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

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Properties and applications of the T copula Outline: Student t distribution T copula Pair-copula decomposition Vines Applications Conclusions & recommendations

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

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Univariate Student t distribution Density function Representation:

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Bivariate Student t distribution

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

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

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T copula From Sklar’s theorem: T copula Density of the T copula:

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T copula

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

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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

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

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Pair Copula Decomposition

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Vines Regular vines: canonical and D-vine 23| |1 T |1 24|1 34|12 D-vineCanonical vine Sampling procedure: 1 24| | |224|3 14|23 T2 T3

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Normal vine has a joint normal distribution Conditional correlation = partial correlation Rank correlation specification: –Spearman’s –Kendall’s T-vine degrees of freedom

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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.

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

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Case study Foreign exchange rates: Canadial dollar vs American dollar, German mark vs American dollar, Swiss franc vs American dollar , M=2909 Log returns:

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Case study Exchange/ statlocationscaleskewnesskurtosis Can vs. U.S.9.24e DM vs. U.S.-3.37e Sw vs. U.S.-1.48e Can vs. U.S.DM vs. U.S.Sw vs. U.S. v Accepted v:[3.7,7.6][3,4.5][3.8,6.2] Degrees of freedom parameter v estimated using bootstrap improved Hill estimator- tail index estimator -standarized data -Kolmogorov-Smirnov test

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Case study Estimating bivariate T copulas:

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Case study Take under consideration: - Choice of the decomposition; 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 - Choice of the copula type; - Estimation of the parameters; - 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:

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Case study 4.4, , , Max log likelihood = AIC = Max log likelihood = AIC = Max log likelihood = AIC = , , , , , ,

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Case study T copula for pesudo- sample AIC performance: –Sample n=3000 from vine: AIC for copula: > AIC for vine: –Sample from copula: AIC for copula: < AIC for vine: And v = 8.2, Max log likelihood = AIC = > AIC for all 3 vines 4, 0.814, , 0.068

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Case study Sample n=3000 from vine I: –Estimated bivariate T copulas: –Tail dependence coefficients:

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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;

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