MGARCH Notes

Multivariate GARCH ModelsNovember 2005

Luc Bauwens

[email protected]

Universite catholique de Louvain

MGARCH – p.1/106

Outline

Multivariate GARCH Models

Introduction (p 3)

Overview of Models (p 14)

Estimation (p 50)

Diagnostic Checking (p 70)

Financial and Economic Applications (p 81)

Conclusion (p 104)

References: see forthcoming survey by Bauwens,Laurent and Rombouts (JAE 21/1, 2006).

MGARCH – p.2/106

Introduction

MGARCH – p.3/106

MGARCH: why?

Understanding and predicting the temporal dependencein the second order moments of asset returns isimportant for many issues in financial econometrics andmanagement.

It is now widely accepted that financial volatilities movetogether more or less closely over time across assetsand markets.

Recognizing this feature through a multivariate modellingframework should lead to more relevant empiricalmodels than working with separate univariate models.

MGARCH – p.4/106

MGARCH: why?

From a financial point of view, it opens the door to betterdecision tools in various areas such as asset pricingmodels, portfolio selection, hedging, and Value-at-Riskforecasts. Several institutions have developed thenecessary skills to use econometric models in a financialperspective.

Although there is a huge literature on univariate modelsdealing with time-varying variance, asymmetry andfat-tails, much less papers are concerned with theirmultivariate extensions, but the field is expanding...

MGARCH – p.5/106

Two series of financial returns

Daily returns, DOW Jones (DJ) and NASDAQ (NQ) indices, 03/26/1990 - 03/23/2000

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200

−5.0

−2.5

0.0

2.5

5.0 DJ

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200

−5

0

5 NQ

MGARCH – p.6/106

Co-movement

2000 2050 2100 2150 2200 2250 2300 2350

−5.0

−2.5

0.0

2.5

5.0DJ

2000 2050 2100 2150 2200 2250 2300 2350

−5

0

5 NQ

MGARCH – p.7/106

ACF of returns and squared returns

0 10 20 30

0.0

0.1

0.2

Returns

Dow Jones

0 10 20 30

0.0

0.1

0.2Nasdaq

0 10 20 30

0.0

0.1

0.2

Squared returns

Dow Jones

0 10 20 30

0.0

0.1

0.2 Nasdaq

MGARCH – p.8/106

Densities

−8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5

0.2

0.4

0.6Density

DJ N(s=0.871)

−9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6

0.1

0.2

0.3

0.4

0.5

DensityNQ N(s=1.06)

MGARCH – p.9/106

Portfolio

Consider a portfolio made up of

�

assets. The euro amountinvested in asset

�

is

��

, where

�

is the total euroamount and � � is the share of asset

�

in the portfolio. Let �

be the vector of shares,

�

the vector of returns, � the vectorof expected returns and

the variance-covariance matrix ofthe returns. Then,

��

(1)

E

��

� � � � � (2)

Var

��

� � � � � � � �� (3)

MGARCH – p.10/106

Value-at-Risk of a portfolio

The VaR at level � of a portfolio worth

�

is the smallest lossvalue that can occur with probability equal to �:

��

� � � � �� (4)

where� � � is determined by

Pr

� ��

� � � �� (5)

For example, if

� � � � � ��

�,

��

� � � � � � �� (6)

with � the �% left-quantile of the N(0,1) distribution.

MGARCH – p.11/106

Conditional VaR with ARCH

Assuming that the mean vector � and the variancematrix

are constant over time is restrictive.

A univariate GARCH model for � �� can be fit for a givenweight vector. If the weight vector changes, the modelhas to be estimated again.

On the contrary, if a MGARCH model is fitted ( � � and

�

instead of � and

), the multivariate distribution of thereturns can be directly used to compute the implieddistribution and VaR at � of any portfolio.

� There is no need to re-estimate the model for adifferent weight vector.

MGARCH – p.12/106

Remarks

It is important to account for the covariances incomputing the VaR! When the correlations are smallerthan 1,

�� is smaller than the sum of the individual

VaR measures, also called the undiversified VaR.

The univariate GARCH approach is directly dependenton the portfolio allocation ( �), and it will require us toredo the volatility modelling every time the portfolio ischanged if we want to study the impact on VaR ofchanging the portfolio allocation.

This approach is appropriate for risk measurement butnot for risk management: to do a sensitivity analysis andassess the benefits of diversification we need modelsthat take account of the dependence between assets.

MGARCH – p.13/106

Overview of Models

MGARCH – p.14/106

Definition

A dynamic model with time-varying means, variances andcovariances for the

�

components of � � � �� :

� � � � � �� (7)

� � � � � � ��

a

� � �matrix (8)

� � �� E

� � � � Var � � � � � (9)

� � � E

� � � �� E �� (10)

� � � � � � ��

� � � ��

� � � Var

� � � �� Var �� (11)

where

� �� is the information available at time ��

, at least

containing

� � ��

.

MGARCH – p.15/106

Remarks

� � � �� is any

� � �

matrix such that

� � is the conditional

variance matrix of � � (e.g.

� � � �� may be obtained by a

Cholesky factorization of

� �).

� � and

� � depend on unknown parameters

�

but areotherwise known (parametric model), hence sometimeswe write explicitly � � � �

and� � � �

. For example, � � maybe a VARMA model.

The " � is IID" assumption may be relaxed to " � is amartingale difference sequence (MDS) with respect to� �� ", e.g. to show that Var �� . However, for MLestimation, the IID assumption is relevant.

MGARCH – p.16/106

Three challenges

State conditions on the parameters

�

such that that� � � � �.

Avoid too many parameters (to keep estimation feasible),but maintain enough flexibility in the dynamics of

� �.Find

� Var

� � � � E

� � � and the conditions for weakstationarity.

For ease of exposition, we make� � � � �� a function of one

lag of � � and one lag of itself, i.e. so-called GARCH(1,1)models.

MGARCH – p.17/106

VEC(1,1) (Bollerslev, Engle, and Wooldridge, 1988)

In this model,

� �� is a linear function of the lagged squarederrors, cross products of errors, and lagged values of all theelements of

� �. The

� � � ��

is defined as:

� � � � � �� (12)

where

� � � vech

� � (13)

� � � vech

� ��

�� (14)

and � is a

� � � �

vector of parameters [with

� � � � � � � ��

]and

�

and

�

are� � � � �

matrices of parameters.

MGARCH – p.18/106

Vech and vec operators

vech is the operator that stacks the lower triangle of a

� � �

matrix as an

� � � � ��

vector:

vech

� � � � � � ��

vec is the operator that stacks a matrix as a columnvector:

vec

� � � � � � ��

A useful property is

vec � � � � � � � � � �

vec

�� (15)

MGARCH – p.19/106

Bivariate VEC(1,1)

��

� � � �

� � � �

� � � ��

��

�

��

� ��

� ��

��

��

��

� � � � � � � � �

� � � � � � � � �

� � � � � � � � ��

��

��

� ��

� ��

� ��

��

�

��

� � � � � � � � �

� � � � � � � � �

� � � � � � � � ��

��

��

� � ��

� � ��

� � ��

�� (16)

� the numbers of parameters is of order

� �

(for

� �2, 3, 4 it is equal to 21, 78, 210 respectively).

MGARCH – p.20/106

Bivariate VEC(1,1)

Equivalently,

� � � � � � � � � � � � ��

� � � � � � ��

� � � � � � � � � � � � ��

� � � � � � ��

� � � � � � � � � � � � ��

� � � � � � ��

MGARCH – p.21/106

Diagonal and Scalar VEC

To reduce the number of parameetrs, BEW (1988)suggest the diagonal VEC (DVEC) model in which the

�

and

�

matrices are diagonal.

Each variance

� � � � depends only on its own past squarederror �

�� and its own lag

� � �� .Each covariance

� �� depends only on its own pastcross-products of errors � �� , and its own lag.

Quite restrictive: no "spillover effect".

Big reduction: 9 parameters instead of 21 when

� �2;18 instead of 78 when

�

=3...

Scalar VEC:� � � �

and

� � � �

where � and � arescalars and

�is a matrix of ones.

MGARCH – p.22/106

Positivity conditions for VEC (Gouriéroux, 1997)

One can write also (see p 21)

� � � � � � � � ��

� � ��

� ��

� ��

�

E ��

��

� � ��

� ��

� ��

since

� � �� E �� (

� � �� ),

and

� � �� E �� .

MGARCH – p.23/106

Positivity conditions for VEC

� � � � � � � � ��

� � ��

� ��

� ��

�

E ��

��

� � ��

� ��

� ��

� � � � � � � � ��

� � ��

� ��

� ��

�

E ��

��

� � ��

� ��

� ��

MGARCH – p.24/106


Putting the different parts together:

� � �

� � � �

� � � �

� � ��

� ��

��

� � � � � ��

� � ��

� � � � � ��

� � ��

��

��

� ��

� ��

� ��

� ��

�

E ��

��

We denote by

� �

the above

� � � matrix, and by

� �

the matrixbuilt in the same way from

�

.MGARCH – p.25/106


A general matrix (rather than vech ) expression of� � in the

VEC(1,1) case is:

� � � � � � � ��

�

E ��

��

Hence, sufficient conditions for positivity of

� � are that

� �

,

� � �

,

� � �

, with at least one strict inequality.

These restrictions are not easy to impose in estimation.Usually they are not imposed, but can be checked afterunrestricted estimation.

MGARCH – p.26/106

BEKK(1,1,K) (Engle and Kroner, 1995)

The BEKK

��

model is defined as:

� � � � � � � � ��

��

�

��

� � � � � �� (17)

where

� �� and

� � �are

� � �matrices of parameters but

� �

is upper triangular. One can write as well

� � � � � � � �

.

Positivity of

� � is automatically guaranteed if

��

.

MGARCH – p.27/106

Bivariate BEKK(1,1,1)

� � � � � � � �

� � � � � � � �

�

� ��

� � � � � � � �

� ��

� � � �

� � ��

� ��

� � ��

� ��

� ��

� ��

� � ��

� ��

��

� � ��

� ��

� �� (18)

� � �

parameters, against 21 in the VEC model.

MGARCH – p.28/106

Bivariate BEKK(1,1,1)

Same linear structure as in VEC model...

� � � � � � � ��

� � � ��

� � � � � � � ��

� � ��

... but constraints on parameters (compare with p 21).

MGARCH – p.29/106

Remarks

Interpretation of the basic parameters not obvious, seeprevious equations.

By increasing

�

, one makes the specification moreflexible (e.g. for

� � � �

, there are 19 parameters,against 21 in the bivariate VEC).

Diagonal BEKK model: take� � �and

� � �as diagonalmatrices. It is a restricted DVEC model (check thecovariance equation to see the restrictions).

One can define a scalar BEKK model:

� � � � � � �

,

� � � � � � �

.

MGARCH – p.30/106

Stationarity conditions: VEC

The VEC(1,1) model

� � � � � �� can be written asa VARMA(1,1) model for � � � vech

� ��

�

:

� � � � � � � � � � ��

where � � � � � � � is a MDS.Consequently, the VEC(1,1) model is weakly stationary if theeigenvalues of

� � �

are less than 1 in modulus. In this case,

vech

� E

vech� � � � � ��

where

� � � � � � � �� .

MGARCH – p.31/106

Stationarity conditions: BEKK

The BEKK(1,1,1) model

� � � � � � � ��

can be written as a VEC model (subject to restrictions) usingformula (15):

vec

� � � vec

� � � � � � � � �

vec

� ��

vec

� ��

Hence, the BEKK model is weakly stationary if theeigenvalues of

� � � � � � � � � � � � �are smaller than 1 in

modulus, and then

vec

� E

vec

� � � � � � � � � � � � � � � � � � � � � � � � �

vec

��

MGARCH – p.32/106

Factor-GARCH(1,1,K) (Engle, Ng, and Rotschild, 1990b)

The Factor-GARCH(1,1,K) model can be viewed as aparticular BEKK(1,1,K) model:

� � � � ��

��

��

� � �� (19)

i.e.

� � �and

� � �are replaced by rank one matrices that areproportional to one another. The

� � �

vectors

� �and � �aresubject to the restrictions:

� � ��

for

� � � �

�for

� � �

,

��

� � � � �� (20)

MGARCH – p.33/106

Factor-GARCH(1,1,1)

Taking

� � �

, the model can be written as:

� � � � � � � � � � � � ��

� � � � � � � � � � �

� � � ��

� � � � � � �� (21)

where

� � � � � � ��

is the GARCH(1,1) conditional variance of the factor

� � � � �� .

MGARCH – p.34/106

Bivariate Factor-GARCH(1,1,1)

� � � � � ��

� � � � � ��

� � � � � ��

where

� � � � � � � � ��

� � � � � � ��

� � � � � � � � � � � � ��

MGARCH – p.35/106

Remarks

The elements of

� � obey the same dynamics,determined by the common element

� �.

If we write � � � � � � � � �� , and assume that

� �, thecommon shock (a scalar r.v.) and

� �, the idiosyncratic shocks (a

� � �vector),

are uncorrelated,with Var ��

� � � � � �

andVar ��

�� ,we get

Var ��

as in eq. (21).

Weak stationarity occurs if � � � � � � � � � �

.

MGARCH – p.36/106

Other Factor-GARCH models

The orthogonal GARCH model (Kariya, 1988, Alexanderand Chibumba, 1997) and the generalized orthogonalGARCH models of van der Weide (2002) and Vrontos etal. (2003) are also Factor-GARCH models.

Lanne and Saikkonen (2005) propose "A MultivariateGeneralized Orthogonal Factor GARCH Model", aninteresting alternative to the previous papers.

MGARCH – p.37/106

Number of parameters

�� # parameters

for

� �2, 3, 4, 5

VEC(1,1)

��

�

�� 21, 78, 210, 465

BEKK(1,1,1)

� � � ��

11, 24, 42, 65

F-GARCH(1,1,1)

� � � � � � � � "! �$# � � � � � � �� # � % �$# � � � � � # & � � ��

# � � � ')( * �,+- � # + � '

7, 12, 18, 25

MGARCH – p.38/106

What next?

In the previous models, we specify the conditionalcovariances, in addition to the variances.

Next, we review models where we specify the conditionalcorrelations, in addition to the variances.

This allows some flexibility in the specifications of thevariances: they need not be the same for eachcomponent. For example a GARCH(1,1) for onecomponent, an EGARCH for another, ...

However, we face the problem of specification of apositive-definite conditional correlation matrix...

For some choices, positivity conditions for

� � are easilyimposed and estimation is facilitated (2 steps).

MGARCH – p.39/106

Conditional correlations

For these models

� � can generally be written as

� � � � � � � � �� (22)

� � � diag

� � � ��

� � � �� (23)

� � � �� with � � � � � �� (24)

� � is the

� � �

matrix of conditional correlations, and

� � � � is defined as a univariate GARCH model. Hence,

� �� (25)

Positivity of

� � follows from positivity of

� � and of each

� � � �.

MGARCH – p.40/106

(Bollerslev, 1990)

In this case,

� � � � � �� (26)

i.e. "constant conditional correlations" (CCC). Hence,

� �� (27)

and thus the dynamics of the covariance is determined onlyby the dynamics of the two conditional variances.

NB: there are

� � � �� parameters in

�

.

MGARCH – p.41/106

DCC of Tse & Tsui (2002)

DCC for "dynamic conditional correlations".� � ��

� � � � � � � � � � � � � � �� (28)

� ��

��

��

� � ��

� (29)

� � � � � �� (30)

with

� � � � � �

and

� � � � � �.�

is like in CCC.

Notice that

� � �� by construction.

MGARCH – p.42/106

Remarks

� �� is the sample correlation matrix of �� for

� � � ��

. A necessary condition toensure positivity of

� �� is that

� � �

.

� � is a weighted average of correlation matrices (

�

,

� �� ,

� �� ). Hence,

� � �

if any of the three components is

�

.

If

� � � � � �

, the CCC model is obtained. Hence onecan test for CCC against

� � � � � �

.

MGARCH – p.43/106

DCC of Engle (2002)

� � ��

diag

� � � � � � � � � diag

� � � � � � �� (31)

� � is a

� � �

matrix, symmetric and � , given by

� � � � � � � � � � � � � �� (32)

where � � � � � � � � � � �, � � � � � ��

�� ,

�

is a

� � �

matrix, symmetric and >0, of parameters,and

� � and

� � positive parameters satisfying

� � � � � �

,

� � � �

and

� � � .

MGARCH – p.44/106

Remarks

� � is the covariance matrix of �, since� � � � is not equal to1 by construction. Then it is transformed into acorrelation matrix by (31).

If

� � � � � �

, and� � � � �

, the CCC model is obtained.Hence one can test for CCC against

� � � � � �

.

In both DCC models, all the correlations obey the samedynamics. This saves a lot of parameters, compared toVEC and BEKK models, but is quite restrictive(especially when

�

is large).

MGARCH – p.45/106

Comparison

The correlation coefficient in the bivariate case:for the

� � � � � �

,

� � � � � � ' � � � � � ��

� ��

* �� - � � � � � � � � �� * �� - � � �� * ��- � � �

��

and for the

� � � � ��

, � � � � �

� ' � � � � � ��

� � � � � � � � � � ��

� � ' � � � � � ��

� � � � � � � � � � � ' � � � � � ��

� � � � ��

� � � ��

MGARCH – p.46/106

Number of parameters

� � � �� # parameters

�� diag

� � � � ��

for

� �2, 3, 4, 5

� � � ��

7, 12, 18, 25

� � �� ' � � � � � ��

�

��

�� 9, 14, 20, 27

� � �! � ')( ' � ��

diag" � �� " � �

diag

" � ��

� ��

" � � � ' � � � � � �� " � � � ��

�

" � � � 9, 14, 20, 27

MGARCH – p.47/106

Extensions of DCC

Recent and ongoing research aims at specifying moreflexible dynamic correlations, avoiding the commondynamics restriction of all correlations.

References:Billio, Caporin, and Gobbo (2003): a block-structure ofDCC.Hafner and Franses (2003), "A Generalized DynamicConditional Correlation Model for Many Asset Retruns".Palandri (2005), "Sequential Conditional Correlations:Inference and Evaluation".

Copula-MGARCH models combine GARCH forvariances and copula for conditional dependence.Patton (2000), Jondeau and Rockinger (2001).

MGARCH – p.48/106

Other topics

Leverage effects in MGARCH models:

-see section 2.4 of survey paper, and a well-doneempirical study:-Peter de Goeij and Wessel Marquering (2004),Modeling the Conditional Covariance Between Stockand Bond Returns: A Multivariate GARCH Approach,Journal of Financial Econometrics 2, 531-564.

Transformations of MGARCH models:

-invariance of model type with respect to lineartransformations;-marginalization;-temporal aggregation.

MGARCH – p.49/106

Estimation

MGARCH – p.50/106

ML Estimation

ML is convenient but it requires an assumption about thedensity of �, denoted � � � � �

, where � is an additionalparameter vector.

� Maximize with respect to

��

the function

� � ��

� �� (33)

with

� � � ��

� ��

� � � � � � ��

� (34)

where the dependence with respect to

�

occurs through

� � and

� �.MGARCH – p.51/106

Gaussian likelihood

In many cases, � � � � � � � is assumed (hence � isempty). Then, neglecting a constant,

� � � � � �

�

� ��

�

� � � � ��

� (35)

This Gaussian log-lik. provides the QML estimator that isconsistent for

�

even if the true density is not

� � � � � (if

� � and

� � are correctly specified).

However, this QML estimator is less efficient than the MLestimator that would be obtained using the log-lik. basedon the true density.

MGARCH – p.52/106

Remarks

For financial returns, normality is not realistic, like forunivariate GARCH models.

For financial applications (such as computing the VaR), itis important to use the most correct assumption aboutthe density. Hence, normality is not useful in someapplications...

Alternative distributions: multivariate Student (to accountfor excess conditional kurtosis), multivariateskewed-Student or mixture of two multivariate Gaussiandensities (for conditional skewness), generalizedhyperbolic distribution.

Danger of this approach: if the assumption is not correct,inconsistency of the estimator results. To what extent?

MGARCH – p.53/106

Student density

The multivariate Student density, denoted

�� ( �

corresponds to � ), is

� � � � �

� ��

�

� ��

� ��

� � �

� �

� � ��

�

� (36)

where

� � � � � � � �� is the Gamma function.

Here we impose � �

, and Var � � � � � while E

� � �

.Although uncorrelated, the elements of � are notindependent.

When � � �,

�� .When � �

, the tails of the density become thicker andthicker.

MGARCH – p.54/106

Skewed-Student density (Bauwens and Laurent, 2005)

The multivariate skewed-Student density, denoted� � �� (

�

and � correspond to � ), is a skewversion of the

�� .

� � �� is a vector of skewness parameters, with

� � �

for all

�

.

� � governs the skewness of � � since

� ��

��

�� .If

� � �

, the marginal of � � is left-skewed, while if

� � � �

, it is right-skewed.

If

� � � � � �

, the skewed-Student reduces to the Studentdensity. Hence, one can test the null hypothesis ofsymmetry.

MGARCH – p.55/106

Univariate skewed-Student densities

−4 −2 0 2 4

0.2

0.4

ν=5 and ξ=1.3Normal Student Skewed Student

−4 −2 0 2 4

0.2

0.4ν=15 and ξ=1.3

Normal Student Skewed Student

−4 −2 0 2 4

0.2

0.4

0.6 ν=5 and ξ=1.5Normal Student Skewed Student

−4 −2 0 2 4

0.2

0.4ν=15 and ξ=1.5

Normal Student Skewed Student

−4 −2 0 2 4

0.25

0.50

ν=5 and ξ=2Normal Student Skewed Student

−4 −2 0 2 4

0.2

0.4

ν=15 and ξ=2Normal Student Skewed Student

MGARCH – p.56/106

SKST

��

��

��

�

z 1

z2

f(z)

0.026

0.052

0.078

0.104

0.13

0.156

0.182

0.208

0.234

0.26

−4

−2

0

2

4

−2.50.0

2.55.0

0.1

0.2

MGARCH – p.57/106

Con

tour

sof

SKST

� �

��

��

��

�

−4

−3

−2

−1

01

23

4

−2−1012

z 1

z2

0.025

0.05

0.07

5

0.1

0.12

5

0.15

0.175 0.2

0.22

5

0.02

5

0.05

0.025

0.05

0.07

50.

10.

125

0.15

0.17

50.

20.

225

0.02

5

0.05

Pan

el A

MG

AR

CH

–p.

58/1

06

Another way to get multivariate distributions

One can also define the density of � as the product ofindependent univariate densities for each element of �:

univariate Student

�� (with their own degrees offreedom, not the same for each marginal);

univariate skewed-Student

� � �� (Bauwensand Laurent, 2005);

GED( � � ).Not much implemented up to now...This allows more flexibility, but may render estimation moredifficult since there more parameters in � .

MGARCH – p.59/106

SKST

-IC

� �

��

��

��

�

�

�

�

−4

−3

−2

−1

01

23

4

−2−1012

z 1

z2

0.023

0.04

6

0.06

9

0.09

2

0.11

5

0.13

8 0.16

1

0.18

40.20

7

0.02

3

0.04

6

0.023

0.046

0.06

90.

092

0.11

5

0.13

8

0.16

10.

184

0.20

7

0.02

3

0.04

6

Pan

el B

MG

AR

CH

–p.

60/1

06

Asymptotic properties of ML & QML

Consistency of QMLE is shown (Bollerslev andWooldridge, 1992; Jeantheau, 1998).

Asymptotic normality "assumed" in practice (or shownusing high level assumptions).

Hence, in practice one does inference as usual(asymptotic �

�

Wald and likelihood ratio tests).

Recent work on these issues in univariate GARCHmodels has shown that usual asymptotics does notnecessarily hold if � does not have moments of loworder (4 at least). See Hall and Yao (2003).

MGARCH – p.61/106

Two step estimation of DCC models

This approach uses the Gaussian likelihood (ML underthe normality assumption, or QML otherwise).

Substituting

� � � � � � � � � in the Gaussian log-likelihoodfunction (35) gives:

� � � � � �

�

� ��

� � �

(37)

where � � � � ��

� � � � � , so that

� � ��

� � � ��

��

� � ��

� � � � � �

MGARCH – p.62/106


Hence, we can write:

� � � � � �

�

� ��

� � �

� �

�

� ��

�

�

� ��

� � � � � � � ��

� �� : parameters of the conditional variances (

� �),

� �� : parameters of the conditional correlations (

� �).

MGARCH – p.63/106


� � � � � � � � ��

� � � � � � � ��

��

First step: estimate

� �� by

�� argmax

� � � � � ��

Easy: estimate separate univariate GARCH models ifthere is no spillover effects in conditional variances.

Second step: estimate

� �� by

�� argmax

� � � ��

��

Easy, since many parameters are fixed in this step.

MGARCH – p.64/106

Remarks

These estimators are consistent but not efficientasymptotically, since some information in sacrificed(about

� �� in the first step).

The variance matrix of the estimator�

� �� has to be

adjusted to take account of the first step (see Engle,2002è and Newey and McFadden, 1994) but this is notimportant for VaR forecasts.

MGARCH – p.65/106

Example of Bauwens and Laurent (2005)

2 datasets of daily returns:-3 stocks: Alcoa (AA), Caterpillar Inc. (CAT), and WaltDisney Company (DIS), from January 1990 to May 2002.-3 exchange rates with respect to US dollar: euro (DMbefore euro period), yen, and British pound, fromJanuary 1989 to February 2001.

Conditional means: AR(0) or AR(1) with constant.

Conditional variances: GARCH(1,1) for exchange rates,and GJR(1,1) for stocks:

� � � � � �� .

DCC model of Engle (2002) for conditional correlationmatrix, with skewed-Student distribution.

MGARCH – p.66/106

Partial estimation resultsAA-CAT-DIS EUR-YEN-GBP

1 step 2 steps 1 step 2 steps� � 0.0088 0.0095 0.0303 0.0294(0.0021) (0.0033)�

� 0.9846 0.9837 0.9684 0.9689(0.0047) (0.0037)� � � � � 0.1050 0.0977 -0.0875 -0.0724(0.0257) (0.0242)� � � �

� 0.0786 0.0698 0.0987 0.0983(0.0263) (0.0253)� � � �� 0.0667 0.0591 -0.0677 -0.0353(0.0276) (0.0238)

� 7.2858 7.4020 6.1928 6.4896(0.5335) (0.3960)

Sample size 3113 3066� ��

58.41 895.62� ��

34.58 33.45

Note: For each parameter, the table reports the one step ML estimate and its standard error (inparantheses). The estimate of the two-step approach is also reported.� ��

and

� �� are respectively � � ��

and � � � �

likelihood ratio statistics for thehypotheses of constant correlations and symmetry with respect to the Student density.

MGARCH – p.67/106

Variance and correlation targeting

The constant part of

� �, if unrestricted, contains

� � � � ��

parameters, a number that increases fastlywith

�

.

This constant part, or a function of it, can sometimes beestimated consistently without doing ML or QML. Thenthis consistent estimate can be substituted for thecorresponding parameter matrix in the(quasi-)log-likelihood function, rendering maximizationeasier by the reduction in the number of parameters.

These estimators are consistent but not efficientasymptotically, since some information is sacrificed.

Correlation targeting: a similar argument can be appliedto estimate

�in (28) and

� �

in (32).MGARCH – p.68/106

Example of variance targeting

In the VEC model, we know thatvech

� E

vech

� � � � � ��

Hence we can write (12) as

� � � � ��

vech

� � ��

A consistent estimator of

is

� ��

��

� ��

where

�� , with� � � a consistent estimator of � �

(usually easily available, e.g. by OLS).

Hence we estimate

�

and

�

from

� � � � ��

vech

� � � ��

MGARCH – p.69/106

Diagnostic Checking

MGARCH – p.70/106

Principles

After estimation, it is a standard practice to assess thespecification of the model.

This is done using diagnostic tests (also calledspecification tests) and related procedures, that aredesigned to indicate possible failures of someassumptions.

Important departures from the basic assumptions shouldbe remedied, if possible.

Assumptions are: functional specification of � �, of

� �,and the assumptions about � (independence and theselected distribution).

MGARCH – p.71/106

Principles

Some tests use the estimated �, i.e. the residuals. InMGARCH models,

� � ��

�

�� (38)

where a ‘hat’ denotes an estimated value (by QML). SeeDing and Engle (2001).

Other tests use the residuals standardized to have unitvariance, but still correlated:

� � � � ��

�� (39)

See Tse (2002).

MGARCH – p.72/106

Principles

One can distinguish several kinds of specification tests:

univariate tests applied separately to each� � � or

� � �,

univariate tests applied separately to products

� � � � � �, totest the covariance specification,

multivariate tests applied to the vector

� � as a whole.

All this is still in development...

MGARCH – p.73/106

Univariate tests

Several tests are those used for univariate GARCH models.They are applied to each series

� � � individually:

�

-statistics on

� � � or

� � �,

�

-statistics on

� �� or

� �� ,

Jarque-Bera test of marginal normality,

goodness-of-fit test (for the marginal density),

...

They are very useful but they don’t tell us anything about themultivariate aspect of the specification.

MGARCH – p.74/106

Tests of Tse (2002)

Conditional variance test: for each

�

,

regress

� ��

on a few lags of

� �� (and of� �� for spillover

effects), but no constant term;

estimate this by OLS and test for nullity of the regressioncoefficients;

the test statistic is a quadratic form in the OLS estimatedcoefficient vector, with weighting matrix adapted to takeaccount that the regressors are actually estimatedresiduals (not the inverse of the usual OLS variancematrix);

the test is ��

��

asymptotically, where � is the number ofcoefficients tested to be equal to 0.

MGARCH – p.75/106

Tests of Tse (2002)

Conditional covariance test: for each pair

��

,

regress

� � � � � � �� on a few lags of

� � � � � � (no constantterm), where

�� is the estimated conditional correlationimplied by

� � �;estimate this by OLS and test for nullity of the regressioncoefficients;

the test statistic is like in the previous case;

a MC simulation shows that the finite sample size of thetest is close to the nominal size even with only 200observations, and that the test has reasonably goodpower properties.

MGARCH – p.76/106

Tests of Engle and Ding (2001)

These tests check some implications of a correctspecification of the dynamics of the first two conditionalmoments. Specifically, if the elements of � are mutuallyindependent, then

Cov

��

��

� � � � � � � �� (40)

and if � is i.i.d.

Cov

��

�� for

� � � (41)

For example, if � � �� , and � � �

,

Cov

��

��

� � � ��

for

� � � �

.

MGARCH – p.77/106

Tests of condition (40)

Notice that Cov

��

��

� � E

� ��

.

To test condition (40), let � � be a

� � � �� vector

with typical element

��

.

Condition (40) is then equivalent to the momentcondition E

� � � �

.

� the sample moments

� � � �

��

should be closeto 0 in large samples, where

� � � is defined like � �, butusing estimated residuals

� �, i.e. with typical element

� ��

.

MGARCH – p.78/106

Test of condition (40)

The moment condition E

� � � �

can be tested byapplying the conditional moment test principle (Newey,1985; Tauchen, 1985).

Let

�� denote the score vector at date �.

The

�

test statistic is simply

� � � �,

where

� �

is the uncentered

�

-squared of a regression of1 on

� � � and

�� .

The statistic is distributed asymptotically as ��

with

� � � ��

degrees of freedom under the nullhypothesis.

Other tests can be designed by adding other relevantmoment conditions.

MGARCH – p.79/106

Tests of condition (41)

One can use the same principle as for the previous test:

Use a regression of 1 on

�� and

� ��

� � ��

�

, for

��

, where

� ��

��

� �

.

� � � �

from this regression has a ��

distribution inlarge samples.

MGARCH – p.80/106

Financial and Economic Applications

MGARCH – p.81/106

Topics

MGARCH models have been applied to:

Dynamic asset pricing models

Volatility transmission between assets and markets

Futures hedging

Impact of exchange rate volatility on trade and output

Value-at-Risk.

MGARCH – p.82/106

Futures hedging

Futures contracts are used to hedge the risk incurred byholding a spot (short or long) position.

For example: buy on the spot market one unit at someprice (a long position), and sell in the futures market (ashort position) at the same price to cover the risk ofdepreciation;

� the hedge ratio = 1 (quantity of futures positiondivided by the spot position).

This is the right strategy if spot and futures returns havethe same mean and variance and are perfectlycorrelated.

MGARCH – p.83/106

Futures hedging

However, spot and futures prices are random and notperfectly correlated.

One has therefore to account for this in order to decideabout the hedge ratio, denoted by

�

.

The hedge return is � � �� , where �� is the spotreturn and � � the futures return.

Minimizing the variance of � yields

� � � Cov

��

Var

��

.

This rule can be generalized to take account of theexpected return/risk tradeoff. It remains the same ifexpected utility � E

� � �

� �

Var

� � and E

� � � �

.

MGARCH – p.84/106

Futures hedging

Implicit in the moments Cov

� � � ��

and Var

� � �

that definethe optimal hedge ratio is an information set on whichthey are conditional.

As information accumulates, such as observation ofrealized values of the returns (or prices of spot andfutures), the optimal hedge ratio changes, i.e.

� �

isindexed by �:

� estimate

� �� Cov

��

Var

��

, where

� � isthe information available at time �, and includes theobserved current and lagged values of � � � and � � �.

MGARCH – p.85/106

Futures hedging

One can use a bivariate GARCH model for � � � and � � �,which provides estimates of the conditional momentsrequired to compute

� �� at each �. One can also predict

future values.

Several papers using the GARCH approach to hedginguse a constant correlation specification.

This is restrictive and current technology certainly allowsto use more flexible specifications in the bivariate case.An exception is Bera et al. (1997) who use threespecifications.

MGARCH – p.86/106

Futures hedging

A traditional method consists of estimating a constant

� �

from time series data as the slope of the regression

�� .

When time-varying hedge ratios (TVHR) are computed,one can check the benefits of these compared to aconstant HR. The TVHR are useful if they reduce thevariance of the hedge.

In Sephton (1993), this is found to be the case(in-sample), and in Bera et al. (1997) also, especially forthe diagonal VEC specification (in-sample andout-of-sample). However these comparisons do not takeaccount of the higher costs of using THR.

MGARCH – p.87/106

Asset pricing models

The static (or single period) CAPM model states that inmarket equilibrium, two numbers � � and

� � exist such

that:

E

� � � � � � � �

Cov

� � � � � �� for� � �� (42)

where

� � is the return of asset�

, and

� � �

��

(with � � � � �

, � � being the share of asset

�

in themarket capitalization).

� � is the expected return of a risk-free security.

�

is the ‘market price of risk’: the increase of expectedreturn demanded per additional unit of risk (measured bythe covariance).

MGARCH – p.88/106


In a multi-period context, equation (42) is not compatiblewith non i.i.d. returns. If returns are not i.i.d., momentsvary over time, and unconditional moments must bereplaced by conditional ones.

Thus, the CAPM means that at equilibrium there existstwo processes � � � and

� � such that, at each �,

E �� Cov �� for

� � �� (43)

where

� � � is the return of asset

�

between � �

and �, E ��

is the conditional expectation operator, Cov �� is theconditional covariance operator, and

� � � is the return ofthe market portfolio, i.e.

� � � �

�� .

MGARCH – p.89/106


The system of equations (43) can be written

E ��

� � � � � � �� (44)

where E �� E ��

� � � � � is the vector ofconditional expectations of the returns at �,

� is a vector of ones,� � is the conditional variance matrix of the vector ofreturns

� �, and

� �� is the vector of asset shares at date � �

.

We let � � � � E �� , with

� � � the risk-free rate.

MGARCH – p.90/106


An econometric model compatible with (43) or (44) maybe formulated as

� � � � � � � ��

� � � � � � �� (45)

� � � � � � � � � � � � (46)

E �� Var �� (47)

� � i.i.d. � some distribution (48)

where

� � is specified according to a MGARCHformulation.

Equation (45), the CAPM model, is a GARCH-in-meanmodel (GARCH-M), since the conditional varianceappears in the conditional mean equation.

MGARCH – p.91/106


Notice two features:1. a common intercept � � has been included, and2. the specification of the market price of risk

� �, whichmay be constant ( � � � ) or time dependent (if

� � � �

)through a function of the conditional variance matrix.

Most people use a constant price of risk:

� � � � � .Otherwise

� � may be specified as a function ofVar ��

, like

� � � � � � � � � � � �� .Other risk factors may be included in (45), likemacroeconomic factors (multifactor models).

MGARCH – p.92/106


By aggregating, for the market portfolio, relation (43)implies

E �� Var �� (49)

hence:

� � � E� � � � � ��

Var� � � � � ��

Therefore,

E �� Cov ��

Var ��

E ��

� � � � E �� (50)

MGARCH – p.93/106


The

� � � coefficient of asset

�

measures the systematicrisk of asset

�

in relation with the market, during period �.

These coefficients are of interest to investors, who mayrely on them to choose their investments according tothe asset riskiness relative to the market portfolio, sinceobviously

� � � � �

.

The CAPM with GARCH model is useful to generatetime varying betas, instead of constant betas in theunconditional CAPM. Once

� � has been estimated,

� � �

can be easily computed.

MGARCH – p.94/106


Data requirement: excess returns (stocks, bonds) listedin a single country.

The asset list may be extended to include foreigncurrencies. This is relevant when stocks or bonds arenot denominated in the same currency, since there is aforeign currency risk in addition to the local market risk,unless purchasing power parity holds (internationalCAPM.

DeSantis et al. (1998) find that the exchange rate riskvaries over time, and that in some periods, a negativepremium for foreign exchange risk more than offsets apositive premium for equity market risk.

MGARCH – p.95/106

Volatility transmission

This is the most obvious application of MGARCHmodels: the study of the relations between the volatilitiesand co-volatilities of several markets.

Is the volatility of an asset transmitted to another directly(if the lagged conditional variance of the asset issignificantly present in the conditional variance of theother asset) or indirectly (if the lagged conditionalcovariance between the asset and another enters in theother asset equation)?

Does a shock on a market increase the volatility onanother market, and by how much? Is the impact thesame for negative and positive shocks of the sameamplitude?

MGARCH – p.96/106


Another issue is whether the correlations between thereturns of different markets change over time.

Are the correlations higher during periods of highervolatility (sometimes associated with financial crises)?

Are they increasing in the long run, perhaps because ofthe globalisation of financial markets?

MGARCH – p.97/106


Main problem: the large number of parameters that mustbe estimated.

One cannot use too restrictive models, like diagonalmodels...

No more than 5 assets in practice.

Bollerslev (1990) uses a CCC model for 5 Europeancurrencies and finds that conditional correlations weresignificantly higher after the start of the EuropeanMonetary System (3.79-8.85) than before (7.73-3.79).Only the levels may be compared with a CCC model.

MGARCH – p.98/106


A second-best approach is to estimate several small sizemodels bearing on different combinations of assets.

For example, Kearney and Patton (2000) estimate 3-, 4-and 5-variable models of returns on the most importantcurrencies linked by the former EMS, instead of a12-variable system (for all the currencies of the EMS)that is infeasible to estimate.

The 3-variable system bears on the European currencyunit (ECU), the mark (DM) and the French franc; the4-variable model adds the lira; and the 5-variable modeladds the pound.

MGARCH – p.99/106


The conditional mean vector is constant (implying nodynamics), and the conditional variance uses the BEKKformulation.

This requires 70 parameters in the 5-variable model.

Concerning volatility transmission using daily data, somerobust conclusions emerge from the estimation of thethree models: for example, all models indicate that theDM does not receive volatility directly from the othercurrencies (except the ECU in the 3-variable model), andwith few exceptions, that the DM transmits its volatilitydirectly to the other currencies.

MGARCH – p.100/106


Koutmos and Booth (1995) focus on the volatilityspillovers across the London, New York and Tokyo stockmarkets around the October 1987 crash (September 86to November 90).

Conditional mean specification:-VMA(1) model with a constant term.-This enables to measure shock impacts in any marketon the next expected return in all three markets.Such impacts are significant (at the 0.05 level) from NewYork to Tokyo (positive), as well as from Tokyo and NewYork to London (both negative).

MGARCH – p.101/106


Conditional variance specification:-An EGARCH equation for each variance, where theshock in each market enters the next conditionalvariance of every other market.-For example, a shock in New York, can increase thenext conditional variance of New York (as in anunivariate model), London and Tokyo.These effects are empiricallyvsignificant and work in alldirections.-Moreover the impact of negative and positive shocks ofequal absolute values can be larger for negative shocksthan for positive ones. This is also implied by theestimates.

MGARCH – p.102/106


CCC for the correlations:-Because trading hours are not the same on the threemarkets, conditional correlations do not reflectcontemporaneous correlations (in calendar time), butthey capture (partly) intraday lead/lag relationships,rendering the interpretation of the moving averagecoefficients difficult.

Estimations are repeated for the pre-crash andpost-crash periods, and reveal that interdependencebefore the crash (which covers however a rather shortspell of 13 months) was less important than after it(estimation for a period of about three years).

MGARCH – p.103/106

Conclusion

MGARCH – p.104/106

Limits of MGARCH models

Curse of dimensionality, but Palandri (2005) estimates amodel for 69 series.

Estimation software not yet enough developed.MG@RCH under development by Laurent andRombouts (similar to G@RCH, belongs to OxMetrics).Otherwise, RATS, FinMetrics in S+, Fanpac in GAUSSinclude some models.

MGARCH – p.105/106

Other approaches to multivariate volatility

Stochastic volatility models.

Realized volatility.

MGARCH – p.106/106

Documents

MGARCH Notes