Analysis of contagion from the constant conditional correlation model with Markov regime switching

Working

Paper

340

Analysis of contagion from the

constant conditional correlation

model with Markov regime switching

Pedro Nielsen Rotta

Pedro Luiz Valls Pereira

CEQEF - Nº08

W_ORKING P_{APER 340}–CEQEF Nº 08•DEZEMBRO DE 2013• 1

Os artigos dos Textos para Discussão da Escola de Economia de São Paulo da Fundação Getulio Vargas são de inteira responsabilidade dos autores e não refletem necessariamente a opinião da

FGV-EESP. É permitida a reprodução total ou parcial dos artigos, desde que creditada a fonte.

❆♥❛❧②s✐s ♦❢ ❈♦♥t❛❣✐♦♥ ❢r♦♠ t❤❡ ❈♦♥st❛♥t

❈♦♥❞✐t✐♦♥❛❧ ❈♦rr❡❧❛t✐♦♥ ▼♦❞❡❧ ✇✐t❤ ▼❛r❦♦✈

❘❡❣✐♠❡ ❙✇✐t❝❤✐♥❣

P❡❞r♦ ◆✐❡❧s❡♥ ❘♦tt❛ ✭❊❊❙P✲❋●❱✮ ❛♥❞ P❡❞r♦ ▲✳ ❱❛❧❧s P❡r❡✐r❛ ✭❊❊❙P✲❋●❱ ❛♥❞ ❈❊◗❊❋✲❋●❱✮

✶✽t❤ ❏❛♥✉❛r② ✷✵✶✸

❆❜str❛❝t

✷

✶ ■♥tr♦❞✉❝t✐♦♥

●❧♦❜❛❧✐③❛t✐♦♥✱ ❞❡r❡❣✉❧❛t✐♦♥ ❛♥❞ t❡❝❤♥♦❧♦❣✐❝❛❧ ❛❞✈❛♥❝❡s ❤❛✈❡ ❞❡❡♣❧② ❝❤❛♥❣❡❞ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ✜♥❛♥❝✐❛❧ ♠❛r❦❡ts ♦❢ ❞✐✛❡r❡♥t ❝♦✉♥tr✐❡s✳ ❚❤❡r❡ ✐s s✉✣❝✐❡♥t ❡✈✐❞❡♥❝❡ s✉♣♣♦rt✐♥❣ t❤❛t t❤❡ ✐♥❝r❡❛s❡ ♦❢ ✈❡❧♦❝✐t② ♦❢ ✐♥❢♦r♠❛t✐♦♥ tr❛♥s♠✐ss✐♦♥ ✇❛s ♣❛rt✐❛❧❧② r❡s♣♦♥s✐❜❧❡ t♦ ✐♥❞✉❝❡ str♦♥❣❡r st♦❝❦ ♠❛r❦❡t ✐♥t❡❣r❛t✐♦♥✳ ❖♥ ❝❧♦s❡r ✐♥s♣❡❝t✐♦♥✱ t❤❡r❡ ✇❡r❡ q✉❡st✐♦♥s ❛❜♦✉t ✇❤❛t ✇♦✉❧❞ ❜❡ t❤❡ ❞✐s❛❞✈❛♥t❛❣❡s ♦❢ t❤✐s ♣r♦❝❡ss✳ ❆♠♦♥❣ t❤❡ ♠❛♥② ♥❡❣❛t✐✈❡ ❛s♣❡❝ts ♣♦✐♥t❡❞ ♦✉t✱ ♦♥❡ ♦❢ t❤❡♠ r❡❧❛t❡s t♦ t❤❡ ✐♥t❡♥s✐✜❝❛t✐♦♥ ♦❢ t❤❡ ✜♥❛♥❝✐❛❧ ❝♦♥t❛❣✐♦♥ ♣❤❡♥♦♠❡♥♦♥ ❛♥❞ t❤❡ ❧♦ss❡s ❧✐♥❦❡❞ t♦ ❞✐s❝♦♥t✐♥✉✐t✐❡s ✐♥ t❤❡ ♠❡❝❤❛♥✐s♠s ♦❢ s❤♦❝❦s ♣r♦♣❛❣❛t✐♦♥✳ ❚❤❡ ❝❧♦s❡r r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ ♠❛r❦❡ts ❝❛♥ ❧❡❛❞ t♦ ❛ ❡①tr❡♠❡❧② ✐♠♣♦rt❛♥t ✐♥❝r❡❛s❡ ✐♥ ✈✉❧✲ ♥❡r❛❜✐❧✐t② ♦❢ ❡❝♦♥♦♠✐❡s ✇❤❡♥ ❢❛❝✐♥❣ ❡①t❡r♥❛❧ ✜♥❛♥❝✐❛❧ s❤♦❝❦s✳ ❚❤✉s✱ ✜♥❛♥❝✐❛❧ ❝♦♥t❛❣✐♦♥ r❛✐s❡s ❡ss❡♥t✐❛❧ ✐ss✉❡s ❜♦t❤ ❢♦r ❡❝♦♥♦♠✐❝ ♣♦❧✐❝② ♠❛❦❡rs ❛♥❞ ✐♥t❡r♥❛✲ t✐♦♥❛❧ ✐♥✈❡st♦rs s❡❡❦✐♥❣ t♦ ❞✐✈❡rs✐❢② r✐s❦s✳ ❉✉r✐♥❣ t❤❡ ❧❛st t✇♦ ❞❡❝❛❞❡s✱ t❤❡ ❛♥❛❧②s✐s ♦❢ ♣❛tt❡r♥s ♦❢ ✐♥t❡r♥❛t✐♦♥❛❧ s♣r❡❛❞ ♦❢ ✜♥❛♥❝✐❛❧ ❡✈❡♥ts ❜❡❝❛♠❡ t❤❡ s✉❜✲ ❥❡❝t ♦❢ ♠❛♥② ❛❝❛❞❡♠✐❝ st✉❞✐❡s✱ s♣❡❝✐❛❧❧② ❡♠♣✐r✐❝❛❧ r❡s❡❛r❝❤ ❢♦❝✉s❡❞ ♦♥ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s✳

■♥ t❤✐s ❝♦♥t❡①t✱ t❤❡ ❣♦❛❧ ♦❢ t❤✐s st✉❞② ✐s t♦ ❡✈❛❧✉❛t❡ t❤❡ ✜♥❛♥❝✐❛❧ ❝♦♥t❛✲ ❣✐♦♥ ❜❡t✇❡❡♥ st♦❝❦ ♠❛r❦❡t r❡t✉r♥s✳ ❚❤❡ ❡❝♦♥♦♠❡tr✐❝ ❛♣♣r♦❛❝❤ ❡♠♣❧♦②❡❞ ✇❛s ♦r✐❣✐♥❛❧❧② ♣r❡s❡♥t❡❞ ❜② P❡❧❧❡t✐❡r ✭✷✵✵✻✮✱ ♥❛♠❡❞ ❘❡❣✐♠❡ ❙✇✐t❝❤✐♥❣ ❉②♥❛♠✐❝ ❈♦rr❡❧❛t✐♦♥ ✭❘❙❉❈✮✱ ✇❤✐❝❤ ✐♥✈♦❧✈❡s t❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❈♦♥st❛♥t ❈♦♥❞✐t✐♦♥❛❧ ❈♦rr❡❧❛t✐♦♥ ▼♦❞❡❧ ✭❈❈❈✮ ♣r♦♣♦s❡❞ ❜② ❇♦❧❧❡rs❧❡✈ ✭✶✾✾✵✮ ✇✐t❤ ▼❛r❦♦✈ ❘❡❣✐♠❡ ❙✇✐t❝❤✐♥❣ ▼♦❞❡❧ s✉❣❣❡st❡❞ ❜② ❍❛♠✐❧t♦♥ ❛♥❞ ❙✉s♠❡❧ ✭✶✾✾✹✮✳ ❆ ♠♦❞✐✜❝❛t✐♦♥ ✇❛s ♠❛❞❡ ✐♥ t❤❡ ♦r✐❣✐♥❛❧ ♠♦❞❡❧ ❘❙❉❈✿ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ t❤❡ ●❏❘✲●❆❘❈❍ ●❧♦st❡♥ ♠♦❞❡❧✱ ❢♦r♠✉❧❛t❡❞ ✐♥ ●❧♦st❡♥✱ ❏❛❣❛♥♥❛t❤❛♥ ❡ ❘✉♥❦❧❡ ✭✶✾✾✸✮✱ ♦♥ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ✉♥✐✈❛r✐❛t❡ ✈❛r✐❛♥❝❡s t♦ ❛❧❧♦✇ ❛s②♠♠❡tr✐❝ ❡✛❡❝ts ✐♥ ✈♦❧❛t✐❧✐t② ❜❡ ❝❛♣t✉r❡❞✳ ■t ✐s ✇♦rt❤ ♥♦t✐♥❣ t❤❛t ✐s ♥♦t ♣❛rt ♦❢ t❤❡ s❝♦♣❡ ♦❢ t❤✐s ✇♦r❦ t♦ ✐❞❡♥t✐❢② ♣♦ss✐❜❧❡ ❝❛✉s❡s ♦r ♣r❡❞✐❝t t❤❡ ❡❝♦♥♦♠✐❝ ❝♦♥s❡q✉❡♥❝❡s ♦❢ t❤❡ ✜♥❛♥❝✐❛❧ ❝♦♥t❛❣✐♦♥✳

❚❤❡ ❘❙❉❈ ♠♦❞❡❧ str✉❝t✉r❡ ✐♥❝♦r♣♦r❛t❡s ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❝♦✈❛r✐❛♥❝❡ ♠❛✲ tr✐❝❡s ✐♥t♦ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥s ❛♥❞ ❝♦rr❡❧❛t✐♦♥s✳ ❚❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ❢♦❧❧♦✇s ❛ r❡❣✐♠❡ s✇✐t❝❤✐♥❣ ♠♦❞❡❧ ❛♥❞✱ t❤❡r❡❢♦r❡✱ st❛②s ❝♦♥st❛♥t ✇✐t❤✐♥ ❛ r❡❣✐♠❡✱ ❜✉t ❞✐✛❡r❡♥t ❛❝r♦ss t❤❡ r❡❣✐♠❡s✳ ❚r❛♥s✐t✐♦♥s ❜❡t✇❡❡♥ t❤❡s❡ r❡❣✐♠❡s ❛r❡ ❣♦✈❡r♥❡❞ ❜② ❛ ▼❛r❦♦✈ ❝❤❛✐♥✳ ❚❤❡ ❝❡♥tr❛❧ ❝♦♥❝❡♣t ✐♥ t❤✐s t②♣❡ ♦❢ ♠♦❞❡❧ ✐s t♦ ❞❡❝♦♠♣♦s❡ ❛ s❡r✐❡s ✐♥ ❛ ✜♥✐t❡ s❡q✉❡♥❝❡ ♦❢ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s✱ t❤❛t ✐s✱ ❞✐✈✐❞❡ t❤❡ s❡r✐❡s ✐♥ r❡❣✐♠❡s✳ ❈♦♥s✐❞❡r✐♥❣ t❤✐s ❝❤❛r❛❝t❡r✐st✐❝✱ t❤✐s ♠♦❞❡❧ ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s ♣✐❡❝❡✲ ✇✐s❡ ❧✐♥❡❛r s✐♥❝❡ t❤❡ ♣r♦❝❡ss ✐s ❧✐♥❡❛r ✐♥ ❡❛❝❤ r❡❣✐♠❡✱ ❤♦✇❡✈❡r ✇✐t❤ ❛ ♥♦♥✲❧✐♥❡❛r str✉❝t✉r❡ ✇❤❡♥ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❡♥t✐r❡ ♣r♦❝❡❞✉r❡✳

✸

❉❈❈ ✐s ❛❧r❡❛❞② ✈❡r② ✇✐❞❡s♣r❡❛❞ ❛♥❞ ✉s❡❞ ❢♦r ❝♦♥t❛❣✐♦♥ t❡sts ♣❡r❢♦r♠❡❞ ✐♥ t❤❡ ❧❛st t❡♥ ②❡❛rs✳ ❚♦ ❝♦♠♣❧❡♠❡♥t t❤❡ ❛♥❛❧②s✐s ♦❢ t❤✐s ✇♦r❦✱ ❛ ❝♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t❤❡ r❡s✉❧ts ♦❢ ❜♦t❤ ♠♦❞❡❧s ✐s ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✺✳

❚❤❡ ❞❛t❛❜❛s❡ ✇❛s ❜✉✐❧t ✇✐t❤ t❤❡ s❡r✐❡s ♦❢ ❞❛✐❧② ❝❧♦s✐♥❣ st♦❝❦ ♠❛r❦❡t ✐♥❞✐❝❡s ✐♥ t❤❡ ❯♥✐t❡❞ ❙t❛t❡s ✭❙P✺✵✵✮✱ ❯♥✐t❡❞ ❑✐♥❣❞♦♠ ✭❋❚❙❊✶✵✵✮✱ ❇r❛③✐❧ ✭■❇❖❱❊❙P❆✮ ❛♥❞ ❙♦✉t❤ ❑♦r❡❛ ✭❑❖❙P■✮ ❢♦r t❤❡ ♣❡r✐♦❞ ❢r♦♠ ✵✷✴✵✶✴✷✵✵✸ t♦ ✵✾✴✷✵✴✷✵✶✷✳ ❚❤❡ r❡❛s♦♥ ❢♦r ❝❤♦♦s✐♥❣ t❤✐s ❣r♦✉♣ ♦❢ ❝♦✉♥tr✐❡s ✇❛s t❤❡ ✐❞❡❛ ♦❢ ❤❛✈✐♥❣ t✇♦ r❡♣✲ r❡s❡♥t❛t✐✈❡s ❢r♦♠ ❞✐✛❡r❡♥t r❡❣✐♦♥s ♦❢ ❞❡✈❡❧♦♣❡❞ ♠❛r❦❡ts ✭❊✉r♦♣❡ ❛♥❞ ◆♦rt❤ ❆♠❡r✐❝❛✮ ❛♥❞ t✇♦ ❢r♦♠ ❡♠❡r❣✐♥❣ ♠❛r❦❡ts ✭❙♦✉t❤ ❆♠❡r✐❝❛✱ ❛♥❞ ❆s✐❛✮✳

❚❤❡ r❡st ♦❢ t❤❡ ♣❛♣❡r ✐s ♦r❣❛♥✐③❡❞ ❛s ❢♦❧❧♦✇s✳ ❈❤❛♣t❡r ✷ ♣r♦✈✐❞❡s ❧✐t❡r❛t✉r❡ r❡✈✐❡✇ ♦♥ ❞❡✜♥✐t✐♦♥s ♦❢ ❝♦♥t❛❣✐♦♥ ❛♥❞ ❡❝♦♥♦♠❡tr✐❝ ❛♣♣r♦❛❝❤❡s✳ ❈❤❛♣t❡r ✸ ♦✉t✲ ❧✐♥❡s t❤❡ ❡❝♦♥♦♠❡tr✐❝ ♠❡t❤♦❞♦❧♦❣② ❡♠♣❧♦②❡❞✳ ❚❤❡ ❈❤❛♣t❡rs ✹ ❛♥❞ ✺ ❞❡s❝r✐❜❡✱ r❡s♣❡❝t✐✈❡❧②✱ t❤❡ ❞❛t❛❜❛s❡ ❛♥❞ t❤❡ ♠❛✐♥ r❡s✉❧ts✳ ❋✐♥❛❧❧②✱ ❈❤❛♣t❡r ✻ s✉♠♠❛r✐③❡s t❤❡ ♠❛✐♥ ❝♦♥❝❧✉s✐♦♥s ♦❢ t❤✐s ✇♦r❦✳

✷ ▲✐t❡r❛t✉r❡ ❘❡✈✐❡✇

✷✳✶ ❈♦♥t❛❣✐♦♥ ❉❡✜♥✐t✐♦♥

■♥ t❤❡ ❧❛st t✇♦ ❞❡❝❛❞❡s✱ t❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ tr❛❞✐t✐♦♥❛❧ ♣❛tt❡r♥ ♦❢ ✐♥t❡r✲ ♥❛t✐♦♥❛❧ s♣r❡❛❞ ♦❢ ✜♥❛♥❝✐❛❧ ❡✈❡♥ts ❜❡❝❛♠❡ ❛ t❛r❣❡t ♦❢ ❛❝❛❞❡♠✐❝ ♣r♦❞✉❝t✐♦♥ ❢♦❝✉s❡❞ ♦♥ ♠♦❞❡❧✐♥❣ ✈♦❧❛t✐❧✐t②✳ ❆❧♠♦st ❛❧❧ ♦❢ t❤❡s❡ ✇♦r❦s ❛r❡ ❢❛❝❡❞ ✇✐t❤ ❛ ♣r✐♠❛r② ❞✐✣❝✉❧t✱ t❤❡ ❛❜s❡♥❝❡ ♦❢ ❛ ❝♦♠♠♦♥ ❞❡✜♥✐t✐♦♥ ♦❢ ✜♥❛♥❝✐❛❧ ❝♦♥t❛❣✐♦♥✳

❚❤❡ ❡✈✐❞❡♥❝❡ ♦❢ ❤❡t❡r♦❣❡♥❡✐t② ♦❢ ❞❡✜♥✐t✐♦♥s ❝❛♥ ❜❡ ✐❧❧✉str❛t❡❞ ❜② t❤❡ ✇♦r❦ ♦❢ P❡r✐❝♦❧✐ ❛♥❞ ❙❜r❛❝✐❛ ✭✷✵✵✶✮ t❤❛t s✉❣❣❡st❡❞ ✜✈❡ ❞✐✛❡r❡♥t ❞❡✜♥✐t✐♦♥s ♦❢ ❝♦♥t❛✲ ❣✐♦♥✿ s✐❣♥✐✜❝❛♥t❧② ✐♥❝r❡❛s❡ ✐♥ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛ ❝r✐s✐s ✐♥ ❛ ❝♦✉♥tr② ❝♦♥❞✐t✐♦♥❛❧ ♦♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ❝r✐s✐s ✐♥ ❛♥♦t❤❡r ❝♦✉♥tr②❀ ♦✈❡r✢♦✇ ✭s♣✐❧❧♦✈❡r✮ ♦❢ ✈♦❧❛t✐❧✐t② ♦❢ ❛ ❝♦✉♥tr② ✐♥ ❝r✐s✐s ❢♦r t❤❡ ✜♥❛♥❝✐❛❧ ♠❛r❦❡ts ♦❢ ♦t❤❡r ❝♦✉♥tr✐❡s❀ str♦♥❣ ❣r♦✇t❤ ♦❢ ❝♦✲♠♦✈❡♠❡♥ts ♦❢ ♣r✐❝❡s ❛♥❞ q✉❛♥t✐t✐❡s ❛❝r♦ss ♠❛r❦❡ts✱ ❝♦♥❞✐t✐♦♥❛❧ ♦♥ ❛ ❝r✐✲ s✐s ✐♥ ♦♥❡ ♠❛r❦❡t ♦r ❣r♦✉♣ ♦❢ ♠❛r❦❡ts❀ ❝❤❛♥❣❡s ✐♥ t❤❡ tr❛♥s♠✐ss✐♦♥ ❝❤❛♥♥❡❧ ♦❢ s❤♦❝❦s ❛❝r♦ss ♠❛r❦❡ts❀ ❡①❝❡ss ♦❢ ❝♦✲♠♦✈❡♠❡♥ts ♥♦t ❡①♣❧❛✐♥❡❞ ❜② ❡❝♦♥♦♠✐❝ ❢✉♥❞❛♠❡♥t❛❧s✳

✹

❝❛t❡❣♦r②✱ r❡❢❡rr✐♥❣ t♦ t❤❡ t❤❡♦r② ♦❢ ♣✉r❡ ❝♦♥t❛❣✐♦♥ ❞❡✜♥❡s t❤❛t ❛ ❝r✐s✐s ✐♥ ♦♥❡ ❝♦✉♥tr② ❝❛♥ ❝❛✉s❡ ❛ ❝r✐s✐s ✐♥ ❛♥♦t❤❡r ❝♦✉♥tr② ✇✐t❤♦✉t ❛✛❡❝t✐♥❣ t❤❡ ❡❝♦♥♦♠✐❝ ❢✉♥❞❛♠❡♥t❛❧s ❜❡t✇❡❡♥ t❤❡♠✳

❚❤❡ ❲♦r❧❞ ❇❛♥❦ ✶ _{♦✛❡rs t❤r❡❡ ❞✐✛❡r❡♥t ❞❡✜♥✐t✐♦♥s ♦❢ ❝♦♥t❛❣✐♦♥✿ ❜r♦❛❞✱} r❡str✐❝t✐✈❡ ❛♥❞ ✈❡r② r❡str✐❝t✐✈❡✳ ■♥ t❤❡ ❜r♦❛❞ ❞❡✜♥✐t✐♦♥✱ t❤❡ ❝♦♥t❛❣✐♦♥ ✐s t❤❡ ❝r♦ss✲❝♦✉♥tr② tr❛♥s♠✐ss✐♦♥ ♦❢ s❤♦❝❦s ♦r t❤❡ ❣❡♥❡r❛❧ ❝r♦ss✲❝♦✉♥tr② s♣✐❧❧♦✈❡r ❡❢✲ ❢❡❝ts✳ ❈♦♥t❛❣✐♦♥ ❝❛♥ t❛❦❡ ♣❧❛❝❡ ❜♦t❤ ❞✉r✐♥❣ ❣♦♦❞ t✐♠❡s ❛♥❞ ❜❛❞ t✐♠❡s✳ ❚❤❡♥✱ ❝♦♥t❛❣✐♦♥ ❞♦❡s ♥♦t ♥❡❡❞ t♦ ❜❡ r❡❧❛t❡❞ t♦ ❝r✐s❡s✳ ❇② t❤❡ r❡str✐❝t✐✈❡ ❞❡✜♥✐t✐♦♥✱ ❝♦♥t❛❣✐♦♥ ✐s t❤❡ tr❛♥s♠✐ss✐♦♥ ♦❢ s❤♦❝❦s t♦ ♦t❤❡r ❝♦✉♥tr✐❡s ♦r t❤❡ ❝r♦ss✲❝♦✉♥tr② ❝♦rr❡❧❛t✐♦♥✱ ❜❡②♦♥❞ ❛♥② ❢✉♥❞❛♠❡♥t❛❧ ❧✐♥❦ ❛♠♦♥❣ t❤❡ ❝♦✉♥tr✐❡s ❛♥❞ ❜❡②♦♥❞ ❝♦♠♠♦♥ s❤♦❝❦s✳ ❚❤✐s ❞❡✜♥✐t✐♦♥ ✐s ✉s✉❛❧❧② r❡❢❡rr❡❞ ❛s ❡①❝❡ss ❝♦✲♠♦✈❡♠❡♥t✱ ❝♦♠♠♦♥❧② ❡①♣❧❛✐♥❡❞ ❜② ❤❡r❞✐♥❣ ❜❡❤❛✈✐♦r✳ ❚❤❡ ✈❡r② r❡str✐❝t✐✈❡ ❞❡✜♥✐t✐♦♥ s❛②s t❤❛t ❝♦♥t❛❣✐♦♥ ♦❝❝✉rs ✇❤❡♥ ❝r♦ss✲❝♦✉♥tr② ❝♦rr❡❧❛t✐♦♥s ✐♥❝r❡❛s❡ ❞✉r✐♥❣ ❝r✐s✐s t✐♠❡s r❡❧❛t✐✈❡ t♦ ❝♦rr❡❧❛t✐♦♥s ❞✉r✐♥❣ tr❛♥q✉✐❧ t✐♠❡s✳ ❚❤✐s ❧❛st ❞❡✜♥✐t✐♦♥ ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s t❤❡ ❜r❡❛❦❞♦✇♥ ✐♥ t❤❡ tr❛♥s♠✐ss✐♦♥ ♠❡❝❤❛♥✐s♠ t❤❛t ♦❝❝✉rs ❞✉r✐♥❣ ❛ ♣❡r✐♦❞ ♦❢ t✉r❜✉❧❡♥❝❡✱ ❛❧s♦ ❝♦♠♠♦♥❧② ❝❛❧❧❡❞ s❤✐❢t✲❝♦♥t❛❣✐♦♥✳

❆❢t❡r r❡✈✐❡✇✐♥❣ t❤❡ ❧✐t❡r❛t✉r❡ ❞❡s❝r✐❜❡❞✱ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥t❛❣✐♦♥ ❛❞♦♣t❡❞ ✇✐❧❧ ❜❡ t❤❡ s❛♠❡ ❛s ❋♦r❜❡s ❛♥❞ ❘✐❣♦❜♦♥ ✭✷✵✵✷✮✳ ❚❤✐s ❞❡✜♥✐t✐♦♥ ✐s t❤❡ s❛♠❡ ❛s t❤❛t ❝❛❧❧❡❞ ✈❡r② r❡str✐❝t✐✈❡ ❜② t❤❡ ❲♦r❧❞ ❇❛♥❦✳ ❚❤✉s✱ ✇❡ ❝❛♥ ❞❡✜♥❡ ❝♦♥t❛❣✐♦♥ ❛s ❛ s❤✐❢t ✐♥ t❤❡ ♣❛tt❡r♥ ♦❢ ❝♦rr❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ r❡t✉r♥s ♦❢ t❤❡ st♦❝❦ ♠❛r❦❡t ✐♥❞✐❝❡s ♦❢ ❞✐✛❡r❡♥t ❝♦✉♥tr✐❡s ✇❤❡♥ ❝♦♠♣❛r✐♥❣ ♣❡r✐♦❞s ♦❢ ❝❛❧♠ ❛♥❞ ♣❡r✐♦❞s ♦❢ ✜♥❛♥❝✐❛❧ ❝r✐s❡s✳

❲❤❡r❡❛s t❤❛t ✐♥ ❡❝♦♥♦♠❡tr✐❝ ♠❡t❤♦❞♦❧♦❣②✱ t♦ ❜❡ ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✸✱ ✐s ♥♦t ✉s❡❞ ❛♥② ♠❛❝r♦❡❝♦♥♦♠✐❝ ✈❛r✐❛❜❧❡ ♦r ❡❝♦♥♦♠✐❝ ❢✉♥❞❛♠❡♥t❛❧s✱ ✐s ❜❡✐♥❣ ✉♥❞❡rt❛❦❡♥ ✐♥ s♦♠❡ ✇❛② ❛s ❛ ♣r❡♠✐s❡✱ t❤❛t t❤❡s❡ ❢❛❝t♦rs ♦r ♠❡❝❤❛♥✐s♠s ✇♦✉❧❞ ♥♦t ❜❡ ❛❜❧❡ t♦ tr❛♥s♠✐t t❤❡ ❡✛❡❝ts ♦❢ ❝r✐s❡s ♦❜s❡r✈❡❞ ❢r♦♠ ✷✵✵✸ ✉♥t✐❧ ✷✵✶✷ s♦ q✉✐❝❦❧② ❛♥❞ ❛❜r✉♣t❧②✳ ■♥ ❈❤❛♣t❡r ✻ ✭❈♦♥❝❧✉s✐♦♥✮ ❛ r❡❢❡r❡♥❝❡ ✐s ♠❛❞❡ t♦ t❤✐s q✉❡st✐♦♥✱ ✇❤✐❝❤ ✐s ❞✐r❡❝t❧② r❡❧❛t❡❞ t♦ t❤❡ ❞✐st✐♥❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ✐♥t❡r❞❡♣❡♥❞❡♥❝❡ ❛♥❞ ❝♦♥t❛❣✐♦♥✱ ❛s ❛ s✉❣❣❡st✐♦♥ ❢♦r ❢✉t✉r❡ ❡①t❡♥s✐♦♥ ♦❢ t❤✐s st✉❞②✳

✷✳✷ ❊❝♦♥♦♠❡tr✐❝ ❆♣♣r♦❛❝❤❡s

▲✐❦❡✇✐s❡ t❤❡r❡ ✐s ♥♦ ✉♥✐❢♦r♠✐t② ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ✜♥❛♥❝✐❛❧ ❝♦♥t❛❣✐♦♥✱ t❤❡r❡ ✐s ♥♦ ❝♦♥s❡♥s✉s ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ❛❜♦✉t t❤❡ ❡❝♦♥♦♠❡tr✐❝ ♠❡t❤♦❞♦❧♦❣② t♦ ❜❡ ❡♠♣❧♦②❡❞✳ ❆♠♦♥❣ t❤❡ ♠❛❥♦r ✇♦r❦s ♦♥ t❤❡ s✉❜❥❡❝t✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❤✐❣❤❧✐❣❤t t❤❡ ✇♦r❦ ♦❢ ❋♦r❜❡s ❛♥❞ ❘✐❣♦❜♦♥ ✭✷✵✵✷✮ ✇❤♦ ❛♣♣❧✐❡❞ ❛ ❱❡❝t♦r ❆✉t♦r❡❣r❡ss✐✈❡ ▼♦❞❡❧ ✭❱❆❘✮ ❛♥❞ ❛ ♠❡❛s✉r❡ ♦❢ ❛❞❥✉st❡❞ ❝♦rr❡❧❛t✐♦♥ ❢♦r t❤❡ ❞❡t❡❝t✐♦♥ ♦❢ ❝♦♥✲ t❛❣✐♦♥ ❞✉r✐♥❣ t❤❡ ❝r✐s❡s ✐♥ ▼❡①✐❝♦ ✐♥ ✶✾✾✹ ❛♥❞ ❆s✐❛ ✐♥ ✶✾✾✼ ❜❛s❡❞ ♦♥ ❞❛t❛ ♦❢ ✷✾ ❝♦✉♥tr✐❡s✳ ❯s✐♥❣ t❤❡ s❛♠❡ ❞❛t❛❜❛s❡✱ ❈♦rs❡tt✐ ❡t ❛❧ ✭✷✵✵✺✮ ♣r❡s❡♥t❡❞ ❛ ❝r✐t✲ ✐❝❛❧ r❡✈✐❡✇ ♦❢ t❤❡ ❛❞❥✉st❡❞ ❝♦rr❡❧❛t✐♦♥ ♠❡❛s✉r❡ ❛♥❞ ❛♣♣❧✐❡❞ ❛ st❛♥❞❛r❞ ❢❛❝t♦r ♠♦❞❡❧ ❢♦r r❡t✉r♥s t❤❛t ❞♦ ♥♦t ✐♠♣♦s❡ ❛♥② r❡str✐❝t✐♦♥s ♦♥ t❤❡ ✈❛r✐❛♥❝❡ ♦❢ t❤❡

✺

❝♦♠♠♦♥ ❢❛❝t♦rs✳ ❲✐t❤ r❡s♣❡❝t t♦ ❢❛❝t♦r ♠♦❞❡❧s✱ t❤❡ ❛♣♣r♦❛❝❤ ♦❢ ▲♦♣❡s ❛♥❞ ▼✐❣♦♥ ✭✷✵✵✷✮ ❝♦♥s✐st❡❞ ♦❢ tr❛❞✐t✐♦♥❛❧ t❡❝❤♥✐q✉❡s ♦❢ ❢❛❝t♦r ♠♦❞❡❧s ❝♦♠❜✐♥❡❞ ✇✐t❤ st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s t♦ st✉❞② t❤❡ ❞❡♣❡♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ✐♥❞✐❝❡s ♦❢ st♦❝❦ ♣r✐❝❡s ✐♥ ▲❛t✐♥ ❆♠❡r✐❝❛ ❛♥❞ ◆♦rt❤ ❆♠❡r✐❝❛✳ ❚❤❡ ✇♦r❦ ♦❢ ❉✉♥❣❡② ❡t ❛❧ ✭✷✵✵✹✮ ♣r♦✈✐❞❡s ❛ ❞❡t❛✐❧❡❞ r❡✈✐❡✇ ♦❢ ♠❡t❤♦❞♦❧♦❣✐❡s t❤❛t ❛r❡ ❜❛s❡❞ ♦♥ ✈❡r② ❝❧♦s❡ ❞❡✜♥✐t✐♦♥s ♦❢ ❝♦♥t❛❣✐♦♥ s♦ ✐t ✇❛s ♣♦ss✐❜❧❡ t♦ ❜✉✐❧❞ ❛ ❤♦♠♦❣❡♥❡♦✉s ❢r❛♠❡✇♦r❦ t♦ ❤✐❣❤❧✐❣❤t t❤❡ ♠❛✐♥ s✐♠✐❧❛r✐t✐❡s ❛♥❞ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ t❤❡ ✈❛r✐♦✉s ❛♣♣r♦❛❝❤❡s✳ ●r❛❞✉❛❧❧②✱ ♠❡t❤♦❞♦❧♦❣✐❡s ❝♦♥✈❡r❣❡❞ ❢♦r ❝♦♥❞✐t✐♦♥❛❧ ❤❡t❡r♦s❝❡❞❛st✐❝✐t② ♠♦❞✲ ❡❧s✱ ❛❢t❡r t❤❡ s❡♠✐♥❛❧ ✇♦r❦ ♦❢ ❊♥❣❧❡ ✭✶✾✽✷✮ ❛♥❞ ❇♦❧❧❡rs❧❡✈ ✭✶✾✽✻✮✳ ❙✉❝❤ ♠♦❞❡❧s ✇❡r❡✱ ✐♥ s❤♦rt t✐♠❡✱ ❣❡♥❡r❛❧✐③❡❞ t♦ ♠✉❧t✐✈❛r✐❛t❡ ✈❡rs✐♦♥s ❢♦r t✇♦ ♠❛✐♥ ❢♦r♠✉✲ ❧❛t✐♦♥s✳ ❚❤❡ ❱❊❈ ♠♦❞❡❧ ✐♥ ✇❤✐❝❤ ❇♦❧❧❡rs❧❡✈✱ ❊♥❣❧❡ ❛♥❞ ❲♦♦❧❞r✐❞❣❡ ✭✶✾✽✽✮ ❡①t❡♥❞❡❞ t❤❡ ●❆❘❈❍✲▼ ♠♦❞❡❧ ❢♦r ♠✉❧t✐✈❛r✐❛t❡ ❝♦♥t❡①t ✉s✐♥❣ t❤❡ ♦♣❡r❛t♦r ❱❊❈❍✳ ❆♥❞ t❤❡ ♦t❤❡r ❢♦r♠✉❧❛t✐♦♥ ✇❛s t❤❡ ❇❊❑❑ ♠♦❞❡❧ ❞❡✈❡❧♦♣❡❞ ❜② ❊♥✲ ❣❧❡ ❛♥❞ ❑r♦♥❡r ✭✶✾✾✺✮✳ ❚❤❡ ♥❛♠❡ ❇❊❑❑ r❡❢❡rs t♦ t❤❡ ✐♥✐t✐❛❧s ♦❢ ❡❛❝❤ ♦❢ t❤❡ ❛✉t❤♦rs ❇❛❜❛✱ ❊♥❣❧❡✱ ❑r❛❢t ❛♥❞ ❑r♦♥❡r✳

❚❤❡ ♠✉❧t✐✈❛r✐❛t❡ ●❆❘❈❍ ♠♦❞❡❧s ❤❛✈❡ ❛ttr❛❝t❡❞ ❝♦♥s✐❞❡r❛❜❧❡ ✐♥t❡r❡st ❜❡✲ ❝❛✉s❡ ♦❢ ✐ts ❞✐r❡❝t ❛♣♣❧✐❝❛t✐♦♥ ❜♦t❤ ✐♥ ❡❝♦♥♦♠✐❝ ❛♥❞ ✜♥❛♥❝❡ ❡♠♣✐r✐❝❛❧ r❡s❡❛r❝❤✳ ◆❡✈❡rt❤❡❧❡ss✱ t❤❡ ✜rst ♦r❞❡r ♦❢ ♦❜st❛❝❧❡s ❛r♦s❡ ✇❤❡♥ ❝♦♥s✐❞❡r❡❞ ❧❛r❣❡ ❞✐♠❡♥s✐♦♥ ❡s♣❡❝✐✜❝❛t✐♦♥s ❞✉❡ t♦ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❝♦♠♣✉t❛t✐♦♥❛❧ ♣r♦❝❡❞✉r❡s t♦ ❡st✐♠❛t❡ ♣❛r❛♠❡t❡rs r❡s✉❧t✐♥❣ ❢r♦♠ t❤❡ ❤✐❣❤ ♥✉♠❜❡r ♦❢ ❝♦❡✣❝✐❡♥ts✳ ❙✉❝❤ ❞✐✣❝✉❧t✐❡s ❡♥❝♦✉r❛❣❡❞ ♠❛♥② ❛❝❛❞❡♠✐❝ st✉❞✐❡s t♦ s✉❜♠✐t ❡♥❤❛♥❝❡♠❡♥ts ❛♥❞ ❛❧t❡r♥❛t✐✈❡s t♦ tr❛❞✐t✐♦♥❛❧ ♠✉❧t✐✈❛r✐❛t❡ ●❆❘❈❍ ♠♦❞❡❧s✳

❖♥❡ ♦❢ t❤❡ ♠♦st ✇✐❞❡s♣r❡❛❞ ♠♦❞❡❧s ♦❢ ♠✉❧t✐✈❛r✐❛t❡ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧ ✐s ❝❡r✲ t❛✐♥❧② t❤❡ ❈♦♥st❛♥t ❈♦♥❞✐t✐♦♥❛❧ ❈♦rr❡❧❛t✐♦♥ ✭❈❈❈✮ ♦❢ ❇♦❧❧❡rs❧❡✈ ✭✶✾✾✵✮✱ ✐♥ ✇❤✐❝❤ t❤❡ ❝♦✈❛r✐❛♥❝❡ ♦❢ ❛ ✈❡❝t♦r ♦❢ r❡t✉r♥s ❛r❡ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ ✈❛r✐❛♥❝❡s ❛♥❞ ❝♦rr❡❧❛t✐♦♥s ✳ ❚❤❡ ♠❛✐♥ ❛ss✉♠♣t✐♦♥ ✐♥ t❤✐s ♠♦❞❡❧ ✐s t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❝♦r✲ r❡❧❛t✐♦♥s ❛r❡ ❝♦♥st❛♥t ♦✈❡r t✐♠❡✳ ❚❤❡ ❛❞✈❛♥t❛❣❡ ✐s t❤❛t t❤❡ ❈❈❈ ✇❛s ❞❡s✐❣♥❡❞ t♦ ❤❛✈❡ t❤❡ ✢❡①✐❜✐❧✐t② ♦❢ ❛ ✉♥✐✈❛r✐❛t❡ ●❆❘❈❍ ❜✉t ♥♦t t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❛ tr❛❞✐t✐♦♥❛❧ ♠✉❧t✐✈❛r✐❛t❡ ●❆❘❈❍✳ ❊♥❣❧❡ ✭✷✵✵✷✮ ❛♥❞ ❚s❡ ❛♥❞ ❚s✉✐ ✭✷✵✵✷✮ ♣r❡✲ s❡♥t❡❞ ❛♥ ❡①t❡♥s✐♦♥ t♦ t❤❡ ❈❈❈ ♠♦❞❡❧ t❤❛t ✉s❡s t❤❡ s❛♠❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✱ ❜✉t ✐♥st❡❛❞ ♦❢ ❛ss✉♠✐♥❣ ❝♦♥st❛♥t ❝♦rr❡❧❛t✐♦♥✱ t❤❡ ❛✉t❤♦rs ♣r♦♣♦s❡ ❛ ●❆❘❈❍ ❞②♥❛♠✐❝ ❢♦r ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐❝❡s✳ ❙✉❝❤ ♠♦❞❡❧s✱ ♥❛♠❡❞ ❉②✲ ♥❛♠✐❝ ❈♦♥❞✐t✐♦♥❛❧ ❈♦rr❡❧❛t✐♦♥ ✭❉❈❈✮ ❤❛✈❡ ♥✉♠❡r✐❝❛❧ st❛❜✐❧✐t② ❡✈❡♥ ❢♦r ❧❛r❣❡ ❞✐♠❡♥s✐♦♥ ♣r♦❜❧❡♠s ❜❡❝❛✉s❡✱ ❛s ✇❡❧❧ ❛s t❤❡ ❈❈❈✱ ❡st✐♠❛t✐♦♥ ♣r♦❝❡❞✉r❡ ♠❛② ❜❡ ❞♦♥❡ ✐♥ t✇♦ st❛❣❡s✳ ❚❤❡ ♠❛✐♥ ❜❡♥❡✜t ♦❢ t❤❡ ❝♦rr❡❧❛t✐♦♥s ♠♦❞❡❧s ♦✈❡r t❤❡ ❇❊❑❑ ❛♥❞ ❱❊❈ ❡s♣❡❝✐✜❝❛t✐♦♥s ✐s t❤❡ ♣❛rs✐♠♦♥② ✐♥ ♣❛r❛♠❡t❡r✐③❛t✐♦♥✱ ✇❤✐❝❤ ❛❧❧♦✇s t♦ ♦✈❡r❝♦♠❡ t❤❡ ❜❛rr✐❡rs t♦ ✐♠♣❧❡♠❡♥t ♠♦❞❡❧s ✐♥ ❞✐♠❡♥s✐♦♥s ❧❛r❣❡r t❤❛♥ t❤❡ ❜✐✈❛r✐❛t❡ ♦♥❡✳ ❆♠♦♥❣ t❤❡ st✉❞✐❡s t❤❛t ✉s❡❞ t❤❡s❡ ♠✉❧t✐✈❛r✐❛t❡ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s ❢♦r ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ ❝♦♥t❛❣✐♦♥✿ ▲♦♠❜❛r❞✐ ❡t ❛❧ ✭✷✵✵✹✮ ✱ ▼❛r❝❛❧ ❛♥❞ ❱❛❧❧s P❡r❡✐r❛ ✭✷✵✵✽✮ ❛♥❞ ❋✐❧❧❡t✐ ✱ ❍♦tt❛ ❛♥❞ ❩❡✈❛❧❧♦s ✭✷✵✵✽✮✳

✻

✐③❡❞ ❜② t❤❡ ▼❛r❦♦✈ ❝❤❛✐♥✳ ❆♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❘❙❉❈ ♠♦❞❡❧ ✇❛s ♣r❡s❡♥t❡❞ ✐♥ ❇✐❧❧✐♦ ❛♥❞ ❈❛♣♦r✐♥ ✭✷✵✵✺✮✳ ❚❤✐s st✉❞② ❡❧❛❜♦r❛t❡s ❛ ♠♦r❡ ❣❡♥❡r❛❧ ❢r❛♠❡✇♦r❦ ❜✉✐❧❞✐♥❣ t❤❡ r❡❣✐♠❡ s✇✐t❝❤✐♥❣ ✉♣♦♥ t❤❡ ❉❈❈ ✭▼❙✲❉❈❈✮✱ ✇❤✐❝❤ ❡♥❛❜❧❡s ❢♦r t✐♠❡✲✈❛r②✐♥❣ ❝♦rr❡❧❛t✐♦♥ ✇✐t❤✐♥ ❡❛❝❤ ❝♦rr❡❧❛t✐♦♥ r❡❣✐♠❡✳ ❚❤❡ ❛✉t❤♦rs ♣r♦✈✐❞❡ ❛♥ ❡♠♣✐r✐❝❛❧ ❛♥❛❧②s✐s ♦❢ t❤❡ ♣❤❡♥♦♠❡♥♦♥ ♦❢ ❝♦♥t❛❣✐♦♥ ❜② ❝♦♠♣❛r✐♥❣ t❤❡ r❡s✉❧ts ✇✐t❤ tr❛❞✐t✐♦♥❛❧ r❡♣r❡s❡♥t❛t✐♦♥s ❧✐❦❡ ❈❈❈ ❛♥❞ ❉❈❈✳

❋✐♥❛❧❧②✱ ✐t ✐s ✇♦rt❤ ♠❡♥t✐♦♥✐♥❣ t❤❡ ✇♦r❦ ♦❢ ❈❤❡♥ ✭✷✵✵✾✮ t❤❛t ❛❧s♦ ❛♣♣❧✐❡s t❤❡ ❘❙❉❈ ♠♦❞❡❧ ♣r♦♣♦s❡❞ ❜② P❡❧❧❡t✐❡r ✭✷✵✵✻✮ ❢♦r t❤❡ s❡r✐❡s ♦❢ ❯❙❆ st♦❝❦s ❛♥❞ ❜♦♥❞s ❜❡t✇❡❡♥ ✶✾✾✽ ❛♥❞ ✷✵✵✵✳ ❚❤❡ ❛✉t❤♦r ♣♦✐♥ts ♦✉t t❤❛t ❛ ❉❈❈ str✉❝t✉r❡ ✐♥st❡❛❞ ♦❢ ❈❈❈ ✐♥ ❛ ●❆❘❈❍ ♠♦❞❡❧ ✇✐t❤ ▼❛r❦♦✈ r❡❣✐♠❡ s✇✐t❝❤ ✇♦✉❧❞ ❤✉❣❡❧② ❝♦♠♣❧✐❝❛t❡ t❤❡ ❛❧r❡❛❞② ❝♦♠♣❧✐❝❛t❡❞ ❡st✐♠❛t✐♦♥ ♣r♦❝❡ss✳ ❚❤❡ ❛♠♦✉♥t ♦❢ ♦❜s❡r✲ ✈❛t✐♦♥s ✐♥ ❡❛❝❤ r❡❣✐♠❡ ✇♦✉❧❞ ♥♦t ❜❡ ❡♥♦✉❣❤ t♦ ❛❝❤✐❡✈❡ ❛ r♦❜✉st ❡st✐♠❛t✐♦♥ ♦❢ ❉❈❈✳ ■♥ ■❞✐❡r ✭✷✵✵✾✮ ✐s ❛❧s♦ ♠❛❞❡ r❡❢❡r❡♥❝❡ t♦ t❤❡ ❢❛❝t t❤❛t ❛❞❞✐♥❣ ♠♦r❡ t❤❛♥ ✶✵ ❡①tr❛ ♣❛r❛♠❡t❡rs ✇♦✉❧❞ ♥♦t ❧✐❦❡❧② ♣r♦❞✉❝❡ ❛ ❣r❡❛t ✐♠♣r♦✈❡♠❡♥t ❛♥❞✱ t❤✉s✱ t❤✐s ❢♦r♠✉❧❛t✐♦♥ ✭▼❙✲❉❈❈✮ ✇❛s ♥♦t ❝❤♦s❡♥ ❛s t❤❡ ❝❡♥tr❛❧ ❝♦♥❞✉❝t♦r ♦❢ t❤✐s st✉❞②✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ♠♦❞❡❧ ❘❙❉❈ ✭♦r ▼❙✲❈❈❈✮ ✇♦✉❧❞ ❜❡ s✉✣❝✐❡♥t t♦ ❝❛♣t✉r❡ t❤❡ ❞❡♣❡♥❞❡♥❝② ♦♥ t❤❡ ❝♦rr❡❧❛t✐♦♥✳

✸ ▼❡t❤♦❞♦❧♦❣②

✸✳✶ ❘❙❉❈ ▼♦❞❡❧

❚❤❡ ♠❡t❤♦❞♦❧♦❣② ♦❢ t❤✐s st✉❞② ✇❛s ❜❛s❡❞ ♦♥ t❤❡ ❘❡❣✐♠❡ ❙✇✐t❝❤✐♥❣ ❉②♥❛♠✐❝ ❈♦rr❡❧❛t✐♦♥ ✭❘❙❉❈✮ ♠♦❞❡❧ ♣r♦♣♦s❡❞ ❜② P❡❧❧❡t✐❡r ✭✷✵✵✻✮✳ ■t ✐s t❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❈♦♥st❛♥t ❈♦♥❞✐t✐♦♥❛❧ ❈♦rr❡❧❛t✐♦♥ ✭❈❈❈✮ ♠♦❞❡❧ ♣r❡s❡♥t❡❞ ❜② ❇♦❧❧❡rs❧❡✈ ✭✶✾✾✵✮ ✇✐t❤ ▼❛r❦♦✈ ❘❡❣✐♠❡ ❙✇✐t❝❤✐♥❣ s✉❣❣❡st❡❞ ❜② ❍❛♠✐❧t♦♥ ❛♥❞ ❙✉s♠❡❧ ✭✶✾✾✹✮✳ ❲❡ ♠❛❞❡ ❛ ♠♦❞✐✜❝❛t✐♦♥ ✐♥ t❤❡ ♦r✐❣✐♥❛❧ ❘❙❉❈✱ ✐♥tr♦❞✉❝✐♥❣ t❤❡ ●❏❘✲●❆❘❈❍✷ _{♦♥ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ✉♥✐✈❛r✐❛t❡ ✈❛r✐❛♥❝❡s t♦} ❛❧❧♦✇ ❛s②♠♠❡tr✐❝ ❡✛❡❝ts ✐♥ ✈♦❧❛t✐❧✐t② ❜❡ ❝❛♣t✉r❡❞✳ ❚❤❡ ●❏❘✲●❆❘❈❍ ♠♦❞❡❧ ✇❛s ❢♦r♠✉❧❛t❡❞ ❜② ●❧♦st❡♥✱ r✉♥❦❧❡ ❛♥❞ ❏❛❣❛♥♥❛t❤❛♥ ✭✶✾✾✸✮✳

■♥ t❤❡ ❈❈❈ ♠♦❞❡❧ ♦❢ ❇♦❧❧❡rs❧❡✈ ✭✶✾✾✵✮✱ ❝♦♥❞✐t✐♦♥❛❧ ❝♦✈❛r✐❛♥❝❡ ❛r❡ ♣❛r❛♠✲ ❡t❡r✐③❡❞ t♦ ❜❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ st❛♥❞❛r❞ ❞❡✲ ✈✐❛t✐♦♥s✳ ❚❤❡ ❝♦✈❛r✐❛♥❝❡ ♦❢ ❛ ✈❡❝t♦r ♦❢ r❡t✉r♥s ❛r❡ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥s ❛♥❞ ❝♦rr❡❧❛t✐♦♥s✳ ❚❤✉s✱ ❜♦t❤ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ♣r♦❝❡ss✐♥❣ r❡q✉✐r❡❞ ❢♦r ❡st✐♠❛t✐♦♥ ✇❛s r❡❞✉❝❡❞ ❛♥❞ t❤❡ ✐♠♣♦s✐t✐♦♥ ♦❢ ❝♦♥❞✐t✐♦♥s t♦ ❡♥s✉r❡ t❤❡ ❝♦✲ ✈❛r✐❛♥❝❡ ♠❛tr✐① ✐s ♣♦s✐t✐✈❡ s❡♠✐✲❞❡✜♥✐t❡ ❜❡❝♦♠❡ s✐♠♣❧❡r✳ ◆❡✈❡rt❤❡❧❡ss✱ ❛ ♠❛❥♦r

✼

❛ss✉♠♣t✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧ ✐s t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❝♦rr❡❧❛t✐♦♥s ❛r❡ ❝♦♥st❛♥t ♦✈❡r t✐♠❡ ❛♥❞ t❤✐s s✉♣♣♦s✐t✐♦♥ ✐s ♥♦t ❛❧✇❛②s s✉♣♣♦rt❡❞ ❜② ❞❛t❛ ❛❝❝♦r❞✐♥❣ t♦ ❊♥❣❧❡ ❛♥❞ ❙❤❡♣♣❛r❞ ✭✷✵✵✶✮✳

●✐✈❡♥ t❤✐s ❝♦♥str❛✐♥t✱ P❡❧❧❡t✐❡r ✭✷✵✵✻✮ ♣r♦♣♦s❡❞ ❛ ♥❡✇ ♠✉❧t✐✈❛r✐❛t❡ ✈♦❧❛t✐❧✲ ✐t② ♠♦❞❡❧ ✭❘❙❉❈✮✳ ❚❤❡ str✉❝t✉r❡ ♦❢ t❤✐s ♠♦❞❡❧ ❛❧s♦ ✐♥❝♦r♣♦r❛t❡s t❤❡ ❞❡❝♦♠✲ ♣♦s✐t✐♦♥ ♦❢ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s ✐♥t♦ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥s ❛♥❞ ❝♦rr❡❧❛t✐♦♥s✱ ❜✉t t❤❡ ❝♦rr❡❧❛t✐♦♥s ❤❛✈❡ s♦♠❡ ❞②♥❛♠✐s♠✳ ❚❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ❢♦❧❧♦✇s ❛ r❡❣✐♠❡ s✇✐t❝❤✐♥❣ ♠♦❞❡❧ ❛♥❞✱ t❤❡r❡❢♦r❡✱ st❛②s ❝♦♥st❛♥t ✇✐t❤✐♥ ❛ r❡❣✐♠❡✱ ❜✉t ❞✐✛❡r✲ ❡♥t ❛❝r♦ss t❤❡ r❡❣✐♠❡s✳ ❚r❛♥s✐t✐♦♥s ❜❡t✇❡❡♥ t❤❡s❡ r❡❣✐♠❡s ❛r❡ ❣♦✈❡r♥❡❞ ❜② ❛ ▼❛r❦♦✈ ❝❤❛✐♥✳ ❚❤❡ ❈❈❈ ♠♦❞❡❧ ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ ❘❙❉❈✱ ✇❤❡r❡ t❤❡ ♥✉♠❜❡r ♦❢ r❡❣✐♠❡s ✐s ❡q✉❛❧ t♦ ✶✳

❆♥ ❛❧t❡r♥❛t✐✈❡ ❛♣♣r♦❛❝❤ t♦ t❤❡ ♦♥❡ ❝❤♦s❡♥ ✐♥ t❤✐s st✉❞② ❝♦✉❧❞ ❜❡ t❤❡ ❉②✲ ♥❛♠✐❝ ❈♦♥❞✐t✐♦♥❛❧ ❈♦rr❡❧❛t✐♦♥ ✭❉❈❈✮ ♠♦❞❡❧ ❢♦r♠✉❧❛t❡❞ ❜② ❊♥❣❧❡ ✭✷✵✵✷✮ ❚s❡ ❛♥❞ ❚s✉✐ ✭✷✵✵✷✮✱ t❤❛t ✉s❡s t❤❡ s❛♠❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ♦❢ ❇♦❧❧❡rs❧❡✈ ✭✶✾✾✵✮✱ ❜✉t ✐♥st❡❛❞ ♦❢ ❛ss✉♠✐♥❣ ❝♦♥st❛♥t ❝♦rr❡❧❛t✐♦♥✱ t❤❡ ❛✉t❤♦rs ♣r♦♣♦s❡ t❤❛t t❤❡s❡ ❛r❡ ❝❤❛r❛❝t❡r✐③❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❛ ●❆❘❈❍ t②♣❡ ❞②♥❛♠✐❝✳ ❆❝❝♦r❞✐♥❣ t♦ P❡❧❧❡t✐❡r ✭✷✵✵✻✮✱ ✐t ✐s r❡❛s♦♥❛❜❧❡ t♦ q✉❡st✐♦♥ ✇❤❡t❤❡r ❛ ●❆❘❈❍ t②♣❡ ❛♣♣r♦❛❝❤ ❢♦r ❝♦rr❡❧❛t✐♦♥s ✇♦✉❧❞ ❜❡ ♠♦r❡ ❛♣♣r♦♣r✐❛t❡ ❜❡❝❛✉s❡ t❤❡ ❞②♥❛♠✐❝ ♦❢ t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛② ❜❡ ✐♥❤❡r❡♥t❧② ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ❜❡❤❛✈✐♦r ♦❢ ❛ ❝♦✈❛r✐❛♥❝❡✳ ❋♦r ✐♥st❛♥❝❡✱ t❤❡ ❝♦rr❡❧❛t✐♦♥ ❤❛✈❡ ✉♣♣❡r ❛♥❞ ❧♦✇❡r ❧✐♠✐ts ✇❤✐❧❡ t❤❡ ❝♦✈❛r✐❛♥❝❡ ❞♦❡s ♥♦t✳ ❆♥♦t❤❡r ✐ss✉❡ ✐s t❤❡ ❤✐❣❤ ♣❡rs✐st❡♥❝❡ ♦❢ t❤❡ ●❆❘❈❍ ♠♦❞❡❧s✳ ❉✐❡❜♦❧❞ ✭✶✾✽✻✮ ❡✈❛❧✉❛t❡s t❤❛t t❤✐s ❤✐❣❤ ♣❡rs✐st❡♥❝❡ ♠❛② ❜❡ ❝❛✉s❡❞ ❜② t❤❡ ❢❛✐❧✉r❡ t♦ ♦❜✲ s❡r✈❡ ❝❤❛♥❣❡s ♦❢ t❤❡ ✉♥❝♦♥❞✐t✐♦♥❛❧ ✈♦❧❛t✐❧✐t②✳ ■♥ ❈❤❛♣t❡r ✺ ✭❘❡s✉❧ts✮ t❤❡r❡ ✐s ❛♥ ❡①❡r❝✐s❡ ♣❡r❢♦r♠❡❞ ✇✐t❤ ❞✉♠♠✐❡s t♦ ♦✛❡r ❡✈✐❞❡♥❝❡ ♦♥ t❤❡ ✐ss✉❡ ♦❢ ♣❡rs✐st❡♥❝❡✳ ■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ♣♦✐♥ts r❛✐s❡❞✱ ❛s ❛❧s♦ st❛t❡❞ ✐♥ t❤❡ ✐♥tr♦❞✉❝t✐♦♥✱ t❤❡r❡ ❛r❡ t✇♦ ❢✉♥❞❛♠❡♥t❛❧ ❛s♣❡❝ts t♦ ❝❤♦♦s❡ t❤❡ ❘❙❉❈ ▼♦❞❡❧ ✐♥st❡❛❞ ♦❢ ❉❈❈✳ ❚❤❡ ✜rst ❝♦♥❝❡r♥s t❤❡ ❛❜s❡♥❝❡ ♦❢ t❤❡ ♥❡❡❞ t♦ ❞❡✜♥❡ ❛ ♣r✐♦r✐ ✇❤✐❝❤ ❛r❡ t❤❡ ♣❡r✐♦❞s ♦❢ ✜♥❛♥❝✐❛❧ ❝r✐s❡s✱ ❜❡❝❛✉s❡ t❤❡ ❘❙❉❈ ♠♦❞❡❧ ✐♥❝♦r♣♦r❛t❡s r❡❣✐♠❡ ❝❤❛♥❣❡s t♦ ✐ts ❞②♥❛♠✐❝s✳ ❚❤✐s ♣r♦❝❡❞✉r❡ ✐s ♠❛❞❡ ❡♥❞♦❣❡♥♦✉s❧②✱ s✐♥❝❡ t❤❡ ❧❛t❡♥t ▼❛r❦♦✈ st❛t❡ ✈❛r✐❛❜❧❡ ❞❡t❡r♠✐♥❡s ✇✐❝❤ r❡❣✐♠❡ ♣r❡✈❛✐❧s ❛t ❡❛❝❤ ✐♥st❛♥t ♦❢ t✐♠❡✳ ❚❤❡ s❡❝♦♥❞ ✐s t❤❡ ✐❞❡❛ t❤❛t ✭♣r♦❜❛❜❧②✮ t❤❡ ❝♦♥tr✐❜✉t✐♦♥ t♦ t❤❡ ❧✐t❡r❛t✉r❡ ❝♦✉❧❞ ❜❡ ❤✐❣❤❡r✱ ❣✐✈❡♥ t❤❛t t❤❡ ❉❈❈ ✐s ❛❧r❡❛❞② ✈❡r② ✇✐❞❡s♣r❡❛❞ ❛♥❞ ✉s❡❞ ❢♦r ❝♦♥t❛❣✐♦♥ t❡sts ♣❡r❢♦r♠❡❞ ✐♥ t❤❡ ❧❛st t❡♥ ②❡❛rs✳ ❚♦ ❝♦♠♣❧❡♠❡♥t t❤❡ ❛♥❛❧②s✐s ♦❢ t❤✐s ✇♦r❦✱ ❛ ❝♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t❤❡ r❡s✉❧ts ♦❢ ❜♦t❤ ♠♦❞❡❧s ✐s ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✺✳

❋♦❧❧♦✇✐♥❣ t❤❡ ♠❡t❤♦❞♦❧♦❣② ♣r❡s❡♥t❡❞ ✐♥ P❡❧❧❡t✐❡r ✭✷✵✵✻✮✱ ❛ ✜❧t❡r❡❞ ♣r♦❝❡ssY_t

✇✐t❤ ❑ ✈❛r✐❛❜❧❡s ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❢♦❧❧♦✇s✿

Yt=Ht1/2Ut ✭✶✮

✇✐t❤ Ut ❛♥ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ♣r♦❝❡ss ✭✵✱Ik ✮✳ ❚❤❡ t✐♠❡✲✈❛r②✐♥❣ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① Ht ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦✿

✽

✇✐t❤ S_t ❛ ❞✐❛❣♦♥❛❧ ♠❛tr✐① ❝♦♠♣♦s❡❞ ♦❢ t❤❡ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥s s_mt ✇✐t❤

♠❂✶✱✳✳✳✱❦ ❛♥❞ Γt ❛ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐①✳

St=

       

s1t 0 0 · · · 0

0 s2t 0 · · · 0

✳✳✳ ✳✳✳ ✳✳✳

0 · · · 0 ✳✳✳ 0

0 · · · 0 0 s_❦t

        ✭✸✮

❊❛❝❤ ✈❛r✐❛♥❝❡s2_mt✐s ♠♦❞❡❧❡❞ ❜② ❛ ✉♥✐✈❛r✐❛t❡ ●❆❘❈❍✳ ■♥ t❤❡ ♣r❡s❡♥t st✉❞②✱

✉♥❧✐❦❡ P❡❧❧❡t✐❡r✱ ✇❡ ✇✐❧❧ ✉s❡ t❤❡ ●❏❘✲●❆❘❈❍ ♠♦❞❡❧ t♦ ❝❛♣t✉r❡ t❤❡ ♥❡❣❛t✐✈❡ s❦❡✇♥❡ss✱ ❛s ❢♦❧❧♦✇s✿

s2_mt=ω+

q

X

i=1

α_iǫ2_t−i+

q

X

z=1

γ_zǫ2_t−zI(ǫ_t−z) +

p

X

j=1

β_js2_t−j ✭✹✮

I(ǫt−z) =

(

1, s❡ǫ_t−z<0

0, ❝❛s♦ ❝♦♥trár✐♦ ✭✺✮

❚❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ t❤❡ ▼❛r❦♦✈ r❡❣✐♠❡ ❞②♥❛♠✐❝s ✐s ♠❛❞❡ ✐♥ t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐①✳ ❚❤❡r❡❢♦r❡✱Γtt❛❦❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✿

Γn=

       

1 ρ12 ρ13 · · · ρ1k

ρ21 1 ρ23 · · · ρ2k

ρ31 ρ32 ✳✳✳ ✳✳✳

✳✳✳ ✳✳✳ ✳✳✳ ρk−1k

ρ_❦✶ ρ_❦✷ · · · ρ_❦❦−1 1

        ✭✻✮

Γt= N

X

n=1

I_{∆t=n}Γn ✭✼✮

✇❤❡r❡ ∆t ✐s ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✐♥❞❡♣❡♥❞❡♥t ❢r♦♠ Ut ✇❤✐❝❤ ✐s ❞❡✜♥❡❞ ❛s ❛ st❛t❡ ✈❛r✐❛❜❧❡✳ ❚❤✐s st❛t❡ ✈❛r✐❛❜❧❡ ❢♦❧❧♦✇s ❛ ✜rst ♦r❞❡r ▼❛r❦♦✈ ♣r♦❝❡ss ❛♥❞ ❝❛♥ t❛❦❡ ♦♥❧② ✐♥t❡❣❡r ✈❛❧✉❡s ④✶✱✳✳✳✱◆⑥✳ ❚❤✐s r❡♣r❡s❡♥t❛t✐♦♥ ♦❢Γts❤♦✇s ❛♥♦t❤❡r ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ❘❙❉❈ ♠♦❞❡❧✳ ❆t t❤❡ ❡st✐♠❛t✐♦♥ st❛❣❡✱ t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐①

Γt✱ ♠❛② ❤❛✈❡ ♦♥❧② ◆ ❞✐✛❡r❡♥t ❝♦♥✜❣✉r❛t✐♦♥s ❛♥❞ ❤❡♥❝❡ ♥❡❡❞ t♦ ❜❡ r❡✈❡rs❡❞ ♦♥❧②

✾

❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ∆t ❜❡ ❡q✉❛❧ t♦ ❥ ✐s ❣✐✈❡♥ ♦♥❧② ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢∆t−1✿

P(∆t=j|∆t−1 =i) =p✐❥ ✭✽✮

❲✐t❤ t❤❡ ❝♦♥str❛✐♥t✿

p_✐✶+p_✐✷+✳✳✳+p_✐◆ = 1 ✭✾✮

❚❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❡❛❝❤ s②st❡♠ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ ♠❛tr✐① ✭◆ ① ◆✮ ✉s✉❛❧❧② ❦♥♦✇♥ ❛s t❤❡ tr❛♥s✐t✐♦♥ ♠❛tr✐①✿

Π = 

   

p11 p12 ✳✳✳ p1N

p21 p22 ✳✳✳ p2N ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

p_◆✶ p_◆✷ ✳✳✳ p_◆◆



   

✭✶✵✮

❚❤❡ r❡❣✐♠❡ s✇✐t❝❤✐♥❣ ♠♦❞❡❧ ❛❧❧♦✇s t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❡①♣❡❝t❡❞ ❞✉r❛t✐♦♥ ❉ ✐♥ ❡❛❝❤ r❡❣✐♠❡✳ ❚❤✐s ✐s t❤❡ ❛✈❡r❛❣❡ ❧❡♥❣t❤ ♦❢ ❡❛❝❤ r❡❣✐♠❡ ❝♦♥❞✐t✐♦♥❡❞ ♦♥ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ♦❢ ❜❡✐♥❣ ✐♥ ❛ s♣❡❝✐✜❝ r❡❣✐♠❡✳ ❚❤❡ ✜♥❛❧ ❢♦r♠✉❧❛ ♦❢ ❊✭❉✮ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❜②✿

D= 1,s❡∆t=j ❡ ∆=6 j;P[D= 1] = (1−p❥❥)

D= 2,s❡∆t= ∆t+1=j ❡∆=6 j;P[D= 2] =p❥❥(1−p❥❥)

D= 3,s❡∆t= ∆t+1= ∆t+2=j ❡∆t+3 6=j;P[D= 3] =p2_❥❥(1−p❥❥)

E(D) =P∞

j=1jP[D=j] =

P(∆t+1 6=j|∆t=j) + 2P[∆t+1 =j,∆t+2 6=j|∆t=j] +✳✳✳

E(D) = (1−p_❥❥) + 2p_❥❥(1−p_❥❥) + 3p2_❥❥(1−p_❥❥) +✳✳✳

E(D) = 1

✶✵

❆♥♦t❤❡r r✐❝❤ ✐♥❢♦r♠❛t✐♦♥ t❤❛t ❝❛♥ ❜❡ ❡①tr❛❝t❡❞ ❢r♦♠ t❤❡ ♠♦❞❡❧ ✐s t❤❡ ❡r❣♦❞✐❝ ♣r♦❜❛❜✐❧✐t② ♦❢ ❡❛❝❤ r❡❣✐♠❡✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❡❛❝❤ st❛t❡ ✐♥ ❡q✉✐❧✐❜r✐✉♠✳ ■t ❝❛♥ ❜❡ ❞♦♥❡ ❛s ❢♦❧❧♦✇s✿

π = Ππ

        π1 π2 . . . πk         = Π         π1 π2 . . . πk         ✭✶✷✮ n X i=1

πi = 1 ✭✶✸✮

π_i≥0

✸✳✷ ❊st✐♠❛t✐♦♥

❆❝❝♦r❞✐♥❣ t♦ P❡❧❧❡t✐❡r ✭✷✵✵✻✮✱ t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ❘❙❉❈ ♠♦❞❡❧ ❝❛♥✱ ✐♥ t❤❡♦r②✱ ❜❡ ❞♦♥❡ ✐♥ ♦♥❡ st❡♣ ❜② ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣r❡ss✐♦♥ t♦ ♠❛①✐♠✐③❡ t❤❡ ❧✐❦❡❧✐❤♦♦❞✿

◗▲(θ;Y) =

T

X

t=1

logfY_t|Y_t−1

✭✶✹✮

✇❤❡r❡Yt−1 ={Yt−1, Yt−2,✳✳✳}❛♥❞ θ✐s t❤❡ ✈❡❝t♦r ♦❢ ♣❛r❛♠❡t❡rs✳ ❚❤❡ ❝♦♠✲

♣✉t❛t✐♦♥ ♦❢ ✭✶✹✮ ✐s ❞♦♥❡ ✉s✐♥❣ t❤❡ ❍❛♠✐❧t♦♥✬s ✜❧t❡r ♣r❡s❡♥t❡❞ ✐♥ ❍❛♠✐❧t♦♥ ✭✶✾✽✾✮✱ ❜❡❝❛✉s❡ ❞❡ st❛t❡ ✈❛r✐❛❜❧❡ ∆t ✐s ❛♥ ✉♥♦❜s❡r✈❡❞ ❝♦♠♣♦♥❡♥t✳ ❍♦✇❡✈❡r✱ ✇❤❡♥ ✇♦r❦✐♥❣ ✇✐t❤ ♠♦r❡ t❤❛♥ ❛ ❢❡✇ s❡r✐❡s t❤❡ ♦♥❡✲st❡♣ ♣r♦❝❡❞✉r❡ ❜❡❝♦♠❡s ✈❡r② ❝♦♠♣❧❡① ❞✉❡ t♦ t❤❡ ❤✐❣❤ ♥✉♠❜❡r ♦❢ ♣❛r❛♠❡t❡rs t♦ ❜❡ ❡st✐♠❛t❡❞✳ ❚❤❡r❡❢♦r❡✱ ✐♥ t❤❡ ♣r❡s❡♥t st✉❞② ✇✐❧❧ ♦♥❧② ❜❡ ❛♣♣❧✐❡❞ t❤❡ t✇♦✲st❡♣s ❡st✐♠❛t✐♦♥ ♣r♦❝❡❞✉r❡✳

❚❤❡ t♦t❛❧ s♣❛❝❡ ♦❢ ♣❛r❛♠❡t❡rs θ❝❛♥ ❜❡ s❡♣❛r❛t❡❞ ✐♥ s♣❛❝❡ θ1✱ r❡❢❡rr✐♥❣ t♦

✶✶

❛♥❞ θ2✱ ❢♦r t❤❡ ♣❛r❛♠❡t❡rs ✐♥ t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠♦❞❡❧✳ ❉❡♥♦t✐♥❣QL1 t❤❡ ❧✐❦❡❧✐✲

❤♦♦❞ ♦❢ ♣❛r❛♠❡t❡r s♣❛❝❡θ1 ❛♥❞ QL2 t❤❡ ❧✐❦❡❧✐❤♦♦❞ ♦❢ ♣❛r❛♠❡t❡r s♣❛❝❡ θ2✱ ✇❡

❤❛✈❡✿

◗▲1(θ1;Y) =−

1 2

T

X

t=1

❑❧♦❣(2π) + 2 log (|St|) +U

′

tUt

✭✶✺✮

◗▲2(θ2;Y, θ1) =−

1 2

T

X

t=1

❑❧♦❣(2π) + 2 log (|Γt|) +U

′

tΓ−T1Ut

✭✶✻✮

❚❤❡ ❧✐❦❡❧✐❤♦♦❞ QL1 ❤❛s t✇♦ ✐♠♣♦rt❛♥t ❢❡❛t✉r❡s✳ ❋✐rst✱ ✐t ✐s t❤❡ s✉♠ ♦❢ ❑

✉♥✐✈❛r✐❛t❡ ❧♦❣✲❧✐❦❡❧✐❤♦♦❞s ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❡q✉✐✈❛❧❡♥t t♦ ♠❛①✐♠✐③✐♥❣ ❡❛❝❤ ✉♥✐✲ ✈❛r✐❛t❡ ❧♦❣✲❧✐❦❡❧✐❤♦♦❞ s❡♣❛r❛t❡❧②✳ ❙❡❝♦♥❞✱ t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ❡❛❝❤ ❧♦❣✲❧✐❦❡❧✐❤♦♦❞ ✐s s✐♠♣❧❡✱ s✐♥❝❡ ✐t ❞♦❡s ♥♦t ✐♥✈♦❧✈❡ t❤❡ ✉s❡ ♦❢ ✜❧t❡r ❍❛♠✐❧t♦♥✳

▼❛①✐♠✐③✐♥❣QL2✱ ✉♥❧✐❦❡ QL1✱ r❡q✉✐r❡s t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❍❛♠✐❧t♦♥✬s

✶✷

✹ ❉❛t❛s❡t

✹✳✶ ❉❡s❝r✐♣t✐♦♥

❚❤❡ ❞❛t❛❜❛s❡ ❝♦♥str✉❝t✐♦♥ ✐♥✈♦❧✈❡❞ t❤❡ ❝♦❧❧❡❝t✐♦♥✸ _{♦❢ ❞❛✐❧② ❝❧♦s✐♥❣ st♦❝❦} ♠❛r❦❡ts ✐♥❞✐❝❡s ♦❢ t❤❡ ❯♥✐t❡❞ ❙t❛t❡s ✭❙P✺✵✵✮✱ ❯❑ ✭❋❚❙❊ ✶✵✵✮✱ ❇r❛③✐❧ ✭❇♦✈❡s♣❛ ■♥❞❡①✮ ❛♥❞ ❙♦✉t❤ ❑♦r❡❛ ✭❑❖❙P■✮ ❢r♦♠ t❤❡ ♣❡r✐♦❞ ✵✷✴✵✶✴✷✵✵✸ t♦ ✵✾✴✷✵✴✷✵✶✷✳ ❚❤❡ r❡❛s♦♥ ❢♦r ❝❤♦♦s✐♥❣ t❤✐s ❣r♦✉♣ ♦❢ ❝♦✉♥tr✐❡s ✇❛s t❤❡ ✐❞❡❛ ♦❢ ❤❛✈✐♥❣ t✇♦ r❡♣r❡s❡♥t❛t✐✈❡s ❢r♦♠ ❞✐✛❡r❡♥t r❡❣✐♦♥s ♦❢ ❞❡✈❡❧♦♣❡❞ ♠❛r❦❡ts ✭◆♦rt❤ ❆♠❡r✐❝❛ ❛♥❞ ❊✉r♦♣❡✮ ❛♥❞ t✇♦ ❢r♦♠ ❡♠❡r❣✐♥❣ ♠❛r❦❡ts ✭❙♦✉t❤ ❆♠❡r✐❝❛ ❛♥❞ ❆s✐❛✮✳

600 800 1,000 1,200 1,400 1,600

2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

SP500 0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000

2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

IBOV 3,200 3,600 4,000 4,400 4,800 5,200 5,600 6,000 6,400 6,800

2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

FTSE100 400 800 1,200 1,600 2,000 2,400

2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

KOSPI

❋✐❣✉r❡ ✶✿ ❙t♦❝❦ ▼❛r❦❡t ■♥❞✐❝❡s

❘❡❣❛r❞✐♥❣ t❤❡ s❛♠♣❧❡ ❞❡✜♥✐t✐♦♥✱ t❤❡ ✐♥t❡♥t✐♦♥ ✇❛s t♦ s❡❧❡❝t ❛♥ ❡①t❡♥s✐✈❡ s❡t ♦❢ ❤✐st♦r✐❝❛❧ ❞❛t❛ ✇✐t❤ ❛♣♣r♦①✐♠❛t❡❧② ❛ ✶✵ ②❡❛r ♣❡r✐♦❞✱ ✇❤✐❝❤ ❛♠♦✉♥t❡❞ t♦ ✷✺✸✻ ♦❜s❡r✈❛t✐♦♥s ❢♦r ❡❛❝❤ s❡r✐❡s✱ ❛♥❞ ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② t❤❡ ♣r❡s❡♥❝❡ ♦❢ ✜♥❛♥❝✐❛❧ ❝r✐s✐s ❡✈❡♥ts✳ ❖♥ t❤✐s ♣♦✐♥t✱ ✇❡ ❝❛♥ ♠❡♥t✐♦♥ ♠❛✐♥❧② t❤❡ ❙✉❜♣r✐♠❡ ❝r✐s❡s ♦❢ ✷✵✵✽ ❛♥❞ ❞❡❜t ❝r✐s✐s ✐♥ ❊✉r♦♣❡ ✇♦rs❡♥❡❞ ✐♥ ✷✵✶✶✳ ❍♦✇❡✈❡r✱ ❛s ♣r❡✈✐♦✉s❧② ♠❡♥t✐♦♥❡❞✱ ✐t ✐s ♥♦t ♥❡❝❡ss❛r② t♦ ❞❡✜♥❡ ❛ ♣r✐♦r✐ t❤❡ ♠♦♠❡♥ts ♦❢ ❤✐❣❤❡r ✈♦❧❛t✐❧✐t② ❣❡♥❡r❛t❡❞ ❜② ✜♥❛♥❝✐❛❧ ❝r✐s❡s✳ ❚❤❡ ▼❛r❦♦✈ ❘❡❣✐♠❡ ❙✇✐t❝❤ ▼♦❞❡❧ ✐s ❛❜❧❡ t♦ ❞❡t❡r♠✐♥❡ ❡♥❞♦❣❡♥♦✉s❧② s✉❝❤ ♣❡r✐♦❞s✳ ❚❤❡ st♦❝❦ ♠❛r❦❡t ✐♥❞✐❝❡s ❣r❛♣❤s ✐❧❧✉str❛t❡ t❤❡ ❜❡❤❛✈✐♦r ♦❢ ❡❛❝❤ s❡r✐❡s ❛♥❞ r❡✈❡❛❧s ❛ ❣r❡❛t❡r r❛♥❣❡ ♦❢ ✈❛❧✉❡s ❢♦r t❤❡ s❡❝♦♥❞

✶✸

❤❛❧❢ ♦❢ t❤❡ s❛♠♣❧❡✳ ❋✉rt❤❡r♠♦r❡✱ ❛ ♣r❡❧✐♠✐♥❛r② ✈✐s✉❛❧ ❛♥❛❧②s✐s s❤♦✇s ❛ ❥♦✐♥t ❞❡❝r❡❛s❡ ♦❢ ❛❧❧ s❡r✐❡s ❞✉r✐♥❣ t❤❡ ❝r✐s✐s ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡✳

✹✳✷

■♥✐t✐❛❧ ❆♥❛❧②s✐s

❇❡❢♦r❡ t❤❡ ❡st✐♠❛t✐♦♥ st❛❣❡✱ t❤❡ ✐♥✐t✐❛❧ ❞❛t❛❜❛s❡ s✉✛❡r❡❞ s♦♠❡ tr❡❛t✲ ♠❡♥ts✳ ❚❤❡ ✜rst ✇❛s t♦ ❡st✐♠❛t❡ ❛ ▲♦❝❛❧ ▲❡✈❡❧ ♠♦❞❡❧ ❢♦r ❡❛❝❤ s❡r✐❡s ✐♥ ♦r❞❡r t♦ ✐♠♣✉t❡ ❛♥ ❡st✐♠❛t❡ ❢♦r ♠✐ss✐♥❣ ✈❛❧✉❡s ✭♠✐ss✐♥❣ ❞❛t❛✮ r❡❧❛t❡❞ t♦ t❤❡ ❞❛②s ✇❤❡♥ t❤❡r❡ ✇❛s ♥♦ ♥❡❣♦t✐❛t✐♦♥✹_{✳ ❚❤❡ ♠✐ss✐♥❣ ✈❛❧✉❡s ✇❡r❡ r❡♣❧❛❝❡❞ ❜② t❤❡ s♠♦♦t❤❡❞} ❡st✐♠❛t❡s ♦❢ t❤❡ ❝♦♠♣♦♥❡♥t ❧❡✈❡❧✳ ❚❤✐s ♣r♦❝❡❞✉r❡ ✇❛s ❞♦♥❡ ✇✐t❤ t❤❡ s♦❢t✇❛r❡ ♣❛❝❦❛❣❡ ❖①▼❡tr✐❝s ❙❚❆▼P ✽✳✷ ❛s s✉❣❣❡st❡❞ ❜② ❑♦♦♣♠❛♥ ❡t ❛❧ ✭✷✵✵✾✮✳

❚❤❡ s❡❝♦♥❞ ❞❛t❛ tr❡❛t♠❡♥t ✇❛s t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ r❡t✉r♥ s❡r✐❡s ❜② t❛❦✐♥❣ ✶✵✵ t✐♠❡s t❤❡ ✜rst ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ♥❛t✉r❛❧ ❧♦❣❛r✐t❤♠ ♦❢ ❡❛❝❤ s❡r✐❡s✳

-10 -5 0 5 10 15

2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

SP500

-15 -10 -5 0 5 10 15

2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

IBOV

-10 -5 0 5 10

2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

FTSE100

-12 -8 -4 0 4 8 12

2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

KOSPI

❋✐❣✉r❡ ✷✿ ❙t♦❝❦ ▼❛r❦❡t ❘❡t✉r♥s

❚❤❡ t❤✐r❞ tr❡❛t♠❡♥t ✐♥✈♦❧✈❡❞ t❤❡ tr❛❞✐♥❣ ❞❛②s s②♥❝❤r♦♥✐③❛t✐♦♥ ❜❡❝❛✉s❡ ♦❢ t✐♠❡ ③♦♥❡ ❞✐✛❡r❡♥❝❡s✳ ❲✐t❤ r❡s♣❡❝t t♦ ●r❡❡♥✇✐❝❤ t✐♠❡ ✭●▼❚✮✱ ❙❡♦✉❧ ✐s ●▼❚ ✰✾✱ ❛♥❞ t❤✉s ♦♣❡r❛t✐♦♥❛❧❧② ♦♥❡ ❞❛② ❛❤❡❛❞ ♦❢ ♥❡❣♦t✐❛t✐♦♥s ✐♥ ♦t❤❡r r❡❣✐♦♥s ♦❢ t❤❡ s❡r✐❡s ❝❤♦s❡♥✳ ❋♦❧❧♦✇✐♥❣ ❙❛♥t♦s ❛♥❞ ❱❛❧❧s P❡r❡✐r❛ ✭✷✵✶✶✮✱ ❞❛t❛ s②♥❝❤r♦♥✐③❛t✐♦♥ r❡s♣❡❝t❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ r✉❧❡✿[SP500t, IBOVt, F T SEt, KOSP It+1]✳

❚❤❡ ❛♥❛❧②s✐s ♦❢ ❞❡s❝r✐♣t✐✈❡ st❛t✐st✐❝s✱ ❣r❛♣❤s ❛♥❞ ❤✐st♦❣r❛♠s ♦❢ t❤❡ r❡t✉r♥s ❝♦♥✜r♠ t❤❡ st②❧✐③❡❞ ❢❛❝ts ♦❢ ✜♥❛♥❝✐❛❧ t✐♠❡ s❡r✐❡s✳ ❚❤❡ s❛♠♣❧✐♥❣ ❞✐str✐❜✉t✐♦♥

✶✹

♦❢ t❤❡ ❢♦✉r r❡t✉r♥s s❡r✐❡s ❛r❡ ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❤❡❛✈✐❡r t❛✐❧s ✭❧❡♣t♦❦✉rt✐❝✮ t❤❛♥ ❛ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✱ ♠❡❛♥ ❝❧♦s❡ t♦ ③❡r♦ ❛♥❞ ✈♦❧❛t✐❧✐t② ❝❧✉st❡r✐♥❣✳ ❆❧s♦✱ ❛❧❧ r❡t✉r♥s r❡♣♦rt❡❞ ♥❡❣❛t✐✈❡ ❛s②♠♠❡tr②✳

✹✳✸ ❋✐❧t❡r✐♥❣

❲❤❡r❡❛s ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s ❤❛✈❡ ❛s ✐ts ♣r❡♠✐s❡ t❤❡ ❛❜s❡♥❝❡ ♦❢ s❡r✐❛❧ ❝♦rr❡❧❛t✐♦♥ ❛s ♠❡♥t✐♦♥❡❞ ♣r❡✈✐♦✉s❧②✱ ✐t ✇❛s ♥❡❝❡ss❛r② t♦ ❡①❛♠✐♥❡ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤✐s ❤②♣♦t❤❡s✐s✳ ❚❤❡ ❛✉t♦❝♦rr❡❧❛t✐♦♥s ❢♦r t❤❡ s❡r✐❡s ♦❢ r❡t✉r♥s✱ ❞❡s♣✐t❡ ♥♦t ❤❛✈✐♥❣ ❤✐❣❤ ❡st✐♠❛t❡s st❛t✐st✐❝❛❧❧② ❞✐✛❡r❡♥t ❢r♦♠ ③❡r♦✳ ■♥ ♦r❞❡r t♦ ♠♦❞❡❧ t❤❡ ❛✉t♦❝♦rr❡❧❛t✐♦♥ str✉❝t✉r❡ ♦❢ t❤❡ ❞❛t❛✱ ✇❡ ♣❡r❢♦r♠❡❞ ❛ ✜❧t❡r ❢♦r t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♠❡❛♥ t❤r♦✉❣❤ t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ ❆❘▼❆ ✭♣✱ q✮ ♠♦❞❡❧s✳

❚❤❡ ❞❡t❡r♠✐♥❛t✐♦♥ ♦❢ t❤❡ ♦r❞❡r ♣ ❛♥❞ q ♦❢ t❤❡s❡ ♠♦❞❡❧s ✇❛s ❜❛s❡❞ ♦♥ t❤❡ ❆❦❛✐❦❡ ■♥❢♦r♠❛t✐♦♥ ❈r✐t❡r✐♦♥ ✭❆■❈✮✱ ❍❛♥♥❛♥✲◗✉✐♥♥ ✭❍◗✮ ❛♥❞ ❙❝❤✇❛r③ ✭❙❈ ♦r ❇■❈✮✱ t❡sts ♦♥ r❡s✐❞✉❛❧s ❛♥❞ t❤❡ ♣❛r❛♠❡t❡rs s✐❣♥✐✜❝❛♥❝❡✳ ❲✐t❤ r❡s♣❡❝t t♦ r❡s✐❞✉❛❧s✱ ❜♦t❤ t❤❡ ▲▼ t❡st ❛♥❞ ❉✉r❜✐♥ ❲❛ts♦♥ t❡st ❝♦rr♦❜♦r❛t❡ t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ ♥♦ s❡r✐❛❧ ❛✉t♦❝♦rr❡❧❛t✐♦♥✳ ❋✐♥❛❧❧②✱ ✇❡ ❤❛✈❡ ❝❛❧❝✉❧❛t❡❞ t❤❡ st❛t✐st✐❝s ❢♦r t❤❡ ❆❘❈❍ ❍❡t❡r♦s❝❡❞❛st✐❝✐t② t❡st ❛♥❞ t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s ♦❢ ♥♦ ❤❡t❡r♦s❦❡❞❛st✐❝✐t② ✐s r❡❥❡❝t❡❞ ❥✉st✐❢②✐♥❣ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s ✐♥ t❤✐s st✉❞②✳

✶✺

(p, q) ❆✉t♦❝♦rr❡❧❛t✐♦♥✯ DW ❍❡t❡r♦s❝❡❞❛st✐❝✐t②✯✯ L(1) L(3) L(6) L(9) L(1) L(3) L(6) L(9)

❙P (3,3) 0.01 0.73 0.70 1.88 2.0006 99.0 201.8 165.3 120.5

✭✵✳✾✶✮ ✭✵✳✺✸✮ ✭✵✳✻✺✮ ✭✵✳✵✺✮ ✭✵✳✵✵✮ ✭✵✳✵✵✮ ✭✵✳✵✵✮ ✭✵✳✵✵✮ ■❇ (2,3) 1.20 1.37 1.05 1.20 1.9987 83.6 176.2 115.5 99.5

✭✵✳✷✼✮ ✭✵✳✷✹✮ ✭✵✳✸✾✮ ✭✵✳✷✾✮ ✭✵✳✵✵✮ ✭✵✳✵✵✮ ✭✵✳✵✵✮ ✭✵✳✵✵✮ ❋❚ (3,3) 1.15 0.95 1.16 1.34 2.0046 138.4 157.7 104.0 73.7

✭✵✳✷✽✮ ✭✵✳✹✶✮ ✭✵✳✸✷✮ ✭✵✳✷✵✮ ✭✵✳✵✵✮ ✭✵✳✵✵✮ ✭✵✳✵✵✮ ✭✵✳✵✵✮ ❑❖ (3,3) 0.10 0.88 1.33 1.54 2.0053 111.5 84.8 123.6 85.0

✭✵✳✼✺✮ ✭✵✳✹✹✮ ✭✵✳✷✸✮ ✭✵✳✶✷✮ ✭✵✳✵✵✮ ✭✵✳✵✵✮ ✭✵✳✵✵✮ ✭✵✳✵✵✮ ✯❋ ❊st❛t✐st✐❝ ❛♥❞ Pr♦❜ ♦❢ ▲▼ ❙❡r✐❛❧ ❆✉t♦❝♦rr❡❧❛t✐♦♥ ❚❡st

✯✯❋ ❊st❛t✐st✐❝ ❛♥❞ Pr♦❜ ♦❢ ❆❘❈❍ ❍❡t❡r♦s❝❡❞❛st✐❝✐t② ❚❡st

❚❛❜❧❡ ✶✿ ❘❡s✐❞✉❛❧ ❚❡sts

✺ ❘❡s✉❧ts

✺✳✶ ❘❙❉❈ ▼♦❞❡❧

✺✳✶✳✶ ▼❛r❣✐♥❛❧ ❉✐str✐❜✉t✐♦♥

❆❢t❡r ♣❡r❢♦r♠✐♥❣ t❤❡ ✜❧t❡r✐♥❣ ❢♦r t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♠❡❛♥ ♦❢ t❤❡ s❡r✐❡s t❤r♦✉❣❤ t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ❆❘▼❆ ✭♣✱ q✮ ♠♦❞❡❧s ♣r❡s❡♥t❡❞ ✐♥ s❡❝t✐♦♥ ✹✳✸✱ ✇❡r❡ ❡st✐♠❛t❡❞ ●❏❘✲●❆❘❈❍ ♠♦❞❡❧s ❢♦r t❤❡ ✉♥✐✈❛r✐❛t❡ ✈❛r✐❛♥❝❡ ❡q✉❛t✐♦♥ ♦❢ ❡❛❝❤ s❡r✐❡s✳ ❆s ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡✱ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ ●❏❘✲●❆❘❈❍ ♠♦❞❡❧ ❛t t❤✐s st❛❣❡ ❝♦♥s✐sts ♦❢ ❛ ♠♦❞✐✜❝❛t✐♦♥ ♦♥ t❤❡ ♦r✐❣✐♥❛❧ ❘❙❉❈ ❞❡✈❡❧♦♣❡❞ ❜② P❡❧✲ ❧❡t✐❡r ✭✷✵✵✻✮✳ ❚❤❡ ♣✉r♣♦s❡ ♦❢ t❤✐s ❝❤❛♥❣❡ ✇❛s t♦ ❛❧❧♦✇ ❝❛♣t✉r❡ t❤❡ ❛s②♠♠❡tr✐❝ ❡✛❡❝ts ♠♦r❡ ❛❞❡q✉❛t❡❧② ✺_{✳ ❚❤❡ ❝♦❡✣❝✐❡♥t ❡st✐♠❛t❡s ❛♥❞ st❛♥❞❛r❞ ❡rr♦rs ❛r❡} r❡♣♦rt❡❞ ✐♥ ❚❛❜❧❡ ✸ ❛s ✇❡❧❧ ❛s t❤❡ ♣❡rs✐st❡♥❝❡ λ❛♥❞ t❤❡ ❤❛❧❢✲❧✐❢❡ ✭▼❱✮ ♦❢ t❤❡

s❡r✐❡s ❝♦♠♣✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢♦r♠✉❧❛s ❜❡❧♦✇✳ ❚❤❡ t❡r♠γ ✐s t❤❡ ❝♦❡✣❝✐❡♥t

♦❢ s❦❡✇♥❡ss s♣❡❝✐✜❡❞ ✐♥ ❡q✉❛t✐♦♥s ✹ ❛♥❞ ✺✳

λ=α+β+γ

2 ✭✶✼✮

▼❱= 1−

▲♦❣ [2]

▲♦❣[λ]

✭✶✽✮

✶✻

ω α γ β λ M V

❙P✺✵✵ 0.0143 −0.0130 0.1381 0.9296 0.986 49

✭✵✳✵✵✶✼✮ ✭✵✳✵✵✻✹✮ ✭✵✳✵✶✶✸✮ ✭✵✳✵✵✼✺✮

■❇❖❱ 0.1061 0.0055 0.1301 0.8924 0.963 19

✭✵✳✵✶✺✺✮ ✭✵✳✵✵✾✺✮ ✭✵✳✵✶✻✷✮ ✭✵✳✵✶✷✶✮

❋❚❙❊✶✵✵ 0.0152 0.0000 0.1584 0.9086 0.988 58

✭✵✳✵✵✷✸✮ ✭✵✳✵✵✾✷✮ ✭✵✳✵✶✸✽✮ ✭✵✳✵✵✾✵✮

❑❖❙P■ 0.0617 −0.0131 0.1791 0.8897 0.966 21

✭✵✳✵✵✻✽✮ ✭✵✳✵✵✼✽✮ ✭✵✳✵✶✻✻✮ ✭✵✳✵✵✾✶✮

❚❛❜❧❡ ✷✿ ❯♥✐✈❛r✐❛t❡ ●❏❘✲●❆❘❈❍ ✭✶✱✶✮

❚❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❡st✐♠❛t❡❞ ❝♦♥❞✐t✐♦♥❛❧ ✈❛r✐❛♥❝❡s ✐♥ t❤✐s ✜rst st❛❣❡ ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❜❡❧♦✇✳

0 5 10 15 20 25 30

2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

SP500

0 10 20 30 40 50

2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

IBOV

0 4 8 12 16 20 24 28 32

2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

FTSE100

0 10 20 30 40 50

2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

KOSPI

❋✐❣✉r❡ ✸✿ ❈♦♥❞✐t✐♦♥❛❧ ❱❛r✐❛♥❝❡s

✶✼

t❤❡ ✉♥❝♦♥❞✐t✐♦♥❛❧ ✈♦❧❛t✐❧✐t② ♦✈❡r t✐♠❡✳ ❚❤❡ r❡❧❡✈❛♥❝❡ ♦❢ t❤✐s ♣♦✐♥t ✐s t❤❛t t❤❡ ❝❤❛♥❣❡ ✐♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ✈♦❧❛t✐❧✐t② ✇♦✉❧❞ ❜❡ ❛♥ ❡✈✐❞❡♥❝❡ ♦❢ ❝❤❛♥❣❡s ✐♥ t❤❡ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❝♦rr❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ s❡r✐❡s ♦✈❡r t❤❡ s❛♠♣❧❡✱ ✇❤✐❝❤ ❥✉st✐✜❡s t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ r❡❣✐♠❡ s✇✐t❝❤✐♥❣ ♠♦❞❡❧ t♦ ❜❡ ❞✐s❝✉ss❡❞ ❧❛t❡r✳

❚❤❡ s❛♠❡ ✇❛② ❛s ✐♥ ❆❧♠❡✐❞❛ ❛♥❞ ❱❛❧❧s P❡r❡✐r❛ ✭✶✾✾✾✮✱ ✇❛s ❝r❡❛t❡❞ ❛♥ ❛❞ ❤♦❝ ❞✉♠♠②d1✇✐t❤ ✈❛❧✉❡ ✶ ❢♦r t❤❡ ♣❡r✐♦❞ ❢r♦♠ ❏❛♥✴✷✵✵✽ t♦ ❏✉♥✴✷✵✵✾ ❛♥❞ ✵ ❢♦r

t❤❡ r❡st✱ ❜❛s❡❞ ♦♥❧② ♦♥ ✈✐s✉❛❧ ✐♥s♣❡❝t✐♦♥✳ ❚❤❡ ✐❞❡❛ ✇❛s ❥✉st t♦ s❤♦✇ t❤❛t t❤❡ ✐♥❝❧✉s✐♦♥ ♦❢ ❛♥ ✐♥t❡r❝❡♣t ❞✉♠♠② ✐♥ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ✉♥✐✈❛r✐❛t❡ ✈❛r✐❛♥❝❡s ✇♦✉❧❞ ❜❡ ❝❛♣❛❜❧❡ ♦❢ ❝❤❛♥❣✐♥❣ t❤❡ ♣❡rs✐st❡♥❝❡ ❛♥❞ t❤❡ ❤❛❧❢✲❧✐❢❡ ♦❢ t❤❡ s❡r✐❡s ♣r♦✈✐❞✐♥❣✱ ✐♥ t❤✐s ✇❛②✱ ❢✉rt❤❡r ❡✈✐❞❡♥❝❡ ♦❢ ❝❤❛♥❣❡ ✐♥ t❤❡ ✉♥❝♦♥❞✐t✐♦♥❛❧ ✈♦❧❛t✐❧✐t② ♦✈❡r t✐♠❡✳ ❲✐t❤ t❤✐s ♦❜❥❡❝t✐✈❡✱ t❤❡ r❡❣r❡ss✐♦♥s ✇❡r❡ r❡❝❛❧❝✉❧❛t❡❞ ❢♦r t❤❡ ♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥s✱ ❜✉t t❤✐s t✐♠❡ ✐♥❝♦r♣♦r❛t✐♥❣ t❤❡ ❞✉♠♠② ✐♥ ❡❛❝❤ ❡q✉❛t✐♦♥✳ ❚❤❡ r❡s✉❧ts ✐♥ ❚❛❜❧❡ ✹✱ s❤♦✇s ❛ r❡❞✉❝t✐♦♥ ♦❢ ♣❡rs✐st❡♥❝❡ ❛♥❞ ❤❛❧❢✲❧✐❢❡ ❢♦r t❤❡ ❢♦✉r s❡r✐❡s✳ ❖♥ ❛✈❡r❛❣❡✱ t❤❡ ♣❡rs✐st❡♥❝❡ ❞❡❝r❡❛s❡ ❢r♦♠ ✵✳✾✽ t♦ ✵✳✾✻ ❛♥❞ ❢r♦♠ ✸✼ t♦ ✷✺ ❞❛②s ✐♥ ❤❛❧❢✲❧✐❢❡✳ ▼♦r❡♦✈❡r✱ ✐♥ ❛❧❧ ❝❛s❡s t❤❡ ❞✉♠♠② ✇❛s st❛t✐st✐❝❛❧❧② s✐❣♥✐✜❝❛♥t✳ ❘❡t✉r♥✐♥❣ t♦ ❡q✉❛t✐♦♥ ✹✱ t❤❡ ●❏❘✲●❆❘❈❍ ✭✶✱✶✮ ♠♦❞❡❧ s♣❡❝✐✜❝❛t✐♦♥ ✇✐t❤ ❞✉♠♠② ❛♥❞ q✱ ♥ ❛♥❞ ♣ ❡q✉❛❧ t♦ ✶ ❢♦❧❧♦✇❡❞✿

s2_mt=ω0+ω1d1+

q

X

i=1

αiǫ2t−i+ n

X

z=1

γzǫ2t−zI(ǫt−z) + p

X

j=1

βjs2t−j ✭✶✾✮

ω d1 α γ β λ M V

❙P✺✵✵ 0.0168 0.0338 −0.0130 0.1395 0.9237 0.980 36

✭✵✳✵✵✷✵✮ ✭✵✳✵✶✹✷✮ ✭✵✳✵✵✻✸✮ ✭✵✳✵✶✶✾✮ ✭✵✳✵✵✽✺✮

■❇❖❱ 0.1438 0.1664 −0.0050 0.1466 0.8727 0.941 12

✭✵✳✵✷✶✵✮ ✭✵✳✵✺✶✶✮ ✭✵✳✵✵✾✽✮ ✭✵✳✵✶✾✵✮ ✭✵✳✵✶✺✹✮

❋❚❙❊✶✵✵ 0.0211 0.0600 −0.0051 0.1832 0.8915 0.978 32

✭✵✳✵✵✸✷✮ ✭✵✳✵✶✼✽✮ ✭✵✳✵✶✵✻✮ ✭✵✳✵✶✽✶✮ ✭✵✳✵✶✶✾✮

❑❖❙P■ 0.0684 0.0423 −0.0151 0.1849 0.8819 0.959 18

✭✵✳✵✵✼✾✮ ✭✵✳✵✶✾✽✮ ✭✵✳✵✵✼✽✮ ✭✵✳✵✶✼✺✮ ✭✵✳✵✶✵✹✮

✶✽

✺✳✶✳✷ ✷ ❘❡❣✐♠❡s

❆❢t❡r t❤❡ ❡♥❞ ♦❢ ♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥s ♠♦❞❡❧✐♥❣ st❛❣❡✱ ✇❡ ❡st✐♠❛t❡ t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐❝❡s ✇✐t❤ t❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ♦♥❧② t✇♦ r❡❣✐♠❡s✳ ■♥ t❤❡ ♥❡①t s❡❝t✐♦♥ ✇❡ ❡st✐♠❛t❡ t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐❝❡s ✇✐t❤ t❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ t❤❡ ♣r❡s❡♥❝❡ ♦❢ t❤r❡❡ r❡❣✐♠❡s✳ ❚❤❡ r❡❣✐♠❡ ✐❞❡♥t✐✜❡❞ ❛s st❛t❡ ✶ ✐s ❞❡✜♥❡❞ ❛s t❤❡ ❤✐❣❤ ✈♦❧❛t✐❧✐t② ✭t✉r❜✉❧❡♥❝❡✮ ♦♥❡ ❛♥❞ t❤❡ r❡❣✐♠❡ r❡❢❡rr❡❞ t♦ ❛s st❛t❡ ✷ ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❧♦✇ ✈♦❧❛t✐❧✐t② ✭❝❛❧♠✮✳ ❚❤❡ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t② ♠❛tr✐① Π

✐♥❞✐❝❛t❡s t❤❛t t❤❡ r❡❣✐♠❡s ❛r❡ ♣❡rs✐st❡♥t ❜❡❝❛✉s❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❜❡✐♥❣ ✐♥ st❛t❡ ✶ ❛t t✐♠❡ t ✰✶ ❝♦♥❞✐t✐♦♥❛❧ ♦♥ ❜❡✐♥❣ ❛t t❤❛t s❛♠❡ st❛t❡ ❛t t✐♠❡ t ✐s ✵✳✾✾ ❛♥❞ t❤❡ s✐♠✐❧❛r st❛t✐st✐❝ ❢♦r t❤❡ s❡❝♦♥❞ st❛t❡ ✐s ✵✳✾✽✳ ❘❡t✉r♥✐♥❣ t♦ s❡❝t✐♦♥ ✷ ✭▼❡t❤♦❞♦❧♦❣②✮✱ t❤❡s❡ ✈❛❧✉❡s ❛r❡ ❣✐✈❡♥✱ r❡s♣❡❝t✐✈❡❧②✱ ❜② t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s p11

❡ p22 ❞❡✜♥❡❞ ❜② ❡q✉❛t✐♦♥ ✽✳ ❯s✐♥❣ t❤❡ ❞❡r✐✈❛t✐♦♥ ♦❢ ❡q✉❛t✐♦♥ ✶✷✱ t❤❡ ❤✐❣❤❡r

♣❡rs✐st❡♥❝❡ ♦❢ st❛t❡ ✶ r❡✢❡❝ts ❛ ❣r❡❛t❡r ❛✈❡r❛❣❡ ❡①♣❡❝t❡❞ ❞✉r❛t✐♦♥ ❊✭❉✮ ♦❢ t❤✐s r❡❣✐♠❡ ✇❤✐❝❤ ✐s ✶✹✼ ❞❛②s ✈❡rs✉s ✼✺ ❞❛②s ♦❢ st❛t❡ ✷✳ ❲✐t❤ r❡s♣❡❝t t♦ t❤❡ ♥✉♠❜❡r ♦❢ ♦❜s❡r✈❛t✐♦♥s ✐♥ ❡❛❝❤ r❡❣✐♠❡✱ t❤❡ st❛t❡ ✶ ❤❛s ❛❧s♦ ❛ ❣r❡❛t❡r ♥✉♠❜❡r ♦❢ ✶✻✹✻ ♣♦✐♥ts ❛❣❛✐♥st ✽✽✺ ♦♥ t❤❡ ♦t❤❡r r❡❣✐♠❡✳ ❋r♦♠ t❤❡ ❡r❣♦❞✐❝ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ❡❛❝❤ r❡❣✐♠❡ ❝❛❧❝✉❧❛t❡❞ ❢r♦♠ ❡q✉❛t✐♦♥ ✶✷✱ ✐t ❝❛♥ ❜❡ ♦❜s❡r✈❡❞ t❤❛t t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ st❛t❡ ✶ ✐s ✵✳✻✻ ❛♥❞ ❤✐❣❤❡r t❤❛♥ ♦❢ t❤❡ ❙t❛t❡ ✷ ✭✵✳✸✹✮✳

Π ∆t= 1 ∆t= 2 Obs. π E(D)

∆t= 1 0.993 0.007 ∆t= 1 ✶✻✹✻ 0.66 147

(0.004)

∆t= 2 0.013 0.987 ∆t= 2 ✽✽✺ 0.34 75

(0.008)

❚❛❜❧❡ ✹✿ ❚r❛♥s✐t✐♦♥ ▼❛tr✐① ✇✐t❤ ✷ ❘❡❣✐♠❡s

❚❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐❝❡s ❞❡✜♥❡❞ ✐♥ ❡q✉❛t✐♦♥s ✻ ❛♥❞ ✼✱ r❡♣♦rt❡❞ ✐♥ ❚❛✲ ❜❧❡ ✻ ♣r♦✈✐❞❡ s♦♠❡ ❡✈✐❞❡♥❝❡ ♦❢ ✜♥❛♥❝✐❛❧ ❝♦♥t❛❣✐♦♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡✜♥✐t✐♦♥ ❛❞♦♣t❡❞ ✐♥ t❤✐s st✉❞②✳ ❚❤❡ ♠❛tr✐①Γ1✱ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❤✐❣❤ ✈♦❧❛t✐❧✐t② st❛t❡✱

♣r❡s❡♥ts s✐❣♥✐✜❝❛♥t❧② ❤✐❣❤❡r ❝♦rr❡❧❛t✐♦♥ ❝♦❡✣❝✐❡♥ts t❤❛♥Γ2 r❡❧❛t❡❞ t♦ ❝❛❧♠ ♣❡✲

✶✾

Γ1 ❙P ■❇ ❋❚ ❑❖ Γ2 ❙P ■❇ ❋❚ ❑❖

❙P 1.000 ❙P 1.000

■❇ 0.754 1.000 ■❇ 0.473 1.000

(0.049) (0.043)

❋❚ 0.644 0.589 1.000 ❋❚ 0.403 0.294 1.000

(0.061)(0.045) (0.034)(0.055)

❑❖ 0.448 0.433 0.346 1.000 ❑❖ 0.302 0.226 0.228 1.000

(0.051)(0.058)(0.086) (0.022)(0.034)(0.027)

❚❛❜❧❡ ✺✿ ❈♦rr❡❧❛t✐♦♥ ▼❛tr✐❝❡s ❢♦r ✷ ❘❡❣✐♠❡s

❚❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ s♠♦♦t❤❡❞ ♣r♦❜❛❜✐❧✐t✐❡s ❣r❛♣❤s r❡✈❡❛❧s t❤❛t t❤❡s❡ ❞♦ ♥♦t ♠♦✈❡ ✈❡r② ♦❢t❡♥ ❜❡t✇❡❡♥ t❤❡ r❡❣✐♠❡s ❡①❤✐❜✐t✐♥❣ r❡❧❛t✐✈❡ st❛❜✐❧✐t② ❛♥❞ t❤❡r❡ ✐s ❧♦✇ ✉♥❝❡rt❛✐♥t② t♦ ❞❡t❡r♠✐♥❡ ✇✐❝❤ r❡❣✐♠❡ ♣r❡✈❛✐❧s ❛t ❡❛❝❤ ✐♥st❛♥t ♦❢ t✐♠❡✱ ❣✐✈❡♥ t❤❛t t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ❛r❡ ❝♦♥❝❡♥tr❛t❡❞ ❛r♦✉♥❞ ✵ ♦r ✶ ❛t ❡❛❝❤ ♦❜s❡r✈❛t✐♦♥✳ ❚❤✐s ❝♦♥❝❧✉s✐♦♥ s❤♦✇s t❤❛t t❤❡ tr❛♥s✐t✐♦♥ ❜❡t✇❡❡♥ r❡❣✐♠❡s ♦❝❝✉rs ♥♦t ❣r❛❞✉❛❧❧② ❜✉t ❛❜r✉♣t❧②✳

✷✵

❋✐❣✉r❡ ✹✿ ❙♠♦♦t❤❡❞ Pr♦❜❛❜✐❧✐t✐❡s ✇✐t❤ ✷ ❘❡❣✐♠❡s

✺✳✶✳✸ ✸ ❘❡❣✐♠❡s

❚❤✐s s❡❝t✐♦♥ ✇✐❧❧ ♣r❡s❡♥t t❤❡ r❡s✉❧ts ❢♦r t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐❝❡s ✇✐t❤ t❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ t❤❡ ♣r❡s❡♥❝❡ ♦❢ t❤r❡❡ r❡❣✐♠❡s✳ ❆s ✇✐❧❧ ❜❡ ❞❡t❛✐❧❡❞✱ t❤❡ ♠♦❞❡❧ ✇✐t❤ ✸ r❡❣✐♠❡s s❤♦✇❡❞ ❣r❡❛t❡r ✐♥st❛❜✐❧✐t② ❛♥❞ ✉♥❝❡rt❛✐♥t② ❛❜♦✉t t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ r❡❣✐♠❡ ♣r❡✈❛✐❧✐♥❣ ✐♥ ❡❛❝❤ ♣♦✐♥t ✐♥ t✐♠❡✳ ■♥ ❛❞❞✐t✐♦♥✱ ❡st✐♠❛t❡s st❛♥❞❛r❞ ❡rr♦rs ✐♥❝r❡❛s❡❞ ❛♥❞ ❞✐❞ ♥♦t r❡s✉❧t ✐♥ t❤❡ s❛♠❡ str♦♥❣ ❝♦♥❝❧✉s✐♦♥s ♦❢ t❤❡ ❝❛s❡ ✇✐t❤ ♦♥❧② t✇♦ r❡❣✐♠❡s✳ ●✐✈❡♥ t❤✐s ❝♦♥t❡①t✱ t❤❡ ✜♥❛❧ ❝♦♥❝❧✉s✐♦♥s ♦❢ t❤✐s ✇♦r❦ ✇✐❧❧ ❜❡ ❜❛s❡❞✱ ✐♥ ♠♦st ♣❛rt✱ ♦♥ t❤❡ r❡s✉❧ts ♦❢ t❤❡ ♠♦❞❡❧ ✇✐t❤ t✇♦ r❡❣✐♠❡s✳ ❚❤✐s ✇✐❧❧ ❜❡ ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳

❚❤❡ r❡❣✐♠❡ ✐❞❡♥t✐✜❡❞ ❛s st❛t❡ ✶ ✐s ❞❡✜♥❡❞ ❛s t❤❡ ♠❡❞✐✉♠ ✈♦❧❛t✐❧✐t②✱ t❤❡ r❡❣✐♠❡ ❞❡✜♥❡❞ ❛s ❙t❛t❡ ✷ ✐s t❤❡ ❤✐❣❤ ✈♦❧❛t✐❧✐t② r❡❣✐♠❡ ✭t✉r❜✉❧❡♥❝❡ ♣❡r✐♦❞✮ ❛♥❞✱ ✜♥❛❧❧②✱ t❤❡ ✏❙t❛t❡ ✸ ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❧♦✇ ✈♦❧❛t✐❧✐t② ✭❝❛❧♠ ♣❡r✐♦❞✮✳ ❚❤❡ ♠❛tr✐①

Πtr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s s❤♦✇s t❤❛t t❤❡ r❡❣✐♠❡s ♦❢ ❙t❛t❡ ✶✱ ❙t❛t❡ ✷ ❛♥❞ ❙t❛t❡

✸ ❛r❡ ❝❤❛r❛❝t❡r✐③❡❞✱ r❡s♣❡❝t✐✈❡❧②✱ ❜② ❧♦✇✱ ♠♦❞❡r❛t❡ ❛♥❞ ❤✐❣❤ ♣❡rs✐st❡♥❝❡✱ ❣✐✈❡♥ t❤❛t p11❂✵✳✹✻✱ p22❂✵✳✽✼ ❛♥❞ p33❂✵✳✾✽✳ ❚❤❡ ❞✐✛❡r❡♥❝❡s ✐♥ ♣❡rs✐st❡♥❝❡ ✐♠♣❛❝t

✷✶

❚❤❡ ▼❛r❦♦✈ ❝❤❛✐♥ ✐s s♣❡♥❞✐♥❣ ♠♦st ♦❢ ✐ts t✐♠❡ ✐♥ t❤❡ t✉r❜✉❧❡♥t r❡❣✐♠❡✱ ✇✐❝❤ ❤❛s ✶✺✽✺ ♦❜s❡r✈❛t✐♦♥s✱ ✈❡rs✉s ✽✶✹ ♦❢ ❙t❛t❡ ✸ ❛♥❞ ✶✸✷ ♦❢ ❙t❛t❡ ✶✳ ❘❡❣❛r❞✐♥❣ t❤❡ ❡r❣♦❞✐❝ ♣r♦❜❛❜✐❧✐t✐❡s ❢♦r ❡❛❝❤ r❡❣✐♠❡✱ ✐t ✐s ♥♦t❡❞ t❤❛t t❤❡ st❡❛❞② st❛t❡ ♣r♦❜❛❜✐❧✐t② ♦❢ st❛t❡ ✷ ✭π2❂ ✵✳✺✺✮ ✐s ❛❧s♦ ❤✐❣❤❡r t❤❛♥ t❤❡ ♦t❤❡rs ✭π1❂ ✵✳✶✸ ❛♥❞

π3❂ ✵✳✸✷✮✳

Π ∆t= 1 ∆t= 2 ∆t= 3 Obs. π E(D)

∆t= 1 0.462 0.488 0.050 ∆t= 1 ✶✸✷ 0.13 2

(0.343) (0.080)

∆t= 2 0.128 0.872 0.000 ∆t= 2 ✶✺✽✺ 0.55 8

(0.046) (0.017)

∆t= 3 0.008 0.013 0.979 ∆t= 3 ✽✶✹ 0.32 48

(0.017) (0.030)

❚❛❜❧❡ ✻✿ ❚r❛♥s✐t✐♦♥ ▼❛tr✐① ✇✐t❤ ✸ ❘❡❣✐♠❡s

❚❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐❝❡s✱ r❡♣♦rt❡❞ ✐♥ ❚❛❜❧❡ ✽✱ ❣✐✈❡ s♦♠❡ ❡✈✐❞❡♥❝❡ ♦❢ ✜✲ ♥❛♥❝✐❛❧ ❝♦♥t❛❣✐♦♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡✜♥✐t✐♦♥ ❛❞♦♣t❡❞ ✐♥ t❤✐s st✉❞②✱ ❜✉t ❧❡ss ✐♥❝✐s✐✈❡ t❤❛♥ ✐♥ t❤❡ ❝❛s❡ ♦❢ ✷ r❡❣✐♠❡s✳ ❚❤❡ ♠❛tr✐①Γ2✱ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❤✐❣❤

✈♦❧❛t✐❧✐t② st❛t❡✱ ♣r❡s❡♥ts s✐❣♥✐✜❝❛♥t❧② ❤✐❣❤❡r ❝♦rr❡❧❛t✐♦♥ ❝♦❡✣❝✐❡♥ts t❤❛♥ Γ3✱

r❡❧❛t❡❞ t♦ tr❛♥q✉✐❧ ♣❡r✐♦❞s✳ ◆♦♥❡t❤❡❧❡ss✱ t❤❡ ❞✐st✐♥❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ♠❛tr✐①

Γ1 ✭♠❡❞✐✉♠ ✈♦❧❛t✐❧✐t②✮ ❝♦❡✣❝✐❡♥ts ✇✐t❤ t❤❡ ♦t❤❡r ♠❛tr✐❝❡s ✐s ♥♦t s✐❣♥✐✜❝❛♥t

❢♦r ❛❧❧ ♣❛✐rs ♦❢ ❝♦✉♥tr✐❡s✳ ■❧❧✉str❛t✐♥❣ t❤✐s ♣♦✐♥t✱ t❤❡ ♣❛✐r ❙P✺✵✵ ❛♥❞ ❇♦✈❡s♣❛ ■♥❞❡①✱ ❤❛s ❛ ❝♦rr❡❧❛t✐♦♥ ❝♦❡✣❝✐❡♥t ♦❢ ✵✳✹✼ ❞✉r✐♥❣ t❤❡ ❝❛❧♠ r❡❣✐♠❡ ❛❣❛✐♥st ✵✳✼✾ ✐♥ t❤❡ t✉r❜✉❧❡♥❝❡ ♣❡r✐♦❞✱ ❛♥❞ t❤❡② ❛r❡ st❛t✐st✐❝❛❧❧② ❞✐✛❡r❡♥t ❝♦♥s✐❞❡r✐♥❣ t❤❡ st❛♥❞❛r❞ ❡rr♦rs✳ ◆❡✈❡rt❤❡❧❡ss✱ ❝♦♠♣❛r✐♥❣ ❜♦t❤ ♥✉♠❜❡rs ✇✐t❤ t❤❡ ❝♦❡✣❝✐❡♥t ✵✳✺✽ ✭st❛♥❞❛r❞ ❡rr♦r ✵✳✶✼✮ ♦❢ t❤❡ ♠❡❞✐✉♠ ✈♦❧❛t✐❧✐t② r❡❣✐♠❡ ✇❡ ❝❛♥ ♥♦t s✉st❛✐♥ t❤❡ s❛♠❡ ❛ss❡rt✐♦♥✳ ■♥ ❝❡rt❛✐♥ ❝❛s❡s ✇❡ ❝❛♥ ♦❜s❡r✈❡ ❡✈❡♥ ❛♥ ✐♥✈❡rs✐♦♥✳ ❚❤❡ ♣❛✐rs ❑❖❙P■ ✇✐t❤ ■❇❖❱❊❙P❆ ❛♥❞ ❑❖❙P■ ✇✐t❤ ❋❚❙❊✶✵✵ ✐♥ ♠❛tr✐①Γ1 ❤❛✈❡

❤✐❣❤❡r ❝♦❡✣❝✐❡♥ts t❤❛♥ t❤❡ Γ2✱ ✇❤❛t ✐s ♥♦t ❡①♣❡❝t❡❞ ♦r✐❣✐♥❛❧❧②✳ ❆ ♣♦ss✐❜❧❡