Three-Stage Semi-parametric Estimation of T-Copulas: Asymptotics, Finite-Samples Properties and Computational Aspects

8/14/2019 Three-Stage Semi-parametric Estimation of T-Copulas: Asymptotics, Finite-Samples Properties and Computational Aspects

1/37

Three-Stage Semi-parametric Estimation of T-Copulas:

Asymptotics, Finite-Samples Properties and ComputationalAspects

Dean Fantazzini

Moscow School of Economics

EEA-ESEM 2008 Milan: Thursday August 28


2/37

Overview of the Presentation


Asymptotics, Finite-Samples Properties and Computational Aspects2


3/37


Semi-Parametric Copula Estimators: A Review


Asymptotics, Finite-Samples Properties and Computational Aspects2-a


4/37



The Three-Stage KME-CML Method


Asymptotics, Finite-Samples Properties and Computational Aspects2-b


5/37




Asymptotic Properties Of The Three-Stage Method


Asymptotics, Finite-Samples Properties and Computational Aspects2-c


6/37




Asymptotic Properties Of The Three-Stage Method Finite-Sample Properties And Computational Aspects


Asymptotics, Finite-Samples Properties and Computational Aspects2-d


7/37




Asymptotic Properties Of The Three-Stage Method Finite-Sample Properties And Computational Aspects

Conclusions


Asymptotics, Finite-Samples Properties and Computational Aspects2-e


8/37


Genest et al. (1995) were the rst to analyze a semi-parametric estimationof a bivariate Copula with i.i.d. observations.

Their Canonical Maximum Likelihood (CML) method differs from fullMaximum Likelihood methods because no assumptions are made about theparametric form of the marginal distributions.

Let us consider a multivariate random sample represented by the timeseries X = ( x1t , . . . , xnt ), and t = 1 , . . . , T , and let f h be the density of the joint distribution of X . Then, by using Sklars theorem (1959)

f h (x i ; 1 , . . . , n , ) = c(F 1 (x1; 1 ), . . . , F 1 (xn ; n ); )

n

i =1 f i (x i ; i ) (1)

where f i is the univariate density of the marginal distribution F i , c is thecopula density, i , i = 1 , . . . , n is the vector of parameters of the marginaldistribution F i , while is the vector of the copula parameters.




9/37


The CML estimation process is performed in two steps:

Denition 1.1 (CML copula estimation). 1. Transform the dataset

(x1t , x 2t , . . . , x nt ), t = 1 , . . . , T into uniform variates ( u1t , u2t , . . . , unt )using the empirical distributions F iT () dened as follows:

F iT (x it ) =1T

T

t =1

1l { x it x i ) , i = 1 . . . n (2)

where 1l { x } represents the indicator function.

2. Estimate the copula parameters by maximizing the log-likelihood:

CML = arg maxT

t =1

log(c(F 1T (x1t ), . . . , F nT (xnt )); ) (3)




10/37


Under some regularity conditions, Genest et al. (1995) show that thesemiparametric estimator CML (for bivariate copula) has the followingasymptotic distribution:

T ( CML 0 )d

N 0,2

h2 (4)

where

2 = var [l (F 1 (X 1 ), F 2 (X 2 ); ) + W 1 (X 1 ) + W 2 (X 2 )]

W i (x i ) = 1l F i ( X i ) u i l,i (u1 , u 2 ; ) c(u1 , u 2 ; )du 1 du 2 i = 1 , 2h = E [l, (F 1 (x1t ), F 2 (x2t ); )]

and where W i (x it ) can have this alternative expression too, uponintegrating by parts with respect to u i (i = 1 , 2):

W i (x it ) = 1l F i ( X i ) u i l (u1 , u 2 ; ) li (u1 , u 2 ; ) c(u1 , u 2 ; )du 1 du 2Three-Stage Semi-parametric Estimation of T-Copulas:



11/37


Since the seminal work by Genest et al. (1995), it has become commonpractice to use semi-parametric methods with high-dimensional ellipticalStudents T copulas too (see, e.g. Cherubini et al. (2004) and Mcneil etal. (2005)):

c(t (x 1 ) , . . . , t (x n )) = | | 1/ 2 + n2

2

2 +12

n 1 + 1

+ n2

n

i =11 +

2i2

+12

Particularly, after the marginal empirical distribution functions arecomputed in a rst stage , the correlation matrix is estimated in a secondstage using a method-of-moment estimator based on Kendalls tau , whilethe degrees of freedom are estimated in a third stage using MaximumLikelihood methods .

Despite the widespread use of this procedure, its asymptotics andnite-sample properties have not been developed yet.




12/37


Denition 1.2 (Kendalls tau). If we have ( X 1 ; X 2 ) and ( X 1 ; X 2 ) twoindependent and identically distributed random vectors, the population version of Kendalls tau (X 1 ; X 2 ) is, (see Kruskal (1958)):

(X 1 , X 2 ) = E sign (X 1 X 1 )( X 2 X 2 ) (5)Besides, Kendalls tau can be expressed in terms of copulas, thussimplifying calculus, see, e.g., Nelsen (1999), p.127.

(X 1 , X 2 ) = 41

01

0 C (u1 , u 2 )dC (u1 , u 2 ) 1 (6)Lindskog, McNeil, and Schmock (2002) proved that Kendalls tau forelliptical distributions is given by

(X 1 , X 2 ) =2

arcsin X 1 X 2 (7)

where X 1 X 2 is the copula correlation parameter.




13/37


The Kendalls tau moment estimator can now be dened:

Denition 1.3 (Copula estimation with Kendalls tau). Let usconsider the population version of Kendalls (5) and its relationship with

copula parameters (6) to build a moment function of the typeE [ (X 1 , X 2 ; 0 )] = 0 (8)

Then we can construct an empirical estimate of the Kendalls tau pairwise

correlation matrix and use relationship (6) to infer an estimate of therelevant parameters of the copula.

This is a method of moments estimate because the true moment (5) isreplaced by its empirical analogue,

T

2

1

1 t


14/37


There are cases when the copula parameter vector has different kinds of parameters and only some of them can be expressed as a function of theKendalls tau. This is the case for the T-copula . In this situation, Bouyeet al. (2001) and McNeil et al. (2005) have suggested the following

estimation procedure:

Denition 1.4 (Three-stage KME - CML copula estimation).

1. Transform the dataset (x1t , x 2t , . . . x nt ), into uniform variates(F 1T (x1t ), F 2T (x2t ), . . . , F nT (xnt )) , using the empirical distribution function.

2. Collect all pairwise estimates of the sample Kendalls tau given by (9)in an empirical Kendalls tau matrix R dened by R jk = (F jT (X j ), F kT (X k )) , and then construct the correlation matrix using this relationship j,k = sin( 2 R

j,k ), where the estimated

parameters are the q = n

(n

1)/ 2 correlations [1 , . . . q ] .




15/37


Since there is no guarantee that this componentwise transformation of the empirical Kendalls tau matrix is positive denite, when needed, can be adjusted to obtain a positive denite matrix using a proceduresuch as the eigenvalue method of Rousseeuw and Molenberghs (1993)or other methods.

3. Look for the CML estimator of the degrees of freedom CML by maximizing the log-likelihood function of the T-copula density:

CML = arg max

T

t =1log cT copula (F 1T (x1t ), . . . , F nT (xnt ); , ) (10)




16/37


The second step in the previous denition 1.4 corresponds to amethod-of-moments estimation based on q moments and Kendall tau rankcorrelations estimated with empirical distribution functions.

We can therefore build a q 1 moments vector for the parameter vector0 = [ 1 , . . . , q ] as reported below:

(F 1 (X 1 ), . . . , F n (X n ); 0 ) =

E [1 (F 1 (X 1 ), F 2 (X 2 ); 1 )]...

E [q (F n 1 (X n 1 ), F n (X n ); q )]

= 0

(11)




17/37


Then these theorems follow:Theorem 1.1 (Consistency of ). Let assume that (x1t , . . . , x nt ) arei.i.d random variables with dependence structure given by c(u1,t , . . . , u n,t ; 0 , 0 ). Suppose that

(i) the parameter space is a compact subset of R q ,

(ii) the q-variate moment vector (F 1 (X 1 ), . . . , F n (X n ); 0 ) is continuousin 0 for all X i ,

(iii) (F 1 (X 1 ), . . . , F n (X n ); ) is measurable in X i for all in ,(iv) E [ (F 1 (X 1 ), . . . , F n (X n ); )] = 0 for all = 0 in ,

(v) E sup (F 1 (X 1 ), . . . , F n (X n ); ) < ,Then

p

0 as n .Theorem 1.2 (Consistency of CML ). Let the assumptions of theprevious theorem hold, as well as the regularity conditions reported in

Proposition A.1 in Genest et al.(1995). Then CMLp

0 as n .Three-Stage Semi-parametric Estimation of T-Copulas:



18/37


The asymptotic normality is not straightforward, since we use a 3-stepprocedure where we perform a different kind of estimation at the secondand third stage.

A possible solution is to consider the CML used in the 3rd stage as a

special method-of-moment estimator.

Just note that the CML estimator is dened by the derivative of thelog-likelihood function with respect to the degrees of freedom:

l(; )

=T

t =1

l F 1T (x1,t ), . . . , F nT (xn,t ); , = 0 (12)

Dividing both sides by T yields the denition of the method of momentsestimator:

1T

T

i =1

l F 1 T (x 1 ,t ) , . . . , F nT (x n,t ); , =1T

n

i =1

(F 1 T (x 1 ,t ) , . . . , F nT (x n,t ); , ) = 0




19/37


Let dene the sample moments vector KME CML for the parameter

vector = [ 1 , . . . q , ] as follows:

KME CML F 1T (x 1,t ) , . . . , F nT (x n,t ); =

=

1T T i =1 1 (F 1T (x 1,t ) , F 2T (x 2,t ); 1 )...1T

T i =1 q F n 1,T (x n 1,t ) , F nT (x n,t ); q

1

T

T

i =1 F

1T (x 1,t ) , . . . , F

nT (x n,t ); ,

= 0




20/37


Let also dene the population moments vector with a correction to takethe non-parametric estimation of the marginals into account, together withits variance (see Genest et al. (1995), 4):

0 =

1 (F 1 (X 1 ) , F 2 (X 2 ); 1 )

...

q (F n 1 (X n 1 ) , F n (X n ); q )

(F 1 (X 1 ) , . . . , F n (X n ); 0 , 0 ) +n

i =1W i, (X i )

= 0 (13)

0 var [ 0 ] = E KME CML KME CML (14)

where

W i, (X i ) =

1l F i ( X i ) u i

2

u ilog c(u 1 , . . . u n )dC (u 1 , . . . u n ) (15)

Note that the population moments used to estimate the correlations arenot affected by the marginals empirical d.f., since the Kendalls tau isinvariant under strictly increasing marginal transformations




21/37


Theorem 1.3 (Asymptotic Distribution 3-stages KME-CMLMethod). Let the assumptions of the previous theorems hold. Assume further that

KM E CM L ( ; ) is O(1) and uniformly negative denite,

while 0 is O(1) and uniformly positive denite. Then, the three-stagesKME-CML estimator veries the properties of asymptotic normality:

T ( 0 )d

N 0, E KME CML

1

0 E KME CML

1

(16)

Theorem 1.4 (Asymptotic Distribution 3-stages KME-CMLMethod for multivariate heteroscedastic time series models).

Let the regularity conditions (i)-(v) reported in theorem 1.1 hold, together with conditions A.1 and A.9 in Gunky et al. (2007). Then, the three-stagesKME-CML estimator veries the properties of asymptotic normality dened in (16).




22/37

Finite-Sample Properties And Computational Aspects

We consider the following possible DGPs:

1. Bivariate Students T copula , 0.9, . . . 0.9 (step 0 .1); 3, . . . 30(step 1).

We consider two possible data situations: n = 50 and n = 500.

2. We examine the case that ten variables have a multivariate Students

T copula, with the copula correlation matrix equal to:1 -0.15 -0.15 -0.15 -0.15 -0.14 -0.09 -0.03 0.05 0.13

-0.15 1 -0.15 -0.15 -0.15 -0.13 -0.08 -0.02 0.06 0.14

-0.15 -0.15 1 -0.15 -0.15 -0.12 -0.07 -0.01 0.07 0.15

-0.15 -0.15 -0.15 1 -0.15 -0.11 -0.06 0.01 0.08 0.15

-0.15 -0.15 -0.15 -0.15 1 -0.10 -0.05 0.02 0.09 0.15

-0.14 -0.13 -0.12 -0.11 -0.10 1 -0.04 0.03 0.10 0.15

-0.09 -0.08 -0.07 -0.06 -0.05 -0.04 1 0.04 0.11 0.15

-0.03 -0.02 -0.01 0.01 0.02 0.03 0.04 1 0.12 0.15

0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 1 0.15

0.13 0.14 0.15 0.15 0.15 0.15 0.15 0.15 0.15 1




23/37


We choose this correlation matrix because its lowest eigenvalue isvery close to zero (0.0786) and it allows us to study the effect that theeigenvalue method by Rousseeuw and Molenberghs (1993) has on the

limiting distribution of the KME-CML estimator.

Furthermore, we consider 3, . . . 30 (step 1), as well as two possibledata situations: n = 50 and n = 500.




24/37


3. We examine the case that ten variables have a multivariate StudentsT copula, with the copula correlation matrix equal to:

1 0.21 0.33 0.22 0.36 0.30 0.37 0.34 0.31 0.47

0.21 1 0.20 0.15 0.27 0.18 0.18 0.31 0.20 0.21

0.33 0.20 1 0.16 0.32 0.28 0.40 0.33 0.17 0.42

0.22 0.15 0.16 1 0.20 0.16 0.18 0.20 0.27 0.20

0.36 0.27 0.32 0.20 1 0.32 0.33 0.55 0.33 0.35

0.30 0.18 0.28 0.16 0.32 1 0.28 0.32 0.26 0.31

0.37 0.18 0.40 0.18 0.33 0.28 1 0.35 0.23 0.40

0.34 0.31 0.33 0.20 0.55 0.32 0.35 1 0.31 0.35

0.31 0.20 0.17 0.27 0.33 0.26 0.23 0.31 1 0.30

0.47 0.21 0.42 0.20 0.35 0.31 0.40 0.35 0.30 1

This is the correlation matrix of the returns of the rst 10 stocksbelonging to the Dow Jones Industrial Index, observed between the

18/11/1988 and the 20/11/2003.

Furthermore, we consider 3, . . . 30 (step 1), as well as two possibledata situations: n = 50 and n = 500.




25/37


The estimators considered are

- the 3-stage KME-CML method ,

- and the Maximum Likelihood estimator computed with given marginals , inorder to assess the loss in efficiency associated with absence of knowledgeof the marginals.

We also considered the 2-stage CML method which delivered resultsin-between the KME-CML and ML methods, as expected.




26/37

Coverage rate (95%) for and . % of convergence failures.(Bivariate T-copula estimated with the KME-CML method)



C (95%) f d % f f il


27/37

Coverage rate (95%) for and . % of convergence failures.(Bivariate T-copula estimated with the ML method)



C (95%) f d (Ill ifi d i


28/37

Coverage rate (95%) for and . (Ill-specified ten-variateT-copula estimated with the KME-CML method)



C g t (95%) f d (Ill ifi d t i t


29/37

Coverage rate (95%) for and . (Ill-specified ten-variateT-copula estimated with the ML method)



% f g f il % f ti h th l ti t i t


30/37

% of convergence failures. % of times when the correlation matrix was notpositive definite. Mean / Median biases (in %), and R-RMSE of

(Ill-specied ten-variate T-copula estimated with the KME-CML method)



% of convergence failures % of times when the correlation matrix was not


31/37


(Ill-specied ten-variate T-copula estimated with the ML method)



Coverage rate (95%) for and (Dow-Jones returns


32/37

Coverage rate (95%) for and . (Dow-Jones returns,ten-variate T-copula estimated with the KME-CML method)



Coverage rate (95%) for and (Dow-Jones returns


33/37

Coverage rate (95%) for and . (Dow Jones returns,ten-variate T-copula estimated with the ML method)



% of convergence failures % of times when the correlation matrix was not


34/37


Dow-Jones returns, ten-variate T-copula estimated with the KME-CML m.



% of convergence failures. % of times when the correlation matrix was not


35/37


(Dow-Jones returns, ten-variate T-copula estimated with the ML method)




36/37

Conclusions

We examined the asymptotics and the nite-sample properties of arecent semi-parametric estimation method used in the nancialliterature with the multivariate Students T-copula.

We found that the KME-CML estimator was more efficient and lessbiased than the one-stage ML estimator when small samples andt-copulas with low degrees of freedom were of concern.

When small samples were of concern and was high, the number of times when the numerical maximization of the log-likelihood failed toconverge was much higher for the ML method than for the KME-CMLmethod.

Yet, while the coverage rates at the 95% level for the ML estimates for did not show any particular bias or trend, the KME-CML estimatesshowed very low rates when became close to 30 and the correlationswere not too strong .




37/37

Conclusions

However, this drop in the coverage rates for was large with bivariatet-copulas, only, while it was much lower with ten-variate copulas .

The coverage rates for the correlations were quite close to the truevalues .

Finally, we found that the eigenvalue method by Rousseeuw and Molenberghs (1993) has to be used to obtain a positive denitecorrelation matrix only when dealing with very small samples(n < 100) and when the true underlying process has the lowesteigenvalue close to zero.

This x induces a positive mean bias in the estimate of , but theeffects on the coverage rates are rather limited . Besides, the number of times when this method has to be used quickly decreases when increases.



Documents

Three-Stage Semi-parametric Estimation of T-Copulas: Asymptotics, Finite-Samples Properties and Computational Aspects