A C T A U N I V E R S I T A T I S L O D Z I E N S I S
F O L IA O E C O N O M IC A 194, 2005
K r z y s z t o f J a ju ^ a *
TAIL D E PE N D E N C E IN BIVARIATE DIST R IBU TIO N S
Abstract
In the p a p er the p ro b lem o f tail d ependence fo r bivariate d a ta is considerod. T h e review o f d ifferent ap p ro ach es is given. T h e p a rtic u la r em phasis is p u t on the co n d itio n al co rre latio n coefficients and tail d ependence coefficients. It is show n how the latter can be analyzed th ro u g h c o p u la analysis.
Key words: tail d ep en d en ce, co p u la analysis, b ivariate d istrib u tio n .
* — * I. IN T R O D U C T IO N C Q
T h e analysts o f the dependence (relationship) between variables is one o f the m ost im portant tasks in m ultivariate (particularly - bivariate) statistical analysis. One usually considers either the so called jo in t relation ship or the so callcd conditional relationship. In the first type o f relationship all variables arc regarded as whole set, in the second type o f relationship one (or m ore) variable is regarded as the dep en d en t variable and the other variables are considered as the independent variables.
T h e analysis o f the relationship is usually perform ed th ro u g h tw o different q u an titativ e approaches:
m odeling the relationship by a function - for exam ple - regression function;
m easuring the relationship by a num ber - for exam ple - co rrelatio n coefficient.
Very often, how ever, these tw o ap p roach es are strictly connected, as it is in the case o f regression function and co rrelatio n coefficient.
* P rofessor, D e p a rtm e n t o f F in an cial Investm ents and In su ran ce, W rocław U niversity o f E conom ics.
W hen one assum es stochastic ap p ro ach , the variables arc treated as random variables, then all inform ation a b o u t the relationship (dependence) is contained in the cum ulative d istrib u tio n function. F o r exam ple, if we consider tw o variables, X and У, then this in form ation is given as:
- for jo in t relationship:
P ( X < x , Y < y ) ,
- for conditional relationship:
Р ( У < у |А г < х ) .
O f course, in the applications, the simplification is made. Instead o f cum ulative d istribution function one takes into account only som e p aram eters, usually m om ents of the distrib u tio n . T hen we get for example:
- for jo in t relationship:
C O V (X , Y) = E ( X Y ) - E (X )E (Y ),
- for conditional relationship:
Е (У |Х ).
Such a sim plification, how ever, m ay n o t cap tu re the p articu lar properties o f the relationship, for exam ple the tail relationship, th a t is the relationship existing between the very large (or very sm all) values o f two variables. I his is sim ilar problem as in univariate analysis, where the classical “m ean-b ased” analysis does n o t ca p tu re the extrem e peculiarities.
In this paper we discuss the problem o f tail dependence. F o r sim plicity, we consider the case o f b ivariate distrib u tio n s (two variables: X and У).
II. M O D ELU N G T A IL D E P E N D E N C E - D IF F E R E N T A P P R O A C H E S
T here are different ap p ro ach es th a t can be used in the m odeling o f tail dependence. W e divide them into three classes:
- separate m odeling o f center and tails o f distribution; - conditional dependence m easures;
Sep arate modeling o f center and tails o f distribution
T he first ap p ro ach consists in the separation o f d a ta set in to (usually) two classes. T h e first class co n tain s the “ cen ter” (the “co re” ) o f the m ultivariate d istrib u tio n , here the m odeling o f the relationships is done for the “ typical” observations. I he second class con tains the tails (the “ o u tlie rs” ) o f the m u ltiv a ria te d istrib u tio n , here th e m od eling o f the relationships is do n e for the extrem e values. It m ay also h ap p en th a t the d a ta set is separated into m ore th an tw o classes (when m ore th an one tail is considered). In this ap p ro ach we can distinguish tw o groups of m ethods:
- clustering m ethods; - m ixture m odels.
C lustering m eth o d s aim a t classifying the d a ta set into classes, in such a way th a t the observations in the sam e class are as sim ilar as possible, and the observations in different classes are as dissim ilar as possible. In m any m ethods, the clustering optim ization criterion is defined. T his crite rio n depend s on the goal o f classification and th e u n d e rsta n d in g o f sim ilarity o f the o b serv atio n s. F o r exam ple, in the on e o f the m o st p o p u lar m ethods, к -m eans m eth o d , the sim ilarity is m easured th ro u g h the Euclidean distance between the observations. T his m eans th a t fo r the p urpose o f the m odeling o f the relationship one has to apply the suitable criterion.
T h e second group, m ixture m odels, assum es stochastic ap p ro ach . H ere the m ultivariate d istrib u tio n is treated as a m ixture o f d istrib u tio n s, where the respective com ponents o f the m ixture correspond to the center and tails o f the distribution. M ixture m odels are described for exam ple by M cL achlan and Peel (2000).
C onditional dcpcndcncc m easures
H ere one considers the co nditional distrib u tio n o f two variables given th a t one o f these variables takes the value from the tail. As the n atural dependence m easure the so called co nditional co rrelatio n coefficient can be used. It is given by the follow ing form ula (w ith o u t the loss o f generality we consider the up p er tail):
C O V (X , y | Z > s )
P c
= ——
•
ПЛ
H ere s denotes the large value o f the variable. T herefore this con ditio n al correlation coefficient is defined given th a t one variable takes large (extrem e) value. It can be proved th a t for som e bivariate d istrib ution s the co n ditio nal co rrelatio n coefficient is related to correlatio n coefficient.
F o r bivariate (stan d a rd ) norm al d istrib u tio n we have:
Pc -
\
2 • (2)V ' n * i * > S)
In th e lim iting case - if s goes to infinity we get:
P 1 /14
PC ( )
Therefore the conditional co rrelatio n coefficient converges to zero regardless o f the value o f (unconditional) co rrelatio n coefficient.
O n the o ther h an d , for b iv ariate t distrib utio n with degrees o f freedom we have in lim iting case:
P c ~ * __________ j— z--- ‘ (4 )
v V + ( v - 1) - p 2)
T able 1 presents the lim it o f co n d itio n al correlatio n coefficient for b ivariate t distribution - different nu m ber degrees o f freedom in colum ns and different unconditional correlatio n coefficient in rows.
T abic 1. C o n d itio n a l co rre latio n coefficient - lim it in the case o f b iv ariate í d istrib u tio n
3 4 10 20 30 -0 .9 -0 .7 9 -0 .6 8 -0 .4 3 -0.31 -0 .2 5 -0 .5 -0 .4 5 -0 .3 5 -0 .1 9 -0.13 -0 .1 0 0 0 0 0 0 0 0.5 0.45 0.35 0.19 0.13 0.10 0.9 0.79 0.68 0.43 0.31 0.25
In all presented cases - cxcept for 0 - th e conditional correlation coefficient is lower th an (u nconditional) co rrelatio n coefficient. It also gets low er when the num ber o f degrees o f freedom increases - but it is different from 0.
Som etim es the o th e r concept o f co nditio nal co rrelatio n coefficient is used, where instead o f one variable, condition ing is o n bo th variables. T h en we get the follow ing version o f co n d itio n al correlatio n coefficient:
COV(X, y | X > s ,
Y > s )Q c c — F - --- - ■ ... ■ ■ --- - —= * ( 5 ) J V { X \ X > s t
y > s ) F ( y | X > s ,
Y > s )It can be proved th a t in the case o f bivariate n orm al d istrib u tio n we get in the lim iting case:
1 + p 1 1 — p s2
P c ^ - P - , — (6)
So also here the conditional correlation coefficient converges to zero regardless o f the value o f (uncon d itio n al) correlatio n coefficient. A lso the prop erties o f this version o f co n d itio n al co rrelatio n coefficient are sim ilar to the one in the previous version. M o re detailed description o f con ditio nal co rrelatio n coefficients is given by M alevergne and S ornette (2002).
It should be n o ted , th a t in b o th version o f co nditional co rrelatio n coefficient we still are lim ited to linear relationship, which is th e m ain draw back o f this ap p ro ach . T his d raw b ack (as well as som e others) does not exist in the o th er ap p ro ach , tail dependence m easures, described below.
III. TAIL DEPENDENCE MEASURES
T his is the o th er g roup o f dependence m easures, where o ne looks directly into tails o f the bivariate d istrib u tio n . T here are tw o coefficients o f tail dependence, namely:
- coefficient o f low er tail dependence, given as:
Ai = l i m P ( y < G - 1( u ) | A '< F - 1(«)), (7) u->0
- coefficient o f upper tail dependence, given as:
= \imP(Y> G~l (u)\X > F-'iu )),
H ere F and G d en o te the cum ulative m arginal d istribu tio n function o f X and Y, respectively, and и denotes the probability.
Both tail dependence coefficients have ra th e r clear interpretation . T hey show the probability th a t one variable takes extrem ely large value (case o f upper tail dependence) or extremely small value (case o f lower tail dependence) given the o ther variables takes extrem ely large value (case o f u pp er tail dcpcndcncc) o r extrem ely small value (ease o f lower tail dependence). T his probability is taken as lim iting probability and in fact one speaks ab o u t asym ptotic tail dependence (or independence).
As one can sec from (7) and (8) the extrem ely large o r extrem ely small values arc taken as high o r low quan tile and in the limit these quantiles converge to (plus or m inus) infinity. In addition, it should be m entioned that:
the tail dependence coefficient falls into interval [0;1];
we speak o f asym ptotic tail independence if tail dependence coefficient is equal to 0;
- we speak o f asym ptotic tail dependence if tail dependence coefficient is higher than 0.
In practice, the calculation o f tail dependence coefficient is n o t easy. T here is one im p o rtan t case, w hen this coefficient is given th ro u g h the analytical form ula. T his refers to bivariate clliptically sym m etric d istribu tio ns. T he density o f clliptically sym m etric distribution is given as (sec e.g. Jajuga, 1993):
f ( x ) = c | i r ° - 5h [ ( x - / * ) r I - 1C *:-/')]. (9)
A m ong the m em bers o f this family are the following m ultivariate distributions: norm al distribution, an d m o re general - K o tz type d istrib utio n; - C auchy distrib u tio n , an d m o re general t d istrib utio n and even m ore general P earson type VII distrib ution;
- Pearson type II distributio n; - logistic distribution, etc.
T h e upper tail dependence coefficient for bivariate clliptically sym m etric distributions depends on the correlation coefficient and is given as (Em brcchts, M cN cil, S trau m an n , 1999): n/2 j cos “tdt i (ж/ 2 - aresinp) / 2 , , AU — „/г (Ю) J" cosatdt 0
H ere a. denotes the tail index o f the distrib utio n (for exam ple, in the case o f t d istrib ution is equal to the n u m ber o f degrees o f freedom ).
F rom the form ula (10) it can he proved th a t for the n o rm al d istrib u tio n we have the asym ptotic tail independence, if the correlation coefficient is different from + 1 and -1 . T herefo re, bivariate norm al d istrib u tio n is not the suitable m odel to ca p tu re tail de; endencc. O n the o th er h a n d , for bivariate t d istrib u tio n , if the correlation coefficient is different from m inus 1, we have asym ptotic tail dependence.
T ab le 2 (taken from E m brcchts, M cN eil, S trau m an n , 1999) presents the upper tail dependence coefficients for bivariate t d istribu tio n - different num ber degrees o f freedom in colum ns and different unconditional correlation coefficient in rows.
T abic 2. U p p e r tail dependence coefficient - case o f b iv ariate t d istrib u tio n
-0 .5 0 0.5 0.9
2 0.06 0.18 0.39 0.72
4 0.01 0.08 0.25 0.63
10 0 0.01 0.08 0.46
In general situation the analytical form ulas for tail dependence coefficients arc no t given. H ow ever, in som e cases on still can arrive at the so lution by applying the so called cop u la analysis. T h e presen tation o f this idea is given below.
IV. C O P U L A A N A L Y S IS A N D T A IL D E P E N D E N C E
T h e idea o f cop u la analysis lies in the decom position o f the m ultiv ariate distribution into two com ponents. T he first com ponent consists o f the m arginal distributions. The second com ponent - the crucial one - is the function linking these m arginal distributions in m ultivariate distribution. This function reflects th e stru ctu re of the relationship betw een the com ponents o f the m u ltiv ariate random vector. F o r sim plicity, we consider the bivariate case.
T his idea is reflected in S klar theorem , given thro u g h th e follow ing form ula:
H ( x t , x 2) = C (F l ( x l), F 2( x 2)), (11) where:
I I - the m ultiv ariate d istrib u tio n function;
F,. - the d istrib u tio n function o f the i-th m arginal d istrib ution ; С - cop ula function.
T h u s the bivariate d istrib u tio n function is the function o f the univariate (m arginal) d istrib u tio n functions. T his function is called co p u la function and it reflects the stru ctu re o f the relationships between the univariate com ponents. In the case o f b ivariate continu ou s distribution the presentation given by (11) is unique.
T h e p resen tation given by (11) can be reverted. H ere the co p u la function is given as bivariate distrib u tio n function defined for the quan tiles o f the m arginal distributions. It is given as:
C (u i> u2.) = H (F i 1(u l ), p21(u2)). (12)
A m ong the p articu lar cases arc (already discussed) bivariate n orm al dist ribu tion and bivariate t d istrib u tio n . W hen this d istrib u tio n is decom posed according to Sklar theorem , we get the so called norm al copula and t copula. T h eir analytical form is given as:
- norm al copula: ч Ф"г(“,) Ф” г“2) 1 ( x 2 - 2 p x y + y 2\
J. J. W
T
<13) - t copula: í (ui) * (“2) J / — 2 . 0 X V -4-- L J . 2* j r z A ' + - ^ ) ~ v ' n 4 x J y - (l4 )So we see th at in b o th cases, num erical procedures are needed to calculate the values o f copula function.
A m ong the o ther interesting types o f copulas, it is w orth to m en tio n the so called A rchim edean copulas. T hey are defined for strictly decreasing and convex function in the follow ing way:
c ( u i >«2) = ^ + (15)
where:
¥ : [0; 1] —► [0; со) V'(l) = 0.
T he m o st p o p u lar case o f A rchim edean cop u la is G um bel copula, where:
H ere p aram eter ß is interpreted as a m easure o f dependence, tak in g values from 1 to infinity. T he value equal to 1 m eans independence, and the closer is this value to infinity, the closer is to the strict positive dependence.
G um bel cop u la can be w ritten in the follow ing form:
C (u t , u2) = exp( — (( — In U j / + ( — ln u2) ß) i,ß). (17)
T h e crucial property o f the co p u la function refers to the tail dependence coefficients. It tu rn s o u t th a t b o th , upper tail and lower tail dependence coefficients can be expressed th ro u g h the copu la function, in the follow ing way:
the low er tail dependence coefficient:
h = lim [C(u, u))/u\, (18)
и- 0
- the upper tail dependence coefficient:
Xv = lim [(l — 2u + C(u, u ))/(l - и ) ] . (19)
14-1
Using (19), it can be proved th a t for the G um bel cop ula we get asym ptotic tail dependence, if:
ß > \ .
T hen the upper tail dependence coefficient is equal to:
Xa = 2 — 2 ^ . (20)
In practice, the im p o rta n t issue is, o f course, the identification o f suitable cop u la function for given b ivariate d a ta sets - this helps to d eterm ine the asym ptotic tail dependence coefficients.
REFERENCES
Em brechts P., M cN eil A., Stxaum ann D . (1999), Correlation and Dependence in R isk M anagement:
Properties and P itfalls, E T H Z w ork in g p a p er, Z urich.
Jajuga K . (1993), Statystyczna analiza wielowymiarowa (in Polish), W yd. N auk. P W N , W arszaw a. M alevergne Y ., S o rn ette D . (2002), Investigating E xtrem e Dependences: C oncepts and Tools,
m an u scrip t, w w w .glo riam u n d i.o rg
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Krzysztof Jajuga
Z A L E Ż N O Ś Ć W O G O N IK D L A R O Z K Ł A D Ó W D W U W Y M IA R O W Y C H Streszczenie
W artykule rozpatryw any jest problem zależności w ogonie d la rozkładów dw uw ym iarow ych. P rzedstaw iono przegląd ró ż n y ch podejść d o an alizo w a n ia tej zależności. Szczególna uw aga pośw ięcona zo stała w aru n k o w y m w spółczynnikom korelacji o raz w spółczynnikom zależności w ogonie. W skazano, ja k te w spółczynniki m ogą być analizow ane za p o m o cą tzw. analizy połączeń.
