st t st ts r - - PowerPoint PPT Presentation

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍

❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍✲t②♣❡ ♣r♦❝❡ss❡s

❈❤r✐st✐❛♥ ❋r❛♥❝q ❏❡❛♥✲▼✐❝❤❡❧ ❩❛❦♦ï❛♥

❈❘❊❙❚ ❛♥❞ ❯♥✐✈❡rs✐t② ♦❢ ▲✐❧❧❡✱ ❋r❛♥❝❡

◆❡✇ ❘❡s✉❧ts ♦♥ ❚✐♠❡ ❙❡r✐❡s ❛♥❞ t❤❡✐r ❙t❛t✐st✐❝❛❧ ❆♣♣❧✐❝❛t✐♦♥s

❈■❘▼✱ ✶✹✲✶✽ ❙❡♣t❡♠❜❡r✱ ✷✵✷✵

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍

❋✐♥❛♥❝✐❛❧ t✐♠❡ s❡r✐❡s

❆ st❛♥❞❛r❞ ❛ss✉♠♣t✐♦♥ ✐s t❤❛t ♣r✐❝❡s ❛r❡ ♥♦♥st❛t✐♦♥❛r② ✇❤✐❧❡ r❡t✉r♥s ✭♦r ❧♦❣ r❡t✉r♥s✮ ❛r❡ ✭str✐❝t❧②✮ st❛t✐♦♥❛r②✳ ■t ✐s ❣❡♥❡r❛❧❧② ❛❞♠✐tt❡❞ t❤❛t ♠❛♥② ✜♥❛♥❝✐❛❧ r❡t✉r♥s s❡r✐❡s ❤❛✈❡ ❤❡❛✈② t❛✐❧❡❞ ♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥s✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ✐s ♥♦ ❝♦♠♠♦♥❧② ❛❝❝❡♣t❡❞ ❛ss✉♠♣t✐♦♥ ❝♦♥❝❡r♥✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ♦❢ s✉❝❤ r❡t✉r♥s✳

▼❛♥② s❡❛r❝❤❡rs ❛r❣✉❡ t❤❛t st♦❝❦ r❡t✉r♥s ♠✐❣❤t ♥♦t ❛❞♠✐t ✹t❤✲♦r❞❡r ♠♦♠❡♥ts ✭s❡❡ ❡✳❣✳ P♦❧✐t✐s ✭✷✵✵✼✮✮✱ ✇❤✐❧❡ s♦♠❡ ❡✈❡♥ q✉❡st✐♦♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ s❡❝♦♥❞✲♦r❞❡r ♠♦♠❡♥ts✳

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍

❊①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ✐s ❝❡♥tr❛❧ t♦ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s

■♥ ♣r❡s❡♥❝❡ ♦❢ ❤❡❛✈② t❛✐❧s✱ ♠❛♥② st❛t✐st✐❝❛❧ t♦♦❧s ❞❡✈❡❧♦♣❡❞ ❢♦r t❤❡ ❛♥❛❧②s✐s ♦❢ ✜♥❛♥❝✐❛❧ t✐♠❡ s❡r✐❡s ❜❡❝♦♠❡ ✐♥✈❛❧✐❞✳ ❋♦r ✐♥st❛♥❝❡✱ ✉s✐♥❣ t❤❡ ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧ ✐♥ r✐s❦ ❛♥❛❧②s✐s r❡q✉✐r❡s ✜♥✐t❡♥❡ss ♦❢ t❤❡ ✜rst ❛❜s♦❧✉t❡ ♠♦♠❡♥t✳ ▲♦♥❣✲r✉♥ ❤♦r✐③♦♥s ♣r❡❞✐❝t✐♦♥s ♦❢ t❤❡ sq✉❛r❡❞ r❡t✉r♥s r❡q✉✐r❡ ✜♥✐t❡ ✉♥❝♦♥❞✐t✐♦♥❛❧ ✈❛r✐❛♥❝❡ ♦❢ t❤❡ r❡t✉r♥s✱ ❛♥❞ t❤❡✐r ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s r❡q✉✐r❡ ✜♥✐t❡ ❢♦✉rt❤✲♦r❞❡r ♠♦♠❡♥ts✳ ❊st✐♠❛t✐♦♥ ♠❡t❤♦❞s ♠❛② ❛❧s♦ r❡q✉✐r❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ s♦♠❡ ♠♦♠❡♥ts✳

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍

❚❡st✐♥❣ st❛t✐♦♥❛r✐t② ♦❢ ✜♥❛♥❝✐❛❧ s❡r✐❡s

❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts✿ t❛❝❦❧❡❞ ✐♥ ❞✐✛❡r❡♥t ✇❛②s ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✱ ❛♠♦♥❣ ♦t❤❡rs

▲♦r❡t❛♥ ❛♥❞ P❤✐❧❧✐♣s ✭✶✾✾✹✮✿ ♥♦♥♣❛r❛♠❡tr✐❝ ♠❡t❤♦❞s ❢♦r t❡st✐♥❣ t❤❡ ❝♦♥st❛♥❝② ♦❢ t❤❡ ✉♥❝♦♥❞✐t✐♦♥❛❧ ✈❛r✐❛♥❝❡ ✇❤❡♥ t❤❡ ❢♦✉rt❤ ✉♥❝♦♥❞✐t✐♦♥❛❧ ♠♦♠❡♥t ✐s ✐♥✜♥✐t❡✳ ❊st✐♠❛t✐♦♥ ♦❢ t❤❡ t❛✐❧ ✐♥❞❡① ❢♦r ❞❡♣❡♥❞❡♥t ♦❜s❡r✈❛t✐♦♥s❀ ❡✳❣✳ ❍✐❧❧ ✭✷✵✶✺✮✳ ❉✇✐✈❡❞✐ ❛♥❞ ❙✉❜❜❛ ❘❛♦ ✭✷✵✶✶✮✿ ❆ t❡st ❢♦r s❡❝♦♥❞✲♦r❞❡r st❛t✐♦♥❛r✐t② ♦❢ ❛ t✐♠❡ s❡r✐❡s ❜❛s❡❞ ♦♥ t❤❡ ❞✐s❝r❡t❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠✳ ❚r❛♣❛♥✐ ✭✷✵✶✻✮✿ ❛ t❡st ❢♦r ✜♥✐t❡♥❡ss ♦❢ t❤❡ k✲t❤ ♠♦♠❡♥t ♦❢ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡❀ ❜❛s❡❞ ♦♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡✴❞✐✈❡r❣❡♥❝❡ ♦❢ s❛♠♣❧❡ ♠♦♠❡♥ts✳

❚❡st✐♥❣ str✐❝t st❛t✐♦♥❛r✐t② ✐♥ ●❆❘❈❍✲t②♣❡ ♠♦❞❡❧s✿

❏❡♥s❡♥ ❛♥❞ ❘❛❤❜❡❦ ✭✷✵✶✹❛✱ ✷✵✶✹❜✮✱ ❋❩ ✭✷✵✶✷✱ ✷✵✶✸✮✱ P❡❞❡rs❡♥ ❛♥❞ ❘❛❤❜❡❦ ✭✷✵✶✻✮✱ ▲✐✱ ❩❤❛♥❣✱ ❩❤✉ ❛♥❞ ▲✐♥❣ ✭✷✵✶✽✮✳

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍

❖✉t❧✐♥❡

✶

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

✷

❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

✶

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

✷

❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

▼♦❞❡❧✿ ❙t❛♥❞❛r❞ ●❆❘❈❍✭p, q✮

   ǫt = σtηt, (ηt) ✐✳✐✳❞✳, Eηt = 0, Eη2

t = 1,

σ2

t = ω0 + q i=1 α0iǫ2 t−i + p j=1 β0jσ2 t−j

◆❙❈ ❢♦r t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❡✈❡♥✲♦r❞❡r ♠♦♠❡♥ts ❞❡♣❡♥❞ ♦♥ t❤❡ ♠♦♠❡♥ts ηt ✭❡①❝❡♣t ❢♦r t❤❡ ✷♥❞ ♦r❞❡r✮✳

❬❙❡❡ ▲✐♥❣ ❛♥❞ ▼❝❆❧❡❡r ✭✷✵✵✷✮✱ ❈❤❡♥ ❛♥❞ ❆♥ ✭✶✾✾✽✮✱ ❍❡ ❛♥❞ ❚❡räs✈✐t❛ ✭✶✾✾✾✮❪

❙♦♠❡ ♦❢ t❤❡s❡ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡①♣❧✐❝✐t ✭❛❧❣❡❜r❛✐❝ ❢♦r♠✮✿ ✷♥❞✲♦r❞❡r ✭❢♦r ❛❧❧ p ❛♥❞ q✮❀ 2m✲t❤ ♦r❞❡r ✭p = q = 1✮

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

▼♦♠❡♥t r❡str✐❝t✐♦♥s ❢♦r t❤❡ ●❆❘❈❍✭✶✱✶✮

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Gaussian innovations α β E(ε2) < ∞ E(ε4) < ∞ E(ε6) < ∞ 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Student innovations β E(ε2) < ∞ E(ε4) < ∞ E(ε6) < ∞ ❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

• ❛✉ss✐❛♥ ◗▼▲❊ ♦❢ t❤❡ ●❆❘❈❍✭p, q✮
• ✐✈❡♥ ♦❜s❡r✈❛t✐♦♥s ǫ1, . . . , ǫn✱ ❛♥❞ ❛r❜✐tr❛r② ✐♥✐t✐❛❧ ✈❛❧✉❡s t❤❡
• ❛✉ss✐❛♥ ◗▼▲❊ ✐s ❞❡✜♥❡❞ ❜②

ˆ θn = arg min

θ∈Θ

1 n

n

• t=1

˜ ℓt(θ), ✇❤❡r❡ ˜ ℓt(θ) = ǫ2

t

˜ σ2

t (θ) + log ˜

σ2

t (θ).

❆ss✉♠♣t✐♦♥s ❢♦r t❤❡ ❈❆◆ ♦❢ t❤❡ ●❛✉ss✐❛♥ ◗▼▲❊✿

❆✶✿ θ0 ∈

• Θ ❛♥❞ Θ ✐s ❝♦♠♣❛❝t

❆✷✿ γ(A0) < 0✱ ❛♥❞ ❢♦r ❛❧❧ θ ∈ Θ✱ p

j=1 βj < 1

[γ(A0)✿ t♦♣✲▲②❛♣✉♥♦✈ ❡①♣♦♥❡♥t ♦❢ t❤❡ ●❆❘❈❍ ♠♦❞❡❧❪ ❆✸✿ η2

t ❤❛s ❛ ♥♦♥❞❡❣❡♥❡r❛t❡ ❞✐str✐❜✉t✐♦♥ ❛♥❞ Eη2 t = 1 ❛♥❞ Eη4 t < ∞

❆✹✿ ❚❤❡ ❧❛❣ ♣♦❧②♥♦♠✐❛❧s ✈❡r✐❢② st❛♥❞❛r❞ ❝♦♥❞✐t✐♦♥s

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

ˆ ηt = ǫt/ˆ σt

ˆ µr = 1 n

n

• t=1

|ˆ ηt|r, µr = E|ηt|r, ˆ µm = (ˆ µ2, ˆ µ4, . . . , ˆ µ2m)′, µm = (µ2, µ4, . . . , µ2m)′

❏♦✐♥t ❛s②♠♣t♦t✐❝ ♥♦r♠❛❧✐t② ♦❢ t❤❡ ♣❛r❛♠❡t❡r ❡st✐♠❛t♦r ❛♥❞ ❛ ✈❡❝t♦r ♦❢ r❡s✐❞✉❛❧s s❛♠♣❧❡ ♠♦♠❡♥ts

❯♥❞❡r ❆✶✲❆✹ ❛♥❞ ✐❢ µ4m < ∞ √n

• ˆ

θn − θ0

• √n(ˆ

µm − µm)

• L

→ N

• 0, Σm :=

(µ4 − 1)J−1 −θ0b′

m

−bmθ

′

Am

• ,

θ0 = (ω0, α01, . . . , α0q, 0, . . . , 0)′, J = E

• 1

σ4

t

∂σ2

t (θ0)

∂θ ∂σ2

t (θ0)

∂θ′

• ,

Am = (aij)1≤i,j≤m✱ bm = (bi)1≤i≤m✱ ✇✐t❤ aij = µ2(i+j) + µ2iµ2j[i + j + (µ4 − 1)ij − 1] −iµ2iµ2(j+1) − jµ2jµ2(i+1), 1 ≤ i, j ≤ m, bi = µ2i − µ2(i+1) + (µ4 − 1)iµ2i, 1 ≤ i ≤ m.

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

❘❡♠❛r❦s

✶ ❚❤❡ ❛s②♠♣t♦t✐❝ ✈❛r✐❛♥❝❡✲❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① Am ♦❢ t❤❡ ✈❡❝t♦r

♦❢ ❡♠♣✐r✐❝❛❧ ♠♦♠❡♥ts ♦❢ t❤❡ r❡s❝❛❧❡❞ r❡t✉r♥s ✐s ♠♦❞❡❧ ❢r❡❡ ✭❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ θ0✮ ❜✉t ♥♦t ❡st✐♠❛t✐♦♥ ❢r❡❡✳ ❚❤✐s ✐s ❞✉❡ t♦ t❤❡ r❡❧❛t✐♦♥ Ω′J−1Ω = 1

Ω = E

• 1

σ2

t

∂σ2

t (θ0)

∂θ

• , J = E
• 1

σ4

t

∂σ2

t (θ0)

∂θ ∂σ2

t (θ0)

∂θ′

• ✭s❡❡ ❋❩ ✭✷✵✶✸✮✮✳

✷ ❈❛s❡ m = 1 ❞❡❣❡♥❡r❛t❡ ˆ µ1 = 1 ✇❤❡♥❝❡ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡s ❛r❡ s✉❝❤ t❤❛t ❢♦r ❛♥② K > 0✱ K˜ σ2

t (ˆ

θn) = ˜ σ2

t (ˆ

θ

∗ n) ❢♦r s♦♠❡ ˆ

θ

∗ n ∈ Θ✳

❋♦r ♠♦r❡ ❣❡♥❡r❛❧ ✐♥✐t✐❛❧ ✈❛❧✉❡s✱ √n(ˆ µ2 − 1) → 0, ✐♥ ♣r♦❜❛❜✐❧✐t② ❛s n → ∞.

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

• ❆❘❈❍✭✶✱✶✮ ❝❛s❡✿

σ2

t = ω0 + α0ǫ2 t−1 + β0σ2 t−1

■❢ m ≥ 1 ✐s ❛♥ ✐♥t❡❣❡r✱ E(ǫ2m

t

) < ∞ ⇔

m

• i=0

m i

• αi

0βm−i

µ2i < 1 ▲❡t G(θ, µ) = m

i=0

m

i

• αiβm−iµ2i✳ ❯♥❞❡r t❤❡ ♣r❡✈✐♦✉s

❛ss✉♠♣t✐♦♥s √n{G(ˆ θ, ˆ µm) − G(θ0, µm)} L → N(0, σ2

m),

✇❤❡r❡ σ2

m = ∂G(θ0, µm)

∂(θ′, µ′) Σm ∂G(θ0, µm) ∂ θ

µ

• .

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

❚❡st✐♥❣ ♣r♦❜❧❡♠s

❈♦♥s✐❞❡r t❤❡ 2m✲t❤ ♦r❞❡r st❛t✐♦♥❛r✐t② ♣r♦❜❧❡♠s H0 : E(ǫ2m

t

) < ∞ ❛❣❛✐♥st H1 : E(ǫ2m

t

) = ∞, ❛♥❞ H∗

0 :

E(ǫ2m

t

) = ∞ ❛❣❛✐♥st H∗

1 :

E(ǫ2m

t

) < ∞.

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

❚❡st ♦❢ 2m✲t❤ ♦r❞❡r ♠♦♠❡♥t ❢♦r t❤❡ ●❆❘❈❍✭✶✱✶✮

❚❡st ♦❢ H0 ✭r❡s♣✳ H∗

0✮ ❛t t❤❡ ❛s②♠♣t♦t✐❝ ❧❡✈❡❧ α ∈ (0, 1)

❉❡✜♥❡❞ ❜② t❤❡ r❡❥❡❝t✐♦♥ r❡❣✐♦♥ {Tn > Φ−1(1 − α)}, (r❡s♣✳ {Tn < Φ−1(α)}), ✇❤❡r❡ Tn = √n m

i=0

m

i

• ˆ

αi

n ˆ

βm−i

n

ˆ µ2i − 1

• ˆ

σm , ˆ σ2

m = ∂G(ˆ

θn, ˆ µm) ∂(θ′, µ′) ˆ Σm ∂G(ˆ θn, ˆ µm) ∂ θ

µ

• ❛♥❞ ˆ

Σm ✐s ❛ ❝♦♥s✐st❡♥t ❡st✐♠❛t♦r ♦❢ Σm.

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

❘❡♠❛r❦s

❚❤❡ t❡st ✐s ❝♦♥str✉❝t❡❞ ❢♦r t❤❡ ❝❧♦s✉r❡ ♦❢ t❤❡ ♥✉❧❧ ❛ss✉♠♣t✐♦♥ H0 : m

i=0

m

i

• αi

0βm−i

µ2i ≤ 1✳ ❚❤❡ ❛s②♠♣t♦t✐❝ r❡❣✐♦♥ s❛t✐s✜❡s sup

H0

lim

n→∞ P{Tn > Φ−1(1 − α)} = α

❚❡st✐♥❣ t❤❡ ✷♥❞✲♦r❞❡r ♠♦♠❡♥t ❝♦♥❞✐t✐♦♥✿ α0 + β0 < 1 ■♥ t❤✐s ❝❛s❡✱ ✇✐t❤ e = (0, 1, 1)′✱ Tn = √n(ˆ α + ˆ β − 1) {(ˆ µ4 − 1)e′ ˆ J

−1e}1/2 .

❆ ❜♦♦tstr❛♣ ♣r♦❝❡❞✉r❡ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❛✈♦✐❞ ❡st✐♠❛t✐♥❣ t❤❡ ❛s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥✳

❘❡s❛♠♣❧✐♥❣ s❝❤❡♠❡ ❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

✶

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

✷

❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

❘❡♣❛r❛♠❡tr✐③❛t✐♦♥

Pr♦✈✐❞❡❞ t❤❛t E|ηt|r < ∞✱ t❤❡ ●❆❘❈❍✭p, q✮ ♠♦❞❡❧ ❝❛♥ ❜❡ ❡q✉✐✈❛❧❡♥t❧② r❡✇r✐tt❡♥ ❛s ǫt = σt(θ(r)

0 )η(r) t ,

E|η(r)

t |r = 1,

✇❤❡r❡ η(r)

t

= ηt/{E|ηt|r}1/r✳ ▲✐♥❦ ✇✐t❤ t❤❡ ♦r✐❣✐♥❛❧ ♣❛r❛♠❡t❡rs✿ θ0 = B(r)θ(r)

0 ,

B(r) =

• µ−2/r

r

Iq+1 Ip

• =
• µ(r)

2 Iq+1

Ip

• ,

✇❤❡r❡ µ(r)

s

= E|η(r)

t |s ❢♦r ❛♥② s > 0✳

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

• ❡♥❡r❛❧✐③❡❞ ◗▼▲❊ ♦❢ θ(r)

❋♦r Θ(r) s✉❝❤ t❤❛t Θ = {B(r)θ, θ ∈ Θ(r)}

• θ

(r) n

= ❛r❣♠✐♥

θ∈Θ(r)

˜ In(θ), ✇❤❡r❡ ❢♦r θ ∈ Θ(r)✱ ˜ In(θ) = 1 n

n

• t=1

˜ lt(θ) ✇✐t❤ ˜ lt(θ) = log ˜ σ2

t (θ) + 2

r |ǫt|r ˜ σr

t (θ).

❘❡♠❛r❦✿ ✉♥❞❡r t❤❡ ✐❞❡♥t✐✜❛❜✐❧✐t② ❝♦♥str❛✐♥t E|η(r)

t |r = 1✱ t❤❡ ♦♥❧②

◗▼▲❊ ✇❤✐❝❤ ✐s str♦♥❣❧② ❝♦♥s✐st❡♥t ✭t♦ θ(r)

0 ✱ ♥♦t t♦ θ0✮ ✇❤❛t❡✈❡r

t❤❡ ❡rr♦r ❞✐str✐❜✉t✐♦♥ ✐s ♦❢ t❤❡ ❛❜♦✈❡ ❢♦r♠ ✭❝❢ ❋❩✱ ✷✵✶✸✮✳

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

❚✇♦✲st❛❣❡ ◗▼▲❊

θ0 = B(r)θ(r) B(r) ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ❢r♦♠ ❡♠♣✐r✐❝❛❧ ♠♦♠❡♥ts ♦❢ t❤❡ st❛♥❞❛r❞✐③❡❞ r❡t✉r♥s η(r)

t

= ǫt/˜ σt( θ

(r) n ).

❆s②♠♣t♦t✐❝ ❧❛✇ ♦❢ t❤❡ t✇♦✲st❛❣❡ ◗▼▲❊ ♦❢ θ0 ✭❋▲❩✱ ✷✵✶✶✮ ▲❡t r > 0✳ ❯♥❞❡r ❆ss✉♠♣t✐♦♥s ❆✶✲❆✹✱ ❛♥❞ ✐❢ µ2r < ∞✱ √n

• B(r)

n

θ

(r) n − θ0

L → N

• 0, Σ(r)

, Σ(r) = g(r)J−1 + {µ4 − 1 − g(r)} θ0θ

′ 0, g(r) = 2 r 2 µ2r µ2

r

− 1

• ,

θ0 = (ω0, α01, . . . , α0q, 0, . . . , 0)′ ❬❋♦r t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ✭r = 2) ✇❡ ❤❛✈❡ Σ(2) = (µ4 − 1)J−1✳❪

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

❚❡st✐♥❣ s❡❝♦♥❞✲♦r❞❡r st❛t✐♦♥❛r✐t② ✉s✐♥❣ t❤❡ ✷◗▼▲❊ ♦❢ θ0

▲❡t H0 :

q

• i=1

α0i +

p

• j=1

β0j < 1, ♦r✱ ❡q✉✐✈❛❧❡♥t❧② H0 : c′θ0 < 1, ✇❤❡r❡ c = (0, 1, . . . , 1) ∈ Rp+q+1. ❚❡st ♦❢ H0 ❬r❡s♣✳ H∗

0 : c′θ0 ≥ 1❪ ❛t ❧❡✈❡❧ α ∈ (0, 1)

❉❡✜♥❡❞ ❜② t❤❡ r❡❥❡❝t✐♦♥ r❡❣✐♦♥ Cr = {Tn,r > Φ−1(1 − α)}, [r❡s♣✳ C∗

r = {Tn,r < Φ−1(α)}].

✇❤❡r❡ Tn,r = √n(c′ B(r)

n

θ

(r) n − 1)

c′ Σ(r)c .

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

▲♦❝❛❧ ❛❧t❡r♥❛t✐✈❡s

❆r♦✉♥❞ θ0 s✉❝❤ t❤❛t c′θ0 = 1✱ ❧❡t ❛ s❡q✉❡♥❝❡ ♦❢ ❧♦❝❛❧ ♣❛r❛♠❡t❡rs θn = θ0 + τ √n, τ ∈ Rp+q+1. ❘❡❣✉❧❛r✐t② ❛ss✉♠♣t✐♦♥s ♦♥ t❤❡ ❞❡♥s✐t② f ♦❢ ηt✿

f > 0, lim

|y|→∞ yf(y) = 0,

lim

|y|→∞ y2f′(y) = 0,

❛♥❞ ❢♦r K > 0 ❛♥❞ δ > 0✱ |y|

• f′

f (y)

• + y2
• f′

f ′ (y)

• + y2
• f′

f ′′ (y)

• ≤ K
• 1 + |y|δ

, E |η1|2δ < ∞.

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

▲♦❝❛❧ ❛❧t❡r♥❛t✐✈❡s

▲♦❝❛❧ ❆s②♠♣t♦t✐❝ P♦✇❡rs ▲♦❝❛❧ ❛s②♠♣t♦t✐❝ ♣♦✇❡rs ♦❢ t❤❡ ✷♥❞✲♦r❞❡r st❛t✐♦♥❛r✐t② t❡sts✿ lim

n→∞ Pn,τ (Cr) = Φ

• Φ−1(α) + c′τ

σ(r)

• ❢♦r c′τ ≥ 0,

lim

n→∞ Pn,τ (C∗ r) = Φ

• Φ−1(α) − c′τ

σ(r)

• ❢♦r c′τ ≤ 0.

❈♦♠♣❛r✐s♦♥ ✇❤❡♥ r ✈❛r✐❡s t❤✉s ❜♦✐❧s ❞♦✇♥ t♦ ❝♦♠♣❛r✐♥❣ t❤❡ ❝♦❡✣❝✐❡♥ts σ(r) =

• c′Σ(r)c

1/2 .

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

▲♦❝❛❧ ❝♦♠♣❛r✐s♦♥s

❖♣t✐♠❛❧ r ▲❡t [r, r] s✉❝❤ t❤❛t r0 ✐s ✇❡❧❧ ❞❡✜♥❡❞✱ ✇❤❡r❡ r0 = arg min

[r,r] g(r),

g(r) = 2 r 2 µ2r µ2

r

− 1

• .

❚❤❡♥✱ ✇✐t❤✐♥ t❤❡ ❢❛♠✐❧② {Cr, r ∈ [r, r]} ❢♦r t❡st✐♥❣ H0 t❤❡ t❡st Cr0 ❤❛s t❤❡ ❤✐❣❤❡st ❧♦❝❛❧ ❛s②♠♣t♦t✐❝ ♣♦✇❡r✱ ✉♥✐❢♦r♠❧② ✐♥ τ✳ ❘❡♠❛r❦s✿

✶ r0 ❞❡♣❡♥❞s ♦♥ t❤❡ ❡rr♦rs ❞✐str✐❜✉t✐♦♥✱ ❛♥❞ ✐s ❛❧s♦ ♦♣t✐♠❛❧ ❢♦r t❤❡ ❡st✐♠❛t♦r θn,r ♦❢ θ0✳ ■❢ ηt ∼ N(0, 1)✱ r0 = 2✱ ❜✉t ✐♥ ❣❡♥❡r❛❧ t❡sts ❜❛s❡❞ ♦♥ t❤❡ ●◗▼▲❊ ❛r❡ ♥♦t ♦♣t✐♠❛❧✳ ■❢ ηt ∼ t(ν)✿ r0 < 1 ❢♦r s♠❛❧❧ ✈❛❧✉❡s ♦❢ ν✱ ❛♥❞ ✐♥❝r❡❛s❡s t♦ ✷ ❛s ν → ∞✳ ✷ ❆ ♠✐♥✐♠✉♠ ♦❢ g ♦✈❡r R+ ♠❛② ♥♦t ❡①✐st ❢♦r ♣❛rt✐❝✉❧❛r ❞✐str✐❜✉t✐♦♥s ♦❢ ηt✳ ✸ r0 ✐s ♥♦t ❦♥♦✇♥ ❜✉t ❝❛♥ ❜❡ ❝♦♥s✐st❡♥t❧② ❡st✐♠❛t❡❞✳

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

✶

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

✷

❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

P❡r❢♦r♠❛♥❝❡ ♦❢ t❡sts ♦❢ ❡①✐st❡♥❝❡ ♦❢ 2mt❤✲♦r❞❡r ♠♦♠❡♥ts

N = 1000 ✐♥❞❡♣❡♥❞❡♥t tr❛❥❡❝t♦r✐❡s ♦❢ s✐③❡ n = 2000, 4000, 8000 ♦❢ ❛ ●❆❘❈❍✭✶✱✶✮✿    ǫt = σtηt, (ηt) ✐✳✐✳❞✳N(0, 1) σ2

t

= 0.5 + 0.105ǫ2

t−1 + 0.87σ2 t−1 m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m

i=0

m

i

• αi

0βm−i

µ2i − 1 ✲✵✳✵✷✺ ✲✵✳✵✷✼ ✵✳✵✵✶ ✵✳✵✼✸ ✵✳✷✶✻ ✵✳✹✽✷

❚❤✉s✱ ❢♦r ✐♥t❡❣❡rs m✱ E|ǫt|2m < ∞ ⇔ m ≤ 2

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

❋✐♥✐t❡✲s❛♠♣❧❡ ♣❡r❢♦r♠❛♥❝❡

❚❛❜❧❡✿ ❘❡❧❛t✐✈❡ ❢r❡q✉❡♥❝② ♦❢ r❡❥❡❝t✐♦♥ ♦❢ H0 : Eǫ2m

t

< ∞ ❛❣❛✐♥st H1 : Eǫ2m

t

= ∞ ❬♦r ♦❢ H∗

0 : Eǫ2m t

= ∞ ❛❣❛✐♥st H∗

1 : Eǫ2m t

< ∞❪ ❛t t❤❡ ♥♦♠✐♥❛❧ ❧❡✈❡❧ 5% ♦r 10%✳

◆✉❧❧ n ❧❡✈❡❧ m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 H0 ✷✵✵✵ ✺✪ ✵✳✵ ✵✳✵ ✶✳✷ ✶✹✳✹ ✸✺✳✽ ✹✽✳✾ ✶✵✪ ✵✳✵ ✵✳✵ ✹✳✺ ✸✵✳✻ ✻✵✳✺ ✽✵✳✻ ✹✵✵✵ ✺✪ ✵✳✵ ✵✳✵ ✷✳✹ ✸✺✳✾ ✼✼✳✶ ✾✸✳✶ ✶✵✪ ✵✳✵ ✵✳✵ ✻✳✹ ✺✸✳✹ ✾✵✳✵ ✾✽✳✺ ✽✵✵✵ ✺✪ ✵✳✵ ✵✳✵ ✸✳✵ ✻✻✳✽ ✾✾✳✵ ✾✾✳✾ ✶✵✪ ✵✳✵ ✵✳✵ ✻✳✾ ✼✾✳✻ ✾✾✳✻ ✶✵✵✳✵ H∗ ✷✵✵✵ ✺✪ ✾✼✳✺ ✹✽✳✶ ✼✳✾ ✵✳✼ ✵✳✶ ✵✳✶ ✶✵✪ ✾✾✳✽ ✻✺✳✾ ✶✺✳✼ ✶✳✽ ✵✳✶ ✵✳✶ ✹✵✵✵ ✺✪ ✶✵✵✳✵ ✼✷✳✼ ✼✳✸ ✵✳✶ ✵✳✵ ✵✳✵ ✶✵✪ ✶✵✵✳✵ ✽✺✳✸ ✶✹✳✼ ✵✳✹ ✵✳✵ ✵✳✵ ✽✵✵✵ ✺✪ ✶✵✵✳✵ ✾✹✳✶ ✻✳✼ ✵✳✵ ✵✳✵ ✵✳✵ ✶✵✪ ✶✵✵✳✵ ✾✼✳✸ ✶✹✳✺ ✵✳✵ ✵✳✵ ✵✳✵

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

❯s✐♥❣ ❇♦♦tstr❛♣

❚❛❜❧❡✿ ❯s✐♥❣ t❤❡ r❡s❛♠♣❧✐♥❣ ❛❧❣♦r✐t❤♠ ✐♥st❡❛❞ ♦❢ t❤❡ ❛s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥

◆✉❧❧ n ❧❡✈❡❧ m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 H0 ✷✵✵✵ ✺✪ ✵✳✵ ✵✳✶ ✸✳✻ ✷✹✳✽ ✺✵✳✷ ✼✷✳✾ ✶✵✪ ✵✳✵ ✵✳✶ ✽✳✸ ✸✽✳✹ ✻✼✳✻ ✽✻✳✽ ✹✵✵✵ ✺✪ ✵✳✵ ✵✳✵ ✻✳✸ ✹✷✳✾ ✽✶✳✺ ✾✹✳✼ ✶✵✪ ✵✳✵ ✵✳✶ ✶✶✳✵ ✻✵✳✷ ✽✾✳✼ ✾✽✳✻ ✽✵✵✵ ✺✪ ✵✳✵ ✵✳✵ ✹✳✸ ✻✽✳✸ ✾✼✳✾ ✾✾✳✽ ✶✵✪ ✵✳✵ ✵✳✵ ✾✳✶ ✽✶✳✺ ✾✾✳✹ ✶✵✵✳✵ H∗ ✷✵✵✵ ✺✪ ✽✸✳✸ ✸✶✳✷ ✹✳✸ ✵✳✻ ✵✳✵ ✵✳✵ ✶✵✪ ✾✺✳✶ ✹✽✳✾ ✾✳✼ ✶✳✸ ✵✳✶ ✵✳✵ ✹✵✵✵ ✺✪ ✾✽✳✾ ✺✶✳✾ ✹✳✺ ✵✳✶ ✵✳✵ ✵✳✵ ✶✵✪ ✶✵✵✳✵ ✻✾✳✽ ✶✵✳✷ ✵✳✼ ✵✳✵ ✵✳✵ ✽✵✵✵ ✺✪ ✶✵✵✳✵ ✽✶✳✽ ✺✳✸ ✵✳✵ ✵✳✵ ✵✳✵ ✶✵✪ ✶✵✵✳✵ ✾✸✳✸ ✶✵✳✷ ✵✳✶ ✵✳✵ ✵✳✵

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

❊♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ Sn = ˆ α + ˆ β ✇❤❡♥ α + β = 1✳

0.98 0.99 1.00 1.01 20 40 60 n=2000 0.990 0.995 1.000 1.005 50 100 150 n=8000

❋✐❣✉r❡✿ ❇❛s❡❞ ♦♥ ✶✱✵✵✵ s✐♠✉❧❛t✐♦♥s ♦❢ ❛ ●❆❘❈❍✭✶✱✶✮ ✇✐t❤ α = 0.1✱ β = 0.9 ❛♥❞ ηt ∼ N(0, 1)✳

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

❚❡sts ❜❛s❡❞ ♦♥ ♥♦♥✲●❛✉ss✐❛♥ ◗▼▲

▲❡t t❤❡ ❡♠♣✐r✐❝❛❧ ❢✉♥❝t✐♦♥ r → ˆ g(r) = 2 r 2 ˆ µ2r ˆ µ2

r

• ❢♦r r ∈ [r, r] ✇❤❡♥ ηt ∼ N(0, 1)✳

❚❤❡ ♦♣t✐♠❛❧ ✈❛❧✉❡ r0 = 2 ♦❢ r ♠✐♥✐♠✐③❡s t❤❡ t❤❡♦r❡t✐❝❛❧ ❢✉♥❝t✐♦♥ g(r)✳

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

2 4 6 8 2.0 2.5 3.0 3.5 n = 1000, r = 0.2, r = 8 r g ^(r) 10 20 30 40 50 60 1 2 3 4 5 n = 1000, r = 0.01, r = 60 r g ^(r) 2 4 6 8 2 3 4 5 6 n = 10000, r = 0.2, r = 8 r g ^(r) 10 20 30 40 50 60 5 10 15 20 n = 10000, r = 0.01, r = 60 r g ^(r)

❋✐❣✉r❡✿ ❊♠♣✐r✐❝❛❧ ❡st✐♠❛t❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ g(r) ✇❤❡♥ t❤❡ ●❆❘❈❍ ✐♥♥♦✈❛t✐♦♥ ηt ∼ N(0, 1)✳

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

❋✐♥✐t❡✲s❛♠♣❧❡ ♣❡r❢♦r♠❛♥❝❡ ❢♦r ♥♦♥✲●❛✉ss✐❛♥ ❡rr♦rs

❚❛❜❧❡✿ ❘❡❧❛t✐✈❡ ❢r❡q✉❡♥❝② ♦❢ r❡❥❡❝t✐♦♥ ♦❢ H0 : Eǫ2

t < ∞ ❛❣❛✐♥st

H1 : Eǫ2

t = ∞ ♦r ♦❢ H∗ 0 : Eǫ2 t = ∞ ❛❣❛✐♥st H∗ 1 : Eǫ2 t < ∞ ❛t t❤❡ ♥♦♠✐♥❛❧

❧❡✈❡❧ 5% ♦r 10%✱ ✉s✐♥❣ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ♦r t❤❡ ❣❡♥❡r❛❧✐③❡❞ ◗▼▲ ♠❡t❤♦❞s✳ ηt ∼ ●❊❉✭✵✳✸✮

(α0, β0) (0.1, 0.8) (0.105, 0.87) (0.105, 0.895) (0.145, 0.88) α0 + β0 0.9 0.975 1 1.025 ◆✉❧❧ n ❧❡✈❡❧ ◗▼▲ ❣◗▼▲ ◗▼▲ ❣◗▼▲ ◗▼▲ ❣◗▼▲ ◗▼▲ ❣◗▼▲ ◗▼▲ ❣◗▼▲ H0 ✷✵✵✵ ✺✪ ✵✳✵ ✵✳✵ ✵✳✷ ✵✳✵ ✵✳✹ ✶✳✽ ✷✳✻ ✽✳✾ ✾✳✾ ✹✶✳✸ ✶✵✪ ✵✳✷ ✵✳✵ ✵✳✽ ✵✳✹ ✷✳✽ ✺✳✸ ✾✳✼ ✷✷✳✹ ✷✼✳✶ ✻✸✳✷ ✹✵✵✵ ✺✪ ✵✳✵ ✵✳✵ ✵✳✶ ✵✳✵ ✶✳✷ ✶✳✻ ✻✳✸ ✷✶✳✵ ✸✸✳✵ ✼✻✳✼ ✶✵✪ ✵✳✵ ✵✳✵ ✵✳✽ ✵✳✷ ✹✳✺ ✻✳✸ ✶✾✳✸ ✸✼✳✶ ✺✻✳✾ ✽✽✳✸ ✽✵✵✵ ✺✪ ✵✳✵ ✵✳✵ ✵✳✷ ✵✳✵ ✷✳✶ ✸✳✶ ✶✹✳✽ ✹✸✳✸ ✻✼✳✺ ✾✻✳✹ ✶✵✪ ✵✳✶ ✵✳✵ ✵✳✽ ✵✳✶ ✻✳✷ ✼✳✽ ✸✶✳✷ ✻✶✳✻ ✽✸✳✶ ✾✽✳✻ H∗ ✷✵✵✵ ✺✪ ✻✳✺ ✽✹✳✹ ✷✳✷ ✸✸✳✺ ✵✳✼ ✶✵✳✹ ✵✳✺ ✷✳✾ ✵✳✹ ✵✳✹ ✶✵✪ ✷✺✳✻ ✾✶✳✶ ✶✻✳✻ ✹✼✳✹ ✻✳✽ ✶✺✳✻ ✹✳✷ ✺✳✵ ✶✳✹ ✵✳✽ ✹✵✵✵ ✺✪ ✸✺✳✶ ✾✽✳✹ ✶✸✳✼ ✹✹✳✶ ✺✳✺ ✶✵✳✶ ✶✳✸ ✶✳✸ ✵✳✶ ✵✳✵ ✶✵✪ ✻✾✳✽ ✾✽✳✼ ✸✺✳✻ ✺✻✳✶ ✶✼✳✺ ✶✻✳✷ ✹✳✸ ✶✳✾ ✵✳✺ ✵✳✵ ✽✵✵✵ ✺✪ ✽✼✳✷ ✶✵✵✳✵ ✸✶✳✸ ✺✽✳✼ ✽✳✷ ✼✳✻ ✶✳✹ ✵✳✷ ✵✳✶ ✵✳✵ ✶✵✪ ✾✹✳✻ ✶✵✵✳✵ ✹✻✳✺ ✻✾✳✵ ✶✺✳✹ ✶✸✳✸ ✷✳✺ ✵✳✾ ✵✳✷ ✵✳✵ ❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

❋✐♥✐t❡✲s❛♠♣❧❡ ♣❡r❢♦r♠❛♥❝❡ ✇✐t❤ ❜♦♦tstr❛♣

❚❛❜❧❡✿ ❯s✐♥❣ r❡s❛♠♣❧✐♥❣ ❛❧❣♦r✐t❤♠s ✐♥st❡❛❞ ♦❢ t❤❡ ❛s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥s✳

(α0, β0) (0.1, 0.8) (0.105, 0.895) (0.15, 0.9) α0 + β0 0.9 1 1.05 ◆✉❧❧ n α ◗▼▲ ❣◗▼▲ ◗▼▲ ❣◗▼▲ ◗▼▲ ❣◗▼▲ H0 ✷✵✵✵ ✺✪ ✵✳✸ ✵✳✵ ✷✳✼ ✹✳✸ ✷✶✳✵ ✹✶✳✵ ✶✵✪ ✶✳✵ ✵✳✶ ✻✳✼ ✽✳✽ ✹✵✳✵ ✺✾✳✻ H∗ ✷✵✵✵ ✺✪ ✶✹✳✷ ✸✶✳✾ ✸✳✻ ✸✳✶ ✵✳✷ ✵✳✺ ✶✵✪ ✸✵✳✼ ✺✶✳✶ ✽✳✶ ✼✳✶ ✵✳✼ ✵✳✻ ❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

❊♠♣✐r✐❝❛❧ ❛♣♣♣❧✐❝❛t✐♦♥

❉❛✐❧② st♦❝❦ r❡t✉r♥s ♦❢ ❚♦t❛❧ ❙❆ ✭✷✵✵✶✲✵✼✲✶✻ t♦ ✷✵✶✽✲✵✾✲✷✶✮ ❊st✐♠❛t❡❞ ●❆❘❈❍✭✶✱✶✮ ♠♦❞❡❧✿

ˆ ω = 0.035(0.009), ˆ α = 0.083(0.011), ˆ β = 0.903(0.011) m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 Tn ✲✷✳✾✻ ✲✵✳✻✾ ✶✳✶✺ ✶✳✻✷ ✶✳✹✺ ✶✳✶✾

price 1000 2000 3000 4000 30 35 40 45 50 55 60 return 1000 2000 3000 4000 −10 −5 5 10

❋✐❣✉r❡✿ ❚♦t❛❧ st♦❝❦ ♣r✐❝❡ ❛♥❞ r❡t✉r♥ ❢r♦♠ ✷✵✵✶✲✵✼✲✶✻ t♦ ✷✵✶✽✲✵✾✲✷✶✳

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

❊st✐♠❛t♦r ♦❢ t❤❡ ❞❡♥s✐t② ♦❢ Sn ✉♥❞❡r t❤❡ ♥✉❧❧ t❤❛t S = 1

❱❛❧✉❡ ♦❢ Sn = m

i=0

m

i

• ˆ

αi ˆ βm−iˆ µ2i ❝♦♠♣✉t❡❞ ❢r♦♠ t❤❡ ♦❜s❡r✈❛t✐♦♥s ❂ ✈❡rt✐❝❛❧ ❧✐♥❡

0.985 0.995 1.005 20 40 60 80 100 m = 1 0.94 0.98 1.02 5 10 15 20 25 30 m = 2 0.90 0.95 1.00 1.05 1.10 5 10 15 m = 3 0.8 0.9 1.0 1.1 1.2 1.3 2 4 6 8 10 m = 4 0.8 1.0 1.2 1.4 2 4 6 m = 5 1.0 1.5 2.0 2.5 1 2 3 4 5 6 m = 6

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

✶

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧

✷

❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❆✉❣♠❡♥t❡❞ ●❆❘❈❍ ♣r♦❝❡ss❡s

▼❛♥② ●❆❘❈❍✭✶✱✶✮✲t②♣❡ ♠♦❞❡❧s ❛r❡ ♦❢ t❤❡ ❢♦r♠ ǫt = σtηt, σδ

t = ω(ηt−1) + a(ηt−1)σδ t−1, δ > 0✱ ω : R → [ω, +∞) ❛♥❞ a : R → [a, +∞)✱ ❢♦r s♦♠❡ ω > 0 ❛♥❞ a ≥ 0✳ ❬❉✉❛♥ ✭✶✾✾✼✮✱ ❍❡ ❛♥❞ ❚❡räs✈✐rt❛ ✭✶✾✾✾✮✱ ❆✉❡✱ ❇❡r❦❡s ❛♥❞ ❍♦r✈át❤ ✭✷✵✵✻✮❪ ❊①❛♠♣❧❡ ❬st❛♥❞❛r❞ ●❆❘❈❍❪✿ ω(η) = ω✱ a(η) = αη2 + β ❛♥❞ δ = 2✳

❙tr✐❝t st❛t✐♦♥❛r✐t② ❝♦♥❞✐t✐♦♥✿ γ = E log a(η1) < 0

✭❛ss✉♠✐♥❣ E log+ a(η1) < ∞ ❛♥❞ E log+ ω(η1) < ∞✮

❊①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❝♦♥❞✐t✐♦♥✿ ❢♦r u > 0✱ E(σuδ

t ) < ∞

⇔ E[au(η1)] < 1 ❛♥❞ E[ωu(η1)] < ∞. u → E[au(η1)] ❝❛♥ ❜❡ ❝❛❧❧❡❞ ▼♦♠❡♥t ●❡♥❡r❛t✐♥❣ ❋✉♥❝t✐♦♥ ✭▼●❋✮ ♦❢ t❤❡ ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ♠♦❞❡❧✳

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❆✉❣♠❡♥t❡❞ ●❆❘❈❍ ♠♦❞❡❧s

❢♦r δ0 > 0 ❛♥❞ θ0 ∈ Θ ⊂ Rd✱ ǫt = σt(θ0)ηt, σδ0

t (θ0) = ω(ηt−1; θ0) + a(ηt−1; θ0)σδ0 t−1

❋♦r ❛♥② θ ∈ Θ✱ ω(·; θ) : R → [ω, +∞) ❛♥❞ a(·; θ) : R → [a, +∞). (ǫt)✿ str✐❝t❧② st❛t✐♦♥❛r②✱ ♥♦♥✲❛♥t✐❝✐♣❛t✐✈❡ ❛♥❞ ❡r❣♦❞✐❝ s♦❧✉t✐♦♥

• ✐✈❡♥ ♦❜s❡r✈❛t✐♦♥s ǫ1, . . . , ǫn✱ ❛♥❞ ❛r❜✐tr❛r② ✐♥✐t✐❛❧ ✈❛❧✉❡s ˜

ǫ0 ❛♥❞ ˜ σ0 > 0 ❧❡t✱ ❢♦r t = 1, . . . , n ❛♥❞ ❛♥② θ ∈ Θ✱ ˜ σδ

t (θ) = ω

• ǫt−1

˜ σt−1(θ); θ

• + a
• ǫt−1

˜ σt−1(θ); θ

• ˜

σδ

t−1

❙❘❊✿ σδ

t (θ)

= ω

• ǫt−1

σt−1(θ); θ

• + a
• ǫt−1

σt−1(θ); θ

• σδ

t−1,

t ∈ Z.

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

▼❛✐♥ ❛ss✉♠♣t✐♦♥s

❇✶✿ ❬❙tr✐❝t st❛t✐♦♥❛r✐t②❪ E log+ ω(η1, θ0) < ∞✱ E log a(η1, θ0) < 0 ❛♥❞ E[as(η1, θ0)] < ∞ ❢♦r s♦♠❡ s > 0✳ ❇✷✿ ❬❊①✐st❡♥❝❡ ♦❢ ❛ s♦❧✉t✐♦♥ t♦ t❤❡ ❙❘❊❪ ❋♦r ❛♥② θ ∈ Θ✱ t❤❡r❡ ❡①✐sts z0 > 0 s✉❝❤ t❤❛t E log+ ω

• ǫt

z1/δ ; θ

• + log+ a
• ǫt

z1/δ ; θ

• < ∞,

E log sup

z≥ω

• ∂

∂z

• ω

ǫt z1/δ ; θ

• + a

ǫt z1/δ ; θ

• z
• < 0.

❇✸✿ ❬■♥✈❡rt✐❜✐❧✐t②❪ ❚❤❡ Ft−1✲♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥ θ → (σt(θ), ˜ σt(θ)) ✐s ❛✳s✳ C1✳ sup

θ∈Θ

|σt(θ) − ˜ σt(θ)| +

• ∂σt(θ)

∂θ − ∂˜ σt(θ) ∂θ

• ≤ Ktρt

✇❤❡r❡ Kt ∈ Ft−1 ❛♥❞ supt E(Kr

t ) < ∞ ❢♦r s♦♠❡ r > 0✳

❇✹✿ ❬❇❛❤❛❞✉r ❡①♣❛♥s✐♦♥❪ √n

• θn − θ0
• =

1 √n

n

t=1 ∆t−1V (ηt) + oP (1),

∆t−1 ∈ Ft−1 ✇✐t❤ EV (ηt) = 0, ✈❛r{V (ηt)} = Υ ✐s ♥♦♥s✐♥❣✉❧❛r✱ E∆t = Λ ✐s ❢✉❧❧ r♦✇ r❛♥❦✳

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❆s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❡♠♣✐r✐❝❛❧ ▼●❋

❘❡❝❛❧❧✐♥❣ t❤❡ ♠♦♠❡♥t ❝♦♥❞✐t✐♦♥ E[au(η1)] < 1✱ ❛ t❡st st❛t✐st✐❝ ❝❛♥ ❜❡ ❜❛s❡❞ ♦♥ t❤❡ ❡♠♣✐r✐❝❛❧ ▼●❋✿ S(u)

n

= 1

n

n

t=1 au(ˆ

ηt; θn) ▲❡t S(u)

∞ = E[au(ηt; θ0)]✳

❆s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥ ♦❢ S(u)

n

❋♦r 0 < u ≤ s/2✱ ✇❡ ❤❛✈❡ √n

• S(u)

n

− S(u)

∞

L → N

• 0, υ2

u := g′ uΣgu + ψu + 2g′ uξu

• .

✇❤❡r❡ Σ = E(∆tΥ∆′

t)✱ ψu = ❱❛r[au(η1; θ0)]✱ ξu = ΛE[V (ηt)au(ηt; θ0)]✱

gu = E

• gu,t
• ✇❤❡r❡

gu,t = ∂ ∂θ au{ηt(θ); θ}

• θ=θ0

.

▼♦r❡♦✈❡r υ2

u > 0 ✐❢ ❱❛r {au(ηt; θ0), V ′(ηt)} ✐s ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✳

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

P❛rt✐❝✉❧❛r ❝❛s❡s

◗▼▲ ❛♥❞ ▼▲ ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍✭✶✱✶✮

Mx,y = E[η2x

t (α0η2 t + β0)y]✱ x, y ∈ R✱ mu = (0, M1,u−1, M0,u−1)′,

J = E 1 σ4

t

∂σ2

t (θ0)

∂θ ∂σ2

t (θ0)

∂θ′

• .

❆s②♠♣t♦t✐❝ ✈❛r✐❛♥❝❡s ♦❢ S(u)

n

= 1

n

n

t=1 au(ˆ

ηt; θn)✿ υ2

u,QML

= u2(κ4 − 1)

• m′

uJ−1mu − α2 0M2 1,u−1

• + M0,2u − M2

0,u,

υ2

u,ML

= = 4u2 ιf

• m′

uJ−1mu − α2 0M2 1,u−1

• + M0,2u − M2

0,u

✇❤❡r❡ κ4 = Eη4

t ❛♥❞ ιf =

• {1 + yf′(y)/f(y)}2 f(y)dy ✐s t❤❡ ❋✐s❤❡r ✐♥❢♦r♠❛t✐♦♥ ❢♦r

s❝❛❧❡✳

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ♦❢ ❣✐✈❡♥ ♦r❞❡r

H0,u : E{au(ηt)} < 1 ❛❣❛✐♥st H1,u : E{au(ηt)} ≥ 1 H∗

0,u :

E{au(ηt)} ≥ 1 ❛❣❛✐♥st H∗

1,u :

E{au(ηt)} < 1 ❚❡st st❛t✐st✐❝ ❜❛s❡❞ ♦♥ t❤❡ ❡♠♣✐r✐❝❛❧ ▼●❋ T (u)

n

= √n

• S(u)

n

− 1

• ˆ

υu , ✇❤❡r❡ ˆ υ2

u = ˆ

g′

u ˆ

Σˆ gu + ˆ ψu + 2ˆ g′

uˆ

ξu, ❚❡st ♦❢ H0,u ❬r❡s♣✳ H∗

0,u❪ ❛t t❤❡ ❛s②♠♣t♦t✐❝ ❧❡✈❡❧ α ∈ (0, 1)

C(u)

T

= {T (u)

n

> Φ−1(1 − α)}, [r❡s♣✳ {T (u)

n

< Φ−1(α)}],

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

▼❛①✐♠❛❧ ▼♦♠❡♥t ❡①♣♦♥❡♥t

❆✉❣♠❡♥t❡❞ ●❆❘❈❍✿ ǫt = σtηt, σδ

t = ω(ηt−1) + a(ηt−1)σδ t−1

❚❤❡ ▼❛①✐♠❛❧ ▼♦♠❡♥t ❊①♣♦♥❡♥t ✭▼▼❊✮✱ ✇❤❡♥ ❡①✐st✐♥❣✱ ✐s t❤❡ ♠❛①✐♠❛❧ ♦r❞❡r u0 ❛t ✇❤✐❝❤ ♠♦♠❡♥ts ♦❢ σδ

t ❡①✐st✿

u0 = sup{u > 0; Eσδu

t

< ∞} = sup{u > 0; E{au(ηt)} < 1}, ❛ss✉♠✐♥❣ Eωu(ηt) < ∞ ❢♦r ❛❧❧ u > 0✳ ❇❡r❦❡s✱ ❍♦r✈át❤ ❛♥❞ ❑♦❦♦s③❦❛ ✭✷✵✵✸✮ ♣r♦♣♦s❡❞ ❛♥ ❡st✐♠❛t♦r ♦❢ t❤✐s ❝♦❡✣❝✐❡♥t ❢♦r st❛♥❞❛r❞ ●❆❘❈❍✭✶✱✶✮ ♠♦❞❡❧s✳

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

▼●❋ ❛♥❞ ▼▼❊ ❢♦r ❛ ●❆❘❈❍✭✶✱✶✮ ❛♥❞ ❙t✉❞❡♥t ❞✐str✐❜✉t✐♦♥s

1 2 3 4 0.95 1.00 1.05 1.10 u Ea

u(ηt)

2.04 2.73 3.12 ν = 10 ν = 15 ν = 20 ❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❈♦♥❞✐t✐♦♥ ❢♦r ❛ ✜♥✐t❡ ▼▼❊

❙✉♣♣♦s❡ γ = E log a(η1) < 0 ■❢ P[a(η1) ≤ 1] = 1✱ t❤❡♥ ∀u > 0✱ E[au(η1)] < 1✱ ❛♥❞ E(σuδ

t ) < ∞ ♣r♦✈✐❞❡❞ E[ωu(η1)] < ∞✳ ❲❡ s❡t u0 = ∞✳

■❢ P[a(η1) ≤ 1] < 1✱ ❛♥❞ 1 ≤ E[as(η1)] < ∞ ❢♦r s♦♠❡ s > 0✱ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ u0 > 0 s✉❝❤ t❤❛t E[au0(η1)] = 1. ■❢ E[ωu0(η1)] < ∞, E(σuδ

t ) < ∞,

∀u < u0, E(σuδ

t ) = ∞,

u ≥ u0.

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❊♠♣✐r✐❝❛❧ ▼▼❊

❙✉♣♣♦s❡ γn := 1

n

n

t=1 log a(ˆ

ηt; θn) < 0 ■❢ a(ˆ ηt; θn) ≤ 1 ❢♦r t = 1, . . . , n✱ t❤❡♥ S(u)

n

< 1, ❢♦r ❛❧❧ u > 0✳ ■❢ a(ˆ ηt; θn) > 1 ❢♦r ❛t ❧❡❛st ♦♥❡ 1 ≤ t ≤ n✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ un > 0 s✉❝❤ t❤❛t S(un)

n

= 1. ▲❡tt✐♥❣ ˆ un = sup{u > 0; S(u)

n

≤ 1}, ✇❡ ❤❛✈❡ ˆ un = ∞ ✇❤❡♥ a(ˆ ηt; θn) ≤ 1 ❢♦r ❛❧❧ 1 ≤ t ≤ n✱ ❛♥❞ ˆ un = un ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❝❛s❡✳

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❆s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❡♠♣✐r✐❝❛❧ ▼▼❊

❯♥❞❡r t❤❡ ♣r❡✈✐♦✉s ✭❛♥❞ ❛❞❞✐t✐♦♥❛❧✮ ❛ss✉♠♣t✐♦♥s ■❢ P[a(η1) > 1] > 0✱ ❛♥❞ 1 < E[as(η1)] < ∞ ❢♦r s♦♠❡ s > 0✱ t❤❡♥ ˆ un → u0, a.s. ❛♥❞ √n(ˆ un − u0) L → N

• 0, w2

u0 := {D(u0) ∞ }−2υ2 u0

• ,

✇❤❡r❡ D(u0)

∞

:=

∂ ∂uS(u0) ∞ . ❘❡♠❛r❦✿ ❚❤❡ ♣r♦♦❢ ✐s ❜❛s❡❞ ♦♥ t❤❡ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❡♠♣✐r✐❝❛❧ ▼●❋✱ ♦♥ t❤❡ s♣❛❝❡ C ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ✉♥✐❢♦r♠ ❞✐st❛♥❝❡✳ ❋♦r [u1, u2] ⊂ (0, s/2) √n

• S(u)

n

− S(u)

∞

C[u1,u2] = ⇒ Γ(u), ✇❤❡r❡ Γ(u) ✐s ❛ ❝❡♥t❡r❡❞ ●❛✉ss✐❛♥ ♣r♦❝❡ss✳

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❚❡st ❜❛s❡❞ ♦♥ t❤❡ ❡♠♣✐r✐❝❛❧ ▼▼❊

◆♦t✐♥❣ t❤❛t t❤❡ ♥✉❧❧ ❛ss✉♠♣t✐♦♥ ♦❢ ✜♥✐t❡ uδ✲t❤ ♠♦♠❡♥t ❝❛♥ ❜❡ ✇r✐tt❡♥ H0,u : u < u0, ❧❡t t❤❡ t❡st st❛t✐st✐❝✱ U (u)

n

= √n {u − ˆ un}

• wˆ

un

, ✇❤❡r❡ w2

u =

• 1

n

n

t=1 aˆ un(ˆ

ηt; θn) log{a(ˆ ηt; θn)} −2 ˆ υ2

u.

❚❡st ♦❢ H0,u ❬r❡s♣✳ H∗

0,u❪ ❛t t❤❡ ❛s②♠♣t♦t✐❝ ❧❡✈❡❧ α ∈ (0, 1)

C(u)

U

= {U (u)

n

> Φ−1(1 − α)}, [r❡s♣✳ {U (u)

n

< Φ−1(α)}],

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

P✉r❡❧② ♣❛r❛♠❡tr✐❝ ❡st✐♠❛t♦r ♦❢ t❤❡ ▼▼❊

❲❤❡♥ t❤❡ ❞❡♥s✐t② f ♦❢ ηt ✐s ❦♥♦✇♥✱ t❤❡ ▼▼❊ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❜② s♦❧✈✐♥❣

• au0(x; θ)f(x)dx = 1,

✇✐t❤ s♦❧✉t✐♦♥ u0 = u0,f(θ) ✭✉♥✐q✉❡ ❜② t❤❡ ❝♦♥✈❡①✐t② ♦❢ t❤❡ ▼●❋✮✳ ▲❡t ˆ un,f = u0,f( θn,ML) ✇❤❡r❡ θn,ML ✐s t❤❡ ▼▲❊ ♦❢ θ0✳ ❆ss✉♠❡ t❤❛t t❤❡ ▼▲❊ s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣❛♥s✐♦♥ √n( θn,ML − θ0) = 2J−1 ιf √n

n

• t=1

1 σ2

t

∂σ2

t

∂θ g1(ηt) + oP (1). ✭s❡❡ ❇❡r❦❡s✱ ❍♦r✈át❤ ❛♥❞ ❑♦❦♦s③❦❛ ✭✷✵✵✹✮✮

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

P✉r❡❧② ♣❛r❛♠❡tr✐❝ ❡st✐♠❛t♦r ♦❢ t❤❡ ▼▼❊

▲❡t t❤❡ t❡st st❛t✐st✐❝ V (u)

n

=

√n(u−ˆ un,f) ˆ σf

✇❤❡r❡ ˆ σf ✐s ❛ ❝♦♥s✐st❡♥t ❡st✐♠❛t♦r ♦❢ σf =

• 4

ιf ∂u0 ∂θ′ J−1 ∂u0 ∂θ

1/2 ✳ ❯♥❞❡r t❤❡ ♣r❡✈✐♦✉s ❛ss✉♠♣t✐♦♥s ❛♥❞ ✐❢ ∂u0

∂θ = 0✱

❛ t❡st ♦❢ H0,u ❬r❡s♣✳ H∗

0,u❪ ❛t t❤❡ ❛s②♠♣t♦t✐❝ ❧❡✈❡❧ α ∈ (0, 1) ✐s

❞❡✜♥❡❞ ❜② C(u)

V

= {V (u)

n

> Φ−1(1 − α)}, [r❡s♣✳ {V (u)

n

< Φ−1(α)}].

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

✶

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ●❛✉ss✐❛♥ ◗▼▲ ❊✣❝✐❡♥❝② ❣❛✐♥s ✈✐❛ ●❡♥❡r❛❧✐③❡❞ ◗▼▲ ◆✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s

✷

❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❚❡st st❛t✐st✐❝s ❢♦r H0,u : E{au(ηt)} < 1

❇❛s❡❞ ♦♥ t❤❡ ❡♠♣✐r✐❝❛❧ ▼●❋✿ T (u)

n

= √n

• S(u)

n

− 1

• ˆ

υu ❇❛s❡❞ ♦♥ t❤❡ ❡♠♣✐r✐❝❛❧ ▼▼❊✿ U (u)

n

= √n {u − ˆ un}

• wˆ

un

❋✉❧❧② ♣❛r❛♠❡tr✐❝✿ V (u)

n

= √n(u − ˆ un,f) ˆ σf

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❆s②♠♣t♦t✐❝ ♣♦✇❡r ✉♥❞❡r ❧♦❝❛❧ ❛❧t❡r♥❛t✐✈❡s

❆r♦✉♥❞ θ0 ∈

• Θ✱ ❧❡t ❛ s❡q✉❡♥❝❡ ♦❢ ❧♦❝❛❧ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ❢♦r♠

θn = θ0 + τ/√n, ✇❤❡r❡ τ ∈ Rd✳ ▲❡t Pn,τ ✭r❡s♣✳ P0✮ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♦❜s❡r✈❛t✐♦♥s ✇❤❡♥ t❤❡ ♣❛r❛♠❡t❡r ✐s θ0 + τ/√n ✭r❡s♣✳ θ0✮✳ ❯♥❞❡r ❛♣♣r♦♣r✐❛t❡ ❛ss✉♠♣t✐♦♥s ♦♥ τ✱ t❤❡ ♣❛r❛♠❡t❡r θn ❜❡❧♦♥❣s t♦ t❤❡ ❛❧t❡r♥❛t✐✈❡ ❢♦r t❡st✐♥❣ H0,u0✳

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

▲♦❝❛❧ ❛s②♠♣t♦t✐❝ ♣♦✇❡rs

▲❆P ♦❢ t❤❡ t❡sts T✱ U ❛♥❞ V

lim

n→∞ Pn,τ

• C(u0)

T

• = lim

n→∞ Pn,τ

• C(u0)

U

• = Φ
• cf,u0(θ0) − Φ−1(1 − α)
• ,

lim

n→∞ Pn,τ

• C(u0)

V

• = Φ
• df,u0(θ0) − Φ−1(1 − α)
• ,

✇❤❡r❡✱ ✉s✐♥❣ g1(y) = 1 + y f ′

f (y) ❛♥❞ ru0 = ∂ ∂θSu0 ∞ (θ0)✱

cf,u0(θ0) = − τ ′ υu0

• E

1 σt ∂σt(θ0) ∂θ

• E{au0(η1)g1(η1)}

+E 1 σt ∂σt(θ0) ∂θ g′

u0∆t−1

• E{V (η1)g1(η1)}
• ,

df,u0(θ0) = r′

u0τ

• 4

ιf r′ u0J −1ru0

.

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

▲❆Ps ♦❢ t❤❡ t❡st T, U ✭❜❧✉❡ ❧✐♥❡✮ ❛♥❞ V ✭❞♦tt❡❞ r❡❞ ❧✐♥❡✮ ❢♦r ❛ ●❆❘❈❍✭✶✱✶✮ ✇✐t❤ ❙t✉❞❡♥t ❡rr♦rs

1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 α = 0.1, β = 0.85, ν = ∞, u0 = 4.536 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 α = 0.1, β = 0.85, ν = 30, u0 = 3.552 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 α = 0.1, β = 0.85, ν = 20, u0 = 3.12 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 α = 0.1, β = 0.85, ν = 5, u0 = 0.598

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❈♦♠♣❛r✐s♦♥s ❜❛s❡❞ ♦♥ ❇❛❤❛❞✉r s❧♦♣❡s

❚♦ ❜❡ ❛❜❧❡ t♦ ❞✐st✐♥❣✉✐s❤ t❤❡ t❡sts T ❛♥❞ U✱ t❤❡ ❇❛❤❛❞✉r ❛♣♣r♦❛❝❤ ❝❛♥ ❜❡ ✉s❡❞✳ s❧♦♣❡ = ❛✳s ❧✐♠✐t ♦❢ −2/n× t❤❡ ❧♦❣❛r✐t❤♠ ♦❢ t❤❡ p✲✈❛❧✉❡ ✉♥❞❡r Pθ ❆s②♠♣t♦t✐❝ s❧♦♣❡s ♦❢ t❤❡ t❡sts✿ cT (u) =

• S(u)

∞ − 1

2 υ2

u

❛♥❞ cU(u) = {u − u0}2 wu02 . ■♥ t❤❡ ❇❛❤❛❞✉r s❡♥s❡✱ T (u)

n

✐s ♠♦r❡ ❡✣❝✐❡♥t t❤❛♥ U (u)

n

✐✛ cT (u) cU(u) =

• S(u)

∞ − 1

2 {u − u0}2 υ2

u0

{E[au0(η1; θ0) log{a(η1; θ0)}]}2υ2

u

> 1.

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❇❛❤❛❞✉r s❧♦♣❡s ♦❢ t❤❡ t❡sts T ❛♥❞ U ❢♦r ●❆❘❈❍✭✶✱✶✮ ♠♦❞❡❧s ✇✐t❤ ●❛✉ss✐❛♥ ❡rr♦rs 2 3 4 5 6 7 8 0.000 0.004 0.008 α = 0.1, β= 0.86 u Slope u0 = 4.05 cT(u) cU(u) 5.5 6.5 7.5 8.5 0.0000 0.0004 0.0008 0.0012 α = 0.09, β= 0.8 u Slope u0 = 7.31 cT(u) cU(u)

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❈♦♥❝❧✉s✐♦♥s

❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ◗▼▲ ❛r❡ ✈❛❧✐❞ ✇❤❛t❡✈❡r t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ✐♥♥♦✈❛t✐♦♥s✳ ❍♦✇❡✈❡r✱ ❝❤♦♦s✐♥❣ t❤❡ ❛♣♣r♦♣r✐❛t❡ ✈❡rs✐♦♥ ♦❢ t❤❡ ◗▼▲ ❝❛♥ ❜r✐♥❣ ❡✣❝✐❡♥❝② ❣❛✐♥s ✇✐t❤♦✉t ♠✉❝❤ ❛❞❞✐t✐♦♥❛❧ ❝♦st✳ ❚❤❡ ❜♦♦tstr❛♣ ✈❡rs✐♦♥s ♦❢ ♦✉r t❡sts ❜r✐♥❣ s✐❣♥✐✜❝❛♥t ✐♠♣r♦✈❡♠❡♥ts ✐♥ t❡r♠s ♦❢ s✐③❡ ❜✉t✱ ❛s ❡①♣❡❝t❡❞✱ ❞♦ ♥♦t ✐♠♣r♦✈❡ ♣♦✇❡rs✳ ▲♦❝❛❧❧② ♦♣t✐♠❛❧ t❡sts ❛r❡ ✇♦rt❤ ❝♦♥s✐❞❡r✐♥❣ ❜♦t❤ ✐♥ t❡r♠s ♦❢ s✐③❡ ❛♥❞ ♣♦✇❡r✱ ❜✉t ♠❛② ❜❡ ✐♥❝♦♥❝❧✉s✐✈❡ ❢♦r ♠♦❞❡r❛t❡ s❛♠♣❧❡ s✐③❡s✳ ❆✉❣♠❡♥t❡❞ ●❆❘❈❍✭✶✱✶✮ ❝❛♥ ❛❧s♦ ❜❡ t❡st❡❞ ❛t ❛♥② ♣♦✇❡r✳ ❚❡sts ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❛ ♣❛r❛♠❡tr✐③❡❞ ❡rr♦r ❞❡♥s✐t② ✭♥♦t s❤♦✇♥✮✳ ◆✉♠❡r✐❝❛❧ r❡s✉❧ts s✉❣❣❡st t❤❛t ♦♥❡ ❤❛s t♦ ❜❡ ❝❛✉t✐♦✉s ✐♥ ❛ss❡ss✐♥❣ t❤❡ ❡①✐st❡♥❝❡✱ ♦r ♥♦♥✲❡①✐st❡♥❝❡✱ ♦❢ ♠♦♠❡♥ts ♦❢ ✜♥❛♥❝✐❛❧ t✐♠❡ s❡r✐❡s✳

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❙♦♠❡ r❡❢❡r❡♥❝❡s

❆✉❡ ❆✳✱ ❇❡r❦❡s✱ ■✳ ❛♥❞ ▲✳ ❍♦r✈át❤ ❙tr♦♥❣ ❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r t❤❡ s✉♠s ♦❢ sq✉❛r❡s ♦❢ ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ s❡q✉❡♥❝❡s✳ ❇❡r♥♦✉❧❧✐ ✶✷✱ ✺✽✸✕✻✵✽✱ ✷✵✵✻✳ ❇❡r❦❡s✱ ■✳✱ ❍♦r✈át❤✱ ▲✳ ❛♥❞ P✳❙✳ ❑♦❦♦s③❦❛ ❊st✐♠❛t✐♦♥ ♦❢ t❤❡ ▼❛①✐♠❛❧ ▼♦♠❡♥t ❊①♣♦♥❡♥t ♦❢ ❛ ●❆❘❈❍✭✶✱✶✮ s❡q✉❡♥❝❡✳ ❊❝♦♥♦♠❡tr✐❝ ❚❤❡♦r② ✶✾✱ ✺✻✺✕✺✽✻✱ ✷✵✵✸✳ ❋r❛♥❝q✱ ❈✳✱ ▲❡♣❛❣❡✱ ●✳ ❛♥❞ ❏✲▼✳ ❩❛❦♦ï❛♥ ❚✇♦✲st❛❣❡ ♥♦♥ ●❛✉ss✐❛♥ ◗▼▲ ❡st✐♠❛t✐♦♥ ♦❢ ●❆❘❈❍ ▼♦❞❡❧s ❛♥❞ t❡st✐♥❣ t❤❡ ❡✣❝✐❡♥❝② ♦❢ t❤❡

• ❛✉ss✐❛♥ ◗▼▲❊✳

❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠❡tr✐❝s ✶✻✺✱ ✷✹✻✕✷✺✼✱ ✷✵✶✸✳ ❋r❛♥❝q✱ ❈✳ ❛♥❞ ❏✳▼✳ ❩❛❦♦ï❛♥ ❖♣t✐♠❛❧ ♣r❡❞✐❝t✐♦♥s ♦❢ ♣♦✇❡rs ♦❢ ❝♦♥❞✐t✐♦♥❛❧❧② ❤❡t❡r♦s❦❡❞❛st✐❝ ♣r♦❝❡ss❡s✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❘♦②❛❧ ❙t❛t✐st✐❝❛❧ ❙♦❝✐❡t② ✲ ❙❡r✐❡s ❇ ✼✺✱ ✸✹✺✕✸✻✼✱ ✷✵✶✸✳ ❋r❛♥❝q✱ ❈✳ ❛♥❞ ❏✳▼✳ ❩❛❦♦ï❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s✳ ❋♦rt❤❝♦♠✐♥❣ ❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠❡tr✐❝s✱ ✷✵✷✵✳ ▲✐♥❣✱ ❙✳ ❛♥❞ ▼✳ ▼❝❆❧❡❡r ❙t❛t✐♦♥❛r✐t② ❛♥❞ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ♦❢ ❛ ❢❛♠✐❧② ♦❢ ●❆❘❈❍ ♣r♦❝❡ss❡s✳ ❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠❡tr✐❝s ✶✵✻✱ ✶✵✾✕✶✶✼✱ ✷✵✵✷✳ ❙tr❛✉♠❛♥♥✱ ❉✳✱ ❛♥❞ ❚✳ ▼✐❦♦s❝❤ ◗✉❛s✐✲♠❛①✐♠✉♠✲❧✐❦❡❧✐❤♦♦❞ ❡st✐♠❛t✐♦♥ ✐♥ ❝♦♥❞✐t✐♦♥❛❧❧② ❤❡t❡r♦s❝❡❞❛st✐❝ t✐♠❡ s❡r✐❡s✿ ❛ st♦❝❤❛st✐❝ r❡❝✉rr❡♥❝❡ ❡q✉❛t✐♦♥s ❛♣♣r♦❛❝❤✳ ❚❤❡ ❆♥♥❛❧s ♦❢ ❙t❛t✐st✐❝s ✸✹✱ ✷✹✹✾✕✷✹✾✺✱ ✷✵✵✻✳ ❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❚❤❛♥❦ ②♦✉✦

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❘❡s❛♠♣❧✐♥❣ s❝❤❡♠❡ ❢♦r m = 1 ✭✷♥❞✲♦r❞❡r st❛t✐♦♥❛r✐t②✮

■♥ t❤❡ ●❆❘❈❍✭✶✱✶✮ ❝❛s❡✿

✶ ❈♦♠♣✉t❡ t❤❡ ❝♦♥str❛✐♥❡❞ ◗▼▲❊ ˆ θ

′ c = (ˆ

ωc, ˆ αc, 1 − ˆ αc) = arg min

θ∈Θc n

• t=1

˜ ℓt(θ) ❛♥❞ t❤❡ st❛♥❞❛r❞✐③❡❞ r❡s✐❞✉❛❧s ˆ ηt = ˜ ηt/sn✱ ✇❤❡r❡ ˜ ηt = ǫt/˜ σt(ˆ θc) ❛♥❞ s2

n = n−1 n t=1 ˜

η2

t ✳ ❉❡♥♦t❡ ❜② F ∗ n t❤❡ ❡♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡s❡ r❡s✐❞✉❛❧s✳

✷ ❙✐♠✉❧❛t❡ ❛ tr❛❥❡❝t♦r② ♦❢ ❧❡♥❣t❤ n ♦❢ ❛ ●❆❘❈❍ ♠♦❞❡❧ ✇✐t❤ t❤❡ ♣❛r❛♠❡t❡r ˆ θc ❛♥❞ ❞✐str✐❜✉t✐♦♥ F ∗

n ❢♦r t❤❡ ✐✳✐✳❞✳ ♥♦✐s❡ η∗ t ✱ ❝♦♠♣✉t❡ t❤❡ ✉♥❝♦♥str❛✐♥❡❞ ◗▼▲❊

ˆ θ

∗ = (ˆ

ω∗, ˆ α∗, ˆ β∗)′ ❛♥❞ t❤❡ st❛t✐st✐❝ S∗

n = ˆ

α∗ + ˆ β∗ ✸ ❖♥ t❤❡ ♦❜s❡r✈❛t✐♦♥s ǫ1, . . . , ǫn✱ ❝♦♠♣✉t❡ t❤❡ ✉♥❝♦♥str❛✐♥❡❞ ◗▼▲❊ ˆ θ = (ˆ ω, ˆ α, ˆ β) ❛♥❞ t❤❡ st❛t✐st✐❝ Sn = ˆ α + ˆ β ✹ ❘❡♣❡❛t B t✐♠❡s st❡♣ ✷✱ ❛♥❞ ❞❡♥♦t❡ ❜② S∗1

n , . . . , S∗B n

t❤❡ ❜♦♦tstr❛♣ t❡st st❛t✐st✐❝✳ ❆♣♣r♦①✐♠❛t❡ t❤❡ ♣✲✈❛❧✉❡ ♦❢ t❤❡ t❡st ♦❢ H0 : Eǫ2

t < ∞ ❛❣❛✐♥st H1 : Eǫ2 t = ∞ ❜②

#{S∗j

n ≥ Sn; j = 1, . . . , B}/B✳ ❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❘❡s❛♠♣❧✐♥❣ s❝❤❡♠❡ ❢♦r m = 1 ✉s✐♥❣ ◆❡✇t♦♥✲❘❛♣❤s♦♥

❚❤❡ ♥✉♠❡r✐❝❛❧ ♦♣t✐♠✐③❛t✐♦♥ ✐♥ ❙t❡♣ ✷✱ r❡♣❡❛t❡❞ ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ t✐♠❡s B✱ ✐s t❤❡ ♠♦st t✐♠❡✲❝♦♥s✉♠✐♥❣ ♣❛rt ♦❢ t❤❡ ❛❧❣♦r✐t❤♠✳ ■♥st❡❛❞✱ ♦♥❡ ❝❛♥ ♠✐♠✐❝ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ◗▼▲❊ ❜② ✉s✐♥❣ ❛ ◆❡✇t♦♥✲❘❛♣❤s♦♥ t②♣❡ ✐t❡r❛t✐♦♥✳ ❙❡t ˆ θ

∗ = ˆ

θc + J−1

n

1 n

n

• t=1
• η∗ 2

t

− 1 ˜ φt(ˆ θc), ✇❤❡r❡

˜ φt(θ) = 1 ˜ σt(θ) ∂˜ σt(θ) ∂θ , Jn = 1 n

n

• t=1
• φt ˜

φ′

t(ˆ

θc)

❛♥❞ η∗

1, . . . , η∗ n ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ F ∗ n✲❞✐str✐❜✉t❡❞✳

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❱❛❧✐❞✐t② ♦❢ t❤❡ r❡s❛♠♣❧✐♥❣ s❝❤❡♠❡ ❢♦r t❡st✐♥❣ H0 : Eǫ2

t < ∞

✭♦r H∗

0 : Eǫ2 t = ∞✮

▲❡t θ0 s✉❝❤ t❤❛t c′θ0 = 1 ✇✐t❤ c′ = (0, 1, . . . , 1)✳ ❆s②♠♣t♦t✐❝ ✈❛❧✐❞✐t② ♦❢ t❤❡ ❜♦♦tstr❛♣ ♣r♦❝❡❞✉r❡ ❢♦r t❤❡ ●❆❘❈❍✭p, q✮ ❆ss✉♠❡ ❆✶✲❆✹ + ❛ ❜♦✉♥❞❡❞ ❞❡♥s✐t② ❢♦r ηt✳ ▲❡t ˆ θ

∗ ♦❜t❛✐♥❡❞ ✐♥ ❙t❡♣ ✷ ✭♦r ❜② ❛ ◆❘ ✐t❡r❛t✐♦♥✮✳

❋♦r ❛❧♠♦st ❛❧❧ r❡❛❧✐③❛t✐♦♥ (ǫt)✱ ❛s n → ∞ ✇❡ ❤❛✈❡✱ ❣✐✈❡♥ (ǫt)✱ √n (S∗

n − 1) L

→ N(0, σ2), σ2 = (µ4 − 1)c′J−1c. ⇒ t❤❡ ❧❛✇ ♦❢ S∗

n ❣✐✈❡♥ (ǫt) ✇❡❧❧ ♠✐♠✐❝s t❤❡ ✭✉♥❝♦♥❞✐t✐♦♥❛❧✮ ❧❛✇ ♦❢

Sn ❛t t❤❡ ❜♦✉♥❞❛r② ♦❢ H0✱ ❛t ❧❡❛st ❢♦r ❧❛r❣❡ n✳ ■♥ ✜♥✐t❡ s❛♠♣❧❡s✱ t❤❡ ❜♦♦tstr❛♣ ❞✐str✐❜✉t✐♦♥ ♦❢ S∗

n ✐s ❡①♣❡❝t❡❞ t♦

❜❡tt❡r ❛♣♣r♦❛❝❤ t❤❡ ❧❛✇ ♦❢ Sn t❤❛♥ ✐ts ❛s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥✳

❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s

❚❡sts ❢♦r t❤❡ st❛♥❞❛r❞ ●❆❘❈❍ ♠♦❞❡❧ ❚❡sts ❢♦r ❛✉❣♠❡♥t❡❞ ●❆❘❈❍ ❚❡sts ❜❛s❡❞ ♦♥ t❤❡ ▼●❋ ❛♥❞ ▼▼❊ P♦✇❡r ❝♦♠♣❛r✐s♦♥s

❇♦♦tstr❛♣ ♣r♦❝❡❞✉r❡ ❢♦r t❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ Eǫ2m

t

✇❤❡♥ m > 1

✺ ❊st✐♠❛t❡ ❛ ●❆❘❈❍✭✶✱✶✮ ♠♦❞❡❧ ❛♥❞ ❝♦♠♣✉t❡ ˆ µ2i = n−1 n

t=1 ˆ

η2i

t

♦♥ t❤❡ r❡❝❡♥tr❡❞ ❛♥❞ r❡s❝❛❧❡❞ r❡s✐❞✉❛❧s✳ ✻ ❊st✐♠❛t❡ ❛ ●❆❘❈❍✭✶✱✶✮ ♠♦❞❡❧ ♦❢ ♣❛r❛♠❡t❡r θc = (ωc, αc, βc) ✉♥❞❡r t❤❡ ❝♦♥str❛✐♥t H0 : m

i=0

m

i

• αi

cβm−i c

ˆ µ2i = 1✳ ✼ ❙✐♠✉❧❛t❡ ❛ tr❛❥❡❝t♦r② ♦❢ ❧❡♥❣t❤ n ♦❢ ❛ ●❆❘❈❍ ♠♦❞❡❧ ✇✐t❤ t❤❡ ♣❛r❛♠❡t❡r ˆ θc ♦❢ t❤❡ ♣r❡✈✐♦✉s st❡♣✱ ❛♥❞ t❤❡ ❡♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ✉♥❝♦♥str❛✐♥❡❞ r❡s✐❞✉❛❧s ❢♦r t❤❡ ✐✳✐✳❞✳ ♥♦✐s❡✳ ❈♦♠♣✉t❡ t❤❡ ✉♥❝♦♥str❛✐♥❡❞ ◗▼▲❊ ˆ θ

∗ = (ˆ

ω∗, ˆ α∗, ˆ β∗)′ ❛♥❞ t❤❡ st❛t✐st✐❝ S∗

n = m i=0

m

i

• ˆ

α∗ i ˆ β∗ m−iˆ µ∗

2i ✇❤❡r❡ ˆ

µ∗

2i ✐s ❝♦♠♣✉t❡❞ ♦♥ t❤❡

r❡s✐❞✉❛❧s ❜❛s❡❞ ♦♥ ˆ θ

∗✳

✽ ❈♦♠♣✉t❡ Sn = m

i=0

m

i

• ˆ

αi ˆ βm−iˆ µ2i✳ ✾ ❆s ❙t❡♣ ✹✳ ❘❡♠❛r❦✿ ❍❡✐♥❡♠❛♥♥ ✭✷✵✶✾✮ ❡st❛❜❧✐s❤❡s t❤❡ ✈❛❧✐❞✐t② ♦❢ ❛ ✜①❡❞✲❞❡s✐❣♥ r❡s✐❞✉❛❧ ❜♦♦tstr❛♣ ✭❛s ✐♥ ❈❛✈❛❧✐❡r❡✱ P❡❞❡rs❡♥ ❛♥❞ ❘❛❤❜❡❦ ✭✷✵✶✽✮✮ ❢♦r t❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍(p, q) ♣r♦❝❡ss❡s✳

❘❡t✉r♥ ❋r❛♥❝q✱ ❩❛❦♦✐❛♥ ❚❡st✐♥❣ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♠♦♠❡♥ts ❢♦r ●❆❘❈❍ ♣r♦❝❡ss❡s