MULTIVARIATE GARCH MODELS - Dipartimento di …€¦ · · 2012-08-31Outline 1 Introduction ... pricing of derivative contracts based on a more than one underlying asset ... Three

MULTIVARIATE GARCH MODELS

Eduardo Rossi

University of PaviaITALY

September 2012

Rossi MGARCH CIdE - 2012 1 / 90

Outline

1 Introduction

2 Multivariate Volatility Models

3 Multivariate GARCHCovariance TargetingVech modelVec modelBEKKBEKK & VechUnconditional Covariance MatrixCovariance stationarityAsymmetric MGARCH-in-mean model

4 Estimation procedureWald Test

5 Factor-GARCH

6 Orthogonal-GARCH model

7 The Constant Conditional Correlations Model

8 The Dynamic Conditional Correlation (DCC) GARCH Model


Introduction

Economics and financial economics present problems whose solutions need thespecification 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)


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


Introduction

Vector of returns:yt = (y1t , . . . , yNt)

′ (N × 1)

yt − µt = εt = H−1/2t zt

Let {zt} be a sequence of (N × 1) i.i.d. random vector with the followingcharacteristics:

E [zt ] = 0

E [ztz′t ] = IN

zt ∼ G (0, IN)

with G continuous density function.


Introduction

Et−1 (εt) = 0

Et−1 (εtε′t) = Ht

E (εtε′t) = Σ

Et−1[·] = E [·|Φt−1]

Φt−1 is the σ-field generated by past values of observable variables. where Ht is amatrix (N × N) positive definite and measurable with respect to the informationset Φt−1, that is the σ-field generated by the past observations: {εt−1, εt−2, . . .}.The correlation matrix:

Corrt−1(εt) = Rt = D−1/2t HtD

−1/2t

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


Multivariate Volatility Models

MVMs provide a parametric structure for the dynamic evolution of Ht . MVMsmust 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.


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


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) orthogonalmodels and latent factor models.)

3 nonlinear combinations of univariate GARCH models; (constant and dynamicconditional correlation models, copula-GARCH models)


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 bythe E [Ht ] or E [Rt ].

The long-run solution is replaced by a consistent estimator.



GARCH(1,1):σ2t = ω + αε2t−1 + βσ2

t−1

Long-run variance (if (α + β) < 1:

σ2 = E [σ2t ] = ω(1− α− β)−1

Variance targeting:

σ2t = σ2(1− α− β) + αε2t−1 + βσ2

t−1

σ2 = T−1∑t

ε2t

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

1 σ2

2 α, β



N assets: N variances + 12N(N − 1) covariances = N

2 (N + 1).Two alternative approaches:

Models of Ht

Models of Dt and Rt

The parametrization of Ht as a multivariate GARCH, which means as a functionof the information set Φt−1, allows each element of Ht to depend on q lagged ofthe squares and cross-products of εt , as well as p lagged values of the elements ofHt . So the elements of the covariance matrix follow a vector of ARMA process insquares and cross-products of the disturbances.


Multivariate GARCH Vech model

Let vech denote the vector-half operator, which stacks the lower triangularelements of an N × N matrix as an [N (N + 1) /2]× 1 vector.Let A be (2× 2), then vech(A)

vech(A) =

a11a21a22

Since the conditional covariance matrix Ht is symmetric,vech (Ht) , (N(N + 1)/2× 1) contains all the unique elements in Ht .



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

vech (Ht) = W +

q∑i=1

A∗i vech(εt−iε

′t−i)

+

p∑j=1

B∗j vech (Ht−j)

= W + A∗ (L) vech (εtε′t) + B∗ (L) vech (Ht)

A∗ (L) = A∗1L + . . .+ A∗qLq

B∗ (L) = B∗1L + . . .+ B∗qLp



N∗ ≡ N(N + 1)

2

W : [N (N + 1) /2]× 1

A∗i , B∗j : [N∗ × N∗]

N = 2, Vech-GARCH(1,1):

h11,th21,th22,t

=

w∗1w∗2w∗3

+

a∗11 a∗12 a∗13a∗21 a∗22 a∗23a∗31 a∗32 a∗33

ε21,t−1ε1,t−1ε2,t−1

ε22,t−1

+

b∗11 b∗12 b∗13b∗21 b∗22 b∗23b∗31 b∗32 b∗33

h11,t−1

h21,t−1h2,t−1


Multivariate GARCH Vec model

This general formulation is termed vec representation by Engle and Kroner(1995).

The number of parameters is[1 + (p + q) [N (N + 1) /2]2

].

Even for low dimensions of N and small values of p and q the number ofparameters 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(N4): the curse of dimensionality.

For any parametrization to be sensible, we require that Ht be positive definite forall values of εt in the sample space in the vech representation this restriction canbe difficult to check, let alone impose during estimation.



A natural restriction is the diagonal representation, in which each element of thecovariance matrix depends only on past values of itself and past values of εjtεkt .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: h11,t

h21,t

h22,t

=

w1

w2

w3

+

a∗11 0 00 a∗22 00 0 a∗33

ε21,t−1ε1,t−1ε2,t−1ε22,t−1

+

b∗11 0 00 b∗22 00 0 b∗33

h11,t−1h21,t−1h22,t−1

hij,t = w∗i + a∗iiεi,t−1εj,t−1 + b∗iihij,t−1



Thus the (i , j) th element in Ht depends on the corresponding (i , j) th element inεt−1ε

′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(N2).The diagonal vech is equivalent to:

Ht = W + A� εt−1ε′t−1 + B�Ht−1

where A and B are symmetric matrices. � is the Hadamard product.



Given thatεtε′t = Ht + Vt

with Et−1 (Vt) = 0.

vech (εtε′t) = vech (Ht) + vech (Vt)

vech (Vt) vector m.d.s.

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



For a GARCH(1,1), the unconditional covariance matrix, when it exists, is given by

vech (εtε′t) = W + A∗1vech

(εt−1ε

′t−1)

+B∗1[vech

(εt−1ε

′t−1)− vech (Vt−1)

]+ vech (Vt)

E (εtε′t) = Σ

vech (E (εtε′t)) = W + (A∗1 + B∗1)

[vech

(E(εt−1ε

′t−1))]

vech (Σ) = [IN∗ − A∗1 − B∗1 ]−1 W.

For a GARCH(p,q) model

vech (Σ) = [IN∗ − A∗ (1)− B∗ (1)]−1 W



Targeting cannot easily introduced in the model.MGARCH(1,1)-vech, unconditional var-cov matrix

[IN∗ − A∗1 − B∗1 ]vech(Σ)

Σ = T−1∑t

εt ε′t

Targeting allows reducing by N∗ the parameters to be estimated. The totalnumber of parameters is still O(N4).

vech (Ht) = vech(

Σ)

+ A∗1vech[εt−1ε

′t−1 − vech

(Σ)]

+B∗1

[vech (Ht−1)− vech

(Σ)]


Multivariate GARCH BEKK

Engle and Kroner (1995) propose a parametrization that imposes positivedefiniteness restrictions.

Consider the following model

Ht = CC′ +K∑

k=1

q∑i=1

Aikεt−iε′t−iA

′ik +

K∑k=1

p∑i=1

BikHt−iB′ik (1)

where C, Aik and Bik are (N × N).

The intercept matrix is decomposed into CC′, where C is a lower triangularmatrix.

Without any further assumption CC′ is positive semidefinite.

This representation is general, it includes all positive definite diagonalrepresentations and nearly all positive definite vech representations.



For exposition simplicity we will assume that K = 1:

Ht = CC′ +

q∑i=1

Aiεt−iε′t−iA

′i +

p∑i=1

BiHt−iB′i

Consider the simple GARCH(1,1) model:

Ht = CC′ + A1εt−1ε′t−1A′1 + B1Ht−1B′1 (2)

BEKK (Engle and Kroner (1995))

Suppose that the diagonal elements in C are restricted to be positive and that a11and 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.



The purpose of the restrictions is to eliminate all other observationally equivalentstructures.

For example, as relates to the term A1εt−1ε′t−1A′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 fora given i and j , as this condition is also sufficient to eliminate −A1 from the setof admissible structures.



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

Ht = CC′ +

[a11 a12a21 a22

] [ε21t−1 ε1t−1ε2t−1ε2t−1ε1t−1 ε22t−1

] [a11 a12a21 a22

]′+

[b11 b12

b21 b22

] [h11t−1 h12t−1h21t−1 h22t−1

] [b11 b12

b21 b22

]′



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 fullrank matrix or if any Bi i = 1, . . . , p is a full rank matrix.



For simplicity consider the GARCH(1,1) model. The BEKK parametrization is

Ht = CC′ + +A1εt−1ε′t−1A′1 + B1Ht−1B′1

The proof proceeds by induction.First Ht is p.d. for t = 1: The term A1ε0ε

′0A′1 is positive semidefinite because

ε0ε′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′ + B1H0B′1

is positive definite. This is true if C or B1 has full rank.



To show that the null space condition is sufficient CC′ + B1H0B′1 is p.d. if andonly if

x ′ (CC′ + B1H0B′1) x > 0 ∀x 6= 0

or

(C′x)′(C′x) +

(H

1/20 B′1x

)′ (H

1/20 B′1x

)> 0 ∀x 6= 0

where H0 = H1/2′0 H

1/20 and H

1/20 is full rank.



Defining N (P) to be the null space of the matrix P, (28) is true if and only if

N (C′) ∩ N(

H1/20 B′1

)= ∅.

N(

H1/20 B′1

)= N (B′1) because H

1/20 is full rank. This implies that

CC′ + B1H0B′1 is positive definite if and only if N (C′) ∩ N(

H1/20 B′1

)= ∅. Now

suppose that Ht is positive definite for t = τ .Then,

Hτ+1 = CC′ + A1ετε′τA′1 + B1HτB′1

is positive definite if and only if, given that A1ετε′τA′1 is positive semidefinite, the

null space condition holds, because Hτ is positive definite by the inductionassumption.



Consider MGARCH(1,1)-BEKK, N = 2 with

A1 = diag(a11, a22) B1 = diag(b11, b22)

the model reduces to

Ht = CC′ +

[a11 00 a22

] [ε21t−1 ε1t−1ε2t−1ε2t−1ε1t−1 ε22t−1

] [a11 00 a22

]′+

[b11 00 b22

] [h11t−1 h12t−1h21t−1 h22t−1

] [b11 00 b22

]′

h11,t = c211 + a211ε

21t−1 + b2

11h11t−1

h12,t = c21c11 + a11a22ε1t−1ε2t−1 + b11b22h12t−1

h22,t = c21c11 + c222 + a222ε

21t−1 + b2

22h11t−1

This model is equivalent to the Hadamard BEKK:

Ht = CC′ + aa′ � εt−1ε′t−1 + bb′ �Ht−1

positive definiteness is not guaranteed. Positive semidefiniteness is obtained byimposing p.s.d. of all terms.



MGARCH(1,1) - Scalar BEKK

A1 = αIN , B1 = βIN

Ht = CC′ + α2(εt−1ε′t−1) + β2Ht−1



1 BEKKHt = Σ + A1

(εt−1ε

′t−1 −Σ

)A′1 + B1 (Ht−1 −Σ) B1

or

Ht = (Σ− A1ΣA1 − B1ΣB′1) + A1

(εt−1ε

′t−1)

A′1 + B1Ht−1B′1

To have p.d-ness of Ht , (Σ− A1ΣA1 − B1ΣB′1) must be p.d..

2 Hadamard BEKK

Ht = Σ + A1 �(εt−1ε

′t−1 −Σ

)+ B1 � (Ht−1 −Σ)(

εt−1ε′t−1 −Σ

)must be p.s.d., while (Ht−1 −Σ) must be p.s.d.

3 Scalar BEKK

Ht = Σ + α2(εt−1ε

′t−1 −Σ

)+ β2 (Ht−1 −Σ)

for α + β < 1.


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 befound simply vectorizing the BEKK equation:

vec(Ht) = vec(CC′) +

q∑i=1

vec(Aiεt−iε′t−iA

′i ) +

p∑i=1

vec(BiHt−iB′i )

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) = (C′ ⊗ A) vec (B)



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 isdenoted Dn:

vec (A) = Dnvech (A)

where Dn is the duplication matrix.



Then

vec(Ht) = vec(CC′) +

q∑i=1

(Ai ⊗ Ai ) vec(εt−iε′t−i )

+

p∑i=1

(Bi ⊗ Bi ) vec(Ht−i )

DNvech (Ht) = DNvech(CC′) +

q∑i=1

(Ai ⊗ Ai ) DNvech(εt−iε′t−i )

+

p∑i=1

(Bi ⊗ Bi ) DNvech(Ht−i )



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

D+N = (D′NDN)

−1D′N

that is a (N (N + 1) /2)×(N2)

matrix, where

D+NDN = IN



This implies that premultiplying by D+N

vech (Ht) = vech(CC′) + D+N

(q∑

i=1

(Ai ⊗ Ai )

)DNvech(εt−iε

′t−i )

+D+N

(p∑

i=1

(Bi ⊗ Bi )

)DNvech(Ht−i )

The vech model implied by any given BEKK model is unique, while theconverse is not true.

The transformation from a vech model to a BEKK model (when it exists) isnot unique, because for a given A∗1 the choice of A1 is not unique.



This can be seen recognizing that (Ai ⊗ Ai ) = (−Ai ⊗−Ai ) so whileA∗i = D+

N (Ai ⊗ Ai ) DN is unique, the choice of Ai is not unique. It can alsobe shown that all positive definite diagonal vech models can be written in theBEKK framework.

Given Ai diagonal matrix, then D+N (Ai ⊗ Ai ) DN is also diagonal, with diagonal

elements given by aiiajj (1 ≤ j ≤ i ≤ N) (See Magnus).



Given the vech model

vech (Ht) = W + A∗ (L) vech (εtε′t) + B∗ (L) vech (Ht)

the necessary and sufficient condition for covariance stationary of {εt} is that allthe eigenvalues of A∗ (1) + B∗ (1) are less than one in modulus. But defining

A∗ (1) = D+N

(q∑

i=1

(Ai ⊗ Ai )

)DN

B∗ (1) = D+N

(q∑

i=1

(Bi ⊗ Bi )

)DN



This implies also that in the BEKK model, {εt} is covariance stationary if andonly if all the eigenvalues of

D+N

(q∑

i=1

(Ai ⊗ Ai )

)DN + D+

N

(p∑

i=1

(Bi ⊗ Bi )

)DN

are less than one in modulus.

Let λ1, . . . , λN be the eigenvalues of Ai , the eigenvalues of

D+N

(q∑

i=1

(Ai ⊗ Ai )

)DN

are λiλj (1 ≤ j ≤ i ≤ N) (Magnus).


Multivariate GARCH Unconditional Covariance Matrix

BEKK model

vec (εtε′t) = vec (CC′) + (A1 ⊗ A1) vec

(εt−1ε

′t−1)

+ (B1 ⊗ B1)[vec(εt−1ε

′t−1)− vec (Vt−1)

]+ vec (Vt)

E [vec (εtε′t)] = vec (CC′) + [(A1 ⊗ A1) + (B1 ⊗ B1)] E

[vec(εt−1ε

′t−1)]

vec (Σ) = [IN2 − (A1 ⊗ A1)− (B1 ⊗ B1)]−1 vec (CC′)

or in vech representation as

DNvech (E (εtε′t)) = DNvech (CC′) + (A1 ⊗ A1) DNvech

(E(εt−1ε

′t−1))

+ (B1 ⊗ B1) DNvech(E(εt−1ε

′t−1))

vech (Σ) =[IN∗ −D+

N (A1 ⊗ A1) DN −D+N (B1 ⊗ B1) DN

]−1vech (CC′)

N∗ = N (N + 1) /2.


Multivariate GARCH Covariance stationarity

The diagonal vech model is stationary if and only if the sum a∗ii + b∗ii < 1 forall i .

In the diagonal BEKK model the covariance stationary condition is thata2ii + b2

ii < 1.

Only in the case of diagonal models the stationarity properties are determinedsolely by the diagonal elements of the Ai and Bi matrices.


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) + εt (3)

yt : (N × 1)

Π (L) = Π1 + Π2L + · · ·+ ΠkLk−1 (N × N)

Ψ : (N × L)

Λ : (N × N (N + 1) /2)

xt : (L× 1)



xt−1 contains predetermined variables. εt is the vector of innovation with respectto the information set formed exclusively of past realizations of yt .

Ht = Et−1 (εtε′t)

Ht = CC′ +

q∑i=1

Ai (εt−i + γ) (εt−i + γ)′A′i +

p∑j=1

BjHt−jB′j



We can consider a multivariate generalization of the size effect and sign effect:

Ht = CC′ + A1εt−1ε′t−1A′1 + B1Ht−1B′1 + Dvt−1v′t−1D′ + Gεt−1ε

′t−1G′

where vt = |zt | − E |zt |, with zit = εit/√

hii,t and

G =

I (ε1t−1 < 0) g11 0 . . . 0

0. . .

......

. . . 00 . . . 0 I (εNt−1 < 0) gNN



When N = 2

vt−1v′t−1 =

∣∣∣ε1t−1/√

h11,t−1

∣∣∣− E∣∣∣ε1t−1/

√h11,t−1

∣∣∣∣∣∣ε2t−1/√

h22,t−1

∣∣∣− E∣∣∣ε2t−1/

√h22,t−1

∣∣∣×

∣∣∣ε1t−1/√

h11,t−1

∣∣∣− E∣∣∣ε1t−1/

√h11,t−1

∣∣∣∣∣∣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

]



Gεt−1ε′t−1G′ =

[g∗211 ε

21t−1 g∗11g∗22ε1t−1ε2t−1

g∗11g∗22ε1t−1ε2t−1 g∗222 ε22t−1

]=

[I (ε1t−1 < 0) g2

11ε21t−1 δ12g11g22ε1t−1ε2t−1

δ12g11g22ε1t−1ε2t−1 I (ε2t−1 < 0) g222ε

22t−1

]δ12 = I (ε1t−1 < 0) I (ε2t−1 < 0)


Estimation procedure

Given the model (3)-(44), the log-likelihood function for {εT , . . . , ε1} obtainedunder the assumption of conditional multivariate normality is:

log LT (εT , . . . , ε1;θ) = −1

2

[TN log (2π) +

T∑t=1

(log |Ht |+ ε′tH

−1t εt

)]

The assumption of conditional normality can be quite restrictive.

The symmetry imposed under normality is difficult to justify, and the tails ofeven conditional distributions often seem fatter than that of normaldistribution.



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 containscontemporaneous ”exogenous” variables.

wt = (xt , yt−1, xt−1, . . . , y1, x1) .

The conditional mean and variance functions are jointly parameterized by a finitedimensional 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.



The validity of most of the inference procedures is proven under the nullhypothesis that the first two conditional moments are correctly specified, for someθ0 ∈ Θ,

E (yt |wt ) = µt (wt ,θ0)

Var (yt |wt ) = Ht (wt ,θ0) t = 1, 2, . . .

The procedure most often used to estimate θ0 is the maximization of a likelihoodfunction that is constructed under the assumption that

yt |wt ∼ N (µt ,Ht) .

The approach taken here is the same, but the subsequent analysis does notassume that yt has a conditional normal distribution.



For observation t the quasi-conditional log-likelihood is

lt (θ; yt ,wt) = −N

2ln (2π)− 1

2ln |Ht (wt ,θ)|

−1

2(yt − µt (wt ,θ))′H−1t (wt ,θ) (yt − µt (wt ,θ))

Lettingεt (yt ,wt ,θ0) ≡ yt − µt (wt ,θ) : (N × 1)

denote the residual function

lt (θ) = −N

2log (2π)− 1

2log |Ht (θ)| − 1

2ε′t (θ) H−1t (θ) εt (θ)

log LT (θ) =T∑t=1

lt (θ)



If µt (wt ,θ) and Ht (wt ,θ) are differentiable on Θ for all relevant wt , and ifHt (wt ,θ) is nonsingular with probability one for all θ ∈ Θ, then thedifferentiation of the loglik yields the (1× P) score function st (θ):

st (θ)′ = ∇θ lt (θ)′ −∇θµt (θ)′H−1t (θ) εt (θ) +

1

2∇θHt (θ)′

[H−1t (θ)⊗H−1t (θ)

]vec

[εt (θ) εt (θ)′ −Ht (θ)

]where

∇θµt (θ) : (N × P)

∇θHt (θ) :(N2 × P

)



If the first conditional two moments are correctly specified, the true error vector isdefined as

ε0t ≡ εt (θ0) = yt − µt (wt ,θ0)

and E(ε0t |wt

)= 0,

E(ε0t ε

0′t |wt

)= Ht (wt ,θ0)

It follows that under correct specification of the first two conditional moments ofyt given wt :

E [st (θ0) |wt ] = 0

The score evaluated at the true parameter is a vector of martingale differencewith respect to the σ − fields {σ (yt ,wt) : t = 1, 2, . . .}. This result can be usedto establish weak consistency of the quasi-maximum likelihood estimator(QMLE).



For robust inference we also need an expression for the hessian ht (θ) of lt (θ).Define the positive semidefinite matrix

at (θ0) = −E [∇θst (θ0) |wt ] = E [−ht (θ0) |wt ] : (P × P)

at (θ0) = ∇θµt (θ0)′H−1t (θ0)∇θµt (θ0)

+1

2∇θHt (θ)′

[H−1t (θ)⊗H−1t (θ)

]∇θHt (θ)

When the normality assumption holds the matrix at (θ0) is the conditionalinformation matrix. However, if yt does not have a conditional normal distributionthen

Var [st (θ0) |wt ] 6= at (θ0)

and the information matrix equality is violated.



The QMLE has the following properties:[A0−1

T B0TA0−1

T

]−1/2√T(θT − θ0

)d→ N (0, IP)

where

A0T ≡ −

1

T

T∑t=1

E [ht (θ0)] =1

T

T∑t=1

E [at (θ0)]

and

B0T ≡ Var

[T−1/2ST (θ0)

]=

1

T

T∑t=1

E[st (θ0)′ st (θ0)

]in addition

AT − A0T

p→ 0

BT − B0T

p→ 0



The matrix A−1T BT A−1T is a consistent estimator od the robust asymptotic

covariance matrix of√

T(θT − θ0

).

In practice,

θT ≈ N(θ, A−1T BT A−1T /T

)Under normality, the variance estimator can be replaced by A−1T /T (Hessian form)

or B−1T /T (outer product of the gradient form).


Estimation procedure Wald Test

The null hypothesis isH0 : r (θ0) = 0

where r : Θ→ RQ is continuously differentiable on int (Θ) and Q < P. Let

R (θ) = ∇θr (θ) : (Q × P)

be the gradient of r on int (Θ). If θ0 ∈ int (Θ) and rank (R (θ0)) = Q then theWald statistic

ξW = Tr(θT

)′ [R(θT

)A−1T BT A−1T R

(θT

)′]−1r(θT

)d→H0

χ2Q .


Factor-GARCH

The Factor GARCH model, introduced by Engle et al. (1990), can be thought ofas 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 theK × 1 vector ft and a time invariant factor loadings given by the N ×K matrix B:

yt = Bft + εt


Factor-GARCH

Assume that the idiosyncratic shocks εt have conditional covariance matrix Ψwhich is constant in time and positive semidefinite, and that the common factorsare characterized by

Et−1 (ft) = 0

Et−1 (ftf′t) = Λt

Λt = diag (λ1, . . . , λK ) and positive definite. The conditioning set is{yt−1, ft−1, . . . , y1, f1}. Also suppose that E (ftε′t) = 0. The conditionalcovariance matrix of yt equals

Et−1 (yty′t) = Ht = Ψ + BΛtB

′ = Ψ +K∑

k=1

βkβ′kλkt

where βk denotes the kth column in B. Thus, there are K statistics whichdetermine the full covariance matrix.


Factor-GARCH

Forecasts of the variances and covariances or of any portfolio of assets, will bebased only on the forecasts of these K statistics.

Factor-representing portfolios

Portfolio weights are orthogonal to all but one set of factor loadings:

rkt = φ′kyt

φ′kβj =

{1 k = j0 otherwise

the vector of factor-representing portfolios is

rt = Φ′yt

where the columns of matrix Φ are the φk vectors.


Factor-GARCH

The conditional variance of rkt is given by

Vart−1 (rkt) = φ′kEt−1 (yty′t)φk = φ′kHtφk

= φ′k (Ψ + BΛtB′)φk

= ψk + λkt

where ψk = φ′kΨφk . The portfolio has the exact time variation as the factors,which is why they are called factor-representing portfolios. In order to estimatethis model, the dependence of the λkt ’s upon the past information set must alsobe parameterized:

θkt ≡ φ′kHtφk = Vart−1 (rkt) = ψk + λkt

So we get thatK∑

k=1

βkβ′kθkt =

K∑k=1

βkβ′kψk +

K∑k=1

βkβ′kλkt

K∑k=1

βkβ′kλkt =

K∑k=1

βkβ′kθkt −

K∑k=1

βkβ′kψk

Ht = Ψ +K∑

k=1

βkβ′kλkt = Ψ +

K∑k=1

βkβ′kθkt −

K∑k=1

βkβ′kψk

= Ψ∗ +K∑

k=1

βkβ′kθkt

where Ψ∗ =

(Ψ−

K∑k=1

βkβ′kψk

).


Factor-GARCH

The simplest assumption is that there is a set of factor-representing portfolioswith univariate GARCH(1,1) representations. The conditional variance θkt followsa GARCH(1,1) process

θkt = ωk + αk

(φ′kεt−1

)2+ γkEt−2

(r2kt−1

)= ωk + αkφ

′k

(εt−1ε

′t−1)φk + γkEt−2

[(φ′kyt

) (φ′kyt

)]= ωk + αkφ

′k

(εt−1ε

′t−1)φk + γk

[φ′kEt−2 (yty

′t)φk

]= ωk + αkφ

′k

(εt−1ε

′t−1)φk + γk

[φ′kHt−1φk

]


Factor-GARCH

The conditional variance-covariance matrix of yt can be written as

Ht = Ψ∗ +K∑

k=1

βkβ′kθkt

= Ψ∗ +K∑

k=1

βkβ′k

{ωk + αk

[φ′k(εt−1ε

′t−1)φk

]+ γk

[φ′kHt−1φk

]}=

(Ψ∗ +

K∑k=1

βkβ′kωk

)

+K∑

k=1

βkβ′k

{αk

[φ′k(εt−1ε

′t−1)φk

]+ γk

[φ′kHt−1φk

]}

Ht = Γ +K∑

k=1

βkβ′kθkt

where Γ = Ψ∗ +K∑

k=1

βkβ′kωk .


Factor-GARCH

Therefore

Ht = Γ +K∑

k=1

αk

[βkφ

′k

(εt−1ε

′t−1)φkβ

′k

]+

K∑k=1

γk[βkφ

′kHt−1φkβ

′k

]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 likelihoodestimation. It is often convenient to assume that the factor-representing portfoliosare known a priori.


Orthogonal-GARCH model

The orthogonal models are particular factor models (Kariya (1988) andAlexander and Chibumba (1997)).

They are based on the assumption that the observed data can be obtained bya linear transformation of a set of uncorrelated components by means of anorthogonal matrix.

The (N × N) time-varying variance matrix Ht is generated by (m × N) univariateGARCH models.The components are the principal components of the data, or a subset of them.



The diagonal matrix V contains the population variances of yt :

V = diag{v21 , . . . , v

2N}

the standardized returns areut = V−1/2yt

whereut = Lft

E [ut ] = 0 E [utu′t ] = R



The population correlation matrix can be decomposed as:

R = PΛP′

P is the orthogonal eigenvectors matrix, Λ is the diagonal matrix of theeigenvalues:

Λ = diag{λ1, . . . , λN}

ranked in descending order.



P satisfies:P′ = P−1 P′P = IN PP′ = IN

It follows thatR = PΛ1/2Λ1/2P′ = LL′

The factor loading matrix is obtained as

L = PΛ1/2

such thatft = L−1ut

with

E [ftf′t ] = L−1E [utu

′t ]L−1′ = L−1RL−1′

= L−1LL′L′−1 = IN



AssumingEt−1[ftf

′t ] = Qt = diag(σ2

f1,t , . . . , σ2fN,t )

Qt is a diagonal matrix.

σ2fi,t = (1− αi,1 − βi,1) + αi,1f 2

i,t−1 + βi,1σ2fi,t−1

i = 1, 2, . . . ,N

Et−1[utu′t ] = Et−1[Lftf

′tL′] = LQtL

′

Et−1[yty′t ] = Et−1[V1/2utu

′tV

1/2] = V1/2LQtL′V1/2

The number of parameters is N(N + 5)/2.



In practice, V and L are replaced by their sample counterparts, and m is chosenby principal component analysis applied to the standardized residuals, ut .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 isreplaced by a matrix (m × N):

Λ−1/2m P′m

Pm is a matrix (N ×m) containing the m eigenvectors of P corresponding to them largest eigenvalues.

fmt = Λ−1/2m P′mut

where fmt = [f1t , . . . , fmt ]

Et−1[fmt ] = 0

Et−1[fmt fm′

t ] = Qm,t = diag(σ2f1,t , . . . , σ

2fm,t )



Alexander (2001, section 7.4.3) emphasizes that using a small number of principalcomponents 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 onthe inverse of Ht .


The Constant Conditional Correlations Model

These models are based on a decomposition of the Ht .

The conditional var-cov matrix is expressed as

Ht = DtRtDt

where Rt is possibly time-varying.

Conditional correlations and variances are separately modeled.



Bollerslev (1990) Constant Conditional Correlations model:The time-varying conditional covariances are parameterized to be proportional tothe product of the corresponding conditional standard deviations. The modelassumptions are:

Et−1 [εtε′t ] = Ht

{Ht}ii = hit i = 1, . . . ,N

{Ht}ij = hijt = ρijh1/2it h

1/2jt i 6= j i , j = 1, . . . ,N

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



The conditional covariance matrix can be written as:

Ht = D1/2t RD

1/2t

Ht =

h1/21t · · · 0

.... . .

...

0 · · · h1/2Nt

1 ρ12 . . . ρ1N

ρ21 1 . . ....

...... . . . ρN−1N

ρN1 . . . ρNN−1 1

h1/21t · · · 0

.... . .

...

0 · · · h1/2Nt

.When N = 2

Ht =

[h1/21t 0

0 h1/22t

] [1 ρ12ρ21 1

][h1/21t 0

0 h1/22t

]

=

[h1t ρ12h

1/21t h

1/22t

ρ12h1/21t h

1/22t h2t

].



The sequence of conditional covariance matrices {Ht} is guaranteed to bepositive definite a.s. for all t, If the conditional variances along the diagonalin the Dt matrices are all positive, and the conditional correlation matrix R ispositive definite

Furthermore the inverse of Ht is given by

H−1t = D−1/2t R−1D

−1/2t .

When calculating the log-likelihood function only one matrix inversion isrequired for each evaluation.

CCC is generally estimated in two steps:1 conditional variances are estimated employing the marginal likelihoods2 R is estimated using the sample estimator of standardized residuals D−1

t yt

(assuming µt = 0).



The CCC solves the curse of dimensionality problem of MGARCH models

The number of parameters is O(N2) but these are not jointly estimated. Thetwo-step estimation procedure impacts on the computational issues.

Asymptotic properties of QMLE estimators verified in McAleer and Ling(2003).


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 ofEngle (2002). The DCC is an extension of the Bollerslev’s CCC Model.



The conditional correlation between two random variables, Xt and Yt is defined as:

ρYX ,t =Covt−1(XtYt)√

Et−1(Xt − µx,t)2Et−1(Yt − µY ,t)2

Assets returns conditional distribution:

yt |Φt−1 ∼ N(0,Ht)

Ht = D1/2t RtD

1/2t .

Dt = diag(Vart−1(y1t), . . . ,Vart−1(yNt))

where the Vart−1(yit), i = 1, . . . ,N are modeled as univariate GARCH processes.



The standardized returns are:ηt = D

−1/2t yt

Et−1(ηtη′t) = D

−1/2t HtD

−1/2t = Rt = {ρij,t}

we can use the conditional variance of ηt to describe the conditional correlation ofyt .The conditional correlation estimator is

ρij,t =qij,t√

qii,tqjj,t.

Where qij,t are assumed to follow a GARCH(1,1) model

qij,t = ρij + α(ηi,t−1ηj,t−1 − ρij) + β(qij,t−1 − ρij) (4)

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



Engle (2002) assumes that ρij ' qij .

Aielli (2006) and Engle et al. (2008) suggest to modify the standard DCC inorder to correct the asymptotic bias which is due to the fact that 1

T

∑t εtε

′t

does not converge to Q.

It is known though that the impact of this is very small (see Engle andSheppard (2001)).



The conditional covariance matrix is positive definite, Qt , as long as it is aweighted 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(ηt−1η′t−1) + β(Qt−1)

where Q is the unconditional covariance matrix of ηt .



DCC model has correlation targeting, when α + β < 1

E [Qt ] = R

E [Qt ] = E [Qt−1]

E [ηtη′t ] = E [Qt ]

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

E [Qt ] = R(1− α− β) + αE [Qt ] + βE [Qt ]



Clearly more complex positive definite multivariate GARCH models could be usedfor the correlation parametrization as long as the unconditional moments are setto the sample correlation matrix.For example, the MARCH family of Ding and Engle (2001) can be expressed infirst order form as:

Qt = Q� (ιι′ − A− B) + A� ηt−1η′t−1 + B�Qt−1 (5)

where � denotes the Hadamard product ({A� B}ij = aijbij).



The Generalized-DCC model specification:

Dt = diag{ωi}+ diag{κi} � yt−1y′t−1 + diag{λi} �Dt−1

ηt = D−1/2t yt (6)

Qt = S� (ιι′ − A− B) + A� ηt−1η′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 assetsfollows a univariate GARCH process.



A real square matrix A, is positive definite if and only if B = A∗−1AA∗−1 ispositive definite, with A∗ = diag{A}.In order to ensure that Ht is positive definite we must have that

D−1/2t HtD

−1/2t is positive definite.



Ht is positive definite ∀t ∈ T , if the following restrictions on the univariateGARCH 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 D2ii,0 > 0

4 The roots of 1− κiZ − λiZ 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)



The log-likelihood function can be written as:

log LT = −1

2

T∑t=1

(N log(2π) + log |Ht |+ y′tH−1t yt)

= −1

2

T∑t=1

(N log(2π) + log |D1/2t RtD

1/2t |+ y′tD

−1/2t R−1t D

−1/2t yt)

= −1

2

T∑t=1

(N log(2π) + log |Dt |+ log |Rt |+ η′tR−1t ηt)

Adding and subtracting y′tD−1/2t D

−1/2t yt = η′tηt

log LT = −1

2

T∑t=1

(N log(2π) + log |Dt |+ y′tD−1/2t D

−1/2t yt

−η′tηt + log |Rt |+ η′tR−1t ηt)

= −1

2

T∑t=1

(N log(2π) + log |Dt |+ r′tD−1t rt)

−1

2

T∑t=1

(η′tR−1t ηt − η′tηt + log |Rt |)



Volatility component:

LV (θ) ≡ log LV ,T (θ) = −1

2

T∑t=1

(N log(2π) + log |Dt |+ y′tD−1t yt)

Correlation component:

LC (θ,φ) ≡ log LC ,T (θ,φ) = −1

2

T∑t=1

(η′tR−1t ηt − η′tηt + log |Rt |)

θ denotes the parameters in Dt and φ the parameters in Rt .

L(θ,φ) = LV (θ) + LC (θ,φ)

LV (θ) = −1

2

T∑t=1

N∑i=1

(log(2π) + log(hi,t) +

r2i,thi,t

).

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



Two-step procedure:

1

θ = arg max{LV (θ)}2

maxφ{LC (θ,φ)}.

Under regularity conditions, consistency of the first step will ensure consistency ofthe second step. The maximum of the second step will a function of the first stepparameter estimates. If the first step is consistent then the second step will be tooas long as the function is continuous in a neighborhood of the true parameters.



Two step GMM problem (Newey and McFadden, 1994). Consider the momentcondition corresponding to the first step

∇θLV (θ) = 0

The moment corresponding to the second step is

∇φL(θ, φ) = 0

Under regularity conditions the parameter estimates will be consistent, andasymptotically normal, with asymptotic covariance matrix

V (φ) =

[E (∇φφLC )

]−1[E({∇φLC − E (∇φθLC )[E (∇θθLV ]−1∇θLV }

{∇φLC − E (∇φθLC )[E (∇θθLV ]−1∇θLV }′)][

E (∇φφLC )

]−1


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