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Properties and Applications of the T copula. Master thesis presentation Joanna Gatz TU Delft 29 of July 2007. Properties and applications of the T copula. Outline: Student t distribution T copula Pair-copula decomposition Vines Applications Conclusions & recommendations. - PowerPoint PPT Presentation
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Master thesis presentationJoanna Gatz
TU Delft29 of July 2007
Properties and Applications of the T copula
Properties and applications of theT copula
Outline:
• Student t distribution• T copula• Pair-copula decomposition• Vines• Applications• Conclusions & recommendations
Multivariate Student t distribution
• Random vector
• As then density is p-variate normal
with mean and correlation matrix .
TpXXX ),...,( 1
Rv
2/)(1
2/12/, )()(1
1||)2/(
2)(
pvT
ptRv xRx
vRvv
pv
xf
Univariate Student t distribution
• Density function
• Representation:
-3 -2 -1 0 1 2 30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
2/)1(2
,
11
)2/(
2
1
)(
vtPv x
vvv
v
xf
),2( v
YS
vT
)1,0(~ NY2~ vS
Bivariate Student t distribution
2/)2(
2122
212, )2(
)1(
11
)2/()(
2
2
)(
v
tv xxxx
vvv
v
xf
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
-3-2
-10
12
3
-3-2
-10
12
30
0.05
0.1
0.15
0.2
Properties of the Student t distribution
• Symmetric,• Does not posses independence property,• Family of the elliptical distributions:
– Explicit relation between and Kendall’s
– Partial correlation = Conditional correlation
2sin
kijkij |.
Properties of the Student t distribution• Upper tail dependence coefficient:
• Bivariate Student t distribution with and
))(|)((lim: 11
1uFXuGYP
uU
))1/11(1(2 221 vtvLU
v
2 4 6 8 10 12 14 16 18 200
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2Tail dependence coefficient for bivariate t distribution with =0
degrees of freedom
U=
L
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u
Tail dependence
T copula
• From Sklar’s theorem:
• T copula
• Density of the T copula:
))(),...,((),...,( 111 ppp xFxFCxxF
))(),...,((),..,( 11
1,1, pvvtRvp
tRv ututFuuC
dxxRx
vRvv
pv
uCpv
Tp
ututtRv
pvv 2/)(1
2/12/
)()(
,
11
||)2/(
2...)(
11
1
p
iiv
tv
pvvtRvt
Rv
utf
ututfuc
1
1
11
1,
,
))((
))(),...,(()(
T copula
Contour plot of the T copula density with =0.5 and v=4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
10
0.2
0.4
0.6
0.8
1
T Copula cumulative distribution function with =0.5 and v=4
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
10
1
2
3
4
5
T copula density with =0.5 and v=4
Contour plot of the T coula density with =0.5 and v=4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Sampling T copula
Generate T ~ random variable:• Choleski decom. A of R;• Simulate
• Simulate• Set
• Set
Return
tRvF ,
Azy zs v 2~
))(),...,(( 1 pvv xtxtU
),,...,( 1 pzzz
)1,0(~ Nzi0 0.5 1
0
0.2
0.4
0.6
0.8
11000 sample from T copula,-0.5,v=2
0 0.5 10
0.2
0.4
0.6
0.8
11000 sample from normal copula,\rhp=0.5
-4 -2 0 2 4-4
-2
0
2
4
N1
N2-4 -2 0 2 4
-4
-2
0
2
4
N2
N1ys
vx
Properties of the T copula
• Symmetric,• Elliptical copula,• Does not posses independence property,
– Explicit relation between and Kendall’s
– Partial correlation = Conditional correlation
– Tail dependence coefficient
2sin
kijkij |.
))1/11(1(2 221 vtvLU
Estimation of the T copula
• Semi parametric pseudo likelihood:• Transformation of the observations pseudo sample :
• Pseudo likelihood function
• Relation between and
),,( 3,2,1, iiii XXXX ni ,...,1
),( vR
n
ixj jiI
nF
1}{
^
,
1)( 3,2,1j
)(),(),( 3,
^
32,
^
21,
^
1
^
iiii xFxFxFU
n
i
in UcL1
^
);()(
1
)2
sin(1
)2
sin()2
sin(1
23
1312
^
R
n
i
iv
RvUcv1
^^
),2(
^
)),,(log(maxarg
Pair Copula Decomposition
)|()|()(),,( 321323321 xxxfxxfxfxxxf 23|13|23123 ffff
),,( 321123123 FFFCF
321321123123 ),,( fffFFFcf
32322323 ),( ffFFcf 232233|2 ),( fFFcf
2|13|23|13|1223|1 ),( fFFcf
3|12|32|12|1323|1 ),( fFFcf v
CF uv
vu
|
131133|1 ),( fFFcf
),(),(),( 2|32|12|1332233113321123 FFcFFcFFcffff
Vines
• Regular vines: canonical and D-vine
23|1
14
1312
4
3
2
1
24|1
T1
1213
14
23|1
24|1
34|12
D-vine Canonical vine
)1,0(~,...,1 Uww n
),...,|(
...
)|(
111
121
2
11
nnn xxwFx
xwFx
wx
Sampling procedure:
1
24|3
3
12
13|2
23
4
34
212 23 34
13|2 24|3
14|23
T2
T3
Normal vine
• has a joint normal distribution
• Conditional correlation = partial correlation• Rank correlation specification:
– Spearman’s – Kendall’s T-vine degrees of freedom
)6/sin(2 ;ijrij
v
)2/sin( ijij
2312223
2122|1313 11
),,( 321 XXXX
1 2 3
2312
12 23
2|13
1
1
1
23
1312
R
r
12;r 23;r
2|13;r
1223
2|13
Inference for a vine
• Observe n variables at M time points, ,• Log-likelihood function for canonical vine:
• Cascade estimation procedure:• Estimate parameters for tree 1;• Compute observations for tree 2;• Estimate parameters for tree 2;• Compute observations for tree 3;• Estimate parameters for tree 3;• Etc.
),...,( ,1, nii xx ),...,( ,1, nii uuMi ,...,1
1
1 1 1),1(),...,,1(),(,1,...,,1,1,...,1|, }]|(),|({log[
n
j
jn
i
M
ttjttijtjttjjijj uuFuuFc
Inference for three dimensional vine
• Observed data: ,),,( 3,2,1, iii xxx ),,( 3,2,1, iii uuu
M
ttttttt wwcuucuuc
11,22,1,2|132,13,2,231,12,1,12 ))],,(log(),,(log(),,([log(
Mi ,...,1
1.Estimate and for tree 1
2.Compute observations and for tree 2
3. Estimate
12c12c
23c23c
2|13c
1w2w
1u 2u 3u
2
21121
),(
u
uuCw
2
23232
),(
u
uuCw
2|13c),( 212|13 wwc
21 ,uu 32uu
Case studyForeign exchange rates:• Canadial dollar vs American
dollar,
• German mark vs American dollar,
• Swiss franc vs American
dollar
• 1973-1984, M=2909• Log returns:
1X
2X
3X
MiXXR iii ,...,1,loglog 1
0 1000 2000 30000.5
1
1.5Exchange rates
Can.
vs
U.S.
0 1000 2000 30001
2
3
4
DM v
s U.
S.
0 1000 2000 30001
2
3
4Sw
vs
U.S.
Time points
0 1000 2000 3000-0.02
0
0.02Exchange returns
0 1000 2000 3000-0.1
0
0.1
0 1000 2000 3000-0.1
0
0.1
Time points
Case study
Exchange/ stat location scale skewness kurtosis
Can vs. U.S. 9.24e-005 0.0022 0.32 5.02
DM vs. U.S. -3.37e-005 0.0064 -0.015 9.3
Sw vs. U.S. -1.48e-004 0.0076 0.23 5.76
Can vs. U.S. DM vs. U.S. Sw vs. U.S.
v 4.3 0.19 3.2 0.22 4.2 0.2
Accepted v: [3.7,7.6] [3,4.5] [3.8,6.2]
Degrees of freedom parameter v estimated using bootstrap improved Hill estimator- tail index estimator
-standarized data
-Kolmogorov-Smirnov test
Case study
• Estimating bivariate T copulas:
14
2384.0
1533.0
12
12
12
v
p
6.14
2344.0
1506.0
12
12
13
v
p
4.4
8789.0
6835.0
23
23
23
v
p
Case study
Take under consideration:
- Choice of the decomposition;
•Comparing all max log-likelihoods of all decompositions - infeasible for large dimensions;
•Determine the most important bivariate relations and let them determine the decomposition
•In case of the T copula, since low v indicates strong tail dependence, copulas in tree 1 should be ordered in increasing order with respect to v
- Choice of the copula type;
- Estimation of the parameters;
- Model comparison criteria - AIC:
•Kulback-Leibler information
•Akaike (1973,1974) found a relation between K-L information and max log-likelihood value of model
•Akaike Information Crierion:
dx
xg
xfxfgfI
)|(
)(log)(),(
KdataLAIC 2))|(log(2
Case study
1u2u3u
1u 2u3u
1u2u 3u
3|12c
1|23c
2|13c
4.4, 0.878 14, 0.2384
42.2, 0.053Max log likelihood = 2210.2
AIC = -4408.4
Max log likelihood = 2205.3
AIC = -4398.6
Max log likelihood = 2201.1
AIC = -4390.2
14.6, 0.2344 4.4, 0.878
68.4, -0.028
14, 0.2384 14.6, 0.2344
4.4, 0.872421 ,uu
21 ,uu
32uu
32uu
31uu
31uu
Case study• T copula for pesudo- sample
• AIC performance:
– Sample n=3000 from vine: • AIC for copula: - 3314 > AIC for vine: - 3324
– Sample from copula:• AIC for copula: -3409.8 < AIC for vine: -3377.6
),,( 321 UUUU
1
878.01
2344.02384.01
RAnd v = 8.2,
Max log likelihood = 2166.6
AIC = -4341.2 > AIC for all 3 vines
1u2u3u
2|13c
4, 0.8 14, 0.2
42.2, 0.06821 ,uu32uu
Case study
• Sample n=3000 from vine I:– Estimated bivariate T copulas:
– Tail dependence coefficients:
6.23
18.0
23
23
23
v
p
c
2.17
24.0
13
13
13
v
p
c
8.4
,82.0
12
12
12
v
p
c
1 2 34.4
.878
14
.238
Conclusions & Recommendations
• T-copula can be used to model financial data– E.g. Modeling joint extreme co-movements;
• Copula-vine decomposition of the multivariate distribution captures complex dependence structures;– Hierarchical structure, where copulas as building blocks capture
pair-wise interactions;– Cascade inference;
• It is possible to construct decomposition using different types of copulas the best fit pairs of data;
• Algorithms for finding the best decompositions;• Criteria to compare copula-vine decompositions;