Directory UMM :Data Elmu:jurnal:J-a:Journal of Econometrics:Vol99.Issue2.Dec2000:
Journal of Econometrics 99 (2000) 349}372
On estimation and testing goodness of "t for
m-dependent stable sequences
Rohit S. Deo*
8-57 MEC, 44 West 4th Street, New York, NY 10012, USA
Received 5 June 2000; accepted 25 June 2000
Abstract
A class of estimators of the characteristic index of m-dependent stable sequences is
proposed. The estimators are shown to be consistent and asymptotically normal. In
addition, a class of goodness-of-"t tests for stability is also obtained. The performance of
the estimators and the goodness-of-"t tests is evaluated through a simulation study. One
of the estimators is shown to have a reasonably high relative e$ciency which is uniformly
superior to that of the regression estimator, which is currently most widely used. Our
results have signi"cance for modelling "nancial data like stock returns which have thick
tailed distributions and exhibit non-linear behaviour. ( 2000 Elsevier Science S.A.
All rights reserved.
JEL classixcation: C12; C13; C14
Keywords: Thick-tailed distributions; Characteristic index; U-statistics
1. Introduction
Since the seminal papers by Mandelbrot (1963) and Fama (1965), stable
distributions have been proposed as models for data from such diverse areas as
economics, astronomy, physics and "nance. A prime obstacle in the estimation
* Tel.: #1-212-998-0469; fax: #1-212-995-4003.
E-mail address: [email protected] (R.S. Deo).
0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 3 0 4 - 4 0 7 6 ( 0 0 ) 0 0 0 3 9 - 7
350
R.S. Deo / Journal of Econometrics 99 (2000) 349}372
of stable distributions has been the lack of closed-form analytical expressions
for the densities of all but a few members of the family of stable distributions.
Series expansions for the densities have been obtained (Feller, 1971, Vol. II,
Chapter XVII), but these are too complex to be of any practical use. As a
result, stable distributions are de"ned via their characteristic functions which
are given by
/(t)"expMidt!cDtDa[1!ib sgn(t)w(t, a)]N
(1)
where w(t, a)"tan(na/2) if aO1, w(t, a)"!(2/n) logDtD if a"1, and the parameters satisfy the conditions 0(a)2, !1)b)1, c'0, and d is real. The
parameters c and d are measures of scale and location, respectively. The
parameter b is a measure of skewness, the density function being symmetric
around d when b"0. The density is right skewed when b'0 and left skewed
when b(0. The parameter a is called the characteristic exponent (or index) and
governs the tail behaviour of the distribution. The smaller the value of a, the
thicker the tail of the distribution. As a matter of fact, the stable distribution
only has moments of order r(a, except when a"2, in which case all moments
exist (Feller, 1971, Chapter XVII, Theorem 1). When a"2, the normal distribution is obtained with mean d and variance 2c and the parameter b becomes
redundant.
There are several alternative representations of the characteristic function of
a stable distribution, obtained by various reparametrisations of expression (1),
which have been used by researchers. We will use expression (1) throughout
since it seems to have been the one used most often (McCulloch, 1986).
Maximum likelihood estimation of the stable distribution parameters is
problematic due to the lack of tractable forms for the density functions of stable
distributions. Alternative estimators which have been proposed include the
regression based estimators of Koutrouvelis (1980) and the quantile method
estimators of McCulloch (1986).
The most promising estimator of a seems to be the regression estimator of
Koutrouvelis (1980). See Koutrouvelis (1980) and Akgiray and Lamoureux
(1989). Koutrouvelis and Bauer (1982) have shown consistency and asymptotic
normality when the estimator is computed from a regression with a "xed
number of points and without random standardisation of the data. However,
the regression estimator used in simulation studies (Koutrouvelis, 1980; Akgiray
and Lamoureux, 1989) is one computed from standardised data and involves
a regression in which the number of points used depends on the sample size and
the value of a. Thus, the asymptotic normality result of Koutrouvelis and Bauer
(1982) cannot be used for inference in such cases.
A second issue, as important as the problem of parameter estimation is that of
testing goodness of "t. Formal statistical tests for the goodness of "t for stable
distributions have been proposed by Hsu et al. (1974), Koutrouvelis and Kellermeier (1981) and CsoK rgo (1987). All three tests have standard s2 limiting
R.S. Deo / Journal of Econometrics 99 (2000) 349}372
351
distributions but are computationally tedious. Furthermore, they all require the
speci"cation of some constants, whose choice will a!ect the size and power of
the tests. There does not seem to have been any study so far about the choice of
these constants.
In this paper, we consider the twin issues of estimating the characteristic
exponent a and of testing a null hypothesis which includes the composite
hypothesis of a stable distribution. We propose a class of estimators of a and
corresponding studentised statistics, all of which are asymptotically normal. In
addition, a class of formal goodness of "t test statistics are proposed, which
asymptotically have a s2 distribution. Moreover, the asymptotic theory for all
these estimators and test statistics is valid for m-dependent stable sequences.
This generalisation, which incorporates dependence in the observed series
without imposing a linear structure is important from the point of view of
modelling "nancial time series like stock returns, which show non-linear behaviour. Furthermore, the estimators and test statistics do not require knowledge of
m. It should be noted that our theory holds only for "xed and "nite values of m.
Hence, it would not hold for processes with an in"nite amount of dependence
like GARCH processes, which have been used to model returns. We also note
that some of the current estimation procedures for stable distributions estimate
all the parameters, while our procedure estimates only a.
The layout of the paper is as follows. In Section 2, we suggest a family of
estimators of a and in Section 3 give their asymptotic behaviour. In Section 4,
feasible versions of our estimators are described and their asymptotic distributions obtained. A consistent estimator of the asymptotic variance covariance
matrix of the family of estimators is suggested in Section 5 and a class of
goodness-of-"t tests obtained from it. The performance of our estimators and
the goodness-of-"t tests based on them is evaluated in Section 6 through Monte
Carlo studies. We summarize the results of the paper in Section 7. All proofs are
relegated to the Appendix.
2. Estimating the tail index
Assume that X ,2, X is a stationary m dependent time series with a mar1
n
ginal distribution function F ( ) ) characterised by the characteristic function /(t)
X
of (1) and where m is a "xed and "nite number. In this section, we de"ne a class of
estimators for the characteristic exponent a.
Let k and s be positive integers such that n'2k'2s and let
I ,(i (2(i )L(1,2, n), I ,( j (2(j )L(i ,2, i ). (2)
k
1
2k
s
1
2s
1
2k
The set I consists of a collection of 2k integers between 1 and n, while the set
k
I consists of a subcollection of 2s integers taken from I . Using the fact that the
s
k
characteristic function of a sum of independent variables is the product of their
352
R.S. Deo / Journal of Econometrics 99 (2000) 349}372
individual characteristic functions, from (1) we can see that for c"a~1,
X 1 #2#X k !X k`1 !2!X 2k
i
i
i
i
k c
D
MX 1 #2#X s !X s`1 !2!X 2s N
]&
j
j
j
j
s
AB
(3)
when
min Di !i D'm.
(4)
p
q
pEq
We will "nd it convenient to work with the new parameter c. From (3) it follows
that logDX 1 #2#X k !X k`1 !2!X 2k D has the same distribution as
i
i
i
i
c log(k/s)#logDX 1 #2#X s !X s`1 !2!X 2s D when (4) is satis"ed and
j
j
j
j
hence, we conclude that
E(logDX 1 #2#X k !X k`1 !2!X 2k D)
i
i
i
i
"c log(k/s)#E(logDX 1 #2#X s !X s`1 !2!X 2s D).
j
j
j
j
Now, using the fact that for any two positive random variables Z and >,
E(log Z)!E(log >)"E(log Z!log >)"E[log (Z/>)], we see that
G A BH
C
k
UH (X 1 ,2, X 2k )" log
k,s i
i
s
~1
D
DX 1 #2#X k !X k`1 2!X 2k D
i
i
i
i
(5)
] log
DX 1 #2#X s !X s`1 !2!X 2s D
j
j
j
j
is an unbiased estimator of c when the elements of I satisfy (4). We show in
k
Lemma A.3 in the Appendix that the expected value of UH is indeed "nite. The
k,s
fact that UH is an unbiased estimator of c now motivates us to consider
k,s
U statistics based on the kernel function UH as estimators of c.
k,s
Let the sets I and I be de"ned as in (2) and let U (X 1 ,2, X 2k ) be the
i
k
s
k,s i
symmetric version of UH (X 1 ,2, X 2k ) (i.e. invariant to all permutations of its
k,s i
i
arguments). Note that since U is invariant to all permutations of its arguk,s
ments, it will depend only on I , k and s but not on the set I LI . We now
k
s
k
de"ne a class of estimators of c by
AB
n ~1
(6)
c( "
+ U (X 1 ,2, X 2k ),
i
k,s
k,s i
2k
C
where the summation in (6) is over all ( n ) choices of I from (1,2, n). The sum in
2k
k
(6) will not depend on I , since as noted above, U does not depend on I but
s
k,s
s
only on s. Note that c( is no longer unbiased for c due to the fact that we no
k,s
longer impose condition (4) on the elements of I . However, it will be shown (see
k
Theorem 1 below) that c( is asymptotically unbiased for c, due to the fact that
k,s
the number of terms in the average (6) which are biased for c is negligible
R.S. Deo / Journal of Econometrics 99 (2000) 349}372
353
compared to the total number of terms. Though it is possible to construct an
unbiased U statistic for c, such an estimator will require assuming knowledge of
the degree of dependence through m, an assumption we would rather not make.
Furthermore, such an unbiased U statistic would be computationally cumbersome and so we will work with the asymptotically unbiased estimator. In the
next section, we study the asymptotic behaviour of the class of estimators c( .
k,s
3. Asymptotic theory
While studying the asymptotic theory, we will concentrate on the class
c( a , a"1,2, p, where 2s (2s (2(2s (2k and k is "xed. We could
1
2
p
k,s
also, of course, vary k but will not do so to avoid introducing more subscripts.
We now de"ne some quantities which are useful in studying the asymptotic
behaviour of c( a . Let
k,s
U a (x)"EMU a (X 1 ,2, X 2k ) D X 1 "xN!c, a"1,2, p
(7)
k,s ,1
k,s i
i
i
where the elements of I satisfy (4). Also let
k
f
"EMU a (X 1 )U b (X 1 )N, h"0,2, m, a, b"1,2, p
1>h,ab
k,s ,1 i k,s ,1 i `h
and
m
f "f
#2 + f
a, b"1,2, p.
(8)
1,ab
1>0,ab
1>h,ab
h/1
Furthermore, let
A
B
n!1 ~1
< "
+ U a (X 1 ,2, X 2k )
(9)
j,a
k,s i
i
2k!1
Cj
where the sum in (9) is over all the sets in I which contain the index j. We now
k
state a theorem about the limiting behaviour of c( .
k,s
Theorem 1. Let the sequence X ,2, X satisfy the assumptions of Section 2.
1
n
(i) If
EU2 (X 1 ,2, X 2k )(R for all I
i
k
k,s i
then EMc( a N"c#O(n~1), a"1,2, p.
k,s
(ii) If (10) holds then
(10)
D N(0, R),
(11)
(2k)~1Jn(bK !c1)P
k
where bK "(c( 1 ,2, c( p )@, 1"(1,2, 1)@ and the (a, b)th element of R is
k
k,s
k,s
p "f .
ab
1,ab
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R.S. Deo / Journal of Econometrics 99 (2000) 349}372
(iii) If (10) holds then
(12)
EM< !U a (X )!cN2"O(n~1)
j,a
k,s ,1 j
uniformly in j"1, 2,2, n for all a"1,2, p.
(iv) If m"0 (i.e. the sequence MX N is independent), then the diagonal elements of
t
R are strictly positive.
Remark 1. Using standard Taylor series arguments, one can establish a limiting
normal distribution for the class of estimators of a de"ned by c( ~1a .
k,s
Remark 2. The asymptotic variance covariance matrix R depends on the unknown quantities U a #c and result (12) shows that they are approximated in
k,s ,1
mean square by the observable sequence M< N. This result is useful in solving
j,a
the problem of estimating R, which we discuss in Section 5 below.
In the following lemma, the proof of which is given in the Appendix, we
provide examples of linear and non-linear m-dependent stable processes which
satisfy condition (10) and for which Theorem 1 would hold.
Lemma 1. (i) Let Me N be a sequence of independent stable random variables with
t
marginal distribution function G ( ) ) characterised by the characteristic function in
e
(1) and tail index a. Let X "e #+m a e , where the a are constants. Then
t
t
j/1 j t~j
j
X is an m-dependent linear stable process with tail index a which satisxes (10).
t
(ii) Let Mv N be a sequence of independent strictly positive stable random variables
t
with tail index h where 0(h(1. Let Mu N be a sequence of independent normal
t
random variables with mean zero and variance p2. Let Mu N be independent of the
t
Mv N. Dexne X "(v #+m b v )1@2u , where the b are positive constants. Then
j/1 j t~j
t
j
t
t
t
X is a non-linear m-dependent process with a stable distribution and tail index 2h
t
and satisxes (10).
The non-linear stable process de"ned in (ii) of Lemma 1 above is related to the
non-linear process that was proposed by de Vries (1991) to model both conditional heteroscedasticity and thick tails in "nancial returns. However, de Vries
(1991) allowed for an in"nite amount of dependence in his model as opposed to
the m-dependence here.
Though we have de"ned a class of estimators for c (and a), these estimators
are computationally infeasible. The number of items which are averaged in the
estimator (6) are of the order of n2k. For example, a sample of size 100 and k"2
(the smallest possible value for k if no assumptions are made about the symmetry and location parameters of the distribution of X ) would require averi
aging over about 108 quantities. However, there is a fairly well-developed theory
for incomplete U statistics, which were designed precisely to handle this problem
and we consider them in the following section.
R.S. Deo / Journal of Econometrics 99 (2000) 349}372
355
4. Incomplete U statistics
As we noted above, a U statistic can be computationally infeasible even for
moderate values of the sample size. However, note that in the average (6) which
de"nes our class of estimators, there are also terms which have several arguments in common. Thus, these terms will be highly correlated and not provide
a signi"cant amount of new information, which suggests that they may be left
out without increasing the asymptotic variance. We therefore will consider the
class of estimators based on incomplete U statistics of the form
c8
"(N)~1+ U a (X 1 ,2, X 2k ), a"1,2, p.
(13)
k,s i
i
D
where the sum in (13) is now over a subset of N elements of the set
S comprising the collection of all sets of the kind I . The set D is called the
n,k
k
design of the incomplete U statistic and there are several ways in which it can be
chosen. (See Section 4.3 of Lee (1990) for a detailed discussion of incomplete
U statistics and choices of D.) Throughout this paper we will restrict our
attention to the class de"ned by (13), where the N elements of D are chosen with
replacement from S . Along with de"ning an incomplete estimator of c, we
n,k
de"ne an incomplete version of < as follows. Let
j,a
k,sa
n
+ U a (X 1 ,2, X 2k )
(14)
. From Theorem 3, Chapter
XV, Vol II of Feller (1971), it follows that the probability density of > is bounded
everywhere. Hence, there exists a K(R such that f (y)(K for all y. Furthermore, by Theorem 1, Chapter XVII, Vol. II of Feller (1971), we also have
ED>De(R for any 0(e(a. Also, since lim
y~e(log y)r"0 and
y?=
lim
y1@2(log y)r"0 for any e'0, there exists a C(R such that
y?0
sup
DyD1@2[logDyD]r(C and sup
DyD~e[logDyD]r(C. Using all of these
@y@:1
@y@w1
facts, we get
P
P
P
E[logD>D]r"
=
[logDyD]rf (y) dy
~=
P
[logDyD]rf (y) dy#
"
@y@:1
[logDyD]rf (y) dy
@y@w1
DyD~1@2(DyD1@2[logDyD]r)f (y) dy
"
@y@:1
P
DyDe(DyD~e[logDyD]r) f (y) dy
#
@y@w1
P
DyD~1@2(DyD1@2[logDyD]r)K dy
)
@y@:1
P
DyDe(DyD~e[logDyD]r)f (y) dy
#
@y@w1
P
P
DyDeCf(y) dy
@y@w1
=
DyD~1@2 dy#C
)CK
DyDef (y) dy
@y@:1
~=
)4CK#ED>De(R.
DyD~1@2CK dy#
)
@y@:1
P
P
370
R.S. Deo / Journal of Econometrics 99 (2000) 349}372
Proof of Lemma 1. We "rst prove part (i). It is easy to see from the very de"nition
of X that it is an m-dependent process. The fact that X has a stable distribution
t
t
with tail index a follows easily by using the characteristic function (1) and the
fact that the e are independent.
t
We next show that (10) holds. To do this, we "rst show that any arbitrary
linear combination of X also has a stable distribution with tail index a.
t
is some arbitrary linear combination
Suppose that > "+s d X
j/0 j t~j
t
of (X ,2, X ) for some integer s and arbitrary coe$cients d . Then we can
t
t~s
j
as a linear combination of the e
also express > in the form > "+p a e
j/0 j t~j
t
t
t
for some p and some coe$cients a . Hence, by using the independence of the e ,
j
t
we observe that > is a stable random variable with tail index a. Next we note
t
that
1
+ UH (X 1 ,2, X 2k ),
U (X 1 ,2, X 2k )"
k,s l
i
l
k,s i
(2k)!
C
(A.5)
where (l ,2, l ) is some permutation of (i ,2, i ) and the summation is over
1
2k
1
2k
all (2k)! permutations of I . Since by the Cauchy}Schwarz inequality we have
k
E(+s Z )2)+s E(Z2)#2++ JE(Z2)E(Z2) for any set of random varij
i
i
iEj
i/1
i/1 i
ables (Z ,2, Z ), from (A.5) it su$ces for us to show that
1
n
E[UH (X 1 ,2, X 2k )]2(R for any (l ,2, l ). It is easy to see that we may write
k,s l
l
1
2k
UH (X 1 ,2, X 2k ) in the form UH (X 1 ,2, X 2k )"Mlog(k/s)N~1(log > !log > ),
k,s l
l
l
1
2
k,s l
where > and > are each some linear combination of (X 1 ,2, X 2k ). By the
1
2
l
l
Cauchy}Schwarz inequality, it is thus enough to show that both
E(log > )2(R and E(log > )2(R. But as shown above, > and > being
1
2
1
2
linear combinations of (X 1 ,2, X 2k ) will both be stable random variables with
l
l
tail index a. Thus, by Lemma A.4 above, E(log > )2(R and E(log > )2(R
1
2
and the proof is complete.
Proof of (ii). It is obvious that X as de"ned is an m-dependent non-linear
t
has a stable distriprocess. Since the v are independent, p ,v #+m b v
j/1 j t~j
t
t
t
bution with tail index h. Also, p is independent of e . Thus, by Example h,
t
t
Chapter VI.2, Vol. II of Feller (1971), X has a stable distribution with tail index
t
2h.
We now show that (10) holds. To do this, we "rst show that any arbitrary
for some integer s and
linear combination of X , given by > "+s d X
j/0 j t~j
t
t
arbitrary coe$cients d , also has a stable distribution with tail index 2h. Let
j
F denote the sigma algebra generated by Mv , v ,2N and / the characteristic
t
t t~1
Y
function of > . Then
t
A
B
G A
s
s
"EE exp il + d p1@2 e
/ (l)"E exp il + d p1@2 e
j t~j t~j
Y
j t~j t~j
j/0
j/0
BK H
F .
t
R.S. Deo / Journal of Econometrics 99 (2000) 349}372
371
But, using the characteristic function of normal random variables and the
independence of v and e , we have
t
t
s
s
F "exp !0.5p2l2 + d2p
E exp il + d p1@2 e
j t~j
t
j t~j t~j
j/0
j/0
s`m
"exp !0.5p2l2 + a v
j t~j
j/0
for some positive coe$cients a . Note that by the independence of the
j
itself has a stable distribution with tail index h. Hence, by
v , +s`ma v
j/0 j t~j
t
Theorem 1, Chapter XIII.6, Vol. II of Feller (1971) we get
G A
BK H
A
B
B
B
A
A
A
B
s`m
s`m
"exp ![0.5p2]hl2h + ah
E exp !0.5p2l2 + a v
j
j t~j
j/0
j/0
which is the characteristic function of a stable random variable with tail index
2h. Thus, any linear combination of X is also stably distributed. The rest of the
t
proof now follows exactly as in the proof of (i) above.
References
Akgiray, V
On estimation and testing goodness of "t for
m-dependent stable sequences
Rohit S. Deo*
8-57 MEC, 44 West 4th Street, New York, NY 10012, USA
Received 5 June 2000; accepted 25 June 2000
Abstract
A class of estimators of the characteristic index of m-dependent stable sequences is
proposed. The estimators are shown to be consistent and asymptotically normal. In
addition, a class of goodness-of-"t tests for stability is also obtained. The performance of
the estimators and the goodness-of-"t tests is evaluated through a simulation study. One
of the estimators is shown to have a reasonably high relative e$ciency which is uniformly
superior to that of the regression estimator, which is currently most widely used. Our
results have signi"cance for modelling "nancial data like stock returns which have thick
tailed distributions and exhibit non-linear behaviour. ( 2000 Elsevier Science S.A.
All rights reserved.
JEL classixcation: C12; C13; C14
Keywords: Thick-tailed distributions; Characteristic index; U-statistics
1. Introduction
Since the seminal papers by Mandelbrot (1963) and Fama (1965), stable
distributions have been proposed as models for data from such diverse areas as
economics, astronomy, physics and "nance. A prime obstacle in the estimation
* Tel.: #1-212-998-0469; fax: #1-212-995-4003.
E-mail address: [email protected] (R.S. Deo).
0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 3 0 4 - 4 0 7 6 ( 0 0 ) 0 0 0 3 9 - 7
350
R.S. Deo / Journal of Econometrics 99 (2000) 349}372
of stable distributions has been the lack of closed-form analytical expressions
for the densities of all but a few members of the family of stable distributions.
Series expansions for the densities have been obtained (Feller, 1971, Vol. II,
Chapter XVII), but these are too complex to be of any practical use. As a
result, stable distributions are de"ned via their characteristic functions which
are given by
/(t)"expMidt!cDtDa[1!ib sgn(t)w(t, a)]N
(1)
where w(t, a)"tan(na/2) if aO1, w(t, a)"!(2/n) logDtD if a"1, and the parameters satisfy the conditions 0(a)2, !1)b)1, c'0, and d is real. The
parameters c and d are measures of scale and location, respectively. The
parameter b is a measure of skewness, the density function being symmetric
around d when b"0. The density is right skewed when b'0 and left skewed
when b(0. The parameter a is called the characteristic exponent (or index) and
governs the tail behaviour of the distribution. The smaller the value of a, the
thicker the tail of the distribution. As a matter of fact, the stable distribution
only has moments of order r(a, except when a"2, in which case all moments
exist (Feller, 1971, Chapter XVII, Theorem 1). When a"2, the normal distribution is obtained with mean d and variance 2c and the parameter b becomes
redundant.
There are several alternative representations of the characteristic function of
a stable distribution, obtained by various reparametrisations of expression (1),
which have been used by researchers. We will use expression (1) throughout
since it seems to have been the one used most often (McCulloch, 1986).
Maximum likelihood estimation of the stable distribution parameters is
problematic due to the lack of tractable forms for the density functions of stable
distributions. Alternative estimators which have been proposed include the
regression based estimators of Koutrouvelis (1980) and the quantile method
estimators of McCulloch (1986).
The most promising estimator of a seems to be the regression estimator of
Koutrouvelis (1980). See Koutrouvelis (1980) and Akgiray and Lamoureux
(1989). Koutrouvelis and Bauer (1982) have shown consistency and asymptotic
normality when the estimator is computed from a regression with a "xed
number of points and without random standardisation of the data. However,
the regression estimator used in simulation studies (Koutrouvelis, 1980; Akgiray
and Lamoureux, 1989) is one computed from standardised data and involves
a regression in which the number of points used depends on the sample size and
the value of a. Thus, the asymptotic normality result of Koutrouvelis and Bauer
(1982) cannot be used for inference in such cases.
A second issue, as important as the problem of parameter estimation is that of
testing goodness of "t. Formal statistical tests for the goodness of "t for stable
distributions have been proposed by Hsu et al. (1974), Koutrouvelis and Kellermeier (1981) and CsoK rgo (1987). All three tests have standard s2 limiting
R.S. Deo / Journal of Econometrics 99 (2000) 349}372
351
distributions but are computationally tedious. Furthermore, they all require the
speci"cation of some constants, whose choice will a!ect the size and power of
the tests. There does not seem to have been any study so far about the choice of
these constants.
In this paper, we consider the twin issues of estimating the characteristic
exponent a and of testing a null hypothesis which includes the composite
hypothesis of a stable distribution. We propose a class of estimators of a and
corresponding studentised statistics, all of which are asymptotically normal. In
addition, a class of formal goodness of "t test statistics are proposed, which
asymptotically have a s2 distribution. Moreover, the asymptotic theory for all
these estimators and test statistics is valid for m-dependent stable sequences.
This generalisation, which incorporates dependence in the observed series
without imposing a linear structure is important from the point of view of
modelling "nancial time series like stock returns, which show non-linear behaviour. Furthermore, the estimators and test statistics do not require knowledge of
m. It should be noted that our theory holds only for "xed and "nite values of m.
Hence, it would not hold for processes with an in"nite amount of dependence
like GARCH processes, which have been used to model returns. We also note
that some of the current estimation procedures for stable distributions estimate
all the parameters, while our procedure estimates only a.
The layout of the paper is as follows. In Section 2, we suggest a family of
estimators of a and in Section 3 give their asymptotic behaviour. In Section 4,
feasible versions of our estimators are described and their asymptotic distributions obtained. A consistent estimator of the asymptotic variance covariance
matrix of the family of estimators is suggested in Section 5 and a class of
goodness-of-"t tests obtained from it. The performance of our estimators and
the goodness-of-"t tests based on them is evaluated in Section 6 through Monte
Carlo studies. We summarize the results of the paper in Section 7. All proofs are
relegated to the Appendix.
2. Estimating the tail index
Assume that X ,2, X is a stationary m dependent time series with a mar1
n
ginal distribution function F ( ) ) characterised by the characteristic function /(t)
X
of (1) and where m is a "xed and "nite number. In this section, we de"ne a class of
estimators for the characteristic exponent a.
Let k and s be positive integers such that n'2k'2s and let
I ,(i (2(i )L(1,2, n), I ,( j (2(j )L(i ,2, i ). (2)
k
1
2k
s
1
2s
1
2k
The set I consists of a collection of 2k integers between 1 and n, while the set
k
I consists of a subcollection of 2s integers taken from I . Using the fact that the
s
k
characteristic function of a sum of independent variables is the product of their
352
R.S. Deo / Journal of Econometrics 99 (2000) 349}372
individual characteristic functions, from (1) we can see that for c"a~1,
X 1 #2#X k !X k`1 !2!X 2k
i
i
i
i
k c
D
MX 1 #2#X s !X s`1 !2!X 2s N
]&
j
j
j
j
s
AB
(3)
when
min Di !i D'm.
(4)
p
q
pEq
We will "nd it convenient to work with the new parameter c. From (3) it follows
that logDX 1 #2#X k !X k`1 !2!X 2k D has the same distribution as
i
i
i
i
c log(k/s)#logDX 1 #2#X s !X s`1 !2!X 2s D when (4) is satis"ed and
j
j
j
j
hence, we conclude that
E(logDX 1 #2#X k !X k`1 !2!X 2k D)
i
i
i
i
"c log(k/s)#E(logDX 1 #2#X s !X s`1 !2!X 2s D).
j
j
j
j
Now, using the fact that for any two positive random variables Z and >,
E(log Z)!E(log >)"E(log Z!log >)"E[log (Z/>)], we see that
G A BH
C
k
UH (X 1 ,2, X 2k )" log
k,s i
i
s
~1
D
DX 1 #2#X k !X k`1 2!X 2k D
i
i
i
i
(5)
] log
DX 1 #2#X s !X s`1 !2!X 2s D
j
j
j
j
is an unbiased estimator of c when the elements of I satisfy (4). We show in
k
Lemma A.3 in the Appendix that the expected value of UH is indeed "nite. The
k,s
fact that UH is an unbiased estimator of c now motivates us to consider
k,s
U statistics based on the kernel function UH as estimators of c.
k,s
Let the sets I and I be de"ned as in (2) and let U (X 1 ,2, X 2k ) be the
i
k
s
k,s i
symmetric version of UH (X 1 ,2, X 2k ) (i.e. invariant to all permutations of its
k,s i
i
arguments). Note that since U is invariant to all permutations of its arguk,s
ments, it will depend only on I , k and s but not on the set I LI . We now
k
s
k
de"ne a class of estimators of c by
AB
n ~1
(6)
c( "
+ U (X 1 ,2, X 2k ),
i
k,s
k,s i
2k
C
where the summation in (6) is over all ( n ) choices of I from (1,2, n). The sum in
2k
k
(6) will not depend on I , since as noted above, U does not depend on I but
s
k,s
s
only on s. Note that c( is no longer unbiased for c due to the fact that we no
k,s
longer impose condition (4) on the elements of I . However, it will be shown (see
k
Theorem 1 below) that c( is asymptotically unbiased for c, due to the fact that
k,s
the number of terms in the average (6) which are biased for c is negligible
R.S. Deo / Journal of Econometrics 99 (2000) 349}372
353
compared to the total number of terms. Though it is possible to construct an
unbiased U statistic for c, such an estimator will require assuming knowledge of
the degree of dependence through m, an assumption we would rather not make.
Furthermore, such an unbiased U statistic would be computationally cumbersome and so we will work with the asymptotically unbiased estimator. In the
next section, we study the asymptotic behaviour of the class of estimators c( .
k,s
3. Asymptotic theory
While studying the asymptotic theory, we will concentrate on the class
c( a , a"1,2, p, where 2s (2s (2(2s (2k and k is "xed. We could
1
2
p
k,s
also, of course, vary k but will not do so to avoid introducing more subscripts.
We now de"ne some quantities which are useful in studying the asymptotic
behaviour of c( a . Let
k,s
U a (x)"EMU a (X 1 ,2, X 2k ) D X 1 "xN!c, a"1,2, p
(7)
k,s ,1
k,s i
i
i
where the elements of I satisfy (4). Also let
k
f
"EMU a (X 1 )U b (X 1 )N, h"0,2, m, a, b"1,2, p
1>h,ab
k,s ,1 i k,s ,1 i `h
and
m
f "f
#2 + f
a, b"1,2, p.
(8)
1,ab
1>0,ab
1>h,ab
h/1
Furthermore, let
A
B
n!1 ~1
< "
+ U a (X 1 ,2, X 2k )
(9)
j,a
k,s i
i
2k!1
Cj
where the sum in (9) is over all the sets in I which contain the index j. We now
k
state a theorem about the limiting behaviour of c( .
k,s
Theorem 1. Let the sequence X ,2, X satisfy the assumptions of Section 2.
1
n
(i) If
EU2 (X 1 ,2, X 2k )(R for all I
i
k
k,s i
then EMc( a N"c#O(n~1), a"1,2, p.
k,s
(ii) If (10) holds then
(10)
D N(0, R),
(11)
(2k)~1Jn(bK !c1)P
k
where bK "(c( 1 ,2, c( p )@, 1"(1,2, 1)@ and the (a, b)th element of R is
k
k,s
k,s
p "f .
ab
1,ab
354
R.S. Deo / Journal of Econometrics 99 (2000) 349}372
(iii) If (10) holds then
(12)
EM< !U a (X )!cN2"O(n~1)
j,a
k,s ,1 j
uniformly in j"1, 2,2, n for all a"1,2, p.
(iv) If m"0 (i.e. the sequence MX N is independent), then the diagonal elements of
t
R are strictly positive.
Remark 1. Using standard Taylor series arguments, one can establish a limiting
normal distribution for the class of estimators of a de"ned by c( ~1a .
k,s
Remark 2. The asymptotic variance covariance matrix R depends on the unknown quantities U a #c and result (12) shows that they are approximated in
k,s ,1
mean square by the observable sequence M< N. This result is useful in solving
j,a
the problem of estimating R, which we discuss in Section 5 below.
In the following lemma, the proof of which is given in the Appendix, we
provide examples of linear and non-linear m-dependent stable processes which
satisfy condition (10) and for which Theorem 1 would hold.
Lemma 1. (i) Let Me N be a sequence of independent stable random variables with
t
marginal distribution function G ( ) ) characterised by the characteristic function in
e
(1) and tail index a. Let X "e #+m a e , where the a are constants. Then
t
t
j/1 j t~j
j
X is an m-dependent linear stable process with tail index a which satisxes (10).
t
(ii) Let Mv N be a sequence of independent strictly positive stable random variables
t
with tail index h where 0(h(1. Let Mu N be a sequence of independent normal
t
random variables with mean zero and variance p2. Let Mu N be independent of the
t
Mv N. Dexne X "(v #+m b v )1@2u , where the b are positive constants. Then
j/1 j t~j
t
j
t
t
t
X is a non-linear m-dependent process with a stable distribution and tail index 2h
t
and satisxes (10).
The non-linear stable process de"ned in (ii) of Lemma 1 above is related to the
non-linear process that was proposed by de Vries (1991) to model both conditional heteroscedasticity and thick tails in "nancial returns. However, de Vries
(1991) allowed for an in"nite amount of dependence in his model as opposed to
the m-dependence here.
Though we have de"ned a class of estimators for c (and a), these estimators
are computationally infeasible. The number of items which are averaged in the
estimator (6) are of the order of n2k. For example, a sample of size 100 and k"2
(the smallest possible value for k if no assumptions are made about the symmetry and location parameters of the distribution of X ) would require averi
aging over about 108 quantities. However, there is a fairly well-developed theory
for incomplete U statistics, which were designed precisely to handle this problem
and we consider them in the following section.
R.S. Deo / Journal of Econometrics 99 (2000) 349}372
355
4. Incomplete U statistics
As we noted above, a U statistic can be computationally infeasible even for
moderate values of the sample size. However, note that in the average (6) which
de"nes our class of estimators, there are also terms which have several arguments in common. Thus, these terms will be highly correlated and not provide
a signi"cant amount of new information, which suggests that they may be left
out without increasing the asymptotic variance. We therefore will consider the
class of estimators based on incomplete U statistics of the form
c8
"(N)~1+ U a (X 1 ,2, X 2k ), a"1,2, p.
(13)
k,s i
i
D
where the sum in (13) is now over a subset of N elements of the set
S comprising the collection of all sets of the kind I . The set D is called the
n,k
k
design of the incomplete U statistic and there are several ways in which it can be
chosen. (See Section 4.3 of Lee (1990) for a detailed discussion of incomplete
U statistics and choices of D.) Throughout this paper we will restrict our
attention to the class de"ned by (13), where the N elements of D are chosen with
replacement from S . Along with de"ning an incomplete estimator of c, we
n,k
de"ne an incomplete version of < as follows. Let
j,a
k,sa
n
+ U a (X 1 ,2, X 2k )
(14)
. From Theorem 3, Chapter
XV, Vol II of Feller (1971), it follows that the probability density of > is bounded
everywhere. Hence, there exists a K(R such that f (y)(K for all y. Furthermore, by Theorem 1, Chapter XVII, Vol. II of Feller (1971), we also have
ED>De(R for any 0(e(a. Also, since lim
y~e(log y)r"0 and
y?=
lim
y1@2(log y)r"0 for any e'0, there exists a C(R such that
y?0
sup
DyD1@2[logDyD]r(C and sup
DyD~e[logDyD]r(C. Using all of these
@y@:1
@y@w1
facts, we get
P
P
P
E[logD>D]r"
=
[logDyD]rf (y) dy
~=
P
[logDyD]rf (y) dy#
"
@y@:1
[logDyD]rf (y) dy
@y@w1
DyD~1@2(DyD1@2[logDyD]r)f (y) dy
"
@y@:1
P
DyDe(DyD~e[logDyD]r) f (y) dy
#
@y@w1
P
DyD~1@2(DyD1@2[logDyD]r)K dy
)
@y@:1
P
DyDe(DyD~e[logDyD]r)f (y) dy
#
@y@w1
P
P
DyDeCf(y) dy
@y@w1
=
DyD~1@2 dy#C
)CK
DyDef (y) dy
@y@:1
~=
)4CK#ED>De(R.
DyD~1@2CK dy#
)
@y@:1
P
P
370
R.S. Deo / Journal of Econometrics 99 (2000) 349}372
Proof of Lemma 1. We "rst prove part (i). It is easy to see from the very de"nition
of X that it is an m-dependent process. The fact that X has a stable distribution
t
t
with tail index a follows easily by using the characteristic function (1) and the
fact that the e are independent.
t
We next show that (10) holds. To do this, we "rst show that any arbitrary
linear combination of X also has a stable distribution with tail index a.
t
is some arbitrary linear combination
Suppose that > "+s d X
j/0 j t~j
t
of (X ,2, X ) for some integer s and arbitrary coe$cients d . Then we can
t
t~s
j
as a linear combination of the e
also express > in the form > "+p a e
j/0 j t~j
t
t
t
for some p and some coe$cients a . Hence, by using the independence of the e ,
j
t
we observe that > is a stable random variable with tail index a. Next we note
t
that
1
+ UH (X 1 ,2, X 2k ),
U (X 1 ,2, X 2k )"
k,s l
i
l
k,s i
(2k)!
C
(A.5)
where (l ,2, l ) is some permutation of (i ,2, i ) and the summation is over
1
2k
1
2k
all (2k)! permutations of I . Since by the Cauchy}Schwarz inequality we have
k
E(+s Z )2)+s E(Z2)#2++ JE(Z2)E(Z2) for any set of random varij
i
i
iEj
i/1
i/1 i
ables (Z ,2, Z ), from (A.5) it su$ces for us to show that
1
n
E[UH (X 1 ,2, X 2k )]2(R for any (l ,2, l ). It is easy to see that we may write
k,s l
l
1
2k
UH (X 1 ,2, X 2k ) in the form UH (X 1 ,2, X 2k )"Mlog(k/s)N~1(log > !log > ),
k,s l
l
l
1
2
k,s l
where > and > are each some linear combination of (X 1 ,2, X 2k ). By the
1
2
l
l
Cauchy}Schwarz inequality, it is thus enough to show that both
E(log > )2(R and E(log > )2(R. But as shown above, > and > being
1
2
1
2
linear combinations of (X 1 ,2, X 2k ) will both be stable random variables with
l
l
tail index a. Thus, by Lemma A.4 above, E(log > )2(R and E(log > )2(R
1
2
and the proof is complete.
Proof of (ii). It is obvious that X as de"ned is an m-dependent non-linear
t
has a stable distriprocess. Since the v are independent, p ,v #+m b v
j/1 j t~j
t
t
t
bution with tail index h. Also, p is independent of e . Thus, by Example h,
t
t
Chapter VI.2, Vol. II of Feller (1971), X has a stable distribution with tail index
t
2h.
We now show that (10) holds. To do this, we "rst show that any arbitrary
for some integer s and
linear combination of X , given by > "+s d X
j/0 j t~j
t
t
arbitrary coe$cients d , also has a stable distribution with tail index 2h. Let
j
F denote the sigma algebra generated by Mv , v ,2N and / the characteristic
t
t t~1
Y
function of > . Then
t
A
B
G A
s
s
"EE exp il + d p1@2 e
/ (l)"E exp il + d p1@2 e
j t~j t~j
Y
j t~j t~j
j/0
j/0
BK H
F .
t
R.S. Deo / Journal of Econometrics 99 (2000) 349}372
371
But, using the characteristic function of normal random variables and the
independence of v and e , we have
t
t
s
s
F "exp !0.5p2l2 + d2p
E exp il + d p1@2 e
j t~j
t
j t~j t~j
j/0
j/0
s`m
"exp !0.5p2l2 + a v
j t~j
j/0
for some positive coe$cients a . Note that by the independence of the
j
itself has a stable distribution with tail index h. Hence, by
v , +s`ma v
j/0 j t~j
t
Theorem 1, Chapter XIII.6, Vol. II of Feller (1971) we get
G A
BK H
A
B
B
B
A
A
A
B
s`m
s`m
"exp ![0.5p2]hl2h + ah
E exp !0.5p2l2 + a v
j
j t~j
j/0
j/0
which is the characteristic function of a stable random variable with tail index
2h. Thus, any linear combination of X is also stably distributed. The rest of the
t
proof now follows exactly as in the proof of (i) above.
References
Akgiray, V