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Arthur CHARPENTIER - Unifying copula families and tail dependence Unifying standard multivariate copulas families (with tail dependence properties) Arthur Charpentier charpentier.arthur@uqam.ca http ://freakonometrics.hypotheses.org/ inspired

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Arthur CHARPENTIER - Unifying copula families and tail dependence

Unifying standard multivariate copulas families (with tail dependence properties)

Arthur Charpentier

charpentier.arthur@uqam.ca http ://freakonometrics.hypotheses.org/

inspired by some joint work (and discussion) with A.-L. Fougères, C. Genest, J. Nešlehová, J. Segers January 2013, H.E.C. Lausanne 1

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Arthur CHARPENTIER - Unifying copula families and tail dependence

Agenda

  • Standard copula families
  • Elliptical distributions (and copulas)
  • Archimedean copulas
  • Extreme value distributions (and copulas)
  • Tail dependence
  • Tail indexes
  • Limiting distributions
  • Other properties of tail behavior

2

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Arthur CHARPENTIER - Unifying copula families and tail dependence

Copulas

Definition 1

A copula in dimension d is a c.d.f on [0, 1]d, with margins U([0, 1]).

Theorem 1 1. If C is a copula, and F1, ..., Fd are univariate c.d.f., then

F(x1, ..., xn) = C(F1(x1), ..., Fd(xd)) ∀(x1, ..., xd) ∈ Rd (1) is a multivariate c.d.f. with F ∈ F(F1, ..., Fd).

  • 2. Conversely, if F ∈ F(F1, ..., Fd), there exists a copula C satisfying (1). This copula

is usually not unique, but it is if F1, ..., Fd are absolutely continuous, and then, C(u1, ..., ud) = F(F −1

1

(u1), ..., F −1

d (ud)), ∀(u1, , ..., ud) ∈ [0, 1]d

(2) where quantile functions F −1

1

, ..., F −1

n

are generalized inverse (left cont.) of Fi’s. If X ∼ F, then U = (F1(X1), · · · , Fd(Xd)) ∼ C. 3

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Arthur CHARPENTIER - Unifying copula families and tail dependence

Survival (or dual) copulas

Theorem 2 1. If C⋆ is a copula, and F 1, ..., F d are univariate s.d.f., then

F(x1, ..., xn) = C⋆(F 1(x1), ..., F d(xd)) ∀(x1, ..., xd) ∈ Rd (3) is a multivariate s.d.f. with F ∈ F(F1, ..., Fd).

  • 2. Conversely, if F ∈ F(F1, ..., Fd), there exists a copula C⋆ satisfying (3). This

copula is usually not unique, but it is if F1, ..., Fd are absolutely continuous, and then, C⋆(u1, ..., ud) = F(F

−1 1 (u1), ..., F −1 d (ud)), ∀(u1, , ..., ud) ∈ [0, 1]d

(4) where quantile functions F −1

1

, ..., F −1

n

are generalized inverse (left cont.) of Fi’s. If X ∼ F, then U = (F1(X1), · · · , Fd(Xd)) ∼ C and 1 − U ∼ C⋆. 4

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Arthur CHARPENTIER - Unifying copula families and tail dependence

Benchmark copulas

Definition 2

The independent copula C⊥ is defined as C⊥(u1, ..., un) = u1 × · · · × ud =

d

  • i=1

ui.

Definition 3

The comonotonic copula C+ (the Fréchet-Hoeffding upper bound of the set of copulas) is the copuladefined as C+(u1, ..., ud) = min{u1, ..., ud}. 5

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Arthur CHARPENTIER - Unifying copula families and tail dependence

Spherical distributions

Definition 4

Random vector X as a spherical distribution if X = R · U where R is a positive random variable and U is uniformly dis- tributed on the unit sphere of Rd, with R ⊥ ⊥ U. E.g. X ∼ N(0, I).

−2 −1 1 2 −2 −1 1 2

  • ●
  • ●
  • −2

−1 1 2 −2 −1 1 2

  • 0.02

0.04 . 6 0.08 . 1 2 . 1 4

Those distribution can be non-symmetric, see Hartman & Wintner (AJM, 1940)

  • r Cambanis, Huang & Simons (JMVA, 1979))

6

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Arthur CHARPENTIER - Unifying copula families and tail dependence

Elliptical distributions

Definition 5

Random vector X as a elliptical distribution if X = µ + R · A · U where R is a positive random variable and U is uniformly dis- tributed on the unit sphere of Rd, with R ⊥ ⊥ U. , and where A satisfies AA′ = Σ. E.g. X ∼ N(µ, Σ).

−2 −1 1 2 −2 −1 1 2

  • ●
  • ●
  • ●
  • −2

−1 1 2 −2 −1 1 2

  • .

2 0.04 0.06 0.08 . 1 2 . 1 4

Elliptical distribution are popular in finance, see e.g. Jondeau, Poon & Rockinger (FMPM, 2008) 7

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Arthur CHARPENTIER - Unifying copula families and tail dependence

Archimedean copula

Definition 6

If d ≥ 2, an Archimedean generator is a function φ : [0, 1] → [0, ∞) such that φ−1 is d-completely monotone (i.e. ψ is d-completely monotone if ψ is continuous and ∀k = 0, 1, ..., d, (−1)kdkψ(t)/dtk ≥ 0).

Definition 7

Copula C is an Archimedean copula is, for some generator φ, C(u1, ..., ud) = φ−1(φ(u1) + ... + φ(ud)), ∀u1, ..., ud ∈ [0, 1].

Exemple1

φ(t) = − log(t) yields the independent copula C⊥. 8

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Arthur CHARPENTIER - Unifying copula families and tail dependence

Stochastic representation of Archimedean copulas

Consider some striclty positive random variable R independent of U, uniform on the simplex of Rd. The survival copula of X = R·U is Archimedean, and its generator is the inverse of Williamson d- transform, φ−1(t) = ∞

x

  • 1 − x

t d−1 dFR(t). Note that R

L

= φ(U1) + · · · + φ(Ud)

0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5

  • See Nešlehová & McNeil (AS, 2009).

9

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Archimedean copula, exchangeability and frailties

Consider residual life times X = (X1, · · · , Xd) condition- ally independent given some latent factor Θ, and such that P(Xi > xi|Θ) = Bi(xi)θ. Then F(x) = P(X > x) = ψ

  • −

n

  • i=1

log F i(xi)

  • where ψ is the Laplace transform of Θ, ψ(t) = E(e−tΘ).

Thus, the survival copula of X is Archimedean, with gener- ator = ψ−1. See Oakes (JASA, 1989).

20 40 60 80 100 20 40 60 80 100

Conditional independence, continuous risk factor

!3 !2 !1 1 2 3 !3 !2 !1 1 2 3

Conditional independence, continuous risk factor

10

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Arthur CHARPENTIER - Unifying copula families and tail dependence

Nested Archimedean copula, and hierarchical structures

Consider C(u1, · · · , ud) defined as φ−1

1 [φ1[φ−1 2 (φ2[· · · φ−1 d−1[φd−1(u1) + φd−1(u2)] + · · · + φ2(ud−1))] + φ1(ud)]

where φi’s are generators. Then C is a copula if φi ◦ φ−1

i−1 is the inverse of a

Laplace transform, and is called fully nested Archimedean copula. Note that partial nested copulas can also be considered, U1 U2 U3 U4 U5 φ4 φ3 φ2 φ1 U1 U2 U3 U4 U5 φ2 φ1 φ3 φ4 11

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Arthur CHARPENTIER - Unifying copula families and tail dependence

(Univariate) extreme value distributions

Central limit theorem, Xi ∼ F i.i.d. Xn − bn an

L

→ S as n → ∞ where S is a non-degenerate random variable. Fisher-Tippett theorem, Xi ∼ F i.i.d., Xn:n − bn an

L

→ M as n → ∞ where M is a non-degenerate random variable. Then P Xn:n − bn an ≤ x

  • = F n(anx + bn) → G(x) as n → ∞, ∀x ∈ R

i.e. F belongs to the max domain of attraction of G, G being an extreme value distribution : the limiting distribution of the normalized maxima. − log G(x) = (1 + ξx)−1/ξ

+

12

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Arthur CHARPENTIER - Unifying copula families and tail dependence

(Multivariate) extreme value distributions

Assume that Xi ∼ F i.i.d., F n(anx + bn) → G(x) as n → ∞, ∀x ∈ Rd i.e. F belongs to the max domain of attraction of G, G being an (multivariate) extreme value distribution : the limiting distribution of the normalized componentwise maxima, Xn:n = (max{X1,i}, · · · , max{Xd,i}) − log G(x) = µ([0, ∞)\[0, x]), ∀x ∈ Rd

+

where µ is the exponent measure. It is more common to use the stable tail dependence function ℓ defined as ℓ(x) = µ([0, ∞)\[0, x−1]), ∀x ∈ Rd

+

13

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Arthur CHARPENTIER - Unifying copula families and tail dependence

i.e. − log G(x) = ℓ(− log G1(x1), · · · , log Gd(xd))∀x ∈ Rd Note that there exists a finite measure H on the simplex of Rd such that ℓ(x1, · · · , xd) =

  • Sd

max{ω1x1, · · · , ωdxd}dH(ω1, · · · , ωd) for all (x1, · · · , xd) ∈ Rd

+, and

  • Sd ωidH(ω1, · · · , ωd) = 1 for all i = 1, · · · , n.

Definition 8

Copula C : [0, 1]d → [0, 1] is an multivariate extreme value copula if and only if there exists a stable tail dependence function such that ℓ C(u1, · · · , ud) = exp[−ℓ(− log u1, · · · , − log ud)] Assume that U i ∼ C i.i.d., Cn(u

1 n ) = Cn(u 1 n

1 , · · · , u

1 n

d ) → Γ(u) as n → ∞, ∀x ∈ Rd

i.e. C belongs to the max domain of attraction of Γ, Γ being an (multivariate) extreme value copula. 14

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Arthur CHARPENTIER - Unifying copula families and tail dependence

What do we have in dimension 2 ?

C is an Archimedean copula if C = Cφ Cφ(u, v) = φ−1 [φ(u) + φ(v)] C is an extreme value copula if C = CA CA(u, v) = exp

  • log[uv]A

log[v] log[uv]

  • where A[0, 1] → [1/2, 1] is a convex function such that

max{ω, 1 − ω} ≤ A(ω) ≤ 1, ∀ω ∈ [0, 1].

Exemple2

A(ω) = 1 yields the independent copula, C⊥. 15

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Arthur CHARPENTIER - Unifying copula families and tail dependence

What do we have in dimension 2 ?

C is an Archimax copula (from Capéerà, Fougères & Genest (JMVA, 2000)) if C = Cφ,A Cφ,A(u, v) = φ−1

  • [φ(u) + φ(v)]A
  • φ(u)

φ(u) + φ(v)

  • Note that there is a frailty type construction, see C. (K, 2006) : given Θ, X has

(survival) copula CA, Θ has Laplace transform φ−1. 16

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Quantifying tail dependence, in dimension 2 ?

Venter (2002) suggested to visualize tail concentration functions,

Definition 9

For the lower tail, define L(z) = P(U < z, V < z) z = C(z, z) z = P(U < z|V < z) = P(V < z|U < z), and for the upper tail R(z) = P(U > z, V > z) 1 − z = P(U > z|V > z). Joe (JMVA, 1999) defined tail dependence coefficients from lower and upper limits, respectively (if those limits exist) λU = R(1) = lim

z→1 R(z) et λL = L(0) = lim z→0 L(z).

17

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Arthur CHARPENTIER - Unifying copula families and tail dependence

Quantifying tail dependence, in dimension 2 ?

Definition 10

Let (X, Y ) denote a random vector in R2. Define tail dependence indices in the lower (L) and upper (U) tails as λL = lim

u↓0 P

  • X ≤ F −1

X (u) |Y ≤ F −1 Y

(u)

  • ∈ [0, 1],

and λU = lim

u↑1 P

  • X > F −1

X (u) |Y > F −1 Y

(u)

  • ∈ [0, 1].

Proposition 1

Let (X, Y ) denote a random vector with copula C, then λL = lim

u↓0

C(u, u) u and λU = lim

u↓0

C⋆(u, u) u . 18

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Arthur CHARPENTIER - Unifying copula families and tail dependence

Quantifying tail dependence, in dimension 2 ?

Exemple3

For Archimedean copulas (see Nelsen (2007), C. & Segers (JMVA, 2008)), λU = 2 − lim

x→0

1 − φ−1(2x) 1 − φ−1(x) and λL = lim

x↓0

φ−1(2φ(x)) x = lim

x↓∞

φ−1(2x) φ−1(x) . Ledford and Tawn (B, 1996) suggested an alternative approach : assume that X

L

= Y . – assuming independence, P(X > t, Y > t) = P(X > t) × P(Y > t) = P(X > t)2, – assuming comonotonicity, P(X > t, Y > t) = P(X > t) = P(X > t)1, Thus, assume that one has P(X > t, Y > t) ∼ P(X > t)η as t → ∞, where η ∈ [1, 2] will be a tail dependence index. 19

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Arthur CHARPENTIER - Unifying copula families and tail dependence

Quantifying tail dependence, in dimension 2 ?

Following Coles, Heffernan & Tawn (E, 1999) define

Definition 11

Let χU(z) = 2 log(1 − z) log C⋆(z, z) − 1 et χL(z) = 2 log(1 − z) log C(z, z) − 1 Then ηU = (1 + limz→0 χU(z))/2 and ηL = (1 + limz→0 χL(z))/2 are respectively tail indices in the upper and lower tail, respectively.

Exemple4

If (X, Y ) has a Gumbel copula, with (unit) Fréchet margins P(X ≤ x, Y ≤ y) = exp(−(x−α + y−α)1/α), where α ≥ 0, ∀x, y ≥ 0 then ηU = 1 while ηL = 1/2α. For a Gaussian copula with correlation r ηU = ηL = (1 + r)/2. 20

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Arthur CHARPENTIER - Unifying copula families and tail dependence

Quantifying tail dependence, in dimension 2 ?

Gaussian copula

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

  • 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

L and R concentration functions

L function (lower tails) R function (upper tails)

GAUSSIAN

  • Student t copula

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

  • 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

L and R concentration functions

L function (lower tails) R function (upper tails)

STUDENT (df=3)

  • Clayton copula

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

  • 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

L and R concentration functions

L function (lower tails) R function (upper tails)

CLAYTON

  • Gumbel copula

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

  • 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

L and R concentration functions

L function (lower tails) R function (upper tails)

GUMBEL

  • 21
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Arthur CHARPENTIER - Unifying copula families and tail dependence

Quantifying tail dependence, in dimension 2 ?

Gaussian copula

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

  • 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Chi dependence functions

lower tails upper tails

GAUSSIAN

  • Student t copula

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

  • 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Chi dependence functions

lower tails upper tails

STUDENT (df=3)

  • Clayton copula

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

  • ●
  • 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Chi dependence functions

lower tails upper tails

CLAYTON

  • Gumbel copula

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

  • 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Chi dependence functions

lower tails upper tails

GUMBEL

  • 22
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Arthur CHARPENTIER - Unifying copula families and tail dependence

Can describe tail dependence in dimension d ≥ 2 ?

Oh & Patton (2012) defined a crash de- pendence index (related to a measure in Embrechts, et al., 2000) : let Nu = d

i=1 1(Xi ≤ F −1 i

(u)), define πu,k = E[Nn|Nu ≥ k] − k d − k (Source : Oh & Patton (2012)) 23