MULTIVARIATE GARCH MODELS Eduardo Rossi University of Pavia ITALY

September 2012

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Outline 1 2 3

4

5 6 7 8

Introduction Multivariate Volatility Models Multivariate GARCH Covariance Targeting Vech model Vec model BEKK BEKK & Vech Unconditional Covariance Matrix Covariance stationarity Asymmetric MGARCH-in-mean model Estimation procedure Wald Test Factor-GARCH Orthogonal-GARCH model The Constant Conditional Correlations Model The Dynamic Conditional Correlation (DCC) GARCH Model Rossi

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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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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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Introduction

Vector of returns: yt = (y1t , . . . , yNt )0 yt − µt = t =

(N × 1)

−1/2 Ht zt

Let {zt } be a sequence of (N × 1) i.i.d. random vector with the following characteristics: E [zt ] = 0 E [zt z0t ] = IN zt ∼ G (0, IN ) with G continuous density function.

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Introduction

Et−1 (t ) = 0 Et−1 (t 0t )

=

Ht

(t 0t )

=

Σ

E

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: {t−1 , t−2 , . . .}. The correlation matrix: −1/2

Corrt−1 (t ) = Rt = Dt

−1/2

Ht Dt

Dt = diag(h11,t , . . . , hNN,t )

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Multivariate Volatility Models

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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Multivariate Volatility Models

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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Multivariate GARCH

Three approaches for constructing multivariate GARCH models: 1

direct generalizations of the univariate GARCH model of Bollerslev (1986); (VEC, BEKK and factor models)

2

linear combinations of univariate GARCH models; ((generalized) orthogonal models and latent factor models.)

3

nonlinear combinations of univariate GARCH models; (constant and dynamic conditional correlation models, copula-GARCH models)

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Multivariate GARCH

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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Multivariate GARCH

Covariance Targeting

GARCH(1,1): 2 σt2 = ω + α2t−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) + α2t−1 + βσt−1

σ b2 = T −1

X

b 2t

t

Introduction of targeting transforms the model estimation into a two-step estimation approach: 1

σ b2

2

α, β

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Multivariate GARCH

Covariance Targeting

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 t , 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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Multivariate GARCH

Vech model

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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Multivariate GARCH

Vech model

A natural multivariate extension of the univariate GARCH(p,q) model is vech (Ht )

= W+

q X i=1 ∗

= W+A

Rossi

p
X A∗i vech t−i 0t−i + B∗j vech (Ht−j ) j=1

(L) vech (t 0t )

∗

+ B (L) vech (Ht )

A∗ (L)

= A∗1 L + . . . + A∗q Lq

B∗ (L)

= B∗1 L + . . . + B∗q Lp

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Multivariate GARCH

N∗ ≡

A∗i ,

Vech model

N(N + 1) 2

W

:

[N (N + 1) /2] × 1

B∗j

:

[N ∗ × N ∗ ]

N = 2, Vech-GARCH(1,1):   

h11,t h21,t h22,t



   = 

Rossi

w1∗ w2∗ w3∗





+

∗ a11 ∗ a21 ∗ a31

∗ a12 ∗ a22 ∗ a32

∗ a13 ∗ a23 ∗ a33





  

2 1,t−1 1,t−1 2,t−1 2 2,t−1

MGARCH



  + 

∗ b11 ∗ b21 ∗ b31

∗ b12 ∗ b22 ∗ b32

∗ b13 ∗ b23 ∗ b33

  

h11,t−1 h21,t−1 h2,t−1

  

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Multivariate GARCH

Vec model

This general formulation is termed vec representation by Engle and Kroner (1995). i h 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 t in the sample space in the vech representation this restriction can be difficult to check, let alone impose during estimation.

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Multivariate GARCH

Vec 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 jt εkt . In the diagonal model the A∗i and B∗j matrices are all taken to be diagonal. For N = 2 and p = q = 1, the diagonal model is written as:  2       ∗ ε1,t−1 h11,t w1 0 a11 0 ∗  h21,t  =  w2  +  0 0   ε1,t−1 ε2,t−1  a22 ∗ ε22,t−1 0 0 a33 h22,t w3  ∗   b11 0 0 h11,t−1 ∗ b22 0   h21,t−1  + 0 ∗ 0 0 b33 h22,t−1 hij,t = wi∗ + aii∗ i,t−1 j,t−1 + bii∗ hij,t−1

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Multivariate GARCH

Vec model

Thus the (i, j) th element in Ht depends on the corresponding (i, j) th element in εt−1 ε0t−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 t−1 0t−1 + B Ht−1 where A and B are symmetric matrices. is the Hadamard product.

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Multivariate GARCH

Vec model

Given that t 0t = Ht + Vt with Et−1 (Vt ) = 0. vech (t 0t ) = vech (Ht ) + vech (Vt ) vech (Vt )

vector m.d.s.

with E (vech (Vt )) = vech(E (Vt )) = 0.

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Multivariate GARCH

Vec model

For a GARCH(1,1), the unconditional covariance matrix, when it exists, is given by
vech (t 0t ) = W + A∗1 vech t−1 0t−1 
 +B∗1 vech t−1 0t−1 − vech (Vt−1 ) + vech (Vt ) E (t 0t ) = Σ 

 vech (E (t 0t )) = W + (A∗1 + B∗1 ) vech E t−1 0t−1 −1

vech (Σ) = [IN ∗ − A∗1 − B∗1 ]

W.

For a GARCH(p,q) model −1

vech (Σ) = [IN ∗ − A∗ (1) − B∗ (1)]

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Multivariate GARCH

Vec model

Targeting cannot easily introduced in the model. MGARCH(1,1)-vech, unconditional var-cov matrix [IN ∗ − A∗1 − B∗1 ]vech(Σ) X b = T −1 b Σ t b 0t t ∗

Targeting allows reducing by N the parameters to be estimated. The total number of parameters is still O(N 4 ).

vech (Ht )

Rossi

=

  h  i b + A∗ vech t−1 0 − vech Σ b vech Σ 1 t−1 h  i b +B∗1 vech (Ht−1 ) − vech Σ

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Multivariate GARCH

BEKK

Engle and Kroner (1995) propose a parametrization that imposes positive definiteness restrictions. Consider the following model p q K X K X X X 0 0 Aik t−i t−i Aik + Bik Ht−i B0ik Ht = CC + 0

k=1 i=1

(1)

k=1 i=1

where C, Aik and Bik are (N × N). The intercept matrix is decomposed into CC0 , where C is a lower triangular matrix. Without any further assumption CC0 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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Multivariate GARCH

BEKK

For exposition simplicity we will assume that K = 1: Ht = CC0 +

q X

Ai t−i 0t−i A0i +

i=1

p X Bi Ht−i B0i i=1

Consider the simple GARCH(1,1) model: Ht = CC0 + A1 t−1 0t−1 A01 + B1 Ht−1 B01

(2)

BEKK (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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Multivariate GARCH

BEKK

The purpose of the restrictions is to eliminate all other observationally equivalent structures. For example, as relates to the term A1 t−1 0t−1 A01 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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Multivariate GARCH

BEKK

MGARCH(1,1)-BEKK, N = 2: Ht

=

0



CC +  +

Rossi

b11 b21

a11 a21 b12 b22

a12 a22 



ε21t−1 ε2t−1 ε1t−1

h11t−1 h21t−1

h12t−1 h22t−1

MGARCH

 ε1t−1 ε2t−1 a11 ε22t−1 a21  0 b11 b12 b21 b22

a12 a22

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Multivariate GARCH

BEKK

BEKK-GARCH(p,q) model (Engle and Kroner (1995)):

Sufficient condition for positive definiteness of Ht 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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Multivariate GARCH

BEKK

For simplicity consider the GARCH(1,1) model. The BEKK parametrization is Ht = CC0 + +A1 t−1 0t−1 A01 + B1 Ht−1 B01 The proof proceeds by induction. First Ht is p.d. for t = 1: The term A1 0 00 A01 is positive semidefinite because 0 00 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 CC0 + B1 H0 B01 is positive definite. This is true if C or B1 has full rank.

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Multivariate GARCH

BEKK

To show that the null space condition is sufficient CC0 + B1 H0 B01 is p.d. if and only if x 0 (CC0 + B1 H0 B01 ) x > 0 ∀x = 6 0 or

 0   0 1/2 1/2 H0 B01 x > 0 (C0 x) (C0 x) + H0 B01 x 1/20

where H0 = H0

Rossi

1/2

H0

1/2

and H0

∀x 6= 0

is full rank.

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Multivariate GARCH

BEKK

Defining N (P) to be the null space of the matrix P, (28) is true if and only if   1/2 N (C0 ) ∩ N H0 B01 = ∅.   1/2 1/2 N H0 B01 = N (B01 ) because H0 is full rank. This implies that   1/2 CC0 + B1 H0 B01 is positive definite if and only if N (C0 ) ∩ N H0 B01 = ∅. Now suppose that Ht is positive definite for t = τ . Then, Hτ +1 = CC0 + A1 τ 0τ A01 + B1 Hτ B01 is positive definite if and only if, given that A1 τ 0τ A01 is positive semidefinite, the null space condition holds, because Hτ is positive definite by the induction assumption.

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Multivariate GARCH

BEKK

Consider MGARCH(1,1)-BEKK, N = 2 with A1 = diag(a11 , a22 ) B1 = diag(b11 , b22 ) the model reduces to   2  a11 0 ε1t−1 ε1t−1 ε2t−1 a11 Ht = CC0 + 0 a22 ε2t−1 ε1t−1 ε22t−1 0    0 b11 0 h11t−1 h12t−1 b11 0 + 0 b22 h21t−1 h22t−1 0 b22 h11,t

2 2 2 2 = c11 + a11 1t−1 + b11 h11t−1

h12,t

= c21 c11 + a11 a22 1t−1 2t−1 + b11 b22 h12t−1

h22,t

2 2 2 2 = c21 c11 + c22 + a22 1t−1 + b22 h11t−1

0 a22

0

This model is equivalent to the Hadamard BEKK: Ht = CC0 + aa0 t−1 0t−1 + bb0 Ht−1 positive definiteness is not guaranteed. Positive semidefiniteness is obtained by imposing p.s.d. of all terms. Rossi

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Multivariate GARCH

BEKK

MGARCH(1,1) - Scalar BEKK A1 = αIN ,

B1 = βIN

Ht = CC0 + α2 (t−1 0t−1 ) + β 2 Ht−1

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Multivariate GARCH

1

BEKK

BEKK
Ht = Σ + A1 t−1 0t−1 − Σ A01 + B1 (Ht−1 − Σ) B1 or
Ht = (Σ − A1 ΣA1 − B1 ΣB01 ) + A1 t−1 0t−1 A01 + B1 Ht−1 B01 To have p.d-ness of Ht , (Σ − A1 ΣA1 − B1 ΣB01 ) must be p.d..

2

Hadamard BEKK
Ht = Σ + A1 t−1 0t−1 − Σ + B1 (Ht−1 − Σ)
t−1 0t−1 − Σ must be p.s.d., while (Ht−1 − Σ) must be p.s.d.

3

Scalar BEKK
Ht = Σ + α2 t−1 0t−1 − Σ + β 2 (Ht−1 − Σ) for α + β < 1.

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Multivariate GARCH

BEKK & Vech

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(CC0 ) +

q X

vec(Ai t−i 0t−i A0i ) +

i=1

p X vec(Bi Ht−i B0i ) i=1

where vec
() is an operator such that given a matrix A (n × n), vec(A) is a n2 × 1 vector. The vec () satisfies vec (ABC) = (C0 ⊗ A) vec (B)

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Multivariate GARCH

BEKK & Vech

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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Multivariate GARCH

BEKK & Vech

Then vec(Ht )

0

=

vec(CC ) +

q X

(Ai ⊗ Ai ) vec(εt−i 0t−i )

i=1

+

p X

(Bi ⊗ Bi ) vec(Ht−i )

i=1

DN vech (Ht )

0

= DN vech(CC ) +

q X

(Ai ⊗ Ai ) DN vech(t−i 0t−i )

i=1

+

p X

(Bi ⊗ Bi ) DN vech(Ht−i )

i=1

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Multivariate GARCH

BEKK & Vech

If DN is a full column rank matrix we can define the generalized inverse of DN as: −1

0 D+ D0N N = (DN DN )
that is a (N (N + 1) /2) × N 2 matrix, where

D+ N DN = IN

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Multivariate GARCH

BEKK & Vech

This implies that premultiplying by D+ N vech (Ht )

=

0

vech(CC ) +

D+ N

q X

! (Ai ⊗ Ai ) DN vech(t−i 0t−i )

i=1

+D+ N

p X

! (Bi ⊗ Bi ) DN vech(Ht−i )

i=1

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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Multivariate GARCH

BEKK & Vech

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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Multivariate GARCH

BEKK & Vech

Given the vech model vech (Ht ) = W + A∗ (L) vech (εt ε0t ) + 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) = D+ (Ai ⊗ Ai ) DN N i=1 ∗

B (1)

=

D+ N

q X

! (Bi ⊗ Bi ) DN

i=1

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Multivariate GARCH

BEKK & Vech

This implies also that in the BEKK model, {t } 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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Multivariate GARCH

Unconditional Covariance Matrix

BEKK model vec (t 0t )


vec (CC0 ) + (A1 ⊗ A1 ) vec t−1 0t−1 
 + (B1 ⊗ B1 ) vec t−1 0t−1 − vec (Vt−1 ) + vec (Vt )
  E [vec (t 0t )] = vec (CC0 ) + [(A1 ⊗ A1 ) + (B1 ⊗ B1 )] E vec t−1 0t−1 =

vec (Σ) = [IN 2 − (A1 ⊗ A1 ) − (B1 ⊗ B1 )]

−1

vec (CC0 )

or in vech representation as DN vech (E (t 0t ))

= DN vech (CC0 ) + (A1 ⊗ A1 ) DN vech E t−1 0t−1

+ (B1 ⊗ B1 ) DN vech E t−1 0t−1





 −1 + vech (Σ) = IN ∗ − D+ vech (CC0 ) N (A1 ⊗ A1 ) DN − DN (B1 ⊗ B1 ) DN N ∗ = N (N + 1) /2.

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Multivariate GARCH

Covariance stationarity

The diagonal vech model is stationary if and only if the sum aii∗ + bii∗ < 1 for all i. In the diagonal BEKK model the covariance stationary condition is that aii2 + bii2 < 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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Multivariate GARCH

Asymmetric MGARCH-in-mean model

A general multivariate model can be written as: yt = µ + Π (L) yt−1 + Ψxt−1 + Λvech (Ht ) + t

(3)

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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Multivariate GARCH

Asymmetric MGARCH-in-mean model

xt−1 contains predetermined variables. t is the vector of innovation with respect to the information set formed exclusively of past realizations of yt . Ht = Et−1 (t 0t ) Ht = CC0 +

q X

0

Ai (t−i + γ) (t−i + γ) A0i +

i=1

Rossi

p X

Bj Ht−j B0j

j=1

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Multivariate GARCH

Asymmetric MGARCH-in-mean model

We can consider a multivariate generalization of the size effect and sign effect: 0 Ht = CC0 + A1 t−1 0t−1 A01 + B1 Ht−1 B01 + Dvt−1 vt−1 D0 + Gt−1 0t−1 G0

where vt = |zt | − E |zt |, with zit = εit /    G=  

I (ε1t−1 < 0) g11 0 .. . 0

Rossi

p hii,t and

0 .. .

... ..

...

MGARCH

0

.

0 .. . 0 I (εNt−1 < 0) gNN

     

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Multivariate GARCH

Asymmetric MGARCH-in-mean model

When N = 2 0 vt−1 vt−1

=

=

Rossi

  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 0  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 |) 2 (|z2t | − E |z2t |) (|z1t | − E |z1t |) (|z2t | − E |z2t |)

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Multivariate GARCH

Gt−1 0t−1 G0

∗ ∗ g22 ε1t−1 ε2t−1 g11 ∗2 2 g22 ε2t−1



∗2 2 ε1t−1 g11 ∗ ∗ g22 ε1t−1 ε2t−1 g11



2 2 ε1t−1 I (ε1t−1 < 0) g11 δ12 g11 g22 ε1t−1 ε2t−1

= =

Asymmetric MGARCH-in-mean model



δ12 g11 g22 ε1t−1 ε2t−1 2 2 ε2t−1 I (ε2t−1 < 0) g22



δ12 = I (ε1t−1 < 0) I (ε2t−1 < 0)

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Estimation procedure

Given the model (3)-(44), the log-likelihood function for {εT , . . . , ε1 } obtained under the assumption of conditional multivariate normality is: " # T X
1 0 −1 log |Ht | + t Ht t log LT (T , . . . , 1 ; θ) = − TN log (2π) + 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.

Rossi

MGARCH

CIdE - 2012

48 / 90

Estimation procedure

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} P

where Θ ⊂ R and µt and Ht are known functions of wt and θ.

Rossi

MGARCH

CIdE - 2012

49 / 90

Estimation procedure

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 , θ 0 ) Var (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.

Rossi

MGARCH

CIdE - 2012

50 / 90

Estimation procedure

For observation t the quasi-conditional log-likelihood is lt (θ; yt , wt )

N 1 ln (2π) − ln |Ht (wt , θ)| 2 2 1 0 − (yt − µt (wt , θ)) H−1 t (wt , θ) (yt − µt (wt , θ)) 2

= −

Letting t (yt , wt , θ 0 ) ≡ yt − µt (wt , θ) : (N × 1) denote the residual function lt (θ) = −

1 1 N log (2π) − log |Ht (θ)| − 0t (θ) H−1 t (θ) t (θ) 2 2 2 log LT (θ) =

T X lt (θ) t=1

Rossi

MGARCH

CIdE - 2012

51 / 90

Estimation procedure

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)

0

=

0

0

∇θ lt (θ) − ∇θ µt (θ) H−1 t (θ) t (θ) +     1 0 0 −1 ∇θ Ht (θ) H−1 t (θ) ⊗ Ht (θ) vec t (θ) t (θ) − Ht (θ) 2

where ∇θ µt (θ) : (N × P)
∇θ Ht (θ) : N 2 × P

Rossi

MGARCH

CIdE - 2012

52 / 90

Estimation procedure

If the first conditional two moments are correctly specified, the true error vector is defined as 0t ≡ t (θ 0 ) = yt − µt (wt , θ 0 )
and E 0t |wt = 0,
E 0t 00 t |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 − fields {σ (yt , wt ) : t = 1, 2, . . .}. This result can be used to establish weak consistency of the quasi-maximum likelihood estimator (QMLE).

Rossi

MGARCH

CIdE - 2012

53 / 90

Estimation procedure

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) at (θ 0 )

=

0

∇θ µt (θ 0 ) H−1 t (θ 0 ) ∇θ µt (θ 0 )  1 0 −1 + ∇θ Ht (θ) H−1 t (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 Var [st (θ 0 ) |wt ] 6= at (θ 0 ) and the information matrix equality is violated.

Rossi

MGARCH

CIdE - 2012

54 / 90

Estimation procedure

The QMLE has the following properties:   0−1 0 0−1 −1/2 √  d bT − θ 0 → AT BT AT T θ N (0, IP ) where T

A0T ≡ −

T

1X 1X E [ht (θ 0 )] = E [at (θ 0 )] T t=1 T t=1

and T h i  1X  0 B0T ≡ Var T −1/2 ST (θ 0 ) = E st (θ 0 ) st (θ 0 ) T t=1

in addition

p b T − A0 → 0 A T p b T − B0 → B 0 T

Rossi

MGARCH

CIdE - 2012

55 / 90

Estimation procedure

b −1 B bT A b −1 is a consistent estimator od the robust asymptotic The matrix A T T   √ bT − θ 0 . covariance matrix of T θ In practice,   bT ≈ N θ, A b −1 B bT A b −1 /T θ T T b −1 /T (Hessian form) Under normality, the variance estimator can be replaced by A T −1 b /T (outer product of the gradient form). or B T

Rossi

MGARCH

CIdE - 2012

56 / 90

Estimation procedure

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  0     0 −1   d −1 b b −1 b b b bT bT → ξW = Tr θ T R θ T AT BT AT R θ r θ χ2Q . H0

Rossi

MGARCH

CIdE - 2012

57 / 90

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.

Factor 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 + t

Rossi

MGARCH

CIdE - 2012

58 / 90

Factor-GARCH

Assume that the idiosyncratic shocks t 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 ft0 ) = Λt Λt = diag (λ1 , . . . , λK ) and positive definite. The conditioning set is {yt−1 , ft−1 , . . . , y1 , f1 }. Also suppose that E (ft 0t ) = 0. The conditional covariance matrix of yt equals Et−1 (yt yt0 ) = Ht = Ψ + BΛt B0 = Ψ +

K X

β k β 0k λkt

k=1

where β k denotes the kth column in B. Thus, there are K statistics which determine the full covariance matrix.

Rossi

MGARCH

CIdE - 2012

59 / 90

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.

Factor-representing portfolios Portfolio weights are orthogonal to all but one set of factor loadings: rkt = φ0k yt  1 k =j 0 φk β j = 0 otherwise the vector of factor-representing portfolios is rt = Φ0 yt where the columns of matrix Φ are the φk vectors.

Rossi

MGARCH

CIdE - 2012

60 / 90

Factor-GARCH

The conditional variance of rkt is given by Vart−1 (rkt )

=

φ0k Et−1 (yt yt0 ) φk = φ0k Ht φk

=

φ0k (Ψ + BΛt B0 ) φk

=

ψk + λkt

φ0k Ψφk .

where ψk = 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 ≡ φ0k Ht φk = Vart−1 (rkt ) = ψk + λkt So we get that K X

β k β 0k θ kt =

k=1

k=1

K X

K X

β k β 0k λkt =

k=1

Ht

=

K X

Ψ+

β k β 0k ψk +

β k β 0k λkt

k=1

β k β 0k θ kt −

k=1

K X

β k β 0k ψk

k=1

K K K X X X β k β 0k ψk β k β 0k λkt = Ψ + β k β 0k θ kt − k=1

Rossi

K X

k=1 MGARCH

k=1 CIdE - 2012

61 / 90

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 θ kt = ωk + αk φ0k t−1 = ωk + αk φ0k = ωk + αk φ0k = ωk + αk φ0k

Rossi


2


2 + γk Et−2 rkt−1


 t−1 0t−1 φk + γk Et−2 φ0k yt φ0k yt
  t−1 0t−1 φk + γk φ0k Et−2 (yt yt0 ) φk
  t−1 0t−1 φk + γk φ0k Ht−1 φk

MGARCH

CIdE - 2012

62 / 90

Factor-GARCH

The conditional variance-covariance matrix of yt can be written as Ht = Ψ∗ +

K X β k β 0k θ kt k=1

K X  
   = Ψ∗ + β k β 0k ωk + αk φ0k t−1 0t−1 φk + γk φ0k Ht−1 φk k=1 K X Ψ + β k β 0k ωk

!

∗

=

k=1

+

K X

 
   β k β 0k αk φ0k t−1 0t−1 φk + γk φ0k Ht−1 φk

k=1

Ht = Γ +

K X β k β 0k θ kt k=1

where Γ = Ψ∗ +

K P

β k β 0k ωk .

k=1 Rossi

MGARCH

CIdE - 2012

63 / 90

Factor-GARCH

Therefore Ht = Γ +

K K X 
 X   αk β k φ0k t−1 0t−1 φk β 0k + γk β k φ0k Ht−1 φk β 0k k=1

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.

Rossi

MGARCH

CIdE - 2012

64 / 90

Orthogonal-GARCH model

The orthogonal models are particular factor models (Kariya (1988) and Alexander and Chibumba (1997)). They are based on the assumption that the observed data can be obtained by a linear transformation of a set of uncorrelated components by means of an orthogonal matrix. The (N × N) time-varying variance matrix Ht is generated by (m × N) univariate GARCH models. The components are the principal components of the data, or a subset of them.

Rossi

MGARCH

CIdE - 2012

65 / 90

Orthogonal-GARCH model

The diagonal matrix V contains the population variances of yt : V = diag{v12 , . . . , vN2 } the standardized returns are ut = V−1/2 yt where ut = Lft E [ut ] = 0

Rossi

E [ut u0t ] = R

MGARCH

CIdE - 2012

66 / 90

Orthogonal-GARCH model

The population correlation matrix can be decomposed as: R = PΛP0 P is the orthogonal eigenvectors matrix, Λ is the diagonal matrix of the eigenvalues: Λ = diag{λ1 , . . . , λN } ranked in descending order.

Rossi

MGARCH

CIdE - 2012

67 / 90

Orthogonal-GARCH model

P satisfies: P0 = P−1

P0 P = IN

PP0 = IN

It follows that R = PΛ1/2 Λ1/2 P0 = LL0 The factor loading matrix is obtained as L = PΛ1/2 such that ft = L−1 ut with E [ft ft0 ] = L−1 E [ut u0t ]L−10 = L−1 RL−10 = L−1 LL0 L0−1 = IN

Rossi

MGARCH

CIdE - 2012

68 / 90

Orthogonal-GARCH model

Assuming Et−1 [ft ft0 ] = Qt = diag(σf21,t , . . . , σf2N,t ) Qt is a diagonal matrix. 2 σf2i,t = (1 − αi,1 − βi,1 ) + αi,1 fi,t−1 + βi,1 σf2i,t−1

i = 1, 2, . . . , N

Et−1 [ut u0t ] = Et−1 [Lft ft0 L0 ] = LQt L0 Et−1 [yt yt0 ] = Et−1 [V1/2 ut u0t V1/2 ] = V1/2 LQt L0 V1/2 The number of parameters is N(N + 5)/2.

Rossi

MGARCH

CIdE - 2012

69 / 90

Orthogonal-GARCH model

In practice, V and L are replaced by their sample counterparts, and m is chosen bt . by principal component analysis applied to the standardized residuals, u 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 P0m m Pm is a matrix (N × m) containing the m eigenvectors of P corresponding to the m largest eigenvalues. ftm = Λ−1/2 P0m ut m where ftm = [f1t , . . . , fmt ] Et−1 [ftm ] = 0 0

Et−1 [ftm ftm ] = Qm,t = diag(σf21,t , . . . , σf2m,t )

Rossi

MGARCH

CIdE - 2012

70 / 90

Orthogonal-GARCH model

Alexander (2001, section 7.4.3) emphasizes that using a small number of principal components compared to the number of assets is the strength of the approach. However, note that the conditional variance matrix has reduced rank (if m < N), which may be a problem for applications and for diagnostic tests which depend on the inverse of Ht .

Rossi

MGARCH

CIdE - 2012

71 / 90

The Constant Conditional Correlations Model

These models are based on a decomposition of the Ht . The conditional var-cov matrix is expressed as Ht = Dt Rt Dt where Rt is possibly time-varying. Conditional correlations and variances are separately modeled.

Rossi

MGARCH

CIdE - 2012

72 / 90

The Constant Conditional Correlations Model

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 [t 0t ] = Ht {Ht }ii = hit {Ht }ij = hijt =

i = 1, . . . , N

1/2 1/2 ρij hit hjt

i 6= j

i, j = 1, . . . , N

Dt = diag {h1t , . . . , hNt }

Rossi

MGARCH

CIdE - 2012

73 / 90

The Constant Conditional Correlations Model

The conditional covariance matrix can be written as: 1/2

1/2

Ht = Dt RDt



1/2 h1t

 . Ht =   .. 0

··· .. . ···





1

ρ12

0    ρ21 ..  .  .  .. 1/2 hNt ρN1

1 .. . ...

"

h1t 0

#

"

h1t 1/2 1/2 ρ12 h1t h2t

... ...

ρ1N .. .

... ρNN−1

ρN−1N 1



     

1/2

h1t .. . 0

··· .. . ···

 0  .. . .  1/2 hNt

When N = 2 Ht

=

=

Rossi

1/2

0 1/2 h2t

1 ρ21

ρ12 1

1/2 1/2

ρ12 h1t h2t h2t

MGARCH

"

1/2

h1t 0

0 1/2 h2t

#

# .

CIdE - 2012

74 / 90

The Constant Conditional Correlations Model

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 2

conditional variances are estimated employing the marginal likelihoods b −1 R is estimated using the sample estimator of standardized residuals D t yt (assuming µt = 0).

Rossi

MGARCH

CIdE - 2012

75 / 90

The Constant Conditional Correlations Model

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).

Rossi

MGARCH

CIdE - 2012

76 / 90

The Dynamic Conditional Correlation (DCC) 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.

Rossi

MGARCH

CIdE - 2012

77 / 90

The Dynamic Conditional Correlation (DCC) GARCH Model

The conditional correlation between two random variables, Xt and Yt is defined as: ρYX ,t = p

Covt−1 (Xt Yt ) 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 = Dt Rt Dt . Dt = diag(Vart−1 (y1t ), . . . , Vart−1 (yNt )) where the Vart−1 (yit ), i = 1, . . . , N are modeled as univariate GARCH processes.

Rossi

MGARCH

CIdE - 2012

78 / 90

The Dynamic Conditional Correlation (DCC) GARCH Model

The standardized returns are:

−1/2

η t = Dt −1/2

Et−1 (η t η 0t ) = Dt

yt

−1/2

Ht Dt

= Rt = {ρij,t }

we can use the conditional variance of η t to describe the conditional correlation of yt . The conditional correlation estimator is ρij,t = √

qij,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 )

(4)

The term ρij is not the unconditional correlation between ηit and ηjt ; the unconditional correlation between ηit and ηjt has no closed form.

Rossi

MGARCH

CIdE - 2012

79 / 90

The Dynamic Conditional Correlation (DCC) GARCH Model

Engle (2002) assumes that ρij ' q ij . Aielli (2006) and Engle et al. (2008) suggest to modify the standardP DCC in order to correct the asymptotic bias which is due to the fact that T1 t t 0t does not converge to Q. It is known though that the impact of this is very small (see Engle and Sheppard (2001)).

Rossi

MGARCH

CIdE - 2012

80 / 90

The Dynamic Conditional Correlation (DCC) GARCH Model

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 0t−1 ) + β(Qt−1 ) where Q is the unconditional covariance matrix of η t .

Rossi

MGARCH

CIdE - 2012

81 / 90

The Dynamic Conditional Correlation (DCC) GARCH Model

DCC model has correlation targeting, when α + β < 1 E [Qt ]

= R

E [Qt ]

= E [Qt−1 ]

E [η t η 0t ]

Rossi

= E [Qt ]

E [Qt ]

= Q(1 − α − β) + αE [η t−1 η 0t−1 ] + βE [Qt−1 ]

E [Qt ]

= R(1 − α − β) + αE [Qt ] + βE [Qt ]

MGARCH

CIdE - 2012

82 / 90

The Dynamic Conditional Correlation (DCC) GARCH Model

Clearly more complex positive definite multivariate GARCH models could be used for the correlation parametrization 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 (ιι0 − A − B) + A η t−1 η 0t−1 + B Qt−1

(5)

where denotes the Hadamard product ({A B}ij = aij bij ).

Rossi

MGARCH

CIdE - 2012

83 / 90

The Dynamic Conditional Correlation (DCC) GARCH Model

The Generalized-DCC model specification: 0 Dt = diag{ωi } + diag{κi } yt−1 yt−1 + diag{λi } Dt−1 −1/2

η t = Dt

yt

(6)

0

Qt = S (ιι − A − B) + A Rt = diag{Qt }

−1/2

η t−1 η 0t−1 −1/2

Qt diag{Qt }

+ B Qt−1

.

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.

Rossi

MGARCH

CIdE - 2012

84 / 90

The Dynamic Conditional Correlation (DCC) GARCH Model

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 −1/2 Dt Ht Dt is positive definite.

Rossi

MGARCH

CIdE - 2012

85 / 90

The Dynamic Conditional Correlation (DCC) GARCH Model

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

3

2 >0 Dii,0

4

The roots of 1 − κi Z − λi Z are outside the unit circle.

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)

Rossi

MGARCH

CIdE - 2012

86 / 90

The Dynamic Conditional Correlation (DCC) GARCH Model

The log-likelihood function can be written as: T

log LT

1X (N log(2π) + log |Ht | + yt0 H−1 t yt ) 2 t=1

=

−

=

1X 1/2 1/2 −1/2 −1 −1/2 − (N log(2π) + log |Dt Rt Dt | + yt0 Dt Rt Dt yt ) 2 t=1

=

−

T

T

1X (N log(2π) + log |Dt | + log |Rt | + η 0t R−1 t ηt ) 2 t=1 −1/2

Adding and subtracting yt0 Dt

−1/2

Dt

yt = η 0t η t

T

log LT

=

−

1X −1/2 −1/2 (N log(2π) + log |Dt | + yt0 Dt Dt yt 2 t=1

−η 0t η t + log |Rt | + η 0t R−1 t ηt ) T

=

−

1X (N log(2π) + log |Dt | + rt0 Dt−1 rt ) 2 t=1

−

1 X 0 −1 (η t Rt η t − η 0t η t + log |Rt |) 2 MGARCH

T

Rossi

CIdE - 2012

87 / 90

The Dynamic Conditional Correlation (DCC) GARCH Model

Volatility component: T

LV (θ) ≡ log LV ,T (θ) = −

1X (N log(2π) + log |Dt | + yt0 D−1 t yt ) 2 t=1

Correlation component: T

LC (θ, φ) ≡ log LC ,T (θ, φ) = −

1 X 0 −1 (η R η − η 0t η t + log |Rt |) 2 t=1 t t t

θ denotes the parameters in Dt and φ the parameters in Rt . L(θ, φ) = LV (θ) + LC (θ, φ) T N 2 ri,t 1 XX LV (θ) = − log(2π) + log(hi,t ) + 2 t=1 hi,t

! .

i=1

The likelihood is apparently the sum of individual GARCH likelihoods, which will be jointly maximized by separately maximizing each term. Rossi

MGARCH

CIdE - 2012

88 / 90

The Dynamic Conditional Correlation (DCC) GARCH Model

Two-step procedure: 1

ˆ = arg max{LV (θ)} θ 2

ˆ φ)}. max{LC (θ, 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.

Rossi

MGARCH

CIdE - 2012

89 / 90

The Dynamic Conditional Correlation (DCC) GARCH Model

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 }0

Rossi

MGARCH



−1 E (∇φφ LC )

CIdE - 2012

90 / 90