Time-localized wavelet multiple regression and correlation

Abstract

This paper extends wavelet methodology to handle comovement dynamics of multivariate time series via moving weighted regression on wavelet coefficients. The concept of wavelet local multiple correlation is used to produce one single set of multiscale correlations along time, in contrast with the large number of wavelet correlation maps that need to be compared when using standard pairwise wavelet correlations with rolling windows. Also, the spectral properties of weight functions are investigated and it is argued that some common time windows, such as the usual rectangular rolling window, are not satisfactory on these grounds.
The method is illustrated with a multiscale analysis of the comovements of Eurozone stock markets during this century. It is shown how the evolution of the correlation structure in these markets has been far from homogeneous both along time and across timescales featuring an acute divide across timescales at about the quarterly scale. At longer scales, evidence from the long-term correlation structure can be interpreted as stable perfect integration among Euro stock markets. On the other hand, at intramonth and intraweek scales, the short-term correlation structure has been clearly evolving along time, experiencing a sharp increase during financial crises which may be interpreted as evidence of financial ‘contagion’.

Full text

Time-Localized Wavelet Multiple Regression and Correlation
Javier Fernández-Machoa,∗
aDpt. of Econometrics & Statistics,
University of the Basque Country, Bilbao, Spain
Abstract
This paper extends wavelet methodology to handle comovement dynamics of multi-
variate time series via moving weighted regression on wavelet coefficients. The concept
of wavelet local multiple correlation is used to produce one single set of multiscale
correlations along time, in contrast with the large number of wavelet correlation maps
that need to be compared when using standard pairwise wavelet correlations with rolling
windows. Also, the spectral properties of weight functions are investigated and it is
argued that some common time windows, such as the usual rectangular rolling window,
are not satisfactory on these grounds.
The method is illustrated with a multiscale analysis of the comovements of Eurozone
stock markets during this century. It is shown how the evolution of the correlation
structure in these markets has been far from homogeneous both along time and across
timescales featuring an acute divide across timescales at about the quarterly scale. At
longer scales, evidence from the long-term correlation structure can be interpreted as
stable perfect integration among Euro stock markets. On the other hand, at intramonth
and intraweek scales, the short-term correlation structure has been clearly evolving along
time, experiencing a sharp increase during financial crises which may be interpreted as
evidence of financial ‘contagion’.
Keywords: Comovement dynamics, Euro zone, local regression, multiscale analysis,
multivariate time series, non-stationary time series, stock markets, wavelet transform,
weighted least squares.
1. Introduction
One important aspect in the analysis of economic and financial time series is the study
of the degree and behavior of their comovements at different periods and frequencies.
Whilst traditional time series analysis (TSA) is mostly based on cross-correlation functions
in the time domain, that is, on an observation-by-observation basis, Fourier analysis
∗Corresponding author: phone: +34 946013729, fax: +34 946017100.
Email address: javier.fernandezmacho@ehu.eus (Javier Fernández-Macho)
Preprint submitted to Physica A November 19,2017

allows for the visualization of the data in the frequency domain, that is, on a frequency-
by-frequency basis. In other words, traditional TSA provides full time resolution of such
relationships but does not expose frequency information while Fourier analysis has full
frequency resolution but does not preserve information in time. These types of analysis
implicitly assume stationarity, possibly after differences, as the main characteristic of the
time series under study [21, p.435].
The more recent wavelet analysis emerges as a compromise between both approaches,
with partial resolution in both time and frequency domains. Thus, the continuous wavelet
transform (CWT) allows for a visualization of the spectral features of the time series and
their comovements but as a function of both time and scale (frequency), separating their
different periodic components as they evolve over time [27,2].
The CWT is highly redundant on both time and scale dimensions, the latter not being
a desirable feature for regression/correlation decomposition over timescales. On the
other hand, the discrete wavelet transform (DWT) selects a minimal subsample of time-
frequency values from the CWT without loosing any information present in the original
data. This is an important feature for some engineering applications like signal and
image compression, but for most economic applications time redundancy is somehow
desirable as long as it allows for data features to be properly aligned and compared across
all scales/frequencies. In this sense, the maximal overlap discrete wavelet transform
(MODWT) is the most popular wavelet transform as it is redundant in the time dimension
but non-redundant in the scale/frequency dimension. MODWT is known to have several
important advantages, including energy preservation which is particularly important for
the main objective of this paper (28,27;20, p.135).
MODWT based wavelet multiple correlations (WMC) and cross-correlations (WMCC)
were introduced by Fernández-Macho
[15]
. These statistics distribute among the different
timescales the overall statistical relationship that might exist between several time series
and they have received some attention in the recent literature [see e.g.
5
,
2
,
37
,
6
,
7
,
23
,
24
,
1
,
34
,
36
, among others]. However, as with standard bivariate wavelet correlations,
it is implicitly assumed that the time series follow difference-stationary processes and,
therefore, that there exists a sufficiently long wavelet filter that eliminates this type of
nonstationarity from the data. In consequence, only one single global correlation per
scale needs be produced.
This notwithstanding, with economic and financial time series, such data features are
probably non-stationary in nature and a regression/correlation analysis must be able to
handle and visualize a changing structure that evolves along time. Therefore, the present
paper generalizes the previous global WMC to a local multiple regression framework
where comovement dynamics across the different scales/frequencies can be analyzed
along time by using weighted or windowed wavelet coefficients.
The proposed method is justified on several grounds. First of all, the alternative of
combining standard bivariate wavelet correlation analysis with rolling time windows
needs to calculate, plot and compare a large number of wavelet correlation graphs that
now would require an additional time dimension [
31
,
10
,
32
,
5
,etc.]. More specifically,
with
n
time series, a total of
n(n−1)/2
wavelet correlation maps would be required
2

each of dimension
J×T
, where
J=blog2(T/(L−1) + 1)c
is the order of the wavelet
transform,
L
is the wavelet filter length and
T
is the time series length.
1
Not surprisingly,
when dealing with all possible pairwise comparisons in a multiscale context, the analyst
may end up with a vast amount of potentially conflicting information that can be very
difficult to process and even lead to the typical experimentwise error rate inflation
and the spurious detection of correlations at some wavelet scales [
15
]. In contrast,
the proposed method based on local multiple regression, consists in one single set of
multiscale relationships which can be expressed in a single scale-time correlation map.
This is not only easier to handle and interpret but also may provide a better insight of
the overall statistical comovement dynamics within the multivariate time series under
scrutiny. Also, in the proposed method the number of feasible scales remains the same
as in global wavelet analysis and does not depend on the length of the time window.
Therefore, long wavelet filters and windows of long length or even with infinite support
like the gaussian weight function can be used. Finally, as discussed in section 4, the
spectral properties of the weight function need to be taken into account as, for example,
the usual rectangular rolling window will not be satisfactory on these grounds.
All this will be illustrated with the application of the proposed Wavelet Local Multiple
Correlation (WLMC) in the multiscale analysis of daily returns obtained from a set of
eleven Eurozone stock markets during the 17-year period of the present century during
which several financial and debt crises have occurred. In this relation, we may point
out that correlation among European stock markets is a common measure of market
integration in the economic and financial literature [see, e.g.,
18
,
42
,
22
,
38
,
4
, and others].
However, these studies do not usually take into account the fact that stock markets
involve heterogenous agents that make decisions over different time horizons (
20
, p.10,
19
), or that such comovement structure across different timescales may be evolving along
time. On the other hand, the relatively large but not uncommon number of markets to be
analyzed will render, as already mentioned, pairwise multiscale comparisons pointless
in practice, which is the reason why this type of market analysis may find useful the
wavelet local multiple regression tools proposed here.
The paper is organized as follows. Sections 2and 3set up the framework for the
proposed wavelet local multiple regression tools and extend such framework with the
definition of the WLMC considering the decomposition of the time series structure
across different timescales. Sample estimators of these quantities and their approximate
confidence intervals based on their large sample theory are also provided for estimation
and testing purposes. Section 4discusses the spectral properties of common weight
functions in local regression that can be used in practice and Section 5presents some
examples: the first two with simulated changes along time in the correlation structure
across different timescales that serve to illustrate the validity of the proposed statistics
and, finally, a case study that shows the results of an empirical application using Eurozone
stock markets. Section 6summarizes the main conclusions.
1
For example, the empirical analysis in Section 5.3would need to handle a total of 55 wavelet correlation
maps of dimension 9×4542 each.
3

2. Outline of the method
The proposed method consists of the following steps: a) discrete wavelet transform
(MODWT), Section 3.1; b) weight function or rolling time window, i.e. moving averages,
Section 4; c) local least-squares regression and d) local multiple-correlation coefficients,
Section 3; or a-b-d if only correlations are needed.
The use of rolling-window correlations in bivariate wavelet analysis has been proposed
by Ranta
[31]
, Dajcman et al.
[10]
, Ranta
[32]
, Benhmad
[5]
among others. However, many
studies seem to reverse the above order to b-a-d so that they calculate wavelet correlations
on a rectangular rolling window of data. In principle, since both a) and b) are linear
filters, it would appear that reversing this order does not matter. Indeed, this would be
the case if boundary conditions on the rolling window take into account surrounding
values from the complete series, but this is not done. Instead, each subsample selected
by the rolling window is taken as if it were a complete series with the usual standard
periodic or reflecting boundary conditions and wavelet coefficients affected by such
boundaries customarily rejected. This explains why these authors complain of running
out of data points as the wavelet level increases and, trying to mitigate this boundary
problems, they are forced to use a filter with the shortest length such as the Haar filter,
which is not recommended due to its poor band-pass properties [
32
, p.142].
2
Using the
a-b-d order solves these problems without further ado.
3. Local multiple regression and correlation
Let
X∈Rn×R
be a
n
-variate time series observed at times
t
=1. . .
T
, and let
X−i=
X\ {xi}
for some
xi∈X
. We wish to obtain a linear function
fs(X−i)
that for a fixed
s∈[1, . . . , T], minimizes a weighted sum of squared errors
Ss=min
fs
∑
t
θ(t−s)fs(X−i,t)−xit2, (1)
where
θ(x)
is a given moving average weight function, such as those in Table 1, that
depends on the time lag between observations Xtand Xs[cf. e.g. 26].
Thus, the local weighted least squares approximation around scan be written as
fs(X−i) = Ziβs,Zi=X−i−X−i, (2)
where
X−i
is the vector of values of
X−i
at fixed
s
, and the unknown coefficients
βs
can
be estimated as b
βs=∑
t
θ(t−s)ZitZ0
it−1∑
t
θ(t−s)Zit xit. (3)
2
Benhmad
[5]
seems to apply the preferred a-b-d order. Nevertheless, he uses DWT instead of MODWT
for the scale decomposition and, therefore, he also runs out of data points as the wavelet level increases.
4

The corresponding variance-covariance matrix can be estimated as
Vb
βs=σ2
s∑
t
θ(t−s)ZitZ0
it−1, (4)
where
σ2
s
is the error variance within a local neighborhood of
Xs
that is approximately
stationary.
Finally, letting
s
move along time we obtain a local regression
b
fs(X−i) = Zib
βs
and
the corresponding residual weighted sum of squares
RwSSs=∑tθ(t−s)Z0
it b
βs−
xit2.
These residuals could then be used to calculate a series of local coefficients of
determination
R2
s=1−RwSSs
TwSSs
,s=1 . . . T, (5)
where
TwSSs=∑tθ(t−s)x2
it
is the total weighted sum of squares at time
s
. Alternatively,
they can also be obtained directly from the data correlation matrix as in [
15
] (see
Section 3.1below).
3.1. Wavelet local multiple correlation
Let
Wjt = (w1jt ,w2jt, . . . , wnjt )
be the scale
λj
wavelet coefficients obtained by applying
the maximal overlap discrete wavelet transform (MODWT) [
20
,
27
] to each time series
xi∈X
in Section 3. Fernández-Macho
[15]
used a MODWT based global multiple
regression context to define the wavelet multiple correlation (WMC)
ϕX(λj)
, and its
companion wavelet multiple cross-correlation (WMCC), as single sets of multiscale
correlations and cross-correlations calculated for the multivariate time series
X
. However,
the evolution of nonstationary dynamics cannot be studied with these global measures
and therefore they need to be extended to a local multiple regression framework where
comovement dynamics can be properly analyzed.
Following [
15
], at each wavelet scale
λj
we calculate a series of local multiple cor-
relation coefficients as the square roots of the regression coefficients of determination
(Eq. (5)) for that linear combination of variables
wijt
,
i
=1. . .
n
, where such coefficients of
determination are maxima. In practice, none of these auxiliary regressions need to be run
since, as it is well known, the coefficient of determination corresponding to the regression
of a variable
zi
on a set of regressors
{zk
,
k6=i}
, can be obtained as
R2
i=1−1/ρii
,
where
ρii
is the
i
-th diagonal element of the inverse of the complete correlation matrix
P
.
Therefore, ϕX,s(λj)is obtained as
ϕX,s(λj) = s1−1
max diag P−1
j,s
,s=1 . . . T, (6)
where
Pj,s
is the
(n×n)
weighted correlation matrix of
Wjt
with weights
θ(t−s)
, and the
max diag(·)operator selects the largest element in the diagonal of the argument.
5

Since a regression coefficient of determination can also be obtained as the square of
the correlation between observed values and fitted values, we have that
ϕX,s(λj)
can also
be expressed as
ϕX,s(λj) = Corr(θ(t−s)½wijt,θ(t−s)½b
wijt)
=Cov(θ(t−s)½wijt ,θ(t−s)½b
wijt)
qVar(θ(t−s)½wijt )Var(θ(t−s)½b
wijt)
,s=1 . . . T, (7)
where
wij
is chosen so as to maximize
ϕX,s(λj)
and
b
wij
are the fitted values in the local
regression of
wij
on the rest of wavelet coefficients at scale
λj
. Hence the adopted name
of ‘wavelet local multiple correlation’ (WLMC) for this new statistic. Equation (7) will be
useful later in determining the statistical properties of an estimator of ϕX,s(λj).
Applying a MODWT of order
J
to each of the univariate time series in
X
we would
obtain
J
length-
T
vectors of MODWT coefficients
e
Wj={e
Wj0. . . e
Wj,T−1}
, for
j
=1. . .
J
.
From Eq. (6) the WLMC of scale
λj
is a nonlinear function of all the
n(n−1)/2
weighted
correlations of
Wjt
. Alternatively, it can also be expressed in terms of all the weighted
covariances and variances of
Wjt
as in Eq. (7). Therefore, a consistent estimator of the
WLMC based on the MODWT is given by
e
ϕX,s(λj) = s1−1
max diag e
P−1
j,s
=Corr(θ(t−s)½e
wijt,θ(t−s)½b
e
wijt)
=Cov(θ(t−s)½e
wijt,θ(t−s)½b
e
wijt)
qVar(θ(t−s)½e
wijt)Var(θ(t−s)½b
e
wijt)
,s=1 . . . T, . (8)
The weighted wavelet covariances and variances can be estimated as
Cov(e
wijt,b
e
wijt) = γj,s=
T−1
∑
t=Lj−1
θ(t−s)e
wijt b
e
wijt,s=1 . . . e
T, (9a)
Var(e
wijt) = δ2
j,s=
T−1
∑
t=Lj−1
θ(t−s)e
w2
ijt,s=1 . . . e
T, (9b)
Var(b
e
wijt) = ζ2
j,s=
T−1
∑
t=Lj−1
θ(t−s)b
e
w2
ijt,s=1 . . . e
T, (9c)
where
e
wij
is such that the local regression of
e
wij
on the set of regressors
{e
wkj
,
k6=i}
maximizes the corresponding coefficient of determination,
b
e
wij
denotes the fitted values
and
Lj= (2j−1)(L−1) + 1
is the number of wavelet coefficients affected by the boundary
associated with a wavelet filter of length Lat scale λj.
6

The large-sample distribution of the sample WLMC
e
ϕX,s(λj)
can be established along
similar lines as for the standard pairwise wavelet correlation [
20
]. In our present local
multivariate case, we note from Eq. (8) that
e
ϕX,s(λj)
is a nonlinear function of all the
sample weighted wavelet variances and covariances which, in turn, are just sample
moments of vectors of MODWT coefficients. Therefore, the estimator can be written as a
function of the three local wavelet moments in Eq. (9):
e
ϕX,s(λj) = f(γj,s,δj,s,ζj,s) = γj,s
δj,sζj,s
. (10)
We may now apply the continuous mapping theorem to establish that
qe
Tjhe
ϕX,s(λj)−ϕX,s(λj)i∼ N(0, Vj,s), (11)
where
e
Tj=T−Lj+1
is the number of coefficients unaffected by the boundary conditions
and
Vj,s=d f 0
j,sSj,s(0)d fj,s, (12)
with d fj,sas the gradient vector of f(γj,s,δj,s,ζj,s)and
Sj,s(0) = 

Sγ2,j,s(0)Sδγ,j,s(0)Sζγ,j,s(0)
Sδγ,j,s(0)Sδ2,j,s(0)Sζδ,j,s(0)
Sζγ,j,s(0)Sζδ,j,s(0)Sζ2,j,s(0)
(13)
where e.g.
Sδγ,j,s(0)
is the spectral density function of the product of scale
λj
local wavelet
moments δj,sγj,sevaluated at the zero frequency, etc.[cf. 40].
In practice, obtaining the spectral density functions involved in the computation of
a consistent estimate of
Vj,s
can be very tiresome. As in [
20
] or [
15
], a more feasible
alternative can be obtained by using Fisher’s’s transformation [
16
]. In the present case,
applying the transformation to the sample wavelet multiple correlation coefficient
e
ϕX(λj)
we obtain:
Theorem 1.
Let
X={X1. . . XT}
be a realization of a multivariate Gaussian stochastic process
Xt= (x1t,x2t, . . . , xnt)
and let
e
Wj={e
Wj0. . . e
Wj,T−1}
=
{(e
w1j0. . . e
wnj0)
,
. . .
,
(e
w1j,T/2j−1. . .
e
wnj,T/2j−1)}
,
j
=1. . .
J
, be vectors of wavelet coefficients obtained by applying a MODWT of order
J
to each of the univariate time series
{xi1. . . xiT }
for
i
=1. . .
n
. Let
e
ϕX,s(λj)
be the sample wavelet
local multiple correlation (WLMC) obtained from Eq. (6). Then,
e
zj,s
a
∼ FN(zj,s,(T/2j−3)−1). (14)
where zj,s=arctanh(ϕX,s(λj)) and FN stands for the folded normal distribution.3
3
That is, the probability distribution of
abs(e)
such that
e
is normally distributed with the given mean
and variance.
7

Table 1: Weight functions
solid long dash short dash
blue lines: Uniform window: Cleveland’s tricube window: Epanechnikov’s parabolic
window:
u(x) = 1/(2M)c(x) = 701−x
M33/(81 M)e(x) = 31− | x
M|2/(4M)
red lines: Bartlett’s triangular window: Wendland’s truncated power window: Gaussian window:
b(x) = 1−| x
M|/M w(x) = 31−x
M44x
M+1/(2M)g(x) = e−(x/M)2/(√πM)
All windows θ(x)with compact support |x| ≤ Mand 0 value otherwise with M>0, except the Gaussian window that takes values
for x∈(−∞,∞). The line color and format refer to their graphical representation in Fig. 1.
The demonstration is straightforward along similar lines as in Fernández-Macho
[15]
.
X
being Gaussian implies that, at each scale
λj
, the sample wavelet coefficients in
e
Wj
are
also Gaussian and, in turn, this means that
b
e
wij
, which is a linear combination of
e
w1j
,
. . .
,
e
wnj
, must also be Gaussian. Therefore, we have from Eq. (7) that
e
ϕX,s(λj)
is a correlation
coefficient between weighted observations from two Gaussian variates, of which
T/2j
are
(asymptotically) serially uncorrelated.4Applying Fisher’s result the theorem follows. 
Therefore,
CI(1−α)(ϕX,s(λj)) = tanh he
zj,s±φ−1
1−α/2/pT/2j−3i, (15)
where
φ−1
p
is the 100
p
% point of the standard normal distribution, can be used in practice
to construct a confidence interval for the wavelet multiple correlation coefficient, as well
as for testing hypothesis about the value of the wavelet correlation amongst a multivariate
set of observed variables X.
4. Weight functions and their spectral properties
In principle, any weight function satisfying the properties suggested by Cleveland
[8]
could be used and in fact many choices have been proposed in the literature [see e.g.
30
]. Table 1shows a selection of the six most widely used functions for averaging and
smoothing that are graphically depicted in Fig. 1. It can be shown that for all of them
R∞
−∞θ(x)dx =1, so that they can be described as moving averages.
However, the spectral properties of these moving averages differ much as shown by
the corresponding Fourier transforms in Figs. 2and 3. For example, the Fourier transform
of the variances needed to calculate the wavelet local correlations
ϕX(λj)
in Eq. (7) can
4
Note that this is the number of wavelet coefficients from a DWT that can be shown to decorrelate a
wide range of stochastic processes [9].
8

-M
-M
2
0
M
2
M
1
2M
3
4M
1
M
3
2M
Figure 1: Some weight functions for local least-squares regression (see Table 1). Blue lines:
u(x)
solid,
c(x)
long-dash, e(x)short-dash; Red lines: b(x)solid, w(x)long-dash, g(x)thick short-dash.
be obtained from Eq. (9b) as
τ(ω) = 1
2πe
T
∑
s=1
e−iωsδ2
j,s=1
√2πe
T
∑
s=1
e−iωs1
√2π
T−1
∑
t=Lj−1
θ(t−s)e
w2
ijt
=1
√2π
T−1
∑
t=Lj−1
e−iωte
w2
ijth1
√2π
T−t
∑
s=1−t
e−iωsθ(s)i=ξ(ω)φ(ω),ω∈(−π,π), (16)
where
φ(ω)
, the Fourier transform of the weight function
θ(x)
, must be non-negative to
ensure that
τ(ω)≥0
. This is certainly a desirable property for the spectral decomposition
of a variance [cf. 30, p.438] and similarly for ζ2
j,sand γj,sin Eq. (9).
However, common weight functions such as the uniform window
u(x)
, Cleveland’s
tricube window
c(x)
[
8
] and Epanechnikov’s parabolic window
e(x)
[
14
] are not rec-
ommended because of the presence of negative values in their corresponding spectral
windows. This means that, within a particular timescale, some frequencies may be
negatively weighted, which is not desirable. Even more so, those frequencies that are
negatively weighted are not invariant to the choice of bandwidth
2π/M
so that, in
the present case, the sign of correlation at frequencies corresponding to a particular
timescale may change depending of the choice of bandwidth. This, as can be seen in
Fig. 2, is particularly true for the uniform window which is a common choice in the
rolling-window wavelet correlation literature so far.
On the other hand, some other weight functions such as the Gaussian window,
Bartlett’s triangular window [
3
] and Wendland’s truncated power window [
39
] are
favored in the signal extraction and smoothing literature because their spectral windows
are non-negative everywhere which in the present case ensures that the sign of correlation
9

-15 Π
2M
-11 Π
2M
-7Π
2M
-3Π
2M
3Π
2M
7Π
2M
11 Π
2M
15 Π
2M
-0.1
0.1
0.2
0.3
0.4
Figure 2: Spectral windows corresponding to the blue weight functions in Fig. 1:
u(x)
solid,
c(x)
long-dash,
e(x)short-dash. Note that they take negative values at some frequencies.
at frequencies corresponding to a particular timescale will be invariant to the choice of
bandwidth.
If computational power is an issue Bartlett and Wendland windows are probably good
choices that are not difficult to implement. In what follows, we will use the Gaussian
window because of some desirable properties. In particular, it is the closest to uniform
weights in the time domain within a given bandwidth, its Fourier transform is also a
Gaussian function, it has near-compact support in the frequency domain, and its spectral
window is positive everywhere, therefore frequency/scale weights are always positive
and correlation sign is invariant to the choice of bandwidth.
5. Examples
In order to demonstrate the validity of the proposed tool this section shows first its
performance with simulated data sets generated by two nonstationary statistical models
that try to mimic some relevant features relating to the evolving correlation structure at
different frequencies of many multivariate financial time series of interest, especially at
the shortest timescales (short-run transient comovements typical of market ‘contagion’
[
17
,
13
,
32
,
6
]), while possibly maintaining a high correlation at the longest timescales
(long-run permanent comovements typical of market integration [
18
,
42
,
22
,
15
,
25
,
12
]).
In both cases the proposed WLMC is shown to clearly capture these changing correlation
structures both along time as well as across the different timescales. The last example
illustrates the usage of the WLMC with real data carrying out a multiscale analysis of
the comovements of Eurozone stock markets during a recent period.
10

-6Π
M
-4Π
M
-2Π
M
0
2Π
M
4Π
M
6Π
M
0.1
0.2
0.3
0.4
Figure 3: Spectral windows corresponding to the red weight functions in Fig. 1:
b(x)
solid,
w(x)
long-dash,
g(x)thick short-dash. Note that they are non-negative everywhere.
5.1. Simulation: detecting breaks in the correlation structure.
Figure 4shows the WLMC of a simulated bivariate time series
Xt= (xt,yt)
of length
T=512
with varying correlation both along time and across timescales/frequencies. The
two series were obtained from the following data generating process (dgp) (17):
xt=η1t, (17a)
yt(ω1) = (η2t,T/3 <t<2T/3,
0.9 η1t, otherwise, ,ω1∈(0, π/8), (17b)
yt(ω2) = (0.9 η1t,T/3 <t<2T/3,
η2t, otherwise, ,ω2∈[π/8, π), (17c)
where
ηt= (η1t,η2t)
is a zero-mean bivariate normal random variable with covariance
matrix Σ=[(0.1,0)0,(0,0.1)0]and ωdenotes the angular frequency.
Notice that since the dgp for
yt
differs depending on the angular frequency
ω
the
implied correlation structure will not be the same across different timescales. That
is,
xt
and
yt
are highly correlated at long timescales (low frequencies up to
π/8
) but
uncorrelated at short timescales (frequencies higher than
π/8
). However, during a period
of time spanning the second third of the sample (
T/3 <t<2T/3
) that behavior is
reversed so that data become highly correlated at short timescales but uncorrelated at
low frequencies. Altogether, the correlation structure is quite complex involving several
sudden changes both along time and across the different timescales. We also note that
this bivariate time series, even though it appears to be stationary in the means and
variances, it is intrinsically nonstationary due to its changing correlation structure.
In spite of the complexity of the changes in the correlation structure both along time
and across timescales generated by dgp (17), the heatmap in the middle of Fig. 4shows
11

time
0 100 200 300 400 500
-3 -2 -1 0 1 2 3
Wavelet Local Multiple Correlation
time
periods
0.2
0.4
0.6
0.8
j
10 30 50 70 90 120 150 180 210 240 270 300 330 360 390 420 450 480 510
(2-4]
(4-8]
(8-16]
(16-32]
(32-64]
(64-128]
smooth
-1.0
-0.5
0.0
0.5
1.0
level6
-1.0
-0.5
0.0
0.5
1.0
level5
-1.0
-0.5
0.0
0.5
1.0
level4
-1.0
-0.5
0.0
0.5
1.0
level3
-1.0
-0.5
0.0
0.5
1.0
level2
10 50 90 130 180 230 280 330 380 430 480
-1.0
-0.5
0.0
0.5
1.0
level1
10 50 90 130 180 230 280 330 380 430 480
time
Wavelet Local Multiple Correlation
Figure 4: Wavelet local multiple correlation of two simulated time series (top) with sudden breaks in the
correlation structure (dgp 17). The heatmap (middle) shows how the WLMC clearly detects the complex
changes implicit in the simulated data. The red lines (bottom) correspond to the upper and lower bounds
of the 95% confidence interval at different timescales.
12

how the WLMC clearly picks up these changes without any problem. The graphs at the
bottom complete the statistical analysis obtained from the WLMC showing the actual
correlation values together with their 95% confidence intervals at each individual scale.
As expected from the dgp, for the three shortest timescales the WLMC estimates
are very low, some even not statistically significant, at both ends of the sample while
showing high correlation during the middle part of the sample. In contrast, for the three
longest timescales the WLMC gives high correlation values at both ends but does not
detect a statistically significant correlation during the middle part. These results confirm
the validity of the proposed WLMC in detecting complex evolving correlation structures,
both along time and across different frequencies/timescales, that may be present in the
data.
5.2. Simulation: detecting gradual change in correlation structure of nonstationary process.
The second example (see Figure 5) illustrates the WLMC with a simulated bivariate
nonstationary time series
Xt= (xt,yt)
of length
T=512
with constant high correlation
in the long run and increasing correlation at shorter timescales. More specifically, the
two series were now generated as follows (dgp 18):
xt=µ1t+ε1t, (18a)
yt(ω1) = µ1t+ε2t,ω1∈(0, π/32), (18b)
yt(ω2) = ctµ1t+ (1−ct)µ2t+ε2t,ω2∈[π/32, π), (18c)
where
ct
are a set of slowly-increasing plus rapidly-decreasing weights in the interval
[0, 1]5and µt= (µ1t,µ2t)is a bivariate random-walk trend series
µt=µt−1+ηt, (18d)
with
ηt
as in the previous example and
εt= (ε1t,ε2t)
as a new zero-mean bivariate normal
random variable with the same covariance matrix Σand uncorrelated with ηt.
That is,
xt
and
yt
are now two nostationary ARIMA(0,1,1) processes [
21
] that share a
common trend and, therefore, they are highly correlated at long timescales (frequencies
up to
π/32
) for the whole span of the sample. This is similar to the long-run permanent
comovements that are typical of market integration in, for example, close financial
markets. On the other hand, at shorter timescales (frequencies higher than
π/32
) their
correlation structure evolves in such a manner that they gradually become more and
more correlated as time increases until nearing the last stretch of the sample when the
correlations decrease. This is similar to the short-run transient comovements that are
typical of market ‘contagion’ due to, for example, financial crises.
We also note that this bivariate time series is nostationary not only in the means and
variances but also in its correlation structure. However, using a sufficiently long wavelet
5More specifically, ct={(t−1
t0−1)3/2|t=1 . . . t0;(T−t
T−t0)3/2|t=t0. . . T}, with t0=7T/8.
13

time
0 100 200 300 400 500
96 98 100 102 104
Wavelet Local Multiple Correlation
time
periods
0.2
0.4
0.6
0.8
1.0
j
10 30 50 70 90 120 150 180 210 240 270 300 330 360 390 420 450 480 510
(2-4]
(4-8]
(8-16]
(16-32]
(32-64]
(64-128]
smooth
-1.0
-0.5
0.0
0.5
1.0
level6
-1.0
-0.5
0.0
0.5
1.0
level5
-1.0
-0.5
0.0
0.5
1.0
level4
-1.0
-0.5
0.0
0.5
1.0
level3
-1.0
-0.5
0.0
0.5
1.0
level2
10 50 90 130 180 230 280 330 380 430 480
-1.0
-0.5
0.0
0.5
1.0
level1
10 50 90 130 180 230 280 330 380 430 480
time
Wavelet Local Multiple Correlation
Figure 5: Wavelet local multiple correlation of two nonstationary I(1) simulated time series (top) highly
correlated in the long run and with increasing correlation at shorter timescales (dgp 18). The heatmap
(middle) shows how the WLMC clearly picks up the evolving correlation structure at different timescales
implicit in the simulated data. The red lines (bottom) correspond to the upper and lower bounds of the
95% confidence interval at different timescales. 14

filter, in this case the usual Daubechies wavelet filter of length
L
=4[
11
], this type of
combined nonstationarity poses no problem for the WLMC.
As can be seen in the middle of Fig. 5the heatmap shows how the WLMC clearly
picks up the evolving correlation structure at different timescales implicit in dgp (18)
while the graphs at the bottom show the 95% confidence intervals for the correlations at
each individual scale.
All this can be taken as evidence confirming the validity of the proposed tool in
detecting evolving correlation structures both along time and across different frequen-
cies/timescales.
5.3. Eurozone stock markets.
In this section we illustrate the usage of the advocated wavelet local multiple cor-
relation (WLMC) with data from the eleven main Eurozone stock markets as follows
(ordered by nominal GDP of the country where they operate): DAX (Germany), CAC40
(France), FTSE/MIB (Italy), IBEX35 (Spain), AEX (Netherlands), BEL20 (Belgium), ATX
(Austria), ISEQ (Ireland), OMXH25 (Finland), PSI20 (Portugal) and FTSE/ASE (Greece).
The data were collected daily (closing prices) from January 3,2000 (Monday) to May 31,
2017 (Wednesday)
6
and the analysis was conducted using daily stock market returns, i.e.,
Rit =log(Sit /Si,t−1) = ∆log Sit
, where
Sit
,
i
=1...11, are the corresponding stock market
index values for
t
=2. . . 4543 trading days. Therefore, the total number of observations
used is 49962 returns, thus containing a large amount of information that may not be
easy to convey using standard procedures.
In order to calculate the proposed WLMC we decomposed the daily stock market
returns applying the MODWT with a Daubechies wavelet filter of length
L
=4[
11
].
7
The
maximum decomposition level
J
is given by
blog2(T)c
[
27
], which, in the present case,
means a maximum level of 12. Since the number of feasible wavelet coefficients gets
critically small for high levels, we chose to carry out the wavelet analysis with
J
=9so that
ten vectors (nine of wavelet coefficients and one of scaling coefficients) were produced
for each daily returns series, i.e. e
wi1, . . . , e
wi9and e
vi9,i=1...11, respectively.
We may note that, since a MODWT based on Daubechies wavelets approximates
an ideal band-pass filter with bandpass given by the frequency interval
[2−jπ, 21−jπ)
for
j
=1. . .
J
, after inverting that frequency range the corresponding periods are within
(2j, 2j+1]
time units intervals [
41
]. This means that, with 5daily data per week, the scales
λj
,
j
=1...9, of the wavelet coefficients are associated to periods of, respectively, 2–4days
(which includes most intraweek scales), 4–8days (including the weekly scale), 8–16 days
(fortnightly scale), 16–32 days (monthly scale), 32–64 days (quarterly scale), 64–128 days
(quarterly to biannual scale), 128–256 days (biannual scale), 256–512 days (annual scale)
and 512–1028 days (two to four year scale).
6
As published by Yahoo! https://finance.yahoo.com, Investing https://www.investing.com/indices/
and Financial Times https://markets.ft.com/data/indices.
7
Results with the Daubechies LA(8) filter were practically the same except for the two longest timescales
(one year and longer) that were indistinguishable from 1up to three decimal points for the whole length of
the sample period.
15

5
6
7
8
9
10
11
time
logs
2001 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17
DAX
CAC40
MIB
IBEX35
AEX
BEL20
ATX
FTSE.ASE20
OMXH25
PSI20
ISEQ
Wavelet Local Multiple Correlation
time
periods
0.90
0.95
0.96
0.97
0.98
0.99
1
j
2001 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17
(2-4]
(4-8]
(8-16]
(16-32]
(32-64]
(64-128]
(128-256]
(256-512]
(512-1024]
smooth
0.986
0.988
0.990
0.992
0.994
0.996
0.998
1.000
level 9
CAC40
0.96
0.97
0.98
0.99
1.00
level 8
CAC40
0.96
0.97
0.98
0.99
1.00
level 7
CAC40
0.96
0.97
0.98
0.99
1.00
level 6
CAC40
0.94
0.95
0.96
0.97
0.98
0.99
1.00
level 5
CAC40
0.94
0.95
0.96
0.97
0.98
0.99
1.00
level 4
CAC40
0.92
0.94
0.96
0.98
1.00
level 3
CAC40
0.92
0.94
0.96
0.98
1.00
level 2
2001 03 05 07 09 11 13 15 17
CAC40
0.90
0.92
0.94
0.96
0.98
1.00
level 1
2001 03 05 07 09 11 13 15 17
CAC40
time
Wavelet Local Multiple Correlation
Figure 6: EUro11 stocks (top) and WMLC of their daily returns. The heatmap (middle) shows how the
WLMC evolves along time revealing perfect market integration at long timescales and evidence of financial
‘contagion’ at short timescales during 2007-2012. The red lines (bottom) correspond to the upper and lower
bounds of the 95% confidence interval at different timescales.
16

Figure 6shows the wavelet local multiple correlations obtained as a measure of
comovement dynamics among Eurozone stock markets. As can be seen, the correlation
structure of these markets during this century is far from homogeneous both along time
and across timescales/frequencies. One of the most striking features that the WLMC
brings to light is the sharp divide in the correlation structure across timescales at about
the quarterly scale.
Above the quarterly scale, the long-term correlation structure appears to be quite
stable in time with multiple correlations that are all very high starting at around 0.98
and reaching values near 1at the longest timescales (year long and above). This can
be interpreted as near integration among Euro stock markets at quarterly horizons and
above in the sense that the returns obtained in any of them can be determined by the
overall performance in the other markets [
38
,
25
]. Also, for time horizons of one year
and longer the existence of an exact linear relationship between Eurozone stock markets
cannot be ruled out. This can be taken as evidence of perfect integration even in spite of
several financial and debt crises occurring during this period.
On the other hand, the WLMC in Fig. 6shows that for intraweek and intramonth
periods the correlation structure is clearly evolving along time. This is specially so during
the financial crises of 2007–2012. Prior to this period, short-term multiple correlations
were around 0.90 but experienced a sharp increase above 0.98 during 2007 which can be
taken as evidence of financial ‘contagion’ among Eurozone stock markets [
17
,
13
,
32
,
6
].
8
Such phenomenon appears to have mitigated after 2012 where short-term multiple
correlations have decreased.
6. Conclusions
This paper presents a new statistical tool, the wavelet local multiple correlation
(WLMC), that may be useful in the analysis of comovements within a set of time series.
The WLMC consists in one single set of multiscale correlations along time, each of
them calculated as the square root of the regression coefficient of determination in that
linear combination of locally weighted wavelet coefficients for which such coefficient
of determination is a maximum. In contrast, the alternative of using standard bivariate
wavelet correlations with rolling windows needs to calculate, plot and compare a large
number of wavelet correlation maps and it may run out of data points at longer timescales.
Also, it has been shown how the spectral properties of the weight function or rolling
window need to be taken into account. Figures 4and 5offer some graphical examples of
this tool as obtained from two simulated data sets with either breaks or gradual changes
in the correlation structure illustrating the validity of the proposed WLMC statistic.
8
On the other hand, it has been suggested by one anonymous reviewer that this result for intra-
EU co-movement relationships contrasts with evidence from Islamic stock market indexes where it is
found that “recession periods were accompanied by a local dissociation at some frequency subbands
between conventional and Islamic market indexes”, especially in emerging markets, with a co-movement
relationship that “appeared to be varying in both time and frequency” [34].
17

Furthermore, Figure 6presents the time-localized wavelet analysis of a set of eleven
Eurozone stock market returns during a recent period of 4542 trading days. The WLMC
analysis reveals the existence of a stable and practically exact linear relationship between
these stock markets for periods of time of one year and longer, which can be interpreted
as perfect integration. On the other hand, the WLMC shows that for intraweek and
intramonth periods the correlation structure is clearly evolving along time, experiencing
a sharp increase during financial crises which some authors interpret as evidence of
financial ‘contagion’.
The scope of suggestions for further research using the WLMC analysis is really
huge. In general, when it is suspected that a multivariate dataset is subject to different
implications depending on the time horizon, WLMC analysis can be very useful. For
example, a similar analysis as in the previous section can be extended to applications
with other types of multivariate data (financial, energy, commodities, etc.) as well as to
other geographical areas where bivariate correlations and global indices have been used
so far. In this sense, WLMC analysis could be applied to unconventional or developing
stock markets to see how their reactions to economic/financial crises across different
timescales compare to those from conventional developed stock markets such as in the
Eurozone [cf.
34
]. Further development of methodological aspects of the WLMC could
consider other contexts such as artificial neural networks [
35
], causality issues [
33
,
29
], or
even the actual local regression coefficients (see Section 3) which could be exploited to
study how the relationship connecting the different markets under study evolves in time
across scales as well as for forecasting purposes.
We may finally point out that all these results and suggestions would be quite hard
to establish using the standard wavelet analysis that relies on the visualization of all
the wavelet correlation maps between pairs of variables and they serve to illustrate the
potential of this new tool in the multiscale analysis of multivariate data along time.
Supplemental material
A new version of the wavemulcor Rcomputer package facilitates the computation
of wavelet local multiple correlations. It can be obtained from The Comprehensive R
Archive Network (CRAN) at https://cran.r-project.org/package=wavemulcor.
Acknowledgments
The author would like to acknowledge research funding received from UPV/EHU
Econometrics Research Group (Basque Government Dpt. of Education grant IT-642-13)
and Spanish Ministry of Economy and Competitiveness (grant MTM2013-40941-P).
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