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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