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5.5 5.5.1 5.5.2 5.5.3
5.5 Fitting Copulas to Data
Presenter: Yen ju Chao
April 10, 2012
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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)).
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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.
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Method-of-Moments using Rank Correlation
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)
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Method-of-Moments using Rank Correlation
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
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Method-of-Moments using Rank Correlation
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.
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Method-of-Moments using Rank Correlation
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.
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Method-of-Moments using Rank Correlation
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).
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Method-of-Moments using Rank Correlation
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)
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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.
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Forming a Pseudo-Sample from the Copula
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))
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Forming a Pseudo-Sample from the Copula
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.
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Forming a Pseudo-Sample from the Copula
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.
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Multivariate Archimedean Copulas
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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.
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Maximum Likelihood Estimation
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.
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Maximum Likelihood Estimation
Example 5.58 (fitting the Gaussian copula).
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=()
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Maximum Likelihood Estimation
Example 5.58 (fitting the Gaussian copula).
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
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