Copula goodness-of-fit testing. An overview and power comparison

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1 : An overview and power comparison DANIEL BERG DnB NOR Asset Management 22nd Nordic Conference on Mathematical Statistics. Vilnius, Lithuania June 2008

2 Outline Preliminaries Proposed approaches Monte Carlo simulation results and recommendations

3 Motivation Figure: Two simulated data sets - both with standard normal margins and correlation coefficient 0.7.

4 Motivation Figure: Nonzero precipitation values in two Norwegian cities and its copula.

5 Definition & Theorem Definition (Copula) A d-dimensional copula is a multivariate distribution function C with standard uniform marginal distributions. Theorem (Sklar, 1959) Let H be a joint distribution function with margins F 1,..., F d. Then there exists a copula C : [0, 1] d [0, 1] such that H(x 1,..., x d ) = C(F 1 (x 1 ),..., F d (x d )).

6 Useful results A general d-dimensional density h can be expressed, for some copula density c, as h(x 1,..., x d ) = c{f 1 (x 1 ),..., F d (x d )}f 1 (x 1 ) f d (x d ). Non-parametric estimate for F i (x i ) commonly used to transform original margins into standard uniform: u ji = F i (x ji ) = R ji n + 1, where R ji is the rank of x ji amongst x 1i,..., x ni. u ji commonly referred to as pseudo-observations and models based on non-parametric margins and parametric copulas are referred to as semi-parametric copulas

7 Preliminaries Proposed approaches H 0 : C C = {C θ ; θ Θ} vs. H 1 : C / C = {C θ ; θ Θ} Univariate Anderson-Darling or QQ-plot, Multivariate fewer alternatives Pseudo-observations no longer independent. In addition, limiting distribution of many copula GoF test depends on null hypothesis copula and parameter value p-value estimation via parametric bootstrap procedures Focus in literature almost exclusively bivariate NOT model selection! Several techniques proposed: binning, multivariate smoothing, dimension reduction

8 Preliminaries Proposed approaches Preliminaries Rosenblatt s transform: Dependent variables independent U[0, 1] variables, given multivariate distribution v = R(z) = (R 1 (z 1 ),..., R d (z d )): v 1 = R 1 (z 1 ) = F 1 (z 1 ) = z 1, v 2 = R 2 (z 2 ) = F 2 1 (z 2 z 1 ),. v d = R d (z d ) = F d 1...d (z d z 1,..., z d ). Inverse of simulation (conditional inversion) GoF: v = R(z) test v for independence d! different permutation orders

9 Preliminaries Proposed approaches Proposed approaches: A 1 (1/9) v = R(z) W 1j = d i=1 Γ{v ji; α}, j = 1,..., n Special case (a): d i=1 Φ 1 (v ji ) 2 Special case (b): d i=1 v ji 0.5 S 1 (t) = P{F 1 (W 1 ) t}, t [0, 1] CvM statistic: T 1 = n 1 0 } 2 {Ŝ1 (t) S 1 (t) ds1 (t) References: Breymann et al. (2003); Malevergne and Sornette (2003); Berg and Bakken (2005)

10 Preliminaries Proposed approaches Proposed approaches: A 2 (2/9) Empirical copula: Ĉ(u) = 1 n + 1 CvM statistic: T 2 = n [0,1] d n I {Z j1 u 1,..., Z jd u d } j=1 {Ĉ(z) Cbθ (z) } 2 d Ĉ(z) References: Fermanian (2005); Genest and Rémillard (2008); Genest et al. (2008)

11 Preliminaries Proposed approaches Proposed approaches: A 3 (3/9) Approach A 2 on v = R(z) CvM statistic: T 3 = n [0,1] d References: Genest et al. (2008) } 2 {Ĉ(v) C (v) d Ĉ(v)

12 Preliminaries Proposed approaches Proposed approaches: A 4 (4/9) Cdf of empirical copula (Kendall s dependence function): CvM statistic: T 4 = n S 4 (t) = P{C(z) t} 1 0 {Ŝ4 (t) S 4, bθ (t) } 2 ds4, bθ (t) References: Genest and Rivest (1993); Wang and Wells (2000); Savu and Trede (2004); Genest et al. (2006)

13 Preliminaries Proposed approaches Proposed approaches: A 5 (5/9) Spearman s dependence function: CvM statistic: T 5 = n S 5 (t) = P{C (z) t} 1 References: Quessy et al. (2007) 0 {Ŝ5 (t) S 5, bθ (t) } 2 ds5, bθ (t)

14 Preliminaries Proposed approaches Proposed approaches: A 6 (6/9) Shih s test for bivariate gamma frailty model (Clayton): T Shih = n { θτ θ } W Extension to arbitrary dimension: T 6 = d 1 d { θτ,ij θ W,ij } 2 i=1 j=i+1 References: Shih (1998); Berg (2007)

15 Preliminaries Proposed approaches Proposed approaches: A 7 (7/9) Inner product of two vectors = 0 iff from the same family Q(z) = z z bθ κ d z z bθ κ a symmetric kernel, e.g. the gaussian kernel: κ d (z, z bθ ) = exp { z z bθ 2 /(2dh 2 ) } Statistic becomes: T 7 = 1 n n n 2 n i=1 j=1 i=1 j=1 n κ d (z i, z j ) 2 n 2 n κ d (z bθ,i, z bθ,j ) References: Panchenko (2005) n i=1 j=1 n κ d (z i, z bθ,j )

16 Preliminaries Proposed approaches Proposed approaches: A 8 (8/9) Approach A 7 on v = R(z) Statistic: T 8 = 1 n n n 2 κ d (v i, v j ) 2 n n 2 n i=1 j=1 i=1 j=1 n κ d (v bθ,i, v bθ,j ) n n κ d (v i, v bθ,j ) i=1 j=1 References: Berg (2007)

17 Preliminaries Proposed approaches Proposed approaches: A 9 (9/9) Each approach may detect deviations from H 0 differently Average approaches: T (a) 9 = 1 9 T (b) 9 = 1 3 References: Berg (2007) { T (a) (b) 1 + T 1 + { T2 + T 3 + T 4 } } 8 T k k=2

18 Test procedure Experimental setup Power comparison Monte Carlso simulations Test procedure 1) x n samples from the d-dimensional H 1 copula with θ(τ). 2) z pseudo-observations (normalized ranks) 3) ˆθ estimated parameter of the H 0 copula 4) T i test statistic i computed under the H 0 copula using ˆθ. 5) Repeat steps 1-4 M times with H 1 = H 0 and θ = ˆθ T 0 i,m 6) ˆp = 1 M M m=1 1( T 0 i,m > T i ) 7) ˆp < 5% reject H 0

19 Test procedure Experimental setup Power comparison Monte Carlso simulations Experimental setup H 0 copula (5 choices: Gaussian, Student, Clayton, Gumbel, Frank), H 1 copula (5 choices: Gaussian, Student (ν = 6), Clayton, Gumbel, Frank), Kendall s tau (2 choices: τ = {0.2, 0.4}), Dimension (3 choices: d = {2, 4, 8}), Sample size (2 choices: n = {100, 500}) Student only considered as null in bivariate case. For each of these 240 cases, 10, 000 repetitions size/power

20 Test procedure Experimental setup Power comparison Monte Carlso simulations Testing the Gaussian copula Power difference (%) A 1 (a) A 1 (b) A 2 A 3 A 4 A 5 A 7 A 8 A 9 (a) A 9 (b)

21 Test procedure Experimental setup Power comparison Monte Carlso simulations Testing the Student copula Power difference (%) A 1 (a) A 1 (b) A 2 A 3 A 4 A 5 A 7 A 8 A 9 (a) A 9 (b)

22 Test procedure Experimental setup Power comparison Monte Carlso simulations Testing the Clayton copula Power difference (%) A 1 (a) A 1 (b) A 2 A 3 A 4 A 5 A 6 A 7 A 8 A 9 (a) A 9 (b)

23 Test procedure Experimental setup Power comparison Monte Carlso simulations Testing the Gumbel copula Power difference (%) A 1 (a) A 1 (b) A 2 A 3 A 4 A 5 A 7 A 8 A 9 (a) A 9 (b)

24 Test procedure Experimental setup Power comparison Monte Carlso simulations Testing the Frank copula Power difference (%) A 1 (a) A 1 (b) A 2 A 3 A 4 A 5 A 7 A 8 A 9 (a) A 9 (b)

25 and recommendations Nominal levels all match prescribed size of 5% Power generally increases with dimension, sample size and dependence Clayton > Gumbel > Frank > Gaussian > Student (>: easier to test) No universally most powerful approach, but A 2, A 4 and A (b) 9 perform very well in most cases A (b) 9 is recommended in general, with special case exceptions: For testing the Gaussian copula, if trying to detect heavy tails for d > 2 and large n then A 1 very powerful For testing the Clayton copula the generalized Shih s test is most powerful Permutational variation of little concern for approaches based on Rosenblatt s transform (see Berg (2007))

26 References Berg, D. (2007). : an overview and power comparison. Technical report, University of Oslo. Statistical research report no. 5, ISSN Berg, D. and H. Bakken (2005). A goodness-of-fit test for copulae based on the probability integral transform. Technical report, University of Oslo. Statistical research report no. 10, ISSN Breymann, W., A. Dias, and P. Embrechts (2003). Dependence structures for multivariate high-frequency data in finance. Quantitative Finance 1, Fermanian, J. (2005). Goodness of fit tests for copulas. Journal of Multivariate Analysis 95, Genest, C., J.-F. Quessy, and B. Rémillard (2006). Goodness-of-fit procedures for copula models based on the probability integral transform. Scandinavian Journal of Statistics 33, Genest, C. and B. Rémillard (2008). Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models. Ann. Henri Poincaré 44. In press. Genest, C., B. Rémillard, and D. Beaudoin (2008). Omnibus goodness-of-fit tests for copulas: A review and a power study. Insurance: Mathematics and Economics 42. In press. Genest, C. and L.-P. Rivest (1993). Statistical inference procedures for bivariate archimedean copulas. Journal of the American Statistical Association, Malevergne, Y. and D. Sornette (2003). Testing the gaussian copula hypothesis for financial assets dependence. Quantitative Finance 3, Panchenko, V. (2005). Goodness-of-fit test for copulas. Physica A 355(1), Quessy, J.-F., M. Mesfioui, and M.-H. Toupin (2007). A goodness-of-fit test based on Spearmans dependence function. Working paper, Université du Québec à Trois-Rivières. Savu, C. and M. Trede (2004). Goodness-of-fit tests for parametric families of archimedean copulas. CAWM discussion paper, No. 6. Shih, J. H. (1998). A goodness-of-fit test for association in a bivariate survival model. Biometrika 85, Wang, W. and M. T. Wells (2000). Model selection and semiparametric inference for bivariate failure-time data. Journal of the American Statistical Association 95,

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