STATISTICAL ISSUES IN MODELING MULTIVARIATE STABLE PORTFOLIOS
4. Estimation of the stable spectral measure
4.1. Tail estimators
A method of estimating the spectral measure of a stable r.v.Ybased on a random sample
X1, . . . ,Xn (56)
from the domain of attraction of Ywas proposed by Rachev and Xin (1993) and Cheng and Rachev (1995). The method, referred to as the Rachev–Xin–Cheng (RXC) method by Nolan and Panorska (1997), is based on the limiting relation in Proposition 2.1. To estimateΓ(D), whereΓ is the (normalized) spectral measure ofY[cf. parameterization (27)], choose a large value ofr and calculate the proportion of theXi’s with the norm exceedingrthat belong to the setDwhen normalized, that is
Γ(D)=2{Xi/Xi ∈DandXi> r}
2{Xi> r} . (57)
Equivalently, we can choose an integerk=k(n)n/2 and consider the set
Xi1, . . . ,Xik (58)
148 T.J. Kozubowski et al.
of theklargest order statistics connected with the corresponding sample of the norms:
X1, . . . ,Xn. (59)
Then, the RXC estimator ofΓ is the discrete measure onSdthat assigns the mass of 1/k to each of the unit vectors
Xi1
Xi1, . . . , Xik
Xik. (60)
The authors suggest taking about 20% of the largest order statistics. Under appropriate technical conditions the estimator is strongly consistent and asymptotically normal.
A similar method was recently proposed by Davydov et al. (2000) and discussed further in Davydov and Paulauskas (1999). We refer to this approach as the Davydov–Paulauskas–
Rackauskas (DPR) method. Assuming that the sample (56) is actually from anα-stable distribution with a zero shift vectormand a symmetric (normalized) spectral measureΓ, and the sample sizenis a perfect squaren=k2for some integerk, the method consists of splitting the data intokgroups ofkvariables each, choosing a vector with the largest norm within each group, leading to a set ofkvectorsXi1, . . . ,Xik, and again estimatingΓ by the empirical measure based on the unit vectors (60). The consistency and asymptotic normality of the resulting estimators,
Γ(D)=1 k
k
j=1
ID
Xij
Xij
, (61)
is established in Davydov and Paulauskas (1999).
Both RXC and DPR methods do not assume any prior knowledge ofαand are well suited for the Sα∗(m, σ,Γ) parameterization, as they provide estimators for thenormal- izedspectral measure. Once the spectral measure and the indexαare estimated, the scale parameterσ can be estimated by methods described in Section 5.
4.2. The empirical characteristic function method
The method described below, proposed in Nolan, Panorska and McCulloch (2001) and investigated in Nolan and Panorska (1997), assumes that the sample comes from an α- stable distribution with shift vectormequal to zero. First, estimate the index of stability and center the data by the sample mean (ifα >1) or sample median (ifα <1). In Nolan, Panorska and McCulloch (2001) the value ofαwas estimated by the average 1dd
j=1αˆj, whereαˆj is an estimate of the index obtained from a univariate sampleX1j, . . . , Xnj (the quantile method of McCulloch (1986) was used to obtain these). Then, the method uses the sample to estimate the exponentIα of the stable ch.f. (2) (withm=0):
Iˆα(t)= −logΦn(t), (62)
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 149
where the quantityΦnis the sample characteristic function, Φn(t)=1
n
n
j=1
eit,Xj. (63)
For some gridt1, . . . ,tk∈Sd, the quantity IˆECF=Iˆn(t1), . . . ,Iˆn(tk)
(64) is the empirical ch.f. (ECF) estimate of Iα. IfΓ is a discrete measure of the form (12), then the exponentIα is given by (14), and we can estimateγ =(γ1, . . . , γk) by solving the following system of linear equations:
I=Aγ , (65)
whereI = ˆIECF is an estimate ofIα given by (64) andAis a k×k (complex) matrix [aij]i,j=1,...,kwith
aij=ωα,1 ti,sj
. (66)
If the grid is chosen so that the inverse ofAexists, then the solution of the system (65) isγ=A−1I.
For a general spectral measure, divide the unit sphere intok non-overlapping patches Aj with some central pointssj, wherej=1, . . . , k, and consider an approximation ofΓ of the form (12), whereγi=Γ(Aj)(which is always possible in view of Proposition 2.4).
Whend=2, it is convenient to take the arcs Aj=
2π(j−3/2)
k ,2π(j−1/2) k
, j=1, . . . , k, (67)
centered at sj=
cos2π(j−1)
k ,sin2π(j−1) k
∈Sd, j=1, . . . , k. (68) We would again estimateIα by (64) and solve the system (65) to obtain the estimates of the weightsγj.
As reported in Nolan and Panorska (1997), in practice there are some problems with the direct implementation of the above method; the matrixAmay be ill-conditioned and the solution of the system (65) may include negative or complex numbers (although the values ofγj must be real and positive). Thus, in practice one should restate the problem as a constrained quadratic programming problem,
minimizeI−Aγ =(I−Aγ )(I−Aγ )subject toγ0, (69)
150 T.J. Kozubowski et al.
which guarantees a nonnegative solutionγ. We refer the readers to Nolan (1998), Nolan and Panorska (1997), Nolan, Panorska and McCulloch (2001) for examples and further discussion of these issues.
4.3. The projection method
The projection (PROJ) method was introduced in McCulloch (1994) and studied in Nolan and Panorska (1997), Nolan, Panorska and McCulloch (2001). As before, assume that the data have been shifted so that the parametermis zero. The method is similar to the ECF method, since we estimate the weightsγj at sj of a discrete spectral measureΓ of the form (12) by solving the linear system of Equations (65). However, the PROJ method uses a different value ofI, the estimate of Iα, obtained from estimators of univariate stable parameters applied to a one-dimensional sample
X1,tj, . . . ,Xn,tj, j=1, . . . , k, (70)
wheret1, . . . ,tk∈Sdis a suitably chosen grid on the unit sphere. More precisely, for each t∈Rd the r.v.X1,tis one-dimensional stable with parameters given by (18)–(20) and ch.f.
ψ(u)=Eeiut,X=Eeiut,X=Φ(ut)=e−Iα(ut), (71) whereIα is the characteristic exponent of theXj’s. Now, we can estimate the scaleσ (tˆ j) and skewnessβ(tˆ j)(and also the shiftà(tˆ j)ifα=1) of the univariate stable law corre- sponding to the sample (70), and use them to estimate the ch.f. (9) of this univariate law.
Then, we can equate the above estimate with the right-hand side of (71) withu=1 to estimate the quantityIα on the gridt1, . . . ,tk:
Iˆn(tj)=
ˆ σα(tj)
1−iβ(tˆ j)tanπ α 2
forα=1, ˆ
σ (tj)
1−ià(tˆ j)
forα=1.
(72)
For the index α McCulloch (1994) recommend using the pooled estimate obtained by averaging the univariate estimates obtained for each of the univariate samples (70). Thus, the PROJ estimate ofIα on the gridt1, . . . ,tkis the quantity
IˆPROJ=Iˆn(t1), . . . ,Iˆn(tk)
. (73)
Now, the weightsγj of the spectral measure are obtained as before by solving the sys- tem (65). For examples and further discussion, please see McCulloch (1994), Nolan and Panorska (1997), Nolan, Panorska and McCulloch (2001).
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 151