An empirical analysis of multivariate copula models

Transcription

1 An empirical analysis of multivariate copula models Matthias Fischer and Christian Köck Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany Homepage: Berlin 2007 Workshop on Copulas Fischer (FAU Erlangen-Nürnberg) Berlin / 39

2 Outline 1 Multivariate copula models for finance: A literature review 2 Some remarks to specific copulas 3 An empirical comparison of different copula models 4 Mis-specification of the marginals: Some first results Fischer (FAU Erlangen-Nürnberg) Berlin / 39

3 Multivariate copula models for finance: A literature review 1. D-copulas in finance: A review The elliptical world 1 Many practicioners do use the Gaussian copula though often not knowing that they use a copula at all 2 Paul Embrechts recommands the desert island copula Experts on HA-copulas and Pair-copulas sit next to me Details will be skipped Generalized multiplicative Archimedean (GMA) copulas Liebscher - Morillas - Fischer/Köck approaches Koehler-Symanowski-copula: KS(2) KS(3) KS(4) Fischer (FAU Erlangen-Nürnberg) Berlin / 39

4 Some remarks to specific copulas 2. Evolution of GMA copulas Plain Archimedean Copulas derived from a generator function ϕ which requires ϕ : [0, 1] [0, ), ϕ(1) = 0, ϕ completely monotonic of order d, i.e. ( 1) i ϕ (i) (x) 0. C(u 1,..., u d ) = ϕ 1 (ϕ(u 1 ) + ϕ(u 2 ) + + ϕ(u d )) Consider (multiplicative) version setting ϑ(x) exp( ϕ(x)) C(u 1,..., u d ) = ϑ 1 (ϑ(u 1 ) ϑ(u 2 )... ϑ(u d )) (Trivial) re-formulation using the independence copula Π: C(u 1,..., u d ) = ϑ 1 (Π(ϑ(u 1 ),..., ϑ(u d ))) Shortcomings: not very flexible. Fischer (FAU Erlangen-Nürnberg) Berlin / 39

5 2.1 Morillas copulas Some remarks to specific copulas Morillas, P. M. (2005). A method to obtain new copulas from a given one. Metrika 61, Replace Π by an arbitrary copula C : C(u 1,..., u d ) = ϑ 1 (Π(ϑ(u 1 ),..., ϑ(u d ))) C(u 1,..., u d ) = ϑ 1 (C (ϑ(u 1 ),..., ϑ(u d ))) which is a copula iff ϑ 1 is AM(d) ( absolutely monotonic ). Recall: u(x) is AM(n) u (i) 0 for i = 0,..., n. Fischer (FAU Erlangen-Nürnberg) Berlin / 39

6 Some remarks to specific copulas 2.2 Liebscher copulas Liebscher, E. (2006). Modelling and estimation of multivariate copulas. Working paper, University of Applied Sciences, Merseburg. One may also weight different functions ϑ 1,..., ϑ m : C(u 1,..., u d ) = Ψ 1 m ϑ j (u 1 )... ϑ j (u d )), where m j=1 ϑ j (0) = 0, ϑ j (1) = 1, ϑ j is monotonely increasing, Ψ is AM(d) and Ψ 1 m m ϑ j (v) = v. j=1 Fischer (FAU Erlangen-Nürnberg) Berlin / 39

7 Some remarks to specific copulas 2.3 Fischer-Köck copulas Fischer, M; Köck, C. (2007). Constructing and generalizing given multivariate copulas. Working paper, University of Erlangen-Nürnberg, Nürnberg. Replace independence copula by C1,..., C m: C(u 1,..., u d ) = Ψ 1 m Π(ϑ j (u 1 ),..., ϑ j (u d )) m C(u 1,..., u d ) = Ψ 1 m j=1 m Cj (ϑ j (u 1 ),..., ϑ j (u d )) j=1 Fischer (FAU Erlangen-Nürnberg) Berlin / 39

8 Some remarks to specific copulas 2.3 Fischer-Köck copulas: Some special cases C1 =... = C m = Π = Liebscher copulas. m = 1, C1 = Π, Ψ = ϑ 1 = Morillas copulas. m = 1, Ψ(x) = ϑ 1 (x) = x = C = C1. m = 2, Ψ(x) = ϑ 1 (x) = x = C = 0.5(C1 + C 2 ). Fischer (FAU Erlangen-Nürnberg) Berlin / 39

9 Some remarks to specific copulas 2.3 Fischer-Köck copulas A simple construction of Ψ: 1 Ψ(t) AM(d) Ψ(t) = p 0 + p 1 t + p 2 t (Feller, 1950). 2 Ψ(0) = 0 p 0 = 0. 3 Ψ(1) = 1 i p i = 1. Result: Ψ(t) corresponds (up to a some scaling) to a PGF of a discrete random variable X, denoted by G(t) G X (t) = E(t X ): Ψ(t) = G(t) G(0) G(1) G(0). Fischer (FAU Erlangen-Nürnberg) Berlin / 39

10 Some remarks to specific copulas 2.3 Fischer-Köck copulas Some examples for Ψ: Distribution Ψ(t) Parameter Binomial Geometric Poisson ((1 p)+pt) n (1 p) n 1 (1 p) n p (0, 1] (1 q)t 1 qt q (0, 1] e λt 1 e λ 1 λ > 0 Fischer (FAU Erlangen-Nürnberg) Berlin / 39

11 Some remarks to specific copulas 2.3 Fischer-Köck copulas Liebscher s construction of ϑ j : 1 Define ϑ j (t) = h j (Ψ 1 (t)) for j = 1,..., m 2 with an increasing function h j : [0, 1] [0, 1] satisfying h j (0) = 0, h j (1) = 1, m h j (v) = m v. j=1 Liebscher s proposal for h 1,..., h m : 1 h 1,..., h m 1 arbitrary s.t. h j (0) = 0, h j (1) = 1, h j increasing. 2 Setting h m = mv (h h m 1 ): Then h(0) = 0 and h(1) = 1. 3 h m is increasing if h 1 (t) h m 1 (t) < m. Fischer (FAU Erlangen-Nürnberg) Berlin / 39

12 Some remarks to specific copulas 2.3 Fischer-Köck copulas Possible construction of h j (j = 1,..., m 1): Assume that Y is a rv on (a, b) [0, 1] and has cdf F. Define F (u) F (0) h j (u) = F (1) F (0). Fischer (FAU Erlangen-Nürnberg) Berlin / 39

13 Some remarks to specific copulas 2.3 Koehler-Symanowski (KS) copulas Palmitesta, P.; Provasi, C. (2005). Aggregation of Dependent Risks using the Koehler-Symanowski Copula Function. Computational Economics 25, Koehler, K. J.; Symanowski, J. T. (1995). Constructing Multivariate Distributions with Specific Marginal Distributions. Journal of Multivariate Distributions 55, KS copulas are simply constructed by combining several independent Gamma variables. The dependence structure is characterized by different (bivariate, trivariate,...) association parameters. Fischer (FAU Erlangen-Nürnberg) Berlin / 39

14 Some remarks to specific copulas 2.3 KS(2) copulas for d = 4 Restriction to bivariate association terms. Assume that Y 1,..., Y 4 iid Exp(λ = 1), G 11, G 12, G 13, G 14, G 22, G 23, G 24, G 33, G 34, G 44 iid Γ(α ij, 1) Define the KS(2)-copula as copula of the uniform random variables U 1 = 1 + Y 1 G 11 + G 12 + G 13 + G 14 (α11 +α 12 +α 13 +α 14 ) U 2 = 1 + U 3 = 1 + U 4 = 1 + Y 2 G 12 + G 22 + G 23 + G 24 Y 3 G 13 + G 23 + G 33 + G 34 Y 4 G 14 + G 24 + G 34 + G 44 (α12 +α 22 +α 23 +α 24 ) (α13 +α 23 +α 33 +α 34 ) (α14 +α 24 +α 34 +α 44 ) Fischer (FAU Erlangen-Nürnberg) Berlin / 39

15 Some remarks to specific copulas 2.3 KS(3) copulas for d = 4 Restriction to bi- and trivariate association terms. Assume that iid Y 1,..., Y 4 Exp(λ = 1), iid G 1, G 2, G 3, G 4, G 12, G 13, G 14, G 23, G 24, G 34, G 123, G 134, G 124, G 234 Γ(α ijk, 1) {z } Define the KS(3)-copula as copula of the uniform random variables U 1 = 1 + Y 1 G 1 + G 12 + G 13 + G 14 + G G G 134 α1. U 2 = 1 + U 3 = 1 + U 4 = 1 + Y 2 G 2 + G 12 + G 23 + G 24 + G G G 123 Y 3 G 3 + G 13 + G 23 + G 34 + G G G 234 Y 4 G 4 + G 14 + G 24 + G 34 + G G G 234 α2. α3. α4. Fischer (FAU Erlangen-Nürnberg) Berlin / 39

16 Some remarks to specific copulas 2.3 Koehler-Symanowski (KS) copulas Application of multivariate variable transformation reveals 1 the copula in closed form, 2 the copula density in closed form. Several kinds of asymmetries and tail dependence behaviour can be rebuild. The Clayton copula is included as special case. Whereas Palmitesta & Provasi (2005) only use the KS(2)-copula, we also implemented the KS(3) and the KS(4)-copula! Fischer (FAU Erlangen-Nürnberg) Berlin / 39

17 An empirical comparison of different copula models 3. Results from an empirical analysis There are many recent construction schemes of multivariate copulas. Up to now we found no comprehensive comparison between these schemes. Fischer, M.; Köck, C.; Schlüter, S.; Weigert, F. (2007). An empirical analysis of multivariate copula models. Submitted to Quantitative Finance (in revision). Empirical results for stock/fx-markets/commodity markets. About 30 4-copula models have been fitted to each data set. Different goodness-of-fit measures have been calculated. Fischer (FAU Erlangen-Nürnberg) Berlin / 39

18 An empirical comparison of different copula models 3.1 The stock market data set Figure: German stock prices and stock returns ( ). N Stocks µ s 2 S K LB(10) LM(10) 3486 HVB BMW Allianz MunichRe Fischer (FAU Erlangen-Nürnberg) Berlin / 39

19 An empirical comparison of different copula models 3.2 Data filtering 1 Firstly, calculate the (percentual) log-returns R t = 100(log P t log P t 1 ), t = 2,..., N. 2 To account for possible time-dependencies, we also fitted univariate ARMA-GARCH models of the form R t = µ + γ 1 R t γ k R t k + h t ɛ t h 2 t = α 0 + α 1 R 2 t α 1 R 2 t p + β 1 h 2 t β q h 2 t q 3 Finally, the standardized residuals ɛ t are transformed via ecdf into uniform variables: U t = F N ( ɛ t ). 4 The resulting pseudo observations are used to estimate the unknown parameters. Fischer (FAU Erlangen-Nürnberg) Berlin / 39

20 An empirical comparison of different copula models 3.3 Semi-parametric parameter estimation ML estimation is used to estimate the unknown parameters of the copula models. Number of unknown parameters varies between 1 and 15. Estimation time varies between 1 second and 3 days. Software: Matlab 7.0, nlminb from OPTIMIZATION Toolbox (requires no gradient). Fischer (FAU Erlangen-Nürnberg) Berlin / 39

21 An empirical comparison of different copula models 3.4 Goodness-of-fit measures 1 Loglikelihood-value ll. 2 Bayesian information criterion (BIC) to account for the number of parameters. 3 Applying the Rosenblatt s transformation to the pseudo-observations should provide independence observation: T 1 (u) = u T 2 (u) = C U2 U 1 (u) = T 3 (u) = C U3 U 2,U 1 (u) = T 4 (u) = C U4 U 3,U 2,U 1 (u) = u 1 C(u 1, u, 1, 1) 2 u 1 u 2 C(u 1, u 2, u, 1) 2 u 1 u 2 C(u 1, u 2, 1, 1) 3 u 1 u 2 u 3 C(u 1, u 2, u 3, u) 3 u 1 u 2 u 3 C(u 1, u 2, u 3, 1) We use a multivariate version of Spearman s ρ ( ρ S,4 ) to quantify independence (see Schmid & Schmidt, 2006) Fischer (FAU Erlangen-Nürnberg) Berlin / 39

22 An empirical comparison of different copula models 3.4 Goodness-of-fit measures II Following Breymann et al. (2003), transform Z j = (Z j1,..., Z jd ) : χ j = d Φ 1 (Z ji ) 2, j = 1,..., N, i=1 to χ 2 (d)-observations and quantify the distance between the theoretical (χ 2 (d)) and the empirical cdf: Kolmogorov-Smirnov measure KS = N max j=1,...,n Averaged Kolmogorov-Smirnov measure L 2 measure AKS = 1 N N j=1 Fχ2 (d)(χ j ) F N,χ (χ j ), F χ2 (d)(χ j ) F N,χ (χ j ). Fischer (FAU Erlangen-Nürnberg) Berlin / 39

23 An empirical comparison of different copula models 3.5 (Selected) Results from an empirical analysis Copula ll BIC KS AKS L 2 ρ S,4 CLA rogum NORM T PC-T PC-CLA PC-roGUM KS(2) KS(4) MORILLAS-CLA LIEBSCHER-best FK-CLAY HA-CLA HA-GUM Fischer (FAU Erlangen-Nürnberg) Berlin / 39

24 An empirical comparison of different copula models 3.5 Main empirical findings (Summary) At least for the 4-dimensional stock return data set... 1 Student-t copula is still very dominant, BUT pair-copulas based on bivariate Student-t copulas show similar fit. 2 Plain Archimedean copulas offer only a poor fit. 3 Considering GMA s (Morillas, Liebscher, Fischer/Köck) clearly improves the fit. 4 Pair-copula approach dominates the Hierarchical Archimedean approach. 5 The fit of the KS(2)-copula (used by Palmitesta & Provasi, 2005) can be significantly improved if the KS(4)-copula is considered instead. Fischer (FAU Erlangen-Nürnberg) Berlin / 39

25 An empirical comparison of different copula models 3.6 Some remarks The results for the other two data sets (fx-markets and metals) are very similar. The dominance of the pair-copula against the hierarchical approach goes in line with Berg, H.; Aas, K. (2007) Models for construction of multivariate dependence. Working Paper. Our intension was not primarily to test ( goodness-of-fit test ) whether the data stem from a specific copula model. For this purpose we refer to Genest, C.; Remillard B.; Beaudoin, D: (2007). Goodness-of-fit tests for copulas: A review and a power study. Insurance, Mathematics and Economics (to appear). Fischer (FAU Erlangen-Nürnberg) Berlin / 39

26 Mis-specification of the marginals: Some first results 4. What about the marginal distribution? Usually, marginal distributions are unknown. Some of the authors (e.g. Palmitesta & Provasi, 2005 or Berg & Aas, 2007) use parametric specifications (e.g. skew t, skew GED, NIG). Other authors (e.g. Fischer. Köck, Schlüter, Weigert, 2007) consider the marginals as nuisance parameters and apply the empirical cumulative distribution (ecdf) function. Final question: How much does mis-specification of marginal effect the estimator of the copula parameter? Fischer (FAU Erlangen-Nürnberg) Berlin / 39

27 Mis-specification of the marginals: Some first results 4.1 Recent results from a simulation study We present first results from a recent working paper Fischer, M.; Köck, C. (2008) Mis-specification of marginals in the multivariate case - results from a simulation study. Working Paper. The basic setup of our simulation study (4-dimensions): 1 In a first step, data are simulated from a true model (different copulas with skewed Student t margins). 2 Secondly, 10 estimation methods are applied to the simulated data set. 3 Thirdly, different goodness-of-fit measures are calculated. Fischer (FAU Erlangen-Nürnberg) Berlin / 39

28 Mis-specification of the marginals: Some first results 4.2 Estimation procedures One-step maximum-likelihood estimation (EML) 1 Skewed Student-t (Hansen, 1994) 2 Skewed GED (Theodossiou, 2001) 3 Student t 4 Normal Two-step estimation (IFM and CML) 1 Skewed Student-t (Hansen, 1994) 2 Skewed GED (Theodossiou, 2001) 3 Student t 4 Normal 5 Empirical distribution function 6 Empirical distribution function with GPD tails (McNeil & Frey, 2000) Fischer (FAU Erlangen-Nürnberg) Berlin / 39

29 Mis-specification of the marginals: Some first results 4.2 Estimation procedures Introducing skewness by splitting up the scale parameter Skewed Student-t (Hansen, 1994): ( bc f SGT (x) = bc η 2 ( η 2 ( x ) 2 (η+1)/2 1 λ) x 0 ( ) 2 (η+1)/2 x 1+λ) x > 0, Skewed GED (Theodossiou, 2001): { c exp { w1 x f SGED (x) = κ } x 0 c exp { w 2 x κ } x > 0, Fischer (FAU Erlangen-Nürnberg) Berlin / 39

30 Mis-specification of the marginals: Some first results 4.2 The ECDF-GPD estimator was proposed by McNeil & Frey (2000). The aim is to improve the fit in the tails of the distribution. Idea: Use the Generalized Pareto distribution (GDP) for the tail area and the empirical cumulative distribution function (ECDF) otherwise: ( N k N 1 + ξ ) x x (k) 1/ ξ b 1 1 if x < x bβ (k), 1 F 1 (x) = N N j=1 1(x j x) if x (k) x x (N k), ( 1 k N 1 + ξ ) x x (N k) 1/ ξ b 2 2 if x > x bβ (N k). 2 Note that β i and ξ i of the GPD have to be estimated in advance. Rule of thumb: k = N 2/3. Fischer (FAU Erlangen-Nürnberg) Berlin / 39

31 Mis-specification of the marginals: Some first results 4.3 A typical data sample from a SGT... high kurtosis, skewness to the left/right. Fischer (FAU Erlangen-Nürnberg) Berlin / 39

32 Mis-specification of the marginals: Some first results 4.4 How accurate are the estimators? We restrict ourselves to 1 BIAS(ˆθ; θ) = θ E(ˆθ) ) 2 MSE(ˆθ; θ) = E ((ˆθ θ) 2 3 MESE(ˆθ; θ) = MSE(ˆθ;θ) 1 σ(ˆθ) Note that if ˆθ is unbiased, RMSE converges against σ(ˆθ) for N, hence MESE converges to zero. Fischer (FAU Erlangen-Nürnberg) Berlin / 39

33 Mis-specification of the marginals: Some first results 4.5. Results for the Clayton copula (REP= 1000) One-step estimation Two-step estimation True SGED T NV SGT SGED T NV ECDF ECDF-GPD BIAS MSE MESE Table: Sample size N = 750. One-step estimation Two-step estimation True SGED T NV SGT SGED T NV ECDF ECDF-GPD BIAS MSE MESE Table: Sample size N = One-step estimation Two-step estimation True SGED T NV SGT SGED T NV ECDF ECDF-GPD BIAS MSE MESE Table: Sample size N = Fischer (FAU Erlangen-Nürnberg) Berlin / 39

34 Mis-specification of the marginals: Some first results 4.5 Results for the Student-t copula (REP= 1000) One-step estimation Two-step estimation BIAS True SGED T NV SGT SGED T NV ECDF ECDF-GPD ρ ρ ρ ρ ρ ρ ν Table: N = 1000 One-step estimation Two-step estimation MSE True SGED T NV SGT SGED T NV ECDF ECDF-GPD ρ ρ ρ ρ ρ ρ ν Table: N = 1000 Fischer (FAU Erlangen-Nürnberg) Berlin / 39

35 Mis-specification of the marginals: Some first results 4.5 Results for the Student-t copula (REP= 1000) One-step estimation Two-step estimation MESE True SGED T NV SGT SGED T NV ECDF ECDF-GPD ρ ρ ρ ρ ρ ρ ν Table: N = 1000 Fischer (FAU Erlangen-Nürnberg) Berlin / 39

36 Mis-specification of the marginals: Some first results 4.5 Results for the Student-t copula (REP= 1000) One-step estimation Two-step estimation BIAS True SGED T NV SGT SGED T NV ECDF ECDF-GPD ρ ρ ρ ρ ρ ρ ν Table: N = 3000 One-step estimation Two-step estimation MSE True SGED T NV SGT SGED T NV ECDF ECDF-GPD ρ ρ ρ ρ ρ ρ ν Table: N = 3000 Fischer (FAU Erlangen-Nürnberg) Berlin / 39

37 Mis-specification of the marginals: Some first results 4.6 What are the practical implications? 1 If you know the true margins... use them!!! 2 If you have many data and are not interested in the marginal distribution... use the ECDF-GPD estimator rather than the ECDF estimator. 3 If you have not so many data and you are not interested in the marginal distribution... use the ECDF estimator. 4 The choice of the parametric marginal distribution is not so important if the model is able to capture both skewness and kurtosis. 5 For large samples, however, the ECDF-GPD estimator seems to be a better choice. Fischer (FAU Erlangen-Nürnberg) Berlin / 39

38 Mis-specification of the marginals: Some first results Summary and conclusion Several new construction schemes appeared in the recent literature. Some of them seem to be promising, at least for moderate dimensions (PC, HA, FK). We still observe the curse of dimension. The desert-island copula still plays a dominant role. If there are sufficient (financial) data, use the ECDF-GPD estimator (which is not so prominent in the copula literature). Fischer (FAU Erlangen-Nürnberg) Berlin / 39

39 Mis-specification of the marginals: Some first results THANK YOU FOR YOUR ATTENTION & FOR ORGANIZING THIS COPULA-WORKSHOP Fischer (FAU Erlangen-Nürnberg) Berlin / 39