). So the estimators mainly considered here are linear
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1 6 Ioic Ecooică (4/7 Moe Geel Cedibiliy Models Vigii ATANASIU Dee o Mheics Acdey o Ecooic Sudies e-il: vigii_siu@yhooco This couicio gives soe exesios o he oigil Bühl odel The e is devoed o sei-lie cedibiliy whee oe exies ucios o he do vibles eeseig cli ous he h he cli ous heselves The i uose o sei-lie cedibiliy heoy is he esiio o μ ( E [ ] (he e eiu o coc wih isk ee: by lie cobiio o give ucios o he obsevble vibles: ' So he esios ily cosideed hee e lie ucios o sevel ucios o he obsevble do vibles The oxiio o μ ( bsed o escibed oxiig ucios leds o he oil o-hoogeeous lieied esio o he sei-lie cedibiliy odel Also we discuss he cse whe kig o ll: y o id he oil ucio I should be oed h he oxiio o μ ( bsed o uiue oil oxiig ucio is lwys bee h he oe uished i he sei-lie cedibiliy odel bsed o escibed oxiig ucios: The useuless o he le oxiio is h i is esy o ly sice i is suicie o kow esies o he sucul ees eig i he cedibiliy cos Fo his eso we give soe ubised esios o he sucue ees Fo his uose we ebed he coc i collecive o cocs ll ovidig ideede ioio o he sucue disibuio We close his e by givig he sei-lie hiechicl odel used i he licios che Mheics Subec Clssiicio: 6P5 Keywods: cocs ubised esios sucue ees sevel oxiig ucios sei-lie cedibiliy heoy uiue oil ucio ee esiio hiechicl sei-lie cedibiliy heoy I oducio I his icle we is give he sei-lie cedibiliy odel (see Secio which ivolves oly oe isoled coc Ou oble (o Secio is he esiio o μ ( E [ ] (he e eiu o coc wih isk ee: by lie cobiio o give ucios o he obsevble vibles: Mi E μ is he oil o-hoogeeous lieied esio (ely he sei-lie cedibiliy esul I Secio we discuss he cse whe kig o ll: y o id ' So ou oble (o Secio is he deeiio o he lie cobiio o d he do vibles: closes o [ ] ( μ E i he les sues sese whee is he sucue vible The soluio o his oble: he uiue oil ucio I should be oed h he oxiio o μ ( bsed o uiue oil oxiig ucio is lwys bee h he oe uished i whee:
2 Ioic Ecooică (4/7 7 he sei-lie cedibiliy odel bsed o escibed oxiig ucios: The useuless o he le oxiio is h i is esy o ly sice i is suicie o kow esies o he sucul ees: b (wih eig i he cedibiliy cos (whee To obi esies o hese sucue ees o he seilie cedibiliy odel i Secio 3 we ebed he coc i collecive o cocs ll ovidig ideede ioio o he sucue disibuio We close his e by givig he sei-lie hiechicl odel used i he licios che (see Secio 4 Secio (The oxiio o μ I his secio we coside oe coc wih ukow d ixed isk ee: duig eiod o yes The yely cli ous e deoed by: The isk ee is suosed o be dw o soe sucue disibuio ucio: U ( I is ssued h o give: he clis e codiiolly ideede d ideiclly disibued (codiiolly iid wih kow coo disibuio ucio F ( x The do vibles e obsevble d he do vible is cosideed s beig o (ye obsevble We ssue h: hve iie vice Fo: we ke he ucio o we w o oecs bsed o escibed oxiig ucios: We use he oio: μ E ( [ ( ] ( ; This exessio does o deed o We deie he ollowig sucue ees: E[ μ ( ] E{ E[ ]} E[ ] ( E{ Cov[ ]} (3 b Cov[ μ ( μ ( ] (4 c ] (5 d Cov[ μ ( ] (6 o: These exessios do o deed o: The we c lso obi o-hoogeeous cedibiliy esies kig s esios he clss o lie cobiios o give ucios o sucue ees e coeced by he ollowig elios: lowig heoe: he obsevble vibles s show i he ol- c b Theoe (Oil o-hoogeeous (7 lieied esios (8 d b The lie cobiio o d he do o: This ollows o he covice elios obied i he obbiliy he- vibles ; closes o oy whee hey e vey well-kow Jus s μ ( E [ ] d o i i he cse o cosideig lie cobiios o he obsevble vibles heselves he les sues sese euls: M (9
3 8 Ioic Ecooică (4/7 whee [ c ( d ] d is soluio o he lie syse o euios: o o he euivle lie syse o euios: ( b b ( ( ( ( Poo: we hve o exie he soluio o he oble: Mi E μ ( Tkig he deivive wih esec o gives: [ ] [ ( ] E ( E μ o: Iseig his exessio o io ( leds o he ollowig oble: Mi E μ ( ( (3 O uig he deivives wih esec o ' eul o eo we ge he ollowig syse o euios ( ; ' : [ ( ] Cov[ Cov μ ] (4 ' Becuse o he syey i ie clely: Le us oge ow bou his sucue o so usig he covice esuls o his syse o is closes o: μ d look o y ucio such h ( ( I e cosideed oly euios c be wie s: ucios such h hs iie vice he he oil oxiig uc- b [ c ( d ] (5 io esuls o he ollowig heoe: Now (5 d (3 led o (9 wih: Theoe (Oil oxiig ucio is closes o μ ( d o i he les sues sese i d Secio (The oxiio o μ ( oly i is soluio o he euio: bsed o uiue oil oxiig ( E[ ] E[ ] ucio: The esio M o μ ( o Theoe c be dislyed s: M whee: x ( ( x ( ' ( Poo: we hve o solve he ollowig iiiio oble: Mi E g {[ g g ] } (3 Suose h deoes he soluio o his oble he we coside: g h wih h( biy like i viiol clculus Le:
4 Ioic Ecooică (4/7 9 ϕ { } ( E [ h h ] o be oil ' {[ ][ h h } (4 Clely o ϕ so o evey choice o h : E ] (5 us hold This c be ewie s: E h h h (6 [ ( ( ( ( ( ( ] o: h ( E E [ { [ ] [ ]}] E (7 Becuse his euio hs o be sisied o evey choice o he ucio h oe obis he exessio i bckes i (7 us be ideicl o eo which oves ( A licio o Theoe : I c oly ke he vlues d P[ ] o: ( ( ( ( Ideed: ( he is closes o μ ( d o i he les sues sese i d oly i o ( is soluio o he lie syse: (8 : E P ; E P Iseig hese exessios o: E d ; [ ] (( P [ ] (( ( es Fo his uose we coside k cocs k d k ( ideede d [ ] E [ ] io ( leds o (8 ideiclly disibued vecos ' ( ( o k The Secio 3 (Pee esiio coc idexed is do veco cosisig o do sucue ee I should be oed h he oxiio o d μ ( bsed o uiue oil oxiig ucio is lwys bee h he obsevios: whee k Fo oe uished i Secio bsed o escibed oxiig ucios: vibles: evey coc k d o ixed he e codiiolly ideede d ideiclly disibued The useuless o he le oxiio is h i is esy o ly sice i Theoe 3 (Ubised esios o he is suicie o kow esies o he sucul ees sucue ees b (wih Le: k eig i he cedibiliy cos (whee (3 k k Fo his eso we give soe ubised esios o he sucue e- k (3 k(
5 3 Ioic Ecooică (4/7 b k k k k (33 he: E E E b b whee: k k wih ( d o k ( k d such h < Poo: oe h he usul deiiios o he sucue ees ly wih elcig d elcig so: E k ( k E E[ ] k k k ; [ Cov E E Cov E E Cov E E Cov ( E E k ( b b b b ( b b k ; E b k k Cov k E E Cov E E Cov k k k E E Cov E E k k k k k k b b b k k k k k k b b b k b k k k k k k k k b b k k Secio 4 (Alicios o sei-lie cedibiliy heoy We close his e by givig he seilie hiechicl odel used i he licios che Like i Jewell s hiechicl odel we coside oolio o cocs which c be boke u io P secos ech seco cosisig o k gous o cocs Ised o esiig: ( E[ ] μ (he ue e isk eiu o he coc ( E[ ] ν (he ue e isk eiu o he seco we ow esie: E [ ] μ (he ue e isk eiu o he coc E (he ue e isk [ ] ν eiu o he seco whee P d k I sei-lie cedibiliy heoy he ollowig clss o esios is co-
6 Ioic Ecooică (4/7 3 P k sideed: i ( ( whee e ucios give i dvce Le us coside he cse o oe give ucio i ode o oxie o ν ( d μ ( We oule he ollowig heoe: Theoe 4 (Hiechicl sei-lie cedibiliy Usig he se oios s ioduced o he hiechicl odel o Jewell d deoig s s d s s oe obis he ollowig les sues esies o he ue e isk eius: i ( ( ( ( i ν w μ w (3 w whee: w w k w w w d /[ c ( w d] (he cedibiliy co o coc level wih: Cov d Cov d ( ' ( ' ' c Cov V d: D /[ C ( D ] (he cedibiliy co seco level wih: Cov w ' Cov w ' ' V D w D C w Cov w w w Rek 4: he lie cobiio o d he do vibles ( P k d ν i he les sues sese euls o closes o ν ( d he lie cobiio o d he do vibles ( k P closes o μ ( i he les sues sese euls μ Reeeces [] Gooves MJ Ks R V Hewde AE BuwelickxT: Isuce Seies volue 3 Eecive Acuil Mehods Uivesiy o Ased The Nehelds 99 [] Peikäie T Dyki CD d Pesoe M: Pcicl Risk Theoy o Acuies Uivesié Pieé e Mie Cuie 99 [3] Sud B: A Ioducio o No-Lie Isuce Mheics Veoelichuge des Isius ü Vesicheugswissesch de Uivesiä Mhei Bd 8 VVW Klsuhe 984
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