Post on 21-Dec-2015
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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
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')','(
''
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