Copula Parameter Estimation by ML and MD Estimators · margins or IFM method) ... Gumbel...

Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 1

Chair of Banking and FinanceRuhr-Universität Bochum

Gregor Weiß

Copula Parameter Estimation by Maximum-Likelihood and Minimum Distance Estimators –A Simulation Study

Presentation at the workshop “Finance and Insurance”FSU Jena, March 16-20, 2009.



Outline of the presentation

Introduction and related literature

Copula parameter estimators

Design of the simulation studies

Results and empirical example

Conclusion and future work



Introduction and related literature I

Copula models have become a major tool in statistics and risk management for modeling and analysing dependence structures between random variables.

This is mainly due to the fact that in contrast to linear correlation a copula captures the complete dependence structure inherent in a set of random variables.

Copula Parameter estimation in these studies is usually performed by a fully parametric (ML), stepwise parametric (the so called inference function for margins or IFM method) or semiparametric pseudo-maximum-likelihood approach depending on the available information on the marginal distributions.

In the semiparametric approach, the marginal distributions are first substituted by their empirical counterparts with the copula parameters being subsequently estimated via maximum-likelihood.



Introduction and related literature II

Recent research has more or less focused on deriving goodness-of-fit test statistics in a copula setting (see e.g. Fermanian, J. Multiv. Ana., 2005; Savu and Trede, Quant. Finance, 2008 and Genest et al., Insur.: Math. & Econom., 2008).

Consequently, copula parameter estimation can also be achieved by minimising one of the distances initially derived for GoF-testing. Each GoF-test thus yields a minimum-distance (MD) estimator for the copula parameters.



Introduction and related literature III

Previous studies on the performance of minimum-distance estimators are relatively rare:

Mendes et al. (Comm. in Stat.: Sim. Comp., 2007) derive weighted minimum-distance estimators based on the empirical copula process. In their simulation study they show that these MD-estimators are robust against contaminations of the data (but only consider MD-estimators based on the empirical copula).

Tsukahara (Can. J. Stat., 2005) finds that the PML-estimator should be preferred to MD-estimators based on the empirical copula process.



Introduction and related literature IV

Finally, Kim et al. (CSDA, 2007) show in a simulation study that the semiparametric pseudo-maximum-likelihood (PML) approach yields considerably better finite sample results than the fully or stepwise parametric approach when the marginals are misspecified.

Concerning MD-estimators, however, they simply state that just like the PML-estimator, the MD-estimators should perform better than the fully or stepwise ML-estimation as both PML and MD-estimators use rank-transformed pseudo-samples without assuming any parametric distribution for the marginals.



Introduction and related literature V

The following questions remain unanswered:

Do MD-estimators yield finite sample parameter estimates that are comparable to those of the Pseudo-ML-estimator?

Which one of the various MD-estimators yields the best estimation results?

Are the differences in estimation bias and MSE of any practical relevance, e.g. when fitting copula models to financial data?



Copula parameter estimators I

Consider a d-dimensional random vector

with joint cdf G, marginals F1,…, Fd and a d-copula such that

is a decomposition of G.

We are interested in fitting a parametric copula family

parameterised by a finite parameter vector



Copula parameter estimators II

In order to estimate the parameter vector, we can choose one of the following estimators:

Fully parametric standard maximum-likelihood: requires distributional assumptions for the margins. If the margins are specified correctly, this estimator possesses the usual optimality properties of the ML-estimator.

Semiparametric pseudo-maximum-likelihood: replaces the margins by their empirical cdfs, then plugs the empirical cdfs into the copula density yielding

which in turn is maximised numerically. The function arguments equal the (scaled) rank-transformed pseudo-observations



Copula parameter estimators III

Minimum-distance estimators based on the empirical copula process:

As Deheuvels’ empirical copula process

converges uniformly to the true copula, GoF-tests and minimum-distance estimators can be based on the process

The convergence of the process under appropriate regularity conditions on the parametric copula family and the sequence of estimators is established in Genest and Rémillard (2008).



Copula parameter estimators IV

Simple Cramér-von-Mises- and Kolmogorov-Smirnov-statistics are given by

Furthermore, I consider the following L1-variant of the CvM-statistic:

Minimising any of these statistics yields the vector of MD-estimates.



Copula parameter estimators V

Minimum-distance estimators based on Kendall’s transform:

Consider the probability integral transform

Then let K denote the univariate cdf of V. The true cdf of K under C can be approximated by (see Genest and Rivest, 1993)

If we assume Cθ to come from a specific parametric copula family, we get



Copula parameter estimators VI

GoF-tests and minimum-distance estimators can then be based on the process

The convergence of the process under appropriate regularity conditions is established in Genest et al. (Scand. J. of Stat., 2006).



Copula parameter estimators VII

Minimum-distance estimators based on Rosenblatt’s transform:

Consider the (bivariate) probability integral transform

with

The transformed data

are then independent and uniformly distributed on the unit square (see Genest et al., Insur: Math & Econ. 2008).



Copula parameter estimators VIII

Minimum-distance estimators based on Rosenblatt’s transform:

The idea is then to compute a distance between the empirical copula and the independence copula evaluated at the PIT-transformed observations.

Asymptotic convergence of the resulting test statistics / estimators is proved in Ghoudi and Rémillard (2006).



Design of the simulation studies I

Simulation study consisted of two parts:

First part: simulate directly from a given copula, thus we only compare classical ML and MD-estimators without any influence of the marginals.

Second part: simulate from a bivariate joint cdf with normal or t-distributed marginals under a given parametric copula.

Choice of copulas and parameters (first part, n=50,100,300 or 500, 1000 repetitions):

Gaussian and Student‘s t (θ=-0.9,-0.8,…,0.8,0.9 and df=3)

Clayton and Frank (θ=0.1,0.2,…,1.9)

Gumbel (θ=1.1,1.2,…,2.9)

Total of 10 estimators



Results I (Gaussian copula, n=50)



Results II (Gaussian copula, n=500)



Results III (Clayton copula, n=500)



Results IV (Frank copula, n=500)



Design of the simulation studies II

Choice of copulas and parameters (second part, n=50 or 500, 500 repetitions):

Gaussian and Student‘s t (θ=-0.8,-0.6,…,0.6,0.8 and df=3)

Clayton and Frank (θ=0.2,0.4,…,1.8)

Gumbel (θ=1.2,1.4,…,2.8)

Marginals either normal or t-distributed

Total of 22 estimators (with correctly or misspecified marginals).



Results V (Gaussian copula, t-dist. marginals, n=500)



Further results

The PML-estimator yielded the best bias and MSE in all settings.

The IFM- or fully parametric estimators could only match these results if all marginals were correctly specified. The IFM-estimator yielded better results than the MD-estimator if at least one marginal was correctly specified.

Especially for the archimedean copulas, the MD-estimators yielded relative errors of up to 50% even for a sample size of n=500.

Also, the MD-estimators require much more computations than the PML-method (the evaluation of the copula density is computationally less complex than the evaluation of the different copula distances).



Empirical example

The question remains, whether these differences are of any practical importance.

Two stocks listed in the German DAX: E.ON / Siemens, 1000 observations (log returns)

1.745976.345777.0712382.024.0973146.843.65611.54571.4849253.34Gumbel

4.38944.14224.1995382.020.83133.51551.05873.37943.22763.6011Frank

1.16381.159097.13473.00382.99740.72011.17381.07601.00880.8265Clayton

0.51800.55530.52760.46730.35500.45120.50000.52880.51630.5118t

0.55590.62740.63090.47280.54800.53460.53930.53140.51640.4799Gaussian

R:KSR:CvMR:L1K:KSK:CvMK:L1E:KSE:CvME:L1PML



Conclusion and future work

The simulation study showed that compared to the Pseudo-Maximum-Likelihood estimator, all MD-estimators based on different GoF-approaches yielded worse estimates at higher computational cost when using finite samples.

Why would one be interested in using MD-estimators at all? Answer could lie in their possible robustness against contaminated data.

Mendes et al. (2007) raise this question and show that when the data is contaminated by a bivariate normal distribution, weighted MD-estimators in some cases give better estimates than a weighted ML-estimator. However, their results could well depend on their choice of copula parameters in the simulation and/or the systematic contamination.



Thank you very much for your attention!

Copula Parameter Estimation by ML and MD Estimators · margins or IFM method) ... Gumbel...

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