Dipartimento di Metodi Quantitativi per le Scienze Economiche ed Aziendali
Università degli Studi di Milano Bicocca
Via Bicocca degli Arcimboldi 8 - 20126 Milano - Italia
Tel +39/02/64483102/3 - Fax +39/2/64483105
Segreteria di redazione: Andrea Bertolini
Rapporto n. 211
Testing Serial Independence
via Density-Based Measures of Divergence
Autori
Luca Bagnato, Lucio De Capitani, Antonio Punzo
Giugno 2011
Testing Serial Independence via Density-Based
Measures of Divergence
Luca Bagnato · Lucio De Capitani ·
Antonio Punzo
Abstract This article reviews the nonparametric serial independence tests based
on measures of divergence between densities. Among others, t h e well-known Kull-
back-Leibler, Hellinger and Tsallis divergences are a n alyzed. Moreover, the cop u la-
based version of the considered divergence functionals is defined and taken into
account. In order to implement serial independence tests based on these divergence
function als, it is necessary to choose a densi ty estimation technique, a way to com-
pute p-values and other settings . Via a wide simulation study, the performance of
the serial independence tests arisin g from the adoption of the divergence function -
als with different implementation is compared. Both single-lag and multiple-lag
test procedu res are investigated in order to find the best solutions in t erms of size
and power.
Keywords Serial independence · Divergence measures · Nonparametric density
estima tion · Copula
Luca Bagnato
Dipartimento di Metodi Quantitativi per le Scienze Economiche ed Aziendali,
Universit`a di M ilano-Bicocca (Italy)
Tel.: +39-02-64483186
Fax: +39-02-64483105
E-mail: luca.bagnato@unimib.it
Lucio De C apitani
Dipartimento di Metodi Quantitativi per le Scienze Economiche ed Aziendali,
Universit`a di M ilano-Bicocca (Italy)
Tel.: +39-02-64483186
Fax: +39-02-64483105
E-mail: lucio.decapitani1@unimib.it
Antonio Punzo
Dipartimento di Impresa, Culture e Societ`a,
Universit`a di Catania (Italy)
Tel.: +39-095-7537640
Fax: +39-095-7537610
E-mail: antonio.punzo@unict.it
2 Luca Bagnato et al.
1 Introduction
In the analysis of a strictly stationary time series, a primary concern is whether
the observations are serially independent or not. Detection of the presence of serial
depend ence is usually a preliminary step carried out before proceeding with further
analysis (Kendall and Stuart, 1966, p. 35 0) l ike modeling and pr edictio n (Lo, 2000 ).
Since the theoretical models incorp orate independent and identically distributed
(i.i.d.) innovation (noise), the analysis of the dependence is al so useful in model-
checking and in testing important econom ic and financial post ulates, such as the
random walk hypothesis (Delgado, 1996; Darrat and Zhong, 2000). For example,
independence of log-returns is an essential feature of the Black-Scholes option
pricing equati on (Hull, 1999).
The considera tions stat ed above represent only some of the examples justifying
both the importance of testing serial independence and the considerable attention
devoted to this pro b lem in the literature (see, e.g ., the recent work of Diks, 2009 ,
for a com p rehensive review, and an extensive bi bliography, on the subject). Nev-
ertheless, as underlined by Genest et al (2002) and Genest a n d Verret (2005), test
procedures merely based on the serial correlation coefficient (see, e.g., the pro-
posals of Wald and Wolfowitz 1943, Moran 1948, Durbin and Watson 1950, 1951,
Ljung and Box 1978, and Dufour and Roy 19 85) continue to be the most com-
monly used in practi ce (see King, 1987, for a survey). Alt h ough they perform well
when the dependence structure is linear and the innovations are normal, they are
not consistent against alternatives with zero serial correlation (see, e.g., Hall and
Wolff, 1995) and can behave rather poorly, both in terms of level and power, when
applied t o non-Gaussian and nonli near time series (see, e.g. , the simulations re-
ported by Hallin and M´elard, 1988). Nowadays, as highli ghted by Hong (1999), it is
widely recognized that many time series arising in practice, display non-Gaussian
and nonlinear features (see, e.g., Brock et al, 1991; Granger and Andersen, 1978;
Granger and Ter¨asvirta, 1993; Priestley, 1988; Subba Rao and G abr, 198 0, 1984;
Tong, 1990).
This has motivated the development of tests for serial independence which
are powerful against general types of dependence (omnibus tests; s ee Diks, 2009,
pp. 6256–6257 for details). Since the con cept of independence can be characterized
in term s of distributions, a line o f research in these terms consists of basing test s
of serial independence on the fact that the null hypothesis ho lds if, a nd only if,
the joint density equals the product of the marginals . In these procedures, the test
statistic is based on a distance or, more in general, on a divergence measure be-
tween the estima ted joint density and the produ ct of the estimated marginal densi-
ties. Nat urally, both the divergence measure, and the density estimation t echnique
adopted, make the procedures different.
The princi pal literature on this topic can be summarized as fo llows. Chan and
Tran (199 2) propose an estimat ion of the joint and marginal densities by the his-
togram and introduce a statistic based on the L
1
-norm. Bagnato and Punzo (2010)
and Dion´ısio et al (2006) consider the χ
2
-statistic and the entropy of Shannon re-
sp ectively, using a histogram-based estimator (see also Bagnato a n d Punzo, 2 011,
for a contextualization in the model validation phase). Robinson (1991) adopts the
Kullback-Leibler information criterion using smoothi n g kernel density estimators
(see also Gran ger and Lin, 1994; Hong and White, 2005); Skaug and Tjøstheim
(199 3b) extend Ro b inson’s framework to other measures of divergence between
Testing Serial Independence via D ensity-Based Measures of D ivergence 3
densities, including the Hellinger di s tance. Maasoumi and Racine (2002), Granger
et al (2004), and Racine and Maasoumi (2007) contribute to this line of r esearch
considering a normal ization of the Hellinger distance, while Fernandes and N´eri
(201 0) extend all these proposals taking into c onsideration the generalized en-
tropic measure suggested by Tsallis (1998). Rosenblatt (1975) and Ahmad and Li
(199 7), among others, use smoo thing kernel d ens ity esti mators with the von-Mises
L
2
-norm; Robinson (1991) and Skaug and Tjøstheim (1996) also use kernel den-
sity estim ation but they consider, respectively, the Kullback-Leibler inf ormation
criteri on and the Helli nger measure (beyond several other measures).
It is worthwhile to note that in this context a uniformly most powerful test
does not exist. Indeed, different tests will be more powerful against different al-
ternatives. Motivated by this consideration, this paper reviews the nonparametric
serial independence tests based on m easures of divergence between densities. This
comparison is not as simple as i t might seem since, in addition to the diver-
gence measure and the density estimation technique, there are other quantities
to specify (the way p-values are computed, use of trimm ing functio n s , and some
computational aspects). Moreover, the co p ula-based versions of a ll the considered
divergence functional s are defined. Thus, t h e performance of the discussed tests are
investigated by varying the considered quantities through a wide simulation study.
The general aim is to provide a guideline for the use of these testing procedures.
This paper is structured into two main parts: review of existing methodologies
(Sectio n 3 and Section 4) and a comparative simulation study (Sections 5-8). Sec-
tion 3 introduces some general and prelim inary aspects such as the distinction be-
tween single-lag testing procedures, developed to be powerful against dependence
in a pa rticular lag, and multiple-l ag testing procedures c ontrolling dependence on
a finite set of pre-specified lags. All the divergence measures and their correspo n d -
ing copula versions, are introduced here. Focus ing on the single-lag procedures,
Section 4 summarizes the techniques used to estimate densities (histogram-b ased
in Section 3.1, Gaussian kernel in Section 3.2, and jackknife kernel in Sect ion 3.3)
and to estimate the co p ula density. For the latter pr oblem, the Kallenberg (2009)
method is considered (see Section 3.4). Section 4 describes some computational
aspects related to th e divergence functionals estimated through both raw densi-
ties and copula densities (Secti on 4.1). Details o n resampling-based approaches
(bootstrap and permutation) to compute p-values, and extensions to multiple-lag
testing procedures of the discussed methodologies, are given in Section 4.2. The
design of the simulation study is illustrated in Section 5 while results for sing le-lag,
and multiple-lag, procedures are described in Section 6 and Section 7, respectively.
Results on simulations are further summarized in Section 8.
2 Density-Based Measures of Dependence
Let {X
t
}
t∈
+
be a rea l-valued and strictly stationary stochastic process and as-
sume that X
1
is continuous with density g and support S. Mo reover, let l be a
positi ve integer and f
l
the joint distribution function of (X
1
, X
1+l
). U nd er the
assumption of strict stationarity, the components of {X
t
}
t∈
+
are i nd ependent
if, and only if, X
1
and X
1+l
are independent for all the integers l ≥ 1. As a
consequence, the hypothesis of seria l independence can be expressed as fo llows:
H
0
: f
l
(x, y) = g(x)g(y) ∀(x, y) ∈ S
2
and ∀l ≥ 1. (1)
4 Luca Bagnato et al.
From a practical point of view, testing hyp othesis (1) is impossible. Usually, a
maximum value for l, say p, is fixed and the presence of serial independence is
checked by testing the following simpler hypothesis:
H
0
: f
l
(x, y) = g(x)g(y) ∀(x, y) ∈ S
2
and ∀l ∈ {1, . . . , p}. (2)
Naturally, if (2 ) is false for a particular p ≥ 1, then also (1) is false. On the
contrary, (2) may be true even if (1) i s false; this happens if X
1
and X
1+r
are
depend ent only for some lag(s) r > p. However, the afor ementioned problem seems
to have only a little practical relevance sin ce it can b e substantially avoided by
choosing a sufficiently large value of p. In the following (2 ) will be referred to as
multiple-lag testing problem. On the contrary
H
0
: f
l
(x, y) = g(x)g(y) ∀(x, y) ∈ S
2
, (3)
for a particular value of l, will be called single-lag testing problem; the case l = 1
is particularly interesting since, in almost all the cases of practical interest, it
is reasonable to retain that the process exhibits th e higher dependence between
consecutive components. Accordingly, special attention in the literature (see, e.g.,
Robinso n , 1991; Hong and White, 2005; Fernandes and N´eri, 2010) is devoted to
it. Section 4.2 will show how to face the multiple testing problem (2) by using t h e
inform ation arising from the fir s t p single tests in (3). If there is no ambiguity, f
l
will be written as f from here onwards.
Several density-based measures of dependence can be used for testing ( 3).
All the functionals con s idered here evaluate the discrepancy between f(x, y) and
g · g(x, y) = g(x)g(y) as follows:
∆ =
Z
S
2
B{f(x, y), g(x), g(y)}f(x, y)dxdy, (4)
where B is a real-valued f un ction. An example o f these f u n ctiona ls is the Tsallis
(198 8) generalized entropy, which in this context coincides with :
∆
γ
=
1
1 − γ
Z
S
2
"
1 −
g(x)g(y)
f(x, y)
1−γ
#
f(x, y)dxdy γ 6= 1
Z
S
2
log
f(x, y)
g(x)g(y)
f(x, y)dxdy γ = 1.
(5)
It is easy to note that ∆
1/2
coincides with the Hellinger met ric while ∆
1
with the
Kullback-Leibler divergence. Moreover, when γ = 2 it turns out that
∆
2
=
Z
S
2
(f(x, y) − g(x)g(y))
2
g(x)g(y)
dxdy,
which can be interpreted as the “continuous counterpart” of the Pearson Chi-
square of independence.
A further intuitive dependence measure is the L
1
distance:
∆
L
1
=
Z
S
2
|f(x, y) − g(x)g(y)|dxdy. (6)
