Time Varying Hierarchical Archimedean Copulae (HALOC)

Transcription

1 Time Varying Hierarchical Archimedean Copulae () Wolfgang Härdle Ostap Okhrin Yarema Okhrin Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. Center for Applied Statistics and Economics Humboldt-Universität zu Berlin Chair of Statistics Universität Augsburg

2 Motivation 1-1 Simple AC over time Figure 1: Estimated copula dependence parameter θt with the Local Change Point method for 6-dimensional data: DC, VW, Bayer, BASF, Allianz and Münchener Rückversicherung. Clayton Copula. Giacomini et. al (2009)

3 θ θ (((1.4).3).2) = 1.12 Motivation 1-2 Time Varying Structures θ ((JPY.USD).GBP) = 0.3 θ (JPY.USD) = 0.5 USD JPY GBP BIC Figure 2: Film of changing structures over time. Date

4 Motivation 1-3 Main Idea combine interpretability with exibility of copulae determine the structure of HAC for a given time series identify time varying dependencies apply to risk pattern analysis reduce dimension of dependency

5 Motivation 1-4 Outline 1. Motivation 2. Hierarchical Archimedean copulae 3. Local Parametric Modeling by HAC 4. Simulation Study 5. Empirical Part 6. References

6 Hierarchical Archimedean Copulae 2-1 Copula For a distribution function F with marginals F X1..., F Xd. There exists a copula C : [0, 1] d [0, 1], such that F (x 1,..., x d ) = C{F X1 (x 1 ),..., F Xd (x d )} (1) for all x i R, i = 1,..., d. If F X1,..., F Xd are cts, then C is unique. If C is a copula and F X1,..., F Xd are cdfs, then the function F dened in (1) is a joint cdf with marginals F X1,..., F Xd.

7 Hierarchical Archimedean Copulae 2-2 Archimedean Copulae Multivariate Archimedean copula C : [0, 1] d [0, 1] dened as C(u 1,..., u d ) = φ{φ 1 (u 1 ) + + φ 1 (u d )}, (2) where φ : [0, ) [0, 1] is continuous and strictly decreasing with φ(0) = 1, φ( ) = 0 and φ 1 its pseudo-inverse. Example φ Gumbel (u, θ) = exp{ u 1/θ }, where 1 θ < φ Clayton (u, θ) = (θu + 1) 1/θ, where θ [ 1, )\{0} Disadvantages: too restrictive, single parameter, exchangeable

8 Hierarchical Archimedean Copulae 2-3 Hierarchical Archimedean Copulae Simple AC with s=(1234) AC with s=((123)4) C(u 1, u 2,, ) = C 1 (u 1, u 2,, ) C(u 1, u 2,, ) = C 1 {C 2 (u 1, u 2, ), } x 1 x 2 x 3 x 4 z (1234) Fully nested AC with s=(((12)3)4) C(u 1, u 2,, ) = C 1 [C 2 {C 3 (u 1, u 2 ), }, ] x 1 x 2 x 3 x 4 z (123) z ((12)3)4 Partially Nested AC with s=((12)(34)) C(u 1, u 2,, ) = C 1 {C 2 (u 1, u 2 ), C 3 (, )} x 1 x 2 x 3 x 4 z 12 z (12)3 z ((12)3)4 x 1 x 2 x 3 x 4 z 12 z 34 z (12)(34)

9 Hierarchical Archimedean Copulae 2-4 Hierarchical Archimedean Copulae Advantages of HAC: exibility and wide range of dependencies: for d = 10 more than structures dimension reduction: d 1 parameters to be estimated subcopulae are also HAC

10 Hierarchical Archimedean Copulae 2-5 Criteria for grouping based on θ's x 1 x 2 z 12 x 1 x 3 z 13 x 1 x 4 z 14 x 2 x 3 z 23 x 2 x 4 z 24 x 3 x 4 z 34

11 Hierarchical Archimedean Copulae 2-6 Criteria for grouping based on θ's x 1 x 2 z 12 x 1 x 3 z 13 x 1 x 4 z 14 x 2 x 3 z 23 x 2 x 4 z 24 x 3 x 4 z 34 max{ θ 12, θ 13, θ 14, θ 23, θ 24, θ 34 } = θ 13

12 Hierarchical Archimedean Copulae 2-7 Criteria for grouping based on θ's x 1 x 2 z 12 x 1 x 3 z 13 x 1 x 4 z 14 x 2 x 3 z 23 x 2 x 4 z 24 x 3 x 4 z 34 max{ θ 12, θ 13, θ 14, θ 23, θ 24, θ 34 } = θ 13 x 1 x 3 x 2 z 13 z (13)2 x 1 x 3 x 4 z 13 x 2 x 4 z (13)4 z 24

13 Hierarchical Archimedean Copulae 2-8 Criteria for grouping based on θ's x 1 x 2 z 12 x 1 x 3 z 13 x 1 x 4 z 14 x 2 x 3 z 23 x 2 x 4 z 24 x 3 x 4 z 34 max{ θ 12, θ 13, θ 14, θ 23, θ 24, θ 34 } = θ 13 x 1 x 3 x 2 z 13 z (13)2 x 1 x 3 x 4 z 13 x 2 x 4 z (13)4 z 24 max{ θ (13)2, θ (13)4, θ 24 } = θ (13)4

14 Hierarchical Archimedean Copulae 2-9 Criteria for grouping based on θ's x 1 x 2 z 12 x 1 x 3 z 13 x 1 x 4 z 14 x 2 x 3 z 23 x 2 x 4 z 24 x 3 x 4 z 34 max{ θ 12, θ 13, θ 14, θ 23, θ 24, θ 34 } = θ 13 x 1 x 3 x 2 z 13 z (13)2 x 1 x 3 x 4 z 13 x 2 x 4 z (13)4 z 24 max{ θ (13)2, θ (13)4, θ 24 } = θ (13)4 x 1 x 3 x 4 x 2 z 13 z (13)4 z ((13)4)2

15 Hierarchical Archimedean Copulae 2-10 Criteria for grouping based on θ's x 1 x 2 z 12 x 1 x 3 z 13 x 1 x 4 z 14 x 2 x 3 z 23 x 2 x 4 z 24 x 3 x 4 z 34 max{ θ 12, θ 13, θ 14, θ 23, θ 24, θ 34 } = θ 13 x 1 x 3 x 2 z 13 z (13)2 x 1 x 3 x 4 z 13 x 2 x 4 z (13)4 z 24 max{ θ (13)2, θ (13)4, θ 24 } = θ (13)4 x 1 x 3 x 4 x 2 z 13 θ (13) θ ((13)4) z (13)4 z ((13)4)2

16 Hierarchical Archimedean Copulae 2-11 Criteria for grouping based on θ's x 1 x 2 z 12 x 1 x 3 z 13 x 1 x 4 z 14 x 2 x 3 z 23 x 2 x 4 z 24 x 3 x 4 z 34 max{ θ 12, θ 13, θ 14, θ 23, θ 24, θ 34 } = θ 13 x 1 x 3 x 2 z 13 z (13)2 x 1 x 3 x 4 z 13 x 2 x 4 z (13)4 z 24 max{ θ (13)2, θ (13)4, θ 24 } = θ (13)4 x 1 x 3 x 4 x 2 x 1 x 3 x 4 x 2 z 13 θ (13) θ ((13)4) z (13)4 z (134) z ((13)4)2 z ((134)2)

17 Y X θ (((1.4).3).2) = 1.12 Local Parametric Modeling by HAC 3-1 Motivation 1-9 Local Parametric Assumption Local Change Point Detection t m* 300 t Figure 3: Dependence over time for DaimlerChrysler, Volkswagen, Bayer, Figure 3: Parameter θ t (blue), size of homogeneity interval at t (black). BASF, Allianz and Münchener Rückversicherung, Giacomini COPULAEet. IN TEMPORE al (2009) VARIENTES

18 Local Parametric Modeling by HAC 3-2 Adaptive Copula Estimation adaptively estimate largest interval where homogeneity hypothesis is accepted Local Change Point detection (LCP): sequentially test θ t, s t are constants (i.e. θ t = θ, s t = s) within some interval I (local parametric assumption).

19 Local Parametric Modeling by HAC 3-3 Oracle choice: largest interval I = [t 0 m k, t 0 ] where small modelling bias condition (SMB) I (s, θ) = t I K{C( ; s t, θ t ), C( ; s, θ)}. holds for some 0. m k is the ideal scale, (s, θ) is ideally estimated from I = [t 0 m k, t 0 ] and K(, ) is the Kullback-Leibler divergence K{C( ; s t, θ t ), C( ; s, θ)} = IE st,θ t log c( ; s t, θ t ) c( ; s, θ)

20 Local Parametric Modeling by HAC 3-4 Under the SMB condition on I k and assuming that max k k IE s,θ L( s k, θ k ) L(s, θ) r R r (s, θ), we obtain { IE st,θ t log 1 + L( s k, θ k) } L(s, θ) r 1 +, R r (s, θ) { IE st,θ t log 1 + L( s k, θ k) L(ŝ k, θ k) } r ρ +, R r (s, θ) where â I is an adaptive estimator on I and ã I is any other parametric estimator on I.

21 Local Parametric Modeling by HAC 3-5 Local Change Point Detection 1. dene family of nested intervals I 0 I 1 I 2... I K = I K+1 with length m k as I k = [t 0 m k, t 0 ] 2. dene T k = [t 0 m k, t 0 m k 1 ] t 0 m k+1 t 0 m k t 0 m k 1 t 0 T k+1 T k I k 1 I k I k+1

22 Local Parametric Modeling by HAC 3-6 Local Change Point Detection 1. test homogeneity H 0,k against the change point alternative in T k using I k+1 2. if no change points in T k, accept I k. Take T k+1 and repeat previous step until H 0,k is rejected or largest possible interval I K is accepted 3. if H 0,k is rejected in T k, homogeneity interval is the last accepted Î = I k 1 4. if largest possible interval I K is accepted Î = I K

23 Local Parametric Modeling by HAC 3-7 Test of homogeneity Interval I = [t 0 m, t 0 ], T I H 0 : τ T, θ t = θ, s t = s, t J = [τ, t 0 ], t J c = I \ J H 1 : τ T, θ t = θ 1, s t = s 1 ; t J, θ t = θ 2 θ 1 ; s t = s 2 s 1, t J c J c J t 0 m τ t 0 T I

24 Local Parametric Modeling by HAC 3-8 Test of homogeneity Likelihood ratio test statistic for xed change point location: T I,τ = max{l J (θ 1 ) + L J c (θ 2 )} max L I (θ) θ 1,θ 2 θ = ML J + ML J c ML I Test statistic for unknown change point location: Reject H 0 if for a critical value ζ I T I = max τ T I T I,τ T I > ζ I

25 Local Parametric Modeling by HAC 3-9 Selection of I k and ζ k set of numbers m k dening the length of I k and T k are in the form of a geometric grid m k = [m 0 c k ] for k = 1, 2,..., K, m 0 {20, 40}, c = 1.25 and K = 10, where [x] means the integer part of x I k = [t 0 m k, t 0 ] and T k = [t 0 m k, t 0 m k 1 ] for k = 1, 2,..., K (Mystery Parameters)

26 Simulation Study 4-1 Sequential choice of ζ k after k steps are two cases: change point at some step l k and no change points. let B l be the event meaning the rejection at step l B l = {T 1 ζ 1,..., T l 1 ζ l 1, T l > ζ l }, and (ŝ k, θ k ) = ( s l 1, θ l 1 ) on B l for l = 1,..., k. we nd sequentially such a minimal value of ζ l that ensures following inequality max k=l,...,k IE s,θ L( s k, θ k ) L( s l 1, θ l 1 ) r I(B l ) ρr r (s, θ k ) K 1

27 Simulation Study 4-2 Illustration CS k*+1 Stop

28 Simulation Study 4-3 Sequential choice of ζ k 1. pairs of Kendall's τ: {τ 1, τ 2 } {0.1, 0.3, 0.5, 0.7, 0.9} 2, τ 1 τ 2 2. simul. from C θi,θ j (u 1, u 2, ) = C{C(u 1, u 2 ; θ 1 ), ; θ 2 }, θ = θ(τ) 3. run sequential algorithm for each sample λ k of the 3-dimensional HAC kas a function of k with the xed Figure 4: ζ k m 0 = 40, ρ = 0.5, r = 0.5, τ 1 = 0.1 and for dierent τ 2. τ 2 = 0.1 (solid), τ 2 = 0.3 (solid), τ 2 = 0.5 (solid), τ 2 = 0.7 (dashed), τ 2 = 0.9 (dashed). u 3

29 Simulation Study 4-4 Simulation: Change in θ 1, I { C t (u 1, u 2, ; s, θ) = 1. N = 400 and 100 runs C{u 1, C(u 2, ; θ 1 = 3.33); θ 2 = 1.43} for 1 t 200 C{u 1, C(u 2, ; θ 1 = 2.00); θ 2 = 1.43} for 200 < t LCP based on the same critical values τ Figure 5: θ 1 and θ 2 on the intervals of homogeneity (median - dashed line, u mean - solid line). 4 u 3

30 Simulation Study 4-5 Simulation: Change in θ 1, II ml length Figure 6: Intervals of homogeneity and ML on these intervals (median - dashed line, mean - solid line) u 3

31 Simulation Study 4-6 Simulation: Change in θ 2, I { C t (u 1, u 2, ; s, θ) = 1. N = 400 and 100 runs C{u 1, C(u 2, ; θ 1 = 3.33); θ 2 = 1.43} for 1 t 200 C{u 1, C(u 2, ; θ 1 = 3.33); θ 2 = 2.00} for 200 < t LCP based on the same critical values τ Figure 7: θ 1 and θ 2 on the intervals of homogeneity (median - dashed line, mean - solid line). u 3

32 Simulation Study 4-7 Simulation: Change in θ 2, II ml length Figure 8: Intervals of homogeneity and ML on these intervals (median - dashed line, mean - solid line) u 3

33 Simulation Study 4-8 Simulation: Change in the Structure, I { C t (u 1, u 2, ; s, θ) = structure 1. N = 400 and 100 runs C{u 1, C(u 2, ; θ 1 = 3.33); θ 2 = 1.43} for 1 t 200 C{C(u 1, u 2 ; θ 1 = 3.33), ; θ 2 = 1.43} for 200 < t LCP based on the same critical values ((1.2).3) ((1.3).2) ((2.3).1) Figure 9: The structure of the est. HAC on the intervals of homogeneity (median - dashed line, mean - solid line) u 3

34 Simulation Study 4-9 Simulation: Change in the Structure, II ml length Figure 10: Intervals of homogeneity and ML on these intervals (median - dashed line, mean - solid line) u 3

35 Empirical Part 5-1 Data and Copula daily values for the exchange rates JPN/EUR, GBP/EUR and USD/EUR timespan = [ ; ] (n = 2771) {φ = exp( u 1/θ )} - Gumbel generator

36 Empirical Part 5-2 Data and Copula a univariate GARCH(1,1) process on log-returns X j,t = µ j,t + σ j,t ε j,t with σ 2 j,t = ω j + α j σ 2 j,t 1 + β j (X j,t 1 µ j,t 1 ) 2 ε t C{F 1 (x 1 ),..., F d (x d ); θ t } estimated copula from the whole sample s = (JPY USD) GBP µ j ω j α j βj BL KS JPY 4.85e e e-05 (1.15e-04) (1.02e-07) (7.49e-03) (7.64e-03) GBP 6.34e e e-04 (7.39e-05) (5.11e-08) (8.75e-03) (9.12e-03) USD 1.76e e e-03 (1.10e-04) (5.92e-08) (4.14e-03) (4.28e-03) Table 1: Estimation results univariate time series modelling.

37 Empirical Part 5-3 Rolling window n ML = log{f (u i1,..., u id, θ)}, i=1 where f denotes the joint multivariate density function. AIC = 2ML + 2m, BIC = 2ML + 2 log(m), where m is the number of parameters to be estimated. Θ t (d d) - matrix of the pairwise θ based on the 250 days before t Θ t Θ t 1 2 = λ max {( Θ t Θ t 1 )( Θ t Θ t 1 ) }

38 Empirical Part 5-4 Copulae over time BIC Date Figure 11: Time-varying HAC: BIC for the AC, Gaussian copula and HAC. Dierence Matrix and points of the changes of the structure.

39 Empirical Part 5-5 LCP for HAC to real Data τ structure ((1.2).3) ((1.3).2) ((2.3).1) Figure 12: Structure, τ 1 and τ 2 of the HAC on the intervals of homogeneity

40 Empirical Part 5-6 LCP for HAC to real Data ML length Figure 13: Intervals of homogeneity and ML on these intervals

41 Empirical Part 5-7 Data and Copula daily returns values for Dow Jones (DJ), DAX and NIKKEI timespan = [ ; ] (n = 2771) {φ = exp( u 1/θ )} - Gumbel generator estimated copula from the whole sample s = (DAX DJ) NIKKEI µ j ω j α j βj BL KS DAX 6.94e e e-05 (1.39e-04) (5.29e-07) (0.01) (9.39e-03) DJ 5.96e e e-07 (1.11e-04) (3.38e-07) (0.01) (9.40e-03) NIKKEI 5.62e e e-13 (1.45e-04) (5.18e-07) (0.01) (8.71e-03) Table 2: Estimation results univariate time series modelling.

42 Empirical Part 5-8 Copulae over time BIC Date Figure 14: Time-varying HAC: BIC for the AC, Gaussian copula and HAC. Dierence Matrix and points of the changes of the structure.

43 Empirical Part 5-9 LCP for HAC to real Data τ structure ((1.2).3) ((1.3).2) ((2.3).1) Figure 15: Structure, τ 1 and τ 2 of the HAC on the intervals of homogeneity

44 Empirical Part 5-10 LCP for HAC to real Data ML length Figure 16: Intervals of homogeneity and ML on these intervals

45 Time Varying Hierarchical Archimedean Copulae () Wolfgang Härdle Ostap Okhrin Yarema Okhrin Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. Center for Applied Statistics and Economics Humboldt-Universität zu Berlin Chair of Statistics Universität Augsburg

46 Bibliography 6-1 H. Joe Multivariate Models and Concept Dependence Chapman & Hall, 1997 V. Spokoiny Local Parametric Methods in Nonparametric Estimation Springer Verlag, 2009 E. Giacomini, W. Härdle and V. Spokoiny Inhomogeneous Dependence Modeling with Time-Varying Copulae Journal of Business and Economic Statistics, 27(2), 2009 O. Okhrin and Y. Okhrin and W. Schmid On the Structure and Estimation of Hierarchical Archimedean Copulas under revision in Journal of Econometrics, 2009