Slides lausanne-2013-v2
23
Arthur CHARPENTIER - Unifying copula families and tail dependence Unifying standard multivariate copulas families (with tail dependence properties) Arthur Charpentier [email protected] 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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Transcript of Slides lausanne-2013-v2
11pt
11pt
Note Exemple Exemple
11pt
Preuve
Arthur CHARPENTIER - Unifying copula families and tail dependence
Unifying standard multivariate copulas families(with tail dependence properties)
Arthur Charpentier
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
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
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Arthur CHARPENTIER - Unifying copula families and tail dependence
CopulasDefinition 1A 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 copulais usually not unique, but it is if F1, ..., Fd are absolutely continuous, and then,
C(u1, ..., ud) = F (F−11 (u1), ..., F−1
d (ud)), ∀(u1, , ..., ud) ∈ [0, 1]d (2)
where quantile functions F−11 , ..., F−1
n are generalized inverse (left cont.) of Fi’s.
If X ∼ F , then U = (F1(X1), · · · , Fd(Xd)) ∼ C.
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Arthur CHARPENTIER - Unifying copula families and tail dependence
Survival (or dual) copulasTheorem 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). Thiscopula is usually not unique, but it is if F1, ..., Fd are absolutely continuous, andthen,
C?(u1, ..., ud) = F (F−11 (u1), ..., F−1
d (ud)), ∀(u1, , ..., ud) ∈ [0, 1]d (4)
where quantile functions F−11 , ..., 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?.
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Arthur CHARPENTIER - Unifying copula families and tail dependence
Benchmark copulasDefinition 2The independent copula C⊥ is defined as
C⊥(u1, ..., un) = u1 × · · · × ud =d∏i=1
ui.
Definition 3The comonotonic copula C+ (the Fréchet-Hoeffding upper bound of the set of copulas)is the copuladefined as C+(u1, ..., ud) = min{u1, ..., ud}.
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Arthur CHARPENTIER - Unifying copula families and tail dependence
Spherical distributions
Definition 4Random vectorX 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).
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Those distribution can be non-symmetric, see Hartman & Wintner (AJM, 1940)or Cambanis, Huang & Simons (JMVA, 1979))
6
Arthur CHARPENTIER - Unifying copula families and tail dependence
Elliptical distributions
Definition 5Random vectorX 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 AsatisfiesAA′ = Σ.
E.g. X ∼ N (µ,Σ).
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Elliptical distribution are popular in finance, see e.g. Jondeau, Poon & Rockinger(FMPM, 2008)
7
Arthur CHARPENTIER - Unifying copula families and tail dependence
Archimedean copulaDefinition 6If d ≥ 2, an Archimedean generator is a function φ : [0, 1]→ [0,∞) such that φ−1 isd-completely monotone (i.e. ψ is d-completely monotone if ψ is continuous and∀k = 0, 1, ..., d, (−1)kdkψ(t)/dtk ≥ 0).
Definition 7Copula C is an Archimedean copula is, for some generator φ,
C(u1, ..., ud) = φ−1(φ(u1) + ...+ φ(ud)),∀u1, ..., ud ∈ [0, 1].
Exemple1φ(t) = − log(t) yields the independent copula C⊥.
8
Arthur CHARPENTIER - Unifying copula families and tail dependence
Stochastic representation of Archimedean copulas
Consider some striclty positive random variable Rindependent of U , uniform on the simplex of Rd.The survival copula ofX = R ·U is Archimedean,and its generator is the inverse of Williamson d-transform,
φ−1(t) =∫ ∞x
(1− x
t
)d−1dFR(t).
Note that R L= φ(U1) + · · ·+ φ(Ud)
0.0 0.5 1.0 1.5 2.0 2.5
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See Nešlehová & McNeil (AS, 2009).
9
Arthur CHARPENTIER - Unifying copula families and tail dependence
Archimedean copula, exchangeability and frailties
Consider residual life times X = (X1, · · · , Xd) condition-ally independent given some latent factor Θ, and such thatP(Xi > xi|Θ) = Bi(xi)θ. Then
F (x) = P(X > x) = ψ
(−
n∑i=1
logF 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).
0 20 40 60 80 100
020
4060
80100
Conditional independence, continuous risk factor
!3 !2 !1 0 1 2 3
!3!2
!10
12
3
Conditional independence, continuous risk factor
10
Arthur CHARPENTIER - Unifying copula families and tail dependence
Nested Archimedean copula, and hierarchical structuresConsider C(u1, · · · , ud) defined as
φ−11 [φ1[φ−1
2 (φ2[· · ·φ−1d−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−1i−1 is the inverse of a
Laplace transform, and is called fully nested Archimedean copula. Note thatpartial 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
Arthur CHARPENTIER - Unifying copula families and tail dependence
(Univariate) extreme value distributions
Central limit theorem, Xi ∼ F i.i.d. Xn − bnan
L→ S as n→∞ where S is anon-degenerate random variable.
Fisher-Tippett theorem, Xi ∼ F i.i.d., Xn:n − bnan
L→M as n→∞ where M is anon-degenerate random variable.
Then
P(Xn:n − bn
an≤ x
)= Fn(anx+ bn)→ G(x) as n→∞,∀x ∈ R
i.e. F belongs to the max domain of attraction of G, G being an extreme valuedistribution : the limiting distribution of the normalized maxima.
− logG(x) = (1 + ξx)−1/ξ+
12
Arthur CHARPENTIER - Unifying copula families and tail dependence
(Multivariate) extreme value distributionsAssume that Xi ∼ F i.i.d.,
Fn(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 normalizedcomponentwise maxima,
Xn:n = (max{X1,i}, · · · ,max{Xd,i})
− logG(x) = µ([0,∞)\[0,x]),∀x ∈ Rd+where µ is the exponent measure. It is more common to use the stable taildependence function ` defined as
`(x) = µ([0,∞)\[0,x−1]),∀x ∈ Rd+
13
Arthur CHARPENTIER - Unifying copula families and tail dependence
i.e.− logG(x) = `(− logG1(x1), · · · , logGd(xd))∀x ∈ Rd
Note that there exists a finite measure H on the simplex of Rd such that
`(x1, · · · , xd) =∫Sd
max{ω1x1, · · · , ωdxd}dH(ω1, · · · , ωd)
for all (x1, · · · , xd) ∈ Rd+, and∫SdωidH(ω1, · · · , ωd) = 1 for all i = 1, · · · , n.
Definition 8Copula C : [0, 1]d → [0, 1] is an multivariate extreme value copula if and only if thereexists a stable tail dependence function such that `
C(u1, · · · , ud) = exp[−`(− log u1, · · · ,− log ud)]
Assume that U i ∼ C i.i.d.,
Cn(u 1n ) = Cn(u
1n1 , · · · , u
1n
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
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].
Exemple2A(ω) = 1 yields the independent copula, C⊥.
15
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)) ifC = 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
Arthur CHARPENTIER - Unifying copula families and tail dependence
Quantifying tail dependence, in dimension 2 ?Venter (2002) suggested to visualize tail concentration functions,Definition 9For 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 upperlimits, respectively (if those limits exist)
λU = R(1) = limz→1
R(z) et λL = L(0) = limz→0
L(z).
17
Arthur CHARPENTIER - Unifying copula families and tail dependence
Quantifying tail dependence, in dimension 2 ?Definition 10Let (X,Y ) denote a random vector in R2. Define tail dependence indices in the lower(L) and upper (U ) tails as
λL = limu↓0
P(X ≤ F−1
X (u) |Y ≤ F−1Y (u)
)∈ [0, 1],
andλU = lim
u↑1P(X > F−1
X (u) |Y > F−1Y (u)
)∈ [0, 1].
Proposition 1Let (X,Y ) denote a random vector with copula C, then
λL = limu↓0
C(u, u)u
and λU = limu↓0
C?(u, u)u
.
18
Arthur CHARPENTIER - Unifying copula families and tail dependence
Quantifying tail dependence, in dimension 2 ?Exemple3For Archimedean copulas (see Nelsen (2007), C. & Segers (JMVA, 2008)),
λU = 2− limx→0
1− φ−1(2x)1− φ−1(x) and λL = lim
x↓0
φ−1(2φ(x))x
= limx↓∞
φ−1(2x)φ−1(x) .
Ledford and Tawn (B, 1996) suggested an alternative approach : assume thatXL= 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→∞, whereη ∈ [1, 2] will be a tail dependence index.
19
Arthur CHARPENTIER - Unifying copula families and tail dependence
Quantifying tail dependence, in dimension 2 ?Following Coles, Heffernan & Tawn (E, 1999) defineDefinition 11Let
χU (z) = 2 log(1− z)logC?(z, z) − 1 et χL(z) = 2 log(1− z)
logC(z, z) − 1
Then ηU = (1 + limz→0 χU (z))/2 and ηL = (1 + limz→0 χL(z))/2 are respectively tailindices in the upper and lower tail, respectively.
Exemple4If (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
Arthur CHARPENTIER - Unifying copula families and tail dependence
Quantifying tail dependence, in dimension 2 ?Gaussian copula
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0.0 0.2 0.4 0.6 0.8 1.0
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L and R concentration functions
L function (lower tails) R function (upper tails)
GAUSSIAN
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Student t copula
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0.0 0.2 0.4 0.6 0.8 1.0
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1.0
L and R concentration functions
L function (lower tails) R function (upper tails)
STUDENT (df=3)
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Clayton copula
0.0 0.2 0.4 0.6 0.8 1.0
0.0
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1.0
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0.0 0.2 0.4 0.6 0.8 1.0
0.0
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0.6
0.8
1.0
L and R concentration functions
L function (lower tails) R function (upper tails)
CLAYTON
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Gumbel copula
0.0 0.2 0.4 0.6 0.8 1.0
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0.0 0.2 0.4 0.6 0.8 1.0
0.0
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1.0
L and R concentration functions
L function (lower tails) R function (upper tails)
GUMBEL
●
●
21
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
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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
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lower tails upper tails
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0.0 0.2 0.4 0.6 0.8 1.0
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0.0 0.2 0.4 0.6 0.8 1.0
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1.0
Chi dependence functions
lower tails upper tails
CLAYTON
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Gumbel copula
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0.0 0.2 0.4 0.6 0.8 1.0
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Chi dependence functions
lower tails upper tails
GUMBEL
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22
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 inEmbrechts, et al., 2000) :let Nu =
∑di=1 1(Xi ≤ F−1
i (u)),define
πu,k = E[Nn|Nu ≥ k]− kd− k
(Source : Oh & Patton (2012))
23
