STATISTICAL ISSUES IN MODELING MULTIVARIATE STABLE PORTFOLIOS
6. Extensions to other stable models
In this section we briefly discuss two generalizations of multivariate stable laws that often compete with them in modeling financial data: theν-stable laws that arise as limiting dis- tributions in therandom summationscheme andoperator stablelaws arising as limits in ordinary summation (5) but normalized by linear operatorsan.
6.1. ν-stable laws
LetX1,X2, . . .be a sequence of i.i.d. random vectors in Rd and let νp, p∈(0,1), be a family of integer-valued random variables independent of theXi’s. Assuming that νp converges to infinity (in probability) asp→0, we can study the limiting distributions of the random sums
ap
νp
j=1
(Xj+bp), (83)
whereap>0 andbp∈Rd. It follows from transfer theorems [see, e.g., Rosi´nski (1976)]
that if the variablespνp converge in distribution to a positive r.v. Z with the Laplace transformλ(s)=Eexp(−sZ)and theXj’s are in the domain of attraction of someα-stable distribution with ch.f.Φ, then the random sums (83) will converge to a random variable with the ch.f. of the form
Ψ(t)=λ
−logΦ(t)
. (84)
The variables with the ch.f. (84), referred to as the ν-stable laws – see, e.g., Klebanov and Rachev (1996), Kozubowski and Panorska (1998, 1999b), can be described by the same parameters as the corresponding stable laws: the tail indexα, location vectorm, and spectral measureΓ.Strictlyν-stable laws are given by (84) with a strictly stable ch.f.Φ.
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 153
We use the notationνα(m,Γ)for the distribution corresponding to the ch.f. (84) withΦ given by (2).
The ν-stable laws are essentially location-scale mixtures of stable laws [see, e.g., Kozubowski and Panorska (1998)] and for a light-tailed r.v.Z have the same tail behav- ior as the corresponding stable laws. More precisely, the tail behavior of each coordinate of aν-stable r.v.Xis of the formP (Xk > x)=O(x−α)asx→ ∞ under the following conditions [see Kozubowski and Panorska (1996, 1998)]:
• EZ <∞ifXis strictlyν-stable,
• EZ1∨α<∞andα=1 orE|ZlogZ|<∞andα=1, ifXis not strictlyν-stable.
Under the above conditions, the same tail behavior applies to every linear combinations X,bofX, the order statistics of the vectorX(as well as their absolute values), and the normX, see Kozubowski and Panorska (1998) for details. Note that these conditions are satisfied, for example, by the geometric stable laws discussed below.
Remark. Although the tails ofν-stable laws are essentially of the same type as those of stable distributions,ν-stable densities may behave very differentlynear the modethan their stable counterparts (may be more peaked, or even infinite) which may lead to an improved fit when modeling financial data.
Kozubowski and Panorska (1999b) showed that if the spectral measure is discrete, then trulyd-dimensionalν-stable random vectors admit a representation similar to that of stable laws given in Proposition 2.3:
Proposition 6.1. LetY∼να(m,Γ)withΓ of the form(12)and0< α <2. Then
Y=d
Zm+Z1/α
k
j=1
γj1/αVjsj ifα=1,
Zm+Z
k
j=1
Vj+ 2
π log(γjZ)
γjsj ifα=1,
(85)
where the Vj’s are i.i.d. totally skewed, one-dimensional standard stable variables Sα(1,1,0), independent ofZ.
Thus,ν-stable random variates are straightforward to simulate ifΓ is discrete. Distri- butions with generalΓ can be approximated by those with discrete spectral measure [see Kozubowski and Panorska (1999b)] as in the stable case, so that in practice we can restrict attention to the case with discreteΓ.
6.1.1. Geometric stable laws
An important special case are the limiting distributions of (83) when the variablesνp are geometric with mean 1/p in which case the variablespνp converge to a standard expo-
154 T.J. Kozubowski et al.
nential variable with the Laplace transformλ(s)=(1+s)−1. We then obtain the class of geometric stable law(GS) lawsGSα(m,Γ)with the ch.f.
Ψ(t)=
1+Iα(t)−it,m−1
, (86)
wherem∈Rd andIα is given by (3). In financial applications, where these laws have been successfully applied [see, e.g., Kozubowski and Panorska (1999a), Kozubowski and Rachev (1994), Mittnik and Rachev (1991, 1993a)] the r.v.νprepresents the moment when the probabilistic structure governing the returns changes, so that the random sum
νp
j=1
Xj (87)
represents the total return up to this random time. In caseα=2, we obtain the multivariate Laplace distribution [see, e.g., Kozubowski and Podgórski (2000)], which may be partic- ularly well suited for financial applications due to its simplicity and flexibility [see, e.g., Kozubowski and Podgórski (2001)], although the tails of these laws, being heavier than Gaussian tails, are not as heavy as those of stable and geometric stable laws. More infor- mation on theory and applications of GS laws can be found in Kozubowski and Rachev (1999).
6.1.2. Statistical issues
Most estimation procedures for stable laws can be extended to the correspondingν-stable distributions. For simplicity we consider the problem of estimatingαandΓ of a strictly geometric stable distribution given by the ch.f. (86) withm=0 andα=2, based on a random sample
Y1, . . . ,Yn. (88)
For estimatingα, the tail estimators of Section 3.2 can be applied to one-dimensional samples corresponding to (88) by taking the norms of theYi’s or their projectionsYi,b for someb∈Rd. These apply regardless of whether the sample is actually geometric stable or only belongs to a geometric stable domain of attraction. Alternatively, assuming that the Yi’s are geometric stable, one can use estimators for univariate geometric stable parameters [see, e.g., Kozubowski (1983, 2001), Rachev and Mittnik (2000)] applied to the projections Yi,b.
To estimate the spectral measureΓ, one can use the RXC tail estimator discussed in Section 4.1 since geometric stable distributions have the same domains of attraction as the corresponding stable laws (that have the sameαandΓ), see, e.g., Klebanov and Rachev (1996). Alternatively, the empirical characteristic function method discussed in Section 4.2
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 155
can be modified to accommodate the geometric stable case. Assuming that the sample (88) is from a GS distribution, we estimate the exponentIα of the GS ch.f. (86) as follows:
Iˆα(t)= 1
Ψn(t)−1, (89)
whereΨnis the sample characteristic function (63) based on theYi’s. The rest is the same as in the stable case. For some gridt1, . . . ,tk∈Sd, the quantity (64) is the empirical ch.f.
(ECF) estimate ofIα. IfΓ is a discrete measure of the form (12), thenIα is given by (14), and we can estimate γ=(γ1, . . . , γk) by solving the system of linear equations of the form (65), whereI = ˆIECFis an estimate ofIα given by (64) andAis ak×k(complex) matrix with the entries specified in (66). If the inverse ofAexists, then the solution of the system (65) isγ=A−1I. To avoid the same numerical problems as in the stable case, in practice one should restate the problem as a constrained quadratic programming problem (69). The projection method of Section 4.3 can be extended similarly.
6.2. Operator stable laws
If we have a heavy-tail multivariate data with different tail indexes in different directions, then the multivariate stable (as well as the ν-stable) laws are no longer appropriate to model such data. Instead, we can consider the class of multivariate laws with stable mar- ginal distributions, introduced in Resnick and Greenwood (1979), that arise as limiting distributions in the summation scheme (5) where the scaling factors are diagonal matri- ces, diag(an1, . . . , and), for some positiveani’s. The resulting limiting marginally stable random vectorsXpossess a stability property similar to (1),
X1+ ã ã ã +Xn d
=nEX+Dn, (90)
where theXi’s are i.i.d. copies ofX,Eis a diagonal matrix E=diag
1
α1, . . . , 1 αd
, 0< αi2, i=1, . . . , d, (91)
called thecharacteristic exponentofX, andnEdenotes the diagonal matrix nE=diag
n1/α1, . . . , n1/αd
. (92)
Remark. More generaloperator stable(OS) laws arise as the limits in (5) when the sums are normalized by some linear operators an [see Sharpe (1969)]. For a comprehensive review of the theory of OS laws see Jurek and Mason (1993).
156 T.J. Kozubowski et al.
Marginally stable OS laws satisfying (90) with the characteristic exponentEof the form (91) can be described in terms of their characteristic function. If allα’s are strictly between one and two, we have
Φ(t)=exp
C
Sd
∞
0
eit,rEs−1−i
t, rEsdr
r2Γ(ds)+it,m
, (93)
wherem∈Rd is the shift parameter, C >0 controls the scale, and Γ is a probability measure on the unit sphereSd (the normalizedspectral measure, also called themixing measure). If all theα’s of the characteristic exponentEin (91) are equal, then (93) reduces to the stable ch.f. with the same spectral measure. We use the notationOS(m, C, E,Γ)to denote the distributions with the ch.f. (93) withEgiven by (91). Similarly to the stable case, the measureΓ determines the dependence among the components of a marginally stable vector. For example, ifX∼OS(m, C, E,Γ)is positively or negatively associated, then the spectral measureΓ satisfies the condition (29) or (34), respectively [see Mittnik, Rachev and Rüschendorf (1999)].
6.2.1. Statistical issues
Estimating the parameters of anOS(m, C, E,Γ)distribution is similar to the stable case.
Since all marginal distributions are univariate stable, one can obtain estimates of theαi’s by using the methods for univariate stable laws (see Section 3.1) for each of thed samples
X1j, . . . , Xnj, j =1, . . . , d. (94)
For samples from a domain of attraction of an OS law we can again consider univariate samples (94) and apply the methods of Section 3.2, or use the moment estimator ofE based on the sample covariance matrix [see Meerschaert and Scheffler (1999)].
To estimateC andΓ, one can use a generalization of the tail estimator of the spectral measure for stable laws described in Section 4.1 [see Mittnik, Rachev and Rüschendorf (1999), Scheffler (1999)]. First, write each of the data points (different than zero) in the unique form
Xi=τ (Xi)Esi, (95)
whereτ (Xi) >0 is the “radius” ofXi andsi is a point on the unit sphereSd[these are the so-calledJurek coordinates, see Jurek (1984)]. Next, for some integerk=k(n)consider theklargest of theτ (Xi)’s, that is theklargest order statistics
τ (Xi1), . . . , τ (Xik) (96)
corresponding to the random sample
τ (X1), . . . , τ (Xn). (97)
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 157
Then, the estimator ofΓ is the discrete measure onSdthat assigns the mass of 1/kto each of the unit vectors
si1, . . . ,sik (98)
corresponding to these order statistics via (95). Thus, the probability assigned by the esti- mated spectral measureΓ to a setA∈Sdis the fraction of the points (98) falling in the set A.
The corresponding estimator ofCis C=k
nτ Y∗
, (99)
whereY∗is thek-th largest of the values (97). More details regarding the estimation ofΓ (including the asymptotic properties of estimators) can be found in Mittnik, Rachev and Rüschendorf (1999), Scheffler (1999).