Analyzing and Modeling Multivariate Association

Statistical Measures and Pair-Copula Constructions
1. Auflage 2013
ISBN: 978-3-8441-0229-1
Verlag: Josef Eul Verlag GmbH

Buch, Englisch, Band 176, 200 Seiten, Paperback, Format (B × H): 148 mm x 210 mm, Gewicht: 337 g

Reihe: Quantitative Ökonomie

Many applications in quantitative finance, such as the estimation of the Value-at-Risk of a portfolio, require the modeling of dependencies of a large number of random variables. The most popular approach is Pearson’s correlation coefficient, which is based on the covariance, i. e. the mixed second moments of the corresponding random variables. However, the correlation coefficient only captures linear dependencies and solely in case of a few distributions, such as the multivariate normal distribution, completely determines the dependence structure.

Empirical data, unfortunately, is often non-normal and exhibits asymmetric dependence patterns. The multivariate normal distribution, for example, often underestimates the probability of simultaneous extremes, which can lead to incorrect estimates of the risk of a given portfolio. Hence, for many empirical applications, the normal distribution is not suitable.

As an alternative concept of dependence modeling, the theory of copulas has drawn a lot of attention in the past decades. Based on this theory, the author introduces a new class of measures of association between random vectors that are invariant with respect to the marginal distribution functions of the considered random vectors and can distinguish between positive and negative association.

The second part of the thesis focuses on the modeling of high dimensional dependencies with Pair-copula constructions. To this end, a data-driven sequential estimation method for these models is developed. Empirical applications of these models in Value-at-Risk forecasting and the spatial modeling of meteorological data are given.

Autoren/Hrsg.

Schnieders, Julius

Inhaltsverzeichnis

1. Introduction

2. Copulas and Dependence Concepts
2.1. Theory of Copulas
2.2. Parametric Copula Families
2.3. Survival and rotated copulas
2.4. The empirical copula
2.5. Measures of Association

3. Association of Random Vectors
3.1. Notation and definitions
3.2. Established measures of association
3.3. Copula based measures of association
3.4. Statistical estimation of the measures
3.5. Empirical example
3.6. Influence of outliers
3.7. Conclusion

4. Pair-Copula Constructions
4.1. Pair-Copula Constructions
4.2. Conditional distribution functions
4.3. Vines
4.4. Simplified PCCs
4.5. Estimation of Pair-Copula Constructions
4.6. Simulation Techniques
4.7. Conclusion

5. Model Selection for Pair-Copula Constructions
5.1. Model Selection for bivariate copulas
5.2. Goodness-of-Fit
5.3. Model selection for Vine Structures
5.4. Simulation Study
5.5. Application to Empirical Datasets
5.6. Conclusion

6. Application of Pair-Copula Constructions
6.1. Dynamic PCC Models: An Application with Value-at-Risk Forecasting
6.2. Spatial dependence in wind and optimal wind power allocation

Über Autor(innen)

Julius Schnieders, born in 1983, studied Mathematics with a focus on Statistics in Münster and Madrid. He finished his studies in 2008 and became a member of the Graduate School of Risk Management at the University of Cologne. In 2012, he completed his doctoral studies in Statistics at the Chair of Economic and Social Statistics of Prof. Dr. Friedrich Schmid. He has written articles on dependence modeling, and has given presentations on this subject at several international conferences.