Bayesian Inference for Pair-copula Constructions of Multiple Dependence

Transcription

1 Bayesian Inference for Pair-copula Constructions of Multiple Dependence Claudia Czado and Aleksey Min Technische Universität München December 7, 2007

2 Overview 1 Introduction 2 Pair-copula constructions (PCC) 3 Bayesian Analysis of PCC s 4 Application: Financial Returns 5 Bayesian Model Selection among PCC s 6 Euro Swap Rates 7 Summary and Outlook

3 Introduction Introduction Bedford and Cooke (2001) and Bedford and Cooke (2002) gave a probabilistic construction of multivariate distributions based on simple building blocks called pair-copulas. See also Kurowicka and Cooke (2006). Extends work by Joe (1996). Aas, Czado, Frigessi, and Bakken (2007) used the PCC construction to construct flexible multivariate copulas based on pair-copulas such as bivariate Gaussian, t-, Gumbel and Clayton copulas

4 Introduction Introduction Parameters are estimated by maximum likelihood (ML) and PCC s are sucessfully applied to financial return data Berg and Aas (2007) give further example of PCC s and developed an R-package for fitting PCC s using MLE The Fisher information is difficult to handle analytically, so therefore estimated standard errors for parameter estimates are so far absent Here we follow a Bayesian approach where such interval estimates are easy to establish

5 Introduction Multivariate Distributions Consider n random variables X = (X 1,...,X n ) with joint density f (x 1,...,x n ) and marginal densities f i (x i ),i = 1,...,n joint cdf F(x 1,...,x n ) and marginal cdf s F i (x i ),i = 1,...,n factorization f (x 1,..., x n) = f n(x n) f (x n 1 x n) f (x n 2 x n 1, x n)... f (x 1 x 2,...,x n) (1)

6 Introduction Copula A copula is a multivariate distribution on [0,1] n with uniformly distributed marginals. copula cdf C(u 1,...,u n ) copula density c(u 1,...,u n ) Using Sklar s Theorem (1959) we have for absolutely continuous distributions with continuous marginal cdf s f (x 1,...,x n ) = c 12 n (F 1 (x 1 ),... F n (x n )) f 1 (x 1 ) f n (x n ) (2) for some n-variate copula density c 12 n ( ).

7 Pair-copula constructions (PCC) Pair-copula constructions (PCC) n = 3 f 1 23 (x 1 x 2,x 3 ) = f 12 3(x 1,x 2 x 3 ) f 2 3 (x 2 x 3 ) = c 12 3 (F 1 3 (x 1 x 3 ),F 2 3 (x 2 x 3 )) f 1 3 (x 1 x 3 ) general n For v = (v 1,...,v d ) and any j = 1,...,d f (x v) = c xvj v j (F(x v j ),F(v j v j )) f (x v j ) v j = (v 1,...,v j 1,v j+1,...,v d ) c xvj v j ( ) = bivariate copula density Combining this decomposition of the conditional distribution with the factorization (1), we derive at a decomposition of f (x 1,...,x n ) that only consist of pair-copulae. We call this a pair-copula construction (PCC).

8 Pair-copula constructions (PCC) Conditional cdf s Univariate v: Since f (x v) = c xv (F x (x),f v (v))f x (x) we have F(x v) = x c xv (F x (u),f v (v))f x (u)du = C xv(f x (x),f v (v)) F v (v) General v: Under regularity conditions Joe (1996) showed that F(x v) = C x,v j v j (F(x v j ),F(v j v j )), F(v j v j )

9 Pair-copula constructions (PCC) Vines For high-dimensional distributions there are many possible pair-copula constructions. Bedford and Cooke (2001) introduced a graphical model called regular vine to help organize them. The class of regular vines is large and embraces a large number of possible PCC s. We concentrate on two special cases (Kurowicka and Cooke 2004): D-vine Canonical Vine

10 Pair-copula constructions (PCC) Vines(2) An n-dimensional vine is represented by n-1 trees. Tree j has n + 1 j nodes and n j edges. Each edge corresponds to a pair-copula density. Edges in tree j become nodes in tree j + 1. Two nodes in tree j + 1 are joined by an edge if the corresponding edges in tree j share a node. The complete decomposition is defined by the n(n 1) 2 edges (i.e. pair copula densities) and the marginal densities.

11 Pair-copula constructions (PCC) Canonical and D-vines Canonical vine: A regular vine for which each tree has a unique node that is connected to n j edges. D-vine: A regular vine for which no node in any tree is connected to more than two edges Abbreviations: c ij v := c ij v (F i v (x i x v ),F j v (x j x v )) f j := f j (x j ) f v := f v (x v )

12 Pair-copula constructions (PCC) Five dimensional canonical vine f = f 1 f 2 f 3 f 4 f 5 c 12 c 13 c 14 c 15 c 23 1 c 24 1 c 25 1 c c c T T T T 4

13 Pair-copula constructions (PCC) Five dimensional D-vine f = f 1 f 2 f 3 f 4 f 5 c 12 c 23 c 34 c 45 c 13 2 c 24 3 c 35 4 c c c T T T T 4

14 Pair-copula constructions (PCC) General density expressions Canonical vine density D-vine density n k=1 n k=1 n 1 f k n j c j,j+i 1,...,j 1 j=1 i=1 n 1 n j f k c i,i+j i+1,...,i+j 1 j=1 i=1 where index j identifies the trees, while i runs over the edges in each tree.

15 Bayesian Analysis of PCC s Bayesian Analysis of PCC s with t-copula pairs Assume bivariate t-copula for each pair copula term, i.e. θ c(u 1, u 2 θ) = where = (ρ, ν) ρ = correlation, ν = df Γ( ν+2 2 )/Γ( ν 2 ) νπt ν (x 1 )t ν (x 2 ) 1 ρ 2 ( 1 + x2 1 + x2 2 2ρ x 1 x 2 ν(1 ρ 2 ) x 1 = tν 1 1) x 2 = tν 1 (u 2 ) t ν ( ) = pdf of univariate t ν distribution tν 1 ( ) = quantile function of t ν distribution ) ν+2 2,

16 Bayesian Analysis of PCC s Bayesian Analysis of PCC s with t-copula pairs For the PCC s we assume a regular vine structure with edge indices denoted by s := ij v for s S. Let θ s = (ρ s,ν s ) the corresponding parameters of the pair copula associated with edge s. Let θ = {θ s,s S} denote the vector of all parameters of the PCC with t-copula pairs We assume uniform priors for ρ s on ( 1,1) and for ν s on (1, U) for each s and prior independence among all parameters Inference is based on the posterior distribution given by p(θ data) = f (data θ) p(θ) f (data θ) p(θ)dθ where f (data θ) is the likelihood of the PCC and p(θ) the prior for θ

17 Bayesian Analysis of PCC s Bayesian Analysis of PCC s with t-copula pairs Since θ is high dimensional the posterior is not analytically tractable. Posterior distribution is estimated using Markov Chain Monte Carlo (MCMC) methods For each univariate parameter Metropolis-Hastings (MH) updates with symmetric normal random walk proposals are used Proposal variances are determined by pilot runs to achieve acceptance rates between 20% and 80%. Posterior mean, mode, median and density are estimated using the MCMC iterates. Credible intervals are estimated by empirical quantiles of the MCMC iterates.

18 Application: Financial Returns Application: Financial Returns Tail dependence properties are important in finance. n-dimensional Student s t-copula has been often used for modeling financial returns, however it only has a single parameter for tail dependence. Pair-copula constructions allows for multiple parameters for modeling tail dependence.

19 Application: Financial Returns Data set Daily data from Jan. 4, 1999 until July 8, 2003 for T = Norwegian stock index (TOTX) M = MSCI world stock index B = Norwegian bond index (BRIX) S = SSBWG hedged bond index Standardized residuals of a AR(1)-GARCH(1,1) model for the log return x i,t x i,t = c i + α i x i,t 1 + σ i,t z i,t, E[z t,i ] = 0 and Var[z t,i ] = 1, σ 2 i,t = a i,0 + a i ǫ 2 i,t 1 + b i σ 2 i,t 1 where ǫ i,t 1 = σ i,t z i,t are found to be independent and are transformed to uniform margins

20 Application: Financial Returns Data set Fitted degree of freedom for a t-copula to each pair Between M T B S M T 12.60

21 Application: Financial Returns Which PCC? Strongest dependence between S and M, M and T and T and B, which will be used in the top tree of a D-vine SM MT TB S M T B ST M MB T SM MT TB ST M SB MT MB T

22 Application: Financial Returns Bayesian Inference of the Norwegian return data For each pair copula we assume a bivariate t-copula with correlation ρ and ν degree of freedom. For each ρ we assume uniform(-1,1) prior and for ν a uniform(1,1000) prior. Further prior independence between all parameters are assumed. For the likelihood we assume the following PCC c(x S, x M, x T, x B ) = c SM (F(x S ), F(x M )) c MT (F(x M ), F(x T )) c TB (F(x T ), F(x B )) c ST M (F(x S x M ), F(x T x M )) c MB T (F(x M x T ), F(x B x T )) c SB MT (F(x S x M, x T ), F(x B x M, x T ))

23 Application: Financial Returns Bayesian Inference of the Norwegian return data For the MH updates in the MCMC algorithm we used symmetric random walk proposals normal proposals with variances determined by pilot runs to achieve acceptance rates between 25% and 80% MCMC iterations were run MLE estimates were used as starting values Implemented using Daniel Berg s R-package

24 Application: Financial Returns Trace Plots of MCMC iterations ρ SM ν SM ρ MT ν MT ρ Iterations ν Iterations ρ Iterations ν Iterations ρ TB ν TB ρ ST M ν ST M ρ Iterations ν Iterations ρ Iterations ρ MB T ν MB T ρ SB MT ν SB MT ρ ρ ν Iterations ν ν Iterations Iterations Iterations Iterations

25 Application: Financial Returns Autocorrelations of the MCMC iterations ρ SM ν SM ρ MT ν MT ACF ACF ACF ACF Lag Lag Lag Lag ρ TB ν TB ρ ST M ν ST M ACF ACF ACF ACF Lag Lag Lag Lag ρ MB T ν MB T ρ SB MT ν SB MT ACF ACF ACF ACF Lag Lag Lag Lag For the further analysis we used burnin 1000 iteration and only every 20th iteration.

26 Application: Financial Returns Estimated Posterior Densities ρ SM ν SM ρ MT ν MT Density Density Density Density ρ TB ν TB ρ ST M ν ST M Density Density 4e 04 1e Density Density 4e 04 1e ρ MB T ν MB T ρ SB MT ν SB MT Density Density 5e 04 9e Density Density

27 Application: Financial Returns Summary statistics for thinned MCMC 2.5% 50% 97.5% Est. Post. Est. Post. MLE Quantile Quantile Quantile Mean Mode ρ SM ν SM ρ MT ν MT ρ TB ν TB ρ ST M ν ST M ρ MB T ν MB T ρ SB MT ν SB MT red corresponds to uncorrelatedness magenta corresponds to near normality

28 Application: Financial Returns Model fit: Kendall s τ Kendall s τ measures dependence and is defined by τ := P((X 1 X 2 )(Y 1 Y 2 ) > 0) P((X 1 X 2 )(Y 1 Y 2 ) < 0) where (X 1,Y 1 ) and (X 2,Y 2 ) are i.i.d. τ is invariant under monotone transformations Relationship between τ and ρ for bivariate t-distributions τ = 2 π arcsin(ρ) Estimated posterior summaries of τ 2.5% 50% 97.5% Est. Post. Est. Post Empirical Quantile Quantile Quantile Mean Mode τ τ SM τ MT τ TB

29 Application: Financial Returns Model fit: Tail dependence Lower and upper tail dependence: λ l := lim P(X F 1 1 v 0 X (v) Y FY λ u := lim P(X F 1 1 v 1 X (v) Y FY For t distributions we have λ := λ l = λ u = 2t ν+1 ( ) 1 ρ ν ρ Estimated posterior summaries of λ 2.5% 50% 97.5% Est. Post. Est. Post Empirical Quantile Quantile Quantile Mean Mode λ l λ SM λ MT λ TB

30 Application: Financial Returns Model fit: Lambda function The λ function is defined as a shifted copula distribution function given by λ(z,θ) := z K(z,θ) K(z,θ) := P(C(u 1,u 2,θ) z) Here θ denotes the parameters of the bivariate copula pair. For the conditional pairs ST M,MB T and SB MT an empirical estimate of λ(z,θ) is based on transformed data using posterior mode estimates

31 Application: Financial Returns Estimated pointwise posterior mode λ(z, θ) Pair ST Pair MT Pair TB z z z Pair ST M Pair MB T Pair SB MT λ(z) λ(z) λ(z) λ(z) λ(z) λ(z) z z z red = 95% CI, black = est. posterior mode, green = empirical

32 Bayesian Model Selection among PCC s Bayesian Model Selection among PCC s Want to select among K models M 1,,M K, where Model M k has parameter θ k and compare them on the basis of posterior model probabilities given by P( Model M k data),k = 1,,K We distinguish two situations K is small and it is feasible timewise to fit all models (Congdon (2006) and Scott (2002)) K is large and we want only to fit models which are probable using reversible jump MCMC (RJMCMC) (Green (1995))

33 Bayesian Model Selection among PCC s Bayesian Model Selection : K small Congdon (2006) made the following assumptions The distribution of the data is independent of {θ j k } given M k Independence among θ k s given Model M and showed that Posterior distributions of θ k are independent given M = M k and can be sampled individually and used that P(M = M k data, θ) P(data θ k, M = M k )P(θ M = M k )P(M = M k ) (3)

34 Bayesian Model Selection among PCC s Bayesian Model Selection : K small Assume that K independently MCMC runs result in M 1 : θ (r) 1,r = 1,,R. M K : θ (r) K,r = 1,,R approximating p(θ 1 data). p(θ K data) Using {θ (r) := (θ (r) 1,,θ(r) K ),r = 1,,R} P(M data) = P(M θ, data)p(θ data)dθ is approximated by P(M data) := 1 R R r=1 P(M θ(r),data).

35 Bayesian Model Selection among PCC s Bayesian Model Selection : K small Using (3) we can estimate P(M = M k data,θ (r) ) by w (r) k := G (r) k K j=1 G(r) j, where G (r) k L (r) k := exp(l (r) k L max) (r) ( ) := log P(data θ (r) k,m = M k)p(θ (r) k M = M k)p(m = M k ) L (r) max := max k=1,,k L(r) k Therefore 1 R R r=1 w (r) k estimates P(M = M k data).

36 Bayesian Model Selection among PCC s K small: Model selection for financial return data Since zero in 95% credible interval for ρ MB T and the posterior mode of ρ ST M and ρ MB T small we want to use Congdon (2006) s method if these copula pairs needed Model PCC Formula P(Mk data) M 1 : with all pairs c SM c MT c TB c ST M c MB T c SB MT M 2 : without c SM c MT c TB c ST M c SB MT c MB T M 3 : without c SM c MT c TB c SB MT c MB T and c ST M M 4 : without c SM c MT c SB MT c MB T,c ST M and c TB

37 Bayesian Model Selection among PCC s K small: Model selection for financial return data Many of the posterior estimates of the degree of freedom are large, so want to check if we can use for these pairs a Gaussian copula Model PCC P(Mk data) M 1 : All pairs are t copulas M 2 : First pair c SM is t copula all other pairs are Gaussian copulas M 3 : All pairs are Gaussian copulas Only for c SM a t-copula is needed.

38 Bayesian Model Selection among PCC s Bayesian Model Selection: K large (RJMCMC) If a single pair copula type is used in a PCC of dimension n, then all other possible PCC s of dimension n provide a factorization of the same joint density So we want to find reduced PCC s which best fits the data, i.e. let the data discover conditional independence conditions. Identifiability: Use only one decomposition n = 3: Full PCC s: c 12 c 23 c 13 2 Reduced by 1 pair-copula: c 12 c 23,c 12 c 13 2,c 23 c 13 2 Reduced by 2 pair copulas: c 12,c 23,c 13 2

39 Bayesian Model Selection among PCC s Bayesian framework of PCC s with n = 3 Each of the 7 different PCC s with n 3 is identified by a model index m = (m 1,m 2,m 3 ) = (i 1 j 1,i 2 j 2,i 3 j 3 k 3 ). Example: c 12 c 23 1 corresponds to m = (12,00,23 1). Each PCC model m has parameter vector θ m = (θ m1,θ m2,θ m3 ). Example: Model m = (12, 23, 00 0) has parameter vector θ m = (θ 12,θ 23,θ 00 0 ) = (θ 12,θ 23 ). Goal is to estimate the best fitting model m and the corresponding parameter vector θ m using a Bayesian approach, i.e. the model m and θ m are considered random quantities. Inference about m and θ m is done via the joint posterior distribution of (m,θ m ).

40 Bayesian Model Selection among PCC s MCMC algorithm for PCC s with n = 3 Problems: Want to facilitate estimation without having to fit all models Joint posterior is not analytically tractable Approach: Construct a Markov Chain Monte Carlo (MCMC) algorithm which simultaneously estimates m and θ m. Requirement: Need to accommodate varying model dimension, i.e. construct MCMC iterates θ m r = (θ r m r 1,θr m r 2,θr m r 3 ) where mr is the current model at iteration r. Solution: Reversible jump (RJ) MCMC proposed by (Green 1995)

41 Bayesian Model Selection among PCC s General RJ MCMC (1) algorithm stays in current model using a Metropolis Hastings (MH) step algorithm moves to a larger model using a MH step (birth) algorithm moves to a smaller model using a MH step (death)

42 Bayesian Model Selection among PCC s General RJ MCMC (2) Model M 1 M 2 Parameter θ (1) R d 1 θ (2) R d 2 d 1 < d 2 Proposal η (1) ϕ 1 ( ) η (2) ϕ 2 ( ) η (1) R a 1 η (2) R a 2 Dimension matching d 1 + a 1 = d 2 + a ( ) ( 2 ) θ (1) θ (2) Bijection η (1) η (2)

43 Bayesian Model Selection among PCC s Acceptance probability for MH step from M 1 to M 2 α(θ (1),θ (2) ) = min {1,A}, where A := p(2,θ(2) y) p(1,θ (1) y) p2 1 ϕ2(η (2) ) p 1 2 ϕ 1 (η (1) ) ) (θ (2),η (2) ) (θ (1),η (1) p(k,θ (k) y) = joint posterior density of M k and θ (k) p i j = prior switching probability from M i to M j ϕ k (η (k) ) = proposal distribution of η (k) (suitably chosen) θ (2),η (2) θ (1) = Jacobian of bijection,η (1) For moves from M 2 to M 1 we use A 1.

44 Bayesian Model Selection among PCC s RJ MCMC moves for selecting PCC s (n=3) Notation Meaning Example B0D0 stay move c 12 c 12 B1D0 birth of 1 factor c 12 c 12 c 23 D1B0 death of 1 factor c 12 c 23 c 12

45 Bayesian Model Selection among PCC s Graph of all PCC s for n = 3 a = 1 4, b = 1 3 b a c12c23c13 2 a a a a a a a c12c23 c12c13 2 c23c13 2 a a a a a a a b b b b b b c12 c23 c13 2 b b a With probabilities a or b the chain moves to other model or stays in the current model

46 Bayesian Model Selection among PCC s Acceptance probability for B1D0 move Actual state θ o := θ o m o mo = (m1 o,mo 2,mo 3 ) Proposed state θ p := θ p m p mp = (m p 1,mp 2,mp 3 ) birth index {ms p } := m p \m o stay indices {mv o,mo w } := mp m o Example: ( θ p ) 12,θp 23 (θ p 12,θp 23,θp 13 2 ) Bijection: θ p := θ p m p s θ p m o v θ p m o w = η o m p s θ o m o v θ o m o w =: ( η o m p s θ o ) Here η o ms p is bivariate normal centered at θ m p s,last covariance matrix with specified

47 Bayesian Model Selection among PCC s MLE for SMT- data Parameter MLE ρ SM ρ MT 0.47 ρ ST M ν SM 4.22 ν MT ν ST M 300 log.likelih Is the correlation ρ ST M = 0.11 significantly negligible?

48 Bayesian Model Selection among PCC s Model notation c SM c MT c ST M M 1 c SM c MT M 2 c SM c ST M M 3 c MT c ST M M 4 c SM M 5 c MT M 6 c ST M M 7 Would Model M 2 now be preferred to Model M 1?

49 Bayesian Model Selection among PCC s Model M 1 was visited 2872 times out of Model M 2 was visited 7128 times out of Financial returns: Statistics of visited models c ST M c ST M c SM c SM c MT c ST M c MT c ST M c SM c ST M c SM c ST M c SM c MT c SM c MT c SM c MT c ST M c SM c MT c ST M Iterations First 10 Iterations

50 Bayesian Model Selection among PCC s Summary statistics for visited Models Model M 2 : c SM c MT 2.5% 50% 97.5% Est. Post. Est. Post. MLE Quantile Quantile Quantile Mean Mode ρ SM ν SM ρ MT ν MT Model M 1 : c SM c MT c ST M 2.5% 50% 97.5% Est. Post. Est. Post. MLE Quantile Quantile Quantile Mean Mode ρ SM ν SM ρ MT ν MT ρ ST M ν ST M

51 Bayesian Model Selection among PCC s Posterior model probabilities RJMCMC Model PCC Formula P(Mk data) M 1 with all pairs c SM c MT c ST M 0.29 M 2 without c ST M c SM c MT 0.71 M 3 without c MT c SM c ST M 0 M 4 without c SM c MT c ST M 0 M 5 without c MT and c ST M c SM 0 M 6 without c SM and c ST M c MT 0 M 7 without c SM and c MT c ST M 0 Congdon Model PCC Formula P(Mk data) M 1 : with all pairs c SM c MT c ST M 0.22 M 2 : without c ST M c SM c MT 0.78

52 Euro Swap Rates Euro Swap Rates daily swap rates with 2,3,5,7 and 10 year maturity quoted in Euro s considered period is Dec 7, 1988 until June 21, 2001 serial marginal correlation removed with an ARMA(1,1)-GARCH(1,1) model and standardized residuals transformed empirically to data with uniform margins Notation: Swap2, Swap3, Swap5, Swap7 and Swap10 or abbreviated S 1,...,S 5

53 Euro Swap Rates Dependence structure of Euro Swap Rates Swap Swap3 Swap Swap7 Swap

54 Euro Swap Rates Estimated df s of bivariate t-copula margins for swap rates Swap3 Swap5 Swap7 Swap10 Swap Swap Swap Swap7 2.07

55 Euro Swap Rates Chosen D-vine structure for swap rates S 1 S 1 S 2 S2 S 2 S 3 S3 S 3 S 4 S4 S 4 S 5 S5 S 1 S 2 S 1 S 3 S 2 S 2 S 3 S 2 S 4 S 3 S 3 S 4 S 3 S 5 S 4 S 4 S 5 S 1 S 3 S 2 S 1 S 4 S 2 S 3 S 2 S 4 S 3 S 2 S 5 S 3 S 4 S 3 S 5 S 4 S 1 S 4 S 2 S 3 S 1 S 5 S 2 S 3 S 4 S 2 S 5 S 3 S 4

56 Euro Swap Rates Trace of MCMC iterations for swap rates ρ S1 S 2 ν S1 S 2 ρ S2 S 3 ν S2 S 3 ρ S3 S 4 ρ Iterations ν Iterations ρ Iterations ν Iterations ρ Iterations ν S3 S 4 ρ S4 S 5 ν S4 S 5 ρ S1 S 3 S 2 ν S1 S 3 S 2 ν Iterations ρ Iterations ν Iterations ρ Iterations ν Iterations ρ S2 S 4 S 3 ν S2 S 4 S 3 ρ S3 S 5 S 4 ν S3 S 5 S 4 ρ S1 S 4 S 2 S 3 ρ Iterations ν Iterations ρ Iterations ν Iterations ρ Iterations ν S1 S 4 S 2 S 3 ρ S2 S 5 S 3 S 4 ν S2 S 5 S 3 S 4 ρ S1 S 5 S 2 S 3 S 4 ν S1 S 5 S 2 S 3 S 4 ν Iterations ρ Iterations ν Iterations ρ Iterations ν Iterations

57 Euro Swap Rates For the further analysis we used burnin 1000 iteration and only every 20th iteration. Autocorrelations among the MCMC iterations for the swap rates ρ S1 S 2 ν S1 S 2 ρ S2 S 3 ν S2 S 3 ρ S3 S 4 ACF ACF ACF ACF ACF Lag Lag Lag Lag Lag ν S3 S 4 ρ S4 S 5 ν S4 S 5 ρ S1 S 3 S 2 ν S1 S 3 S 2 ACF ACF ACF ACF ACF Lag Lag Lag Lag Lag ρ S2 S 4 S 3 ν S2 S 4 S 3 ρ S3 S 5 S 4 ν S3 S 5 S 4 ρ S1 S 4 S 2 S 3 ACF ACF ACF ACF ACF Lag Lag Lag Lag Lag ν S1 S 4 S 2 S 3 ρ S2 S 5 S 3 S 4 ν S2 S 5 S 3 S 4 ρ S1 S 5 S 2 S 3 S 4 ν S1 S 5 S 2 S 3 S 4 ACF ACF ACF ACF ACF Lag Lag Lag Lag Lag

58 Euro Swap Rates Estimated posterior densities of D-vine parameters ρ S1 S 2 ν S1 S 2 ρ S2 S 3 ν S2 S 3 ρ S3 S 4 Density Density Density Density Density ν S3 S 4 ρ S4 S 5 ν S4 S 5 ρ S1 S 3 S 2 ν S1 S 3 S 2 Density Density Density Density Density ρ S2 S 4 S 3 ν S2 S 4 S 3 ρ S3 S 5 S 4 ν S3 S 5 S 4 ρ S1 S 4 S 2 S 3 Density Density Density Density Density Density ν S1 S 4 S 2 S 3 ρ S2 S 5 S 3 S 4 ν S2 S 5 S 3 S 4 ρ S1 S 5 S 2 S 3 S 4 ν S1 S 5 S 2 S 3 S Density Density Density Density

59 Euro Swap Rates Summary statistics for thinned MCMC 2.5% 50% 97.5% Est. Post. Est. Post. MLE Quantile Quantile Quantile Mean Mode ρ S1 S ν S1 S ρ S2 S ν S2 S ρ S3 S ν S3 S ρ S4 S ν S4 S ρ S1 S 3 S ν S1 S 3 S ρ S2 S 4 S ν S2 S 4 S ρ S3 S 5 S ν S3 S 5 S ρ S1 S 4 S 2 S ν S1 S 4 S 2 S ρ S2 S 5 S 3 S ν S2 S 5 S 3 S ρ S1 S 5 S 2 S 3 S ν S1 S 5 S 2 S 3 S

60 Euro Swap Rates Log likelihoods Parameter values loglikelihood multivariate t with common df MLE Posterior Mode Likelihood Ratio Test rejects a multivariate t copula with common df decisively, estimated common df is Since posterior mode log likelihood is close to MLE likelihood, prior choice for ρ and ν are close to being noninformative in this data set.

61 Euro Swap Rates Model selection for swap rate data Model PCC P(Mk data) M 1 : with all pairs 0.19 M 2 : without c S1 S 5 S 2 S 3 S M 3 : without c S1 S 5 S 2 S 3 S 4 and c S2 S 5 S 3 S M 4 : without c S1 S 5 S 2 S 3 S 4, c S2 S 5 S 3 S and c S1 S 4 S 2 S 3

62 Summary and Outlook Summary and Conclusion PCC s such as canonical and D-Vines allow for very flexible class of multivariate distributions. The Bayesian approach can solve estimation as well as model selection problems. It also gives credible intervals for parameters of interest. The proposed RJMCMC Bayesian algorithm for n > 3 is under implementation. Incorporation of pair-copulas from different parametric families is under consideration. The proposed RJMCMC algorithm is an alternative to DIC and other Bayesian model choice algorithms Congdon s method can be used to make model comparisons for a small number of models

63 References Aas, K., C. Czado, A. Frigessi, and H. Bakken (2007). Pair-copula constructions of multiple dependence. Insurance, Mathematics and Economics, DOI /j.insmatheco , preprint available at SFB 386 Statistical Analysis of Diskrete Structures, muenchen.de/sfb386/. Bedford, T. and R. M. Cooke (2001). Probability density decomposition for conditionally dependent random variables modeled by vines. Annals of Mathematics and Artificial Intelligence 32, Bedford, T. and R. M. Cooke (2002). Vines - a new graphical model for dependent random variables. Annals of Statistics 30(4), Berg, D. and K. Aas (2007). Models for construction of higher-dimensional dependence.

64 References Kurowicka, D. and R. Cooke (2006). Uncertainty analysis with high dimensional dependence modelling. Congdon, P. (2006). Bayesian model choice based on monte carlo estimates of posterior model probabilities. Computational Statistics & Data Analysis 50, Green, P. (1995). Reversible jump markov chain monte carlo computation and Bayesian model determination. Biometrika 82, Joe, H. (1996). Families of m-variate distributions with given margins and m(m-1)/2 bivariate dependence parameters. In L. Rüschendorf and B. Schweizer and M. D. Taylor (Ed.), Distributions with Fixed Marginals and Related Topics.

65 Summary and Outlook Chichester: Wiley. Kurowicka, D. and R. M. Cooke (2004). Distribution - free continuous bayesian belief nets. In Fourth International Conference on Mathematical Methods in Reliability Methodology and Practice, Santa Fe, New Mexico. Scott, S. L. (2002). Bayesian methods for hidden Markov models: Recursive computing in the 21st century. J. Amer. Statist. Assoc. 97,