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Properties and Applications of the T copula

Properties and Applications of the T copula. Master thesis presentation Joanna Gatz TU Delft 29 of July 2007. Properties and applications of the T copula. Outline: Student t distribution T copula Pair-copula decomposition Vines Applications Conclusions & recommendations.

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Properties and Applications of the T copula

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  1. Properties and Applications of the T copula Master thesis presentation Joanna Gatz TU Delft 29 of July 2007

  2. Properties and applications of theT copula Outline: • Student t distribution • T copula • Pair-copula decomposition • Vines • Applications • Conclusions & recommendations

  3. Multivariate Student t distribution • Random vector • As then density is p-variate normal with mean and correlation matrix .

  4. Univariate Student t distribution • Density function • Representation:

  5. Bivariate Student t distribution

  6. Symmetric, Does not posses independence property, Family of the elliptical distributions: Explicit relation between and Kendall’s Partial correlation = Conditional correlation Properties of the Student t distribution

  7. Properties of the Student t distribution • Upper tail dependence coefficient: • Bivariate Student t distribution with and

  8. T copula • From Sklar’s theorem: • T copula • Density of the T copula:

  9. T copula

  10. Sampling T copula Generate T ~ random variable: • Choleski decom. A of R; • Simulate • Simulate • Set • Set Return

  11. Properties of the T copula • Symmetric, • Elliptical copula, • Does not posses independence property, • Explicit relation between and Kendall’s • Partial correlation = Conditional correlation • Tail dependence coefficient

  12. Estimation of the T copula • Semi parametric pseudo likelihood: • Transformation of the observations pseudo sample : • Pseudo likelihood function • Relation between and

  13. Pair Copula Decomposition

  14. Vines 2 12 • Regular vines: canonical and D-vine 13 3 1 Sampling procedure: 14 12 23 34 T1 1 2 3 4 4 13 13|2 24|3 23|1 12 23 34 T2 12 14 24|1 14|23 34|12 13|2 24|3 23|1 24|1 T3 D-vine Canonical vine

  15. Normal vine • has a joint normal distribution • Conditional correlation = partial correlation • Rank correlation specification: • Spearman’s • Kendall’s T-vine degrees of freedom 1 2 3 12 23

  16. Inference for a vine • Observe n variables at M time points, , • Log-likelihood function for canonical vine: • Cascade estimation procedure: • Estimate parameters for tree 1; • Compute observations for tree 2; • Estimate parameters for tree 2; • Compute observations for tree 3; • Estimate parameters for tree 3; • Etc.

  17. Inference for three dimensional vine • Observed data: , 1.Estimate and for tree 1 2.Compute observations and for tree 2 3. Estimate

  18. Case study Foreign exchange rates: • Canadial dollar vs American dollar, • German mark vs American dollar, • Swiss franc vs American dollar • 1973-1984, M=2909 • Log returns:

  19. Case study Degrees of freedom parameter v estimated using bootstrap improved Hill estimator- tail index estimator -standarized data -Kolmogorov-Smirnov test

  20. Case study • Estimating bivariate T copulas:

  21. Case study Take under consideration: • - Choice of the copula type; • - Choice of the decomposition; • - Estimation of the parameters; • Comparing all max log-likelihoods of all decompositions - infeasible for large dimensions; • Determine the most important bivariate relations and let them determine the decomposition • In case of the T copula, since low v indicates strong tail dependence, copulas in tree 1 should be ordered in increasing order with respect to v • - Model comparison criteria - AIC: • Kulback-Leibler information • Akaike (1973,1974) found a relation between K-L information and max log-likelihood value of model • Akaike Information Crierion:

  22. Case study 4.4, 0.878 14, 0.2384 Max log likelihood = 2210.2 AIC = -4408.4 42.2, 0.053 14.6, 0.2344 4.4, 0.878 Max log likelihood = 2205.3 AIC = -4398.6 68.4, -0.028 14, 0.2384 14.6, 0.2344 Max log likelihood = 2201.1 AIC = -4390.2 4.4, 0.8724

  23. Case study And v = 8.2, • T copula for pesudo- sample • AIC performance: • Sample n=3000 from vine: • AIC for copula: - 3314 > AIC for vine: - 3324 • Sample from copula: • AIC for copula: -3409.8 < AIC for vine: -3377.6 Max log likelihood = 2166.6 AIC = -4341.2 > AIC for all 3 vines 4, 0.8 14, 0.2 42.2, 0.068

  24. Case study 4.4 .878 14 .238 • Sample n=3000 from vine I: • Estimated bivariate T copulas: • Tail dependence coefficients: 1 2 3

  25. Conclusions & Recommendations • T-copula can be used to model financial data • E.g. Modeling joint extreme co-movements; • Copula-vine decomposition of the multivariate distribution captures complex dependence structures; • Hierarchical structure, where copulas as building blocks capture pair-wise interactions; • Cascade inference; • It is possible to construct decomposition using different types of copulas the best fit pairs of data; • Algorithms for finding the best decompositions; • Criteria to compare copula-vine decompositions;

  26. Questions?

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