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1SAMSI RISK WS 2007
Heavy TailsHeavy Tails and Financial and Financial Time Series ModelsTime Series Models
Richard A. Davis
Columbia University
www.stat.columbia.edu/~rdavis
Thomas MikoschUniversity of Copenhagen
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2SAMSI RISK WS 2007
Outline
Financial time series modeling
General comments
Characteristics of financial time series
Examples (exchange rate, Merck, Amazon)
Multiplicative models for log-returns (GARCH, SV)
Regular variation
Multivariate case
Applications of regular variation
Stochastic recurrence equations (GARCH)
Stochastic volatility
Extremes and extremal index
Limit behavior of sample correlations
Wrap-up
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3SAMSI RISK WS 2007
Financial Time Series Modeling
One possible goal: Develop models that capture essential features of financial data.
Strategy: Formulate families of models that at least exhibit these key characteristics. (e.g., GARCH and SV)
Linkage with goal: Do fitted models actually capture the desired characteristics of the real data?
Answer wrt to GARCH and SV models: Yes and no. Answer may depend on the features.
Stǎricǎ’s paper: “Is GARCH(1,1) Model as Good a Model as the Nobel Accolades Would Imply?”
Stǎricǎ’s paper discusses inadequacy of GARCH(1,1) model as a “data generating process” for the data.
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4SAMSI RISK WS 2007
Financial Time Series Modeling (cont)
Goal of this talk: compare and contrast some of the features of GARCH and SV models.
• Regular-variation of finite dimensional distributions
• Extreme value behavior
• Sample ACF behavior
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5SAMSI RISK WS 2007
Characteristics of financial time series
Define Xt = ln (Pt) - ln (Pt-1) (log returns)
• heavy tailed
P(|X1| > x) ~ RV(-),
• uncorrelated
near 0 for all lags h > 0
• |Xt| and Xt2 have slowly decaying autocorrelations
converge to 0 slowly as h increases.
• process exhibits ‘volatility clustering’.
)(ˆ hX
)(ˆ and )(ˆ 2|| hhXX
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6SAMSI RISK WS 2007
day
log
retu
rns
(exc
hang
e ra
tes)
0 200 400 600 800
-20
24
lag
AC
F
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
lag
AC
F o
f squ
ares
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
lag
AC
F o
f abs
val
ues
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
Example: Pound-Dollar Exchange Rates (Oct 1, 1981 – Jun 28, 1985; Koopman website)
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7SAMSI RISK WS 2007
Example: Pound-Dollar Exchange Rates Hill’s estimate of alpha (Hill Horror plots-Resnick)
m
Hill
0 50 100 150
12
34
5
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8SAMSI RISK WS 2007
Stǎricǎ Plots for Pound-Dollar Exchange Rates
15 realizations from GARCH model fitted to exchange rates + real exchange rate data. Which one is the real data?
time
exc
ha
ng
e r
etu
rns
-40
24
time
exc
ha
ng
e r
etu
rns
-40
24
time
exc
ha
ng
e r
etu
rns
-40
24
time
exc
ha
ng
e r
etu
rns
-40
24
time
exc
ha
ng
e r
etu
rns
-40
24
time
exc
ha
ng
e r
etu
rns
-40
24
time
exc
ha
ng
e r
etu
rns
-40
24
time
exc
ha
ng
e r
etu
rns
-40
24
time
exc
ha
ng
e r
etu
rns
-40
24
time
exc
ha
ng
e r
etu
rns
-40
24
time
exc
ha
ng
e r
etu
rns
-40
24
time
exc
ha
ng
e r
etu
rns
-40
24
time
exc
ha
ng
e r
etu
rns
-40
24
time
exc
ha
ng
e r
etu
rns
-40
24
time
exc
ha
ng
e r
etu
rns
-40
24
timee
xch
an
ge
re
turn
s
-40
24
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9SAMSI RISK WS 2007
Stǎricǎ Plots for Pound-Dollar Exchange Rates
ACF of the squares from the 15 realizations from the GARCH model on previous slide.
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
LagA
CF
0 10 20 30 40
0.0
0.4
0.8
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12SAMSI RISK WS 2007
Example: Merck log(returns) (Jan 2, 2003 – April 28, 2006; 837 observations)
time
0 200 400 600 800
-0.3
-0.2
-0.1
0.0
0.1
Lag
AC
F
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F o
f sq
ua
res
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F o
f Ab
s V
alu
es
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
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13SAMSI RISK WS 2007
Example: Merck log-returns Hill’s estimate of alpha (Hill Horror plots-Resnick)
m
Hill
0 50 100 150
12
34
5
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15SAMSI RISK WS 2007
Example: Amazon-returns (May 16, 1997 – June 16, 2004)
time
log
re
turn
s
0 500 1000 1500
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
Lag
AC
F
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F o
f sq
ua
res
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F o
f a
bs
valu
es
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
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17SAMSI RISK WS 2007
time
exc
ha
ng
e r
etu
rns
-0.4
0.0
0.4
timee
xch
an
ge
re
turn
s
-0.4
0.0
0.4
time
exc
ha
ng
e r
etu
rns
-0.4
0.0
0.4
time
exc
ha
ng
e r
etu
rns
-0.4
0.0
0.4
time
exc
ha
ng
e r
etu
rns
-0.4
0.0
0.4
time
exc
ha
ng
e r
etu
rns
-0.4
0.0
0.4
time
exc
ha
ng
e r
etu
rns
-0.4
0.0
0.4
time
exc
ha
ng
e r
etu
rns
-0.4
0.0
0.4
time
exc
ha
ng
e r
etu
rns
-0.4
0.0
0.4
time
exc
ha
ng
e r
etu
rns
-0.4
0.0
0.4
time
exc
ha
ng
e r
etu
rns
-0.4
0.0
0.4
time
exc
ha
ng
e r
etu
rns
-0.4
0.0
0.4
time
exc
ha
ng
e r
etu
rns
-0.4
0.0
0.4
time
exc
ha
ng
e r
etu
rns
-0.4
0.0
0.4
time
exc
ha
ng
e r
etu
rns
-0.4
0.0
0.4
time
exc
ha
ng
e r
etu
rns
-0.4
0.0
0.4
Stǎricǎ Plots for the Amazon Data
15 realizations from GARCH model fitted to Amazon + exchange rate data. Which one is the real data?
![Page 14: SAMSI RISK WS 2007 1 Heavy Tails and Financial Time Series Models Richard A. Davis Columbia University rdavis Thomas Mikosch University](https://reader035.vdocuments.mx/reader035/viewer/2022062619/5517750b5503460e6e8b50c2/html5/thumbnails/14.jpg)
18SAMSI RISK WS 2007
Stǎricǎ Plots for Amazon
ACF of the squares from the 15 realizations from the GARCH model on previous slide.
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F0 10 20 30 40
0.0
0.4
0.8
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19SAMSI RISK WS 2007
Multiplicative models for log(returns)
Basic model
Xt = ln (Pt) - ln (Pt-1) (log returns)
= t Zt ,
where
• {Zt} is IID with mean 0, variance 1 (if exists). (e.g. N(0,1) or a t-distribution with df.)
• {t} is the volatility process
• t and Zt are independent.
Properties:
• EXt = 0, Cov(Xt, Xt+h) = 0, h>0 (uncorrelated if Var(Xt) < )
• conditional heteroscedastic (condition on t).
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24SAMSI RISK WS 2007
Two models for log(returns)-cont
Xt = t Zt (observation eqn in state-space formulation)
(i) GARCH(1,1) (General AutoRegressive Conditional Heteroscedastic – observation-driven specification):
(ii) Stochastic Volatility (parameter-driven specification):
)1,0(IID~}{ ,211
2110
2tt tt-t-tt Z σβ Xαα, σZX
Main question:
What intrinsic features in the data (if any) can be used to discriminate between these two models?
)N(0, IID~}{ , log log , 22110
2 ttttttt ZX
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27SAMSI RISK WS 2007
Equivalence:
is a measure on RRm which satisfies for x > 0 and A bounded away
from 0,
(xA) = x(A).
Multivariate regular variation of X=(X1, . . . , Xm): There exists a
random vector Sm-1 such that
P(|X|> t x, X/|X| )/P(|X|>t) v xP( )
(v vague convergence on SSm-1, unit sphere in Rm) .
• P( ) is called the spectral measure
• is the index of X.
)()t |(|
)t (
vP
P
X
X)(
)t |(|
)t (
vP
P
X
X
Regular variation — multivariate case
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28SAMSI RISK WS 2007
Examples:
1. If X1> 0 and X2 > 0 are iid RV(, then X= (X1, X2 ) is multivariate
regularly varying with index and spectral distribution
P( =(0,1) ) = P( =(1,0) ) =.5 (mass on axes).
Interpretation: Unlikely that X1 and X2 are very large at the same
time.
0 5 10 15 20
x_1
010
2030
40
x_2
Figure: plot of (Xt1,Xt2) for realization
of 10,000.
Regular variation — multivariate case (cont)
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29SAMSI RISK WS 2007
2. If X1 = X2 > 0, then X= (X1, X2 ) is multivariate regularly varying
with index and spectral distribution
P( = (1/2, 1/2) ) = 1.
3. AR(1): Xt= .9 Xt-1 + Zt , {Zt}~IID symmetric stable (1.8)
sqrt(1.81), W.P. .9898
,W.P. .0102
-10 0 10 20 30
x_t
-10
010
2030
x_{t
+1}
Figure: scatter plot
of (Xt, Xt+1) for
realization of 10,000
Distr of
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31SAMSI RISK WS 2007
Use vague convergence with Ac={y: cTy > 1}, i.e.,
where t-L(t) = P(|X| > t).
Linear combinations:
X ~RV() all linear combinations of X are regularly varying
),(w:)A()t |(|
)t (
)(
)tA ( T
cX
XcXc
c
P
P
tLt
P
i.e., there exist and slowly varying fcn L(.), s.t.
P(cTX> t)/(t-L(t)) w(c), exists for all real-valued c,
where
w(tc) = tw(c).
Ac),(w:)A(
)t |(|
)t (
)(
)tA ( T
cX
XcXc
c
P
P
tLt
P
Applications of multivariate regular variation (cont)
![Page 21: SAMSI RISK WS 2007 1 Heavy Tails and Financial Time Series Models Richard A. Davis Columbia University rdavis Thomas Mikosch University](https://reader035.vdocuments.mx/reader035/viewer/2022062619/5517750b5503460e6e8b50c2/html5/thumbnails/21.jpg)
32SAMSI RISK WS 2007
Converse?
X ~RV() all linear combinations of X are regularly varying?
There exist and slowly varying fcn L(.), s.t.
(LC) P(cTX> t)/(t-L(t)) w(c), exists for all real-valued c.
Theorem (Basrak, Davis, Mikosch, `02). Let X be a random vector.
1. If X satisfies (LC) with non-integer, then X is RV().
2. If X > 0 satisfies (LC) for non-negative c and is non-integer,
then X is RV().
3. If X > 0 satisfies (LC) with an odd integer, then X is RV().
Applications of multivariate regular variation (cont)
![Page 22: SAMSI RISK WS 2007 1 Heavy Tails and Financial Time Series Models Richard A. Davis Columbia University rdavis Thomas Mikosch University](https://reader035.vdocuments.mx/reader035/viewer/2022062619/5517750b5503460e6e8b50c2/html5/thumbnails/22.jpg)
33SAMSI RISK WS 2007
1. If X satisfies (LC) with non-integer, then X is RV().
2. If X > 0 satisfies (LC) for non-negative c and is non-integer, then X is RV().
3. If X > 0 satisfies (LC) with an odd integer, then X is RV().
Applications of multivariate regular variation (cont)
There exist and slowly varying fcn L(.), s.t.
(LC) P(cTX> t)/(t-L(t)) w(c), exists for all real-valued c.
Remarks:
• 1 cannot be extended to integer (Hult and Lindskog `05)
• 2 cannot be extended to integer (Hult and Lindskog `05)
• 3 can be extended to even integers (Lindskog et al. `07, under
review).
![Page 23: SAMSI RISK WS 2007 1 Heavy Tails and Financial Time Series Models Richard A. Davis Columbia University rdavis Thomas Mikosch University](https://reader035.vdocuments.mx/reader035/viewer/2022062619/5517750b5503460e6e8b50c2/html5/thumbnails/23.jpg)
34SAMSI RISK WS 2007
1. Kesten (1973). Under general conditions, (LC) holds with L(t)=1
for stochastic recurrence equations of the form
Yt= At Yt-1+ Bt, (At , Bt) ~ IID,
At dd random matrices, Bt random d-vectors.
It follows that the distributions of Yt, and in fact all of the finite dim’l
distrs of Yt are regularly varying (no longer need to be non-even).
2. GARCH processes. Since squares of a GARCH process can be
embedded in a SRE, the finite dimensional distributions of a
GARCH are regularly varying.
Applications of theorem
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35SAMSI RISK WS 2007
Example of ARCH(1): Xt=(01 X2t-1)1/2Zt, {Zt}~IID.
found by solving E| Z2| = 1.
.312 .577 1.00 1.57
8.00 4.00 2.00 1.00
Distr of
P( ) = E{||(B,Z)|| I(arg((B,Z)) )}/ E||(B,Z)||
where
P(B = 1) = P(B = -1) =.5
Examples
![Page 25: SAMSI RISK WS 2007 1 Heavy Tails and Financial Time Series Models Richard A. Davis Columbia University rdavis Thomas Mikosch University](https://reader035.vdocuments.mx/reader035/viewer/2022062619/5517750b5503460e6e8b50c2/html5/thumbnails/25.jpg)
36SAMSI RISK WS 2007
Figures: plots of (Xt, Xt+1) and estimated distribution of for realization of 10,000.
Example of ARCH(1): 01=1, =2, Xt=(01 X2t-1)1/2Zt, {Zt}~IID
Examples (cont)
theta
-3 -2 -1 0 1 2 3
0.0
80
.10
0.1
20
.14
0.1
60
.18
x_1
x_2
-40 -20 0 20 40
-40
-20
02
04
0
![Page 26: SAMSI RISK WS 2007 1 Heavy Tails and Financial Time Series Models Richard A. Davis Columbia University rdavis Thomas Mikosch University](https://reader035.vdocuments.mx/reader035/viewer/2022062619/5517750b5503460e6e8b50c2/html5/thumbnails/26.jpg)
39SAMSI RISK WS 2007
Is this process time-reversible?
Figures: plots of (Xt, Xt+1) and (Xt+1, Xt) imply non-reversibility.
Example of ARCH(1): 01=1, =2, Xt=(01 X2t-1)1/2Zt, {Zt}~IID
Examples (cont)
x_1
x_2
-40 -20 0 20 40
-40
-20
02
04
0
x_2
x_1
-40 -20 0 20 40
-40
-20
02
04
0
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40SAMSI RISK WS 2007
x_1
x_2
-2000 0 2000
-20
00
02
00
0
Example: SV model Xt = t Zt
Suppose Zt ~ RV() and
Then Zn=(Z1,…,Zn)’ is regulary varying with index and so is
Xn= (X1,…,Xn)’ = diag(1,…, n) Zn
with spectral distribution concentrated on (1,0), (0, 1).
Figure: plot of
(Xt,Xt+1) for realization
of 10,000.
Examples (cont)
)N(0, IID~}{ , log log , 22110
2 ttttttt ZX )N(0, IID~}{ , log log , 22110
2 ttttttt ZX
![Page 28: SAMSI RISK WS 2007 1 Heavy Tails and Financial Time Series Models Richard A. Davis Columbia University rdavis Thomas Mikosch University](https://reader035.vdocuments.mx/reader035/viewer/2022062619/5517750b5503460e6e8b50c2/html5/thumbnails/28.jpg)
41SAMSI RISK WS 2007
Example: SV model Xt = t Zt
Examples (cont)
theta
-3 -2 -1 0 1 2 3
0.1
20
.14
0.1
60
.18
0.2
0
x_1
x_2
-2000 0 2000
-20
00
02
00
0
x_2
x_1
-2000 0 2000
-20
00
02
00
0
• SV processes are time-reversible if log-volatility is Gaussian.
• Asymptotically time-reversible if log-volatility is nonGaussian
![Page 29: SAMSI RISK WS 2007 1 Heavy Tails and Financial Time Series Models Richard A. Davis Columbia University rdavis Thomas Mikosch University](https://reader035.vdocuments.mx/reader035/viewer/2022062619/5517750b5503460e6e8b50c2/html5/thumbnails/29.jpg)
45SAMSI RISK WS 2007
Setup
Xt = t Zt , {Zt} ~ IID (0,1)
Xt is RV
Choose {bn} s.t. nP(Xt > bn) 1
Then }.exp{)( 11 xxXbP n
n
Then, with Mn= max{X1, . . . , Xn},
(i) GARCH:
is extremal index ( 0 < < 1).
(ii) SV model:
extremal index = 1 no clustering.
},exp{)( 1 xxMbP nn
},exp{)( 1 xxMbP nn
Extremes for GARCH and SV processes
![Page 30: SAMSI RISK WS 2007 1 Heavy Tails and Financial Time Series Models Richard A. Davis Columbia University rdavis Thomas Mikosch University](https://reader035.vdocuments.mx/reader035/viewer/2022062619/5517750b5503460e6e8b50c2/html5/thumbnails/30.jpg)
46SAMSI RISK WS 2007
(i) GARCH:
(ii) SV model:
}exp{)( 1 xxMbP nn
}exp{)( 1 xxMbP nn
Remarks about extremal index.
(i) < 1 implies clustering of exceedances
(ii) Numerical example. Suppose c is a threshold such that
Then, if .5,
(iii) is the mean cluster size of exceedances.
(iv) Use to discriminate between GARCH and SV models.
(v) Even for the light-tailed SV model (i.e., {Zt} ~IID N(0,1), the extremal index is 1 (see Breidt and Davis `98 )
95.~)( 11 cXbP n
n
975.)95(.~)( 5.1 cMbP nn
Extremes for GARCH and SV processes (cont)
![Page 31: SAMSI RISK WS 2007 1 Heavy Tails and Financial Time Series Models Richard A. Davis Columbia University rdavis Thomas Mikosch University](https://reader035.vdocuments.mx/reader035/viewer/2022062619/5517750b5503460e6e8b50c2/html5/thumbnails/31.jpg)
47SAMSI RISK WS 2007
0 20 40 60
time
010
2030
0 20 40 60
time
010
2030
** * * *** ***
Extremes for GARCH and SV processes (cont)
Absolute values of ARCH
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48SAMSI RISK WS 2007
Extremes for GARCH and SV processes (cont)
time
0 10 20 30 40 50
01
23
45
* * * * * ** *
Absolute values of SV process
![Page 33: SAMSI RISK WS 2007 1 Heavy Tails and Financial Time Series Models Richard A. Davis Columbia University rdavis Thomas Mikosch University](https://reader035.vdocuments.mx/reader035/viewer/2022062619/5517750b5503460e6e8b50c2/html5/thumbnails/33.jpg)
51SAMSI RISK WS 2007
Remark: Similar results hold for the sample ACF based on |Xt| and Xt
2.
,)/())(ˆ( ,,10,,1 mhhd
mhX VVh
.)0()(ˆ,,1
1,,1
/21mhhX
dmhX Vhn
.)0()(ˆ,,1
1,,1
2/1mhhX
dmhX Ghn
Summary of results for ACF of GARCH(p,q) and SV models
GARCH(p,q)
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52SAMSI RISK WS 2007
.)0()(ˆ,,1
1,,1
2/1mhhX
dmhX Ghn
.)(ˆln/0
2
1
1h1/1
S
Shnn hd
X
Summary of results for ACF of GARCH(p,q) and SV models (cont)
SV Model
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53SAMSI RISK WS 2007
Sample ACF for GARCH and SV Models (1000 reps)
-0.3
-0.1
0.1
0.3
(a) GARCH(1,1) Model, n=10000-0
.06
-0.0
20
.02
(b) SV Model, n=10000
![Page 36: SAMSI RISK WS 2007 1 Heavy Tails and Financial Time Series Models Richard A. Davis Columbia University rdavis Thomas Mikosch University](https://reader035.vdocuments.mx/reader035/viewer/2022062619/5517750b5503460e6e8b50c2/html5/thumbnails/36.jpg)
54SAMSI RISK WS 2007
Sample ACF for Squares of GARCH (1000 reps)
(a) GARCH(1,1) Model, n=100000
.00
.20
.40
.60
.00
.20
.40
.6
b) GARCH(1,1) Model, n=100000
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55SAMSI RISK WS 2007
Sample ACF for Squares of SV (1000 reps)0.
00.
010.
020.
030.
04
(d) SV Model, n=100000
0.0
0.0
50
.10
0.1
5
(c) SV Model, n=10000
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56SAMSI RISK WS 2007
Example: Amazon-returns (May 16, 1997 – June 16, 2004)
time
log
re
turn
s
0 500 1000 1500
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
Lag
AC
F o
f sq
ua
res
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F o
f a
bs
valu
es
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
![Page 39: SAMSI RISK WS 2007 1 Heavy Tails and Financial Time Series Models Richard A. Davis Columbia University rdavis Thomas Mikosch University](https://reader035.vdocuments.mx/reader035/viewer/2022062619/5517750b5503460e6e8b50c2/html5/thumbnails/39.jpg)
57SAMSI RISK WS 2007
Amazon returns (GARCH model)
GARCH(1,1) model fit to Amazon returns:
0.000024931= .0385, = .957, Xt=(01 X2t-1)1/2Zt, {Zt}~IID
t(3.672)
Lag
AC
F a
bs
valu
es
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F o
f sq
ua
res
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
Simulation from GARCH(1,1) model
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58SAMSI RISK WS 2007
Amazon returns (SV model)
Stochastic volatility model fit to Amazon returns:
Lag
AC
F o
f sq
ua
res
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F o
f a
bs
valu
es
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
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59SAMSI RISK WS 2007
Wrap-up
• Regular variation is a flexible tool for modeling both dependence
and tail heaviness.
• Useful for establishing point process convergence of heavy-tailed
time series.
• Extremal index < 1 for GARCH and for SV.
• ACF has faster convergence for SV.