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MULTIVARIATE GARCH

Rob Engle

UCSD & NYU

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MULTIVARIATE GARCH

• MULTIVARIATE GARCH MODELS• ALTERNATIVE MODELS• CHECKING MODEL ADEQUACY• FORECASTING CORRELATIONS• HEDGING CORRELATIONS• APPLICATIONS AND SOFTWARE

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TIME VARYING CORRELATIONS

All financial practitioners recognize that volatilities are time varying

Evidence is in implied volatilities and other derivative prices, and estimates over different sample periods

Similarly, correlations are time varying– Derivative prices of correlation sensitive

products– Time series estimates

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CORRELATION

22

21

2,12,1

Definition:

Properties: Always between (-1,1) Measure of linear association Conditional Correlation:

2

,212,11

.2,11,2,1

tttt

tttt

rErE

rrE

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COVARIANCE MATRIX

2,

2,2

,1,122,1

1

....

...

...

..

..

'

tn

t

tntt

tttt HrrE

tntt rrrDefine ,,1 ,...,':

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USEFUL RELATIONS

If and only if H is positive definite, all portfolios will have correlations (-1,1)

2211

2121

1

''

')','(

''

wHwwHw

wHwrwrwCorr

wHwrwV

tt

tttt

ttt

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MODELS

Moving Average(n)

Exponential Smoothing ()

ktj

N

kktitji rr

N

,

0,,, 1

1

)()1( 1,1,1,,,, tjtitjitji rr

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REWRITING IN MATRICES

111

1

1'

Smoother lExponentia

'1

Average Moving

tttt

n

kktktt

HrrH

rrn

H

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Positive Definite Matrices

A matrix M is positive definite if

The sum of a positive (semi)definite and positive semidefinite matrix, is positive (semi)definite

Both Moving Average and Exponential Smoothers are positive semidefinite

0 allfor ,0' xMxx

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DIAGONAL MULTIVARIATE GARCH The reason this is “diagonal” will be clear

shortly

In Matrix representation

is a Hadamard Product or element by element multiplication

)( 1,1,,1,,,,,, tjtijitjijijitji rr

BA

HBrrAH tttt

111 '

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Positive Definite Diagonal Models If then H is positive (semi)definite if A is

positive (semi)definite Models with A and B positive definite

are useful restricted diagonal models Scalar, squared diagonals and fuller

matrices are used.

0,' rrrAH

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BEKK: Baba,Engle,Kroner,Kraft Model with guaranteed positive definite

structure

A and B can be diagonal, triangular or full.

If A and B are diagonal, then this is a “diagonal” model as written above

BHBArrAH tttt 111 '''

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VEC

The VEC operator converts a matrix to a vector by stacking its columns

Useful Theorem:

The VEC Model

BvecCAABCvec '

111 ' tttt HvecrrvecvecHvec

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Special Cases

If A and B are diagonal, then we get the diagonal models

If A and B are themselves tensors, then these are BEKK models

Not all VEC models are positive definite Because A and B are n2xn2 there are

many parameters!!

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Constant Conditional CorrelationBollerslev If conditional correlations are constant

then the problem is much simpler

OR but why should conditional correlations

be constant?

tjtijitji ,,,,,

tjtttt hdiagDRDDH ,2/12/1 ,

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COMPONENT BEKK

Permanent and Transitory Components

And can even add in an asymmetric term

FHFRQRQ

BQHBAQAQH

ttttt

ttttttt

1111

11111

''

)(')'('

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FACTOR ARCH

One factor version

In Matrix notation

22,

22, itfifti

ijtfjfiftij 2,,

2,' tftH

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K FACTOR MODEL

or in Matrix notation

K

f

K

f

FACTORtffjfifjitji

1 1',',',,,

' tt FH

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THE APT

In the APT correlations across assets are related to expected returns

t

ott

K

ktkkj

otjt

tj

K

k

otkkj

otj

rrE

rrE

rfrr

1

1,,,1

,1

,,,

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Volatilities in APT

If idiosyncracies have constant variance

If idiosyncracies do not have constant variance, then they need ARCH models too

If idiosyncracies are independent of factors, and each other then univariate is sufficient

' tt FH

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MAXIMUM LIKELIHOOD

T

ttttt

tt

tt

HH

HN

r

1

1'log21

past on the lconditiona ),,0(~

L

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DIAGNOSTIC CHECKING

Test standardized residuals:

Test for own autocorrelation Test for cross asset autocorrelation Test for cross product autocorrelation Test for asymmetries

ttttt rErH 12/1

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FORECASTING AND VARIANCE TARGETING In the VEC model

Forecast recursion is:

And

111 ' tttt HvecrrvecvecHvec

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1111 '

ktt

ktkttktt

HvecE

rrvecEvecHvecE

vecIHvecE t1