WP 093 What if Brazil Hadn’t Floated the Real in 1999

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What if Brazil Hadn't Floated the Real in 1999?

Carlos Viana de Carvalho

André D. Vilela

What if Brazil Hadn't Floated the Real in 1999?

Carlos Viana de Carvalho

André D. Vilela

Carlos Viana de Carvalho

PUC-Rio

André D. Vilela

❲❤❛t ✐❢ ❇r❛③✐❧ ❍❛❞♥✬t ❋❧♦❛t❡❞ t❤❡ ❘❡❛❧ ✐♥ ✶✾✾✾❄

❈❛r❧♦s ❱✐❛♥❛ ❞❡ ❈❛r✈❛❧❤♦ P❯❈✲❘✐♦

❆♥❞ré ❉✳ ❱✐❧❡❧❛ ❇❛♥❝♦ ❈❡♥tr❛❧ ❞♦ ❇r❛s✐❧

◆♦✈❡♠❜❡r✱ ✷✵✶✺

❆❜str❛❝t

❲❡ ❡st✐♠❛t❡ ❛ ❞②♥❛♠✐❝✱ st♦❝❤❛st✐❝✱ ❣❡♥❡r❛❧ ❡q✉✐❧✐❜r✐✉♠ ♠♦❞❡❧ ♦❢ t❤❡ ❇r❛③✐❧✐❛♥ ❡❝♦♥♦♠② t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ tr❛♥s✐t✐♦♥ ❢r♦♠ ❛ ❝✉rr❡♥❝② ♣❡❣ t♦ ✐♥✢❛t✐♦♥ t❛r❣❡t✐♥❣ t❤❛t t♦♦❦ ♣❧❛❝❡ ✐♥ ✶✾✾✾✳ ❚❤❡ ❡st✐♠❛t❡❞ ♠♦❞❡❧ ❡①❤✐❜✐ts q✉✐t❡ ❞✐✛❡r❡♥t ❞②♥❛♠✐❝s ✉♥❞❡r t❤❡ t✇♦ ♠♦♥❡t❛r② r❡❣✐♠❡s✳ ❲❡ ✉s❡ ✐t t♦ ♣r♦❞✉❝❡ ❝♦✉♥t❡r❢❛❝t✉❛❧ ❤✐st♦r✐❡s ♦❢ t❤❡ tr❛♥s✐t✐♦♥ ❢r♦♠ ♦♥❡ r❡❣✐♠❡ t♦ ❛♥♦t❤❡r✱ ❣✐✈❡♥ t❤❡ ❡st✐♠❛t❡❞ ❤✐st♦r② ♦❢ str✉❝t✉r❛❧ s❤♦❝❦s✳ ❖✉r r❡s✉❧ts s✉❣❣❡st t❤❛t ♠❛✐♥t❛✐♥✐♥❣ t❤❡ ❝✉rr❡♥❝② ♣❡❣ ✇♦✉❧❞ ❤❛✈❡ ❜❡❡♥ t♦♦ ❝♦st❧②✱ ❛s ✐♥t❡r❡st r❛t❡s ✇♦✉❧❞ ❤❛✈❡ ❤❛❞ t♦ r❡♠❛✐♥ ❛t ❡①tr❡♠❡❧② ❤✐❣❤ ❧❡✈❡❧s ❢♦r s❡✈❡r❛❧ q✉❛rt❡rs✱ ❛♥❞ ●❉P ✇♦✉❧❞ ❤❛✈❡ ❝♦❧❧❛♣s❡❞✳ ❆❝❝❡❧❡r❛t✐♥❣ t❤❡ ♣❛❝❡ ♦❢ ♥♦♠✐♥❛❧ ❡①❝❤❛♥❣❡ r❛t❡ ❞❡✈❛❧✉❛t✐♦♥s ❛❢t❡r t❤❡ ❆s✐❛♥ ❈r✐s✐s ✇♦✉❧❞ ❤❛✈❡ ❧❡❛❞ t♦ ❤✐❣❤❡r ✐♥✢❛t✐♦♥ ❛♥❞ ✐♥t❡r❡st r❛t❡s✱ ❛♥❞ s❧✐❣❤t❧② ❧♦✇❡r ●❉P✳ ❋✐♥❛❧❧②✱ t❤❡ ✜rst ❤❛❧❢ ♦❢ ✶✾✾✽ ❛r❣✉❛❜❧② ♣r♦✈✐❞❡❞ ❛ ✇✐♥❞♦✇ ♦❢ ♦♣♣♦rt✉♥✐t② ❢♦r ❛ s♠♦♦t❤ tr❛♥s✐t✐♦♥ ❜❡t✇❡❡♥ ♠♦♥❡t❛r② r❡❣✐♠❡s✳

❏❊▲ ❝❧❛ss✐✜❝❛t✐♦♥ ❝♦❞❡s✿ ❊✺✷✱ ❋✹✶

❑❡②✇♦r❞s✿ ♠♦♥❡t❛r② ♣♦❧✐❝②✱ r❡❣✐♠❡ s❤✐❢t✱ ❝✉rr❡♥❝② ♣❡❣✱ ✐♥✢❛t✐♦♥ t❛r❣❡t✐♥❣✱ ❇r❛③✐❧

∗❚❤✐s ♣❛♣❡r ✐s ❜❛s❡❞ ♦♥ ❱✐❧❡❧❛ ✭✷✵✶✹✮✳ ❋♦r ❝♦♠♠❡♥ts ❛♥❞ s✉❣❣❡st✐♦♥s✱ ✇❡ t❤❛♥❦ ❚✐❛❣♦ ❇❡rr✐❡❧✱ ❉✐♦❣♦ ●✉✐❧❧é♥✱ ❛♥❞

s❡♠✐♥❛r ♣❛rt✐❝✐♣❛♥ts ❛t t❤❡ ❈❡♥tr❛❧ ❇❛♥❦ ♦❢ ❇r❛③✐❧✱ ■P❊❆✱ ❛♥❞ t❤❡ ❈♦♥❢❡r❡♥❝❡ ✏❉❙●❊ ♠♦❞❡❧s ❢♦r ❇r❛③✐❧✿ ❙❆▼❇❆ ❛♥❞ ❜❡②♦♥❞✑✱ ❤❡❧❞ ❛t ❊❊❙P ✐♥ ❆✉❣✉st ✷✵✶✹✳ ❚❤❡ ✈✐❡✇s ❡①♣r❡ss❡❞ ✐♥ t❤✐s ♣❛♣❡r ❛r❡ t❤♦s❡ ♦❢ t❤❡ ❛✉t❤♦rs ❛♥❞ ❞♦ ♥♦t ♥❡❝❡ss❛r✐❧② r❡✢❡❝t t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ❈❡♥tr❛❧ ❇❛♥❦ ♦❢ ❇r❛③✐❧✳ ❊✲♠❛✐❧s✿ ❝✈✐❛♥❛❝❅❡❝♦♥✳♣✉❝✲r✐♦✳❜r✱ ❛♥❞r❡✳✈✐❧❡❧❛❅❜❝❜✳❣♦✈✳❜r✳

✶ ■♥tr♦❞✉❝t✐♦♥

✏❘❡s❡r✈❡s ❣r❡✇ q✉✐❝❦❧② ❛♥❞ ❝♦♥✜❞❡♥❝❡ r❡t✉r♥❡❞✳ ▼❛②❜❡ t❤✐s ✐s ✇❤❛t ♠❛❞❡ ✉s ♠✐ss t❤❡ ♦♣♣♦rt✉♥✐t② t♦ r❡✈✐❡✇ t❤❡ ❡①❝❤❛♥❣❡ r❛t❡ ✐ss✉❡ ✐♥ t❤❡ ✜rst ♠♦♥t❤s ♦❢ ✶✾✾✽✱ ✇❤❡♥ ✐t ♠✐❣❤t ❤❛✈❡ ❜❡❡♥ ♣♦ss✐❜❧❡ t♦ ❞♦ ✐t✳✑ ✭❈❛r❞♦s♦✱ ✷✵✵✻✮✶

❚❤❡ tr❛♥s✐t✐♦♥ ❢r♦♠ ❛♥ ❡①❝❤❛♥❣❡ r❛t❡ ❝r❛✇❧✐♥❣ ♣❡❣ t♦ t❤❡ ✐♥✢❛t✐♦♥ t❛r❣❡t✐♥❣ r❡❣✐♠❡ ✇✐t❤ ✢♦❛t✐♥❣ ❡①❝❤❛♥❣❡ r❛t❡s ✇❛s t❤❡ ♠♦st s✐❣♥✐✜❝❛♥t ♠♦♥❡t❛r② ♣♦❧✐❝② ❝❤❛♥❣❡ ✐♥ ❇r❛③✐❧ s✐♥❝❡ t❤❡ P❧❛♥♦ ❘❡❛❧ ✐♥ ✶✾✾✹✳ ❚❤❡ ❝❤❛♥❣❡ ♦❝❝✉rr❡❞ ❞✉r✐♥❣ ❛ tr♦✉❜❧❡❞ ♣❡r✐♦❞✱ ❛❢t❡r ❛ s❡q✉❡♥❝❡ ♦❢ ✐♥t❡r♥❛t✐♦♥❛❧ ❝r✐s❡s✱✷ _{❛♥ ❛❣r❡❡♠❡♥t}

✇✐t❤ t❤❡ ■▼❋✱ ❛♥❞ ♣r❡s✐❞❡♥t✐❛❧ ❡❧❡❝t✐♦♥s ✐♥ ❖❝t♦❜❡r ✶✾✾✽✳ ❆❢t❡r ❛ ♠❛r❦❡❞ ❞❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ❘❡❛❧ ✐♥ ❏❛♥✉❛r② ✶✾✾✾✱ ❢♦❧❧♦✇❡❞ ❜② ❛♥ ✐♥❝r❡❛s❡ ✐♥ ✐♥✢❛t✐♦♥ ✐♥ t❤❡ s❤♦rt r✉♥✱ t❤❡ ❈❡♥tr❛❧ ❇❛♥❦ ♦❢ ❇r❛③✐❧ ✭❈❇❇✮ ✐♥❝r❡❛s❡❞ ✐♥t❡r❡st r❛t❡s t♦ ✹✺✪ ♣✳❛✳✱ ✐♥ ❛♥ ❛tt❡♠♣t t♦ ❦❡❡♣ ✐♥✢❛t✐♦♥ ❡①♣❡❝t❛t✐♦♥s ❢r♦♠ ✉♥❛♥❝❤♦r✐♥❣✳ ❙✉❜s❡q✉❡♥t❧②✱ t❤❡ ❈❇❇ ❜❡❣❛♥ t♦ ♦♣❡r❛t❡ ✉♥❞❡r t❤❡ ✐♥✢❛t✐♦♥ t❛r❣❡t✐♥❣ r❡❣✐♠❡✱ ♠❛❞❡ ♦✣❝✐❛❧ ✐♥ ❏✉♥❡✱ ✶✾✾✾✳

❈❤❛♥❣❡s t♦ t❤❡ ❡①❝❤❛♥❣❡ r❛t❡ ♣❡❣✱ ✐♥❝❧✉❞✐♥❣ t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ❛❜❛♥❞♦♥✐♥❣ t❤❛t r❡❣✐♠❡✱ ✇❡r❡ t❤❡ s✉❜❥❡❝t ♦❢ ✉♥❡♥❞✐♥❣ ❞❡❜❛t❡s ✇❤✐❧❡ ✐t ❧❛st❡❞✳ ❊✈❡♥ t❤❡ ❛❞♦♣t✐♦♥ ♦❢ ✐♥✢❛t✐♦♥ t❛r❣❡t✐♥❣ ✇✐t❤ ✢♦❛t✐♥❣ ❡①❝❤❛♥❣❡ r❛t❡s ✇❛s ♥♦t ❡♥♦✉❣❤ t♦ ♣✉t ❛♥ ❡♥❞ t♦ t❤❛t ❞❡❜❛t❡✳ ❲♦✉❧❞ ✐t ❤❛✈❡ ❜❡❡♥ ♣♦ss✐❜❧❡ ❛♥❞ ❞❡s✐r❛❜❧❡ t♦ ❦❡❡♣ t❤❡ ❡①❝❤❛♥❣❡ r❛t❡ ♣❡❣❣❡❞❄ ❲❤❛t ✇♦✉❧❞ ❤❛✈❡ ❤❛♣♣❡♥❡❞ ✐❢ t❤❡ ❝❤❛♥❣❡ ✐♥ r❡❣✐♠❡ ❤❛❞ ♦❝❝✉rr❡❞ ❡❛r❧✐❡r❄ ❲❤❛t ✇♦✉❧❞ ❤❛✈❡ ❜❡❡♥ t❤❡ ♠♦st ❢❛✈♦r❛❜❧❡ ♠♦♠❡♥t ❢♦r s✉❝❤ ❝❤❛♥❣❡❄

■♥ t❤✐s ♣❛♣❡r ✇❡ ❛♥s✇❡r s♦♠❡ ♦❢ t❤❡s❡ q✉❡st✐♦♥s ✉s✐♥❣ ❛ ❞②♥❛♠✐❝✱ st♦❝❤❛st✐❝✱ ❣❡♥❡r❛❧ ❡q✉✐❧✐❜r✐✉♠ ♠♦❞❡❧ ✭✏❉❙●❊✑✮ ❜❛s❡❞ ♦♥ ●❛❧í ❛♥❞ ▼♦♥❛❝❡❧❧✐ ✭✷✵✵✺✮ ❛♥❞ ❏✉st✐♥✐❛♥♦ ❛♥❞ Pr❡st♦♥ ✭✷✵✶✵✮✱ ❡st✐♠❛t❡❞ ❢♦r t❤❡ ❇r❛③✐❧✐❛♥ ❡❝♦♥♦♠②✳ ❚❤❡ ♠♦❞❡❧ ✐s ❡st✐♠❛t❡❞ ✉s✐♥❣ ♠❛❝r♦❡❝♦♥♦♠✐❝ ❞❛t❛ ❢r♦♠ t❤❡ t❤✐r❞ q✉❛rt❡r ♦❢ ✶✾✾✺ t♦ t❤❡ s❡❝♦♥❞ q✉❛rt❡r ♦❢ ✷✵✶✸✳ ❋♦❧❧♦✇✐♥❣ ❈úr❞✐❛ ❛♥❞ ❋✐♥♦❝❝❤✐❛r♦ ✭✷✵✶✸✮✱ ✇❡ ❡①♣❧✐❝✐t❧② ♠♦❞❡❧ t❤❡ ❝❤❛♥❣❡ ♦❢ ♠♦♥❡t❛r② r❡❣✐♠❡ t❤❛t t♦♦❦ ♣❧❛❝❡ ✐♥ t❤❡ ✜rst q✉❛rt❡r ♦❢ ✶✾✾✾✳ ❚♦ t❤❛t ❡♥❞✱ ✇❡ ❛❧❧♦✇ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ✐♥t❡r❡st r❛t❡ r✉❧❡ ❢♦❧❧♦✇❡❞ ❜② t❤❡ ❈❇❇ t♦ ✈❛r② ❛❝r♦ss r❡❣✐♠❡s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇✐t❤ t❤❡ ❛❞♦♣t✐♦♥ ♦❢ ✐♥✢❛t✐♦♥ t❛r❣❡t✐♥❣ t❤❡ ❈❇❇ ❝❡❛s❡s t♦ r❡❛❝t t♦ ❞❡✈✐❛t✐♦♥s ♦❢ t❤❡ ♥♦♠✐♥❛❧ ❡①❝❤❛♥❣❡ r❛t❡ ❢r♦♠ ❛ ♣r❡✲st❛❜❧✐s❤❡❞ ♣❛r✐t②✱ ❛♥❞ st❛rts t♦ r❡❛❝t t♦ ❞❡✈✐❛t✐♦♥s ♦❢ ✐♥✢❛t✐♦♥ ❢r♦♠ t❛r❣❡t✳ ❋♦r t❤❡ s❛❦❡ ♦❢ s✐♠♣❧✐❝✐t②✱ t❤❡ ❝❤❛♥❣❡ ♦❢ r❡❣✐♠❡ ❝♦♠❡s ❛s ❛ s✉r♣r✐s❡ t♦ ❡❝♦♥♦♠✐❝ ❛❣❡♥ts✳ ❚❤❡ ♦t❤❡r ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ♠♦❞❡❧✱ r❡❧❛t❡❞ t♦ ♣r❡❢❡r❡♥❝❡s✱ t❡❝❤♥♦❧♦❣② ❡t❝✳✱ ❛r❡ ❛ss✉♠❡❞ t♦ ❜❡ ✐♥✈❛r✐❛♥t✳

❲❡ ✉s❡ t❤❡ ❡st✐♠❛t❡❞ ♠♦❞❡❧ t♦ r❡❝♦✈❡r t❤❡ str✉❝t✉r❛❧ s❤♦❝❦s t❤❛t ❤✐t t❤❡ ❇r❛③✐❧✐❛♥ ❡❝♦♥♦♠② ❞✉r✐♥❣ t❤❡ s❛♠♣❧❡ ♣❡r✐♦❞✱ ❛♥❞ s✐♠✉❧❛t❡ ❝♦✉♥t❡r❢❛❝t✉❛❧ ❤✐st♦r✐❡s✳ ❙♣❡❝✐✜❝❛❧❧②✱ ✇❡ ❛♥❛❧②③❡ t❤❡ ❡✛❡❝ts ♦❢ ❛❧t❡r♥❛t✐✈❡ t✐♠✐♥❣s ❢♦r t❤❡ ❛❞♦♣t✐♦♥ ♦❢ ✐♥✢❛t✐♦♥ t❛r❣❡t✐♥❣ ✇✐t❤ ✢♦❛t✐♥❣ ❡①❝❤❛♥❣❡ r❛t❡s✳

❇❡❢♦r❡ ✇❡ s✉♠♠❛r✐③❡ ♦✉r ♠❛✐♥ r❡s✉❧ts✱ ❛ ❢❡✇ ♦❜s❡r✈❛t✐♦♥s ❛r❡ ✐♥ ♦r❞❡r✳ ■♥ ❛♥② ❡①❡r❝✐s❡ ♦❢ t❤✐s ❦✐♥❞✱

✶_{❖r✐❣✐♥❛❧❧② ✐♥ P♦rt✉❣✉❡s❡✿ ✏❘❛♣✐❞❛♠❡♥t❡ ❛s r❡s❡r✈❛s ❝r❡s❝❡r❛♠ ❡ ❛ ❝♦♥✜❛♥ç❛ ✈♦❧t♦✉✳ ❚❛❧✈❡③ t❡♥❤❛ s✐❞♦ ✐ss♦ q✉❡ ♥♦s} ❧❡✈♦✉ ❛ ♣❡r❞❡r ♦♣♦rt✉♥✐❞❛❞❡s ♣❛r❛ r❡✈❡r ❛ q✉❡stã♦ ❝❛♠❜✐❛❧ ♥♦ ♣r✐♠❡✐r♦ q✉❛❞r✐♠❡str❡ ❞❡ ✶✾✾✽✱ q✉❛♥❞♦ ❡✈❡♥t✉❛❧♠❡♥t❡ t❡r✐❛ s✐❞♦ ♣♦ssí✈❡❧ ❢❛③ê✲❧♦✑✳ ❖✉r ♦✇♥ tr❛♥s❧❛t✐♦♥✳

✷_{❆s✐❛♥ ❈r✐s✐s ✐♥ ✶✾✾✼ ❛♥❞ ❘✉ss✐❛♥ ❈r✐s✐s ✐♥ ✶✾✾✽✳}

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

❚❤❡ ❡st✐♠❛t❡❞ ♠♦❞❡❧ ♠❛② ❜❡ s✉✐t❛❜❧❡ ❢♦r st✉❞②✐♥❣ ❛❣❣r❡❣❛t❡ ✢✉❝t✉❛t✐♦♥s ❛♥❞ q✉❡st✐♦♥s r❡❧❛t❡❞ t♦ ♠❛❝r♦❡❝♦♥♦♠✐❝ st❛❜✐❧✐③❛t✐♦♥ ♣♦❧✐❝✐❡s✳ ❍♦✇❡✈❡r✱ ✐t ✐s s✐❧❡♥t ♦♥ ❛♥② ✐ss✉❡ t❤❛t ♣❡rt❛✐♥s t♦ t❤❡ ❧♦♥❣ r✉♥✳ ❚❤✐s ✐s s♦ ❜❡❝❛✉s❡ t❤❡r❡ ✐s ♥♦ ❝❤❛♥♥❡❧ ✐♥ t❤❡ ♠♦❞❡❧ t❤r♦✉❣❤ ✇❤✐❝❤ ♣♦❧✐❝✐❡s ♠❛② ❛✛❡❝t tr❡♥❞ ❣r♦✇t❤✳ ❍❡♥❝❡✱ t❤❡ ♠♦❞❡❧ s❤♦✉❧❞ ♦♥❧② ❜❡ ✉s❡❞ t♦ ❛❞❞r❡ss q✉❡st✐♦♥s t❤❛t ❝❛♥ ❜❡ ❝✐r❝✉♠s❝r✐❜❡❞ t♦ ❜✉s✐♥❡ss ❝②❝❧❡ ❢r❡q✉❡♥❝✐❡s✳

❆s ✐♥ ♠♦st ♦❢ t❤❡ ❧✐t❡r❛t✉r❡✱ ✇❡ ✇♦r❦ ✇✐t❤ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧ ❛r♦✉♥❞ ❛ ③❡r♦ ✐♥✢❛t✐♦♥ st❡❛❞② st❛t❡✳ ❍♦✇❡✈❡r✱ ❞✉r✐♥❣ ♦✉r s❛♠♣❧❡ ♣❡r✐♦❞ t❤❡ ❇r❛③✐❧✐❛♥ ❡❝♦♥♦♠② ❢❛❝❡❞ ✐♠♣♦rt❛♥t s❤♦❝❦s✱ ✐♥✲ ❝❧✉❞✐♥❣ t❤❡ ♠♦♥❡t❛r② r❡❣✐♠❡ ❝❤❛♥❣❡ ✐ts❡❧❢✳ ■♥ ❢✉t✉r❡ ✇♦r❦✱ ✐t ✇♦✉❧❞ ❜❡ ✐♥t❡r❡st✐♥❣ t♦ r❡✈✐❡✇ t❤❡ ♣♦✐♥ts ♠❛❞❡ ✐♥ t❤✐s ♣❛♣❡r ✉s✐♥❣ ♠❡t❤♦❞s t❤❛t ♣r❡s❡r✈❡ t❤❡ ♥♦♥✲❧✐♥❡❛r✐t② ♦❢ t❤❡ ♠♦❞❡❧✳ ❆❞❞✐t✐♦♥❛❧❧②✱ ✐t ✇♦✉❧❞ ❜❡ ❛❞✈✐s❛❜❧❡ t♦ ❛❧❧♦✇ ❢♦r tr❡♥❞ ✐♥✢❛t✐♦♥✱ s♦ ❛s t♦ ❜r✐♥❣ t❤❡ st❡❛❞② st❛t❡ ♦❢ t❤❡ ♠♦❞❡❧ ❝❧♦s❡r t♦ t❤❡ ❞❛t❛✳ ❚❤❡ ❤②♣♦t❤❡s✐s t❤❛t t❤❡ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ❡❝♦♥♦♠② ♦t❤❡r t❤❛♥ t❤♦s❡ ♦❢ t❤❡ ♠♦♥❡t❛r② ♣♦❧✐❝② r✉❧❡ ❛r❡ ✐♥✈❛r✐❛♥t ✭✏str✉❝t✉r❛❧✑✮ ✐s ✐♥❤❡r❡♥t t♦ t❤❡ ✐❞❡❛ t❤❛t t❤❡ ♠♦❞❡❧ ✐s ✇❡❧❧ s♣❡❝✐✜❡❞ ❛♥❞ ✐♠♠✉♥❡ t♦ t❤❡ ▲✉❝❛s ❈r✐t✐q✉❡✳ ❚❤✐s ❤②♣♦t❤❡s✐s ❝❛♥ ❜❡ t❡st❡❞ ❡❝♦♥♦♠❡tr✐❝❛❧❧② ❛♥❞✱ ✐❢ r❡❥❡❝t❡❞✱ t❤❡ ♠♦❞❡❧ s♣❡❝✐✜❝❛t✐♦♥ ❝❛♥ ❜❡ ❝❤❛♥❣❡❞✳✸ _{❙✐♠✐❧❛r❧②✱ ✇❡ ❝♦✉❧❞ ❝♦♥s✐❞❡r ❛ ♠♦❞❡❧ ✇✐t❤ ❛❧t❡r♥❛t✐♥❣ ♠♦♥❡t❛r② r❡❣✐♠❡s✱ ✇✐t❤ tr❛♥s✐t✐♦♥}

♣r♦❜❛❜✐❧✐t✐❡s t❤❛t ❛r❡ ✉♥❞❡rst♦♦❞ ❜② ❡❝♦♥♦♠✐❝ ❛❣❡♥ts✳ ❚❤✐s ✇♦✉❧❞ ❛❧❧♦✇ ✉s t♦ ✐♥❝♦r♣♦r❛t❡ ❡①♣❡❝t❛t✐♦♥s ♦❢ ❝❤❛♥❣❡s ✐♥ t❤❡ ❡①❝❤❛♥❣❡ r❛t❡ r❡❣✐♠❡✱ ✇❤✐❝❤ ❝❡rt❛✐♥❧② ❡①✐st❡❞ t♦ ✈❛r②✐♥❣ ❞❡❣r❡❡s ❜❡❢♦r❡ t❤❡ ❛❞♦♣t✐♦♥ ♦❢ ❛ ✢♦❛t✐♥❣ ❡①❝❤❛♥❣❡ r❛t❡ ✐♥ ❏❛♥✉❛r② ✶✾✾✾✳✹

❚❤❡ q✉❡st✐♦♥s t❤❛t ✇❡ ✜♥❞ ♠♦st r❡❧❡✈❛♥t ♣❡rt❛✐♥ t♦ t❤❡ ✈✐❛❜✐❧✐t② ♦❢ t❤❡ ♠♦♥❡t❛r② ♣♦❧✐❝✐❡s ✉s❡❞ ✐♥ s♦♠❡ ♦❢ t❤❡ ❝♦✉♥t❡r❢❛❝t✉❛❧ ❤✐st♦r✐❡s ✕ ♥♦t❛❜❧② t❤❡ ♦♥❡ t❤❛t s✐♠✉❧❛t❡s t❤❡ ❝♦♥t✐♥✉❛t✐♦♥ ♦❢ t❤❡ ❡①❝❤❛♥❣❡ r❛t❡ ❝r❛✇❧✐♥❣ ♣❡❣ r❡❣✐♠❡✳ ■♥ t❤❡ ♠♦❞❡❧✱ st✐❝❦✐♥❣ t♦ t❤✐s r❡❣✐♠❡ ✐s ❛❧✇❛②s ❛ ✈✐❛❜❧❡ ♦♣t✐♦♥✳ ❚❤❡r❡ ❛r❡ ♥♦ ♣♦❧✐t✐❝❛❧ ♣r❡ss✉r❡s✱ ❝♦♥✜❞❡♥❝❡ ❝r✐s❡s✱ s♣❡❝✉❧❛t✐✈❡ ❛tt❛❝❦s✱ ♥♦r ❧♦ss ♦❢ ✐♥t❡r♥❛t✐♦♥❛❧ r❡s❡r✈❡s✳✺ _{■♥ r❡❛❧✐t②✱ ♦♥❡ ❝❛♥}

❛r❣✉❡ t❤❛t t❤❡ ❞❡❢❡♥s❡ ♦❢ ❛ ❝✉rr❡♥❝② ♣❡❣ ✐s s✐♠♣❧② ♥♦t ✈✐❛❜❧❡ ✐♥ ❝❡rt❛✐♥ ❝✐r❝✉♠st❛♥❝❡s✳ ■♥❝♦r♣♦r❛t✐♥❣ t❤❡ r♦❧❡ ♦❢ ❧✐♠✐t❡❞ ❢♦r❡✐❣♥ ❡①❝❤❛♥❣❡ r❡s❡r✈❡s ❛♥❞ s♣❡❝✉❧❛t✐✈❡ ❛tt❛❝❦s ✐♥ s✉❝❤ ❛ ❉❙●❊ ♠♦❞❡❧ ✐s ❛♥ ✐♥t❡r❡st✐♥❣ ❛✈❡♥✉❡ ❢♦r r❡s❡❛r❝❤✱ ✇❤✐❝❤✱ ❛s ❢❛r ❛s ✇❡ ❦♥♦✇✱ r❡♠❛✐♥s ✉♥❡①♣❧♦r❡❞✳

❚❤❡ ❞✐s❝✉ss✐♦♥ ✐♥ t❤❡ ♣r❡✈✐♦✉s ♣❛r❛❣r❛♣❤ ❜r✐♥❣s ✉s t♦ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ❝❛✈❡❛t✱ ✇❤✐❝❤ ♣❡rt❛✐♥s t♦ ✜s❝❛❧ ♣♦❧✐❝② ✕ s♦♠❡t❤✐♥❣ t❤❛t t❤❡ ♠♦❞❡❧ ❡ss❡♥t✐❛❧❧② ❛❜str❛❝ts ❢r♦♠✳ ❖♥❡ ♠❛② r❡❛s♦♥❛❜❧② ❛r❣✉❡

✸_{❆♥♦t❤❡r ♦♣t✐♦♥ ✇♦✉❧❞ ❜❡ t♦ ❛❧❧♦✇ ❢♦r ❝❤❛♥❣❡s ✐♥ s♦♠❡ ✏str✉❝t✉r❛❧ ♣❛r❛♠❡t❡rs✑✳ ❉❡s♣✐t❡ t❤❡ ❢❛❝t t❤❛t t❤✐s r♦✉t❡ s❡❡♠s} t♦ ✈✐♦❧❛t❡ t❤❡ s♣✐r✐t ♦❢ t❤❡ ▲✉❝❛s ❈r✐t✐q✉❡✱ t❤❡r❡ ✐s ❡✈✐❞❡♥❝❡ t❤❛t s♦♠❡ ♣❛r❛♠❡t❡rs ✉s✉❛❧❧② t❛❦❡♥ t♦ ❜❡ str✉❝t✉r❛❧ ❝❛♥ ✈❛r② ♦✈❡r t✐♠❡ ✐♥ ❛♥ ✐♠♣♦rt❛♥t ♠❛♥♥❡r ✭❡✳❣✳✱ ●✉✐s♦ ❡t ❛❧✳ ✷✵✶✸✮✳ ■t ✐s ✇♦rt❤ ♠❡♥t✐♦♥✐♥❣ t❤❛t t❤✐s t②♣❡ ♦❢ ❡✈✐❞❡♥❝❡ ✐s ♥♦t ✐♥ ❝♦♥✢✐❝t ✇✐t❤ t❤❡ ❡ss❡♥❝❡ ♦❢ t❤❡ ▲✉❝❛s ❈r✐t✐q✉❡✳

✹_{❚❤❡s❡ ❡①♣❡❝t❛t✐♦♥s ❝❛♥ ❜❡ ❡①♣❧✐❝✐t❧② ♠♦❞❡❧❡❞ ❛s ✐♥ ❉❛✈✐❣ ❛♥❞ ▲❡❡♣❡r ✭✷✵✶✵✮✱ ✇❤♦ ✐♥tr♦❞✉❝❡ ❛ ▼❛r❦♦✈ s✇✐t❝❤✐♥❣} ♠♦♥❡t❛r② ♣♦❧✐❝② r✉❧❡ ✐♥ t❤❡ ❜❛s✐❝ ♥❡✇ ❑❡②♥❡s✐❛♥ ♠♦❞❡❧✳ ❍♦✇❡✈❡r✱ ❢♦r ❛ ✜rst ❡①❡r❝✐s❡ ❛♣♣r♦❛❝❤✐♥❣ t❤❡ q✉❡st✐♦♥s ✐♥ t❤✐s ♣❛♣❡r ✕ ❛♥❞ t❛❦✐♥❣ ✐♥t♦ ❝♦♥s✐❞❡r❛t✐♦♥ t❤❛t ♦✉r ❛♥❛❧②s✐s ✐s ❜❛s❡❞ ♦♥ q✉❛rt❡r❧② ❞❛t❛ ✕ t❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ ❛ s✉r♣r✐s❡ ❝❤❛♥❣❡ ✐♥ t❤❡ ♠♦♥❡t❛r② r❡❣✐♠❡ ♠❛② ❜❡ ❧❡ss ♣r♦❜❧❡♠❛t✐❝ t❤❛♥ ✐t s❡❡♠s ❛t ✜rst✳

✺_{❚❤❡ r❡❛❞❡r ✇❤♦ ✐s ♥♦t ❢❛♠✐❧✐❛r ✇✐t❤ t❤❡ r❡❝❡♥t ❧✐t❡r❛t✉r❡ ✐♥ ▼♦♥❡t❛r② ❊❝♦♥♦♠✐❝s ♠❛② ✜♥❞ ✐t s✉r♣r✐s✐♥❣ t❤❛t t❤❡ ♠♦❞❡❧} ❛ss✉♠❡s ❛ ❝❛s❤❧❡ss ❡❝♦♥♦♠②✳ ■♥ t❤✐s r❡s♣❡❝t✱ ✇❡ ❢♦❧❧♦✇ t❤❡ ❛♣♣r♦❛❝❤ ❛❞✈♦❝❛t❡❞ ❜② ❲♦♦❞❢♦r❞ ✭✷✵✵✸✮✱ t♦ ✇❤✐❝❤ ✇❡ r❡❢❡r t❤❡ r❡❛❞❡rs ✇❤♦ ✇✐s❤ t♦ ❞❡❡♣❡♥ t❤❡✐r ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ t♦♣✐❝✳

t❤❛t t❤❡ ♣r❡ss✉r❡ t♦ ✢♦❛t t❤❡ ❘❡❛❧ r❡s✉❧t❡❞ ❧❛r❣❡❧② ❢r♦♠ ❛ ♣❡r❝❡♣t✐♦♥ t❤❛t ❇r❛③✐❧✬s ✜s❝❛❧ ♣♦❧✐❝② ✇❛s ✉♥s✉st❛✐♥❛❜❧❡✳ ❚❤✐s ✇♦✉❧❞ ❤❛✈❡ ✐♠♣♦s❡❞ ❧✐♠✐ts t♦ ♠♦♥❡t❛r② ♣♦❧✐❝② ❛♥❞ r❡♥❞❡r❡❞ t❤❡ ❞❡❢❡♥s❡ ♦❢ t❤❡ ❝✉rr❡♥❝② ♣❡❣ ✐♠♣♦ss✐❜❧❡✳ ❚❤✐s ✐ss✉❡ ❝❛♥ ❜❡ ❛❞❞r❡ss❡❞ ✐♥ ❛ ♠♦❞❡❧ ✇✐t❤ r❡❧❡✈❛♥t ✐♥t❡r❛❝t✐♦♥s ❜❡t✇❡❡♥ ♠♦♥❡t❛r② ❛♥❞ ✜s❝❛❧ ♣♦❧✐❝✐❡s ✕ ♣♦ss✐❜❧② ✇✐t❤ r❡❣✐♠❡ s❤✐❢ts ❛♣♣❧✐❝❛❜❧❡ t♦ ❜♦t❤ ♦❢ t❤❡♠✳ ■♥ t❤❛t ❝♦♥t❡①t✱ ✐t ✇♦✉❧❞ ❜❡ ♥❛t✉r❛❧ t♦ ❛ss✉♠❡ t❤❛t t❤❡ r✐s❦ ♣r❡♠✐✉♠ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❢♦r❡✐❣♥ ✐♥❞❡❜t❡❞♥❡ss ❞❡♣❡♥❞s ♦♥ t❤❡ ❝♦✉♥tr②✬s ✜s❝❛❧ ♣♦s✐t✐♦♥✳ ❆s ❛ r❡s✉❧t✱ t❤❡ ✏str✉❝t✉r❛❧ s❤♦❝❦s✑ r❡❝♦✈❡r❡❞ ❢r♦♠ t❤❡ ❡st✐♠❛t❡❞ ♠♦❞❡❧ ❝♦✉❧❞ ❝❤❛♥❣❡ s✐❣♥✐✜❝❛♥t❧②✱ ❧❡❛❞✐♥❣ t♦ ✐♠♣♦rt❛♥t ❞✐✛❡r❡♥❝❡s ✐♥ s♦♠❡ ♦❢ t❤❡ ❝♦✉♥t❡r❢❛❝t✉❛❧ ❤✐st♦r✐❡s t❤❛t ✇❡ ❝♦♥str✉❝t ✭♠♦r❡ ♦♥ t❤❛t ❧❛tt❡r✮✳

❲✐t❤ t❤♦s❡ ❝❛✈❡❛ts ✐♥ ♠✐♥❞✱ ❧❡t ✉s ♠♦✈❡ t♦ t❤❡ r❡s✉❧ts✳ ❆s ❡①♣❡❝t❡❞✱ t❤❡ ❡st✐♠❛t❡❞ ♣❛r❛♠❡t❡rs ❢♦r t❤❡ ❝r❛✇❧✐♥❣ ♣❡❣ r❡❣✐♠❡ ✐♥❞✐❝❛t❡ t❤❛t ♠♦♥❡t❛r② ♣♦❧✐❝② ✇❛s ❣❡❛r❡❞ t♦✇❛r❞s t❤❡ ♠❛✐♥t❡♥❛♥❝❡ ♦❢ t❤❡ ♥♦♠✐♥❛❧ ❡①❝❤❛♥❣❡ r❛t❡ ❛r♦✉♥❞ t❤❡ ❧❡✈❡❧s ❞❡✜♥❡❞ ❜② t❤❡ ❈❇❇✳ ■♥ t✉r♥✱ ✐♥ t❤❡ ✐♥✢❛t✐♦♥ t❛r❣❡t✐♥❣ r❡❣✐♠❡✱ t❤❡ ❡st✐♠❛t❡❞ ♣❛r❛♠❡t❡rs s✉❣❣❡st t❤❛t ♠♦♥❡t❛r② ♣♦❧✐❝② ❢♦❝✉s❡❞ ♦♥ st❛❜✐❧✐③✐♥❣ ✐♥✢❛t✐♦♥✳✻ _{■♥ ❛❞❞✐t✐♦♥✱}

t❤❡ r❡s✉❧ts s✉❣❣❡st ❛ ♠♦r❡ ♣r❡❞✐❝t❛❜❧❡ ❛♥❞ s②st❡♠❛t✐❝ ❜❡❤❛✈✐♦r ♦♥ t❤❡ ♣❛rt ♦❢ t❤❡ ❈❇❇✱ r❡✢❡❝t❡❞ ✐♥ t❤❡ ❧♦✇❡r ✈❛r✐❛♥❝❡ ♦❢ ♠♦♥❡t❛r② ♣♦❧✐❝② s❤♦❝❦s ❛♥❞ ❛ ❤✐❣❤❡r ❞❡❣r❡❡ ♦❢ ✐♥t❡r❡st r❛t❡ s♠♦♦t❤✐♥❣✳

❚❤❡ ❡st✐♠❛t❡❞ ♠♦♥❡t❛r② ♣♦❧✐❝② r✉❧❡s ❧❡❛❞ t♦ ✈❡r② ❞✐✛❡r❡♥t ♠❛❝r♦❡❝♦♥♦♠✐❝ ❞②♥❛♠✐❝s ✐♥ r❡s♣♦♥s❡ t♦ str✉❝t✉r❛❧ ❞✐st✉r❜❛♥❝❡s ✕ ❡s♣❡❝✐❛❧❧② ❢♦r❡✐❣♥ s❤♦❝❦s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ r❡s✉❧ts s✉❣❣❡st t❤❛t t❤❡ ❝❧❛ss✐❝ r♦❧❡ ♦❢ ✢♦❛t✐♥❣ ❡①❝❤❛♥❣❡ r❛t❡s ✕ ♥❛♠❡❧②✱ ❜✉✛❡r✐♥❣ t❤❡ ❡✛❡❝ts ♦❢ t❤♦s❡ s❤♦❝❦s ✕ ✐s ❦❡② t♦ t❤❡ ♦❜s❡r✈❡❞ ❞✐✛❡r❡♥❝❡s✳

❚❤❡s❡ ❞✐✛❡r❡♥❝❡s ✐♥ ❞②♥❛♠✐❝s ✉♥❞❡r t❤❡ t✇♦ ♠♦♥❡t❛r② r❡❣✐♠❡s ♠❛❦❡ ♦♥❡ ✇♦♥❞❡r ✇❤❛t ✇♦✉❧❞ ❤❛✈❡ ❤❛♣♣❡♥❡❞ ✐❢ t❤❡ tr❛♥s✐t✐♦♥ ❜❡t✇❡❡♥ t❤❡♠ ❤❛❞ ♦❝❝✉rr❡❞ ❛t ❛ ❞✐✛❡r❡♥t t✐♠❡✱ ✉♥❞❡r ❞✐✛❡r❡♥t ❝✐r❝✉♠st❛♥❝❡s✳ ❚♦ ❛♥❛❧②③❡ t❤✐s q✉❡st✐♦♥ ✇❡ ❝♦♥str✉❝t ❝♦✉♥t❡r❢❛❝t✉❛❧ ❤✐st♦r✐❡s t❤❛t s✐♠✉❧❛t❡ ❤♦✇ ❛❧t❡r♥❛t✐✈❡ ♠♦♥❡t❛r② ♣♦❧✐❝② ❝♦♥✜❣✉r❛t✐♦♥s ✇♦✉❧❞ ❤❛✈❡ ❛✛❡❝t❡❞ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ❇r❛③✐❧✐❛♥ ❡❝♦♥♦♠② ✐♥ r❡s♣♦♥s❡ t♦ t❤❡ ❡st✐♠❛t❡❞ str✉❝t✉r❛❧ s❤♦❝❦s✳

❖✉r r❡s✉❧ts s✉❣❣❡st t❤❛t ♠❛✐♥t❛✐♥✐♥❣ t❤❡ ❝✉rr❡♥❝② ♣❡❣ ❛❢t❡r t❤❡ ✜rst q✉❛rt❡r ♦❢ ✶✾✾✾ ✇♦✉❧❞ ❤❛✈❡ ❜❡❡♥ ❝♦♠❡ ❛t ❛ ❣r❡❛t ❝♦st✳✼ _{❆❧t❤♦✉❣❤ ✐t ✐s ❛❧✇❛②s ♣♦ss✐❜❧❡ t♦ ❛✈♦✐❞ ❛♥ ❛❜r✉♣t ❡①❝❤❛♥❣❡ r❛t❡ ❞❡✈❛❧✉❛t✐♦♥ ✐♥}

t❤❡ ♠♦❞❡❧✱ ♦✉r ❝♦✉♥t❡r❢❛❝t✉❛❧ ❛♥❛❧②s✐s s✉❣❣❡sts t❤❛t t❤✐s ✇♦✉❧❞ ❤❛✈❡ r❡q✉✐r❡❞ ❡①tr❡♠❡❧② ❤✐❣❤ ✐♥t❡r❡st r❛t❡s ❢♦r s❡✈❡r❛❧ q✉❛rt❡rs✳ ❆s ❛ r❡s✉❧t✱ ❡❝♦♥♦♠✐❝ ❛❝t✐✈✐t② ✇♦✉❧❞ ❤❛✈❡ ❝♦♥tr❛❝t❡❞ s❤❛r♣❧②✳ ❉❡s♣✐t❡ t❤❡ ❢❛❝t t❤❛t t❤❡ ♠♦❞❡❧ ❛❜str❛❝ts ❢r♦♠ s♦♠❡ ✐♠♣♦rt❛♥t ❞✐♠❡♥s✐♦♥s✱ s✉❝❤ ❛s ✜s❝❛❧ ♣♦❧✐❝②✱ ✐t s❡❡♠s ♣❧❛✉s✐❜❧❡ t♦ ❝♦♥❝❧✉❞❡ t❤❛t ❦❡❡♣✐♥❣ t❤❡ ♣❡❣ ❛❢t❡r t❤❡ ✜rst q✉❛rt❡r ♦❢ ✶✾✾✾ ✇♦✉❧❞ ❤❛✈❡ ❜❡❡♥ ❡ss❡♥t✐❛❧❧② ✐♠♣♦ss✐❜❧❡✳ ■♥ ❛ s❡❝♦♥❞ ❝♦✉♥t❡r❢❛❝t✉❛❧ ❛♥❛❧②s✐s✱ ✇❡ s✐♠✉❧❛t❡ ❛♥ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ t❤❡ ♣❛❝❡ ♦❢ ❞❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ❡①❝❤❛♥❣❡ r❛t❡ ❝r❛✇❧✐♥❣ ♣❡❣ ❛❢t❡r t❤❡ ❆s✐❛♥ ❈r✐s✐s✱ ❢r♦♠ ❛♣♣r♦①✐♠❛t❡❧② ✼✪ t♦ ✶✹✪ ♣❡r ②❡❛r✳ ■♥ t❤✐s

✻_{❋♦r r❡❝❡♥t ♣❛♣❡rs ♦♥ ❝❤❛♥❣❡s ✐♥ ♠♦♥❡t❛r② ♣♦❧✐❝② ✐♥ ❇r❛③✐❧ s✐♥❝❡ t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ✐♥✢❛t✐♦♥ t❛r❣❡t✐♥❣ r❡❣✐♠❡✱} s❡❡ ❇❡rr✐❡❧ ❡t ❛❧✳ ✭✷✵✶✸✮✱ ❈❛r✈❛❧❤♦ ❡t ❛❧✳ ✭✷✵✶✸✮ ❡ ●♦♥ç❛❧✈❡s ✭✷✵✶✺✮✳

✼_{❖✉r ❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ❝♦sts ❛♥❞ ❜❡♥❡✜ts ♦❢ ❛❧t❡r♥❛t✐✈❡ ❤✐st♦r✐❡s ✐s ❞❡❧✐❜❡r❛t❡❧② ✐♥❢♦r♠❛❧ ❛♥❞ ♥❡❡❞ ♥♦t ❝♦✐♥❝✐❞❡ ✇✐t❤} ✇❤❛t ✇♦✉❧❞ ❜❡ ✐♠♣❧✐❡❞ ❜② ❛ ❢♦r♠❛❧ ✇❡❧❢❛r❡ ❛♥❛❧②s✐s ❜❛s❡❞ ♦♥ t❤❡ str✉❝t✉r❡ ♦❢ t❤❡ ♠♦❞❡❧✳ ❲❡ ♣r♦❝❡❡❞ ✐♥ t❤✐s ♠❛♥♥❡r t♦ r❡❧❛t❡ t❤❡ ❡✈❛❧✉❛t✐♦♥ ♦❢ ❞✐✛❡r❡♥t ❛❧t❡r♥❛t✐✈❡s ✇✐t❤ ✇❤❛t ✇❡ t❛❦❡ t♦ ❜❡ t❤❡ ✏❝♦♠♠♦♥ s❡♥s❡✑ ❛♠♦♥❣ ♣❛rt✐❝✐♣❛♥ts ♦❢ t❤✐s ❞❡❜❛t❡ ✐♥ ❇r❛③✐❧✳

❝❛s❡✱ ❇r❛③✐❧ ✇♦✉❧❞ ❤❛✈❡ ❡①♣❡r✐❡♥❝❡❞ ❤✐❣❤❡r ✐♥✢❛t✐♦♥✱ ❤✐❣❤❡r ♥♦♠✐♥❛❧ ❛♥❞ r❡❛❧ ✐♥t❡r❡st r❛t❡s✱ ❛♥❞ ✇❡❛❦❡r ❡❝♦♥♦♠✐❝ ❛❝t✐✈✐t②✳

❋✐♥❛❧❧②✱ r❡s✉❧ts ❢r♦♠ ❛ t❤✐r❞ ❝♦✉♥t❡r❢❛❝t✉❛❧ s✐♠✉❧❛t✐♦♥ s✉❣❣❡st t❤❛t t❤❡ ✜rst s❡♠❡st❡r ♦❢ ✶✾✾✽ ♠❛② ❤❛✈❡ ♦✛❡r❡❞ t❤❡ ✐❞❡❛❧ ✇✐♥❞♦✇ ❢♦r ❛ r❡❧❛t✐✈❡❧② s♠♦♦t❤ tr❛♥s✐t✐♦♥ ❢r♦♠ t❤❡ ❝✉rr❡♥❝② ♣❡❣ r❛t❡ t♦ t❤❡ ✐♥✢❛t✐♦♥ t❛r❣❡t✐♥❣ r❡❣✐♠❡ ✇✐t❤ ❛ ✢♦❛t✐♥❣ ❡①❝❤❛♥❣❡ r❛t❡✳

❚❤❡ ♣❛♣❡r ♣r♦❝❡❡❞s ❛s ❢♦❧❧♦✇s✳ ❙❡❝t✐♦♥ ✷ ♣r❡s❡♥ts t❤❡ ♠♦❞❡❧ ❛♥❞ ❡①♣❧❛✐♥s ❤♦✇ ✐ts s♦❧✉t✐♦♥ ✐s ❝❛st ✐♥ ❛ st❛t❡✲s♣❛❝❡ r❡♣r❡s❡♥t❛t✐♦♥✳ ❚❤❡ s✉❜s❡q✉❡♥t s❡❝t✐♦♥ ♣r♦✈✐❞❡s ❞❡t❛✐❧s ♦❢ t❤❡ ♠❡t❤♦❞♦❧♦❣② ❛♥❞ ❞❛t❛ ✉s❡❞ t♦ ❡st✐♠❛t❡ t❤❡ ♠♦❞❡❧✳ ❙❡❝t✐♦♥s ✹ ❛♥❞ ✺ ♣r❡s❡♥t r❡s✉❧ts ♦❢ t❤❡ ❡st✐♠❛t❡❞ ♠♦❞❡❧ ❛♥❞ t❤❡ s✐♠✉❧❛t❡❞ ❝♦✉♥t❡r❢❛❝t✉❛❧ ❤✐st♦r✐❡s✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ❧❛st s❡❝t✐♦♥ ❝♦♥❝❧✉❞❡s✳ ■♥ ❛♥ ❛tt❡♠♣t t♦ s❤♦rt❡♥ t❤❡ ♣❛♣❡r ❛♥❞ ♠❛❦❡ ✐t ❧❡ss ❛r✐❞✱ ✇❡ ❞❡❢❡r t❤❡ t❡❝❤♥✐❝❛❧ ❛♥❞ ♠❡t❤♦❞♦❧♦❣✐❝❛❧ ❞❡t❛✐❧s t♦ t❤❡ ❆♣♣❡♥❞✐①✱ ✇❤❡♥❡✈❡r ♣♦ss✐❜❧❡✳

✷ ▼♦❞❡❧

❖✉r ♠❛✐♥ r❡❢❡r❡♥❝❡ ✐s t❤❡ ✭s❡♠✐✲✮s♠❛❧❧ ♦♣❡♥ ❡❝♦♥♦♠② ♥❡✇ ❑❡②♥❡s✐❛♥ ♠♦❞❡❧ ♦❢ ❏✉st✐♥✐❛♥♦ ❛♥❞ Pr❡st♦♥ ✭✷✵✶✵✮✳ ❆ r❡♣r❡s❡♥t❛t✐✈❡ ❝♦♥s✉♠❡r ❞❡r✐✈❡s ✉t✐❧✐t② ❛♥❞ ❢♦r♠s ❤❛❜✐ts t❤r♦✉❣❤ t❤❡ ❝♦♥s✉♠♣t✐♦♥ ♦❢ ❞♦♠❡st✐❝ ❛♥❞ ✐♠♣♦rt❡❞ ❣♦♦❞s ❛♥❞ s❡r✈✐❝❡s ✭✏❣♦♦❞s✑ ♦r ✏♣r♦❞✉❝ts✑✮✳ ❙❤❡ ❛❧s♦ ✐♥❝✉rs ❞✐s✉t✐❧✐t② ❢r♦♠ s✉♣♣❧②✐♥❣ ❧❛❜♦r t♦ ❞♦♠❡st✐❝ ♣r♦❞✉❝❡rs✳ ■♥ ♦r❞❡r t♦ s♠♦♦t❤ ❝♦♥s✉♠♣t✐♦♥✱ s❤❡ ❝❛♥ r❡s♦rt t♦ ❞♦♠❡st✐❝ ❜♦♥❞s t❤❛t ②✐❡❧❞ t❤❡ ♥♦♠✐♥❛❧ ✐♥t❡r❡st r❛t❡ s❡t ❜② t❤❡ ❈❇❇ ❛♥❞ t♦ ❜♦♥❞s tr❛❞❡❞ ❛❜r♦❛❞✱ ✇❤✐❝❤ ❡❛r♥ ❛♥ ✐♥t❡r❡st r❛t❡ ❞❡t❡r♠✐♥❡❞ ✐♥ t❤❡ ✐♥t❡r♥❛t✐♦♥❛❧ ♠❛r❦❡t ♣❧✉s ❛ ♣r❡♠✐✉♠ t❤❛t ❞❡♣❡♥❞s ♦♥ ❇r❛③✐❧✬s ♥❡t ❢♦r❡✐❣♥ ❛ss❡t ♣♦s✐t✐♦♥✳

❋✐r♠s ♦♣❡r❛t❡ ✉♥❞❡r ♠♦♥♦♣♦❧✐st✐❝ ❝♦♠♣❡t✐t✐♦♥ ❛♥❞ ❛r❡ ❞✐✈✐❞❡❞ ✐♥t♦ t✇♦ ❣r♦✉♣s✳ ❉♦♠❡st✐❝ ♣r♦❞✉❝❡rs ❡♠♣❧♦② ❧❛❜♦r t♦ ♣r♦❞✉❝❡ t❤❡✐r ❣♦♦❞s ✉s✐♥❣ ❛ t❡❝❤♥♦❧♦❣② t❤❛t ✐s s✉❜❥❡❝t t♦ ♣r♦❞✉❝t✐✈✐t② s❤♦❝❦s✳ ❚❤❡ ♦t❤❡r ❣r♦✉♣ ❝♦♠♣r✐s❡s r❡t❛✐❧ ✜r♠s t❤❛t ✐♠♣♦rt t❤❡✐r ♣r♦❞✉❝ts✱ ❞✐✛❡r❡♥t✐❛t❡ t❤❡♠ ❛t ♥♦ ❡①tr❛ ❝♦st✱ ❛♥❞ s❡❧❧ t❤❡♠ ✐♥ t❤❡ ❞♦♠❡st✐❝ ♠❛r❦❡t✳ ❆❧❧ ✜r♠s r❡❡✈❛❧✉❛t❡ t❤❡✐r ♣r✐❝❡s ✐♥❢r❡q✉❡♥t❧②✱ ❛♥❞ ♦t❤❡r✇✐s❡ ✐♥❞❡① t❤❡✐r ♣r✐❝❡s t♦ ♣❛st ✐♥✢❛t✐♦♥✳

❚❤❡ ♠❛✐♥ ♠♦❞✐✜❝❛t✐♦♥ ✇❡ ♠❛❦❡ t♦ t❤❡ ♠♦❞❡❧ ✐s t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ t✇♦ ❞✐st✐♥❝t ♠♦♥❡t❛r② r❡❣✐♠❡s✱ ✐♥ t❤❡ s♣✐r✐t ♦❢ ❈úr❞✐❛ ❛♥❞ ❋✐♥♦❝❝❤✐❛r♦ ✭✷✵✶✸✮✳ ■♥ t❤❡ ✜rst r❡❣✐♠❡✱ t❤❡ ✐♥t❡r❡st r❛t❡ ❞❡✜♥❡❞ ❜② t❤❡ ❈❇❇ r❡s♣♦♥❞s t♦ ❞❡♣❛rt✉r❡s ♦❢ t❤❡ ♥♦♠✐♥❛❧ ❡①❝❤❛♥❣❡ r❛t❡ ❢r♦♠ ❛ t❛r❣❡t t❤❛t ♠❛② ❡✈♦❧✈❡ ♦✈❡r t✐♠❡✱ ✐♥t❡r♣r❡t❡❞ ❛s t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ❡①❝❤❛♥❣❡ r❛t❡ ❜❛♥❞ s♣❡❝✐✜❡❞ ❜② t❤❡ ❝❡♥tr❛❧ ❜❛♥❦✳ ■♥ t❤❡ s❡❝♦♥❞ r❡❣✐♠❡✱ t❤❡ ❈❇❇ r❡s♣♦♥❞s t♦ ❞❡✈✐❛t✐♦♥s ♦❢ ✐♥✢❛t✐♦♥ ❢r♦♠ ✐ts t❛r❣❡t✱ ✇❤✐❝❤ ♠❛② ❛❧s♦ ✈❛r② ♦✈❡r t✐♠❡✳ ❋♦❧❧♦✇✐♥❣ t❤❡ ❛♣♣r♦❛❝❤ ❛❞✈♦❝❛t❡❞ ❜② ❲♦♦❞❢♦r❞ ✭✷✵✵✸✮✱ ✇❡ ✇♦r❦ ✇✐t❤ t❤❡ ❝❛s❤❧❡ss ❧✐♠✐t ♦❢ ❛ ♠♦♥❡t❛r② ❡❝♦♥♦♠②✳

✷✳✶ ❘❡♣r❡s❡♥t❛t✐✈❡ ❝♦♥s✉♠❡r

❚❤❡ r❡♣r❡s❡♥t❛t✐✈❡ ❇r❛③✐❧✐❛♥ ❝♦♥s✉♠❡r ♠❛①✐♠✐③❡s ❡①♣❡❝t❡❞ ✉t✐❧✐t②

E0

∞

X

t=0

βtΓt

"

(Ct−Ht)1−σ

1−σ −

N_t1+ϕ

1 +ϕ

#

,

s✉❜❥❡❝t t♦ t❤❡ ❜✉❞❣❡t ❝♦♥str❛✐♥t ♣r❡s❡♥t❡❞ ❜❡❧♦✇✳ ❚❤❡ Γt t❡r♠ ✐s ❛ ♣r❡❢❡r❡♥❝❡ s❤♦❝❦✱ Ht ≡ hCt−1 ✐s

t❤❡ ✏st♦❝❦✑ ♦❢ ❝♦♥s✉♠♣t✐♦♥ ❤❛❜✐ts ✭t❛❦❡♥ ❛s ❣✐✈❡♥ ❜② t❤❡ ❛❣❡♥t✮ ❛♥❞Nt✐s ❧❛❜♦r s✉♣♣❧②✳ ❚❤❡ ♣❛r❛♠❡t❡r β <1 ✐s t❤❡ s✉❜❥❡❝t✐✈❡ t✐♠❡ ❞✐s❝♦✉♥t ❢❛❝t♦r✱ ❛♥❞ t❤❡ ♣❛r❛♠❡t❡rsσ ❛♥❞ ϕ ❛r❡✱ r❡s♣❡❝t✐✈❡❧②✱ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❡❧❛st✐❝✐t② ♦❢ ✐♥t❡rt❡♠♣♦r❛❧ s✉❜st✐t✉t✐♦♥ ❛♥❞ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ✭❋r✐s❝❤✮ ❡❧❛st✐❝✐t② ♦❢ ❧❛❜♦r s✉♣♣❧②✳ ❚❤❡ ♦♣❡r❛t♦rEt ❞❡♥♦t❡s ❡①♣❡❝t❛t✐♦♥s ❝♦♥❞✐t✐♦♥❛❧ ♦♥ ✐♥❢♦r♠❛t✐♦♥ ❛✈❛✐❧❛❜❧❡ ❛t t✐♠❡t✳

❆❣❣r❡❣❛t❡ ❝♦♥s✉♠♣t✐♦♥ ✐s ❣✐✈❡♥ ❜②✿

Ct=

(1−α)1η_C η−1

η

D,t +α

1

η_C η−1

η I,t

₁η

−η ,

✇❤❡r❡ η ✐s t❤❡ ❡❧❛st✐❝✐t② ♦❢ s✉❜st✐t✉t✐♦♥ ❜❡t✇❡❡♥ ❞♦♠❡st✐❝ ❛♥❞ ✐♠♣♦rt❡❞ ❣♦♦❞s ❛♥❞ α ✐s t❤❡ s❤❛r❡ ♦❢ ✐♠♣♦rt❡❞ ❣♦♦❞s ✐♥ t♦t❛❧ ❝♦♥s✉♠♣t✐♦♥ ✕ ❛ ♠❡❛s✉r❡ ♦❢ t❤❡ ❞❡❣r❡❡ ♦❢ ♦♣❡♥♥❡ss ♦❢ t❤❡ ❡❝♦♥♦♠②✳ CD,t ❛♥❞ CI,t ❛r❡ t❤❡ ❝♦♠♣♦s✐t❡s ♦❢ ❞♦♠❡st✐❝ ❛♥❞ ✐♠♣♦rt❡❞ ❣♦♦❞s✱ r❡s♣❡❝t✐✈❡❧②✱ ♦❜t❛✐♥❡❞ t❤r♦✉❣❤ t❤❡ ❛❣❣r❡❣❛t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥t ✈❛r✐❡t✐❡sCD,t(i) ❡CI,t(i)✿

CD,t=

Z 1

0

CD,t(i) ε−1

ε di

ε−ε1

, CI,t=

Z 1

0

CI,t(i) ε−1

ε di

ε−ε1

.

❚❤❡ ❡❧❛st✐❝✐t② ♦❢ s✉❜st✐t✉t✐♦♥ ❜❡t✇❡❡♥ ✈❛r✐❡t✐❡s ✇✐t❤ t❤❡ s❛♠❡ ♦r✐❣✐♥ ✐s ❣✐✈❡♥ ❜②ε✳

❚♦ ❤❡r ❞✐s♠❛②✱ t❤❡ r❡♣r❡s❡♥t❛t✐✈❡ ❇r❛③✐❧✐❛♥ ❝♦♥s✉♠❡r ❢❛❝❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❜✉❞❣❡t ❝♦♥str❛✐♥t✿

PtCt+Dt+StBt=Dt−1Rt−1+StBt−1R∗t−1Φt−1(St−1Bt−1

Pt−1Y

) +WtNt+ ΠD,t+ ΠI,t,

✇❤❡r❡St ✐s t❤❡ ♥♦♠✐♥❛❧ ❡①❝❤❛♥❣❡ r❛t❡✱ q✉♦t❡❞ ✐♥ ❞♦♠❡st✐❝ ❝✉rr❡♥❝② ✉♥✐ts ✭✏❘❡❛✐s✑✮ ♣❡r ✉♥✐t ♦❢ ❢♦r❡✐❣♥ ❝✉rr❡♥❝② ✭✏❉♦❧❧❛rs✑✮✱ Dt ❛r❡ ✭✏❞♦♠❡st✐❝✑✮ ❜♦♥❞s ❞❡♥♦♠✐♥❛t❡❞ ✐♥ ❘❡❛✐s ❛♥❞ Bt ❛r❡ ❉♦❧❧❛r✲❞❡♥♦♠✐♥❛t❡❞ ✭✏❢♦r❡✐❣♥✑✮ ❜♦♥❞s ✇✐t❤ ❣r♦ss ✐♥t❡r❡st r❛t❡s ❣✐✈❡♥✱ r❡s♣❡❝t✐✈❡❧②✱ ❜②Rt ❛♥❞R∗tΦt(S_Pt_tB_Yt)✱Wt ✐s t❤❡ ♥♦♠✐♥❛❧ ✇❛❣❡✱ ΠD,t ❛♥❞ΠI,t ❛r❡ t❤❡ ♣r♦✜ts ♦❢ ❞♦♠❡st✐❝ ♣r♦❞✉❝❡rs ❛♥❞ ✐♠♣♦rt❡rs✱ r❡s♣❡❝t✐✈❡❧②✱Y ✐s t❤❡ st❡❛❞②✲ st❛t❡ ❧❡✈❡❧ ♦❢ ♦✉t♣✉t✱ ❛♥❞Pt✐s t❤❡ ♣r✐❝❡ ✐♥❞❡① ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❝♦♥s✉♠♣t✐♦♥ ❛❣❣r❡❣❛t♦r✱ t♦ ❜❡ ❞❡✜♥❡❞ ❜❡❧♦✇✳ ❚❤❡ ❝♦♥s✉♠❡r ❛❧s♦ ❢❛❝❡s ❛ st❛♥❞❛r❞ ✏♥♦✲P♦♥③✐✑ ❝♦♥str❛✐♥t✳

❚❤❡ ✐♥t❡r❡st r❛t❡ ♦♥ ❢♦r❡✐❣♥ ❜♦♥❞s Bt ✐s ❣✐✈❡♥ ❜② t❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ✐♥t❡r♥❛t✐♦♥❛❧ ✐♥t❡r❡st r❛t❡ R∗_t ❛♥❞ ❛ ✇❡❞❣❡ t❤❛t ❞❡♣❡♥❞s ♦♥ ❇r❛③✐❧✬s ♥❡t ❢♦r❡✐❣♥ ❛ss❡t ♣♦s✐t✐♦♥✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ❛ r✐s❦

♣r❡♠✐✉♠ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❡①t❡r♥❛❧ ✐♥❞❡❜t❡❞♥❡ss✳ ❚❤✐s ♣r❡♠✐✉♠ ✐s ❣✐✈❡♥ ❜② t❤❡ ❢✉♥❝t✐♦♥Φt()✿✽

Φt(Zt) = exp[−χZt+φt],

✇❤❡r❡φt✐s ❛ r✐s❦✲♣r❡♠✐✉♠ s❤♦❝❦✳

❚❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t t❤❡ ✐♥t❡r❡st r❛t❡ ♦♥ ❢♦r❡✐❣♥ ❜♦♥❞s ❞❡♣❡♥❞s ♣♦s✐t✐✈❡❧② ♦♥ t❤❡ ❧❡✈❡❧ ♦❢ ❡①t❡r♥❛❧ ✐♥❞❡❜t❡❞♥❡ss ❣✉❛r❛♥t❡❡s t❤❡ st❛t✐♦♥❛r✐t② ♦❢ t❤❡ ♠♦❞❡❧✳✾ _{❋✉rt❤❡r♠♦r❡✱ ✐t ❛❧❧♦✇s ✉s t♦ ✐♥tr♦❞✉❝❡ ❛ s❤♦❝❦}

t❤❛t ✐s ♥❡❡❞❡❞ ❢♦r t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧✱ ❛♥❞ ✇❤✐❝❤ ❤❛s t❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ ❜❡✐♥❣ ✐♥t❡r♣r❡t❡❞ ❛s ❛ ❞❡✈✐❛t✐♦♥ ❢r♦♠ t❤❡ st❛♥❞❛r❞ ✉♥❝♦✈❡r❡❞ ✐♥t❡r❡st r❛t❡ ♣❛r✐t② ❝♦♥❞✐t✐♦♥ ✭s❡❡ ❜❡❧♦✇✮✳

❚❤❡ ♦♣t✐♠❛❧ ❛❧❧♦❝❛t✐♦♥ ♦❢ ❝♦♥s✉♠♣t✐♦♥ ✐♥ ❡❛❝❤ ❝❛t❡❣♦r② ♦❢ ❣♦♦❞s ✭❞♦♠❡st✐❝ ❛♥❞ ✐♠♣♦rt❡❞✮ ✐♠♣❧✐❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡♠❛♥❞s ❢♦r t❤❡ ❛❣❣r❡❣❛t❡ ✐♠♣♦rt❡❞ ❛♥❞ ❞♦♠❡st✐❝ ♣r♦❞✉❝ts✿

CD,t= (1−α)

PD,t Pt

−η

Ct and CI,t=α

PI,t Pt

−η

Ct,

❛♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡♠❛♥❞s ❢♦r ❡❛❝❤ ✈❛r✐❡t② ✿

CD,t(i) =

PD,t(i) PD,t

−ε

CD,t and CI,t(i) =

PI,t(i) PI,t

−ε

CI,t. ✭✶✮

❚❤❡ ✈❛r✐❡t✐❡s ♦❢ ❣♦♦❞s ❛♥❞ s❡r✈✐❝❡s ❛r❡ ✐♠♣❡r❢❡❝t s✉❜st✐t✉t❡s✱ ❛♥❞ s♦ ✜r♠s r❡t❛✐♥ s♦♠❡ ♠❛r❦❡t ♣♦✇❡r✳ ❚❤✐s ✐s r❡✢❡❝t❡❞ ✐♥ ♥❡❣❛t✐✈❡❧② s❧♦♣❡❞ ❞❡♠❛♥❞ ❝✉r✈❡s ✭❡q✉❛t✐♦♥ ✭✶✮✮✳

❚❤❡ ♣r✐❝❡ ✐♥❞✐❝❡s ❜② ♣r♦❞✉❝t ♦r✐❣✐♥ ❛r❡ ❣✐✈❡♥ ❜②✿

PD,t=

Z 1

0

PD,t(i)1−εdi

1−1ε

e PI,t=

Z 1

0

PI,t(i)1−εdi

1−1ε ,

❛♥❞ t❤❡ ❛❣❣r❡❣❛t❡ ♣r✐❝❡ ✐♥❞❡① ❢♦r t❤❡ ❞♦♠❡st✐❝ ❡❝♦♥♦♠② ✐s ❣✐✈❡♥ ❜②✿

Pt=

h

(1−α)P_D,t1−η+αP_I,t1−ηi

1 1−η

.

❚❤❡ r❡♠❛✐♥✐♥❣ ✜rst✲♦r❞❡r ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ r❡♣r❡s❡♥t❛t✐✈❡ ❝♦♥s✉♠❡r✬s ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ❛r❡✿✶✵

Wt/Pt=Ntϕ(Ct−hCt−1)σ, ✭✷✮

Γt(Ct−hCt−1)−σ =βEt

Γt+1(Ct+1−hCt)−σRt Pt Pt+1

, ✭✸✮

Γt(Ct−hCt−1)−σ =βEt

Γt+1(Ct+1−hCt)−σR∗tΦt(StBt PtY

)St+1 St

Pt Pt+1

. ✭✹✮

✽_{❲❡ ❛ss✉♠❡ t❤❛t t❤❡ r❡♣r❡s❡♥t❛t✐✈❡ ❛❣❡♥t t❛❦❡s t❤✐s r✐s❦ ♣r❡♠✐✉♠ ❛s ❣✐✈❡♥ ✇❤❡♥ ♠❛❦✐♥❣ ❤❡r ❝♦♥s✉♠♣t✐♦♥ ❛♥❞ ♣♦rt❢♦❧✐♦} ❞❡❝✐s✐♦♥s✳

✾_{❋♦r ❛♥ ❛♥❛❧②s✐s ♦❢ ❛❧t❡r♥❛t✐✈❡ ✇❛②s t♦ ✐♥❞✉❝❡ st❛t✐♦♥❛r✐t② ✐♥ s♠❛❧❧ ♦♣❡♥ ❡❝♦♥♦♠② ♠♦❞❡❧s✱ s❡❡ ❙❝❤♠✐tt✲●r♦❤❡ ❛♥❞ ❯r✐❜❡} ✭✷✵✵✸✮✳

✶✵_{❚❤❡ ♦♣t✐♠❛❧ ❝❤♦✐❝❡s ♠✉st ❛❧s♦ s❛t✐s❢② ❛ st❛♥❞❛r❞ tr❛♥s✈❡rs❛❧✐t② ❝♦♥❞✐t✐♦♥✳}

❊q✉❛t✐♦♥ ✭✷✮ ❞❡t❡r♠✐♥❡s t❤❡ ❧❛❜♦r s✉♣♣❧②✱ ❛♥❞ ✭✸✮ ❛♥❞ ✭✹✮ ❛r❡ st❛♥❞❛r❞ ❊✉❧❡r ❡q✉❛t✐♦♥s✳ ❚❤❡s❡ ❧❛st t✇♦ ❡q✉❛t✐♦♥s ❝❛♥ ❜❡ ❝♦♠❜✐♥❡❞ t♦ ♦❜t❛✐♥ ❛ r✐s❦✲♣r❡♠✐✉♠✲❛❞❥✉st❡❞ ✉♥❝♦✈❡r❡❞ ✐♥t❡r❡st ♣❛r✐t② ❝♦♥❞✐t✐♦♥✿

Et

Γt+1(Ct+1−hCt)−σ Pt Pt+1

R∗_tΦt( StBt

PtY )St+1

St −Rt

= 0.

✷✳✷ ❉♦♠❡st✐❝ ♣r♦❞✉❝❡rs

❚❤❡r❡ ✐s ❛ ❝♦♥t✐♥✉✉♠ ♦❢ ❞♦♠❡st✐❝ ♣r♦❞✉❝❡rs ♦♣❡r❛t✐♥❣ ✉♥❞❡r ♠♦♥♦♣♦❧✐st✐❝ ❝♦♠♣❡t✐t✐♦♥✱ ✐♥❞❡①❡❞ ❜② i ∈ [0,1]✳ ❊❛❝❤ ❝♦♠♣❛♥② ❡♠♣❧♦②s ❧❛❜♦r t♦ ♣r♦❞✉❝❡ ❛ ❞✐✛❡r❡♥t✐❛t❡❞ ❣♦♦❞✴s❡r✈✐❝❡ yD,t(i)✳ Pr♦❞✉❝t✐♦♥ t❡❝❤♥♦❧♦❣✐❡s ❛r❡ s✉❜❥❡❝t t♦ ❛ ❝♦♠♠♦♥ ♣r♦❞✉❝t✐✈✐t② s❤♦❝❦✱ ❣✐✈❡♥ ❜② At✿

yD,t(i) =AtNt(i).

❋♦r ❡❛s❡ ♦❢ ❡①♣♦s✐t✐♦♥✱ ✇❡ ❞❡♥♦t❡ t❤❡ r❡❛❧ ♠❛r❣✐♥❛❧ ❝♦st✱ ✇❤✐❝❤ ✐s t❤❡ s❛♠❡ ❢♦r ❛❧❧ ❞♦♠❡st✐❝ ♣r♦❞✉❝❡rs✱ ❛s✿

M CD,t= Wt AtPD,t

.

❍❡♥❝❡✱ ✇❡ ❝❛♥ ✇r✐t❡ ✜r♠i✬s ♣r♦✜ts ❛s✿

ΠD,t(i) =yD,t(i)(PD,t(i)−PD,tM CD,t).

❋✐r♠s r❡♦♣t✐♠✐③❡ t❤❡✐r ♣r✐❝❡s ✐♥❢r❡q✉❡♥t❧②✱ ❛s ✐♥ ❈❛❧✈♦ ✭✶✾✽✸✮✳ ❋♦r ❡❛❝❤ ✜r♠✱ t❤✐s ❤❛♣♣❡♥s ✇✐t❤ ♣r♦❜❛❜✐❧✐t②1−θD ♣❡r ♣❡r✐♦❞✱ ✐♥❞❡♣❡♥❞❡♥t❧② ♦❢ ✇❤❛t ❤❛♣♣❡♥s t♦ ♦t❤❡r ✜r♠s✳ ❚❤❡r❡❢♦r❡✱ ✐♥ ❡❛❝❤ ♣❡r✐♦❞ ❛ ❢r❛❝t✐♦♥(1−θD)♦❢ ✜r♠s r❡♦♣t✐♠✐③❡s t❤❡✐r ♣r✐❝❡s✱ ✇❤✐❧❡ t❤❡ r❡♠❛✐♥✐♥❣ ❢r❛❝t✐♦♥(θD)❢♦❧❧♦✇s ❛♥ ✐♥❞❡①❛t✐♦♥ r✉❧❡✳ ❙♣❡❝✐✜❝❛❧❧②✱ ✜r♠s t❤❛t ❞♦ ♥♦t r❡♦♣t✐♠✐③❡ ✐♥ ♣❡r✐♦❞ t❛❞❥✉st t❤❡✐r ♣r❡✈✐♦✉s ♣r✐❝❡s ❛❝❝♦r❞✐♥❣ t♦✿

PD,t(i) =PD,t−1(i)

PD,t−1

PD,t−2

δD ,

✇❤❡r❡ t❤❡ ♣❛r❛♠❡t❡r δD ❞❡t❡r♠✐♥❡s t❤❡ ❞❡❣r❡❡ ♦❢ ✐♥❞❡①❛t✐♦♥ t♦ ♣❛st ✐♥✢❛t✐♦♥✳

❆❧❧ ✜r♠s t❤❛t r❡♦♣t✐♠✐③❡ ✐♥ ♣❡r✐♦❞t ❢❛❝❡ t❤❡ s❛♠❡ ✐♥t❡rt❡♠♣♦r❛❧ ♣r♦❜❧❡♠ ❛♥❞ ❝❤♦s❡ t❤❡ s❛♠❡ ♣r✐❝❡

XD,t(i) =XD,t✳ ❍❡♥❝❡✱ t❤❡ ♣r✐❝❡ ✐♥❞❡① ❢♦r ❞♦♠❡st✐❝ ♣r♦❞✉❝ts ❡✈♦❧✈❡s ❛❝❝♦r❞✐♥❣ t♦✿

PD,t =



(1−θD)X

(1−ε)

D,t +θD PD,t−1

PD,t−1

PD,t−2

δD!1−ε

 

1/(1−ε)

.

❋✐r♠s s❡❧❧ t❤❡✐r ♣r♦❞✉❝ts ✐♥ ❜♦t❤ t❤❡ ❞♦♠❡st✐❝ ❛♥❞ ✐♥t❡r♥❛t✐♦♥❛❧ ♠❛r❦❡ts✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ❡①t❡r✲ ♥❛❧ ❞❡♠❛♥❞ ❤❛s t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥❛❧ ❢♦r♠ ❛s t❤❡ ❞♦♠❡st✐❝ ❞❡♠❛♥❞ ✭✶✮✱ s♦ t❤❛t ❛ ✜r♠ t❤❛t r❡♦♣t✐♠✐③❡❞

✐ts ♣r✐❝❡ ✐♥ t❤❡ ♣❡r✐♦❞t❢❛❝❡s t❤❡ ❢♦❧❧♦✇✐♥❣ s❡q✉❡♥❝❡ ♦❢ ❞❡♠❛♥❞s✿

yD,t+τ|t=

XD,t PD,t+τ

PD,t+τ−1

PD,t−1

δD!−ε

(CD,t+τ +CD,t∗ +τ), ✭✺✮

✇❤❡r❡C_D,t∗ _+τ ✐s t❤❡ ❡①t❡r♥❛❧ ❛❣❣r❡❣❛t❡ ❞❡♠❛♥❞ ❢♦r ❞♦♠❡st✐❝ ♣r♦❞✉❝ts ✭t♦ ❜❡ ❞❡t❛✐❧❡❞ ❜❡❧♦✇✮✳

❚❛❦✐♥❣ ♣r✐❝❡ r✐❣✐❞✐t② ✐♥t♦ ❛❝❝♦✉♥t✱ ❛ ✜r♠ s❡❧❡❝t✐♥❣ t❤❡ ♦♣t✐♠❛❧ ♣r✐❝❡ ✐♥ ♣❡r✐♦❞t♠❛①✐♠✐③❡s t❤❡ ♣r❡s❡♥t ❞✐s❝♦✉♥t❡❞ ✈❛❧✉❡ ♦❢ ✐ts ❡①♣❡❝t❡❞ ♣r♦✜ts✿

Et

∞

X

τ=0

θ_DτΘt,t+τyD,t+τ|t

"

XD,t

PD,t+τ−1

PD,t−1

δD

−PD,t+τM CD,t+τ

#

,

s✉❜❥❡❝t t♦ t❤❡ s❡q✉❡♥❝❡ ♦❢ ❞❡♠❛♥❞s ❣✐✈❡♥ ❜② ❡q✉❛t✐♦♥ ✭✺✮✱ ✇❤❡r❡ Θt,t+τ = βτΓ_Γt+_tτ _PP_t+t_τ U_Uc,t_c,t+τ ✐s t❤❡ ♥♦♠✐♥❛❧ ❞✐s❝♦✉♥t st♦❝❤❛st✐❝ ❢❛❝t♦r ♦❢ t❤❡ r❡♣r❡s❡♥t❛t✐✈❡ ❝♦♥s✉♠❡r✳✶✶

✷✳✸ ■♠♣♦rt✐♥❣ r❡t❛✐❧ ✜r♠s

❘❡t❛✐❧ ✜r♠s ✐♠♣♦rt ❣♦♦❞s ❛❝q✉✐r❡❞ ❛t ♣r✐❝❡s ❞❡t❡r♠✐♥❡❞ ✐♥ t❤❡ ✐♥t❡r♥❛t✐♦♥❛❧ ♠❛r❦❡t✱ ❛♥❞ tr❛♥s❢♦r♠ t❤❡♠ ✐♥t♦ ❞✐✛❡r❡♥t✐❛t❡❞ ❣♦♦❞s t♦ ❜❡ s♦❧❞ ✐♥ t❤❡ ❞♦♠❡st✐❝ ♠❛r❦❡t✳ ❋♦r s✐♠♣❧✐❝✐t②✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤✐s ❞✐✛❡r❡♥t✐❛t✐♦♥ ✐s ❞♦♥❡ ❛t ♥♦ ❝♦st✳ ❚❤❡ r❡t❛✐❧ s❡❝t♦r ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② ♠♦♥♦♣♦❧✐st✐❝ ❝♦♠♣❡t✐t✐♦♥✱ s♦ t❤❛t ❡❛❝❤ ✜r♠ ❤❛s s♦♠❡ ♠❛r❦❡t ♣♦✇❡r t♦ s❡t ♣r✐❝❡s✳ ❚❤❡s❡ ❛r❡ s❡t ✐♥ ❧♦❝❛❧ ❝✉rr❡♥❝②✱ ❛♥❞ ❛r❡ s✉❜❥❡❝t t♦ ✐♥❢r❡q✉❡♥t ❛❞❥✉st♠❡♥ts ❛♥❞ ✐♥❞❡①❛t✐♦♥ t♦ ♣❛st ✐♥✢❛t✐♦♥✳ ❚❤✐s ❧❡❛❞s t♦ ❛♥ ✐♠♣❡r❢❡❝t ♣❛sst❤r♦✉❣❤ ❢r♦♠ ✐♥t❡r♥❛t✐♦♥❛❧ ♣r✐❝❡s ❛♥❞ ♥♦♠✐♥❛❧ ❡①❝❤❛♥❣❡ r❛t❡ ♠♦✈❡♠❡♥ts t♦ ❝♦♥s✉♠❡r ♣r✐❝❡s✳

❆❧❧ ✐♠♣♦rt✐♥❣ ✜r♠s ✇❤✐❝❤ r❡✲♦♣t✐♠✐③❡ ✐♥ ♣❡r✐♦❞ t ❢❛❝❡ t❤❡ s❛♠❡ ✐♥t❡rt❡♠♣♦r❛❧ ♣r♦❜❧❡♠ ❛♥❞ ❝❤♦s❡ t❤❡ s❛♠❡ ♣r✐❝❡XI,t(i) =XI,t✳ ❍❡♥❝❡✱ t❤❡ ❛❣❣r❡❣❛t❡ ♣r✐❝❡ ✐♥❞❡① ❢♦r ✐♠♣♦rt❡❞ ❣♦♦❞s s♦❧❞ ✐♥ t❤❡ ❞♦♠❡st✐❝ ♠❛r❦❡t ❡✈♦❧✈❡s ❛❝❝♦r❞✐♥❣ t♦✿

PI,t=



(1−θ_I)X

(1−ε)

I,t +θI PI,t−1

PI,t−1

PI,t−2

δI!1−ε

 

1/(1−ε)

,

✇❤❡r❡θI ✐s t❤❡ ♣r✐❝❡ r✐❣✐❞✐t② ♣❛r❛♠❡t❡r ❛♥❞ δI ✐s t❤❡ ✐♥❞❡①❛t✐♦♥ ♣❛r❛♠❡t❡r✳

❚❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ♦❢ r❡t❛✐❧ ✜r♠s ✐s ❛❧s♦ ❛♥❛❧♦❣♦✉s t♦ t❤❛t ♦❢ ❞♦♠❡st✐❝ ♣r♦❞✉❝❡rs✳ ❙✉❜❥❡❝t t♦ t❤❡ ❞❡♠❛♥❞ s❡q✉❡♥❝❡

CI,t+τ|t=

XI,t PI,t+τ

PI,t+τ−1

PI,t−1

δI!−ε

CI,t+τ,

✶✶_{❇❡❝❛✉s❡ ✇❡ ❛ss✉♠❡ ✐♥❝♦♠♣❧❡t❡ ♠❛r❦❡ts✱ t❤❡ r❡❛❞❡r ✇❤♦ ✐s ❢❛♠✐❧✐❛r ✇✐t❤ t❤✐s t②♣❡ ♦❢ ♠♦❞❡❧ ♠❛② q✉❡st✐♦♥ t❤❡ ✉s❡ ♦❢ t❤❡} st♦❝❤❛st✐❝ ❞✐s❝♦✉♥t ❢❛❝t♦r ❢♦r t❤❡ ✈❛❧✉❛t✐♦♥ ♦❢ ❢✉t✉r❡ str❡❛♠ ♦❢ ♣r♦✜ts✳ ❆❧t❤♦✉❣❤ ❛r❜✐tr❛r②✱ t❤✐s ❛ss✉♠♣t✐♦♥ ✐s ✐♥♥♦❝✉♦✉s✱ ❜❡❝❛✉s❡ ✐t ❞♦❡s ♥♦t ❛✛❡❝t t❤❡ ✜rst✲♦r❞❡r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧ t❤❛t ✇❡ s❤❛❧❧ r❡❧② ♦♥✳ ❲❡ ✇♦✉❧❞ ♦❜t❛✐♥ t❤❡ s❛♠❡ r❡s✉❧ts ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t ❢✉t✉r❡ ♣r♦✜ts ❛r❡ ❞✐s❝♦✉♥t❡❞ ❛t t❤❡ ♥♦♠✐♥❛❧ ✐♥t❡r❡st r❛t❡✳

✜r♠s ♠❛①✐♠✐③❡ t❤❡ ♣r❡s❡♥t ❞✐s❝♦✉♥t❡❞ ✈❛❧✉❡ ♦❢ ❡①♣❡❝t❡❞ ♣r♦✜ts

Et

∞

X

τ=0

θ_IτΘt,t+τCI,t+τ|t

"

XI,t

PI,t+τ−1

PI,t−1

δI

−St+τPt∗+τ

#

,

✇❤❡r❡P_t∗ ✐s t❤❡ ♣r✐❝❡ ♦❢ ✐♠♣♦rt❡❞ ♣r♦❞✉❝ts ✐♥ t❤❡ ✐♥t❡r♥❛t✐♦♥❛❧ ♠❛r❦❡t✳

✷✳✹ ▲❛✇ ♦❢ ♦♥❡ ♣r✐❝❡✱ ❡①❝❤❛♥❣❡ r❛t❡s ❛♥❞ t❡r♠s ♦❢ tr❛❞❡

❋♦r ❧❛t❡r ✉s❡✱ ❤❡r❡ ✇❡ ❞❡✜♥❡ s♦♠❡ ♦❜❥❡❝ts ♦❢ ✐♥t❡r❡st✳ ❚❤❡ r❡❛❧ ❡①❝❤❛♥❣❡ r❛t❡Qt ✐s ❣✐✈❡♥ ❜② t❤❡ r❛t✐♦ ❜❡t✇❡❡♥ ✐♥t❡r♥❛t✐♦♥❛❧ ❛♥❞ ❞♦♠❡st✐❝ ♣r✐❝❡s ❝♦♥✈❡rt❡❞ t♦ t❤❡ s❛♠❡ ❝✉rr❡♥❝②✿

Qt≡StPt∗/Pt.

❚❤❡ t❡r♠s ♦❢ tr❛❞❡ T oTt ❛r❡ ❞❡✜♥❡❞ ❛s t❤❡ r❡❧❛t✐✈❡ ♣r✐❝❡ ♦❢ ❇r❛③✐❧✬s ✐♠♣♦rts ❛♥❞ ❡①♣♦rts✿✶✷

T oTt=PI,t/PD,t.

▲❛st❧②✱ ✇❡ ❞❡✜♥❡ t❤❡ r❛t✐♦ ❜❡t✇❡❡♥ ✐♥t❡r♥❛t✐♦♥❛❧ ♣r✐❝❡s ❝♦♥✈❡rt❡❞ t♦ ❘❡❛✐s ❛♥❞ t❤❡ ♣r✐❝❡s ♦❢ ✐♠♣♦rt❡❞ ❣♦♦❞s ✐♥ t❤❡ ❞♦♠❡st✐❝ ♠❛r❦❡t✿

ΨI,t=StPt∗/PI,t.

❚❤❡ ✈❛r✐❛❜❧❡ΨI,t ♠❡❛s✉r❡s ❞❡✈✐❛t✐♦♥s ❢r♦♠ t❤❡ ▲❛✇ ♦❢ ❖♥❡ Pr✐❝❡ ❢♦r ✐♠♣♦rt❡❞ ❣♦♦❞s✳

✷✳✺ ▼♦♥❡t❛r② ♣♦❧✐❝②

▼♦♥❡t❛r② ♣♦❧✐❝② ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❛ ❞✐st✐♥❝t ✐♥t❡r❡st r❛t❡ r✉❧❡ ❢♦r ❡❛❝❤ r❡❣✐♠❡✳ ■♥ t❤❡ ✜rst ♣❛rt ♦❢ t❤❡ s❛♠♣❧❡✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❝✉rr❡♥❝② ♣❡❣ r❡❣✐♠❡✱ ✇❡ ❡①♣❧✐❝✐t❧② ♠♦❞❡❧ t❤❡ r❡❛❝t✐♦♥ ♦❢ t❤❡ ❈❇❇ t♦ ❞❡✈✐❛t✐♦♥s ♦❢ t❤❡ ♥♦♠✐♥❛❧ ❡①❝❤❛♥❣❡ r❛t❡ ❢r♦♠ ✐ts ❞❡s✐r❡❞ ❧❡✈❡❧✳ ■♥ t❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤❡ s❛♠♣❧❡✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ✐♥✢❛t✐♦♥ t❛r❣❡t✐♥❣ r❡❣✐♠❡✱ t❤❡ ♠❛✐♥ ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ t❤❡ ✐♥t❡r❡st r❛t❡ r✉❧❡ ✐s t❤❡ r❡s♣♦♥s❡ t♦ ❞❡✈✐❛t✐♦♥s ♦❢ ✐♥✢❛t✐♦♥ ❢r♦♠ t❛r❣❡t✳ ❋♦r ❡❛s❡ ♦❢ ❡①♣♦s✐t✐♦♥✱ ❤❡r❡ ✇❡ ❞❡s❝r✐❜❡ t❤❡ ✐♥t❡r❡st r❛t❡ r✉❧❡ ✐♥ ❛ ❤❡✉r✐st✐❝ ♠❛♥♥❡r✱ ❛♥❞ ♣♦st♣♦♥❡ t❤❡ ♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥s t♦ ❙❡❝t✐♦♥ ✷✳✽✱ ✇❤❡r❡ ✇❡ ♣r❡s❡♥t t❤❡ ✜rst✲♦r❞❡r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧✳

✷✳✺✳✶ ❈r❛✇❧✐♥❣ ♣❡❣ r❡❣✐♠❡

❉✉r✐♥❣ t❤❡ ✜rst ✶✷ ♠♦♥t❤s ❛❢t❡r t❤❡ ❧❛✉♥❝❤✐♥❣ ♦❢ t❤❡ ❘❡❛❧ P❧❛♥ ✐♥ ❏✉❧② ✶✾✾✹✱ t❤❡ ❇r❛③✐❧✐❛♥ ❘❡❛❧ ✇❛s ❛❧❧♦✇❡❞ t♦ ✢♦❛t ✇✐t❤✐♥ r❡❧❛t✐✈❡❧② ✇✐❞❡ ❜❛♥❞s✳ ■♥ ❏✉♥❡ ✶✾✾✺✱ t❤❡ ❈❇❇ ❛❞♦♣t❡❞ ❛ s②st❡♠ ♦❢ t✐❣❤t ❡①❝❤❛♥❣❡ r❛t❡ ❜❛♥❞s ✕ t❤❡ s♦✲❝❛❧❧❡❞ ✏♠✐♥✐✲❜❛♥❞s✑ ✕ ✇❤✐❝❤ ✇❡r❡ t❤❡♥ r❡❛❞❥✉st❡❞ ♣❡r✐♦❞✐❝❛❧❧② ❛t ❛♥ ❡ss❡♥t✐❛❧❧②

✶✷_{◆♦t❡ t❤❛t t❤✐s ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ♠♦r❡ ✉s✉❛❧ t❡r♠s ♦❢ tr❛❞❡ ♠❡❛s✉r❡✳ ❲❡ ❛❞♦♣t t❤✐s ❝♦♥✈❡♥t✐♦♥ t♦ ♠❛❦❡ ✐t ❡❛s✐❡r t♦} ❝♦♠♣❛r❡ ♦✉r r❡s✉❧ts t♦ t❤♦s❡ ♦❢ ❏✉st✐♥✐❛♥♦ ❛♥❞ Pr❡st♦♥ ✭✷✵✶✵✮✳

1.25

1 15 1.20

1.10 1.15

1.05

0.95 1.00

0.90

n-95

g-95 ct-9

5

ec-95 b-96 pr-96 n-96 g-96 ct-9

6

ec-96 b-97 pr-97 n-97 g-97 ct-9

7

ec-97 b-98 pr-98 n-98 g-98 ct-9

8

ec-98

Ju Au Oc De Fe Ap Ju Au Oc De Fe Ap Ju Au Oc De Fe Ap Ju Au Oc De

❋✐❣✉r❡ ✶✿ ❊①❝❤❛♥❣❡ r❛t❡ ♠✐♥✐✲ ❛♥❞ ♠❛❝r♦✲❜❛♥❞s ✭s♦❧✐❞ ❛♥❞ ❞❛s❤❡❞ ❧✐♥❡s✱ r❡s♣❡❝t✐✈❡❧②✮ ❛♥❞ t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ❡①❝❤❛♥❣❡ r❛t❡ ✐♥ ❘✩✴❯❙✩ ✭❜❧✉❡ ❞♦ts✮✳

❞❡t❡r♠✐♥✐st✐❝ ❞❡✈❛❧✉❛t✐♦♥ ♣❛❝❡✳✶✸ _{■♥ ♦r❞❡r t♦ ❦❡❡♣ t❤❡ ♥♦♠✐♥❛❧ ❡①❝❤❛♥❣❡ r❛t❡ ✇✐t❤✐♥ t❤❡ s♣❡❝✐✜❡❞ ❧✐♠✐ts✱}

t❤❡ ❈❇❇ r❡s♦rt❡❞ t♦ ✐♥t❡r✈❡♥t✐♦♥s ✐♥ t❤❡ ❢♦r❡✐❣♥ ❡①❝❤❛♥❣❡ ♠❛r❦❡t ❛♥❞ ❝❤❛♥❣❡s ✐♥ t❤❡ ♣♦❧✐❝② r❛t❡ ✭✐♥ ❛ ❝♦♥t❡①t ♦❢ ✐♠♣❡r❢❡❝t ❝❛♣✐t❛❧ ♠♦❜✐❧✐t②✮✳ ❋✐❣✉r❡ ✶ s❤♦✇s t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ♥♦♠✐♥❛❧ ❡①❝❤❛♥❣❡ r❛t❡ ✐♥ ❘✩✴❯❙✩✱ ♦❢ t❤❡ ✉♣♣❡r ❛♥❞ ❧♦✇❡r ❧✐♠✐ts ♦❢ t❤❡ ♠✐♥✐✲❜❛♥❞s ❛♥❞ ♦❢ t❤❡ ♠❛❝r♦✲❜❛♥❞s ❢r♦♠ ❏✉♥❡ ✶✾✾✺ t♦ ❉❡❝❡♠❜❡r ✶✾✾✽✳

❲❡ ❢♦❧❧♦✇ ❈úr❞✐❛ ❛♥❞ ❋✐♥♦❝❝❤✐❛r♦ ✭✷✵✶✸✮ ✐♥ ❛ss✉♠✐♥❣ t❤❛t ♠♦♥❡t❛r② ♣♦❧✐❝② ❞✉r✐♥❣ t❤❡ ❝r❛✇❧✐♥❣ ♣❡❣ r❡❣✐♠❡ ✐s ❞❡s❝r✐❜❡❞ ❜② ❛ st❛♥❞❛r❞ ✐♥t❡r❡st r❛t❡ r✉❧❡✳ ▼♦r❡ s♣❡❝✐✜❝❛❧❧②✱ ✐t ❢❡❛t✉r❡s t❤❡ ✉s✉❛❧ r❡s♣♦♥s❡s t♦ ✐♥✢❛t✐♦♥ ❛♥❞ ❡❝♦♥♦♠✐❝ ❛❝t✐✈✐t②✱ ❜✉t ✐s ♠♦❞✐✜❡❞ t♦ ✐♥❝❧✉❞❡ ❞❡✈✐❛t✐♦♥s ♦❢ t❤❡ ♥♦♠✐♥❛❧ ❡①❝❤❛♥❣❡ r❛t❡ ❢r♦♠ t❤❡ t❛r❣❡t s♣❡❝✐✜❡❞ ❜② t❤❡ ❈❇❇✳ ❚❤✐s ❛❧❧♦✇s t❤❡ ❈❇❇ t♦ r❡❛❝t t♦ ❡①❝❤❛♥❣❡ r❛t❡ ♣r❡ss✉r❡s✱ ❛♥❞ ✐s ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ♥♦♠✐♥❛❧ ❡①❝❤❛♥❣❡ r❛t❡ ✐♥ t❤❛t ♣❡r✐♦❞✳

✷✳✺✳✷ ■♥✢❛t✐♦♥ t❛r❣❡t✐♥❣ r❡❣✐♠❡

▼♦♥❡t❛r② ♣♦❧✐❝② ✐♥ t❤❡ ✐♥✢❛t✐♦♥ t❛r❣❡t✐♥❣ r❡❣✐♠❡ ❢♦❧❧♦✇s ❛ st❛♥❞❛r❞ ✐♥t❡r❡st r❛t❡ r✉❧❡ ✇✐t❤ ❛ s♠❛❧❧ ♠♦❞✐✜❝❛t✐♦♥✳ ❉❡s♣✐t❡ t❤❡ ✢♦❛t✐♥❣ ❡①❝❤❛♥❣❡ r❛t❡✱ t❤❡ ❈❇❇ ✐s ❛❧❧♦✇❡❞ t♦ r❡s♣♦♥❞ t♦ ❝❤❛♥❣❡s ✐♥ t❤❡ ♥♦♠✐♥❛❧ ❡①❝❤❛♥❣❡ r❛t❡✳ ❖t❤❡r t❤❛♥ t❤❛t✱ t❤❡ ✐♥t❡r❡st r❛t❡ r❡s♣♦♥❞s t♦ ❡❝♦♥♦♠✐❝ ❛❝t✐✈✐t② ❛♥❞ ❞❡✈✐❛t✐♦♥s ♦❢ ✐♥✢❛t✐♦♥ ❢r♦♠ ✐ts t❛r❣❡t✳

✷✳✻ ❋♦r❡✐❣♥ ❜❧♦❝❦

❚❤❡ ❞♦♠❡st✐❝ ❡❝♦♥♦♠② ✐s ❛ss✉♠❡❞ t♦ ❜❡ s♠❛❧❧ ❡♥♦✉❣❤ ♥♦t t♦ ❛✛❡❝t t❤❡ ✇♦r❧❞ ❡❝♦♥♦♠②✳ ❋♦r s✐♠♣❧✐❝✐t②✱ t❤❡ ❧❛tt❡r ✐s ❛ss✉♠❡❞ t♦ ❡✈♦❧✈❡ ❛❝❝♦r❞✐♥❣ t♦ ❛ ✜rst✲♦r❞❡r ✈❡❝t♦r ❛✉t♦r❡❣r❡ss✐✈❡ ♠♦❞❡❧ ✭❱❆❘✮✳✶✹ _❚❤❡

✶✸_{❚❤❡ s♦✲❝❛❧❧❡❞ ✏♠❛❝r♦✲❜❛♥❞s✑ ❝♦♥t✐♥✉❡❞ t♦ ❡①✐st✱ ❜✉t ❧♦st ❛♥② ♣r❛❝t✐❝❛❧ r❡❧❡✈❛♥❝❡✳}

✶✹_{❚❤❡ ❱❆❘✭✶✮ ❝♦❡✣❝✐❡♥ts ❛r❡ ❡st✐♠❛t❡❞ s❡♣❛r❛t❡❧②✱ ❛♥❞ ❦❡♣t ✜①❡❞ ❞✉r✐♥❣ t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ❉❙●❊ ♠♦❞❡❧✳}

✈❛r✐❛❜❧❡s ✐♥❝❧✉❞❡❞ ✐♥ t❤❡ ❱❆❘ ❛r❡ ♦✉t♣✉tY∗

t ✱ ✐♥✢❛t✐♦♥πt∗ ≡log(Pt∗/Pt∗−1) ❛♥❞ t❤❡ ❢♦r❡✐❣♥ ✐♥t❡r❡st r❛t❡

i∗_t ≈log(R∗_t)✳ ❚❤❡ ❱❆❘ s❤♦❝❦s ❛r❡ ❞❡♥♦t❡❞ ❜② ε_y∗✱ε∗_π ❛♥❞ ε∗_i✳

❚♦ ❛❧❧♦✇ ✐❞❡♥t✐✜❝❛t✐♦♥ ♦❢ ❛ ❢♦r❡✐❣♥ ♠♦♥❡t❛r② s❤♦❝❦✱ ✇❡ ✐♠♣♦s❡ t❤❡ ✉s✉❛❧ ❈❤♦❧❡s❦② ♦r❞❡r✐♥❣✱ ✇✐t❤ t❤❡ ❢♦r❡✐❣♥ ✐♥t❡r❡st r❛t❡ ♣❧❛❝❡❞ ❧❛st✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❱❆❘✭✶✮ ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❛s

A0

  

Y_t∗ π∗

t i∗_t

  =A1

  

Y_t∗₋₁ π∗

t−1

i∗_t−1

  +   

ε∗_y ε∗

π ε∗_i

  ,

✇❤❡r❡ t❤❡ ❝♦❡✣❝✐❡♥t ♠❛tr✐❝❡s ❛r❡

A0=

  

1 0 0

a0,πy 1 0

a0,iy a0,iπ 1

 

 A1=   

a1,yy a1,yπ a1,yi a1,πy a1,ππ a1,πi a1,iy a1,iπ a1,ii

  .

✷✳✼ ●❡♥❡r❛❧ ❡q✉✐❧✐❜r✐✉♠

❊q✉✐❧✐❜r✐✉♠ ✐♥ t❤❡ ♠❛r❦❡t ❢♦r ❞♦♠❡st✐❝ ❣♦♦❞s ❛♥❞ s❡r✈✐❝❡s r❡q✉✐r❡s ❡q✉❛❧✐t② ❜❡t✇❡❡♥ ❞♦♠❡st✐❝ ♣r♦❞✉❝t✐♦♥ ❛♥❞ t❤❡ s✉♠ ♦❢ ❞♦♠❡st✐❝ ❝♦♥s✉♠♣t✐♦♥ ❛♥❞ ❡①♣♦rts✿

Yt=CD,t+CD,t∗ ,

✇❤❡r❡ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ✐♥t❡r♥❛t✐♦♥❛❧ ❞❡♠❛♥❞ ❢♦r ❞♦♠❡st✐❝ ♣r♦❞✉❝ts ✐s ❣✐✈❡♥ ❜②

C_D,t∗ =

PD,t/St P∗

t

−η

Y_t∗.

❚❤❡ ❡q✉❛t✐♦♥ ❛❜♦✈❡ r❡✢❡❝ts t❤❡ ❛ss✉♠♣t✐♦♥✱ ❛❧r❡❛❞② ❡♠❜❜❡❞❡❞ ✐♥ ❡q✉❛t✐♦♥ ✭✺✮✱ t❤❛t t❤❡ ❡①♣♦rt ♣r✐❝❡s ♦❢ ❞♦♠❡st✐❝ ♣r♦❞✉❝ts ❛r❡ t❤❡ s❛♠❡ ❛s t❤❡ ♣r✐❝❡s ✐♥ t❤❡ ❞♦♠❡st✐❝ ♠❛r❦❡t✱ ❝♦♥✈❡rt❡❞ t♦ ❢♦r❡✐❣♥ ❝✉rr❡♥❝② ✉s✐♥❣ t❤❡ ♥♦♠✐♥❛❧ ❡①❝❤❛♥❣❡ r❛t❡ ❢♦r ❡❛❝❤ ♣❡r✐♦❞✳

❆❞❞✐t✐♦♥❛❧❧②✱ ✇❡ ❛ss✉♠❡ t❤❛t ❞♦♠❡st✐❝ ❜♦♥❞s ❛r❡ ✐♥ ③❡r♦ ♥❡t s✉♣♣❧②✱ s♦ t❤❛t Dt = 0❢♦r ❛❧❧ ♣❡r✐♦❞s✳ ❚❤❡ r❡♠❛✐♥✐♥❣ ❡q✉✐❧✐❜r✐✉♠ ❝♦♥❞✐t✐♦♥s ❛r❡ st❛♥❞❛r❞✳

✷✳✽ ❋✐rst✲♦r❞❡r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧

❆s ✐♥ ♠♦st ♦❢ t❤❡ ❧✐t❡r❛t✉r❡ ♦♥ ❉❙●❊ ♠♦❞❡❧s✱ ✇❡ ✇♦r❦ ✇✐t❤ ❛ ✜rst✲♦r❞❡r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ❝♦♥❞✐t✐♦♥s ❛r♦✉♥❞ ❛ ♥♦♥✲st♦❝❤❛st✐❝ st❡❛❞② st❛t❡ ❝❤❛r❛❝t❡r✐③❡❞ ❜② ③❡r♦ ✐♥✢❛t✐♦♥ ✕ ❛♥❞ ❜❛❧❛♥❝❡❞ tr❛❞❡✳ ❚❤❡ ❝♦♠♣❧❡t❡ s❡t ♦❢ ❧♦❣✲❧✐♥❡❛r✐③❡❞ ❡q✉❛t✐♦♥s ✐s s❤♦✇♥ ✐♥ t❤❡ ❆♣♣❡♥❞✐①✳ ❆s ❛ ❣❡♥❡r❛❧ r✉❧❡✱ ❧♦✇❡r❝❛s❡ ❧❡tt❡rs ✐♥❞✐❝❛t❡ ❞❡✈✐❛t✐♦♥s ♦❢ t❤❡ r❡s♣❡❝t✐✈❡ ✈❛r✐❛❜❧❡s ❢r♦♠ ✐ts st❡❛❞② st❛t❡ ✈❛❧✉❡✳ ■♥ ♠♦st ❝❛s❡s✱ t❤❡ ❞❡✈✐❛t✐♦♥ ✐s ❧♦❣❛r✐t❤♠✐❝✱ ❜✉t ✐♥ s♦♠❡ ❝❛s❡s ✐t ✐s ✐♥ ❧❡✈❡❧s✳

❘❡❣❛r❞✐♥❣ ♠♦♥❡t❛r② ♣♦❧✐❝②✱ ❛s ❞❡t❛✐❧❡❞ ✐♥ ❙❡❝t✐♦♥ ✷✳✺ t❤❡ ♠♦❞❡❧ s♣❡❝✐✜❡s ♦♥❡ ✐♥t❡r❡st r❛t❡ r✉❧❡ ❢♦r ❡❛❝❤ r❡❣✐♠❡✳ ❚❤❡s❡ ❛r❡ t❤❡ ♦♥❧② ❡q✉❛t✐♦♥s t❤❛t ♠❛② ✈❛r② ❛❝r♦ss r❡❣✐♠❡s✳ ❚❤❡ r✉❧❡ ❢♦r t❤❡ ❝r❛✇❧✐♥❣ ♣❡❣

r❡❣✐♠❡ ✐s ❣✐✈❡♥ ❜②

it=ρF Xi,1 it−1+ρi,F X2 it−2+ (1−ρF Xi,1 −ρF Xi,2 )(λF Xπ πt+λF Xy yt+λF Xs (st−sc,t)) +εF Xi,t ,

✇❤❡r❡ πt ≡ log(Pt/Pt−1) ❡ sc,t ❞❡♥♦t❡s t❤❡ ❡①❝❤❛♥❣❡ r❛t❡ t❛r❣❡t✳ ❲❡ ✐♥❝❧✉❞❡❞ ❛♥ ✐♥t❡r❡st r❛t❡ ❧❛❣ t♦ ❛❧❧♦✇ ❢♦r ♣♦❧✐❝② ✐♥❡rt✐❛✳ ❋✐♥❛❧❧②✱εF X_i,t ✐s ❛ s❤♦❝❦ ✐♥ t❤❡ ✐♥t❡r❡st r❛t❡ r✉❧❡✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ❛ ♥♦♥✲s②st❡♠❛t✐❝ ❝♦♠♣♦♥❡♥t ♦❢ ♠♦♥❡t❛r② ♣♦❧✐❝②✳

❋♦r t❤❡ ✐♥✢❛t✐♦♥ t❛r❣❡t✐♥❣ r❡❣✐♠❡✱ ✇❡ ❞❡✜♥❡❞ t❤❡ r✉❧❡✿✶✺

it=ρITi,1it−1+ρi,IT2it−2+ (1−ρITi,1 −ρITi,2)[λITπ (πt−πm,t) +λyITyt+λITs ∆st] +εITi,t,

✇❤❡r❡∆✐s t❤❡ ✜rst✲❞✐✛❡r❡♥❝❡ ♦♣❡r❛t♦r ❛♥❞πm,t ✐s t❤❡ ✐♥✢❛t✐♦♥ t❛r❣❡t✳

✷✳✾ ❙❤♦❝❦s

❚❤❡ ♠♦❞❡❧ ❢❡❛t✉r❡s ❡✐❣❤t str✉❝t✉r❛❧ s❤♦❝❦s✶✻_{✕ ✜✈❡ ♦❢ t❤❡♠ ✐♥ t❤❡ ❞♦♠❡st✐❝ ❡❝♦♥♦♠② ❛♥❞ t❤r❡❡ r❡❧❛t❡❞ t♦}

t❤❡ ❢♦r❡✐❣♥ ❜❧♦❝❦✳ ❚❤❡ str✉❝t✉r❛❧ s❤♦❝❦s r❡❧❛t❡❞ t♦ t❤❡ ❞♦♠❡st✐❝ ❡❝♦♥♦♠② ❛r❡ r❡❧❛t❡❞ t♦ ♠♦♥❡t❛r② ♣♦❧✐❝② ✭i✮✱ ♣r❡❢❡r❡♥❝❡s ✭γ✮✱ t❡❝❤♥♦❧♦❣② ✭a✮✱ r✐s❦ ♣r❡♠✐✉♠ ✭φ✮✱ ❛♥❞ ❝♦st ♦❢ ✐♠♣♦rt❡❞ ❣♦♦❞s ✭cp✮✳✶✼ _{❚❤❡ ❧❛st ❢♦✉r}

s❤♦❝❦s ❢♦❧❧♦✇ ✜rst✲♦r❞❡r ❛✉t♦r❡❣r❡ss✐✈❡ ♣r♦❝❡ss❡s ✭❆❘✭✶✮✮✱ ✇❤❡r❡❛s t❤❡ ❞♦♠❡st✐❝ ♠♦♥❡t❛r② ♣♦❧✐❝② s❤♦❝❦ ❛♥❞ ❢♦r❡✐❣♥ s❤♦❝❦s ❛r❡ ❛ss✉♠❡❞ t♦ ❜❡ ✐✳✐✳❞✳✿

at=ρaat−1+σaǫa,t,

γt=ργγt−1+σγǫγ,t,

εcp,t=ρcpεcp,t−1+σcpǫcp,t,

φt=ρφφt−1+σφǫφ,t,

εr_i,t=σi,rǫi,t, r=F X, IT,

εy∗_,t=σ_y∗ǫ_y∗_,t, ε_π∗_,t=σ_π∗ǫ_π∗_,t, ε_i∗_,t=σ_i∗ǫ_i∗_,t.

❚❤❡ǫ✐♥♥♦✈❛t✐♦♥s ❛r❡ ❝r♦ss✲s❡❝t✐♦♥❛❧❧② ✐♥❞❡♣❡♥❞❡♥t ✐✳✐✳❞✳ N(0,1)✳

✶✺_{❚❤✐s r✉❧❡ ✐s s✐♠✐❧❛r t♦ t❤♦s❡ ❢♦✉♥❞ ✐♥ ❈úr❞✐❛ ❛♥❞ ❋✐♥♦❝❝❤✐❛r♦ ✭✷✵✶✸✮ ❢♦r t❤❡ ❝❛s❡ ♦❢ ❙✇❡❞❡♥✱ ❉❡❧ ◆❡❣r♦ ❛♥❞ ❙❝❤♦r❢❤❡✐❞❡} ✭✷✵✵✾✮ ❢♦r ❈❤✐❧❡✱ ❛♥❞ ❏✉st✐♥✐❛♥♦ ❛♥❞ Pr❡st♦♥ ✭✷✵✶✵✮ ❢♦r ❆✉str❛❧✐❛✱ ❈❛♥❛❞❛ ❛♥❞ ◆❡✇ ❩❡❛❧❛♥❞✳

✶✻_{❚❤✐s ❝♦✉♥t ❧❡❛✈❡s ❛s✐❞❡ s❤♦❝❦s t♦ t❤❡ ✐♥✢❛t✐♦♥ ❛♥❞ ❡①❝❤❛♥❣❡ r❛t❡ t❛r❣❡ts✱ ✇❤✐❝❤ ❛r❡ ♦♥❧② ✉s❡❞ ✐♥ t❤❡ ❝♦✉♥t❡r❢❛❝t✉❛❧} ❡①♣❡r✐♠❡♥ts✳ ❋♦r ❞❡t❛✐❧s✱ s❡❡ ❙❡❝t✐♦♥s ✸✳✶ ❡ ✺✳

✶✼_{❲❡ ❢♦❧❧♦✇ ❏✉st✐♥✐❛♥♦ ❛♥❞ Pr❡st♦♥ ✭✷✵✶✵✮ ❛♥❞ ❛♣♣❡♥❞ ❛ ❝♦st✲♣✉s❤ s❤♦❝❦ t♦ t❤❡ P❤✐❧❧✐♣s ❝✉r✈❡ ❢♦r ✐♠♣♦rt❡❞ ❣♦♦❞s} ✐♥✢❛t✐♦♥✳

✷✳✶✵ ❙♦❧✉t✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧ ❛♥❞ st❛t❡✲s♣❛❝❡ r❡♣r❡s❡♥t❛t✐♦♥

❚❤❡ ♠♦❞❡❧ ❡q✉❛t✐♦♥s ✐♥ t❤❡✐r ✭❧♦❣✲✮❧✐♥❡❛r ❢♦r♠ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s

Et{fr(Υt+1,Υt,Υt−1, ǫt+1, ǫt;θ)}= 0, r=F X, IT, ✭✻✮

✇❤❡r❡θ✐s ❛ ✈❡❝t♦r ❝♦❧❧❡❝t✐♥❣ ❛❧❧ str✉❝t✉r❛❧ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ❉❙●❊ ♠♦❞❡❧✱ Υt✐s ❛ ✈❡❝t♦r ❝♦♥t❛✐♥✐♥❣ ✐ts ✈❛r✐❛❜❧❡s ❛♥❞ ǫt ✐s ❛ ✈❡❝t♦r ✇✐t❤ t❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ str✉❝t✉r❛❧ ✐♥♥♦✈❛t✐♦♥s✳ ❘❡❣❛r❞✐♥❣ t❤❡ ❛ss✉♠♣t✐♦♥ ❛❜♦✉t ❡①♣❡❝t❛t✐♦♥s ❢♦r♠❛t✐♦♥ ✉♥❞❡r❧②✐♥❣ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✐♥ ✭✻✮✱ r❡❝❛❧❧ t❤❛t t❤❡ tr❛♥s✐t✐♦♥ ❜❡t✇❡❡♥ r❡❣✐♠❡s ✐s ❛ss✉♠❡❞ t♦ ❜❡ ✉♥❛♥t✐❝✐♣❛t❡❞✳

❲❡ s♦❧✈❡ t❤❡ ♠♦❞❡❧ ✉s✐♥❣ t❤❡ ♠❡t❤♦❞ ♣r♦♣♦s❡❞ ❜② ❙✐♠s ✭✷✵✵✷✮✱ r❡str✐❝t✐♥❣ t❤❡ ♣❛r❛♠❡t❡r s♣❛❝❡ t♦ ❡①❝❧✉❞❡ ❝❛s❡s ✇✐t❤ ♠✉❧t✐♣❧❡ s♦❧✉t✐♦♥s ♦r ♥♦ ✭❜♦✉♥❞❡❞✮ s♦❧✉t✐♦♥✳ ❚❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s✿

Υt=Ar(θ)Υt−1+Br(θ)ǫt, r=F X, IT. ✭✼✮

❚❤❡s❡ ❡q✉❛t✐♦♥s ❝♦♠♣r✐s❡ t❤❡ r❡❞✉❝❡❞✲❢♦r♠ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧✱ ✇❤✐❝❤ t❛❦❡s t❤❡ ❢♦r♠ ♦❢ ❛ ❱❆❘✭✶✮✳ ❚❤❡② ❛r❡ t❤❡ s♦✲❝❛❧❧❡❞ tr❛♥s✐t✐♦♥ ❡q✉❛t✐♦♥s✳

❚❤❡ ♥❡①t st❡♣ t♦✇❛r❞ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧ ✐s t♦ ❝♦♥♥❡❝t ✈❛r✐❛❜❧❡s ✐♥ Υt t♦ ♦❜s❡r✈❛❜❧❡s t❤r♦✉❣❤ s♦✲❝❛❧❧❡❞ ♦❜s❡r✈❛t✐♦♥ ✭♦r ♠❡❛s✉r❡♠❡♥t✮ ❡q✉❛t✐♦♥s✳ ❚❤✐s ❝♦♠♣❧❡t❡s t❤❡ st❛t❡✲s♣❛❝❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧✱ ❝♦♠♣♦s❡❞ ♦❢ tr❛♥s✐t✐♦♥ ❛♥❞ ♦❜s❡r✈❛t✐♦♥ ❡q✉❛t✐♦♥s✳

❚❤❡ r❡❣✐♠❡✲s♣❡❝✐✜❝ tr❛♥s✐t✐♦♥ ❡q✉❛t✐♦♥s ❛r❡ ❣✐✈❡♥ ❜② t❤❡ r❡❞✉❝❡❞✲❢♦r♠ ❱❆❘✭✶✮ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧ ✭❡q✉❛t✐♦♥ ✼✮✳ ❚❤❡ ♦❜s❡r✈❛t✐♦♥ ❡q✉❛t✐♦♥s ❛r❡ t❤❡ s❛♠❡ ❢♦r ❜♦t❤ r❡❣✐♠❡s ❛♥❞ ❛r❡ ❣✐✈❡♥ ❜②

Υobs_t =CΥt,

✇❤❡r❡Υobs_t ✐s t❤❡ ✈❡❝t♦r ♦❢ ♦❜s❡r✈❛❜❧❡s✱ t♦ ❜❡ ❞✐s❝✉ss❡❞ ✐♥ ❙❡❝t✐♦♥ ✸✳✶✳

●✐✈❡♥ ❡♥♦✉❣❤ str✉❝t✉r❛❧ s❤♦❝❦s ❛♥❞ ♠❡❛s✉r❡♠❡♥t ❡rr♦rs ✐♥ t❤❡ ♦❜s❡r✈❛t✐♦♥ ❡q✉❛t✐♦♥s✱ t❤❡ ❑❛❧♠❛♥ ✜❧t❡r ❝❛♥ ❜❡ ✉s❡❞ t♦ ❝♦♥str✉❝t t❤❡ ❧✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥ ❢♦r t❤❡ ✈❡❝t♦r ♦❢ ♦❜s❡r✈❛❜❧❡s Υobs_t ✱ ❢♦r ❛ ❣✐✈❡♥ ✈❡❝t♦r ♦❢ ♣❛r❛♠❡t❡rsθ✳✶✽

✸ ❊st✐♠❛t✐♦♥

❖✉r ♠♦❞❡❧ ✐s ❝♦♠♣♦s❡❞ ♦❢ t✇♦ s❡ts ♦❢ ❡q✉❛t✐♦♥s✱ ♦♥❡ ❢♦r t❤❡ ❝r❛✇❧✐♥❣ ♣❡❣ r❡❣✐♠❡ ❛♥❞ t❤❡ ♦t❤❡r ❢♦r t❤❡ ✐♥✢❛t✐♦♥ t❛r❣❡t✐♥❣ r❡❣✐♠❡✳ ❚❤❡ ❣♦❛❧ ♦❢ t❤❡ ❡st✐♠❛t✐♦♥ ✐s t♦ ❡①tr❛❝t ✐♥❢♦r♠❛t✐♦♥ ❢r♦♠ t❤❡ ❞❛t❛ ❛❜♦✉t t❤❡ str✉❝t✉r❛❧ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ❡❝♦♥♦♠②✱ ❣r♦✉♣❡❞ ✐♥t♦ t❤❡ ✈❡❝t♦rθ✱ t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ❝❤❛♥❣❡ ✐♥ r❡❣✐♠❡ t❤❛t ❤❛♣♣❡♥❡❞ ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ ✶✾✾✾✳

■♥ t❤❡ ❡❝♦♥♦♠❡tr✐❝s ❧✐t❡r❛t✉r❡ ♦♥ ✏str✉❝t✉r❛❧ ❜r❡❛❦s✑✱ ❛ ❞✐st✐♥❝t✐♦♥ ✐s ♠❛❞❡ ❜❡t✇❡❡♥ ❛ ✏♣✉r❡ ❜r❡❛❦✑✱

✶✽_{❙❡❡✱ ❢♦r ❡①❛♠♣❧❡✱ ❍❛♠✐❧t♦♥ ✭✷✵✵✹✮✳}