Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Merits and drawbacks of variance targeting in
GARCH models
Christian Francq1
Lajos Horvath2
Jean-Michel Zakoïan3
1 Université Lille 3
http://perso.univ-lille3.fr/~cfrancq
2 University
of Utah
3 CREST
GERAD, Montréal
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Outline
1
Volatility Models and QMLE
Properties of Financial Time Series
Models for the Volatility of Financial Returns
2
Variance Targeting Estimator
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
3
Conclusion
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Outline
1
Volatility Models and QMLE
Properties of Financial Time Series
Models for the Volatility of Financial Returns
2
Variance Targeting Estimator
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
3
Conclusion
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Stylized Facts (Mandelbrot (1963))
4000
2000
price
6000
Non stationarity of the prices
19/Aug/91
11/Sep/01
21/Jan/08
CAC 40, from March 1, 1992 to February 20, 2009
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Stylized Facts (Mandelbrot (1963))
10000
6000
price
14000
Non stationarity of the prices
25/Oct/00
1/Dec/08
S&P TSX, from January 3, 2000 to February 20, 2009
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Stylized Facts
0
−5
−10
Returns
5
10
Possible stationarity of the returns
19/Aug/91
11/Sep/01
21/Jan/08
CAC 40 returns, from March 2, 1992 to February 20, 2009
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Stylized Facts
0
−10
−5
Returns
5
10
Possible stationarity of the returns
25/Oct/00
1/Dec/08
S&P TSX returns, from January 3, 2000 to February 20, 2009
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Stylized Facts
0
−10
−5
Returns
5
10
Volatility clustering
21/Jan/08
06/Oct/08
CAC 40 returns, from January 2, 2008 to February 20, 2009
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Stylized Facts
0
−10
−5
Returns
5
10
Conditional heteroskedasticity (compatible with marginal homoscedasticity and even
stationarity)
01/Dec/08
S&P TSX returns, from January 2, 2008 to February 20, 2009
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Stylized Facts
−0.10
0.00 0.05 0.10
Dependence without correlation (warning: interpretation of the dotted lines)
0
5
10
15
20
25
30
Empirical autocorrelations of the CAC returns
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
35
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Stylized Facts
−0.10
0.00 0.05 0.10
Dependence without correlation
0
5
10
15
20
25
30
Empirical autocorrelations of the S&P TSX returns
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Stylized Facts
−0.2
0.0
0.2
0.4
Correlation of the squares
0
5
10
15
20
25
30
Autocorrelations of the squares of the CAC returns
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
35
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Stylized Facts
−0.2
0.0
0.2
0.4
Correlation of the squares
0
5
10
15
20
25
30
Autocorrelations of the squares of the S&P TSX returns
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Stylized Facts
0.2
0.0
0.1
Density
0.3
Tail heaviness of the distributions
−10
−5
0
5
10
Density estimator for the CAC returns (normal in dotted line)
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Stylized Facts
0.2
0.0
Density
0.4
Tail heaviness of the distributions
−10
−5
0
5
Density estimator for the S&P TSX returns (normal in dotted
line)
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
10
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Stylized Facts
Decreases of prices have an higher impact on the future volatility than increases of the
same magnitude
Table: Autocorrelations of tranformations of the CAC returns h
ρ̂ (h)
ρ̂|| (h)
ρ̂(+
t−h , |t |)
ρ̂(−−
t−h , |t |)
1
-0.01
0.19
0.03
0.18
2
-0.03
0.24
0.07
0.20
Francq, Horvath, Zakoïan
3
-0.05
0.25
0.07
0.22
4
0.04
0.23
0.08
0.18
5
-0.06
0.25
0.08
0.21
6
-0.02
0.23
0.12
0.15
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Stylized Facts
Decreases of prices have an higher impact on the future volatility than increases of the
same magnitude
Table: Autocorrelations of tranformations of the S&P TSX returns h
ρ̂ (h)
ρ̂|| (h)
ρ̂(+
t−h , |t |)
ρ̂(−−
t−h , |t |)
1
-0.05
0.30
0.12
0.23
2
-0.06
0.28
0.10
0.22
Francq, Horvath, Zakoïan
3
0.06
0.26
0.07
0.23
4
0.00
0.28
0.10
0.23
5
-0.09
0.35
0.16
0.26
6
0.00
0.39
0.23
0.24
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Outline
1
Volatility Models and QMLE
Properties of Financial Time Series
Models for the Volatility of Financial Returns
2
Variance Targeting Estimator
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
3
Conclusion
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Classes of Volatility Models
Amost all the models are of the form
t = σt ηt
where
(ηt ) is an iid (0,1) process
(σt ) is a process (volatility), σt > 0
the variables σt and ηt are independent
Two main classes of models:
GARCH-type (Generalized Autoregressive Conditional
Heteroskedasticity): σt ∈ σ(t−1 , t−2 , . . .)
Stochastic volatility
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Classes of Volatility Models
Amost all the models are of the form
t = σt ηt
where
(ηt ) is an iid (0,1) process
(σt ) is a process (volatility), σt > 0
the variables σt and ηt are independent
Two main classes of models:
GARCH-type (Generalized Autoregressive Conditional
Heteroskedasticity): σt ∈ σ(t−1 , t−2 , . . .)
Stochastic volatility
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Definition: GARCH(p, q)
Definition (Engle (1982), Bollerslev (1986))

 t = σt ηt

σt2 = ω0 +
Pq
2
i=1 α0i t−i
+
Pp
2
j=1 β0j σt−j ,
∀t ∈ Z
where
(ηt ) iid, Eηt = 0, Eηt2 = 1,
ω0 > 0,
α0i ≥ 0,
β0j ≥ 0.
θ0 = (ω0 , α01 , . . . , α0q , β01 , . . . , β0p ).
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
0
−10
εt
5
10
GARCH(1,1) simulation
0
1000
2000
3000
4000
2 ,
t = σt ηt , ηt iid St5 , σt2 = 0.033 + 0.0902t−1 + 0.893σt−1
t = 1, . . . , n = 4791
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
5
0
−10
Returns
10
The previous GARCH(1,1) simulation resembles real
financial series
0
1000
2000
3000
4000
CAC returns
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Stricty Stationarity

α01 ηt2

A0t = 
 α01
γ(A0 ) =

· · · α0q ηt2 β01 ηt2 · · · β0p ηt2

Iq−1
0
0
.
···
α0q
β01
···
β0p 
0
Ip−1
0
lim a.s.
t→∞
1
log kA0t A0t−1 . . . A01 k.
t
Theorem (Bougerol & Picard, 1992)
The model has a (unique) strictly stationary non anticipative
solution iff
γ(A0 ) < 0.
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Quasi-Maximum Likelihood Estimation
A QMLE of θ is defined as any measurable solution θ̂n of
θ̂n = arg min l̃n (θ),
θ∈Θ
where l̃n (θ) = n−1
Pn
˜
t=1 `t ,
and `˜t =
2t
σ̃t2
+ log σ̃t2 .
Remark
The constraint σ̃t2 > 0 for all θ ∈ Θ is necessary to compute
l̃n (θ).
The QMLE is always constrained: the "unrestricted" QMLE
does not exist.
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Quasi-Maximum Likelihood Estimation
Theorem (Berkes, Horváth and Kokoszka (2003), FZ (2004))
Under appropriate conditions (in particular strict stationarity
and θ0 > 0)
√
κη =
L
n(θ̂n − θ0 ) → N (0, (κη − 1)J −1 ),
Eηt4 ,
J = Eθ0
Francq, Horvath, Zakoïan
1 ∂σt2 (θ0 ) ∂σt2 (θ0 )
∂θ0
σt4 (θ0 ) ∂θ
.
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Properties of Financial Time Series
Models for the Volatility of Financial Returns
Drawbacks of the QMLE
Require a numerical optimization which is difficult when the
number of parameters is large;
The numerical optimization is sensitive to the choice of the
initial value;
The variance of the estimated model can be far from the
empirical variance.
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Outline
1
Volatility Models and QMLE
Properties of Financial Time Series
Models for the Volatility of Financial Returns
2
Variance Targeting Estimator
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
3
Conclusion
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Variance Targeting Principle
Engle and Mezrich (1996)
Principle of the two-step estimator:
1
2
The unconditional variance is estimated by the sample
variance;
The remaining parameters are estimated by QML.
Advantages:
Facilitates the numerical optimization by reducing the
dimensionality of the parameter space;
Speeds up the convergence of the optimization routines;
Ensures a consistent estimate of the long-run variance
even when the model is misspecified;
Provides reasonable initial values for the QMLE.
Potential drawbacks:
Requires stronger assumptions (existence of the variance);
Is likely to suffer from efficiency loss.
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Variance Targeting Principle
Engle and Mezrich (1996)
Principle of the two-step estimator:
1
2
The unconditional variance is estimated by the sample
variance;
The remaining parameters are estimated by QML.
Advantages:
Facilitates the numerical optimization by reducing the
dimensionality of the parameter space;
Speeds up the convergence of the optimization routines;
Ensures a consistent estimate of the long-run variance
even when the model is misspecified;
Provides reasonable initial values for the QMLE.
Potential drawbacks:
Requires stronger assumptions (existence of the variance);
Is likely to suffer from efficiency loss.
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Variance Targeting Principle
Engle and Mezrich (1996)
Principle of the two-step estimator:
1
2
The unconditional variance is estimated by the sample
variance;
The remaining parameters are estimated by QML.
Advantages:
Facilitates the numerical optimization by reducing the
dimensionality of the parameter space;
Speeds up the convergence of the optimization routines;
Ensures a consistent estimate of the long-run variance
even when the model is misspecified;
Provides reasonable initial values for the QMLE.
Potential drawbacks:
Requires stronger assumptions (existence of the variance);
Is likely to suffer from efficiency loss.
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Objectives
Establish the asymptotic distribution of the VTE in
univariate GARCH models
Provide effective comparisons with the standard QML;
Discuss the relative merits and drawbacks of the variance
targeting method.
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Reparameterization of the Standard GARCH(1,1)
Standard form:
p
t = ht ηt
ht = ω0 + α0 2t−1 + β0 ht−1 ,
where θ 0 = (ω0 , α0 , β0 )0 is the unknown parameter.
Alternative form: (with γ0 = ω0 /(1 − α0 − β0 ) when
α0 + β0 < 1)
p
t = ht ηt ,
ht = κ0 γ0 +α0 2t−1 +β0 ht−1 , κ0 +α0 +β0 = 1
where ϑ0 = (γ0 , α0 , κ0 )0 is the new parameter.
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Definition of the VTE of ϑ0 = (γ0 , λ00 )0
with λ0 := (α0 , κ0 )0
Pn
1
First step: γ̂n = σ̂n2 := n−1
2
Second step: λ̂n = arg minλ∈Λ l̃n (λ), where
l̃n (λ) = n−1
n
X
`t,n ,
2
t=1 t ,
`t,n := `t,n (λ) =
t=1
2t
2
+ log σt,n
,
2
σt,n
with
2
2
2
σt,n
= σt,n
(λ) = κσ̂n2 + α2t−1 + (1 − κ − α)σt−1,n
2 = σ2.
and the initial value σ0,n
0
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Standard QMLE of ϑ0 = (γ0 , λ00 )0
∗
ϑ̂n = arg min n−1
ϑ∈Θ
where
`˜t (ϑ) =
n
X
`˜t (ϑ),
t=1
2t
+ log σ̃t2 (ϑ),
2
σ̃t (ϑ)
with
2
(ϑ)
σ̃t2 (ϑ) = γ + α2t−1 + (1 − κ − α)σ̃t−1
2 .
and the initial value σ̃02 (ϑ) = σ02 . Note that σ̃t2 (σ̂n2 , λ) = σt,n
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Outline
1
Volatility Models and QMLE
Properties of Financial Time Series
Models for the Volatility of Financial Returns
2
Variance Targeting Estimator
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
3
Conclusion
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Assumptions
in addition to ηt iid, Eηt2 = 1, ω0 > 0, α0 ≥ 0, β0 ≥ 0, α0 + β0 < 1
The results are stated for the GARCH(1,1), but remain valid in
the general GARCH(p, q) case.
Let the parameter space Λ ⊂ {(α, κ) | α ≥ 0, κ > 0, α + κ ≤ 1}.
A1:
λ0 belongs to Λ and Λ is compact.
A2:
A3:
α0 6= 0 and ηt2 has a non-degenerate distribution.
α02 Eηt4 − 1 + (1 − κ0 )2 < 1.
A4:
λ0 belongs to the interior of Λ.
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Assumptions
in addition to ηt iid, Eηt2 = 1, ω0 > 0, α0 ≥ 0, β0 ≥ 0, α0 + β0 < 1
The results are stated for the GARCH(1,1), but remain valid in
the general GARCH(p, q) case.
Let the parameter space Λ ⊂ {(α, κ) | α ≥ 0, κ > 0, α + κ ≤ 1}.
A1:
λ0 belongs to Λ and Λ is compact.
A2:
A3:
α0 6= 0 and ηt2 has a non-degenerate distribution.
α02 Eηt4 − 1 + (1 − κ0 )2 < 1.
A4:
λ0 belongs to the interior of Λ.
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Asymptotic properties of the GARCH(1,1) VTE
Theorem
Under assumptions A1-A2 the VTE satisfies ϑ̂n → ϑ0 almost
surely as n → ∞ and, under the additional assumptions A3-A4,
we have
√ d
n ϑ̂n − ϑ0 → N (0, (κη − 1)Σ).
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Form of the VTE asymptotic variance
Σ=
b
−bJ −1 K
−bK 0 J −1
−1
J + bJ −1 K K 0 J −1
is non-singular with
(α0 + κ0 )2 γ 2 (2 − κ0 )
,
κ0 1 − α02 Eηt4 − 1 − (1 − κ0 )2
1 ∂σt2 (ϑ0 ) ∂σt2 (ϑ0 )
,
J = E
∂λ0
σt4 (ϑ0 ) ∂λ
2×2
1 ∂σt2 (ϑ0 ) ∂σt2 (ϑ0 )
K = E
.
∂γ
σt4 (ϑ0 ) ∂λ
2×1
b =
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
VTE of the usual parameter θ 0 = (ω0 , α0 , β0 )0
Corollary
The VTE of θ 0 satisfies
√ d
n θ̂ n − θ 0 → N (0, (Eη04 − 1)L0 ΣL),
with


1 − α0 − β0
0 0
0
1 −1  .
L=
−1
ω0 (1 − α0 − β0 )
0 −1
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
VTE and QMLE Comparison
Theorem
∗
The QMLE ϑ̂n satisfies
n
o
√ ∗
d
n ϑ̂n − ϑ0 → N 0, (Eη04 − 1)Σ∗ ,
where
Σ∗ = Σ − (b − a)CC 0 ,
with
C=
1
−J −1 K
(
,
a=
Francq, Horvath, Zakoïan
κ20
1
E( ) − K 0 J −1 K
(α0 + κ0 )2 ht2
)−1
Variance targeting estimator of GARCH models
.
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
The VTE is never asymptotically more accurate than
the QMLE
Theorem
The asymptotic variance (Eη04 − 1)Σ of the VTE and the
asymptotic variance (Eη04 − 1)Σ∗ of the QMLE are such that
Σ − Σ∗
is positive semidefinite, but not positive definite.
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Proof that the QMLE is asymptotically more efficient
than the VTE
The asymptotic variances of the two estimators are the
variances of linear combinations of a same vector:
−1
Σ∗ = E GS t S 0t G 0
,
Σ = E HS t S 0t H 0
where
G=
and
I2 0

,
H=
eht−1
,

2


S t =  h−1 ∂σt (ϑ0 ) 
t
∂λ
e−1 ht
Francq, Horvath, Zakoïan
0 0
1
0 J −1 −J −1 K
e=
κ0
.
1 − β0
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
End of the Proof
We then easily show that
Σ − Σ∗ = ED t D 0t
where D t = Σ∗ GS t − HS t .
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Cases where VTE and QMLE have the same
asymptotic variance
Theorem
Let φ be a mapping from R2 to R, which is continuously
differentiable in a neighborhood of ϑ0 . If
1 −K 0 J −1
∂φ
(ϑ0 ) = 0,
∂ϑ
then the asymptotic distribution of the VTE of the parameter
φ(ϑ0 ) is the same as that of the QMLE.
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Cases where VTE and QMLE have the same
asymptotic variance
Under the previous condition,
o
√ n
d
n φ(ϑ̂n ) − φ(ϑ0 ) → N 0, s2
and
o
√ n
∗
d
n φ(ϑ̂n ) − φ(ϑ0 ) → N 0, s2 ,
where
s2 = (Eη04 − 1)
Francq, Horvath, Zakoïan
∂φ ∂φ
Σ
.
∂ϑ0 ∂ϑ
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Outline
1
Volatility Models and QMLE
Properties of Financial Time Series
Models for the Volatility of Financial Returns
2
Variance Targeting Estimator
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
3
Conclusion
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Numerical evaluation of Σ and Σ∗
The asymptotic variance of the VTE
b
−bK 0 J −1
Σ=
−bJ −1 K J −1 + bJ −1 K K 0 J −1
and that of the QMLE, Σ∗ , are not numerically computable,
even for the simplest model (the ARCH(1)).
; Approximations of Σ and Σ∗ from N = 1, 000 replications
of simulations of size n = 10, 000.
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Asymptotic variances of the QMLE and VTE
for ϑ0 = (γ0 , α0 ) in ARCH(1) models, γ0 = 1 and ηt ∼ N (0, 1)
α0 = 0.1
QMLE
VTE
2.52
0.51
2.52
0.51
0.51
1.69
0.51
1.69
α0 = 0.55
15.94
7.11
28.78
12.82
Francq, Horvath, Zakoïan
7.11
4.20 12.82
6.74
α0 = 0.7
45.27 14.32
14.32 5.02
∞
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Asymptotic variances of the QMLE and VTE
for θ 0 = (ω0 , α0 ) in ARCH(1) models, ω0 = 1 and ηt ∼ N (0, 1)
α0 = 0.1
QMLE
VTE
3.5 −1.4
−1.4 1.7 3.5 −1.4
−1.4 1.7
α0 = 0.55
5.1 −2.2
−2.2 4.2 5.1 −2.1
−2.1 9.3
Francq, Horvath, Zakoïan
α0 = 0.7
5.6 −2.4
−2.4 5.1
∞
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Sampling distribution of the two estimators
on N = 1, 000 independent ARCH(1) simulations
n = 500
parameter
ω
true value
1.0
α
0.55
ω
1.0
α
0.9
Francq, Horvath, Zakoïan
estimator
QMLE
VTE
QMLE
VTE
bias
0.013
0.012
-0.012
-0.026
RMSE
0.102
0.102
0.092
0.088
QMLE
VTE
QMLE
VTE
0.012
0.036
-0.012
-0.103
0.114
0.111
0.110
0.089
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Sampling distribution of the two estimators
on N = 1, 000 independent ARCH(1) simulations
n = 10, 000
parameter
ω
true value
1.0
α
0.55
ω
1.0
α
0.9
Francq, Horvath, Zakoïan
estimator
QMLE
VTE
QMLE
VTE
QMLE
VTE
QMLE
VTE
bias
0.000
0.000
0.000
0.000
RMSE
0.010
0.010
0.009
0.013
0.000
0.010
0.000
-0.032
0.012
0.015
0.011
0.032
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Comparison on daily stock returns
Index
CAC
DAX
FTSE
Nasdaq
Nikkei
SP500
estimator
QMLE
VTE
QMLE
VTE
QMLE
VTE
QMLE
VTE
QMLE
VTE
QMLE
VTE
ω
0.033 (0.009)
0.033 (0.009)
0.037 (0.014)
0.036 (0.013)
0.013 (0.004)
0.013 (0.004)
0.025 (0.006)
0.025 (0.006)
0.053 (0.012)
0.054 (0.012)
0.014 (0.004)
0.014 (0.003)
Francq, Horvath, Zakoïan
α
0.090 (0.014)
0.090 (0.014)
0.093 (0.023)
0.095 (0.022)
0.091 (0.014)
0.090 (0.013)
0.072 (0.009)
0.072 (0.009)
0.100 (0.013)
0.098 (0.013)
0.084 (0.012)
0.084 (0.011)
β
0.893 (0.015)
0.893 (0.015)
0.888 (0.024)
0.888 (0.024)
0.899 (0.014)
0.899 (0.014)
0.922 (0.009)
0.922 (0.009)
0.880 (0.014)
0.880 (0.015)
0.905 (0.012)
0.905 (0.012)
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
Comparison on misspecified models
Two GARCH(1,1) models are estimated by VTE and by QMLE
and are used to compute prediction intervals for n+h :
q
i
hq
2
2
σ̂n+h|n
F̂η−1 (α/2),
σ̂n+h|n
F̂η−1 (1 − α/2) ,
when
t = ω(∆t )ηt ,
ηt iid N (0, 1),
(∆t ) is a Markov chain, independent of (ηt ), with state-space
{1, 2} and transition probabilities
P(∆t = 1|∆t−1 = 1) = P(∆t = 2|∆t−1 = 2) = 0.9.
; Contrary to the QMLE, TVE should guarantee correct
predictions over long horizons.
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Description of the method
Asymptotic Properties of the VTE
Numerical comparison of the VTE and QMLE
TVE guarantees correct long-term predictions
10
15
20
horizon h
Francq, Horvath, Zakoïan
5
10
5
−5
0
exact
asymp
QMLE
VTE
−10
Prediction interval
5
−5
0
exact
asymp
QMLE
VTE
−10
Prediction interval
10
Prediction intervals of the Markov-switching model with different methods
5
10
15
horizon h
Variance targeting estimator of GARCH models
20
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
VTE can be recommended because it
reduces the computational complexity of GARCH
estimation;
is asymptotically less efficient that the QMLE, but can work
better in finite sample;
requires fourth-order moments for asymptotic normality,
but continues to work well with lower moments;
provides good (first step) estimations of real financial
series;
guarantees a consistent estimation of the long-run
variance;
thus guarantees correct long-horizon predictions;
can be an indicator of misspecification if an important
discrepancy with the QMLE is observed.
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models
Volatility Models and QMLE
Variance Targeting Estimator
Conclusion
Directions for Future Work
VTE for Value at Risk calculation;
Extension to other GARCH formulations;
Extension to multivariate models.
Francq, Horvath, Zakoïan
Variance targeting estimator of GARCH models