fitting copulas to data

8/13/2019 Fitting Copulas to Data

1/19

5.5 5.5.1 5.5.2 5.5.3

5.5 Fitting Copulas to Data

Presenter: Yen ju Chao

April 10, 2012

Y-J, Chao 5.5 Fitting Copulas to Data
http://find/


2/19

5.5 5.5.1 5.5.2 5.5.3

Fitting Copulas to Data

We have data vectors X1, . . . , Xn with identical distributionfunction F, describing financial loss or financial risk factor

returns.We write Xt= (Xt,1, . . . ,Xt,d)

for an individual data vector, andX= (X1, . . . ,Xd)

for a generic random vector with dfF.

Fhas continuous margins F1, . . . , Fdand by Sklars Theorem, aunique representation F(x) =C(F1(x1), . . . , Fd(xd)).

http://goforward/http://find/http://goback/


3/19

5.5 5.5.1 5.5.2 5.5.3

Method-of-Moments using Rank Correlation

It may be easier to use empirical estimates of either Spearmansor Kendalls rank correlation to infer an estimate for the copulaparameter.(For example:Table 5.5)

Recall Definition 5.28:For rvs X1 and X2 with marginal dfs F1 and F2 Spermans rho isgiven bys(X1,X2) =(F1(X1),F2(X2)).

We could estimate S(Xi,Xj) by calculating the usual correlation

coefficient for the pseudo-observations:{(Fi,n(Xt,i),Fj,n(Xt,j)) :t= 1, . . . , n}, where Fi,n denotes thestandard empirical df for the ith margin.

http://find/


4/19

5.5 5.5.1 5.5.2 5.5.3


We use rank (Xt,i) to denote the rank ofXt,i in X1,i, . . . ,Xn,i, wecan calculate the correlation coefficient for the rank data

{(rank(Xt,i), rankXt,j)}, and this gives us the Spermans rankcorrelation coefficient:12

n(n2

1)

nt=1

(rank(Xt,i) 12

(n+ 1))(rank(Xt,j) 12

(n+ 1))

(5.49)


5 5 5 5 1 5 5 2 5 5 3
http://find/


5/19

5.5 5.5.1 5.5.2 5.5.3


Recall Definition 5.27:For rvs X1 and X2 Kendalls tau is given by(X1,X2) =E(sign((X1

X1)(X2

X2))), where (X1,X2) is

independent copy of (X1,X2).

The standard estimator of Kendalls tau (Xi,Xj) is Kendallsrank correlation coefficient :

n2

1 1t


6/19

5.5 5.5.1 5.5.2 5.5.3


Example 5.52(bivariate Archimedean copulas with a singleparameter).

We assumed model is of the form F(x1, x2) =C(F1(x1),F2(x2)),

where is a single parameter to be estimated.We have simple relationships of the form(X1,X2) =f().(asshown in Table 5.5)

We can calculate a sample value r for Kendalls tau first,

Then solving the equation r =f() for .For example, Gumbels copula is calibrated by taking= (1 r)1, provided that r 0.


5 5 5 5 1 5 5 2 5 5 3
http://find/


7/19

5.5 5.5.1 5.5.2 5.5.3


Example 5.53(calibrating Gauss copulas using Spearmansrho).

We assumed a meta-Gaussian model for X with CGap

and we wish toestimate the correlation matrix P. It follows from Theorem 5.36 that

S(Xi,Xj) = (6/)arcsin1

2ij ij,

where the final approximation is very accurate. This suggests weestimate Pby the matrix of pairwise Spearmans rank correlationcoefficient RS.


5 5 5 5 1 5 5 2 5 5 3
http://find/http://goback/


8/19

5.5 5.5.1 5.5.2 5.5.3


Example 5.54(calibrating t copulas using Kendalls tau).

We assumed a meta-t model for Xwith copula Ct,Pand we wish to

estimate the correlation matrix P. It follows

(Xi,Xj) = (2/)arcsinij

so that a possible estimator ofP is the matrix R with component

given by rij =sin(12rij).

http://find/


9/19

5.5 5.5.1 5.5.2 5.5.3


10/19

5.5 5.5.1 5.5.2 5.5.3


Algorithm 5.55(eigenvalue method).

Calculate Q=GLG

, which will be symmetric and positive

definite but not a correlation matrix, since its diagonal elementswill not necessarily equal one.

Return the correlation matrix R=(Q), where denotes thecorrelation matrix operator.,where defined

() = (())1(())1,(()) := diag(11, . . . ,dd)


5.5 5.5.1 5.5.2 5.5.3


11/19

5.5 5.5. 5.5.2 5.5.3

Forming a Pseudo-Sample from the Copula

We now turn to the estimation of parameteric copulas bymaximum likelihood (ML).

In this section we describe brifely some general approaches to thefirst step of estimating margins and constructing apseudo sampleof observations from the copula.In the following section we describe how the copula parametersare estimated by ML from pseudo-sample.


5.5 5.5.1 5.5.2 5.5.3


12/19


Let F1, . . . ,Fddenote estimates of the marginal dfs. The

pseudo-sample from the copula consists of the vectorsU1, . . . ,Un, where

Ut= (Ut,1, . . . ,Ut,d)

= (F1(Xt,1), . . . ,Fd(Xt,d))


5.5 5.5.1 5.5.2 5.5.3
http://find/


13/19


Possible methods for obtaining the marginal estimate Fi includethe following.

Parameteric estimation.We choose an appropriate parametric model for the data in questionand fit it.

Non-parametric estimation with of empirical df.We could estimate Fj using

Fi,n(x) =

1

n+ 1

nt=1

I{Xt,ix}

Extreme value theorey for the tails.Empirical distribution functions are known to be poor estimators of theunderlying distribution in the tails, and the tail are model using ageneralized Pareto distribution, the body of distribution may bemodelled empirically.


5.5 5.5.1 5.5.2 5.5.3
http://find/


14/19


Example 5.57.

Five years of daily log-return data (1996-2000).Intel, Microsoft and General Electric stocks.

The marginal distributions are estimated empirically (method(2)).

The pseudo-sample from copula is shown in Figure 5.14.


5.5 5.5.1 5.5.2 5.5.3
http://find/


15/19

Multivariate Archimedean Copulas


5.5 5.5.1 5.5.2 5.5.3
http://find/


16/19

Maximum Likelihood Estimation

Let C denote a parameter copula, where is the vector ofparameters to be estimated. The MLE is obtained by maximizing

ln L(;U1 , . . . ,Un) =n

t=1

ln c(Ut)

One could envisage Using the two-stage method to decide on the

most appropriate cpula family and then estimating all parameters(marginal and copula) in a final fully parametric round ofestimation.


5.5 5.5.1 5.5.2 5.5.3


17/19


Example 5.58 (fitting the Gaussian copula).

In the case of a Gaussina copula implies that the log-likelihood is

ln L(P;U1 , . . . ,Un)

=n

t=1

ln fP(1(Ut,1), . . . ,

1(Ut,d)) n

t=1

dj=1

ln (1(Ut,j))

where f will be used to denote the joint density of a randomvector with Nd(0, ) distribution.


5.5 5.5.1 5.5.2 5.5.3
http://find/


18/19



P= argmaxP

nt=1ln f(Yt), where Yt,j=

1(Ui,j) forj=1,. . . , d andPdenotes the set of all possible linear correlationmatricses.The setPcan be constructed asP= {P=(Q) :Q=AA , Alower triangular with ones on the diagonal}An approximate solution to the maximization may be obtained as

follows: = (1/n)

nt=1 YtY

t

P=()


5.5 5.5.1 5.5.2 5.5.3


19/19



For example 5.57 by full ML, the estimated correlation matrixhas 0.58(INTC-MSFT), 0.34(INTC-GE) and 0.4(MSFT-GE); thelog- likelihood at the maximum is 376.65, using the alternativemethod gives a log likelihood value of 376.62

http://find/

fitting copulas to data

Documents