✬ ✩ 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¨ oteborg, Sweden ✫ ✪
✬ ✩ Wave heights at two different locations 450 750 ELD EUR 700 400 650 350 600 Height (cm) Height (cm) 300 550 500 250 450 200 400 150 350 ELD EUR 100 300 Time (hour) Time (hour) 650 390 ELD ELD EUR EUR 380 600 370 550 360 500 Height (cm) Height (cm) 350 450 340 400 330 350 320 300 310 ✫ ✪ 250 300 Time (hour) Time (hour) 1
✬ ✩ Pseudo exchange returns Negative Daily Return Divided by Estimated Standard Deviation, 1977−2004 10 5 Neg Log Return 0 −5 −10 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 SPOT EXCHANGE RATE, JPY/USD 10 5 Neg Log Return 0 −5 −10 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 SPOT EXCHANGE RATE, CAD/USD 10 5 Neg Log Return 0 −5 ✫ −10 ✪ 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 SPOT EXCHANGE RATE, GBP/USD 2
✬ ✩ Challenges • “Choose dependence model with desired dependence properties.” – dependence in asymptotic independence – tail dependence – extremal co-movements – extremal impacts, etc. ✫ ✪ 3
✬ ✩ Outline 1. Multivariate extremes 2. M4 class 3. Some dependence measures 4. Asymptotically independent processes ✫ ✪ 4
✬ ✩ 1. Multivariate extremes ✫ ✪ 5
✬ ✩ 1. Multivariate extremes Basic definitions Suppose { X i = ( X i 1 , . . . , X iD ) , i = 1 , 2 , ... } is a D-dimensional i.i.d. random vectors with distribution F . Let M nd = max { X id , 1 ≤ i ≤ n } . If there exist normalizing constants a n > 0 , b n such that P { M nd ≤ a nd x d + b nd , 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 ) . ✫ ✪ 6
✬ ✩ 1. Multivariate extremes • 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? 7
✬ ✩ 2. M4 processes ✫ ✪ 8
✬ ✩ 2. M4 class Max-Stable Processes • Suppose { Y id , 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 { Y id ≤ y } = e − 1 /y for 0 < y < ∞ (unit Fr´ echet assumption). • The process is max-stable if for any i = i 1 , i 1 + 1 , ..., i 2 and any positive set of values { y id , i = i 1 , ..., i 2 , d = 1 , ..., D } , we have P { Y id ≤ y id , i = i 1 , ..., i 2 , d = 1 , ..., D } = P n { Y id ≤ ny id , i = i 1 , ..., i 2 , d = 1 , ..., D } . ✫ ✪ 9
✬ ✩ 2. M4 class 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 Y id = max −∞ <k< ∞ a l,k,d Z l,i − k , −∞ < i < ∞ , d = 1 , ..., D. max l =1 , 2 ,... ✫ ✪ 10
✬ ✩ 2. M4 class • Finite representations: (1) Y id = max − K 1 ld ≤ k ≤ K 2 ld a l,k,d Z l,i − k , −∞ < i < ∞ , max 1 ≤ l ≤ L d where L d , K 1 ld , K 2 ld are finite and the coefficients satisfy � L d � K 1 ld k = − K 1 ld a l,k,d = 1 for each d . l =1 • 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?) ✫ ✪ 11
✬ ✩ 2. M4 class (a) 100 90 80 70 60 50 40 30 20 10 0 0 100 200 300 400 (d) (b) 20 100 18 90 16 80 14 70 12 60 10 50 8 40 6 30 4 20 2 10 0 0 41 42 43 44 45 46 102 103 104 105 106 107 (c) (e) 100 35 90 30 80 25 70 60 20 50 15 40 30 10 20 5 ✫ ✪ 10 0 0 256 257 258 259 260 261 312 313 314 315 316 317 12
✬ ✩ 3. Some tail dependence measures ✫ ✪ 13
✬ ✩ 3. Some tail dependence measures • Definition 1 The bivariate asymptotic dependence index λ = lim u → x F 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 { X 1 , X 2 , . . . , X n } is called lag- k asympt. dep. if λ k = lim u → x F P( X k +1 > u | X 1 > u ) > 0 , u → x F P( X k + j > u | X 1 > u ) = 0 , j > 1 , lim ✫ ✪ 14
✬ ✩ 3. Some tail dependence measures u 4 4 3 3 2 2 u 1 1 N(0,1) N(0,1) 0 0 −1 −1 −2 −2 −3 −3 −4 −4 −4 −2 0 2 4 −4 −2 0 2 4 N(0,1) N(0,1) 4 4 3 3 2 2 1 1 N(0,1) N(0,1) 0 0 −1 −1 −2 −2 −3 −3 ✫ −4 −4 ✪ −4 −2 0 2 4 −4 −2 0 2 4 N(0,1) N(0,1) 15
✬ ✩ 3. Some tail dependence measures Asymptotic dependence in M4 Cross sections: max( L d ,L d ′ ) 1+max( K 1 ld ,K 1 ld ′ ) � � λ dd ′ = 2 − max { a l, 1 − m,d , a l, 1 − m,d ′ } , l =1 m =1 − max( K 2 ld ,K 2 ld ′ ) Lag- k in time: L d 1+ k + K 1 ld � � λ d ( k ) = 2 − max { a l, 1 − m,d , a l, 1+ k − m,d } . l =1 m =1 − K 2 ld ✫ ✪ 16
✬ ✩ 3. Some tail dependence measures Coefficients of tail dependence • Ledford and Tawn (1996, 1997) consider the following model: P( X 1 > x, X 2 > x ) ∼ L( x ) x − 1 /η as x → ∞ , (3) • Connection of λ and η P( X 2 > x | X 1 > x ) ∼ L( x ) x 1 − 1 /η as x → ∞ , (4) when margins are unit Fr´ echet. ✫ ✪ 17
✬ ✩ 3. Some tail dependence measures Coefficients of tail dependence in M4 η = 1 or η = 1 / 2 ✫ ✪ 18
✬ ✩ 4. Asymptotic (In)dependent M4 processes ✫ ✪ 19
✬ ✩ 4. Asymptotic (In)dependent M4 processes Extention I of M4 Replacing Z li in M4 by independent GEV shock variables: a − 1 Y id = max max l,k,d W l,i − k , d = 1 , . . . , D, −∞ < i < ∞ , l k (5) for { a l,k,d > 0 , l ≥ 1 , −∞ < k < ∞} , and { W lk , 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 ) ] − 1 /ξ } (6) σ ✫ ✪ 20
✬ ✩ 4. Asymptotic (In)dependent M4 processes Two conditions C1 Suppose all moving coefficients a lkd satisfy lkd ′ , for all d ′ = 2 , . . . , D. (7) a − 1 /ξ a − 1 /ξ � � � � = lk 1 l k l k C2 Suppose there are numbers a ∗ > 0 and n ∗ such that a ∗ = min a lkd , n ∗ = � � min 1 ( a lkd = a ∗ ) , for all d = 1 , . . . , D, l k l k (8) where 1 ( . ) is an indicator function. ✫ ✪ 21
✬ ✩ 4. Asymptotic (In)dependent M4 processes Result # 3 Under ξ > 0 and the condition C1 , we have ∞ ∞ a − 1 /ξ min( a l, 1 − m,d , a l, 1 − m,d ′ ) − 1 /ξ 2 � � − � � lkd l k l =1 m = −∞ λ dd ′ = , (9) k a − 1 /ξ � � l lkd ∞ ∞ a − 1 /ξ min( a l, 1 − m,d , a l, 1+ r − m,d ) − 1 /ξ 2 � � − � � lkd l k l =1 m = −∞ λ d r = , (10) k a − 1 /ξ � � l lkd and ∞ ∞ a − 1 /ξ min( a l, 1 − m,d , a l, 1+ r − m,d ′ ) − 1 /ξ 2 � � − � � lkd m = −∞ l k l =1 λ dd ′ r = . (11) k a − 1 /ξ � � l lkd ✫ ✪ 22
✬ ✩ 4. Asymptotic (In)dependent M4 processes Result # 4 Under ξ = 0 and the condition C2 , we have � � alks 1 ( alks = a ∗ ) Cs = l k ∞ ∞ � Cs − � � min( al, 1 − m,d, al, 1 − m,d ′ ) 1 (min( al, 1 − m,d, al, 1 − m,d ′ )= a ∗ ) s = d,d ′ m = −∞ l =1 λdd ′ = , Cd (12) ∞ ∞ 2 Cd − � � min( al, 1 − m,d, al, 1+ r − m,d ) 1 (min( al, 1 − m,d, al, 1+ r − m,d )= a ∗ ) l =1 m = −∞ λdr = , Cd (13) and ∞ ∞ � Cs − � � min( al, 1 − m,d, al, 1+ r − m,d ′ ) 1 (min( al, 1 − m,d, al, 1+ r − m,d ′ )= a ∗ ) s = d,d ′ l =1 m = −∞ = λdd ′ . r Cd ✫ ✪ (14) 23
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