Maximum likelihood estimation for generalized autoregressive score models - Andre Lucas, Francisco...

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Maximum Likelihood Estimation for Generalized Autoregressive Score Models SYstemic Risk TOmography: Signals, Measurements, Transmission Channels, and Policy Interventions 1 st IAAE Conference London, June , 2014 Andre Lucas, Francisco Blasques, Siem Jan Koopman VU University Amsterdam

Transcript of Maximum likelihood estimation for generalized autoregressive score models - Andre Lucas, Francisco...

Maximum Likelihood Estimation for

Generalized Autoregressive Score

Models

SYstemic Risk TOmography:

Signals, Measurements, Transmission

Channels, and Policy Interventions

1st IAAE Conference

London, June , 2014

Andre Lucas, Francisco Blasques, Siem Jan Koopman

VU University Amsterdam

The GAS Model: Creal Koopman Lucas (2013)

yt ∼ p(yt | ft), ft+1 = ω + βft + αs(yt, ft),

Example: normal p(yt|ft), ft = σ2t , s(yt, ft) = y2

t − ft →GARCH. But what to do in other cases?

Score: s(yt, ft) = ∂ log p(yt|ft)/∂ftObservation driven, ML estimation easy, general

framework, encompasses previous models, intuitive

t-GAS volatility model ft = σ2/(1− 2ν−1):

s(yt, ft) =(1 + ν−1)

1 + y2t /(νft)

· y2t − ft

1 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

The GAS Model: Creal Koopman Lucas (2013)

yt ∼ p(yt | ft), ft+1 = ω + βft + αs(yt, ft),

Example: normal p(yt|ft), ft = σ2t , s(yt, ft) = y2

t − ft →GARCH. But what to do in other cases?

Score: s(yt, ft) = ∂ log p(yt|ft)/∂ftObservation driven, ML estimation easy, general

framework, encompasses previous models, intuitive

t-GAS volatility model ft = σ2/(1− 2ν−1):

s(yt, ft) =(1 + ν−1)

1 + y2t /(νft)

· y2t − ft

1 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

The GAS Model: Creal Koopman Lucas (2013)

yt ∼ p(yt | ft), ft+1 = ω + βft + αs(yt, ft),

Example: normal p(yt|ft), ft = σ2t , s(yt, ft) = y2

t − ft →GARCH. But what to do in other cases?

Score: s(yt, ft) = ∂ log p(yt|ft)/∂ft

Observation driven, ML estimation easy, general

framework, encompasses previous models, intuitive

t-GAS volatility model ft = σ2/(1− 2ν−1):

s(yt, ft) =(1 + ν−1)

1 + y2t /(νft)

· y2t − ft

1 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

The GAS Model: Creal Koopman Lucas (2013)

yt ∼ p(yt | ft), ft+1 = ω + βft + αs(yt, ft),

Example: normal p(yt|ft), ft = σ2t , s(yt, ft) = y2

t − ft →GARCH. But what to do in other cases?

Score: s(yt, ft) = ∂ log p(yt|ft)/∂ftObservation driven, ML estimation easy, general

framework, encompasses previous models, intuitive

t-GAS volatility model ft = σ2/(1− 2ν−1):

s(yt, ft) =(1 + ν−1)

1 + y2t /(νft)

· y2t − ft

1 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

The GAS Model: Creal Koopman Lucas (2013)

yt ∼ p(yt | ft), ft+1 = ω + βft + αs(yt, ft),

Example: normal p(yt|ft), ft = σ2t , s(yt, ft) = y2

t − ft →GARCH. But what to do in other cases?

Score: s(yt, ft) = ∂ log p(yt|ft)/∂ftObservation driven, ML estimation easy, general

framework, encompasses previous models, intuitive

t-GAS volatility model ft = σ2/(1− 2ν−1):

s(yt, ft) =(1 + ν−1)

1 + y2t /(νft)

· y2t − ft

1 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

White spots on the map

yt ∼ p(yt | ft), ft+1 = ω + βft + αs(yt, ft),

Successful applications:

Creal et al. (JBES, 2011): multivariate volatility

Creal et al. (JAEctr, 2013); Harvey (CUP, 2013)

Lucas et al. (JBES, 2014): sovereign systemic risk

Creal et al. (REStat in press): portfolio credit risk

Harvey Luati (JASA, in press): location and scale models

Patton Oh: multivariate / systemic

Many more: see GASMODEL.COM

2 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

White spots on the map

yt ∼ p(yt | ft), ft+1 = ω + βft + αs(yt, ft),

Missing: THEORY

why the score? (optimality: Blasques et al. (2014),

yesterday on the program)

statistical properties of the process? (Blasques et al. 2012;

this paper)

statistical properties of the MLE? (this paper + companion)

HIGH TIME!!

3 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

White spots on the map

yt ∼ p(yt | ft), ft+1 = ω + βft + αs(yt, ft),

Missing: THEORY

why the score? (optimality: Blasques et al. (2014),

yesterday on the program)

statistical properties of the process? (Blasques et al. 2012;

this paper)

statistical properties of the MLE? (this paper + companion)

HIGH TIME!!

3 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Reformulation of GAS Model

yt =g(ft, ut) , {ut} iid ∼ pu(λ)

ft+1 =ω + αs(yt, ft) + βft.

IMPORTANT:

The dynamic proberties of the �ltered f̃t for given sequence

{yt}f̃t+1 = ω + βf̃t + αs(yt, ft)

are VERY di�erent from the dynamic properties of the true

ft

ft+1 = ω + βft + αs(g(ft, ut) , ft

)for given sequence {ut}We want to know the properties of the likelihood for

{ut, ft} as well as for {yt, f̃t}

4 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Reformulation of GAS Model

yt =g(ft, ut) , {ut} iid ∼ pu(λ)

ft+1 =ω + αs(yt, ft) + βft.

IMPORTANT:

The dynamic proberties of the �ltered f̃t for given sequence

{yt}f̃t+1 = ω + βf̃t + αs(yt, ft)

are VERY di�erent from the dynamic properties of the true

ft

ft+1 = ω + βft + αs(g(ft, ut) , ft

)for given sequence {ut}

We want to know the properties of the likelihood for

{ut, ft} as well as for {yt, f̃t}

4 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Reformulation of GAS Model

yt =g(ft, ut) , {ut} iid ∼ pu(λ)

ft+1 =ω + αs(yt, ft) + βft.

IMPORTANT:

The dynamic proberties of the �ltered f̃t for given sequence

{yt}f̃t+1 = ω + βf̃t + αs(yt, ft)

are VERY di�erent from the dynamic properties of the true

ft

ft+1 = ω + βft + αs(g(ft, ut) , ft

)for given sequence {ut}We want to know the properties of the likelihood for

{ut, ft} as well as for {yt, f̃t}4 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Reformulation of GAS Model: GARCH example

yt =g(ft, ut) , {ut} iid ∼ pu(λ)

ft+1 =ω + αs(yt, ft) + βft.

IMPORTANT:

The dynamic proberties of the �ltered f̃t for given sequence

{yt}f̃t+1 = ω + βf̃t + αs(yt, ft)

are VERY di�erent from the dynamic properties of the true

ft

ft+1 = ω + βft + αs(g(ft, ut) , ft

)for given sequence {ut}We want to know the properties of the likelihood for

{ut, ft} as well as for {yt, f̃t}4 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Reformulation of GAS Model: GARCH example

yt =g(ft, ut) , {ut} iid ∼ pu(λ)

ft+1 =ω + αs(yt, ft) + βft.

IMPORTANT:

The dynamic proberties of the �ltered f̃t for given sequence

{yt}f̃t+1 = ω + βf̃t + α

(y2t − f̃t

)are VERY di�erent from the dynamic properties of the true

ft

ft+1 = ω + βft + α (u2t − 1) · ft

for given sequence {ut}We want to know the properties of the likelihood for

{ut, ft} as well as for {yt, f̃t}4 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Reformulation of GAS Model: GARCH example

yt =g(ft, ut) , {ut} iid ∼ pu(λ)

ft+1 =ω + αs(yt, ft) + βft.

IMPORTANT:

The dynamic proberties of the �ltered f̃t for given sequence

{yt}f̃t+1 = ω + βf̃t + α

(w(y2

t /f̃t) · y2t − f̃t

)are VERY di�erent from the dynamic properties of the true

ft

ft+1 = ω + βft + α (w(u2t ) · u2

t − 1) · ftfor given sequence {ut}We want to know the properties of the likelihood for

{ut, ft} as well as for {yt, f̃t}4 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Estimation of Time-Invariant Parameters

Objective: Find simple conditions on g, pu, Θ, etc.

that ensure consistency and asymptotic normality of MLE:

allowing for nonlinear dynamics in {ft}.

allowing for local and global results

allowing for correct and incorrect model speci�cation

Challenge: Generality comes at a cost... a given condition

might �t some choice of g, pu, etc., but not others.

Solution: Balance between generality and easy applicability!

5 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF Uniform convergence of likelihood function:

supθ∈Θ|LT (θ, f1|yT )− L∞(θ)| a.s.→ 0 ∀ f1 ∈ F as T →∞.

IF Identi�able uniqueness of θ0 :

supθ∈Θ : ‖θ−θ0‖>ε

L∞(θ) < L∞(θ0) ∀ ε > 0

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF Uniform convergence of likelihood function:

supθ∈Θ|LT (θ, f1|yT )− L∞(θ)| a.s.→ 0 ∀ f1 ∈ F as T →∞.

IF Identi�able uniqueness of θ0 :

supθ∈Θ : ‖θ−θ0‖>ε

L∞(θ) < L∞(θ0) ∀ ε > 0

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF Uniform convergence of likelihood function:

supθ∈Θ|LT (θ, f1|yT )− L∞(θ)| a.s.→ 0 ∀ f1 ∈ F as T →∞.

IF Identi�able uniqueness of θ0 :

supθ∈Θ : ‖θ−θ0‖>ε

L∞(θ) < L∞(θ0) ∀ ε > 0

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF Uniform convergence of likelihood function:

supθ∈Θ|LT (θ, f1|yT )− L∞(θ)| a.s.→ 0 ∀ f1 ∈ F as T →∞.

IF Identi�able uniqueness of θ0 :

supθ∈Θ : ‖θ−θ0‖>ε

L∞(θ) < L∞(θ0) ∀ ε > 0

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF Uniform convergence of likelihood function:

supθ∈Θ|LT (θ, f1|yT )− L∞(θ)| a.s.→ 0 ∀ f1 ∈ F as T →∞.

IF Identi�able uniqueness of θ0 :

supθ∈Θ : ‖θ−θ0‖>ε

L∞(θ) < L∞(θ0) ∀ ε > 0

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF {LT (θ, f1|yT )} is SE ∀ (θ, f1) ∈ Θ×F .

IF E|LT (θ, f1|yT )| <∞ ∀ (θ, f1) ∈ Θ×F .

IF LT (θ, f1|yT ) is smooth in θ

IF Compact Θ

IF Identi�able uniqueness of θ0 :

supθ∈Θ : ‖θ−θ0‖>ε

L∞(θ) < L∞(θ0) ∀ ε > 0

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF {LT (θ, f1|yT )} is SE ∀ (θ, f1) ∈ Θ×F .

IF E|LT (θ, f1|yT )| <∞ ∀ (θ, f1) ∈ Θ×F .

IF LT (θ, f1|yT ) is smooth in θ

IF Compact Θ

IF Identi�able uniqueness of θ0 :

supθ∈Θ : ‖θ−θ0‖>ε

L∞(θ) < L∞(θ0) ∀ ε > 0

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF {LT (θ, f1|yT )} is SE ∀ (θ, f1) ∈ Θ×F .

IF E|LT (θ, f1|yT )| <∞ ∀ (θ, f1) ∈ Θ×F .

IF LT (θ, f1|yT ) is smooth in θ

IF Compact Θ

IF Identi�able uniqueness of θ0 :

supθ∈Θ : ‖θ−θ0‖>ε

L∞(θ) < L∞(θ0) ∀ ε > 0

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF {LT (θ, f1|yT )} is SE ∀ (θ, f1) ∈ Θ×F .

IF E|LT (θ, f1|yT )| <∞ ∀ (θ, f1) ∈ Θ×F .

IF LT (θ, f1|yT ) is smooth in θ

IF Compact Θ

IF Identi�able uniqueness of θ0 :

supθ∈Θ : ‖θ−θ0‖>ε

L∞(θ) < L∞(θ0) ∀ ε > 0

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF {LT (θ, f1|yT )} is SE ∀ (θ, f1) ∈ Θ×F .

IF E|LT (θ, f1|yT )| <∞ ∀ (θ, f1) ∈ Θ×F .

IF LT (θ, f1|yT ) is smooth in θ

IF Compact Θ

IF Identi�able uniqueness of θ0 :

supθ∈Θ : ‖θ−θ0‖>ε

L∞(θ) < L∞(θ0) ∀ ε > 0

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF {LT (θ, f1|yT )} is SE ∀ (θ, f1) ∈ Θ×F .

IF E|LT (θ, f1|yT )| <∞ ∀ (θ, f1) ∈ Θ×F .

IF LT (θ, f1|yT ) is smooth in θ

IF Compact Θ

IF Identi�able uniqueness of θ0 :

supθ∈Θ : ‖θ−θ0‖>ε

L∞(θ) < L∞(θ0) ∀ ε > 0

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF {(yt, f̃t(θ, f1))} is SE ∀ (θ, f1) ∈ Θ×F .

IF E|yt|m <∞ and E|f̃t(θ, f1))|m <∞

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ

IF Identi�able uniqueness of θ0 :

supθ∈Θ : ‖θ−θ0‖>ε

L∞(θ) < L∞(θ0) ∀ ε > 0

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF {(yt, f̃t(θ, f1))} is SE ∀ (θ, f1) ∈ Θ×F .

IF E|yt|m <∞ and E|f̃t(θ, f1))|m <∞

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ

IF Identi�able uniqueness of θ0 :

supθ∈Θ : ‖θ−θ0‖>ε

L∞(θ) < L∞(θ0) ∀ ε > 0

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF {(yt, f̃t(θ, f1))} is SE ∀ (θ, f1) ∈ Θ×F .

IF E|yt|m <∞ and E|f̃t(θ, f1))|m <∞

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ

IF Identi�able uniqueness of θ0 :

supθ∈Θ : ‖θ−θ0‖>ε

L∞(θ) < L∞(θ0) ∀ ε > 0

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF {(yt, f̃t(θ, f1))} is SE ∀ (θ, f1) ∈ Θ×F .

IF E|yt|m <∞ and E|f̃t(θ, f1))|m <∞

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ

IF Identi�able uniqueness of θ0 :

supθ∈Θ : ‖θ−θ0‖>ε

L∞(θ) < L∞(θ0) ∀ ε > 0

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF {(yt, f̃t(θ, f1))} is SE ∀ (θ, f1) ∈ Θ×F .

IF E|yt|m <∞ and E|f̃t(θ, f1))|m <∞

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ

IF Identi�able uniqueness of θ0 :

supθ∈Θ : ‖θ−θ0‖>ε

L∞(θ) < L∞(θ0) ∀ ε > 0

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF {(yt, f̃t(θ, f1))} is SE ∀ (θ, f1) ∈ Θ×F .

IF E|yt|m <∞ and E|f̃t(θ, f1))|m <∞

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ

IF Model parameter identi�cation (local or global)

IF Correct speci�cation or unique pseudo-true parameter

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF {(yt, f̃t(θ, f1))} is SE ∀ (θ, f1) ∈ Θ×F .

IF E|yt|m <∞ and E|f̃t(θ, f1))|m <∞

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ

IF Model parameter identi�cation (local or global)

IF Correct speci�cation or unique pseudo-true parameter

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF {(yt, f̃t(θ, f1))} is SE ∀ (θ, f1) ∈ Θ×F .

IF E|yt|m <∞ and E|f̃t(θ, f1))|m <∞

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ

IF Model parameter identi�cation (local or global)

IF Correct speci�cation or unique pseudo-true parameter

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF {(yt, f̃t(θ, f1))} is SE ∀ (θ, f1) ∈ Θ×F .

IF E|yt|m <∞ and E|f̃t(θ, f1))|m <∞

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ

IF Model parameter identi�cation (local or global)

IF Correct speci�cation or unique pseudo-true parameter

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF {(yt, f̃t(θ, f1))} is SE ∀ (θ, f1) ∈ Θ×F .

IF E|yt|m <∞ and E|f̃t(θ, f1))|m <∞

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ

IF Invertibility conditions on g, h, S and pu

IF Restrictions on Θ and Var(st(f̃t)) > 0

IF Correct speci�cation or unique pseudo-true parameter

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF {(yt, f̃t(θ, f1))} is SE ∀ (θ, f1) ∈ Θ×F .

IF E|yt|m <∞ and E|f̃t(θ, f1))|m <∞

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ

IF Invertibility conditions on g, h, S and pu

IF Restrictions on Θ and Var(st(f̃t)) > 0

IF Correct speci�cation or unique pseudo-true parameter

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

IF {(yt, f̃t(θ, f1))} is SE ∀ (θ, f1) ∈ Θ×F .

IF E|yt|m <∞ and E|f̃t(θ, f1))|m <∞

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ

IF Invertibility conditions on g, h, S and pu

IF Restrictions on Θ and Var(st(f̃t)) > 0

IF Correct speci�cation or unique pseudo-true parameter

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Consistency

Consistency Conditions:

• IF {(yt, f̃t(θ, f1))} is SE ∀ (θ, f1) ∈ Θ×F .

• IF E|yt|m <∞ and E|f̃t(θ, f1))|m <∞

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ

IF Invertibility conditions on g, h, S and pu

IF Restrictions on Θ and Var(st(f̃t)) > 0

IF Correct speci�cation or unique pseudo-true parameter

THEN θ̂Ta.s.→ θ0 as T →∞.

6 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Asymptotic Normality

Asymptotic Normality Conditions:

IF CLT for the score:

√TL′T (θ0, f1)

d→ N(0, I(θ0)) ∀ f1 ∈ F as T →∞.

IF Uniform convergence of second derivative:

supθ∈Θ|L′′T (θ, f1|yT )− L′′∞(θ)| a.s.→ 0 ∀ f1 ∈ F as T →∞.

IF I(θ0) is invertible.

IF θ̂Ta.s.→ θ0 ∈ int(Θ) as T →∞.

THEN√T (θ̂T − θ0)

d→ N(0, I(θ0)−1) as T →∞.

7 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Asymptotic Normality

Asymptotic Normality Conditions:

IF CLT for the score:

√TL′T (θ0, f1)

d→ N(0, I(θ0)) ∀ f1 ∈ F as T →∞.

IF Uniform convergence of second derivative:

supθ∈Θ|L′′T (θ, f1|yT )− L′′∞(θ)| a.s.→ 0 ∀ f1 ∈ F as T →∞.

IF I(θ0) is invertible.

IF θ̂Ta.s.→ θ0 ∈ int(Θ) as T →∞.

THEN√T (θ̂T − θ0)

d→ N(0, I(θ0)−1) as T →∞.

7 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Asymptotic Normality

Asymptotic Normality Conditions:

IF CLT for the score:

√TL′T (θ0, f1)

d→ N(0, I(θ0)) ∀ f1 ∈ F as T →∞.

IF Uniform convergence of second derivative:

supθ∈Θ|L′′T (θ, f1|yT )− L′′∞(θ)| a.s.→ 0 ∀ f1 ∈ F as T →∞.

IF I(θ0) is invertible.

IF θ̂Ta.s.→ θ0 ∈ int(Θ) as T →∞.

THEN√T (θ̂T − θ0)

d→ N(0, I(θ0)−1) as T →∞.

7 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Asymptotic Normality

Asymptotic Normality Conditions:

IF CLT for the score:

√TL′T (θ0, f1)

d→ N(0, I(θ0)) ∀ f1 ∈ F as T →∞.

IF Uniform convergence of second derivative:

supθ∈Θ|L′′T (θ, f1|yT )− L′′∞(θ)| a.s.→ 0 ∀ f1 ∈ F as T →∞.

IF I(θ0) is invertible.

IF θ̂Ta.s.→ θ0 ∈ int(Θ) as T →∞.

THEN√T (θ̂T − θ0)

d→ N(0, I(θ0)−1) as T →∞.

7 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Asymptotic Normality

Asymptotic Normality Conditions:

IF CLT for the score:

√TL′T (θ0, f1)

d→ N(0, I(θ0)) ∀ f1 ∈ F as T →∞.

IF Uniform convergence of second derivative:

supθ∈Θ|L′′T (θ, f1|yT )− L′′∞(θ)| a.s.→ 0 ∀ f1 ∈ F as T →∞.

IF I(θ0) is invertible.

IF θ̂Ta.s.→ θ0 ∈ int(Θ) as T →∞.

THEN√T (θ̂T − θ0)

d→ N(0, I(θ0)−1) as T →∞.

7 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Asymptotic Normality

Asymptotic Normality Conditions:

IF CLT for the score:

√TL′T (θ0, f1)

d→ N(0, I(θ0)) ∀ f1 ∈ F as T →∞.

IF Uniform convergence of second derivative:

supθ∈Θ|L′′T (θ, f1|yT )− L′′∞(θ)| a.s.→ 0 ∀ f1 ∈ F as T →∞.

IF I(θ0) is invertible.

IF θ̂Ta.s.→ θ0 ∈ int(Θ) as T →∞.

THEN√T (θ̂T − θ0)

d→ N(0, I(θ0)−1) as T →∞.

7 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Asymptotic Normality

Asymptotic Normality Conditions:

IF CLT for the score:

√TL′T (θ0, f1)

d→ N(0, I(θ0)) ∀ f1 ∈ F as T →∞.

IF Uniform convergence of second derivative:

supθ∈Θ|L′′T (θ, f1|yT )− L′′∞(θ)| a.s.→ 0 ∀ f1 ∈ F as T →∞.

IF I(θ0) is invertible.

IF θ̂Ta.s.→ θ0 ∈ int(Θ) as T →∞.

THEN√T (θ̂T − θ0)

d→ N(0, I(θ0)−1) as T →∞.

7 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Asymptotic Normality

Asymptotic Normality Conditions:

IF{

(yt, f̃t(θ, f1), f̃ ′t(θ, f1), f̃ ′′t (θ, f1))}is SE

∀ (θ, f1) ∈ Θ×F .

IF E|yt|m <∞, E|f̃t(θ, f1))|m <∞ , E|f̃ ′t(θ, f1))|m <∞ and

E|f̃ ′′t (θ, f1))|m <∞.

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ.

IF I(θ0) is invertible.

IF θ̂Ta.s.→ θ0 ∈ int(Θ) as T →∞.

THEN√T (θ̂T − θ0)

d→ N(0, I(θ0)−1) as T →∞.

7 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Asymptotic Normality

Asymptotic Normality Conditions:

IF{

(yt, f̃t(θ, f1), f̃ ′t(θ, f1), f̃ ′′t (θ, f1))}is SE

∀ (θ, f1) ∈ Θ×F .

IF E|yt|m <∞, E|f̃t(θ, f1))|m <∞ , E|f̃ ′t(θ, f1))|m <∞ and

E|f̃ ′′t (θ, f1))|m <∞.

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ.

IF I(θ0) is invertible.

IF θ̂Ta.s.→ θ0 ∈ int(Θ) as T →∞.

THEN√T (θ̂T − θ0)

d→ N(0, I(θ0)−1) as T →∞.

7 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Asymptotic Normality

Asymptotic Normality Conditions:

IF{

(yt, f̃t(θ, f1), f̃ ′t(θ, f1), f̃ ′′t (θ, f1))}is SE

∀ (θ, f1) ∈ Θ×F .

IF E|yt|m <∞, E|f̃t(θ, f1))|m <∞ , E|f̃ ′t(θ, f1))|m <∞ and

E|f̃ ′′t (θ, f1))|m <∞.

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ.

IF I(θ0) is invertible.

IF θ̂Ta.s.→ θ0 ∈ int(Θ) as T →∞.

THEN√T (θ̂T − θ0)

d→ N(0, I(θ0)−1) as T →∞.

7 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Asymptotic Normality

Asymptotic Normality Conditions:

IF{

(yt, f̃t(θ, f1), f̃ ′t(θ, f1), f̃ ′′t (θ, f1))}is SE

∀ (θ, f1) ∈ Θ×F .

IF E|yt|m <∞, E|f̃t(θ, f1))|m <∞ , E|f̃ ′t(θ, f1))|m <∞ and

E|f̃ ′′t (θ, f1))|m <∞.

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ.

IF I(θ0) is invertible.

IF θ̂Ta.s.→ θ0 ∈ int(Θ) as T →∞.

THEN√T (θ̂T − θ0)

d→ N(0, I(θ0)−1) as T →∞.

7 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Asymptotic Normality

Asymptotic Normality Conditions:

IF{

(yt, f̃t(θ, f1), f̃ ′t(θ, f1), f̃ ′′t (θ, f1))}is SE

∀ (θ, f1) ∈ Θ×F .

IF E|yt|m <∞, E|f̃t(θ, f1))|m <∞ , E|f̃ ′t(θ, f1))|m <∞ and

E|f̃ ′′t (θ, f1))|m <∞.

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ.

IF I(θ0) is invertible.

IF θ̂Ta.s.→ θ0 ∈ int(Θ) as T →∞.

THEN√T (θ̂T − θ0)

d→ N(0, I(θ0)−1) as T →∞.

7 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Asymptotic Normality

Asymptotic Normality Conditions:

IF{

(yt, f̃t(θ, f1), f̃ ′t(θ, f1), f̃ ′′t (θ, f1))}is SE

∀ (θ, f1) ∈ Θ×F .

IF E|yt|m <∞, E|f̃t(θ, f1))|m <∞ , E|f̃ ′t(θ, f1))|m <∞ and

E|f̃ ′′t (θ, f1))|m <∞.

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ.

IF I(θ0) is invertible.

IF θ̂Ta.s.→ θ0 ∈ int(Θ) as T →∞.

THEN√T (θ̂T − θ0)

d→ N(0, I(θ0)−1) as T →∞.

7 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Asymptotic Normality

Asymptotic Normality Conditions:

IF{

(yt, f̃t(θ, f1), f̃ ′t(θ, f1), f̃ ′′t (θ, f1))}is SE

∀ (θ, f1) ∈ Θ×F .

IF E|yt|m <∞, E|f̃t(θ, f1))|m <∞ , E|f̃ ′t(θ, f1))|m <∞ and

E|f̃ ′′t (θ, f1))|m <∞.

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ.

IF Previous identi�cation conditions on Θ and g, h, S and pu.

IF θ̂Ta.s.→ θ0 ∈ int(Θ) as T →∞.

THEN√T (θ̂T − θ0)

d→ N(0, I(θ0)−1) as T →∞.

7 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Asymptotic Normality

Asymptotic Normality Conditions:

IF{

(yt, f̃t(θ, f1), f̃ ′t(θ, f1), f̃ ′′t (θ, f1))}is SE

∀ (θ, f1) ∈ Θ×F .

IF E|yt|m <∞, E|f̃t(θ, f1))|m <∞ , E|f̃ ′t(θ, f1))|m <∞ and

E|f̃ ′′t (θ, f1))|m <∞.

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ.

IF Previous identi�cation conditions on Θ and g, h, S and pu.

IF θ̂Ta.s.→ θ0 ∈ int(Θ) as T →∞.

THEN√T (θ̂T − θ0)

d→ N(0, I(θ0)−1) as T →∞.

7 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

ML Asymptotic Normality

Asymptotic Normality Conditions:

• [IF{

(yt, f̃t(θ, f1), f̃ ′t(θ, f1), f̃ ′′t (θ, f1))}is SE

∀ (θ, f1) ∈ Θ×F .

• [IF E|yt|m <∞, E|f̃t(θ, f1))|m <∞ , E|f̃ ′t(θ, f1))|m <∞ and

E|f̃ ′′t (θ, f1))|m <∞.

IF g, h, S and pu are smooth with bounded nth derivative

IF Compact Θ.

IF Previous identi�cation conditions on Θ and g, h, S and pu.

IF θ̂Ta.s.→ θ0 ∈ int(Θ) as T →∞.

THEN√T (θ̂T − θ0)

d→ N(0, I(θ0)−1) as T →∞.

7 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Stationarity, Ergodicity and Moments

What we want: SE and Moments for {(yt, f̃dt (θ, f1))}

Unknown DGP: (Incorrect Speci�cation)

IF: {yt} is SE and E|yt|ny <∞ (unknown DGP, test data)

IF: Contracting update f̃t+1 = ω + αs(yt, f̃t;λ) + βf̃t

THEN: {f̃ (d)t (θ, f1)} is SE and E|f̃ (d)

t (θ, f1)|nf <∞.

8 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Stationarity, Ergodicity and Moments

What we want: SE and Moments for {(yt, f̃dt (θ, f1))}

Unknown DGP: (Incorrect Speci�cation)

IF: {yt} is SE and E|yt|ny <∞ (unknown DGP, test data)

IF: Contracting update f̃t+1 = ω + αs(yt, f̃t;λ) + βf̃t

THEN: {f̃ (d)t (θ, f1)} is SE and E|f̃ (d)

t (θ, f1)|nf <∞.

8 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Stationarity, Ergodicity and Moments

What we want: SE and Moments for {(yt, f̃dt (θ, f1))}

Unknown DGP: (Incorrect Speci�cation)

IF: {yt} is SE and E|yt|ny <∞ (unknown DGP, test data)

IF: Contracting update f̃t+1 = ω + αs(yt, f̃t;λ) + βf̃t

THEN: {f̃ (d)t (θ, f1)} is SE and E|f̃ (d)

t (θ, f1)|nf <∞.

8 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Stationarity, Ergodicity and Moments

What we want: SE and Moments for {(yt, f̃dt (θ, f1))}

Unknown DGP: (Incorrect Speci�cation)

IF: {yt} is SE and E|yt|ny <∞ (unknown DGP, test data)

IF: Contracting update f̃t+1 = ω + αs(yt, f̃t;λ) + βf̃t

THEN: {f̃ (d)t (θ, f1)} is SE and E|f̃ (d)

t (θ, f1)|nf <∞.

8 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Stationarity, Ergodicity and Moments

What we want: SE and Moments for {(yt, f̃ (d)t (θ, f1))}

Unknown DGP: (Incorrect Speci�cation)

IF: {yt} is SE and E|yt|ny <∞ (unknown DGP, test data)

IF: Contracting update f̃t+1 = ω + αs(yt, f̃t;λ) + βf̃t

THEN: {f̃ (d)t (θ, f1)} is SE and E|f̃ (d)

t (θ, f1)|nf <∞.

9 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Stationarity, Ergodicity and Moments

What we want: SE and Moments for {(yt, f̃dt (θ, f1))}

Unknown DGP: (Incorrect Speci�cation)

IF: {yt} is SE and E|yt|ny <∞ (unknown DGP, test data)

IF: E supf∣∣α+ β∂s(yt, f ;λ)/∂f

∣∣nf < 1

THEN: {f̃ (d)t (θ, f1)} is SE and E|f̃ (d)

t (θ, f1)|nf <∞.

9 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Stationarity, Ergodicity and Moments

What we want: SE and Moments for {(yt, fdt (θ, f1))}

Unknown DGP: (Incorrect Speci�cation)

• IF: {yt} is SE and E|yt|ny <∞ (unknown DGP, test data)

IF: E supf∣∣β + α ∂s(yt, f ;λ)/∂f

∣∣nf < 1

THEN: {f̃ (d)t (θ, f1)} is SE and E|f̃ (d)

t (θ, f1)|nf <∞.

Known DGP: (Correct Speci�cation)

DGP: yt = g(h(ft), ut) , ft+1 = ω + αs(yt, ft;λ) + βft

9 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Stationarity, Ergodicity and Moments

What we want: SE and Moments for {(yt, fdt (θ, f1))}

Unknown DGP: (Incorrect Speci�cation)

• IF: {yt} is SE and E|yt|ny <∞ (unknown DGP, test data)

IF: E supf∣∣α+ β∂s(yt, f ;λ)/∂f

∣∣nf < 1

THEN: {f̃ (d)t (θ, f1)} is SE and E|f̃ (d)

t (θ, f1)|nf <∞.

Known DGP: (Correct Speci�cation)

DGP: yt = g(h(ft), ut) , ft+1 = ω + αs(yt, ft;λ) + βft

9 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Stationarity, Ergodicity and Moments

What we want: SE and Moments for {(yt, fdt (θ, f1))}

Unknown DGP: (Incorrect Speci�cation)

• IF: {yt} is SE and E|yt|ny <∞ (unknown DGP, test data)

IF: E supf∣∣α+ β∂s(yt, f ;λ)/∂f

∣∣nf < 1

THEN: {f̃ (d)t (θ, f1)} is SE and E|f̃ (d)

t (θ, f1)|nf <∞.

Known DGP: (Correct Speci�cation)

DGP: yt = g(h(ft), ut) , ft+1 = ω + αsu(ut, ft;λ) + βft

e.g., su(ut, ft;λ) = ftu2t , whereas s(yt, ft;λ) = y2

t

9 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Stationarity, Ergodicity and Moments

What we want: SE and Moments for {(yt, fdt (θ, f1))}

Unknown DGP: (Incorrect Speci�cation)

• IF: {yt} is SE and E|yt|ny <∞ (unknown DGP, test data)

IF: E supf∣∣α+ β∂s(yt, f ;λ)/∂f

∣∣nf < 1

THEN: {f̃ (d)t (θ, f1)} is SE and E|f̃ (d)

t (θ, f1)|nf <∞.

Known DGP: (Correct Speci�cation)

DGP: yt = g(h(ft), ut) , ft+1 = ω + αsu(ut, ft;λ) + βft

e.g., su(ut, ft;λ) = ftu2t , whereas s(yt, ft;λ) = y2

t

9 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Stationarity, Ergodicity and Moments

What we want: SE and Moments for {(yt, fdt (θ, f1))}

Unknown DGP: (Incorrect Speci�cation)

• IF: {yt} is SE and E|yt|ny <∞ (unknown DGP, test data)

IF: E supf∣∣β + α∂s(yt, f ;λ)/∂f

∣∣nf < 1

THEN: {f̃ (d)t (θ, f1)} is SE and E|f̃ (d)

t (θ, f1)|nf <∞.

Known DGP: (Correct Speci�cation)

DGP: yt = g(h(ft), ut) , ft+1 = ω + αsu(ut, ft;λ) + βft

IF: E supf∣∣β + α ∂su(ut, f ;λ)/∂f

∣∣nf < 1

THEN: {yt} is SE and E|yt|ny <∞.

9 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Stationarity, Ergodicity and Moments

What we want: SE and Moments for {(yt, fdt (θ, f1))}

Unknown DGP: (Incorrect Speci�cation)

• IF: {yt} is SE and E|yt|ny <∞ (unknown DGP, test data)

IF: E supf∣∣β + α∂s(yt, f ;λ)/∂f

∣∣nf < 1

THEN: {f̃ (d)t (θ, f1)} is SE and E|f̃ (d)

t (θ, f1)|nf <∞.

Known DGP: (Correct Speci�cation)

DGP: yt = g(h(ft), ut) , ft+1 = ω + αsu(ut, ft;λ) + βft

IF: E supf∣∣β + α ∂su(ut, f ;λ)/∂f

∣∣nf < 1

THEN: {yt} is SE and E|yt|ny <∞.

9 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Example: Time-Varying Duration and Fat Tails

yt = g(ft, ut) = exp(ft) · ut, withpu(ut;λ) = (1 + λ−1ut)

−(λ+1), Creal et al. (2008), Harvey

(2013), Koopman et al. (2012)

β + α∂st(ft;λ)/∂ft = β − α exp(−ft)yt(1 + λ−1 exp(−ft)yt)2

Supremum: max(|β|, |β − αλ/4|), so region shrinks if larger

λ values are allowed

Contraction holds for arbitrary nf as the sup is

independent of yt

Correct speci�cation much easier, as

su(ft, ut;λ) = ut1+λ−1ut

− 1, so β + α ∂su(ft,ut;λ)∂ft

= β

10 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Example: Time-Varying Duration and Fat Tails

yt = g(ft, ut) = exp(ft) · ut, withpu(ut;λ) = (1 + λ−1ut)

−(λ+1), Creal et al. (2008), Harvey

(2013), Koopman et al. (2012)

β + α∂st(ft;λ)/∂ft = β − α exp(−ft)yt(1 + λ−1 exp(−ft)yt)2

Supremum: max(|β|, |β − αλ/4|), so region shrinks if larger

λ values are allowed

Contraction holds for arbitrary nf as the sup is

independent of yt

Correct speci�cation much easier, as

su(ft, ut;λ) = ut1+λ−1ut

− 1, so β + α ∂su(ft,ut;λ)∂ft

= β

10 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Example: Time-Varying Duration and Fat Tails

yt = g(ft, ut) = exp(ft) · ut, withpu(ut;λ) = (1 + λ−1ut)

−(λ+1), Creal et al. (2008), Harvey

(2013), Koopman et al. (2012)

β + α∂st(ft;λ)/∂ft = β − α exp(−ft)yt(1 + λ−1 exp(−ft)yt)2

Supremum: max(|β|, |β − αλ/4|), so region shrinks if larger

λ values are allowed

Contraction holds for arbitrary nf as the sup is

independent of yt

Correct speci�cation much easier, as

su(ft, ut;λ) = ut1+λ−1ut

− 1, so β + α ∂su(ft,ut;λ)∂ft

= β

10 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Example: Time-Varying Duration and Fat Tails

yt = g(ft, ut) = exp(ft) · ut, withpu(ut;λ) = (1 + λ−1ut)

−(λ+1), Creal et al. (2008), Harvey

(2013), Koopman et al. (2012)

β + α∂st(ft;λ)/∂ft = β − α exp(−ft)yt(1 + λ−1 exp(−ft)yt)2

Supremum: max(|β|, |β − αλ/4|), so region shrinks if larger

λ values are allowed

Contraction holds for arbitrary nf as the sup is

independent of yt

Correct speci�cation much easier, as

su(ft, ut;λ) = ut1+λ−1ut

− 1, so β + α ∂su(ft,ut;λ)∂ft

= β

10 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Example: Time-Varying Duration and Fat Tails

yt = g(ft, ut) = exp(ft) · ut, withpu(ut;λ) = (1 + λ−1ut)

−(λ+1), Creal et al. (2008), Harvey

(2013), Koopman et al. (2012)

β + α∂st(ft;λ)/∂ft = β − α exp(−ft)yt(1 + λ−1 exp(−ft)yt)2

Supremum: max(|β|, |β − αλ/4|), so region shrinks if larger

λ values are allowed

Contraction holds for arbitrary nf as the sup is

independent of yt

Correct speci�cation much easier, as

su(ft, ut;λ) = ut1+λ−1ut

− 1, so β + α ∂su(ft,ut;λ)∂ft

= β

10 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Example: Time-Varying Duration and Fat Tails

Figure: Asy. normality regions for Burr duration models

yt = exp(ft)ut , ut ∼ Burr(λ) , Approximate Inverse Info Scaling

11 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Example: t-GAS Volatility model

yt = g(ft, ut) =√ft · ut, with pu(ut;λ) = t(λ), and

b = β − (1 + 3λ−1)α, see Creal et al. (2011, 2013)

β+α∂st(ft;λ)/∂ft = β+α(1+3λ−1)((1 + λ)(y2

t /(λft))2

(1 + y2t /(λft))

2−1)

Supremum for f ↓ 0: β + (λ+ 3) · α: small regions for largeλ

Interesting: discontinuous towards the normal (i.e.,

GARCH)

Correct speci�cation much easier, as su is linear in ft, and

under correct speci�cation and at the true parameter we

have again E supf∣∣β + α∂su(f, ut;λ)/∂f

∣∣2 < 1, which can

be done analytically. Finiteness of moments follows

similarly

12 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Example: t-GAS Volatility model

yt = g(ft, ut) =√ft · ut, with pu(ut;λ) = t(λ), and

b = β − (1 + 3λ−1)α, see Creal et al. (2011, 2013)

β+α∂st(ft;λ)/∂ft = β+α(1+3λ−1)((1 + λ)(y2

t /(λft))2

(1 + y2t /(λft))

2−1)

Supremum for f ↓ 0: β + (λ+ 3) · α: small regions for largeλ

Interesting: discontinuous towards the normal (i.e.,

GARCH)

Correct speci�cation much easier, as su is linear in ft, and

under correct speci�cation and at the true parameter we

have again E supf∣∣β + α∂su(f, ut;λ)/∂f

∣∣2 < 1, which can

be done analytically. Finiteness of moments follows

similarly

12 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Example: t-GAS Volatility model

yt = g(ft, ut) =√ft · ut, with pu(ut;λ) = t(λ), and

b = β − (1 + 3λ−1)α, see Creal et al. (2011, 2013)

β+α∂st(ft;λ)/∂ft = β+α(1+3λ−1)((1 + λ)(y2

t /(λft))2

(1 + y2t /(λft))

2−1)

Supremum for f ↓ 0: β + (λ+ 3) · α: small regions for largeλ

Interesting: discontinuous towards the normal (i.e.,

GARCH)

Correct speci�cation much easier, as su is linear in ft, and

under correct speci�cation and at the true parameter we

have again E supf∣∣β + α∂su(f, ut;λ)/∂f

∣∣2 < 1, which can

be done analytically. Finiteness of moments follows

similarly

12 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Example: t-GAS Volatility model

yt = g(ft, ut) =√ft · ut, with pu(ut;λ) = t(λ), and

b = β − (1 + 3λ−1)α, see Creal et al. (2011, 2013)

β+α∂st(ft;λ)/∂ft = β+α(1+3λ−1)((1 + λ)(y2

t /(λft))2

(1 + y2t /(λft))

2−1)

Supremum for f ↓ 0: β + (λ+ 3) · α: small regions for largeλ

Interesting: discontinuous towards the normal (i.e.,

GARCH)

Correct speci�cation much easier, as su is linear in ft, and

under correct speci�cation and at the true parameter we

have again E supf∣∣β + α∂su(f, ut;λ)/∂f

∣∣2 < 1, which can

be done analytically. Finiteness of moments follows

similarly

12 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Example: t-GAS Volatility model

yt = g(ft, ut) =√ft · ut, with pu(ut;λ) = t(λ), and

b = β − (1 + 3λ−1)α, see Creal et al. (2011, 2013)

β+α∂st(ft;λ)/∂ft = β+α(1+3λ−1)((1 + λ)(y2

t /(λft))2

(1 + y2t /(λft))

2−1)

Supremum for f ↓ 0: β + (λ+ 3) · α: small regions for largeλ

Interesting: discontinuous towards the normal (i.e.,

GARCH)

Correct speci�cation much easier, as su is linear in ft, and

under correct speci�cation and at the true parameter we

have again E supf∣∣β + α∂su(f, ut;λ)/∂f

∣∣2 < 1, which can

be done analytically. Finiteness of moments follows

similarly12 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Example: t-GAS Volatility model

Supremum for f ↓ 0 is too strict to be useful: we let

f ↓ f = ω/(1− β), i.e., asymptotic lower bound w.p. 1

AN condition

E supΘ

(β + α(1 + 3λ−1)

((1 + λ)(y2t /(λf))2

(1 + y2t /(λf))2

− 1))2

< 1

exist if Θ is compact

Estimation

1

T

T∑t=1

supΘ

(β + α(1 + 3λ−1)

((1 + λ)(y2t /(λf))2

(1 + y2t /(λf))2

− 1))2

< 1

13 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Example: t-GAS Volatility model

Supremum for f ↓ 0 is too strict to be useful: we let

f ↓ f = ω/(1− β), i.e., asymptotic lower bound w.p. 1

AN condition

E supΘ

(β + α(1 + 3λ−1)

((1 + λ)(y2t /(λf))2

(1 + y2t /(λf))2

− 1))2

< 1

exist if Θ is compact

Estimation

1

T

T∑t=1

supΘ

(β + α(1 + 3λ−1)

((1 + λ)(y2t /(λf))2

(1 + y2t /(λf))2

− 1))2

< 1

13 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Example: t-GAS Volatility model

Figure: CAN regions for t-Volatility GAS in Creal at al. (2013)

yt =√ftut , ut ∼ t(λ) , Inverse Info Scaling

Diagonal restriction: β > (1 + 3λ−1)α ensures ft > 0.

DGP t GAS, β = 0.8, α = 0.05, T = 1, 000

14 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Example: t-GAS Volatility model

Figure: CAN regions for t-Volatility GAS in Creal at al. (2013)

yt =√ftut , ut ∼ t(λ) , Inverse Info Scaling

Diagonal restriction: β > (1 + 3λ−1)α ensures ft > 0.

DGP t GAS, β = 0.0, α = 0.0, T = 1, 000

15 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Why a discontinuity wrt the normal case?

Figure: Phase diagram (ft, ft+1) for �xed yt; λ varies, y2t = 4

16 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

Concluding remarks

GAS models: not only empirically attractive, but also a

solid theoretical backbone

MLE estimation of static parameters for GAS models can

be given a solid basis

Low-level conditions

Correct and incorrect speci�cation results

GAS models provide intriguing new puzzles for

theoreticians

GASMODEL.COM

17 / 17 F.Blasques, S.J.Koopman and A.Lucas ML Estimation of GAS Models

This project is funded by the European Union

under the 7th Framework Programme

(FP7-SSH/2007-2013) Grant Agreement n°320270

www.syrtoproject.eu