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;