STATISTICAL ISSUES IN MODELING MULTIVARIATE STABLE PORTFOLIOS
3. Estimation of the index of stability
In this section we address the issue of estimating the tail indexα. We start with the case when the sample comes from a univariateα-stable distribution, and then consider a more general situation where the observations are not necessarily stable, but asymptotically have a stable-Pareto tail with indexα, that is
P (X1> x)=1−F (x)≈x−αL(x), (36)
where L is some slowly varying function. Given a multivariate heavy tailed data set X1, . . . ,Xn, one can apply the methods of this section to one-dimensional samples cor- responding to the norms||Xj||or the projectionsXj,bfor someb∈Rd.
142 T.J. Kozubowski et al.
3.1. Estimation of univariate stable parameters
Estimating the parameters of stable distributions is a challenging problem due to the fact that the densities and distributions functions of these laws are not available in closed form.
Various estimation methods have been developed over the last 30 years, most of them requiring numerical approximations.
Since the stable characteristic function can be written in a closed form, several esti- mation techniques are based on fitting the sample characteristic function to its theoretical counterpart. The substantial collection of papers in this area started with Press (1972b), and include Arad (1980), Feuerverger and McDunnough (1977, 1981a, b), Kogon and Williams (1998), Koutrouvelis (1980, 1981), Paulson and Delehanty (1984, 1985), Paul- son, Holcomb and Leitch (1975). As noted by McCulloch (1996), these estimation pro- cedures were reported by practitioners to have high efficiency relative to the maximum likelihood approach. However, some of these methods are quite complex and require the practitioner to choose certain arbitrary parameters. A discussion and comparative study of these approaches can be found in Kogon and Williams (1998).
The maximum likelihood (ML) method for the stable case was first proposed by Du- Mouchel (1971, 1973), who also discussed the asymptotic properties of the estimators. To approximate the loglikelihood function DuMouchel (1971) employed fast Fourier trans- form (FFT) for the central part of the data and series expansions for the tails. See also Du- Mouchel (1975, 1983) for numerical approximation of the Fisher information matrix and further comments on this approach. Since this early work, various numerical procedures for approximating stable densities have been developed, which now permit an efficient com- putation of the likelihood function without the grouping procedure of DuMouchel (1971).
For the ML in the symmetric case, see Brorsen and Yang (1990), McCulloch (1979, 1998).
Asymmetric stable ML was treated in Brorsen and Preckel (1993), Liu and Brorsen (1995), Mittnik et al. (1999), Nolan (2001), Stuck (1976). As noted in Mittnik et al. (1999), one ad- vantage of the ML approach over most other methods is its ability to handle generalizations to dependent or not identically distributed data arising in financial modeling (for example, regression or various time series models with stable disturbances). An implementation of the ML method for such generalizations can be found in Liu and Brorsen (1995) (stable GARCH), Mittnik, Rachev and Paolella (1998) (ARMA models driven by asymmetric sta- ble distributions), and Brorsen and Preckel (1993), McCulloch (1998) (linear regression).
In the last section of our chapter, we utilize the maximum likelihood numerical procedures of Nolan (1998), applicable for the most general i.i.d. stable case (available on the author’s web site).
Numerous other methods of estimating stable parameters have been suggested. Per- haps the most commonly used estimators in empirical work are quantile procedures of Fama and Roll (1971) for the symmetric case and their modifications to the general case obtained by McCulloch (1986). Buckle (1995) proposed sampling based Bayesian in- ference for stable laws, see also Qiou and Ravishanker (1995), Ravishanker and Qiou (1998) for further extensions and discussion of the Bayesian approach. Nikias and Shao (1995) derived moment estimators based on sample fractional moments. Compu- tationally simple estimators based on the modified method of scoring were proposed in
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 143
Klebanov, Melamed and Rachev (1994). For further references on estimating stable para- meters, see, e.g., McCulloch (1996), Rachev and Mittnik (2000). Comparative studies of various estimators for stable parameters include Akgiray and Lamoureux (1989) and more recent Hửpfner and Rỹschendorf (1999), Kogon and Williams (1998).
3.2. Estimation of the tail indexα
Assume that we have a one-dimensional random sampleX1, . . . , Xn satisfying (36) and belonging to the domain of attraction of an α-stable distribution. There is a large body of literature concerning estimation of the tail indexα. Many common estimators ofαare based on a subset of the sample order statistics,
X(1)ã ã ãX(n). (37)
Below we sketch few standard and some recent methods for estimatingαand give refer- ences for many others.
3.2.1. The Hill estimator
The Hill estimator [see Hill (1975)] along with its various modifications is perhaps the most common way of estimating the tail thicknessαof a financial data set [see, e.g., Jansen and de Vries (1991), Koedijk, Schafgans and de Vries (1990), Loretan and Phillips (1994), Phillips (1993)]. The estimator uses theklargest order statistics,
ˆ αHill=
1 k
k
j=1
logX(n+1−j )−logX(n−k)
−1
, (38)
and arises as the conditional maximum likelihood estimator for the Pareto distribution P (X > x)=Cx−α. With the proper choice of the sequencek=k(n), the estimator is con- sistent and asymptotically normal, see, e.g., Beirlant and Teugels (1989), Csửrg˝o and Ma- son (1985), de Haan and Resnick (1998), Deheuvels, Haeusler and Mason (1988), Goldie and Smith (1987), Haeusler and Teugels (1985), Hall (1982), Hall and Welsh (1984, 1985), Mason (1982). For further discussion and extensions, see, e.g., Csửrg˝o, Deheuvels and Ma- son (1985), Csửrg˝o and Viharos (1995), Dekkers and de Haan (1993), Dekkers, Einmahl and de Haan (1989).
An obvious problem with the Hill estimator and its generalizations discussed below is the practical choice ofk. Generally, we must have
k→ ∞ and k
n→0 asn→ ∞ (39)
to achieve strong consistency and asymptotic normality. In practice, one usually plots val- ues of the estimator against the values ofk(obtaining the so-calledHill plot) and looks for
144 T.J. Kozubowski et al.
a stabilization (flat spot) in the graph. An alternative, more informative method of doing a Hill plot, is described in Drees, de Haan and Resnick (2000), Resnick and St˘aric˘a (1997).
We refer the readers to Danielsson, Jansen and de Vries (1996), Embrechts, Klüppelberg and Mikosh (1997), Kratz and Resnick (1996), Mittnik and Paolella (1999), Rachev and Mittnik (2000), Resnick (1998), Resnick and St˘aric˘a (1997) and references therein for more details on this and related tail estimators.
3.2.2. A shifted Hill’s estimator
Noting that the Hill estimator is scale invariant but not shift invariant, Aban and Meer- schaert (2001) proposed the modification that is shift invariant. Their method consists of conditional maximum likelihood estimation for the shifted Pareto distributionP (X > x)= C(x−s)−α, and yields the estimators:
ˆ α=
1 k
k
j=1
log(X(j )∗ − ˆs)−log(X∗(k+1)− ˆs)−1
, (40)
ˆ c=k
n(X∗(k+1)− ˆs)αˆ, (41)
wheresˆis obtained by solving the equation ˆ
α(X∗(k+1)− ˆs)−1=(αˆ+1)k−1
k
j=1
(X(j )∗ − ˆs)−1 (42)
over the sets < Xˆ ∗(k+1). Here the starred variables indicate the order statistics taken in the decreasing order:
X∗(1)ã ã ãX∗(n). (43)
Numerical procedures are required to compute the estimators.
3.2.3. The Pickands estimator and its modifications Pickands (1975) introduced a tail estimator of the form
ˆ
αPick= log 2
log(X(n−k+1)−X(n−2k+1))−log(X(n−2k+1)−X(n−4k+1)), 4k < n, (44) see also Drees (1996), Rosen and Weissman (1996). Noting its poor performance on sam- ples from stable distributions, Mittnik and Rachev (1996) introduced a modification of (44)
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 145
based on Bergstrửm expansion of stable distribution function [see Bergstrửm (1952), and also Janicki and Weron (1994)]. Theirunconditional Pickands estimatoris of the form
ˆ
αUP= log 2
logX(n−k+1)−logX(n−2k+1). (45)
We refer the readers to Rachev and Mittnikl (2000) for further discussion on the practical performance and other modifications of the Pickands estimator.
3.2.4. Least-squares estimators
Taking the logarithm of both sides in relation (36) we observe that for large values ofx the points with abscissa logx and ordinate log(1−F (x))should approximately fall on a straight line with slope−α. Using theklargest order statisticsXn+1−j,j =1, . . . , k, we can examine the plot of logXn+1−j versus
logj n≈log
1−F (Xn+1−j)
(46) and visually estimate the slope of the resulting line. This graphical approach was suggested by Mandelbrot (1963b).
Using these upper order statistics one can estimate the slope by the classical least- squares method [see Kratz and Resnick (1996), Schultze and Steinebach (1996)]. Below we briefly describe the estimators obtained in Schultze and Steinebach (1996). Assuming that in (36) we haveL(x)=ec (which is the case for stable distributions), Schultze and Steinebach (1996) applied the method of least squares to estimate the interceptc/α and the slope 1/αof a straight line fit to
logXn+1−j≈ c α+1
αlogn
j, j=1, . . . , k. (47)
This resulted in the following estimator ofα:
ˆ α(1)LS=
1 k
k
j=1
logn
j logXn+1−j− 1 k2
k
j=1
logn j
k
j=1
logXn+1−j −1
×
1 k
k
j=1
log2n j −
1 k
k
j=1
logn j
2
. (48)
146 T.J. Kozubowski et al.
Another estimator was obtained in Schultze and Steinebach (1996) by the least squares method under the assumption of zero intercept in (47):
ˆ αLS(2)=
k
j=1
logn
j logXn+1−j k
j=1
log2n
j. (49)
Finally, Schultze and Steinebach (1996) proposed yet another estimator ofαresulting from expressing (47) in the form
αlogXn+1−j≈c+logn
j, j=1, . . . , k, (50)
and minimizing the sum of squares
k
j=1
αlogXn+1−j−c−logn j
2
.
This produced:
ˆ αLS(3)=
1 k
k
j=1
logn
j logXn+1−j− 1 k2
k
j=1
logn j
k
j=1
logXn+1−j
× 1
k
k
j=1
log2Xn+1−j− 1
k
k
j=1
logXn+1−j 2−1
. (51)
Consistency and asymptotic normality of the above estimators are established in Schultze and Steinebach (1996) and Csửrg˝o and Viharos (1997), respectively [see also Kratz and Resnick (1996) for similar results on their QQ estimator].
3.2.5. The M–S method
Meerschaert and Scheffler (1998) introduced a simple robust estimator for the tail indexα that is based on the asymptotics of the sum and utilizes the entire sample not just the largest order statistics. The estimator is based on the idea that ifXi’s are i.i.d. and belong to the domain of attraction of anα-stable law with 0< α <2 (and their distribution function satisfies (36)), then their sample variance,
ˆ σ2=1
n
n
j=1
Xj−X2
, (52)
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 147
converges to anα/2-stable (totally skewed) r.v.Y:
n1−2/ασˆ2→d Y. (53)
Taking the logarithm on both sides of (53) we obtain the convergence 2 logn
1 ˆ α−1
α d
→logY, (54)
where 1 ˆ
α=logn+logσˆ2
2 logn (55)
is the Meerschaert–Scheffler (M–S) estimator of 1/α. The estimator is consistent and its asymptotic distribution is that of logY for some totally skewedα/2 positive stable r.v.Y. Moreover, the estimator applies to certain dependent data. Comparing its performance with that of Hill’s estimator, Meerschaert and Scheffler (1998) concluded that it works as well as the latter in most cases, and substantially better when applied to stable data, see Meer- schaert and Scheffler (1998) for further details.