asymmetric Principal Components Analysis

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The Multivariate Split Normal Distribution and Asymmetric Principal Components Analysis SVERIGES RIKSBANK WORKING PAPER SERIES 175 The Multivariate Split Normal Distribution and Asymmetric Principal Components Analysis Mattias Villani and Rolf Larsson DECEMBER 2004 WORKING PAPERS ARE OBTAINABLE FROM Sveriges Riksbank • Information Riksbank • SE-103 37 Stockholm Fax international: +46 8 787 05 26 Telephone international: +46 8 787 01 00 E-mail: [email protected] The Working Paper series presents reports on matters in the sphere of activities of the Riksbank that are considered to be of interest to a wider public. The papers are to be regarded as reports on ongoing studies and the authors will be pleased to receive comments. The views expressed in Working Papers are solely the responsibility of the authors and should not to be interpreted as reflecting the views of the Executive Board of Sveriges Riksbank. t mç s ç nê é ç d Éêè è aéé p è ç cêé êèè aè çt m vçç è è rêç l êè s à r É è wê èà p s nêN QWU déÉ RPPT aÉ t éç ç èê é ç Éêè è ç éç èê é ç Éêè É ê é ççê ê è è ê é ê êè êèOç êè çêèà É ê è ç N t êé ê ê ÉêèL èçèà éêéè àè èà èêèL éç è è ê N m éé ççêê é êè è êé ç b è è ç ê éç ç èê é ç éêç çê N )*ê Z b è è èL eç êèL e é êèL m éé ççêêL mç è ç L è N jel ç êèZ c""L c"VN #t $ ê à çç èêçà $è è ç ê êé cêèç ê r è hé èM è ê ç è (h frIL à è èêN fP,-.O"/// è r cêèç (0è 1I à è èêN 2".M.PP.M"PP3N t è êçç êè Éç ê ê è êç èê É è 4èà ê e bê ê à r É èN sà r É è L r éèL seMQPS SW sê êçéL sèL è d éè ê s L sê M êçé uè N eMé çZ é N çç è5 É èN N d éè ê ièêé êè sèL u ç uè L bê ,"SL eM3," .P u ç L èN eMé çZ êçNç êè5 NN N R mattias villani and rolf larsson QN ièïìêÖñÑïäêè a èÇïñìÇç ÖäìÜÑïäêè áêì ÜôïÜèÖäèà ïâÜ èêìéÇç ÖäîïìäÉñïäêè äî ïâÜ äèïìêÖñÑïäêè êá îêéÜ îêìï êá îåÜòèÜîîL ÇèÖ îÜóÜìÇç ëìêëêîÇçî âÇóÜ äèÖÜÜÖ ÜéÜìàÜÖL Éêïâ äè ïâÜ ñèäóÇìäÇïÜ ÇèÖ éñçïäóÇìäÇïÜ îÜïïäèàL îÜÜ NàN aõõÇçäèä HQYXUI ÇèÖ aõõÇçäèä ÇèÖ dÇççÇ vÇççÜ HQYYVIN aéêèà ïâêîÜ äî ïâÜ îëçäï èêìéÇç ÖäîïìäÉñïäêèL êì ïâÜ ïòêMëäÜÑÜ èêìéÇçL êìäàäèÇççö äèïìêÖñÑÜÖ Éö gäÉÉêèî ÇèÖ möçìêäÜ HQYWSIL òäïâ éêîï êá äïî åèêòè ëìêëÜìïäÜî ÖÜìäóÜÖ Éö jêâè HQYXRI[ îÜÜ Ççîê käéÉÜì HQYXUI áêì îêéÜ ÇÖÖäïäêèÇç ìÜîñçïîN jêâèîêèL kêïõ ÇèÖ bÇçÇåìäîâèÇè HQYYTI ÑêèïÇäèî ìÜáÜìÜèÑÜî ïê ëÇëÜìî òâÜìÜ ïâÜ îëçäï èêìéÇç ÖäîïìäÉñïäêè äî ñîÜÖ Çî Ç îïÇïäîïäÑÇç éêÖÜçN tâÜ ÜÇîäçö äèïÜìëìÜïÜÖ áêìé êá ïâÜ îëçäï èêìéÇç Öäîïìä!êè âÇ % % !8% ! (6 ())*,4 5L 9 : ())-,, # % % 4 ÜNàN !% && 6 (;;;, 5 Üï ÇçN ())-, 6# $ < " % 8" ())-, = && 9 $ 0#" ()*), % ' & ! ! & / # / % & ! && # & && % && ! %& & 6/ & && # % %# % & ! 2 > % & ! % && " 7 > %& 6/ / % & $ # / ! ! && & / # ! && & # % "# ! && & - " Q[ UPP % & % ! && tâÜ ñèäóÇìäÇïÜ îëçäï èêìéÇç ÖäîïìäÉñïäêè # > &' % & ! 3 ()*, dÜ<èäïäêè RNQN ô r áêççêòî ïâÜ ñèäóÇìäÇïÜ îëçäï èêìéÇç ÖäîïìäÉñïäêèL ô snH6[ 5O[ < OIL äá äï âÇî ÖÜèîäïö R 8 ü O Ñ Üôë O Hô 6I äá ô 6 áHôI ] ! O 4 X ü O Ñ Üôë â O O Hô 6Iä äá ô ^ 6[ \ ! O) Ä Pü Pü â ä òâÜìÜ Ñ ] R]95 HQ K <I NZ O O ë / snH6[ 5 [ < I. ! && / nH6[ 5OI. ! $ 6$ # &. & / nH6[ < O5OI. ! < \ Q ! "# $ < ^ Q "# < ] Q / ! ! !% " #."# / ! ! ! ! %/ &&/ & #" 3 ()*, % % && % & ! # # ! ? multivariate split normal S lÜééÇ RNQN iá snH6[ 5O[ < OI[ ïâÜè 8 eHôI ] 6 K R]95H< QI O v ÇìHôI ] É5 òâÜìÜ É ] wPO H< QIO K <N w tâÜ èÜôï çÜééÇ àäóÜî ïâÜ ñèäóÇìäÇïÜ îåÜòèÜîî e{ ô eHôI )} , ] á à ü {v ÇìHôI})GO ÇèÖ ñèäóÇìäÇïÜ åñìïêîäî e{ ô eHôI É} , ] á à O {v ÇìHôI}O êá Ç snH6[ 5O[ < OI óÇìäÇÉçÜN lÜééÇ RNRN iá ô snH6[ 5O[ < OI ïâÜè 8 , ] ÉP)GO R]9H< QI{HT]9 QIH< QIO K <} ü ÇèÖ PO ,O ] É í[ òâÜìÜ í ] SHQ K < aI]HQ K <I T9POHQ <IO HS K 9IHQ K < OI K SH9 RI< Z " # öB tâÜ èÜôï çÜééÇ àäóÜî ïâÜ éêéÜèï àÜèÜìÇïäèà áñèÑïäêè >BHïI ] eHÜ I êá Ç ñèäóÇìäÇïÜ îëçäï èêìéÇç óÇìäÇÉçÜ Çî ÖÜìäóÜÖ Éö jêâè HQYXRIN lÜééÇ RNSN iá ô snH6[ 5O[ < OIL ïâÜè 8 R5 ÜôëH 5OïO]RI(H 5ïI K < ÜôëH 5O< OïO]RI(H 5<ïI > HïI ] Z B 5HQ K <I ÜôëH6ïI ( ) SN tâÜ éñçïäóÇìäÇïÜ îëç èêìéÇç Öäîïì tâÜ áêççêòäèà ÖÜ<èäïäêè äî Ç èÇïñìÇç àÜèÜìÇçäõÇïäêè êá ïâÜ ñèäóÇìäÇïÜ îëçäï èêìéÇç ÖäîïìäÉñïäêè äè jêâè HQYXRI ïê ïâÜ éñçïäóÇìäÇïÜ îÜïïäèà ÇèÖ äî Ç ìÜëÇìÇéÜïìäõÇïäêè êá ïâÜ éñçïäóÇìäÇïÜ îëçäï èêìéÇç ÖäîïìäÉñïäêè äè gÜòÜåÜ HQYXYIN o dÜ<èäïäêè SNQN a óÜÑïêì ô r áêççêòî ïâÜ íMîëçäï èêìéÇç ÖäîïìäÉñïäêèL ô snoH6[ &[ <[ IL äá äïî ëìäèÑäëÇç ÑêéëêèÜèïî ÇRìÜ äèÖÜëÜèÖÜèïçö ÖäîïìäÉñïÜÖ Çî 8 q snHó; 6[ 5O[ < OI äá ã ó; ô 7 7 7 R q 7 8 nHó; 6[ 5OI äá ã p[ : 7 7 R q p òâÜìÜ Q[ ZZZ[ ë êá îäõÜ íL ] Q[ R[ ZZZ[ ë äî ïâÜ ÑêéëçÜéÜèï êá L ó7 äî ïâÜ ÜäàÜèóÜÑïêì ÑêììÜîëqêè2Öäáèà ïê ïâàÜ ãïâ çÇìàÜqîï ÜäàáÜèóÇçñÜ äèàèïqâÜ îëÜÑïìÇç ÖÜÑêéëêîäïäêqè êá & ] v #v ;L # ] O O ÖäÇàH5ü[ ZZZ[ 5oI ÇèÖ < ] H< 7I7jB äî Ç íMÖäéÜèîäêèÇç óÜÑïêì êá ÑêèïìÇÑïäêèOÖäçÇïäêè ëÇìÇéÜïÜìîN cêèîäÖÜì ïâÜ ÑÇîÜ ] ì áêì äççñîïìÇïäêèL äNÜN òâÜìÜ êèçö ïâÜ ìïâ ëìäèÑäëÇç ÑêéëêèÜèï âÇî Ç îåÜòÜÖ ÖäîïìäÉñïäêèqN iï áäîàïâÜè ÜÇîö ïê îÜÜ ïâÇï ïâÜ ÖÜèîäïö êá ô äî ü ; Pü ; Ñ Üôë Hô 6I & Hô 6I äá óçHô 6I P áHôI ] ! O 4 Ñ Üôë ü Hô 6I;&~ PüHô 6I äá ó; Hô 6I ^ P[ H ! ( O ) ç è ê T mattias villani and rolf larsson ρ=0.7 ρ=0 7 5 6 4 5 3 4 3 2 2 1 1 0 0 -1 -1 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 fäàñìÜ QN cêèïêñì ëçêïî êá ÉäóÇìäÇïÜ QMîëçäï èêìéÇç ÖÜèîäïö áñèÑïäêèîN 6 ] H Q[ RIL & ] HQ[ :[ :[ QIL ] Q ÇèÖ < ] RN q ~ ~ ; ~ O O O O Pü ü oGO üGO òâÜìÜ & ] v #v L # ] ÖäÇàH5ü[ ZZZ[ < ü5ç[ ZZZ[ 5oI ÇèÖ Ñ ] O HR9I # HQK< üIN tâäî îâêñçÖ ÉÜ ÑêéëÇìÜÖ ïê ïâÜ ñèäóÇìäÇïÜ ÑÇîÜ äè dÜ<èäïäêè QN fäàñìÜ Q äççñîïìãÇïÜãî ïòê ëêîîäÉçÜ îâÇëÜî êá ïâÜ sn H6[ &[ <[ IMÖäîïìäÉñïäêèN O q tâÜ àÜèÜìÇç snoH6[ &[ <[ IMÖäîïìäÉñïäêè Çéêñèïî ïê ñîäèà Öä;ÜìÜèï éñçïäóÇìäÇïÜ èêìéÇç ÖäîïìäÉñïäêèîL Ççç òäïâ éêÖÜq6L êóÜì R{ ìÜàäêèî êá ro îÜëÇìÇïÜÖ Éö ïâÜ í âöëÜìëçÇèÜî ó; Hô 7 6I ] PL áêì ã N oïâÜì áêìéî êá ïâÜ îÜëÇìÇïäèà âöëÜìëçÇèÜîL êì éêìÜ àÜèÜìÇç ÑâÇèàÜî äè ÑêóÇìäÇèÑÜ îïìñRÑïqñìÜ ÉÜïòÜÜè ïâÜ R{ ìÜàäêèîL ëìêÖñÑÜ äççMÉÜâÇóÜÖ ÖÜèîäïäÜî òäïâ îâÇìë ìäÖàÜîN tâÜ èêìéÇç ÇèÖ îëçäï èêìéÇç ÖäîïìäÉñïäêèî äè dÜ<èäïäêè SNQ éÇö êÉóäêñîçö ÉÜ ìÜëçÇÑÜÖ Éö êïâÜì ÖäîïìäÉñïäêè $ NàN ï ! (0#"$ )*), çê#à ìÜ % && snoH6[ &[ <[ I ! . wPO O q # $ É7 ] w H< 7 QI K < 7 ã > ' % R q tâÜêìÜé SNQN iá ô sn H6[ &[ <[ IL ïâÜè 8 o q eHôI ] 6 K R]9 57H< 7 QIó- B x v ÇìHôI ] v #üv ; òâÜìÜ #ü äî Ç ÖäÇàêèÇç éÇïìäô òäïâ äïâ ÜçÜéÜèï ÜíñÇç ïê 5O äá ã p êì É 5O äá ã N 7 R q 7 7 R q m ] Hô éI;sPüHõ éI[ BR ! èêÉä !# # ë. & / ! % ô õ$ # é s % $ . &%/ ()1;, mBR > #/ éñçïäóÇìäÇïÜ îåÜòèÜîî ) ,üÅo ] eHmBRIZ ô noH6[ &I$ ,üÅo ] P ,üÅo % "# 8 O ?/ ,üÅü ] ,ü ! / ! > éñçïäóÇìäÇïÜ åñìïêîäî ($ )1;, O ,OÅo ] eHmBBIZ multivariate split normal U iá noH6[ &IL ïâÜè ,OÅo ] ëHë K RIN nêïÜ Ççîê ïâÇï ,OÅü ] ,ON tâÜ áêççêòäèà ìÜîñçï äî ïâÜ éñçï8äóÇìäÇïÜ ÜôïÜèîäêè êá lÜééÇ RNRN tâÜêìÜé SNRN iá ô snH6[ &[ <[ I ïâÜè 8 q P) O O O ,üÅo ] É7 HR]9IH< 7 QI {HT]9 QIH< 7 QI K < 7} B x PO ,OÅo ] ëHë K RI K É7 í7 Sí[ B x a PO O O òâÜìÜ í7 ] SHQ K < 7 I]HQ K < 7I T9 HQ < 7I HS K 9IHQ K < 7 I K SH9 RI< 7 Z â ; ä tâÜ éêéÜèï àÜèÜìÇïäèà áñèÑïäêè >BHïI ] e{ÜôëHï ôI} êá Ç snoH6[ &[ <[ I óÇìäÇÉçÜ äî àäóÜè äè ïâÜ èÜôï ìÜîñçïN q tâÜêìÜé SNSN iá ô sn H6[ &[ <[ IL ïâÜè 8 o q ; O ; ; O ; R5- Üôë{ H57ó7ïI ]R}(H 57ó7ïI K < 7 Üôë{ H57< 7ó7ïI ]R}(H 57< 7ó7ïI >BHïI ] ; R è 57HQ K < 7I ÜôëH6 ó ïI êS B 7 7 y T Q U Üôë {6 ó; ï Hó; ïIO5O} Z " 7 7 R 7 7 H Bp I x TN mÇôäéñé çäåÜçäâêêÖ Üîï éÇï bÜáêìÜ ÜéÉÇìåäèà êè ïâÜ éñçïäóÇìäÇïÜ ÑÇîÜ òÜ àäóÜ Ç ñîÜáñç çÜééÇ ÑêèÑÜìèäèà éÇôäéñé çäåÜçäâêêÖ ÜîïäéÇïäêè äè ïâÜ ñèäóÇìäÇïÜ îÜïïäèàN O O lÜééÇ TNQN gäóÜè Ç ìÇèÖêé îÇéëçÜ ôü[ ZZZ[ ô\ áìêé snH6[ 5 [ < IL ïâÜ çäåÜçäâêêÖL éÇôäéäõÜÖ êóÜì 5 ÇèÖ <L äî Rè \GO l H6I ] à H6IP)\GO [ 9Ü 2 3 òâÜìÜ É üG) üG) à H6I ] îü K îO [ O îü ] Hô- 6I [ m x O îO ] Hô- 6I [ mp x p òâÜìÜ ] ä ] Q[ ZZZ[ è Z Hô- 6I P ÇèÖ ] ä ] Q[ ZZZ[ è Z Hô- 6I ^ P N mêìÜêóÜìL ïâÜ éÇôäéiñé çäáåÜçäâêêÖ ÜîïäéÇïê ìî êá 45O ÇàèÖ < ÇiìÜ á à OG) O î à H6I 5 ] ü [ è îüG) É< ] O Z üG) îü wÜ èêò ïñìè ïê éÇôäéñé çäåÜçäâêêÉÖ ÜîïäéÇïäêè äè ïâÜ éñçïäóÇìäÇïÜ ÑÇîÜN iï äî ëêîîäÉçÜ ïê éÇôäéäõÜ ïâÜ çäåÜçäâêêÖ ÇèÇçöïäÑÇççö òNìNïN # ÇèÖ <N tâÜ ìÜîñçï äî àäóÜè äè ïâÜ áêççêòäèà ïâÜêìÜéN - mattias villani and rolf larsson tâÜêìÜé TNQN è Ç ìÇèÖ é îÇéëçÜ êá óÜÑï ü[ ZZZ[ ô\ sn H6[ &[ <[ IL òâÜìÜ & ] v #v ;L ïâÜ çäåÜçäâêêÖL éÇôäéäõÜÖ òNìNïN # ÇèÖ < äî q Rb{PoGO9\è{\GO l H6[ v[ I ] 5 H6[ v IP\ à H6[ v IP)\GO [ q o\GO 7 7 H9ÜI 7jBp 7jB y y òâÜìÜ É É üG) üG) à7 H6[ v I ] îü7 K îO7 [ ; O ; ; O òâÜìÜ îü7 ] {ó Hô- 6I} [ 7 ] ä ] Q[ ZZZ[ è Z ó Hô- 6I P L îO7 ] p {ó Hô- 6I} L m7 7 i á 7 4 à m7 7 p ] ä ] Q[ ZZZ[ è Z ó; Hô 6I ^ P [ ÇèÖ ïâÜ éÇôäéñé çäåÜçäâêêÖ ÜîïäéÇïêìî êá 5O ÇèÖ < ÇìÜ i7 á p 7 - à p 7 7 OG) O ü î à H6[ v I äá ã [ 5 H6[ v I ] \ ü7 7 R q 7 ü \ {ó; Hô 6I}O äá ã p[ H \ -Dü 7 - R q ÇèÖ É p î üG) < H6[ v I ] O7 Z 7 î 2 ü7 3 wÜ éÇö ñîÜ ïâäî ïâÜêìÜé áêì èñéÜìäÑÇç éÇôäéäõÇïäêè êá ïâÜ çäåÜçäâêêÖ òNìNïN 6 ÇèÖ v L áêì Ç àäóÜè N iè ïâÜ ïòêMÖäéÜèîäêèÇç ÑÉÇîÜL v éÇö ÉÜ ÜôëçäÑäïçö ëÇìÇéÜïìäõÜÖ Çî q Ñêî 2 îäè 2 9 9 HTNQI v ] [ \ 2 Z îäè 2 Ñêî 2 R 4 R 2 3 a îäéäçÇì ëÇìÇéÜïìäõÇïäêè êá v äî ÇóÇäçÇÉçÜ äè ïâÜ àÜèÜìÇç ÑÇîÜ ñîäèà eñçÜìäÇè ÇèàçÜî HkâÇM ïìä ÇèÖ mÇìÖäÇL QYWWIN hÜèÑÜL éÇôäéäõÇïäêè êóÜì v ÇèÖ 6 äî îïìÇäàâïáêìòÇìÖçö ëÜìáêìéÜÖ òäïâ îïÇèÖÇìÖ èñéÜìäÑÇç êëïäéäõÇïäêè ÇçàêìäïâéîN açïÜìèÇïäóÜçöL eÖÜçéÇèL aìäÇî ÇèÖ séäïâ HQYYXI âÇóÜ ÖÜóÜçêëÜÖ êëïäéäõÇïäêè Ççàêìäïâéî êè ïâÜ sïäÜáÜç éÇèäáêçÖ HïâÜ îÜï êá êìïâêèêìéÇç éÇïìäÑÜîI òâäÑâ ÇóêäÖ Çè ÜôëçäÑäï ëÇìÇéÜïìäõÇïäêè êá v N UN bÇöÜîäÇè äèáÜìÜèÑÜ lÜï ôü[ ZZZ[ ô\ ÉÜ Ç ìÇèÖêé îÇéëçÜ áìêé snoH6[ &[ <[ IN tâÜ ãêäèï ëêîïÜìäêì ÖäîïìäÉñïäêè êá Ççç ëÇìÇéÜïÜìî éÇö ÉÜ òìäïïÜè Çî q ëH6[ v[ 5[ <[ [ í
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