Post on 12-Apr-2018
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.
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 2
Chair of Banking and FinanceRuhr-Universität Bochum
Outline of the presentation
Introduction and related literature
Copula parameter estimators
Design of the simulation studies
Results and empirical example
Conclusion and future work
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 3
Chair of Banking and FinanceRuhr-Universität Bochum
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.
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 4
Chair of Banking and FinanceRuhr-Universität Bochum
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.
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 5
Chair of Banking and FinanceRuhr-Universität Bochum
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.
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 6
Chair of Banking and FinanceRuhr-Universität Bochum
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.
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 7
Chair of Banking and FinanceRuhr-Universität Bochum
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?
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 8
Chair of Banking and FinanceRuhr-Universität Bochum
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
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 9
Chair of Banking and FinanceRuhr-Universität Bochum
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
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 10
Chair of Banking and FinanceRuhr-Universität Bochum
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).
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 11
Chair of Banking and FinanceRuhr-Universität Bochum
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.
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 12
Chair of Banking and FinanceRuhr-Universität Bochum
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
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 13
Chair of Banking and FinanceRuhr-Universität Bochum
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).
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 14
Chair of Banking and FinanceRuhr-Universität Bochum
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).
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 15
Chair of Banking and FinanceRuhr-Universität Bochum
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).
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 16
Chair of Banking and FinanceRuhr-Universität Bochum
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
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 17
Chair of Banking and FinanceRuhr-Universität Bochum
Results I (Gaussian copula, n=50)
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 18
Chair of Banking and FinanceRuhr-Universität Bochum
Results II (Gaussian copula, n=500)
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 19
Chair of Banking and FinanceRuhr-Universität Bochum
Results III (Clayton copula, n=500)
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 20
Chair of Banking and FinanceRuhr-Universität Bochum
Results IV (Frank copula, n=500)
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 21
Chair of Banking and FinanceRuhr-Universität Bochum
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).
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 22
Chair of Banking and FinanceRuhr-Universität Bochum
Results V (Gaussian copula, t-dist. marginals, n=500)
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 23
Chair of Banking and FinanceRuhr-Universität Bochum
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).
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 24
Chair of Banking and FinanceRuhr-Universität Bochum
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
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 25
Chair of Banking and FinanceRuhr-Universität Bochum
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.
Gregor Weiß – Copula Parameter Estimation: A Simulation Study – Slide 26
Chair of Banking and FinanceRuhr-Universität Bochum
Thank you very much for your attention!