[PPT] - Asymptotically (In)dependent Multivariate Maxima of Moving Maxima PowerPoint Presentation

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Asymptotically (In)dependent Multivariate Maxima of Moving Maxima Processes

Zhengjun Zhang Washington University → University of Wisconsin Co-authors: Janet E. Heffernan, Jonathan A. Tawn August 19, 2005, G¨

teborg, Sweden

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Wave heights at two different locations

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100 150 200 250 300 350 400 450 Time (hour) Height (cm) ELD EUR 300 350 400 450 500 550 600 650 700 750 Time (hour) Height (cm) ELD EUR 250 300 350 400 450 500 550 600 650 Time (hour) Height (cm) ELD EUR 300 310 320 330 340 350 360 370 380 390 Time (hour) Height (cm) ELD EUR

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Pseudo exchange returns

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01/03/77 09/30/79 06/26/82 03/22/85 12/17/87 09/12/90 06/08/93 03/04/96 11/29/98 08/25/01 05/21/04 −10 −5 5 10 SPOT EXCHANGE RATE, JPY/USD Neg Log Return Negative Daily Return Divided by Estimated Standard Deviation, 1977−2004 01/03/77 09/30/79 06/26/82 03/22/85 12/17/87 09/12/90 06/08/93 03/04/96 11/29/98 08/25/01 05/21/04 −10 −5 5 10 SPOT EXCHANGE RATE, CAD/USD Neg Log Return 01/03/77 09/30/79 06/26/82 03/22/85 12/17/87 09/12/90 06/08/93 03/04/96 11/29/98 08/25/01 05/21/04 −10 −5 5 10 SPOT EXCHANGE RATE, GBP/USD Neg Log Return

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Challenges

“Choose dependence model with desired

dependence properties.” – dependence in asymptotic independence – tail dependence – extremal co-movements – extremal impacts, etc.

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Outline

1. Multivariate extremes
2. M4 class
3. Some dependence measures
4. Asymptotically independent processes

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1. Multivariate extremes

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1. Multivariate extremes

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Basic definitions

Suppose {Xi = (Xi1, . . . , XiD), i = 1, 2, ...} is a D-dimensional i.i.d. random vectors with distribution F. Let Mnd = max{Xid, 1 ≤ i ≤ n}. If there exist normalizing constants an > 0, bn such that P{Mnd ≤ andxd + bnd, d = 1, . . . , D} → H(x), then the distribution H is called a D-dimensional multivariate extreme value distribution and F is said to belong to the domain of attraction of H, which we write F ∈ D(H).

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1. Multivariate extremes

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Representations: Pickands, de Haan and Resnick,

Deheuvels (1970s) gave general representation formulae for MEVDs (see Resnicks (1987) book for full description). However these formulae are too general to be useful for statistics.

Statistics: Much work on parametric subfamilies

(Tawn, Coles, ....) and on nonparametric estimation methods but these work well only for small D.

Problem 1: What to do about large D? (e.g.

D ≈ 100 for a typical portfolio)

Problem 2: How to extend these methods to take

into account also time-series dependence within each series?

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2. M4 processes

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2. M4 class

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Max-Stable Processes

Suppose {Yid, i = 0, 1, 2, d = 1, ..., D} is a

D-dimensional time series with discrete time index i.

W.l.o.g. we may assume P{Yid ≤ y} = e−1/y for

0 < y < ∞ (unit Fr´ echet assumption).

The process is max-stable if for any i = i1, i1 + 1, ..., i2

and any positive set of values {yid, i = i1, ..., i2, d = 1, ..., D}, we have P{Yid ≤ yid, i = i1, ..., i2, d = 1, ..., D} = Pn{Yid ≤ nyid, i = i1, ..., i2, d = 1, ..., D}.

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2. M4 class

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Representations

For univariate processes, such a characterization was provided by Deheuvels (1983), Davis and Resnick (1989), Hall, Peng and Yao (2002), Ferreira and de Haan (2005). This was generalized by Smith and Weissman (1996) to the following: any max-stable process with unit Fr´ echet margins may be approximated by a multivariate maxima of moving maxima process, or M4 for short, with the representation Yid = max

l=1,2,...

max

−∞<k<∞ al,k,dZl,i−k, −∞ < i < ∞, d = 1, ..., D. 10

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2. M4 class

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Finite representations:

Yid = max

1≤l≤Ld

max

−K1ld≤k≤K2ldal,k,dZl,i−k, −∞ < i < ∞,

(1) where Ld, K1ld, K2ld are finite and the coefficients satisfy Ld

l=1

K1ld

k=−K1ld al,k,d = 1 for each d.

Estimation: Based on bivariate joint distributions
Applications in finance: Zhang and Smith (2003),

Zhang (AIE vol 20, 2005), Zhang and Huang (2005).

Application in wave height data: Zhang (2003?)

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2. M4 class

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100 200 300 400 10 20 30 40 50 60 70 80 90 100

(a)

41 42 43 44 45 46 2 4 6 8 10 12 14 16 18 20

(d)

102 103 104 105 106 107 10 20 30 40 50 60 70 80 90 100

(b)

256 257 258 259 260 261 10 20 30 40 50 60 70 80 90 100

(c)

312 313 314 315 316 317 5 10 15 20 25 30 35

(e)

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3. Some tail dependence measures

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3. Some tail dependence measures

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Definition 1 The bivariate asymptotic dependence

index λ = lim

u→xF P(X > u|Y > u).

(2) λ > 0 ⇒ aysmpt. dep., otherwise asympt. indep.

Sibuya (1960), de Haan and Resnick (1977), Embrechts, McNeil, and Straumann (2002), Zhang (2004).

Definition 2 A sequence of variables

{X1, X2, . . . , Xn} is called lag-k asympt. dep. if

λk = lim

u→xF P(Xk+1 > u|X1 > u) > 0,

lim

u→xF P(Xk+j > u|X1 > u) = 0, j > 1,

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3. Some tail dependence measures

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−4 −2 2 4 −4 −3 −2 −1 1 2 3 4 N(0,1) N(0,1) −4 −2 2 4 −4 −3 −2 −1 1 2 3 4 N(0,1) N(0,1) −4 −2 2 4 −4 −3 −2 −1 1 2 3 4 N(0,1) N(0,1) −4 −2 2 4 −4 −3 −2 −1 1 2 3 4 N(0,1) N(0,1) u u

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3. Some tail dependence measures

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Asymptotic dependence in M4

Cross sections:

λdd′ = 2 −

max(Ld,Ld′ )

l=1

1+max(K1ld,K1ld′ )

m=1−max(K2ld,K2ld′ )

max{al,1−m,d, al,1−m,d′},

Lag-k in time:

λd(k) = 2 −

Ld

l=1

1+k+K1ld

m=1−K2ld

max{al,1−m,d, al,1+k−m,d}.

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3. Some tail dependence measures

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Coefficients of tail dependence

Ledford and Tawn (1996, 1997) consider the

following model: P(X1 > x, X2 > x) ∼ L(x)x−1/η as x → ∞, (3)

Connection of λ and η

P(X2 > x|X1 > x) ∼ L(x)x1−1/η as x → ∞, (4) when margins are unit Fr´ echet.

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3. Some tail dependence measures

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Coefficients of tail dependence in M4

η = 1

r

η = 1/2

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4. Asymptotic (In)dependent M4 processes

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4. Asymptotic (In)dependent M4 processes

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Extention I of M4

Replacing Zli in M4 by independent GEV shock variables: Yid = max

l

max

k

a−1

l,k,dWl,i−k,

d = 1, . . . , D, −∞ < i < ∞, (5) for {al,k,d > 0, l ≥ 1, −∞ < k < ∞}, and {Wlk, l ≥ 1, −∞ < k < ∞} being an array of independent GEV shock variables which have a unified distribution form H(x; µ, σ, ξ) = exp{−[1 + ξ(x − µ) σ ]−1/ξ} (6)

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4. Asymptotic (In)dependent M4 processes

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Two conditions

C1 Suppose all moving coefficients alkd satisfy

l
k

a−1/ξ

lk1

=

l
k

a−1/ξ

lkd′ , for all d′ = 2, . . . , D. (7)

C2 Suppose there are numbers a∗ > 0 and n∗ such that a∗ = min

l

min

k

alkd, n∗ =

l
k

1(alkd=a∗), for all d = 1, . . . , D, (8) where 1(.) is an indicator function.

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4. Asymptotic (In)dependent M4 processes

✬ ✫ ✩ ✪ Result # 3 Under ξ > 0 and the condition C1, we have

λdd′ = 2

l

k

a−1/ξ

lkd

−

∞

l=1

∞

m=−∞

min(al,1−m,d, al,1−m,d′)−1/ξ

l
k a−1/ξ

lkd

, (9) λdr = 2

l

k

a−1/ξ

lkd

−

∞

l=1

∞

m=−∞

min(al,1−m,d, al,1+r−m,d)−1/ξ

l
k a−1/ξ

lkd

, (10) and λdd′

r =

2 l

k

a−1/ξ

lkd

−

∞

l=1

∞

m=−∞

min(al,1−m,d, al,1+r−m,d′)−1/ξ

l
k a−1/ξ

lkd

. (11)

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4. Asymptotic (In)dependent M4 processes

✬ ✫ ✩ ✪ Result # 4 Under ξ = 0 and the condition C2, we have

Cs =

l
k

alks1(alks=a∗) λdd′ =

s=d,d′

Cs − ∞

l=1

∞

m=−∞

min(al,1−m,d, al,1−m,d′ )1(min(al,1−m,d, al,1−m,d′ )=a∗) Cd , (12) λdr = 2Cd − ∞

l=1

∞

m=−∞

min(al,1−m,d, al,1+r−m,d)1(min(al,1−m,d, al,1+r−m,d)=a∗) Cd , (13) and λdd′ r =

s=d,d′

Cs − ∞

l=1

∞

m=−∞

min(al,1−m,d, al,1+r−m,d′ )1(min(al,1−m,d, al,1+r−m,d′ )=a∗) Cd . (14)

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4. Asymptotic (In)dependent M4 processes

✬ ✫ ✩ ✪ Result # 5 Under ξ < 0 and the condition C2, we have

λdd′ = 2 l

k

1{alkd=a∗} − l

m

1{min(al,1−m,d, al,1−m,d′ )=a∗}

l
k

1{alkd=a∗} , (15) λdr = 2 l

k

1{alkd=a∗} − l

m

1{min(al,1−m,d, al,1+r−m,d)=a∗}

l
k

1{alkd=a∗} , (16) and λdd′ r =

s=d,d′
l
k

1(alks=a∗) − ∞

l=1

∞

m=−∞

1(min(al,1−m,d, al,1+r−m,d′ )=a∗)

l
k 1(alkd=a∗)

. (17)

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3. Some tail dependence measures

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Coeff. of tail dependence in Extension I

η = 1

r

η = 1/2

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4. Asymptotic (In)dependent M4 processes

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Extention II of M4

The new model is as follows: Yid = max(W ′1/α

id

, max

l

max

k

a−1

l,k,dWl,i−k),

d = 1, · · · , D, (18) where α > 0, al,k,d > 0, {Wli, l ≥ 1, −∞ < i < ∞} are an array of independent positive random variables; {W ′

id, −∞ < i < ∞, d = 1, . . . , D} are an array of

independent positive random variables, and they are independent of Wli. These max-shock random variables are identically distributed.

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4. Asymptotic (In)dependent M4 processes

✬ ✫ ✩ ✪ Result # 6 Under the conditions that max-shock variable being unit Fr´ echet and that C1 with the value being 1, we have

λdd′ =    0, if α < 1; 2 −

l

k max(a−1

lk1, a−1 lk2),

if α ≥ 1, (19) λdr =    0, if α < 1; 2 −

l

k max(a−1

lkd, a−1 l,k+r,d),

if α ≥ 1, (20) and λdd′

r =

   0, if α < 1; 2 −

l

k max(a−1

lkd, a−1 l,k+r,d′),

if α ≥ 1. (21)

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4. Asymptotic (In)dependent M4 processes

✬ ✫ ✩ ✪ Result # 7 The coefficients of tail dependence are given by η =    max(1/2, α), if α < 1; 1, if α ≥ 1. (22)

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4. Asymptotic (In)dependent M4 processes

✬ ✫ ✩ ✪ Result # 8 Under the conditions that max-shock variable being unit exponential. In addition, C2 holds, and alkd > 1 , we have

λdd′ =      0, if α ≤ 1;

2n∗−

l,m 1[1(al,1−m,d=a∗)+1(al,1−m,d′ =a∗)=1]

n∗

, if α > 1, (23) λdr =      0, if α ≤ 1;

2n∗−

l,m 1[1(al,1−m,d=a∗)+1(al,1+r−m,d=a∗)=1]

n∗

, if α > 1, (24)

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4. Asymptotic (In)dependent M4 processes

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−5 5 10 −5 5 10 SC1 ξ=1 −5 5 10 −5 5 10 SC2 −5 5 10 −2 2 4 6 SC3 −5 5 10 −2 2 4 6 ξ=0 −5 5 10 −2 2 4 6 −5 5 10 −2 2 4 6 −5 5 10 −2 2 4 6 ξ=−1 −5 5 10 −2 2 4 6 −5 5 10 −5 5 10 −10 10 20 −5 5 10 Frechet −10 10 20 −5 5 10 −10 10 20 −5 5 10 −10 10 20 −5 5 10 Exponential −2 2 4 −2 2 4 −2 2 −2 −1 1 2

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Concluding remarks

1. M4 class of processes are very flexible models for

temporally dependent multivariate extreme value processes.

2. All variables in these M4 models are asymptotically

dependent.

3. The purpose of this paper is to extend this M4 class

in a number of ways to produce classes of models which are also asymptotically independent.

4. Ultimately, all of these models will be explored in

modeling financial data, environmental data, etc.

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The man who introduced the name of copula in statistics

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