Copulas: a Review and Recent Developments (2007)

Example (a pitfall of the conditional copula, Patton (2006)). Consider XjW 1 ,Y jW 2 and C W1 ;W 2(:j W 1 ;W 2 )andspecifyH W1 ;W 2(x; yjW 1 ;W 2 )=C W1 ;W 2(F W1 (xjW 1 );G W2 (yjW 2 ) j W 1 ;W 2 ):Then, H W1 ;W 2(x; 1j W 1 ;W 2 )=F W1 (xjW 1 ) which is the conditional marginal distributionof (X; Y )jW 1 . By analogy, H W1 ;W 2(1;yjW 1 ;W 2 )=G W2 (yjW 2 ), the conditionaldistribution of (X; Y )jW 2 . Thus the function H W1 ;W 2can not be the joint distributionof (X; Y ) j W 1 ;W 2 in general. The only exception is the very special case whenF W1 (xjW 1 )=F W1 (xjW 1 ;W 2 )andG W2 (yjW 1 )=G W2 (yjW 1 ;W 2 ).The conditional bivariate copula de¯nitioncanbereformulatedintermsofsubalgebraA containing the past information on the process of interest as follows.De¯nition (conditional copula, Fermanian and Scaillet (2005)). For everysub-algebra A the conditional copula with respect to A, associated with a vectorX =(X 1 ;::: ;X n ) is a random function C(:jA) :[0; 1] n ! [0; 1] such thatH(xjA) =C(F X1 (x 1 jA);::: ;F Xn (x n jA) jA)almost surely for every x =(x 1 ;::: ;x n ) 2 [¡1; 1] n . Such a function is uniqueon the product of values taken by the conditional marginal cumulative distributionfunctions F Xi (:jA);i=1;::: ;n:Even if this de¯nition is useful it is di±cult to meet a real situation such that certainvariables a®ect the conditional distribution of one variable but not the other. Inpractice, the marginal distributions are usually de¯ned with respect to past marginalvalues. Consider for example the Markovian process S i;t ; i =1;::: ;n: One workseasily with conditional distribution of S i;t knowing S 0;t , than with the conditional distributionof S i;t knowing the full vector S 0 =(S 0;1 ;::: ;S 0;n ), related with past historyof the process. Actually, it is much simpler to model future returns of a stock index,say SI 1 , knowing the past history of this index (by some state-space or stochasticvolatility models) than knowing the past values of SI 1 and an additional one, SI 2say.The practitioners often prefer to implement their marginal models in informationsystems and the idea is to use them for analysis of more complex multivariate models.The necessity to de¯ne the notion of conditional pseudo-copula is motivated by thefact that marginal processes are usually better known than dependence structures. Inorder to formalize this mathematically, let us consider the sub-algebras A 1 ;::: ;A n ,B and denote A =(A 1 ;::: ;A n ). These sub-algebras can not be chosen arbitrarily,but assuming the following condition: Let x and y be n-dimensional vectors. Foralmost every w 2 − the relation P (X i · x i jA i )(w) =P (X i · y i jA i )(w) for everyi =1;::: ;n implies P (X · xjB)(w) =P (X · yjB)(w). For example, this conditionis satis¯ed when all conditional distributions of X 1 ;::: ;X n are strictly increasing, orwhen A 1 = ¢¢¢= A n = B, as in Patton (2006).14