Multivariate GARCH Models

Transcript

Multivariate GARCH Models
Eduardo Rossi
University of Pavia
Introduction
Economics and financial economics present problems whose solutions
need the specification and estimation of a multivariate distribution.
• the standard portfolio allocation problem
• the risk management of a portfolio of assets
• pricing of derivative contracts based on a more than one
underlying asset (e.g., Quanto options)
• Financial contagion (shocks transmission volatility and returns)
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Econometria finanziaria 2010
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Introduction
Stylized facts:
• Volatility clustering
• Time-varying dynamic covariances and dynamic correlations
Financial variables have time-dependent second order moments.
Parametric models:
1. Multivariate GARCH models
2. Multivariate Stochastic volatility models
3. Multifactor models
4. Multifactor realized volatility models
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Multivariate Volatility Models
Vector of returns:
yt = (y1t , . . . , yN t )′
(N × 1)
−1/2
yt − µt = ǫt = Ht
zt
Let {zt } be a sequence of (N × 1) i.i.d. random vector with the
following characteristics:
E [zt ] = 0
E [zt z′t ] = IN
zt ∼ G (0, IN )
with G continuous density function.
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Et−1 (ǫt ) = 0
Et−1 (ǫt ǫ′t )
=
Ht
E (ǫt ǫ′t )
=
Σ
Et−1 [·] = E[·|Φt−1 ]
Φt−1 is the σ-field generated by past values of observable variables.
where Ht is a matrix (N × N ) positive definite and measurable with
respect to the information set Φt−1 , that is the σ-field generated by
the past observations: {ǫt−1 , ǫt−2 , . . .}.
The correlation matrix:
−1/2
Corrt−1 (ǫt ) = Rt = Dt
−1/2
Ht D t
Dt = diag(h11,t , . . . , hN N,t )
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MVMs provide a parametric structure for the dynamic evolution of
Ht . MVMs must satisfy:
1. Diagonal elements of Ht must be strictly positive;
2. Positive definiteness of Ht ;
3. Stationarity: E[Ht ] exists, finite and constant w.r.t. t.
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Ideal characteristics of a MVM:
1. Estimation should be flexible for increasing N
2. It should allow for covariance spillovers and feedbacks;
3. Coefficients should have an economic or financial interpretation
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MGARCH
The extension from a univariate GARCH model to an N -variate
model requires allowing the conditional variance-covariance matrix of
the N -dimensional zero mean random variables ǫt depend on the
elements of the information set.
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Covariance targeting
Caporin and McAleer (2009). Covariance targeting if the conditions
are met:
• The model intercept is an explicit function of the model long-run
covariance (or correlation) The long-run covariance (or
correlation) solution is given by the E[Ht ] or E[Rt ].
• The long-run solution is replaced by a consistent estimator.
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Covariance targeting: example
GARCH(1,1):
2
σt2 = ω + αǫ2t−1 + βσt−1
Long-run variance (if (α + β) < 1:
σ 2 = E[σt2 ] = ω(1 − α − β)−1
Variance targeting:
2
σt2 = σ
b2 (1 − α − β) + αǫ2t−1 + βσt−1
X
2
−1
σ
b =T
b
ǫ2t
t
Introduction of targeting transforms the model estimation into a
two-step estimation approach:
1. σ
b2
2. α, β
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Covariance targeting: example
N assets: N variances + 12 N (N − 1) covariances =
Two alternative approaches:
N
2 (N
+ 1).
• Models of Ht
• Models of Dt and Rt
The parametrization of Ht as a multivariate GARCH, which means
as a function of the information set Φt−1 , allows each element of Ht
to depend on q lagged of the squares and cross-products of ǫt , as well
as p lagged values of the elements of Ht . So the elements of the
covariance matrix follow a vector of ARMA process in squares and
cross-products of the disturbances.
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MGARCH-Vech representation
Let vech denote the vector-half operator, which stacks the lower
triangular elements of an N × N matrix as an [N (N + 1) /2] × 1
vector.
Let A be (2 × 2), then vech(A)


a11



vech(A) =  a21 

a22
Since the conditional covariance matrix Ht is symmetric,
vech (Ht ) , (N (N + 1)/2 × 1) contains all the unique elements in Ht .
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A natural multivariate extension of the univariate GARCH(p,q)
model is
q
p
X
X
vech (Ht ) = W +
A∗i vech ǫt−i ǫ′t−i +
B∗j vech (Ht−j )
i=1
=
W + A∗ (L) vech (ǫt ǫ′t ) + B∗ (L) vech (Ht )
A∗ (L)
=
A∗1 L + . . . + A∗q Lq
B∗ (L)
=
B∗1 L + . . . + B∗q Lp
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j=1
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N (N + 1)
N ≡
2
∗
W
:
A∗i , B∗j
:
[N (N + 1) /2] × 1
[N ∗ × N ∗ ]
N = 2, Vech-GARCH(1,1):

h11,t

 h
21,t

h22,t




 = 


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
 
+
 
a∗
11
a∗
21
a∗
31
a∗
12
a∗
22
a∗
32
a∗
13
a∗
23
a∗
33

ǫ2
1,t−1


ǫ1,t−1 ǫ2,t−1


ǫ2
2,t−1
∗
w1
∗
w2
∗
w3


 
 
+

b∗
11
b∗
21
b∗
31
b∗
12
b∗
22
b∗
32
b∗
13
b∗
23
b∗
33

h11,t−1

 h

21,t−1
h2,t−1
14




• This general formulation is termed vec representation by Engle
and Kroner (1995).
h
i
2
• The number of parameters is 1 + (p + q) [N (N + 1) /2] .
• Even for low dimensions of N and small values of p and q the
number of parameters is very large; for N = 5 and p = q = 1 the
unrestricted version of (1) contains 465 parameters.
• The number of parameters is of order O(N 4 ): the curse of
dimensionality.
For any parametrization to be sensible, we require that Ht be
positive definite for all values of ǫt in the sample space in the vech
representation this restriction can be difficult to check, let alone
impose during estimation.
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MGARCH-Diagonal vech model
A natural restriction is the diagonal representation, in which each
element of the covariance matrix depends only on past values of itself
and past values of ǫjt εkt . In the diagonal model the A∗i and B∗j
matrices are all taken to be diagonal.
For N = 2

h11,t

 h21,t

h22,t
and p = q = 1, the diagonal model is written as:


 

w1
a∗11 0
0
ε21,t−1


 

∗
 =  w2  +  0
  ε1,t−1 ε2,t−1
a
0
22


 

w3
0
0
a∗33
ε22,t−1



∗
b11 0
0
h11,t−1



∗



+ 0
b22 0   h21,t−1 

0
0
b∗33
h22,t−1




hij,t = wi∗ + a∗ii ǫi,t−1 ǫj,t−1 + b∗ii hij,t−1
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MGARCH-Diagonal vech model
Thus the (i, j) th element in Ht depends on the corresponding (i, j) th
element in εt−1 ε′t−1 and Ht−1 . This restriction reduces the number of
parameters to [N (N + 1) /2] (1 + p + q). This model does not allow
for causality in variance, co-persistence in variance and asymmetries.
The number of parameters is of order O(N 2 ).
The diagonal vech is equivalent to:
Ht = W + A ⊙ ǫt−1 ǫ′t−1 + B ⊙ Ht−1
where A and B are symmetric matrices. ⊙ is the Hadamard product.
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MGARCH(1,1)-vech-Unconditional Covariance Matrix
Given that
ǫt ǫ′t = Ht + Vt
with Et−1 (Vt ) = 0.
vech (ǫt ǫ′t ) = vech (Ht ) + vech (Vt )
vech (Vt )
vector m.d.s.
with E (vech (Vt )) = vech(E (Vt )) = 0.
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MGARCH(1,1)-vech-Unconditional Covariance Matrix
For a GARCH(1,1), the unconditional covariance matrix, when it
exists, is given by
′
∗
′
vech (ǫt ǫt ) = W + A1 vech ǫt−1 ǫt−1
∗
′
+B1 vech ǫt−1 ǫt−1 − vech (Vt−1 ) + vech (Vt )
E (ǫt ǫ′t ) = Σ
vech (E
(ǫt ǫ′t ))
=W+
(A∗1
+
vech (Σ) = [IN ∗ − A∗1 − B∗1 ]
B∗1 )
−1
′
vech E ǫt−1 ǫt−1
W.
For a GARCH(p,q) model
vech (Σ) = [IN ∗ − A∗ (1) − B∗ (1)]
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W
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MGARCH-vech representation
Targeting cannot easily introduced in the model.
MGARCH(1,1)-vech, unconditional var-cov matrix
[IN ∗ − A∗1 − B∗1 ]vech(Σ)
X ′
−1
b
b
Σ=T
ǫt b
ǫt
t
Targeting allows reducing by N ∗ the parameters to be estimated.
The total number of parameters is still O(N 4 ).
vech (Ht )
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=
h
i
b + A∗ vech ǫt−1 ǫ′ − vech Σ
b
vech Σ
1
t−1
h
i
∗
b
+B1 vech (Ht−1 ) − vech Σ
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MGARCH-BEKK representation
Engle and Kroner (1995) propose a parametrization that imposes
positive definiteness restrictions.
Consider the following model
q
p
K X
K X
X
X
Ht = CC′ +
Aik ǫt−i ǫ′t−i A′ik +
Bik Ht−i B′ik
k=1 i=1
(1)
k=1 i=1
where C, Aik and Bik are (N × N ).
• The intercept matrix is decomposed into CC′ , where C is a lower
triangular matrix.
• Without any further assumption CC′ is positive semidefinite.
• This representation is general, it includes all positive definite
diagonal representations and nearly all positive definite vech
representations.
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For exposition simplicity we will assume that K = 1:
Ht = CC′ +
q
X
Ai ǫt−i ǫ′t−i A′i +
i=1
p
X
Bi Ht−i B′i
i=1
Consider the simple GARCH(1,1) model:
Ht = CC′ + A1 ǫt−1 ǫ′t−1 A′1 + B1 Ht−1 B′1
(2)
Proposition 1. (Engle and Kroner (1995)) Suppose that the
diagonal elements in C are restricted to be positive and that a11 and
b11 are also restricted to be positive. Then if K = 1 there exists no
other C, A1 , B1 in the model (2) that will give an equivalent
representation.
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The purpose of the restrictions is to eliminate all other
observationally equivalent structures.
For example, as relates to the term A1 ǫt−1 ǫ′t−1 A′1 the only other
observationally equivalent structure is obtained by replacing A1 by
−A1 . The restriction that a11 (b11 ) be positive could be replaced
with the condition that aij (bij ) be positive for a given i and j, as
this condition is also sufficient to eliminate −A1 from the set of
admissible structures.
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MGARCH(1,1)-BEKK,

a11
′

Ht = CC +
a21

b11 b12

+
b21 b22
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N = 2:


a12
ε21t−1
ε1t−1 ε2t−1
a

  11
a22
ε2t−1 ε1t−1 ε22t−1
a21


′
h
h12t−1
b
b12
  11t−1
  11

h21t−1 h22t−1
b21 b22
a12
a22
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′

Sufficient condition for positive definiteness of Ht in
BEKK-GARCH(p,q) model.
Proposition 2. (Engle and Kroner (1995)) (Sufficient condition)
If H0 , H−1 , . . . , H−p+1 are all positive definite, then the BEKK
parametrization (with K = 1) yields a positive definite Ht for all
possible values of εt if C is a full rank matrix or if any Bi
i = 1, . . . , p is a full rank matrix.
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For simplicity consider the GARCH(1,1) model. The BEKK
parameterization is
Ht = CC′ + +A1 ǫt−1 ǫ′t−1 A′1 + B1 Ht−1 B′1
The proof proceeds by induction.
First Ht is p.d. for t = 1: The term A1 ǫ0 ǫ′0 A′1 is positive
semidefinite because ǫ0 ǫ′0 is positive semidefinite. Also if the null
spaces of the matrices of C and B1 intersect only at the origin, that
is at least one of two is full rank then
CC′ + B1 H0 B′1
is positive definite. This is true if C or B1 has full rank.
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To show that the null space condition is sufficient CC′ + B1 H0 B′1 is
p.d. if and only if
x′ (CC′ + B1 H0 B′1 ) x > 0
∀x 6= 0
or
′
(C′ x) (C′ x) +
1/2′
where H0 = H0
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1/2
H0
′ 1/2 ′
H0 B 1 x > 0
1/2
H0 B′1 x
1/2
and H0
∀x 6= 0
(3)
is full rank.
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Defining N (P ) to be the null space of the matrix P , (3) is true if and
only if
1/2 ′
′
N (C ) ∩ N H0 B1 = ∅.
1/2 ′
1/2
N H0 B1 = N (B′1 ) because H0 is full rank. This implies that
CC′ + B1 H0 B′1 is positive
definite if and only if
1/2
N (C′ ) ∩ N H0 B′1 = ∅. Now suppose that Ht is positive definite
for t = τ .
Then,
Hτ +1 = CC′ + A1 ǫτ ǫ′τ A′1 + B1 Hτ B′1
is positive definite if and only if, given that A1 ǫτ ǫ′τ A′1 is positive
semidefinite, the null space condition holds, because Hτ is positive
definite by the induction assumption.
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Diagonal BEKK
Consider MGARCH(1,1)-BEKK, N = 2 with
A1 = diag(a11 , a22 )
B1 = diag(b11 , b22 )
the model reduces to



a11 0
ε21t−1
ε1t−1 ε2t−1
a11
′





Ht = CC +
0
a22
ε2t−1 ε1t−1 ε22t−1
0



′
b11 0
h11t−1 h12t−1
b11 0






+
0
b22
h21t−1 h22t−1
0
b22
h11,t
=
c211 + a211 ǫ21t−1 + b211 h11t−1
h12,t
=
c21 c11 + a11 a22 ǫ1t−1 ǫ2t−1 + b11 b22 h12t−1
h22,t
=
c21 c11 + c222 + a222 ǫ21t−1 + b222 h11t−1
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0
a22
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′

Diagonal BEKK
This model is equivalent to the Hadamard BEKK:
Ht = CC′ + aa′ ⊙ ǫt−1 ǫ′t−1 + bb′ ⊙ Ht−1
positive definiteness is not guaranteed. Positive semidefiniteness is
obtained by imposing p.s.d. of all terms.
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Scalar BEKK
MGARCH(1,1) - Scalar BEKK
A1 = αIN ,
B1 = βIN
Ht = CC′ + α2 (ǫt−1 ǫ′t−1 ) + β 2 Ht−1
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MGARCH(1,1)- BEKK - Covariance targeting
1. BEKK
Ht = Σ +
A1 ǫt−1 ǫ′t−1
− Σ A′1 + B1 (Ht−1 − Σ) B1
or
Ht = (Σ − A1 ΣA1 −
B1 ΣB′1 )+A1
ǫt−1 ǫ′t−1
A′1 +B1 Ht−1 B′1
To have p.d-ness of Ht , (Σ − A1 ΣA1 − B1 ΣB′1 ) must be p.d..
2. Hadamard BEKK
ǫt−1 ǫ′t−1
Ht = Σ + A 1 ⊙
− Σ + B1 ⊙ (Ht−1 − Σ)
′
ǫt−1 ǫt−1 − Σ must be p.s.d., while (Ht−1 − Σ) must be p.s.d.
3. Scalar BEKK
Ht = Σ + α
2
ǫt−1 ǫ′t−1
− Σ + β 2 (Ht−1 − Σ)
for α + β < 1.
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BEKK and vec representations
We now examine the relationship between the BEKK and vech
parameterizations. The mathematical relationship between the
parameters of the two models can be found simply vectorizing the
BEKK equation:
vec(Ht ) = vec(CC′ ) +
q
X
i=1
p
X
vec(Ai ǫt−i ǫ′t−i A′i ) +
vec(Bi Ht−i B′i )
i=1
where vec () is an operator such that given a matrix A (n × n),
2
vec(A) is a n × 1 vector. The vec () satisfies
vec (ABC) = (C′ ⊗ A) vec (B)
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For a symmetric A, (n × n):
• vech (A) contains precisely the n (n + 1) /2 distinct elements of
A;
• the elements of vec (A) are those of vech (A) with some
repetitions;
• There exists a unique n2 × n (n + 1) /2 which transforms, for
symmetric A, vech (A) into vec (A). This matrix is called the
duplication matrix and is denoted Dn :
vec (A) = Dn vech (A)
where Dn is the duplication matrix.
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Then
vec(Ht )
=
vec(CC′ ) +
q
X
i=1
+
p
X
i=1
DN vech (Ht )
=
(Ai ⊗ Ai ) vec(εt−i ǫ′t−i )
(Bi ⊗ Bi ) vec(Ht−i )
DN vech(CC′ ) +
q
X
i=1
+
p
X
i=1
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(Ai ⊗ Ai ) DN vech(ǫt−i ǫ′t−i )
(Bi ⊗ Bi ) DN vech(Ht−i )
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If DN is a full column rank matrix we can define the generalized
inverse of DN as:
′
D+
N = (DN DN )
−1
D′N
that is a (N (N + 1) /2) × N
D+
N DN = IN
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matrix, where
36
This implies that premultiplying by D+
N
vech (Ht )
=
vech(CC′ ) + D+
N
q
X
i=1
+D+
N
p
X
i=1
!
(Ai ⊗ Ai ) DN vech(ǫt−i ǫ′t−i )
!
(Bi ⊗ Bi ) DN vech(Ht−i )
• The vech model implied by any given BEKK model is unique,
while the converse is not true.
• The transformation from a vech model to a BEKK model (when
it exists) is not unique, because for a given A∗1 the choice of A1
is not unique.
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• This can be seen recognizing that (Ai ⊗ Ai ) = (−Ai ⊗ −Ai ) so
while A∗i = D+
N (Ai ⊗ Ai ) DN is unique, the choice of Ai is not
unique. It can also be shown that all positive definite diagonal
vech models can be written in the BEKK framework.
Given Ai diagonal matrix, then D+
N (Ai ⊗ Ai ) DN is also diagonal,
with diagonal elements given by aii ajj (1 ≤ j ≤ i ≤ N ) (See Magnus).
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MGARCH-Covariance Stationarity
Given the vech model
vech (Ht ) = W + A∗ (L) vech (εt ε′t ) + B∗ (L) vech (Ht )
the necessary and sufficient condition for covariance stationary of
{εt } is that all the eigenvalues of A∗ (1) + B∗ (1) are less than one in
modulus. But defining
!
q
X
+
∗
A (1) = DN
(Ai ⊗ Ai ) DN
B∗ (1)
=
D+
N
i=1
q
X
i=1
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!
(Bi ⊗ Bi ) DN
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MGARCH-Covariance Stationarity
This implies also that in the BEKK model, {ǫt } is covariance
stationary if and only if all the eigenvalues of
!
!
q
p
X
X
+
+
DN
(Ai ⊗ Ai ) DN + DN
(Bi ⊗ Bi ) DN
i=1
i=1
are less than one in modulus.
Let λ1 , . . . , λN be the eigenvalues of Ai , the eigenvalues of
!
q
X
+
DN
(Ai ⊗ Ai ) DN
i=1
are λi λj (1 ≤ j ≤ i ≤ N ) (Magnus).
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MGARCH(1,1)-BEKK Unconditional Covariance Matrix
BEKK model
vec (ǫt ǫ′t )
′
ǫt−1 ǫ′t−1
vec (CC ) + (A1 ⊗ A1 ) vec
′
+ (B1 ⊗ B1 ) vec ǫt−1 ǫt−1 − vec (Vt−1 ) + vec (Vt )
′
′
′
E [vec (ǫt ǫt )] = vec (CC )+[(A1 ⊗ A1 ) + (B1 ⊗ B1 )] E vec ǫt−1 ǫt−1
=
vec (Σ) = [IN 2 − (A1 ⊗ A1 ) − (B1 ⊗ B1 )]−1 vec (CC′ )
or in vech representation as
DN vech (E
(ǫt ǫ′t ))
vech (Σ) = IN ∗
′
ǫt−1 ǫ′t−1
DN vech (CC ) + (A1 ⊗ A1 ) DN vech E
′
+ (B1 ⊗ B1 ) DN vech E ǫt−1 ǫt−1
−1
+
+
− DN (A1 ⊗ A1 ) DN − DN (B1 ⊗ B1 ) DN
vech (CC′ )
=
N ∗ = N (N + 1) /2.
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MGARCH(1,1)-Diagonal models - Covariance stationarity
• The diagonal vech model is stationary if and only if the sum
a∗ii + b∗ii < 1 for all i.
• In the diagonal BEKK model the covariance stationary condition
is that a2ii + b2ii < 1.
Only in the case of diagonal models the stationarity properties are
determined solely by the diagonal elements of the Ai and Bi
matrices.
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Asymmetric MGARCH-in-mean model
A general multivariate model can be written as:
yt = µ + Π (L) yt−1 + Ψxt−1 + Λvech (Ht ) + ǫt
(4)
yt : (N × 1)
Π (L) = Π1 + Π2 L + · · · + Πk Lk−1
(N × N )
Ψ : (N × L)
Λ : (N × N (N + 1) /2)
xt : (L × 1)
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43
xt−1 contains predetermined variables. ǫt is the vector of innovation
with respect to the information set formed exclusively of past
realizations of yt .
Ht = Et−1 (ǫt ǫ′t )
Ht = CC′ +
q
X
i=1
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′
Ai (ǫt−i + γ) (ǫt−i + γ) A′i +
p
X
Bj Ht−j B′j (5)
j=1
44
We can consider a multivariate generalization of the size effect and
sign effect:
′
D′ +Gǫt−1 ǫ′t−1 G′
Ht = CC′ +A1 ǫt−1 ǫ′t−1 A′1 +B1 Ht−1 B′1 +Dvt−1 vt−1
p
where vt = |zt | − E |zt |, with zit = εit / hii,t and


I (ε1t−1 < 0) g11 0
... 0


..
..


.
 0

.


G= .

..

 ..
. 0


0
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...
0
I (εN t−1 < 0) gN N
45
When N = 2
′
vt−1 vt−1
=
=
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
 p
p
ε1t−1 / h11,t−1 − E ε1t−1 / h11,t−1  ×
p
p
ε2t−1 / h22,t−1 − E ε2t−1 / h22,t−1 ′
 p
p
ε1t−1 / h11,t−1 − E ε1t−1 / h11,t−1  
p
p
ε2t−1 / h22,t−1 − E ε2t−1 / h22,t−1 

(|z1t | − E |z1t |)2
(|z1t | − E |z1t |) (|z2t | − E |z2t |)


(|z2t | − E |z2t |) (|z1t | − E |z1t |) (|z2t | − E |z2t |)2
46
Gǫt−1 ǫ′t−1 G′
=
=




∗2 2
g11
ε1t−1
∗ ∗
g11
g22 ε1t−1 ε2t−1
∗ ∗
g11
g22 ε1t−1 ε2t−1
∗2 2
g22
ε2t−1
-

2 2
ε1t−1
I (ε1t−1 < 0) g11
δ12 g11 g22 ε1t−1 ε2t−1
δ12 g11 g22 ε1t−1 ε2t−1
2 2
I (ε2t−1 < 0) g22
ε2t−1
δ12 = I (ε1t−1 < 0) I (ε2t−1 < 0)
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

47
Estimation procedure
Given the model (4)-(5), the log-likelihood function for {εT , . . . , ε1 }
obtained under the assumption of conditional multivariate normality
is:
"
#
T
X
1
′ −1
log LT (ǫT , . . . , ǫ1 ; θ) = − T N log (2π) +
log |Ht | + ǫt Ht ǫt
2
t=1
• The assumption of conditional normality can be quite restrictive.
• The symmetry imposed under normality is difficult to justify,
and the tails of even conditional distributions often seem fatter
than that of normal distribution.
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Let {(yt , xt ) : t = 1, 2, . . .} be a sequence of observable random
vectors with yt (N × 1) and xt (L × 1).
The vector yt contains the ”endogenous” variables and xt contains
contemporaneous ”exogenous” variables.
wt = (xt , yt−1 , xt−1 , . . . , y1 , x1 ) .
The conditional mean and variance functions are jointly
parameterized by a finite dimensional vector θ:
{µt (wt , θ) , θ ∈ Θ}
{Ht (wt , θ) , θ ∈ Θ}
where Θ ⊂ RP and µt and Ht are known functions of wt and θ.
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The validity of most of the inference procedures is proven under the
null hypothesis that the first two conditional moments are correctly
specified, for some θ 0 ∈ Θ,
E (yt |wt ) = µt (wt , θ0 )
V ar (yt |wt ) = Ht (wt , θ 0 )
t = 1, 2, . . .
The procedure most often used to estimate θ 0 is the maximization of
a likelihood function that is constructed under the assumption that
yt |wt ∼ N (µt , Ht ).
The approach taken here is the same, but the subsequent analysis
does not assume that yt has a conditional normal distribution.
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For observation t the quasi-conditional log-likelihood is
lt (θ; yt , wt )
=
N
1
ln (2π) − ln |Ht (wt , θ)|
2
2
1
− (yt − µt (wt , θ))′ H−1
t (wt , θ) (yt − µt (wt , θ))
2
−
Letting
ǫt (yt , wt , θ 0 ) ≡ yt − µt (wt , θ) : (N × 1)
denote the residual function
N
1
1 ′
lt (θ) = − log (2π) − log |Ht (θ)| − ǫt (θ) H−1
t (θ) ǫt (θ)
2
2
2
log LT (θ) =
T
X
lt (θ)
t=1
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If µt (wt , θ) and Ht (wt , θ) are differentiable on Θ for all relevant wt ,
and if Ht (wt , θ) is nonsingular with probability one for all θ ∈ Θ,
then the differentiation of the loglik yields the (1 × P ) score function
st (θ):
st (θ)
′
′
where
∇θ µt (θ) : (N × P )
2
∇θ Ht (θ) : N × P
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∇θ lt (θ) − ∇θ µt (θ) H−1
t (θ) ǫt (θ) +
1
′ −1
′
−1
∇θ Ht (θ) Ht (θ) ⊗ Ht (θ) vec ǫt (θ) ǫt (θ) − Ht (θ)
2
=
-
52
If the first conditional two moments are correctly specified, the true
error vector is defined as
ǫ0t ≡ ǫt (θ 0 ) = yt − µt (wt , θ0 )
0
and E ǫt |wt = 0,
0 0′
E ǫt ǫt |wt = Ht (wt , θ 0 )
It follows that under correct specification of the first two conditional
moments of yt given wt :
E [st (θ 0 ) |wt ] = 0
The score evaluated at the true parameter is a vector of martingale
difference with respect to the σ − f ields {σ (yt , wt ) : t = 1, 2, . . .}.
This result can be used to establish weak consistency of the
quasi-maximum likelihood estimator (QMLE).
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For robust inference we also need an expression for the hessian
ht (θ) of lt (θ). Define the positive semidefinite matrix
at (θ 0 ) = −E [∇θ st (θ 0 ) |wt ] = E [−ht (θ 0 ) |wt ] : (P × P )
′
∇θ µt (θ 0 ) H−1
t (θ 0 ) ∇θ µt (θ 0 )
1
′ −1
−1
+ ∇θ Ht (θ) Ht (θ) ⊗ Ht (θ) ∇θ Ht (θ)
2
When the normality assumption holds the matrix at (θ 0 ) is the
conditional information matrix. However, if yt does not have a
conditional normal distribution then
at (θ 0 )
=
V ar [st (θ 0 ) |wt ] 6= at (θ 0 )
and the information matrix equality is violated.
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The QMLE has the following properties:
0−1 0 0−1 −1/2 √ d
bT − θ 0 →
T θ
N (0, IP )
A T BT A T
where
T
T
1X
1X
0
AT ≡ −
E [ht (θ 0 )] =
E [at (θ 0 )]
T t=1
T t=1
and
T
h
i
X
1
′
0
−1/2
BT ≡ V ar T
ST (θ 0 ) =
E st (θ 0 ) st (θ 0 )
T t=1
in addition
p
b T − A0T →
A
0
p
b T − B0 →
B
0
T
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b −1 B
bTA
b −1 is a consistent estimator od the robust
The matrix A
T
T
√ bT − θ 0 .
asymptotic covariance matrix of T θ
In practice,
bT ≈ N θ, A
b −1 B
bTA
b −1 /T
θ
T
T
b −1 /T
Under normality, the variance estimator can be replaced by A
T
−1
b /T (outer product of the gradient form).
(Hessian form) or B
T
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Wald Test
The null hypothesis is
H0 : r (θ 0 ) = 0
where r : Θ → RQ is continuously differentiable on int (Θ) and
Q < P . Let
R (θ) = ∇θ r (θ) : (Q × P )
be the gradient of r on int (Θ). If θ 0 ∈ int (Θ) and
rank (R (θ 0 )) = Q then the Wald statistic
′ −1 ′ d
2
bT →
bT
bT A
bT
b −1 B
bTA
b −1 R θ
θ
r
R θ
ξW = T r θ
χ
Q.
T
T
H0
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Factor-GARCH
The Factor GARCH model, introduced by Engle et al. (1990), can be
thought of as an alternative simple parametrization of the BEKK
model. Suppose that the (N × 1) yt has a factor structure with K
factors given by the K × 1 vector ft and a time invariant factor
loadings given by the N × K matrix B:
yt = Bft + ǫt
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Factor-GARCH
Assume that the idiosyncratic shocks ǫt have conditional covariance
matrix Ψ which is constant in time and positive semidefinite, and
that the common factors are characterized by
Et−1 (ft ) = 0
Et−1 (ft ft′ ) = Λt
Λt = diag (λ1 , . . . , λK ) and positive definite. The conditioning set is
{yt−1 , ft−1 , . . . , y1 , f1 }. Also suppose that E (ft ǫ′t ) = 0. The
conditional covariance matrix of yt equals
Et−1 (yt yt′ ) = Ht = Ψ + BΛt B′ = Ψ +
K
X
β k β ′k λkt
k=1
where β k denotes the kth column in B. Thus, there are K statistics
which determine the full covariance matrix.
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Factor-GARCH
Forecasts of the variances and covariances or of any portfolio of
assets, will be based only on the forecasts of these K statistics.
There exists factor-representing portfolios with portfolio weights that
are orthogonal to all but one set of factor loadings:
rkt = φ′k yt


1
k=j
′
φk β j =
 0
otherwise
the vector of factor-representing portfolios is
rt = Φ′ yt
where the columns of matrix Φ are the φk vectors.
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Factor-GARCH
The conditional variance of rkt is given by
Vart−1 (rkt )
=
φ′k Et−1 (yt yt′ ) φk = φ′k Ht φk
=
φ′k (Ψ + BΛt B′ ) φk
=
ψk + λkt
where ψk = φ′k Ψφk . The portfolio has the exact time variation as the
factors, which is why they are called factor-representing portfolios. In
order to estimate this model, the dependence of the λkt ’s upon the
past information set must also be parameterized:
θ kt ≡ φ′k Ht φk = V art−1 (rkt ) = ψk + λkt
So we get that
K
X
β k β ′k θ kt =
k=1
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X
β k β ′k ψk +
k=1
-
K
X
β k β ′k λkt
k=1
61
Factor-GARCH
K
X
β k β ′k λkt =
k=1
Ht
K
X
β k β ′k θ kt −
k=1
=
Ψ+
K
X
K
X
k=1
β k β ′k λkt = Ψ +
k=1
Ψ∗ +
=
K
X
β k β ′k ψk
K
X
β k β ′k θ kt −
k=1
K
X
β k β ′k ψk
k=1
β k β ′k θ kt
k=1
where Ψ∗ =
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Ψ−
K
P
β k β ′k ψk .
k=1
62
Factor-GARCH
The simplest assumption is that there is a set of factor-representing
portfolios with univariate GARCH(1,1) representations. The
conditional variance θ kt follows a GARCH(1,1) process
2
′
2
θ kt = ωk + αk φk ǫt−1 + γk Et−2 rkt−1
′ ′ ′
′
θ kt = ωk + αk φk ǫt−1 ǫt−1 φk + γk Et−2 φk yt φk yt
′
′
′
′
θ kt = ωk + αk φk ǫt−1 ǫt−1 φk + γk φk Et−2 (yt yt ) φk
′
′
′
θ kt = ωk + αk φk ǫt−1 ǫt−1 φk + γk φk Ht−1 φk
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Factor-GARCH
The conditional variance-covariance matrix of yt can be written as
Ht
=
Ψ∗ +
K
X
β k β ′k θ kt
k=1
=
∗
Ψ +
K
X
β k β ′k
k=1
=
∗
Ψ +
Ht = Γ +
K
X
′
′
′
ωk + αk φk ǫt−1 ǫt−1 φk + γk φk Ht−1 φk
β k β ′k ωk
k=1
K
X
!
+
K
X
β k β ′k
k=1
′
′
′
αk φk ǫt−1 ǫt−1 φk + γk φk Ht−1 φk
β k β ′k θ kt
k=1
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Factor-GARCH
∗
where Γ = Ψ +
K
P
k=1
Ht =
K
X
β k β ′k ωk , therefore
Γ+
αk β k φ′k
k=1
ǫt−1 ǫ′t−1
′
K
X
φk β k +
γk β k φ′k Ht−1 φk β ′k
k=1
so that the factor GARCH model is a special case of the BEKK
parametrization. Estimation of the factor GARCH model is carried
out by maximum likelihood estimation. It is often convenient to
assume that the factor-representing portfolios are known a priori.
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Orthogonal-GARCH model
The orthogonal models are particular factor models. They are based
on the assumption that the observed data can be obtained by a linear
transformation of a set of uncorrelated components.
The components are the principal components of the data, or a
subset of them.
The diagonal matrix V contains the empirical variances of yt :
V = diag{s21 , . . . , s2N }
the standardized returns are
ut = V−1/2 yt
E[ut ] = 0
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E[ut u′t ] = R
66
The sample correlation matrix can be decomposed as:
R = PΛP′
P is the orthonormal eigenvectors matrix, Λ is the diagonal matrix of
the eigenvalues:
Λ = diag{λ1 , . . . , λN }
ranked in descending order.
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P satisfies:
P′ = P−1
P′ P = IN
PP′ = IN
It follows that
R = PΛ1/2 Λ1/2 P′ = LL′
L = PΛ1/2
Principal components
ft = L−1 ut
E[ft ft′ ] = L−1 E[ut u′t ]L−1′ = L−1 RL−1′
= L−1 LL′ L′−1 = IN
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Assuming
Et−1 [ft ft′ ] = Qt = diag(σf21,t , . . . , σf2N,t )
Qt is a diagonal matrix.
σf2i,t ∼ GARCH(p,q)
i = 1, 2, . . . , N
Et−1 [ut u′t ] = Et−1 [Lft ft′ L′ ] = LQt L′
Et−1 [yt yt′ ] = Et−1 [V1/2 ut u′t V1/2 ] = V1/2 LQt L′ V1/2
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We can work with a reduced number m < N of principal components
(eigenvalues), those which explain most of the variation in the data.
L−1 is replaced by a matrix (m × N ):
Λ−1/2
P′m
m
Pm is a matrix (N × m) containing the m eigenvectors of P
corresponding to the m largest eigenvalues.
ftm = Λ−1/2
P′m ut
m
where ftm = [f1t , . . . , fmt ]
Et−1 [ftm ] = 0
′
Et−1 [ftm ftm ] = Qt = diag(σf21,t , . . . , σf2m,t )
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Generalized Orthogonal-GARCH model
Assume that
yt |Φt−1 ∼ N (0, Vt )
The observed economic process yt is governed by a linear
combination of independent economic components {ft }
yt = Zft
ft uncorrelated components, |Z| =
6 0. The unobserved components are
normalized such that:
E[ft ft′ ] = IN
V = E[yt yt′ ] = ZZ′
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Vt = Et−1 [yt yt′ ] = ZEt−1 [ft ft′ ]Z′ = ZHt Z′
Ht = diag{h1,t , . . . , hN,t }
2
hi,t = (1 − α1 − βi ) + αi yi,t−i
+ βi hi,t−1
i = 1, . . . , N
V = E[Vt ] = ZHZ′
H diagonal. The diagonal decomposition of the unconditional
covariance matrix:
V = PΛP′
P: orthogonal matrix, O-GARCH estimator for Z, is only guaranteed
to coincide with Z, when the diagonal elements of H are all distinct.
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Suppose that H = I
V = E[Vt ] = ZIZ′ = ZZ′
The matrix Z is no longer identified by the eigenvector matrix of V
as for every orthogonal matrix Q we have
(ZQ)(ZQ)′ = I
The matrix Z is well identified when conditional information is taken
into account. Based on singular value decomposition:
PΛ1/2 U0 = Z
Let the estimator of U0 be U. We restrict
|U| = 1.
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The matrices P and Λ have N (N2−1) and N parameters, so we have
N 2 parameters for the invertible matrix Z.
The matrices P and Λ will be estimated by means of unconditional
information, as they will extracted from the sample covariance
matrix V ( N (N2+1) parameters). Conditional information is required
to estimate U0 ( N (N2−1) free parameters).
The orthogonal U0 is parameterized by means of rotation matrices.
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Rotation matrix: Grs , (N × N ), r, s ∈ N, r < s ≤ N :
Grs = {gij }
grr = gss = cos(θ)
gii = 1 i = 1, . . . , N i 6= r, s
gsr = − sin(θ)
grs = sin(θ)
and all other elements are zero.

cos(θ) sin(θ)

G12 = 
 − sin(θ) cos(θ)
0
0
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N = 3:

0

0 

1
75
Every N-dimensional orthogonal matrix U with det(U) = 1 can be
represented as a product of N (N2−1) rotation matrices:
Y
U=
Gij (θij )
− π ≤ θij ≤ π
i<j
Gij (θij ) performs a rotation in the plane spanned by the i-th and the
j-th vectors of the canonical basis of RN over an angle δij .
N = 3:
U = G12 G13 G23
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Time-varying conditional correlations:
−1
Rt = D−1
t Vt Dt
Dt = (Vt ⊙ Im )
N =2

Z=
1/2
1
0
cos θ
sin θ


θ measures the extent to which the uncorrelated components are
mapped in the same direction. For θ = 0 the mapping in not
invertible, yielding perfect correlation between the observed variables,
whereas θ = π2 , Z = I so that observed variables are completely
uncorrelated.
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Covt−1 (y1t , y2t )
p
ρt = p
V art−1 (y1t ) V art−1 (y2t )
In this example the conditional correlation is:
ρt
=
=
=
h1t cos (θ)
q
√
h1t h1t cos2 (θ) + h2t sin2 (θ)
cos (θ)
q
cos2 (θ) +
h1t
h2t
sin2 (θ)
1
p
1 + zt tan2 (θ)
where zt = hh1t
.
2t
Note that a constant linkage Zθ gives rise to time-varying correlations
between the observed variables. These correlations rise on average
when the components are mapped more in the same direction.
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Suppose GO-GARCH(1,1) parameters to be estimated by means of
the conditional information are (α′ , β ′ , θ ′ )′
α =
(α1 , . . . , αN )
β = (β1 , . . . , βN )
N (N − 1)
θ = (θ1 , . . . , θm ) m =
2
The log-likelihood can be expressed:
1
′ −1
lt = − N log(2π) + log |Vt | + yt Vt yt
2
1
′
′ ′
′ −1
= − N log(2π) + log |Zθ Ht Zθ | + ft Zθ (Zθ Ht Zθ ) Zθ ft
2
1
′
′ ′
′ −1
= − N log(2π) + log |Zθ Zθ | + log |Ht | + ft Zθ (Zθ Ht Zθ ) Zθ ft
2
1
′
′ −1
= − N log(2π) + log |Zθ Zθ | + log |Ht | + ft Ht ft
2
X
log LT =
lt
t
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79
Two-step estimation
1. Zθ = PΛ1/2 U0 where P and Λ from the sample covariance
matrix:
b =P
bΛ
bP
b′
V
bΛ
b 1/2 U
Zθ = P
where
U=
Y
i<j
Gij (θij )
− π ≤ θij ≤ π
2. Estimate (α′ , β ′ , θ′ )′ by maximizing the log LT .
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80
MGARCH -The Constant Conditional Correlations Model
• These models are based on a decomposition of the Ht .
• The conditional var-cov matrix is expressed as
Ht = D t R t D t
where Rt is possibly time-varying.
• Conditional correlations and variances are separately modeled.
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Bollerslev (1990) Constant Conditional Correlations model:
The time-varying conditional covariances are parameterized to be
proportional to the product of the corresponding conditional
standard deviations. The model assumptions are:
Et−1 [ǫt ǫ′t ] = Ht
{Ht }ii = hit
i = 1, . . . , N
1/2 1/2
{Ht }ij = hijt = ρij hit hjt
i 6= j
i, j = 1, . . . , N
Dt = diag {h1t , . . . , hN t }
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The conditional covariance matrix can be written as:
1/2
1/2
Ht = Dt RDt

1/2
h1t

 .
Ht =  ..

0
When N = 2
Ht
=
=
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
···
..
.
0
..
.
···
hN t

1/2
1/2










h1t
0
0
1/2
h2t

1/2 1/2
ρ12 h1t h2t

h1t
1

ρ12
...
ρ21
..
.
1
..
.
...
ρ1N
..
.
...
ρN −1N
ρN 1
...
ρN N −1
1


1
ρ21
ρ12
1
1/2 1/2
ρ12 h1t h2t
h2t



1/2









h1t
0
0
1/2
h2t
1/2
h1t

..
.
···
..
.
0
..
.
0
···
hN t


.
83
1/2




• The sequence of conditional covariance matrices {Ht } is
guaranteed to be positive definite a.s. for all t, If the conditional
variances along the diagonal in the Dt matrices are all positive,
and the conditional correlation matrix R is positive definite
• Furthermore the inverse of Ht is given by
−1/2
H−1
= Dt
t
−1/2
R−1 Dt
.
When calculating the log-likelihood function only one matrix
inversion is required for each evaluation.
• CCC is generally estimated in two steps:
1. conditional variances are estimated employing the marginal
likelihoods
2. R is estimated using the sample estimator of standardized
b −1
residuals D
t yt (assuming µt = 0).
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• The CCC solves the curse of dimensionality problem of
MGARCH models
• The number of parameters is O(N 2 ) but these are not jointly
estimated. The two-step estimation procedure impacts on the
computational issues.
• Asymptotic properties of QMLE estimators verified in McAleer
and Ling (2003).
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The Dynamic Conditional Correlation GARCH Model
The CCC has two main limitations:
1. No spillover neither feedback effects across conditional variances
2. Correlations are static
The evolution of CCC is the Dynamic Conditional Correlation
(DCC) Model of Engle (2002). The DCC is an extension of the
Bollerslev’s CCC Model.
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The conditional correlation between two random variables, Xt and Yt
is defined as:
ρY X,t
Covt−1 (Xt Yt )
= p
Et−1 (Xt − µx,t )2 Et−1 (Yt − µY,t )2
Assets returns conditional distribution:
yt |Φt−1 ∼ N (0, Ht )
1/2
1/2
Ht = D t R t D t .
Dt = diag(Vart−1 (y1t ), . . . , Vart−1 (yN t ))
where the Vart−1 (yit ), i = 1, . . . , N are modeled as univariate
GARCH processes.
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The standardized returns are:
−1/2
η t = Dt
yt
−1/2
Et−1 (η t η ′t ) = Dt
−1/2
Ht D t
= Rt = {ρij,t }
we can use the conditional variance of η t to describe the conditional
correlation of yt .
The conditional correlation estimator is
qij,t
ρij,t = √
.
qii,t qjj,t
Where qij,t are assumed to follow a GARCH(1,1) model
qij,t = ρij + α(ηi,t−1 ηj,t−1 − ρij ) + β(qij,t−1 − ρij )
(6)
The term ρij is not the unconditional correlation between ηit and ηjt ;
the unconditional correlation between ηit and ηjt has no closed form.
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Engle (2002) assumes that ρij ≃ q ij . However, Aielli (2006) and
Engle et al. (2008) suggest to modify the standard DCC in order to
P
1
correct the asymptotic bias which is due to the fact that T t ǫt ǫ′t
does not converge to Q. It is known though that the impact of this is
very small (see Engle and Sheppard (2001)).
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The conditional covariance matrix is positive definite, Qt , as long as
it is a weighted average of definite matrices and semidefinite matrices.
To ensure p.-d-ness of Qt we must impose α + β < 1 In matrix from:
Qt = Q(1 − α − β) + α(η t−1 η ′t−1 ) + β(Qt−1 )
(7)
where Q is the unconditional covariance matrix of η t .
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The DCC - Correlation targeting
DCC model has correlation targeting, when α + β < 1
E[Qt ]
=
R
E[Qt ]
=
E[Qt−1 ]
E[η t η ′t ]
=
E[Qt ]
E[Qt ]
=
E[Qt ]
=
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Q(1 − α − β) + αE[η t−1 η ′t−1 ] + βE[Qt−1 ]
R(1 − α − β) + αE[Qt ] + βE[Qt ]
91
Clearly more complex positive definite multivariate GARCH models
could be used for the correlation parameterization as long as the
unconditional moments are set to the sample correlation matrix.
For example, the MARCH family of Ding and Engle (2001) can be
expressed in first order form as:
Qt = Q ⊙ (ιι′ − A − B) + A ⊙ ǫt−1 ǫ′t−1 + B ⊙ Qt−1
(8)
where ⊙ denotes the Hadamard product ({A ⊙ B}ij = aij bij ).
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The Generalized-DCC model specification:
′
Dt = diag{ωi } + diag{κi } ⊙ yt−1 yt−1
+ diag{λi } ⊙ Dt−1
Qt = S ⊙ (ιι′ − A − B) + A ⊙ ǫt−1 ǫ′t−1 + B ⊙ Qt−1
Rt = diag{Qt }−1/2 Qt diag{Qt }−1/2 .
A and B are symmetric matrices.
The assumption of normality gives rise to a likelihood function.
Without this assumption, the estimator will still have the QML
interpretation. The second equation simply expresses the assumption
that each of the assets follows a univariate GARCH process.
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A real square matrix A, is positive definite if and only if
B = A∗−1 AA∗−1 is positive definite, with A∗ = diag{A}.
In order to ensure that Ht is positive definite we must have that
−1/2
Dt
Ht D−1/2 is positive definite.
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Ht is positive definite ∀t ∈ T , if the following restrictions on the
univariate GARCH parameters are satisfied for all series
i ∈ [1, . . . , N ] :
1. ωi > 0
2. κi and λi such that Dii,t > 0 with probability 1
2
3. Dii,0
>0
4. The roots of 1 − κi Z − λi Z are outside the unit circle.
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and the parameters in the DCC satisfy:
1. α ≥ 0
2. β ≥ 0
3. α + β ≤ 1
4. The minimum eigenvalue of Q0 > δ > 0 (where Q0 must be
positive definite)
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The log-likelihood function can be written as:
T
log LT
1X
−
(N log(2π) + log |Ht | + r′t H−1
t rt )
2 t=1
=
T
1X
1/2
1/2
−1/2 −1 −1/2
−
(N log(2π) + log |Dt Rt Dt | + r′t Dt
R t Dt
rt )
2 t=1
=
T
1X
−
(N log(2π) + log |Dt | + log |Rt | + ǫ′t R−1
t ǫt )
2 t=1
=
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−1/2
Adding and subtracting r′t Dt
−1/2
Dt
rt = ǫ′t ǫt
T
log LT
=
1X
−1/2 −1/2
−
(N log(2π) + log |Dt | + r′t Dt
Dt
rt
2 t=1
−ǫ′t ǫt + log |Rt | + ǫ′t R−1
t ǫt )
T
=
1X
(N log(2π) + log |Dt | + r′t D−1
−
t rt )
2 t=1
T
1 X ′ −1
(ǫt Rt ǫt − ǫ′t ǫt + log |Rt |)
−
2 t=1
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Volatility component:
T
1X
(N log(2π) + log |Dt | + r′t D−1
LV (θ) ≡ log LV,T (θ) = −
t rt )
2 t=1
Correlation component:
T
1 X ′ −1
LC (θ, φ) ≡ log LC,T (θ, φ) = −
(ǫt Rt ǫt − ǫ′t ǫt + log |Rt |)
2 t=1
θ denotes the parameters in Dt and φ the parameters in Rt .
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L(θ, φ) = LV (θ) + LC (θ, φ)
1
LV (θ) = −
2
T X
N
X
log(2π) + log(hi,t ) +
t=1 i=1
2
ri,t
hi,t
!
.
The likelihood is apparently the sum of individual GARCH
likelihoods, which will be jointly maximized by separately
maximizing each term.
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Two-step procedure:
1.
θ̂ = arg max{LV (θ)}
2.
max{LC (θ̂, φ)}.
φ
Under regularity conditions, consistency of the first step will ensure
consistency of the second step. The maximum of the second step will
a function of the first step parameter estimates. If the first step is
consistent then the second step will be too as long as the function is
continuous in a neighborhood of the true parameters.
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Two step GMM problem (Newey and McFadden, 1994). Consider the
moment condition corresponding to the first step
∇θ LV (θ) = 0
The moment corresponding to the second step is
b =0
∇φ L(θ, φ)
Under regularity conditions the parameter estimates will be
consistent, and asymptotically normal, with asymptotic covariance
matrix
−1 V (φ) = E(∇φφ LC )
E {∇φ LC − E(∇φθ LC )[E(∇θθ LV ]−1 ∇θ LV }
{∇φ LC − E(∇φθ LC )[E(∇θθ LV ]−1 ∇θ LV }′
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E(∇φφ LC )
−1
102

Multivariate GARCH Models

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