Testing for changes in Kendall’s tau∗
Herold Dehling
University of Bochum

Daniel Vogel†
University of Aberdeen

Martin Wendler

arXiv:1203.4871v5 [] 6 Oct 2016

University of Greifswald

Dominik Wied
University of Cologne

∗
The authors wish to thank their colleague Roland Fried for several very stimulating discussions that motivated
this paper. Moreover, we are grateful for helpful comments from the editors and referees, which substantially
improved a previous version of the paper. We are also indebted to Alexander Dürre, who did a thorough
proofreading of the manuscript. The research was supported in part by the Collaborative Research Grant 823
Statistical modelling of nonlinear dynamic processes of the German Research Foundation.
†
Corresponding author: Daniel Vogel, E-Mail: daniel.vogel@abdn.ac.uk.

1

Running Head: Testing for changes in Kendall’s tau
Proofs should be sent to: Daniel Vogel, daniel.vogel@abdn.ac.uk

Abstract
For a bivariate time series ((Xi , Yi ))i=1,...,n we want to detect whether the correlation between
Xi and Yi stays constant for all i = 1, . . . , n. We propose a nonparametric change-point test
statistic based on Kendall’s tau. The asymptotic distribution under the null hypothesis of no
change follows from a new U -statistic invariance principle for dependent processes. Assuming
a single change-point, we show that the location of the change-point is consistently estimated.
Kendall’s tau possesses a high efficiency at the normal distribution, as compared to the normal
maximum likelihood estimator, Pearson’s moment correlation. Contrary to Pearson’s correlation coefficient, it shows no loss in efficiency at heavy-tailed distributions, and is therefore
particularly suited for financial data, where heavy tails are common. We assume the data
((Xi , Yi ))i=1,...,n to be stationary and P -near epoch dependent on an absolutely regular process.
The P -near epoch dependence condition constitutes a generalization of the usually considered
Lp -near epoch dependence allowing for arbitrarily heavy-tailed data. We investigate the test
numerically, compare it to previous proposals, and illustrate its application with two real-life
data examples.

Keywords: Change-point analysis, Kendall’s tau, U -statistic, functional limit theorem, near
epoch dependence in probability

2

1. Introduction
For risk management and portfolio optimization, the dependence between financial asset prices
is of enormous importance. It is often assumed to be constant over the observed time period,
which is a simplifying assumption that is evidently violated for longer observation periods. For
good statistical modeling and successful decision making it is essential to detect changes in
the association of financial price processes and, within reasonable time frames, re-estimate the
correlation parameters. Particularly, in times of global financial crises, the price processes of
most financial assets tend to be highly dependent, united in their common downward trend,
causing the hedging powers of investment diversification to cease — an effect for which the term
diversification meltdown has been coined.
The problem of detecting changes in the distribution of sequential observations has a long history
in statistics, see e.g. Csörgő and Horváth (1997). However, particularly detecting changes in the
dependence structure of multivariate time series has attracted the focus of statistical research
only recently. Examples for such detection procedures are Loretan and Phillips (1994), who
test for covariance stationarity of a possibly heavy-tailed time series, Giacomini, Härdle, and
Spokoiny (2009), who consider tests for homogeneity of time-varying copulae, Aue, Hörmann,
Horváth, and Reimherr (2009), who propose a test for a constant covariance matrix, and Wied,
Krämer, and Dehling (2012), who suggest a change-point test for correlations between two
random variables based on Pearson’s correlation coefficient.
With this paper, we want to contribute to the literature by proposing a new test for constant
Kendall’s tau that can be applied to dependent series. We recommend to use the rank correlation measure Kendall’s tau instead of Pearson’s correlation coefficient because it is almost
as efficient as the moment correlation at normality, but is significantly more efficient at heavytailed distributions. For details see Section 5. This issue is very important in finance and
economics, where many key variables, including financial returns and foreign exchange rates,
are commonly known to be heavy-tailed.
Gombay and Horváth (1999), Quessy, Saı̈d, and Favre (2013) study tests for changes in the
dependence of multivariate time series based on Kendall’s tau, but only consider independent
observations. Moreover, these authors also use a bootstrap approximation for deriving critical
values of a test statistic, while we provide a consistent long-run variance estimator, and do
not need to rely on the bootstrap. A recent reference is Bücher and Kojadinovic (2016), who
propose change-point tests for Kendall’s tau under mixing conditions, but not under the concept
of P -near epoch dependence (see below). Our change-point test does not require information
on the position of potential break points. This is a structural similarity with many other tests
in the econometrics and statistics literature, (e.g. Inoue, 2001). See also Section 5.1 in Stock
(1994) and the references therein.
Kendall’s tau is a U -statistic. The main tool in analyzing the asymptotic behavior of the test
statistic is a new functional limit theorem for sequential U -statistics processes for dependent
data. This theorem is of interest in its own right. Allowing unbounded kernels and placing
no moment requirement on the data process, it is formulated in by far greater generality than
necessary for Kendall’s tau, where the U -kernel is bounded and, further, the whole analysis
can be restricted to bounded data sequences by an invariance argument. This functional limit
theorem provides the basis for constructing change-point tests in the same spirit for any quantity
that may be expressed as a U -statistic.
Several authors have used U -statistics for change-point problems before (e.g. Gombay and
Horváth, 1995; Gombay and Horváth, 1999). The main contribution of the present paper is
the thorough treatment of dependent series. We consider approximating functionals of mixing
processes, where the approximation is in probability, not in an Lp sense as in the usual near
epoch dependence condition. We call this approximation concept P -near epoch dependence.
3

It generalizes Lp -near epoch dependence, not requiring the existence of any moments, and is
hence a fitting framework for nonparametric and robust data analysis.
Another appealing property of our approach in terms of broad applicability is the lack of assumptions on the copula between the two random variables Xi and Yi for a fixed i. Although
Kendall’s tau is a dependence measure that only depends on the copula, there are no conditions
on the existence of partial derivatives. In particular, distributions with non-zero tail dependence
are included in our assumptions, which is important in empirical finance (e.g. Patton, 2006).
See Segers (2012) for a discussion on this issue.
The paper is organized as follows: Section 2 contains the main theoretical results about U statistics, that is, the functional limit theorem for U -statistics and the consistency result for the
estimator of the long-run variance. In Section 3, the asymptotic properties of the test statistic
under the null hypothesis are given. Section 4 deals with estimating the location of a potential
change-point. In Section 5, the test is compared to previous proposals. Section 6 contains
a numerical study, where we observe that the efficiency properties of Kendall’s tau and Pearson’s correlation coefficient translate into similar size and power properties of the corresponding
change-point tests. Section 7 demonstrates applications to financial data examples. Appendix
A contains further background on the concept of P -near epoch dependence, Appendix B investigates the behavior of the U -statistic process under a sequence of local alternatives, and
Appendix C contains the proofs for the theorems of the main text. The proofs of the lemmas
in the appendix and further technical results can be found in the online supplement. Readers
may refer to the supplementary material associated with this article, available at Cambridge
Journals Online (journals.cambridge.org/ect).
We use bold type face to denote vector-valued objects. Throughout, | · |p denotes the p-norm
in Rd , p ∈ [1, ∞), d ∈ N. To denote the Lp norm (E|X|p )1/p of a real-valued random variable
X, we occasionally write ||X||p , p ∈ [1, ∞). All random variables are defined on a common
probability space (Ω, F , P ).

2. Invariance Principle for U -Statistics of dependent series
We treat Kendall’s tau in the framework of asymptotic U -statistic theory. We first give a functional central limit theorem for general U -statistics for multivariate and short-range dependent
time series. We further devise an estimator for the long run variance term and show its consistency. Throughout this section, let (X i )i∈Z be a strictly stationary sequence of d-dimensional
random variables with (d-dimensional) distribution function F . Let further g : Rd × Rd → R
be a measurable, symmetric function. We call
Un = Un (g) =

2
n(n − 1)

X

g (X i , X j )

1≤i<j≤n

the U -statistic with kernel g. The essential tool to treat U -statistics asymptotically is the
Hoeffding decomposition into a linear and degenerate part, i.e.,
n

Un (g) = U +

2X
2
g1 (X i ) +
n
n (n − 1)
i=1

X

g2 (X i , X j ) ,

1≤i<j≤n

where
U = Eg(X, Y ),

g1 (x) = Eg(x, Y ) − U,

g2 (x, y) = g(x, y) − g1 (x) − g1 (y) − U, (1)

and X, Y are independent copies of X 0 .
4

Concerning the serial dependence structure of the process (X i )i∈Z , we assume it to be near
epoch dependent in probability (P -NED) on an absolutely regular process. For the formal
statement of this short-range dependence assumption, which follows below, it is convenient to
let the process (X i ) be indexed by Z. The observed data is then the positive branch of the
doubly infinite sequence. For two sub-σ-fields A, B of F , we define the absolute regularity
coefficient
β(A, B) = E [ess sup {|P (A|B) − P (A)| : A ∈ A}] .
The absolute regularity coefficient is a measure of dependence between the σ-fields A and B, it
lies between 0 and 1, and equals 0 if A and B are independent.
Definition 2.1. Let (X n )n∈Z and (Z n )n∈Z be d- and r-variate stochastic processes on (Ω, F , P ),
respectively, d, r ≥ 1, such that the (d + r)-variate process ((X n , Z n ))n∈Z is stationary. For
k ≤ n, let Fkn = σ(Z k , . . . , Z n ), where also k = −∞ and n = ∞ are permitted.
(i) The process (Z n )n∈Z is called absolutely regular if the absolute regularity coefficients
0
βk = β(F−∞
, Fk∞ ),

k ≥ 1,

converge to zero as k → ∞.
(ii) The process (X n )n∈Z is called near epoch dependent in probability or short P -near epoch
dependent (P -NED) on the process (Z n )n∈Z if there is a sequence of approximating constants (ak )k∈N with ak → 0 as k → ∞, a sequence of functions fk : Rr×(2k+1) → Rd ,
k ∈ N, and a non-increasing function Φ : (0, ∞) → (0, ∞) such that
P (|X 0 − fk (Z −k , . . . , Z k )|1 > ε) ≤ ak Φ(ε)

(2)

for all k ∈ N and ε > 0.
By Lemma A.1 (iii) of Appendix A, Lp -near epoch dependence (p ≥ 1) implies near epoch
dependence in probability. So P -NED can be viewed as a generalization of the more frequently
considered L2 -NED. All the limit theorems in this article may be formulated for Lp -NED sequences as well. We prefer to use P -NED instead of Lp -NED, since we particularly want to
include very heavy tailed data and do not want to assume the existence of even first moments.
Further details and references on the different weak dependence concepts are given in Appendix
A.
For the functional U -statistic limit theorem we require Assumption 2.2, 2.3 and 2.4 to hold.
Assumption 2.2. The process (X i )i∈Z is P -NED on an absolutely regular sequence (Z i )i∈Z ,
and there is a δ > 0 such that
ak Φ(k

−6

) = O(k

−6(2+δ)/δ

)

and

∞
X

δ/(2+δ)

kβk

< ∞.

k=1

Furthermore, a moment condition on g(X i , X j ) is required. Note that we do not impose any
moment conditions on the data sequence (X i )i∈Z itself.
Assumption 2.3. There is a constant M > 0 such that for all k, n ∈ N
E |g (fk (Z −k , . . . , Z k ), fk (Z n−k , . . . , Z n+k ))|2+δ ≤ M

and

E |g(X 0 , X n )|2+δ ≤ M.

Note that Assumptions 2.2 and 2.3 are linked via δ. Weaker moment conditions have to be
paid for by a faster decay of the short-range dependence coefficients and vice versa. The next
5

assumption is also known as the variation condition and was introduced by Denker and Keller
(1986). It can be understood as a form of Lipschitz continuity of the kernel g with respect to
F.
Assumption 2.4. There are constants L, 0 > 0 such that for all  ∈ (0, 0 )
!2
E

|g (x, y) − g (X, Y )|

sup

≤ L,

|x−X|≤,|y−Y |≤

where X, Y are independent with the same distribution as X 0 .
We are now ready to state the following weak invariance principle for the sequential U -process.
The proof is given in Appendix C.
Theorem 2.5. Under Assumptions 2.2, 2.3 and 2.4, we have



[ns]
d
√ U[ns] (g) − U
−→ 2σW
n
s∈[0,1]
in D[0, 1], where W denotes a standard Brownian motion, and the long run variance is given
by
∞
X
σ2 =
Cov (g1 (X 0 ), g1 (X r )) .
r=−∞

Without specific assumptions on the distribution of the whole process (X i )i∈Z , the long run
variance term σ 2 is unknown, and, even if one is willing to make such assumptions, it may
yet be cumbersome to evaluate it. Thus, for statistical applications, an estimator of σ 2 is
desired. For the sample mean, the problem of estimating the long run variance has already
been studied by many authors. Our proposal for an estimate of σ 2 is based upon combining
the HAC (heteroscedasticity and autocorrelation consistent) kernel estimator by de Jong and
Davidson (2000) with an empirical version of the Hoeffding decomposition.
For the kernel g, we define the empirical version ĝ1 of g1 as
n
n
1X
1 X
ĝ1 (x) =
g(x, X i ) − 2
g(X i , X j ),
n
n
i=1

i,j=1

Pn−r
and the empirical covariance for lag r as ρ̂(r) = n1 i=1
ĝ1 (X i )ĝ1 (X i+r ). We then estimate σ 2
by
n−1
X r
σ̂n2 = ρ̂(0) + 2
κ
ρ̂(r),
(3)
bn
r=1

where κ is a weight function (or HAC kernel function) and bn a bandwidth depending on n. In
order achieve consistency, κ and bn have to fulfill some regularity conditions.
Assumption 2.6. The function κ : [0, ∞) → [−1, 1] is continuous at 0 and at all but a finite
number of points and κ(0) = 1. Furthermore, |κ| is dominated by a non-increasing, integrable
function and
Z
Z
∞

∞

κ(t) cos(xt)dt dx < ∞.
0

0

√
The bandwidth bn satisfies bn → ∞ as n → ∞ and bn / n → 0.

6

Assumption 2.6 mainly coincides with Assumption 1 of de Jong and Davidson (2000). It is
satisfied by a large class of kernels, in particular the popular Bartlett kernel κ(t) = (1 −
|t|)1{|t|≤1} . The proof of the following consistency result is also given in Appendix C.
p

Theorem 2.7. Under Assumptions 2.2, 2.3, 2.4 and 2.6, we have σ̂n2 −→ σ 2 as n → ∞, where
σ 2 is as in Theorem 2.5.
Theorems 2.5 and 2.7 give rise to a general test statistic
T̂n =

1
2σ̂n

k
√ |Uk − Un |
2≤k≤n−1
n
max

for CUSUM-type change-point tests based on U -statistics. By combining the continuous mapping theorem (applied to the functional which maps x ∈ D[0, 1] to the real number sup0≤t≤1 |x(t)−
tx(1)|) and Slutsky’s lemma we arrive at the following result.
d

Corollary 2.8. Under Assumptions 2.2, 2.3, 2.4 and 2.6, and if σ 2 > 0, we have T̂n −→
sup0≤λ≤1 |B(λ)|, where B is a standard Brownian bridge on [0, 1].
The assumptions of Theorems 2.5 and 2.7 are very broad and easy to verify. The kernel g as
well as the marginal distribution F are, except for the variation condition, completely arbitrary.
Furthermore, all time series models relevant in financial applications fulfill our short-range
dependence condition with exponential decay of ak and βk . For further details on how P NED
is related to the usual L2 near epoch dependence, see Appendix A.
Invariance principles for U -statistics similar to Theorem 2.5 were established by Yoshihara
(1976) for absolutely regular processes, which do not cover many time series models. Central
limit theorems for U -statistics have been investigated under more general conditions: Denker
and Keller (1986) considered Lipschitz continuous functionals of absolutely regular sequences
and Borovkova, Burton, and Dehling (2001) L1 -NED processes. As far as we know, functional
central limit theorems (invariance principles) for U -statistics have not been studied under more
general dependence conditions.
The potential applications of Corollary 2.8 are manifold. Several U -statistics have gained popularity as estimators that combine high efficiency under normality with appealing invariance and
robustness properties (in the classical sense of robust statistics). The leading example is certainly Kendall’s tau, which we will study in depth in Sections 3 through 7. Another prominent
example is Gini’s mean difference
gn =

2
n(n − 1)

X

|xi − xj |

1≤i<j≤n

for univariate data x1 , . . . , xn . Similarly to Kendall’s tau, Gini’s mean difference is, as a measure
of scale, almost as efficient as the maximum likelihood estimator at normality (the standard
deviation), but is more efficient than the latter at heavy-tailed distributions and less sensitive
to single outlying observations (Gerstenberger and Vogel, 2015). Thus Corollary 2.8 directly
yields the asymptotic null distribution of a Gini’s mean difference based change-point test for
scale, which requires only 2 + δ moments as compared to 4 + δ moments for a sample-variancebased test. Assumption 2.4 is automatically fulfilled in this example as the corresponding kernel
g(x, y) = |x − y| is Lipschitz continuous.
Furthermore, the results can be straightforwardly extended to multivariate U -statistics. The
process convergence of a p-dimensional U -statistic
U n = (Un(1) , . . . , Un(p) ),
7

where p is generally different from the data dimension d, is obtained by means of the Cramér–
P
(i)
Wold device by considering the univariate U -statistic Ũn = i λi Un for arbitrary (λ1 , . . . , λp ) ∈
p
R . Similarly, a multivariate version of Theorem 2.7 follows from an entry-wise consideration
as convergence in probability of a random matrix is implied by the convergence of its marginals.
Besides vector-valued versions of Kendall’s tau or Gini’s mean difference to test for changes in
the rank correlation or scale, respectively, of several time series jointly, we can also consider the
following estimator,
Kn =

2
n(n − 1)

(X i − X j )(X i − X j )>
,
|X i − X j |22
1≤i<j≤n
X

which is sometimes referred to as the spatial Kendall’s tau matrix. It is also known to possess
a rather high efficiency at normality, which has led several authors to consider this estimator
in various contexts (e.g. Fan, Liu, and Wang, 2015). This estimator allows to consistently
estimate the eigenvectors and the ordering of the eigenvalues of the covariance matrix. Thus
Theorems 2.5 and 2.7 also provide the theoretical foundation for a robust change-point test
for, say, detecting changes in the leading eigenvector of the marginal covariance matrix of a
multivariate time series.
Further, these U -statistic results encompass all linear statistics, i.e., U -statistics of order one,
such as the classical CUSUM test based on the sample mean. The classical change-point test
for detecting changes in scale, as studied by Inclán and Tiao (1994), is essentially the CUSUM
test applied to the squares of the centered data. It requires some additional technical effort to
thoroughly deal with the centering for the data, which is ignored by some authors and dealt
with in different ways by other authors. Employing the U -statistic representation of the sample
variance with kernel g(x, y) = (x − y)2 /2, our results provide another elegant method of proof.
A referee raised the question whether the tail dependence coefficient could also be treated in
the current U -statistic framework. In this situation, one would consider a U -statistic of order
one with kernel
r
n
g(FX (xi ), FY (yi )) =
1
(4)
k {FX (xi )≤k/n, FY (yi )≤k/n}
for a bivariate time series ((Xi , Yi ))i∈Z , where FX and FY denote the marginal distributions of
X0 and Y0 , respectively, and k = k(n) converges to ∞ and k/n converges to 0. Using this kernel
in Theorem 2.5, one obtains the process Bn from Bücher, Jäschke, and Wied (2015), which
provides the basis for a CUSUM-type test statistic for detecting changes in tail dependence.
Thus one would need to extend the current setting to a situation where the kernel g may depend
on n. More crucially, one would need a theorem in which boundedness of the second moments
of the kernel is sufficient. While g 2 converges to the tail dependence coefficient, g 2+δ diverges
unless the tail dependence coefficient is zero. This seems difficult to achieve in the current serial
dependence setting. A solution could be to restrict the dependency conditions, e.g. to uniformly
mixing processes. An alternative way of treating tail dependence is by employing the direct link
between the tail dependence coefficient and Kendall’s tau in specific parametric models. By
testing for constant Kendall’s tau and assuming a specific model is true, one automatically tests
for constant tail dependence as well. However, such an approach may be considered not optimal
since, in contrast to an approach based on (4), it strongly relies on the model assumption, see
the discussion in Bücher et al. (2015).

8

3. Change-point Detection for Kendall’s tau
Let ((Xi , Yi ))i∈Z be a strictly stationary series of bivariate random vectors with marginal distribution function F (x, y) = P (X0 ≤ x, Y0 ≤ y). Throughout the remainder of the article, we
assume F to be Lipschitz continuous. This is fulfilled if F possesses a bounded density, but,
e.g., X0 = Y0 almost surely is also allowed. Kendall’s rank correlation coefficient, also known
as Kendall’s tau, is defined as
τ = P ((X 0 − X)(Y 0 − Y ) > 0) − P ((X 0 − X)(Y 0 − Y ) < 0),
where (X, Y ) and (X 0 , Y 0 ) are two independent random variables with distribution function
F . Kendall’s tau is a measure of correlation between the random variables X and Y , where
we understand correlation generally as monotone dependence, which, loosely speaking, carries
information on to what extent one variable on average increases or decreases as the value of the
other increases. Kendall’s tau, as well as the related dependence measure Spearman’s rho, is
a function of the copula only. In particular, it does not depend on the marginal distributions
and is hence invariant to monotone marginal transformations, see Nelsen (e.g. 2006, Chap. 5).
Consequently, its sample version only depends on the ranks of the data, which is the reason
for Kendall’s tau being also referred to as rank correlation measure. The sample version of
Kendall’s tau is defined as
X 

2
τ̂n =
1(0,∞) {(Xj − Xi )(Yj − Yi )} − 1(−∞,0) {(Xj − Xi )(Yj − Yi )} ,
n(n − 1)
1≤i<j≤n

which is a U -statistic with kernel function g : R2 × R2 → R given by

g (x, y), (x0 , y 0 ) = 1(0,∞) {(x0 − x)(y 0 − y)} − 1(−∞,0) {(x0 − x)(y 0 − y)}.

(5)

For dependent data, τ̂n is not necessarily unbiased, but under the weak dependence conditions
we consider, it is (strongly) consistent for τ . We will study a test for change in the dependence
structure of the marginals by the test statistic
k
T̂τ,n = max √ |τ̂k − τ̂n |,
k=1,...,n
n
which rejects the null hypothesis of constant rank correlation if T̂τ,n is too large. By means of
the functional limit theorem for sequential U -statistics (Theorem 2.5), we know that


[tn]
√ (τ̂[tn] − τ )
n
t∈[0,1]
converges under the assumption of no change and for weakly dependent series weakly to a
Brownian motion 2στ W , where
στ2 =

∞
X

E [ψ(X0 , Y0 )ψ(Xj , Yj )]

(6)

j=−∞

with ψ(x, y) = 4F (x, y) − 2FX (x) − 2FY (y) + 1 − τ . The function ψ can be seen to be the linear
part g1 of the Hoeffding decomposition (1) for the specific kernel g in (5) for any continuous

9

distribution function F . By the continuous mapping theorem, we have the weak convergence
k
d
T̂τ,n = max √ |τ̂k − τ̂n | −→ 2στ sup |B(λ)|,
k=1,...,n
n
0≤λ≤1
where B is a standard Brownian Bridge. The distribution of sup0≤λ≤1 |B(λ)| is known and
sometimes referred to as Kolmogorov distribution. We use the estimator for the long run
variance proposed in Section 2. Let Fn , FX,n and FY,n denote the empirical distribution functions of ((Xi , Yi ))i=1,...,n , (Xi )i=1,...,n and (Yi )i=1,...,n , respectively. Then ψ̂n,i = 4Fn (Xi , Yi ) −
2FX,n (Xi ) − 2FY,n (Yi ) + 1 − τ̂n can be seen to equal ĝ1 ((Xi , Yi )), and the variance estimator (3)
can be written as
2
σ̂τ,n

n

n−1

i=1

j=1

1X 2
2X
=
ψ̂n,i +
κ
n
n



j
bn

X
n−j

ψ̂n,i ψ̂n,i+j ,

(7)

i=1

where κ and bn are the same as in Section 2. Corollary 3.1 below gives the asymptotic distribution of the test statistic T̂n /(2σ̂τ,n ) under the null hypothesis of no change.
Corollary 3.1. Let ((Xi , Yi ))i∈Z be a two-dimensional, stationary process with a Lipschitz continuous marginal distribution function. Assume that ((Xi , Yi ))i∈Z is P -NED with approximating
constants (ak )k≥1 on an absolutely regular process with absolute regularity coefficients (βk )k≥1
satisfying Assumption 2.2 for some δ > 0. Let further Assumption 2.6 hold. Then, if στ2 > 0,
T̂τ,n d
−→ sup |B(λ)|,
2σ̂τ,n
0≤λ≤1

(8)

where (B(λ))0≤λ≤1 is, as before, a standard Brownian bridge.

4. Change-point Estimation and Local Power
If the test rejects the null hypothesis of constant correlation, and if it is furthermore reasonable
to assume that there is one sudden change-point, it is of interest to locate this change-point.
An intuitive estimator, which is common when dealing with CUSUM-type change-point tests,
is the position at which the weighted differences take their maximum, that is
k
k̂n = arg max √ |τ̂k − τ̂n |.
n
1≤k≤n
We will show in the following that this is indeed a reasonable estimator. We will assume that
the following model holds.
(n)

(n)

Model 4.1 (Change-point model). Let 0 < λ∗ < 1. For n ∈ N let ((Xi , Yi
and

(n)
(n)
((Xi , Yi ))[λ∗ n]+1≤i≤n

))1≤i≤[λ∗ n]

be two bivariate, stationary stochastic processes with marginal
(n)

(n)

distribution functions F and G, respectively. Let furthermore ((Xi , Yi ))1≤i≤n be P -near
epoch dependent1 on an absolutely regular process with coefficients satisfying Assumption 2.2
uniformly for all n.
1

For non-stationary processes the short-range dependence conditions have to be formulated slightly
more generally than in Definition 2.1. The absolute regularity coefficients (βk )k∈N are defined as βk =
t
∞
supt∈Z β(F−∞
, Ft+k
), and the P -NED approximation coefficients (ak )k∈N must satisfy

sup P |X t − fk,t (Z t−k , . . . , Z t+k )|1 > ε ≤ ak Φt (ε),
t∈Z

10

The goal is to estimate λ∗ . Let τF and τG denote Kendall’s tau of F and G, respectively.
Moreover, let τF,G = Eg((X1 , Y1 ), (X2 , Y2 )), cf. (5), where (X1 , Y1 ) ∼ F and (X2 , Y2 ) ∼ G are
independent.
(n)

(n)

Theorem 4.2. If ((Xi , Yi

))1≤i≤n,n∈N follows Model 4.1, furthermore τF 6= τG and

τF,G − τF
(1 − λ∗ )2
< 1,
≤
∗
2
∗
2 ((1 − λ ) + λ )
τG − τF

(9)

p

then k̂n /n −→ λ∗ as n → ∞.
Condition (9) prohibits τF,G to be too close to τF compared to τG . It is an open research
question which values of τF,G are possible for given τF and τG , and in particular if τF,G may at
all lie outside the interval [τF , τG ].
We further analyze the power of the proposed change-point test under local alternatives. We
give a formula for the asymptotic distribution of the test statistic within a specific class of local
alternatives.
Model 4.3. Let (Xi , Yi )i∈Z be a strictly stationary, bivariate process that satisfies Assumption
2.2, and let 0 < λ∗ < 1 and ∆ > 0. We define for any integer n ≥ 1
(
(Xi , Yi )
1 ≤ i ≤ [nλ∗ ]
(n)
(n)
(Xi , Yi ) =
∆
(Xi + √n Yi , Yi ) [nλ∗ ] ≤ i ≤ n.
We assume that (Xi , Yi ) has an absolutely continuous distribution F with bounded density f
satisfying limx→∞ f (x, y) = 0 and limy→∞ f (x, y) = 0 for all x, y ∈ R and
Z Z
∂
sup |y f (x + αy, y)|dxdy < ∞
0≤|α|≤ε ∂x
(n)

(n)

for some ε > 0. By Fn we denote the distribution of (Xi , Yi

). The density of Fn is given by

∆
fn (x, y) = f (x − √ y, y).
n
Theorem 4.4. If (Xi , Yi )i≥1 follows Model 4.3, we have
d

T̂τ,n −→ sup |2στ B(λ) + ∆(φλ∗ (λ) − λφλ∗ (1))| ,
0≤λ≤1

where στ is defined in (6) and the function φλ∗ : [0, 1] → R is defined as

( R
 2
R∞
∞
4 −∞ −∞ y(2F (y|x) − 1)F (dy|x) fX
(x)dx (s − λ∗ ) s ≥ λ∗ ,
φλ∗ (s) =
0
s ≤ λ∗ .
From Theorem 4.4 we can conclude the consistency of the test against local alternatives of the
type studied in Model 4.3, for which it suffices to observe that the integral occurring in the
limit in Theorem 4.4 is non-zero: For any absolutely continuous distribution function F with
where the functions fk,t and Φt may also depend on t. The underlying process (Z t )t∈Z is not required to be
stationary.

11

density f , and with finite expectation, we have
Z ∞
Z ∞
Z ∞
yf (y)dy
2yF (y)f (y)dy −
y(2 F (y) − 1)f (y)dy =
−∞
−∞
Z−∞
Z
=
ydF 2 (y) − ydF (y) = E (max(Y1 , Y2 )) − E(Y1 ),
where Y1 , Y2 are independent random variables with distribution F . Now, E (max(Y1 , Y2 )) −
E(Y1 ) >R0, unless Y1 is a constant, in which case E (max(Y1 , Y2 )) − E(Y1 ) = 0. Thus, the inner
∞
integral −∞ y(2F (y|x) − 1)F (dy|x) is positive, unless the conditional distribution of Y1 given
X1 = x is degenerate, i.e. Y1 takes only one value. Hence the integral
Z ∞Z ∞
2
y(2F (y|x) − 1)F (dy|x)fX
(x)dx
−∞

−∞

is positive unless Y1 is a deterministic function of X1 .

5. Previous Proposals
Wied, Krämer, and Dehling (2012) consider the test statistic
k
T̂%,n = max √ |%̂k − %̂n |
1≤k≤n
n
based on Pearson’s linear correlation coefficient %̂n . This test will serve as the main benchmark
method for our test. Our motivation for using Kendall’s tau is the wish to efficiently detect
structural changes in arbitrarily heavy-tailed and potentially contaminated data. Both tests
are constructed in a similar way. The differences between the two tests are largely due to the
different properties of the estimators %̂k and τ̂k . It is therefore worthwhile to have a brief look
at these two correlation measures.
Kendall’s tau is invariant with respect to strictly monotonic, componentwise transformations
of F . For continuous F , it can be written as τ = 4EF (X, Y ) − 1, which can be seen to be a
function of the copula of F alone. The same applies to its asymptotic variance at i.i.d. data,
ASV (τ̂n ) = 16Var(ψ(X, Y )) with ψ being defined in Section 3. For further details see, e.g.,
Nelsen (2006, Chap. 5). Consequently, no matter how heavy the tails of the distribution F are,
as long as the marginals are joined by a Gauss copula, the variance of τ̂n is the same as in the
normal model.
In the normal model there is a one-to-one correspondence between τ and Pearson’s moment
correlation %, which is given by
τ = (2/π) arcsin(%),

−1 ≤ % ≤ 1.

(10)

Thus by letting %̂τ,n = sin(π(τ̂n − 1/2)), the estimators %̂n and %̂τ,n are, under normality, both
Fisher-consistent for the same quantity %. Comparing their asymptotic variances
ASV (%̂n ) = (1 − %2 )2 ,

ASV (%̂τ,n ) = (1 − %2 )(π 2 /9 − 4 arcsin2 (%/2))

(11)

(e.g. Croux and Dehon, 2010), allows a prognosis concerning the efficiency relation of the corresponding change-point tests. For example, for two independent random variables we have
ASV (%̂n ) = 1 and ASV (%̂τ,n ) = π 2 /9 = 1.097.
The scope of the identity (10) extends to all elliptical distributions with finite second moments.
(e.g. McNeil, Frey, and Embrechts, 2005, p. 97). For example, at the two-dimensional elliptical
12

tν -distribution with ν degrees of freedom and uncorrelated margins, we have
ASV (%̂n ) = (ν − 2)/(ν − 4),
ASV (%̂t(ν),n ) = (ν + 4)/(ν + 2),

ν > 4,
ν > 0,

where %̂t(ν),n is the maximum likelihood estimator of % at the two-dimensional tν family. (e.g.
Bilodeau and Brenner, 1999, p. 221). Dengler (2010) gives values for the asymptotic variance of
%̂τ,n at uncorrelated tν -distributions. It is a decreasing function of ν, it equals 1.922 and 1.296
for ν = 1 and ν = 5, respectively, and is smaller than ASV (%̂n ) for ν ≤ 16. We further take
note of the remarkable fact that for all uncorrelated t- and normal distributions, the asymptotic
relative efficiency of Kendall’s tau with respect to the respective MLE is above 90% for ν ≥ 2.
It is more than 99% at an uncorrelated t5 distribution.
The other popular nonparametric correlation measure, Spearman’s rho, which is often considered alongside Kendall’s tau, is defined as Pearson’s linear correlation of the ranks of the data.
It can be written as
n

X
12
n+1
r̂n =
Rn (Xi )Rn (Yi ) − 3
,
(n − 1)n(n + 1)
n−1
i=1

where Rn (Xi ) denotes the rank of the ith observation Xi among X1 , . . . , Xn , likewise Rn (Yi ).
The population version of Spearman’s rho, s = 12E (FX (X)FY (Y )) − 3, is also a function of
the copula. Generally, Kendall’s tau and Spearman’s rho have similar statistical properties.
See, e.g., Nelsen (2006, Chap. 5) for details on their relationship. Croux and Dehon (2010)
compare both with respect to robustness and efficiency and arrive at the conclusion, that in
both respects their performance is comparable, but Kendall’s tau is slightly favorable. Wied,
Dehling, van Kampen, and Vogel (2014) propose a nonparametric, robust change-point test for
constant correlation for strongly mixing sequences that is closely related to Spearman’s rho.
They consider the test statistic
T̂s,n

k
= max √ |ŝk − ŝn |,
1≤k≤n
n

where

ŝk = 12n

−3

k
X

Rn (Xi )Rn (Yi ) − 3 − 12/n,

1 ≤ k ≤ n,

i=1

(12)
2 in (7) above. The proof
along with a suitable long-run variance estimator in the same vein as σ̂τ,n
of its convergence is based on an invariance principle for the multivariate sequential empirical
process. Despite the mentioned practical parity of Kendall’s tau and Spearman’s rho, this test
has a low efficiency compared to our Kendall’s tau based test (see Section 6). The reason lies in
the usage of Rn (·) instead of Rk (·) in (12). Recently, Kojadinovic, Quessy, and Rohmer (2016)
proposed a new constancy test for Spearman’s rho which fixes this drawback and has therefore
considerably higher power. Spearman’s rho is asymptotically equivalent to a U -statistic of order
3, and an asymptotic analysis of the related test statistic
k
T̂r,n = max √ |r̂k − r̂n |
1≤k≤n
n
by means of U -statistics theory is mathematically much more involved. Since Spearman’s rho,
on the other hand, exhibits no pronounced advantage over Kendall’s tau, we do not pursue
this further here. Finally, we note that both estimators require a comparable computing effort.
Both can be computed in O(n log n) time. Simple algorithms to compute the test statistics
require O(n2 ).

13

6. Simulation Results
In this section we give some numerical results, comparing the performance of the test to the
previous proposals by Wied et al. (2012) and Wied et al. (2014). These are based on the test
statistics T̂%,n and T̂s,n , respectively (cf. Section 5) and referred to as Pearson test and Spearman
test in the following. We call the new proposal Kendall test. Moreover, we consider the improved
Spearman test by Kojadinovic et al. (2016).
2
Throughout, we estimate the long-run variance στ2 of the Kendall test by the estimator σ̂τ,n
given by (7), where we choose the bandwidth bn = b2n1/3 c and κ to be the quartic kernel
κ(x) = (1 − x2 )2 1[−1,1] (x).
Altogether we found neither the choice of the kernel nor the bandwidth to be very critical.
Generally, choosing the bandwidth too small in strong dependence scenario tends to have a
bigger impact than choosing it too large in the case of little or no dependence. We recommend
the quartic kernel since it is smooth and has more of a flat-top-like behavior than the popular
Bartlett kernel, giving more weight to small-lag autocorrelations. The variance estimation for
T̂%,n and T̂s,n is done according to the authors’ proposals, which both also implement kernel
estimators following de Jong and Davidson (2000). For the improved Spearman test by Kojadinovic et al. (2016), we use implementation in the R-package npcp (Kojadinovic, 2015) with the
Bartlett kernel for the long-run variance estimation.
For all tests and all dependence scenarios, we take the bandwidth bn = b2n1/3 c. This is a
relatively large bandwidth, which is suitable for strong dependence as in our AR(1) example
below. A smaller bandwidth would be more appropriate for no or little serial dependence. The
problem of selecting an optimal, data-adaptive bandwidth is a ubiquitous one. It affects the
long-run variance estimation of all change-point tests in a similar way and is analogous to the
optimal blocklength selection problem for subsampling or bootstrapping, which may also be
employed to obtain critical values. We do not discuss the problem of automatic bandwidth
selection in the present article. Our concern is how the use of different estimators affects the
behavior of the change-point tests. The methods under consideration use the same technique for
long-run variance estimation and, hence, choosing the same bandwidth in all cases allows a fair
comparison of the methods. However, any bandwidth selection procedure previously proposed,
see, e.g., Kojadinovic et al. (2016, Section 3.3) and the references therein, can also be put to
use for the Kendall test and is likely to further improve the power of the test as compared to
the results presented below.
We consider three data models that implement three different types of serial dependence: independent observations, a multivariate AR(1) process and a constant conditional correlation
(CCC) GARCH(1,1) process, as considered, e.g., in Aue et al. (2009). All three models are
based on a sequence of independent and identically distributed innovations (δi , εi ), i ∈ Z, having a bivariate, centered elliptical distribution E2 (0, S) with


1 %
S=
,
% 1
where the parameter |%| ≤ 1 is equal to the usual moment correlation if the second moments of
(δ1 , ε1 ) are finite.
   
Xi
δ
Model 6.1.
= i ,
i ∈ Z,
Yi
εi

14

Model 6.2. The series ((Xi , Yi ))i∈Z follows the AR(1) process
 

  
Xi
Xi−1
δ
= 0.8
+ i ,
i ∈ Z,
Yi
Yi−1
εi
with AR parameter ϕ = 0.8.
Model 6.3. The series ((Xi , Yi ))i∈Z follows the CCC-GARCH(1,1) process
     
 2      2  
  2 
Xi
σi
δi
σi
0.1
0.1
Xi−1
0.84
σ
=
◦
with
=
+
◦
+
◦ i−1
,
2
2
2
Yi
τi
εi
τi
0.1
0.1
Yi−1
0.84
τi−1

i ∈ Z,

where ◦ denotes the Hadamard product, i.e. component-wise vector multiplication.
Most simulation results below are for n = 500. At this sample size, the distribution of the test
statistic T̂τ,n /(2σ̂τ,n ) is well approximated by its limit distribution under the null hypothesis
in all dependence scenarios considered. For the first half of the data, we sample independent
realizations (δi , εi ), i = 1, . . . , 250, with correlation parameter %1 = 0.4. For the second half of
the data, we use the correlation parameter %2 , for which we allow the values 0.4 (null hypothesis),
0.6, 0.8, 0.2, 0, −0.2, −0.4. Thus in Model 6.1, we have a constant correlation of 0.4 at the
beginning and then a sudden jump, whereas in Models 6.2 and 6.3, there is a gradual but
quick change in the correlation of the observed process (Xi , Yi ). Also note that, under the
null, the data process ((Xi , Yi ))i∈Z has the same marginal correlation as the innovation process
((δi , εi ))i∈Z in Models 6.1 and 6.2, but generally not in Model 6.3.
In Models 6.1 and 6.2, we consider five different elliptical distributions for (δi , εi ): the bivariate
normal distribution and bivariate tν -distribution with ν = 20, 5, 3, 1 The parameter ν > 0 is
called the degrees of freedom or the tail index. The t20 distribution has slightly heavier tails
than the normal, whereas t5 , t3 and t1 serve as examples of very heavy-tailed distributions. The
tν distribution possesses finite moments of order α for any α < ν. Thus, the Pearson test by
Wied et al. (2012), which requires finite fourth moments, does not work for ν = 3 and ν = 1.
From an economic point of view, however, the tail indices 3 and 1 are most interesting. There
is evidence that financial returns on many stocks, stock indices and foreign exchange rates in
developed economies typically have tail indices ν in the interval ν ∈ (2, 4) and thus have finite
variances but infinite fourth moments. It has emerged that ν = 3 is an appropriate choice for
financial returns in developed markets and Ibragimov, Ibragimov, and Kattuman (2013) provide
empirical evidence that ν may be even smaller than 2 for foreign exchange rates in emerging
economies.
In Model 6.3, we used the normal distribution and tν distributions with ν = 20, 8, 5. The
CCC-GARCH model generates heavy tails also for normal innovations. For tν innovations with
ν = 3, 1, the process explodes. For ν = 5, the CCC-GARCH process of Model 6.3 generates
very pronounced volatility clusters.
For each combination of model, jump height and marginal distribution we generate 1000 samples
and compute the three test statistics from each sample. The observed rejection frequencies at
the significance level .05 for sample size n = 500 are given in Tables 1, 2 and 3 for Models 6.1,
6.2 and 6.3, respectively. At Table 1 (independence scenario) we note the following.
(1) The Pearson test is slightly better than the Kendall test for the normal distribution. Both
tests lose power with increasing tails, but the loss is much smaller for the Kendall test.
For the t20 distribution, the results are comparable. The Kendall test is clearly better for
heavier tails. These observations are fully in line with our expectations considering the
efficiency comparison of the respective correlation measures in Section 5.
15

Table 1: Efficiency comparison of several correlation change-point tests under Model 6.1. Different marginal distributions, 500 observations, different jump sizes in the middle of the sample.
Empirical rejection frequencies at the asymptotic .05 level based on 1000 repetitions.
Change at n/2:
none
-.2 +.2
-.4
+.4
-.6
-.8
Distribution

Test

normal

Pearson
Spearman
improved Spearman
Kendall

.04
.04
.03
.05

.46
.06
.43
.44

.70
.07
.58
.65

.97
.22
.95
.96

1.00
.20
1.00
1.00

1.00
.47
1.00
1.00

1.00
.78
1.00
1.00

t20

Pearson
Spearman
improved Spearman
Kendall

.04
.03
.02
.04

.42
.07
.42
.46

.65
.08
.56
.63

.97
.22
.94
.97

1.00
.20
1.00
1.00

1.00
.47
1.00
1.00

1.00
.77
1.00
1.00

t5

Pearson
Spearman
improved Spearman
Kendall

.04
.04
.03
.04

.24
.08
.38
.41

.41
.08
.50
.55

.73
.22
.91
.95

.95
.20
1.00
1.00

.95
.46
1.00
1.00

.98
.76
1.00
1.00

t3

Pearson
Spearman
improved Spearman
Kendall

.06
.03
.04
.03

.14
.08
.32
.39

.25
.08
.46
.52

.39
.21
.88
.91

.69
.18
1.00
1.00

.64
.43
1.00
1.00

.79
.72
1.00
1.00

t1

Pearson
Spearman
improved Spearman
Kendall

.47
.03
.03
.04

.48
.06
.25
.29

.50
.07
.33
.38

.49
.17
.74
.83

.56
.17
.96
.98

.52
.38
.98
1.00

.51
.63
1.00
1.00

(2) Throughout, the naive Spearman test has a very low power, which is not true for the
improved Spearman test. The improved Spearman test and Kendall test show comparable
results under normality as well as heavy tails, with advantages for the Kendall test. This
is in line with the efficiency comparison of both estimators by Croux and Dehon (2010).
(3) For the t3 distribution, the Pearson test yields still approximate results, whereas for the t1
distribution it is completely useless.
Analyzing Table 2 (AR scenario) we find that
(4) the power of all tests is lower for the AR(1) process than in the independent case, and
(5) the observations made at Table 1 concerning the comparison of the tests generally also
apply here. The Kendall test is slightly better than the improved Spearman test.
(6) For normal, t20 , t5 and t3 innovations, the performance of the Kendall, the improved Spearman and the Pearson test are rather similar. The effect of the heavy tails is less pronounced
than in the independent case. This is not entirely surprising. In Model 6.2, the marginal
distribution of the process can be expressed as a sum of independent random variables.
Although it is generally not normal, it is, purely heuristically speaking, closer to a normal
distribution than the innovations (if these possess finite second moments).

16

Table 2: Efficiency comparison of several correlation change-point tests. ((Xi , Yi ))i=1,...,n AR(1)
process with AR-parameter ϕ = 0.8. Different innovation distributions, 500 observations, several alternatives. Empirical rejection frequencies at the asymptotic .05 level based on 1000
repetitions.
Change at n/2:
none
-.2 +.2
-.4 +.4
-.6
-.8
Distribution

Test

normal

Pearson
Spearman
improved Spearman
Kendall

.07
.05
.03
.05

.13
.06
.11
.14

.27
.04
.15
.19

.45
.07
.39
.46

.77
.07
.73
.73

.78
.11
.73
.79

.94
.18
.95
.96

t20

Pearson
Spearman
improved Spearman
Kendall

.06
.05
.03
.04

.11
.06
.12
.13

.31
.06
.15
.23

.41
.08
.36
.42

.77
.08
.70
.74

.77
.12
.70
.79

.94
.20
.95
.95

t5

Pearson
Spearman
improved Spearman
Kendall

.08
.06
.03
.05

.12
.05
.13
.15

.26
.06
.15
.19

.34
.08
.35
.40

.67
.09
.67
.67

.67
.12
.68
.73

.89
.18
.90
.95

t3

Pearson
Spearman
improved Spearman
Kendall

.10
.06
.05
.05

.11
.08
.09
.12

.26
.06
.12
.18

.25
.08
.28
.34

.56
.09
.58
.62

.50
.09
.58
.67

.67
.17
.84
.90

t1

Pearson
Spearman
improved Spearman
Kendall

.46
.08
.03
.06

.49
.07
.05
.12

.52
.08
.06
.11

.50
.09
.13
.18

.56
.11
.20
.34

.50
.12
.24
.34

.54
.14
.41
.53

At Table 3 (CCC-GARCH scenario) we observe that
(7) the efficiencies of the test are comparable to the independence case. GARCH processes are
white noise with zero autocorrelations. Serial dependence in the second-order characteristics
appear to influence the tests less.
(8) The Pearson test has difficulties keeping the size. A size-adjusted power comparison shows
a strict superiority of the Kendall test over the Pearson test.
To give an impression of the power of the test in other data situations, we also include some
limited simulation results for sample sizes n = 250, 500, 1000 and change locations ranging from
n/8 to n/2. The data generating process is CCC-GARCH (Model 6.3) with normal innovations
and ρ1 = 0.4 and ρ2 = 0. The parameters for the long-run variance estimation are as before. The
results are summarized in Table 4. We find the general picture mediated by Table 3 concerning
the comparison of the tests confirmed. The power of all change-point tests is lower for changes
occurring nearer to either end of the data sequence.
Altogether the simulation results are favorable for the Kendall test. It is clearly superior to
the Pearson test at heavy-tailed data and qualitatively non-inferior under normality. It is noninferior to the improved Spearman test in all scenarios considered. Kojadinovic et al. (2016)
also propose a bootstrapping procedure to obtain critical values for the improved Spearman
17

Table 3: Efficiency comparison of several correlation change-point tests. ((Xi , Yi ))i=1,...,n multivariate GARCH process. Different innovation distributions, 500 observations, several alternatives. Empirical rejection frequencies at the asymptotic .05 level based on 1000 repetitions.
Change at n/2:
none
-.2 +.2
-.4
+.4
-.6
-.8
Distribution

Test

normal

Pearson
Spearman
improved Spearman
Kendall

.06
.04
.03
.04

.47
.07
.43
.45

.63
.09
.56
.59

.94
.22
.96
.96

.99
.17
1.00
1.00

1.00
.47
1.00
1.00

1.00
.76
1.00
1.00

t20

Pearson
Spearman
improved Spearman
Kendall

.07
.04
.02
.05

.46
.07
.41
.46

.55
.07
.51
.57

.90
.22
.94
.96

.95
.19
1.00
1.00

.98
.46
1.00
1.00

1.00
.78
1.00
1.00

t8

Pearson
Spearman
improved Spearman
Kendall

.13
.05
.03
.04

.42
.07
.38
.45

.42
.08
.50
.52

.76
.21
.90
.94

.78
.19
1.00
.99

.90
.42
1.00
1.00

.94
.72
1.00
1.00

t5

Pearson
Spearman
improved Spearman
Kendall

.30
.06
.03
.07

.44
.09
.34
.40

.36
.10
.39
.41

.66
.20
.88
.89

.62
.23
.99
.94

.75
.41
1.00
.99

.81
.69
1.00
.99

test, which their simulation results suggest improves the power slightly. A similar procedure
applicable to the Kendall test has been proposed recently by Bücher and Kojadinovic (2016).
The above data generating processes are all based on elliptical innovations. Any change in the
generalized correlation coefficient % constitutes a shift of the same magnitude for all dependence
measures considered, and the corresponding tests generally behave similarly. However, in data
situations where a change occurs that affects the various dependence measures to a different
degree, the tests may behave qualitatively differently. A class of examples where a change
occurs in the Pearson correlation but neither in Kendall’s tau nor Spearman’s rho is generated
by leaving the copula constant but changing the marginal distributions. Similarly, one may
change the copula in such a way that Kendall’s tau or Spearman’s rho remain fixed. (A changepoint test for the whole copula has recently been proposed by Bücher, Kojadinovic, Rohmer,
and Segers (2014)). To illustrate this point, consider the following example: Suppose in the
first half of the sequence that the observations are i.i.d., following the distribution of
 
  

X
0
1 0.6
= N2
,
.
0
0.6 1
Y
√
In the second half, the observations are i.i.d. with distribution of (X, Y 3 |Y |/ 105). Thus, before
and after the change, the data have mean zero, marginal variances one, and a Kendall’s tau
coefficient
of τ = 2π −1 arcsin(0.6) ≈ 0.41. However, the Pearson correlation changes from 0.6
p
to 2/(105π)24/5 ≈ 0.37. Consequently, the Kendall test has no power against this change
whereas the Pearson test does. If the distribution of the observations changes
√ from that of
(X, Y ) in
the
first
half
to
that
of
(U,
V
)
in
the
second
half,
where
U
=
2
3(Ũ − 1/2) and
√
V = 42 3/13(Ũ 13 − 1/14) for Ũ ∼ U [0, 1], then, as before, mean and marginal variances

18

Table 4: Efficiency comparison of several correlation change-point tests at Model 6.3 (multivariate GARCH). Innovation correlation changes from %1 = 0.4 to %2 = 0, normal innovations.
Different sample sizes, different jump locations. Empirical rejection frequencies at the asymptotic .05 level based on 1000 repetitions.
Change at :
none n/8
n/4 3n/8
n/2
Sample size

Test

n = 250

Pearson
Spearman
improved Spearman
Kendall

.06
.03
.02
.04

.12
.04
.04
.11

.35
.07
.20
.38

.59
.10
.47
.60

.64
.10
.58
.69

n = 500

Pearson
Spearman
improved Spearman
Kendall

.06
.04
.03
.04

.22
.06
.12
.22

.76
.11
.68
.79

.92
.15
.91
.94

.94
.22
.96
.96

n = 1000

Pearson
Spearman
improved Spearman
Kendall

.06
.05
.03
.04

.55
.10
.44
.56

.99
.26
.99
1.00

1.00
.40
1.00
1.00

1.00
.43
1.00
1.00

Table 5: Empirical rejection frequencies at the asymptotic .05 level based on 1000 repetitions;
Sample size n = 500.
Kendall test
Pearson test
constant copula example
constant correlation example

.23
1.00

.94
.07

remain the same, also the Pearson correlation remains at 0.6, but the Kendall’s tau coefficient
changes from 0.41 to 1. In this case, the Pearson test has no power, whereas the Kendall test
has. In Table 5, we give simulated rejection frequencies (at the 0.05 level) for the two examples
for n = 500 with the settings for the long-run variance estimation as before. Generally, the
differences in the dependence measures tend to be of comparable size in realistic data models
and not as pronounced as in the artificial examples above.

7. Data Examples
Wied et al. (2012) analyze the dependence between the German stock index (DAX) and the
Standard and Poor’s 500 (S&P 500). We apply the Kendall test and the Pearson test (with the
same parameter choices for the variance estimation as in the simulation section) to the daily
log returns of the two financial indices in the years 2006 through 2009 (1043 observations).
The second half of this
 period covers what has been termed the Global Financial Crisis. The
processes √kn |r̂k − r̂n | k=1,...,n and √kn |τ̂k − τ̂n | k=1,...,n are depicted in Figure 1. Their maxima
are the values of the test statistics of the Pearson and the Kendall test, respectively. Both tests
give a p-value below 0.005, and both attain their maximum on July 14, 2008, at the height of
the financial crisis. (Lehman Brothers filed for bankruptcy on September 14, 2008.) The tests
behave similarly and their outcome supports the assumption that the dependence between both
indices considered can not be assumed to be identical before and during the financial crisis of

19

3.0
2.5

Pearson

2.0
1.5

5% significance level

1.0

Kendall

0.5
0.0

2006

2007

2008

2009

Figure 1: Processes √kn |r̂k − r̂n | (grey) and √kn |τ̂k − τ̂n | (black), k = 1, . . . , n, computed from
log returns of DAX and S&P 500 between Jan 1, 2006, and Dec 31, 2009.
2008.
With a second set of data, the differences between both tests become apparent. We consider the
Dow Jones Industrial Average and the Nasdaq Composite in the years 1987 and 1988 (Figure 2).
The most notable feature of both time series is the heavy loss on October 19, 1987, commonly
known as Black Monday. Here we may ask in particular the question if the market conditions
substantially changed after this date. Does Black Monday constitute a break in the correlation
between the two time series? The Pearson test reports a p-value indistinguishable from zero by
machine accuracy. The underlying processes of the Pearson and the Kendall test are shown in
Figure 3. The outcome of the Pearson test is determined by the peak on October 19, 1987, which
is explained as follows. On October 19, both indices suffered heavy losses, suggesting a strong
positive correlation of their log returns. The following day the Dow Jones recovered to some
small degree, whereas the Nasdaq experienced an even larger drop, suggesting strong negative
correlation. Thus the process of successive sample correlations jumps up and immediately down
again.
The Kendall test gives a p-value of 0.24, indicating one can assume the correlation between the
two time series to be constant over the observed time period. The empirical Kendall’s tau is
0.52 prior to Black Monday, and 0.56 afterwards. Indeed, the market conditions turned out to
be not much different from before, the DJIA even closed positive for 1987.
The strong impact of a few or even a single extreme observation on the Pearson test is in line
with Stock and Watson (2010, Ch.2), who illustrate the inappropriateness of normal distribution
assumptions and the necessity of considering heavy-tailed distributions in modeling financial
time series such as the DJIA during the Black Monday crisis.

8. Conclusion
We have presented a fluctuation test for detecting changes in the dependence between two time
series based on Kendall’s rank correlation coefficient. We have demonstrated the non-inferiority
of the test in terms of efficiency and the clear superiority in terms of robustness and applicability
to a similar, previously proposed test, which is based on Pearson’s moment correlation. To allow
arbitrarily heavy-tailed data and very weak assumptions concerning the serial dependence, we
have introduced the concept of near epoch dependence in probability. We have studied the
asymptotic behavior of the test statistic under stationarity by means of limit theorems for

20

2600
2400
2200
2000
1800

Dow Jones

450
400
350

Nasdaq

300

1987

1988

Figure 2: Daily closings of Dow Jones Industrial Average and Nasdaq Composite from Jan 1,
1987, to Dec 31, 1988.
U -statistics for weakly dependent, stationary processes.
We showed simulation results for various dependence scenarios with elliptical innovations. The
elliptical model is of wide-spread use and its intrinsic symmetry allows a meaningful comparison of the two change-point tests. Outside ellipticity, where monotone dependence (which
is measured, e.g., by Kendall’s τ ) and linear dependence (which is measured by the Pearson
correlation) are not necessarily equivalent, it is a matter of debate which of the two types of
dependence is more relevant. A detailed discussion goes beyond the scope of this paper, but
it appears that one is often interested in monotone dependence rather than linear dependence,
and the prevalent use of the sample correlation to measure monotone dependence is presumably
due to its simplicity and historical dominance.
Furthermore, simulations also show that the proposed test possesses a variety of advantageous
features that have not been discussed in this paper. It has power against gradual or fluctuating changes in the correlation, not only sudden jumps, as presented in Section 6. It also
exhibits a much better robustness against heteroscedasticity than the Pearson test. However, a
thorough theoretical assessment of these properties as well as constructing tests that explicitly
allow heteroscedasticity require the study of U -statistics at non-stationary sequences, which is
mathematically rather involved, as our study of the one-change-point model 4.3 illustrates.
Further future research directions include, e.g., the extension to more than two dimensions
or guidelines for an on-line application of the test with results about the detection time of a
change. Another interesting research question, which is related to the one studied here, is to
devise a robust test for detecting changes in the coherence of two time series. For example, a
series of i.i.d. variables, shifted by one observation, is highly coherent to the original series, but
our test, which only compares observations at the same time point, does not detect that type
of dependence.

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23

Appendix
The Appendix is split into three parts. Appendix A provides background on the concept of P -near epoch
dependence. In Appendix B, we study the asymptotic behavior of the U -statistic process under local
alternatives, which provides the basis for the proof of Theorem 4.4. Appendix C contains all proofs of
the results of the main text. Results labeled D.1, D.2, . . . refer to the online supplement (Appendix D),
available at Cambridge Journals Online (journals.cambridge.org/ect).

A. P -NED: Near epoch dependence in probability
In this section we add some background on the newly introduced P -near epoch dependence and connect
it to the usual Lp -near epoch dependence (Lemma A.1). Similar conditions that embody the idea of
approximating functionals in a probability sense are S-mixing considered by Berkes, Hörmann, and
Schauer (2009) and the L0 -approximability of Pötscher and Prucha (1997, Chapter 6).
First note that the P -NED condition (Definition 2.1 (ii)) is equivalent to convergence in probability of
fk (Z −k , . . . , Z k ) to X 0 for k → ∞. If the latter is true, i.e., if
o
n
(k → ∞),
εk = inf ε P (|X 0 − fk (Z −k , . . . , Z k )|1 > ε) ≤ ε → 0
then setting ak = εk and


P (|X 0 − fl (Z −l , . . . , Z l )|1 > εk )
∨1
Φ(ε) = sup
εl
l≤k

for ε ∈ [εk , εk−1 )

fulfills (2). The requirement of the bound on P (|X 0 − fk (Z −k , . . . , Z k )|1 > ε) in (2) to factorize into
an ε-part and a k-part is not a restriction, but facilitates rate computations.
Recall that a process (X n )n∈Z is called Lp near epoch dependent (Lp -NED), p ≥ 1, on the process
p
k
) p )1/p , k ≥ 1, converge to zero as
(Z n )n∈Z if the approximating constants ap,k = (E X 0 − E(X 0 |F−k
k → ∞.
Compared to L2 -NED, the P -NED condition substantially enlarges the class of processes for which the
condition is easily checked by many heavy-tailed distributions. For example, the autoregressive process
Xn =

∞
X

ak Z n−k ,

k=0

where a ∈ (−1, 1), and (Z n )n∈Z is an i.i.d. sequence of Rd -valued random variables, can easily be seen
to be P -NED on (Z n )n∈Z with ak = |a|αk and Φ(ε) = Kε−α for some constant K > 0, as long as
P (|Z n |1 ≥ t) = O(t−α ) is satisfied. This is a very weak condition on the innovation distribution, that
should be compared to analogous conditions for an AR(1) process to be mixing. In particular, the
existence of a density is not required, and all standard examples of discrete or heavy-tailed (e.g. Pareto,
Cauchy, geometric) distributions are permitted. Furthermore, P -NED substantially enlarges the class
of Lipschitz functionals considered by Denker and Keller (1986, p. 74). The above example would
satisfy their condition only if Z n was bounded. To further detail this example, let, e.g., a = 1/2 and
(1)
(2)
Z i = (Zi , Zi ) have the following discrete distribution




(1)
(2)
(1)
(2)
P (Zi , Zi ) = (0, 12 ) = P (Zi , Zi ) = ( 12 , 0) = (1 − %)/4,




(1)
(2)
(1)
(2)
P (Zi , Zi ) = (0, 0) = P (Zi , Zi ) = ( 12 , 12 ) = (1 + %)/4.
The parameter % ∈ [−1, 1] is the moment correlation of this distribution. The Kendall’s tau coefficient
(1)
(2)
(1)
(2)
of (Zi , Zi ) is τ = %/2. The process (X i )i∈Z = ((Xi , Xi ))i∈Z is not strongly mixing (since
X i−1 is a deterministic function of X i ) and hence not absolutely regular, but it is P -NED on (Z i )i∈Z
(1)
(2)
with exponentially decreasing approximation coefficients. The distribution of (Xi , Xi ) has the same
(1)
(2)
moment correlation % as (Zi , Zi ), its Kendall rank correlation is τ = 2%/(3 − %2 ), and the margins
(1)
(2)
(1)
Xi and Xi are uniformly distributed on (0, 1). Then consider the process ((X̃i , Ỹi ))i∈Z with X̃i =

24

(1)

(2)

(2)

H(Xi ), X̃i = H(Xi ), where H denotes the quantile function of a Pareto (type I) distribution with
shape parameter 1/2 and location parameter 1, i.e., H(x) = (1 − x)−2 , x ∈ [0, 1). This strictly increasing
(1)
transformation leaves the P -NED coefficients as well as Kendall’s tau unchanged. The margins X̃i ,
(2)
X̃i are Pareto distributed and have finite moments only up to order less than 1/2. The stationary
(1)
(2)
process ((X̃i , X̃i ))i∈Z is neither mixing nor Lp -NED for any p ≥ 1, but fulfills the assumptions of
Corollary 3.1. The next lemma connects P -NED and Lp -NED.
Lemma A.1. Let ((X n , Z n ))n∈Z be as in Definition 2.1.
(i) If (X n )n∈Z is P -NED on (Z n )n∈Z with functions Φ and fk , k ∈ N, and approximating constants
(ak )k∈N , and g : Rr → Rd is a Lipschitz continuous function with Lipschitz constant L, then the
process (g(X n ))n∈Z is P -NED on (Z n )n∈Z with functions Φ̃(ε) = Φ(ε/L) and g ◦ fk , k ∈ N, and
the same approximating constants (ak )k∈N .
(ii) Let (X n )n∈Z be bounded and P -NED on (Z n )n∈Z with functions Φ and fk , k ∈ N, and approximating constants (ak )k∈N . Then (X n )n∈Z is Lp -NED on (Z n )n∈Z for any p ≥ 1. If there is
further a sequence (sk )k∈N of non-negative numbers such that
ak Φ(sk ) = O(sk )

(k → ∞),

(13)

then the Lp -NED approximating constants (ap,k )k∈N of (X n )n∈Z satisfy app,k = O(sk ) for k → ∞.
(iii) Let (X n )n∈Z be Lp -NED, p ≥ 1, on (Z n )n∈Z with approximating constants (ap,k )k∈N . Then
(X n )n∈Z is P -NED on (Z n )n∈Z . If there is further a non-increasing function Φ : (0, ∞) → (0, ∞)
and a sequence (sk )k∈N of non-negative numbers converging to zero that satisfy
Φ(ε)sk ≥ q p−1 app,k ε−p ,
then (X n )n∈Z is P -NED on (Z n )n∈Z with approximation constants (sk )k∈N and function Φ. The
k
functions fk can be chosen as fk (Z −k , . . . , Z k ) = E(X 0 |F−k
), k ∈ N.
Since Φ is non-increasing, condition (13) puts an upper bound on the speed of decay of (sk )k∈N in the
sense that, if (13) is fulfilled by some sequence (sk )k∈N , then it is also fulfilled by any sequence (s̃k )k∈N
for which s̃k ≤ sk for all k larger than some n ∈ N.
The next lemma shows that the near epoch dependence is preserved under transformations that satisfy
the variation condition.
(2+δ)/δ

Lemma A.2. Let (X n )n∈Z be P -NED on (Z n )n∈Z with ak Φ(sk ) = O(sk
). Furthermore, let g be a
U -statistic kernel with uniform (2+δ) moments for some δ > 0, i.e., E|g(X 0 , X n )|2+δ < M for all n ∈ N,
and let g satisfy the variation condition with respect to the distribution F of X 0 (Assumption 2.4). Then
δ/(2+2δ)
the sequence (g1 (X n ))n∈Z is L2 -NED on (Z n )n∈N with approximation constants ak,2 = O(sk
).

B. U -statistics process under local alternatives
In this section,
we provide the basis for the proof of Theorem 4.4 in Section C. We analyze the U -statistic
P
process 1≤i<j≤[ns] g(Xi , Xj ), 0 ≤ s ≤ 1, in the case of a change point at time [nλ∗ ], and calculate the
asymptotic distribution under local alternatives. We consider the following model

ξi
1 ≤ i ≤ [nλ∗ ]
(n)
Xi = Xi =
(n)
ξi
[nλ∗ ] < i ≤ n,
(n)

where (ξi )i≥1 and (ξi )i≥1 , n ≥ 1, are stationary processes such that for any n ≥ 1 the bivariate processes
(n)
(ξi , ξi )i≥1 are P-NED on an absolutely regular process, with mixing coefficients and NED-coefficients
independent of n.
P
(n)
(n)
For the corresponding U -statistic process 1≤i<j≤[ns] g(Xi , Xj ), we have to distinguish the cases
∗
∗
∗
s ≤ λ and s ≥ λ . When s ≤ λ , we have
X
X
(n)
(n)
g(Xi , Xj ) =
g(ξi , ξj ),
1≤i<j≤[ns]

1≤i<j≤[ns]

25

while for s ≥ λ∗ we obtain
X
(n)
(n)
g(Xi , Xj )
1≤i<j≤[ns]

X

=

g(ξi , ξj ) +

1≤i<j≤[nλ∗ ]

[nλ∗ ]

[ns]
X

X

[ns]
X

(n)

g(ξi , ξj ) +

j=[nλ∗ ]+1 i=1

j−1
X

(n)

(n)

g(ξi , ξj ).

(14)

j=[nλ∗ ]+1 i=[nλ∗ ]+1

Note that of the three terms on the r.h.s., the first and the last term are one-sample U-statistics, while
the second term is a two-sample U -statistic. We now define the constant terms and the first order terms
of the Hoeffding decompositions,
θ = E(g(ξ, η)),
(n)
θ1

(n)

= E(g(ξ (n) , η)),

θ2

= E(g(ξ (n) , η (n) )),

(n)

g1 (x) = E(g(x, ξj )),

(n)

g1 (x) = E(g(x, ξj )).

Here, ξ, η, ξ (n) and η (n) are independent random variables such that ξ and η have the same distribution
(n)
(n)
as ξ1 , while ξ (n) and η (n) have the same distribution as ξ1 . Note that the functions g1 and g1 are not
centered. In fact, they require different centerings for each of the three terms on the r.h.s. of (14).
In this way, we obtain the following Hoeffding decompositions of the summands in (14):
g(ξi , ξj )

=

θ + (g1 (ξi ) − θ) + (g1 (ξj ) − θ) + h(ξi , ξj )

(n)
g(ξi , ξj )

=

θ1

=

θ2

(n)

(n)

g(ξi , ξj )
(n)

(n)

+ (g1 (ξi ) − θ1 ) + (g1 (ξj ) − θ1 ) + h1 (ξi , ξj )

(n)

(n)

+ (g1 (ξi ) − θ2 ) + (g1 (ξj ) − θ2 ) + h2 (ξi , ξj ).

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

The functions h, h1 , and h2 are defined in the obvious way, and a straightforward calculation shows
that they are degenerate, i.e. that the integral with respect to one of the arguments vanishes.
Plugging this into (14), we obtain the Hoeffding decomposition of the U -statistic


 ∗
X
[ns] − [nλ∗ ] (n)
[nλ ]
(n)
(n)
(n)
θ + [nλ∗ ]([ns] − [nλ∗ ])θ1 +
θ2
g(Xi , Xj ) =
2
2
1≤i<j≤[ns]

[nλ∗ ]
∗

+([nλ ] − 1)

X

[nλ∗ ]
∗

(g1 (ξi ) − θ) + ([ns] − [nλ ])

i=1

X

(n)

(n)

(g1 (ξi ) − θ1 )

i=1

[ns]

X

+[nλ∗ ]

(n)

i=[nλ∗ ]+1

+

X
1≤i<j≤[nλ∗ ]

[ns]
X

(n)

(g1 (ξi ) − θ1 ) + ([ns] − [nλ∗ ] − 1)

(n)

(n)

(n)

(g1 (ξi ) − θ2 )

i=[nλ∗ ]+1
[ns]
X

h(ξi , ξj ) +

[nλ∗ ]

X

(n)
(n)
h1 (ξi , ξj )

j=[nλ∗ ]+1 i=1

[ns]
X

+

j−1
X

(n)

(n)

(n)

h2 (ξi , ξj ).

j=[nλ∗ ]+1 i=[nλ∗ ]+1

We make a number of assumptions regarding the asymptotic behaviour of the terms in the Hoeffding
expansion that have to be checked in particular examples.
Assumption B.1. There are real numbers c1 , c2 such that

√

(n)

n(θ1

− θ) → c1 and

Assumption B.2.
k

1 X
(n)
(n)
max √
(g1 (ξi ) − θ) − (g1 (ξi ) − θ1 ) → 0,
1≤k≤n
n i=1
k


1 X
(n)
(n)
max √
(g1 (ξi ) − θ) − (g1 (ξi ) − θ1 ) → 0,
1≤k≤n
n i=1

26

√

(n)

n(θ2

− θ) → c2 .

k


1 X
(n) (n)
(n)
max √
(g1 (ξi ) − θ) − (g1 (ξi ) − θ2 ) → 0.
1≤k≤n
n i=1
Theorem B.3. Under Assumptions B.1 and B.2,
√

2
n([nt] − 1)

(n)

X

(n)

d

(g(Xi , Xj ) − θ) −→ 2σW + φλ∗ (s),

1≤i<j≤[ns]

where σ and W are defined as in Theorem 2.5, and where the function φ∗λ is defined as follows:
( ∗
∗ 2
)
2λ (s−λ∗ )
c1 + (s−λ
c2 s ≥ λ∗ ,
s
s
φλ∗ (s) =
0
s ≤ λ∗
with c1 and c2 being defined in Assumption B.1.
Proof of Theorem B.3. We first consider the Hoeffding decomposition of the U -statistic under the null
hypothesis, i.e.
X

(g(ξi , ξj ) − θ) = ([ns] − 1)

[ns]
X
i=1

1≤i<j≤[ns]

X

(g1 (ξi ) − θ) +

h(ξi , ξj ).

1≤i<j≤[ns]

Comparing this with the Hoeffding decomposition under the local alternative, we obtain for s ≥ λ∗ ,
X
X


(n)
(n)
g(Xi , Xj ) − θ −
g(ξi , ξj ) − θ
1≤i<j≤[ns]

1≤i<j≤[ns]
(n)

= [nλ∗ ]([ns] − [nλ∗ ])(θ1

1
(n)
− θ) + ([ns] − [nλ∗ ])([ns] − [nλ∗ ] − 1)(θ2 − θ)
2
[nλ∗ ]

∗

+([ns] − [nλ ])

(n)

X

(n)

(g1 (ξi ) − θ1 ) − (g1 (ξi ) − θ)



i=1

+([ns] − [nλ∗ ] − 1)

[ns]
X

(n)

(g1 (ξi


(n)
− θ1 ) − (g1 (ξi ) − θ)

i=[nλ∗ ]+1
[ns]
X

+([ns] − [nλ∗ ] − 1)


(n) (n)
(n)
(g1 (ξi ) − θ2 ) − (g(ξi ) − θ)

i=[nλ∗ ]+1

+

[nλ∗ ]

[ns]
X

X

j=[nλ∗ ]+1

−

(n)

i=1

[ns]
X

[ns]
X

[ns]
X

j=[nλ∗ ]+1

i=[nλ∗ ]+1

(n)

h1 (ξi , ξj ) +
[nλ∗ ]

X

h(ξi , ξj ) −

j=[nλ∗ ]+1 i=1

[ns]
X

(n)

(n)

(n)

h2 (ξi , ξj )

[ns]
X

h(ξi , ξj ),

j=[nλ∗ ]+1 i=[nλ∗ ]+1

while for s ≤ λ∗ , the term on the l.h.s. equals zero. Thus we obtain, again for s ≥ λ∗ ,


X
X


2
(n)
(n)
√
g(Xi , Xj ) − θ −
g(ξi , ξj ) − θ − φ∗λ (s)
n([ns] − 1)
1≤i<j≤[ns]

1≤i<j≤[ns]

=

2[nλ∗ ]([ns] − [nλ∗ ]) √ (n)
([ns] − [nλ∗ ])([ns] − [nλ∗ ] − 1) √ (n)
n(θ1 − θ) +
n(θ2 − θ) − φ∗λ (s)
n([ns] − 1)
n([ns] − 1)
∗

[nλ ]

2([ns] − [nλ∗ ]) 1 X
(n)
(n)
√
+
(g1 (ξi ) − θ1 ) − (g1 (ξi ) − θ)
n
n i=1

27

+

2([ns] − [nλ∗ ] − 1) 1
√
[ns] − 1
n

2([ns] − [nλ∗ ] − 1) 1
√
+
[ns] − 1
n
2
+√
n([ns] − 1)

[ns]
X

(n)

(g1 (ξi

X

[ns]
X

(n)

X

j=[nλ∗ ]+1 i=1

(n)



i=[nλ∗ ]+1

(n)
(n)
h1 (ξi , ξj ) + √

[nλ∗ ]

[ns]
X

(n)

(g1 (ξi ) − θ2 ) − (g1 (ξi ) − θ)

j=[nλ∗ ]+1 i=1

2
−√
n([nt] − 1)



i=[nλ∗ ]+1

[nλ∗ ]

[ns]
X

(n)

− θ1 ) − (g1 (ξi ) − θ)

[ns]
X

2
n([ns] − 1)

[ns]
X

(n)

(n)

(n)

h2 (ξi , ξj )

j=[nλ∗ ]+1 i=[nλ∗ ]+1

2
h(ξi , ξj ) − √
n([ns] − 1)

[ns]
X

[ns]
X

h(ξi , ξj ),

j=[nλ∗ ]+1 i=[nλ∗ ]+1

The right hand side converges to zero uniformly in s ≥ λ∗ . For the first four lines on the r.h.s., this
follows from our assumptions. Uniform convergence of the terms involving the degenerate kernels h,
(n)
(n)
h1 , and h2 follows with arguments similar to those employed in the proof of Theorem 2.5.
Returning to the original notation of our paper, we define the U -statistic
(n)

Uk

=

1

k
2

(n)

X

(n)

g(Xi , Xj ), 0 ≤ k ≤ n.

1≤i<j≤k

Using the continuous mapping theorem, we derive the following corollary to the above theorem
Corollary B.4.
1
2σ̂n

1
k
d
√ |Uk(n) − Un(n) | −→ sup |B(λ) +
(φλ∗ (λ) − λφλ∗ (1)) |.
1≤k≤n−1
2σ
n
0≤λ≤1
max

C. Proofs of Sections 2, 3, and 4
In this section we prove the theorems of the main text. These are Theorems 2.5 (invariance principle
for the sequential U -process), 2.7 (consistency of the long run variance estimator), and Corollary 3.1
(asymptotics of the test statistic under the null), Theorem 4.2 (estimation of the change-point), and
Theorem 4.4 (local power analysis).
Proof of Theorem 2.5 (Invariance principle for the sequential U -process). Using the Hoeffding decomposition, we can write the sequential U -process as
[ns]


2 X
2
[ns]
1
√ U[ns] − U = √
g1 (X i ) + √
n
n i=1
n [ns] − 1

X

g2 (X i , X j ).

1≤i<j≤[ns]

For the first summand we find by Assumption 2.2 and Lemma A.2 that the sequence (g1 (X n ))n∈N is
L2 -NED with approximation constants ak,2 = O(k −3 ). So we can apply Corollary 3.2 of Wooldridge
P[ns]
and White (1988), stating that the partial sum process ( √2n i=1 g1 (X i ))t∈[0,1] converges weakly to a
Brownian
motion with variance 4σ 2 . For the second summand, we use Lemma ??, which implies that
P
1
1
| n 1≤i<j≤n g2 (X i , X j )| ≤ Cn 4 log2 (n) almost surely and consequently
1
√
s∈[0,1] [ns] n
sup

X

1

g2 (X i , X j ) ≤ Cn− 4 log2 (n) → 0

1≤i<j≤[ns]

almost surely as n → ∞. Slutsky’s theorem completes the proof.

28

Proof of Theorem 2.7 (consistency of the long run variance estimator). We can write the variance estimator σ̂n2 as
σ̂n2 =

n−1
X
r=−(n−1)


κ

|r|
bn

+



n−|r|
1 X
ĝ1 (X i )ĝ1 (X i+|r| ) =
n i=1

n−1
X
r=−(n−1)

n−1
X


κ

r=−(n−1)

|r|
bn



n−|r|
1 X
g1 (X i )g1 (X i+|r| )
n i=1

n−|r|

1 X
g1 (X i )g1 (X i+|r| ) − ĝ1 (X i )ĝ1 (X i+|r| ) κ(|k|/bn ).
n i=1

2
By
P∞Theorem 2.1 of de Jong and Davidson (2000), we know that the first summand converges to σ =
Cov
(g
(X
),
g
(X
)).
The
second
summand
converges
to
0
by
Lemma
??,
and
the
proof
is
1
0
1
k
k=−∞
complete.

Proof of Corollary 3.1 (asymptotic null distribution of change-point test statistic). Corollary 3.1 is a special case of Corollary 2.8 for the specific kernel g given by (5). It remains to show that, under the
conditions of Corollary 3.1, the assumptions of Corollary 2.8 are met. Since g is bounded, Assumption
2.3 is satisfied for any δ > 0, and it suffices that Assumption 2.2 is fulfilled for some δ > 0.
Proof of Theorem 4.2 (Change-point estimation). Under the assumptions in Theorem 4.2 concerning the
value of τF G , the function |c(λ)|, λ ∈ [0, 1], has a unique maximum at λ = λ∗ , where the function
c : [0, 1] → R is given by
(

(1 − λ∗2 )τF − (1 − λ∗ )2 τG − 2λ∗ (1 − λ∗ )τF,G λ
for 0 ≤ λ < λ∗ ,

(15)
c(λ) =
1
for λ∗ ≤ λ ≤ 1.
2λ∗ (τF,G − τG )(1 − λ) + λ∗2 (τF + τG − 2τF,G ) λ − λ
The proof relies on the fact that

(Cn (λ))0≤λ≤1 =


[λn]
τ̂[λn] − τ̂n
n



d

−→ (c(λ))0≤λ≤1

(16)

0≤λ≤1

in D[0, 1]. With the argmax theorem (van der Vaart and Wellner, 1996, Corollary 3.2.3) we have that
p

λ̂n = arg max |Cn (λ)| −→ arg max |c(λ)| = λ∗ .
0≤λ≤1

0≤λ≤1
(n)

(n)

(n)

It remains to prove (16). To simplify notation, we let m = [λ∗ n], write Z i short for (Xi , Yi ) and
further suppress the subscript n. Assume for an instant that the Z i , i = 1, . . . , n, are independent. Then
we have
(
τF
for k ≤ m,
tn (k) = E(τ̂k ) = m(m−1)
(k−m)(k−m−1)
2m(k−m)
τG + k(k−1) τF,G
for k ≥ m + 1,
k(k−1) τF +
k(k−1)
from where we derive the mean function cn (λ) = E[Cn (λ)] = [λn]n−1 (tn ([λn]) − tn (n)), λ ∈ [0, 1], of
the process of (Cn (λ))0≤λ≤1 and observe that it converges to the function c. Thus it remains to show
that max0≤λ≤1 |Cn (λ) − cn (λ)| converges to zero in probability also under the short-range dependence
assumption of Model 4.1. In the following, let the Z i , i = 1, . . . , n be weakly dependent as specified by
Model 4.1. We will prove that
k
p
max
|τ̂k − tn (k)| −→ 0.
(17)
m<k≤n n
The convergence of max1<k≤m nk |τ̂k − tn (k)| follows along the same lines. Hence the maximum in (17)
can be extended to the range k = 1, . . . , n. We split the difference nk |τ̂k − tn (k)| into three parts: two
one-sample U -statistics and one two-sample U -statistic with the kernel
g ((x1 , y1 ), (x2 , y2 )) = 1(0,∞) ((x2 − x1 )(y2 − y1 )) − 1(−∞,0) ((x2 − x1 )(y2 − y1 ))

29

as in Section 3. By the triangle inequality we get:
2
n(k − 1)

k
|τ̂k − tn (k)| ≤
n

+

2
n(k − 1)

X

X

(g(Z i , Z j ) − τF )

1≤i<j≤m

(g(Z i , Z j ) − τG ) +

m+1≤i<j≤k

2
n(k − 1)

X

(g(Z i , Z j ) − τF,G ) .

(18)

1≤i≤m<j≤k

For the first summand on the right-hand side, we have by Theorem 2.5:
2
(g(Z i , Z j )
m<k≤n n(k−1) 1≤i<j≤m

P

max

− τF ) ≤

2
n(m−1)

P

p

(g(Z i , Z j ) − τF ) −→ 0.

1≤i<j≤m

(n)

Due to our assumptions, (Z i )m+1≤i≤n is a stationary process which satisfies Condition 2.2, so we also
treat the second summand by Theorem 2.5. For the third summand, we apply a two-sample Hoeffding
decomposition g(z 1 , z 2 ) = τF,G + g̃1 (z 1 ) + g̃2 (z 2 ) + g̃3 (z 1 , z 2 ) with
g̃1 (z 1 ) = E (g(z 1 , Z m+1 )) − τF,G ,

g̃2 (z 1 ) = E (g(Z 1 , z 2 )) − τF,G ,

g̃3 (z 1 , z 2 ) = g(z 1 , z 2 ) − g̃1 (z 1 ) − g̃2 (z 2 ) − τF,G ,
where z 1 , z 2 ∈ R2 . We get

=

2
n(k − 1)

X

(g(Z i , Z j ) − τF,G )

1≤i≤m<j≤k

m
k
X
2m
2
2(k − m) X
g̃1 (Z i ) +
g̃2 (Z j ) +
n(k − 1) i=1
n(k − 1) j=m+1
n(k − 1)

X

g̃3 (Z i , Z j ).

(19)

1≤i≤m<j≤k

To obtain a maximal inequality, we use Theorem 2.4.1 of Stout (1974): For random variables R1 , . . . , Rn
Pk+l−1
with E( j=k Rj )2 ≤ C1 l for a constant C1 , we have that

E

max

 Xl
j=1

1≤l≤n

Rj

2 

2

≤ C1 n (log(2n)/ log 2) .

(20)

Pm
We define the random variables Rj = i=1 g̃3 (Z i , Z j+m ). Without loss of generality, we can assume that
the random variables Z i are bounded and thus the process is L1 -NED by Lemma A.1(ii). Furthermore,
the kernel g̃3 is degenerate, so we can apply Proposition A.2 of Dehling and Fried (2012) to obtain the
 Pk+l−1 Pm
2
Pk+l−1 2
moment bound E
Rj = E
≤ Cml. Applying (20), we find
j=1
j=1
i=1 g̃3 (Z i , Z j+m )
2


E  max

m<k≤n

2
n(k − 1)

X

g̃3 (Z i , Z j ) 


≤ m−2 E 

max

1<k≤n−m

1≤i≤m<j≤k

2
k
C log2 (2n)
2X
Rj  ≤
n j=1
mn log2 2

converges to zero as n → 0. Thus the third summand in (19) converges to zero
hPin probability.i As for the
 Pk+l−1
2
k+l−1
−1
first two summands, we have that E
g̃2 (Z j ) ≤ Cl, since l Var
g̃2 (Z j ) converges
j=k
j=k
to a finite limit as l → ∞. Hence, (20) applied to Rj = g̃2 (Z j+m ) leads to
max

m<k≤n

2 Xk
p
g̃2 (Z j ) −→ 0
j=m+1
n

Pm
p
as n → ∞. Finally, 2/n i=1 g̃1 (Z i ) −→ 0, as its variance converges to zero. We have thus shown (17),
which completes the proof.
Proof of Theorem 4.4. It remains to calculate the quantities c1 , c2 introduced above for the change-point

30

test based on Kendall’s τ , under the local alternative as specified in Model 4.3. Observe that, in this
(n)
(n)
case, θ = τF , θ1 = τF,Fn , and θ2 = τFn ,Fn , where F denotes the joint distribution of (Xi , Yi ) before
(n)
(n)
the change, and Fn denotes the joint distribution of (Xi , Yi ) after the change.
We now provide a formula for τF,G in the case when both distributions F, G are absolutely continuous,
with densities f (x, y) and g(x, y). Let (X1 , Y1 ) and (X2 , Y2 ) be random variables with densities f (x, y),
and g(x, y), respectively, and let (X1 , Y1 ) be independent of (X2 , Y2 ). Then we have
(I)

(II)

τF,G = 2(τF,G + τF,G ) − 1,

(21)

where
∞

Z

(I)
τF,G

=

(II)

=

∞

Z

Z

∞

P (X1 ≤ X2 , Y1 ≤ Y2 ) =

f (x1 , y1 )g(x2 , y2 )dy2 dy1 dx2 dx1
−∞
Z ∞

τF,G

∞

Z

x1
Z ∞

−∞
Z ∞

y
Z 1∞

P (X1 ≥ X2 , Y1 ≥ Y2 ) =

f (x1 , y1 )g(x2 , y2 )dy2 dy1 dx2 dx1 .
−∞

(I)

x2

−∞

y2

(II)

(I)

(II)

Noting that τF = τF,F , we obtain τF = 2(τF,F + τF,F ) − 1, where τF,F and τF,F are given by the above
integrals with f = g. Finally, we obtain


(I)
(I)
(II)
(II)
τF,G − τF = 2 (τF,G − τF,F ) + (τF,G − τF,F )
Under Model 4.3, we have G = Fn , where Fn is the distribution of (X1 + √∆n Y1 , Y1 ). By the transformation
formula for densities, Fn has the density
∆
fn (x2 , y2 ) = f (x2 − √ y2 , y2 ).
n
Thus, we obtain
(I)

Z

(I)

τF,Fn − τF,F =

∞

Z

−∞

∞

Z

∞

Z

−∞

x1

∞

y1



∆
f (x1 , y1 ) f (x2 − √ y2 , y2 ) − f (x2 , y2 ) dy2 dy1 dx2 dx1 .
n

Under the assumptions made in Model 4.3 on the densities, we then obtain
Z ∞ Z ∞Z ∞ Z ∞
√
(I)
(I)
lim n(τF,Fn − τF,F ) = −∆
f (x1 , y1 )y2 f1 (x2 , y2 )dy2 dy1 dx2 dx1 ,
n→∞

−∞

∂
∂x f (x, y).

where f1 (x, y) =

x1

−∞

y1

Furthermore,
Z ∞
f1 (x2 , y2 )dx2 = −f (x1 , y2 ),
x1

and thus
lim

√

n→∞

(I)

Z

(I)

n(τF,Fn − τF,F ) = ∆
Z

∞

Z

∞

Z

y2

= ∆
−∞

Z

∞

Z

∞

= ∆
−∞

−∞

∞

Z

∞

Z

∞

f (x1 , y1 )y2 f (x1 , y2 )dy2 dy1 dx1
−∞

−∞

y1


2
y2 f (y1 |x)f (y2 |x)dy1 dy2 fX
(x)dx

−∞



2
y2 F (y2 |x)f (y2 |x)dy2 fX
(x)dx = ∆

−∞

Z

∞

−∞

Z

∞


2
y2 F (y2 |x)F (dy2 |x) fX
(x)dx,

−∞

where fX denotes the marginal density of X, and where f (y|x) and F (y|x) denote the conditional density,
respectively the conditional distribution function of Y given X = x.
With similar calculations, we obtain

Z ∞ Z ∞
√
(II)
(II)
2
y2 (F (y2 |x) − 1)F (dy2 |x) fX
(x)dx.
lim n(τF,Fn − τF,F ) = ∆
n→∞

−∞

−∞

31

Finally, one can show that limn→∞

√

n(τFn ,Fn − τF,F ) = 2 limn→∞

32

√

n(τF,Fn − τF,F ).

