Computational Statistics and Data Analysis 56 (2012) 3491–3497
Contents lists available at SciVerse ScienceDirect
Computational Statistics and Data Analysis
journal homepage: www.elsevier.com/locate/csda
A wavelet-based approach to test for financial market contagion
Marco Gallegati ∗
Department of Economics, Faculty of Economics ‘‘G. Fuá’’, Universitá Politecnica delle Marche, Piazzale Martelli 8, 60121 Ancona, Italy
article info abstract
Article history: A wavelet-based approach to test whether contagion occurred during the US subprime
Received 20 January 2010 crisis of 2007 is proposed. After separately identifying contagion and interdependence
Received in revised form 22 October 2010 through wavelet decomposition of the original returns series, the presence of contagion
Accepted 3 November 2010
is assessed using a simple graphical test based on non-overlapping confidence intervals of
Available online 21 November 2010
estimated wavelet coefficients in crisis and non-crisis periods. The results indicate that all
stock markets have been affected by the US subprime crisis and that Brazil and Japan are
Keywords:
Wavelet decomposition
the only countries in which contagion is observed at all scales.
Contagion © 2010 Elsevier B.V. All rights reserved.
Interdependence
Wavelet correlation
1. Introduction
The repeated occurrence of turmoils in international financial markets has drawn a striking interest in the linkages
between financial markets during times of crisis. Indeed, in turbulent periods we observe an increase in cross-market
linkages due to the propagation of shocks from one country to other countries as a consequence of the high integration
of financial and goods markets. The increase in cross-market linkages from the pre-crisis period to the crisis period may
take the form of interdependence or contagion. The issue of discriminating between contagion and interdependence during
periods of high volatility in financial markets has important implications for the asset allocation strategy of risk managers
and for policymakers’ optimal policy response to a crisis. In effect, in turbulent periods the usefulness of hedging operations is
called into question, as correlations between financial time series in these periods may differ markedly from those in quiet
or normal periods. Such a changing correlation pattern, known as ‘‘correlation breakdown’’, suggests that the benefits of
international diversification for asset allocation and portfolio composition may be substantially reduced in stressful market
situations, just when these benefits are needed most.
Despite the large number of papers on financial market contagion, there is no agreement in the literature on the exact
definition of what constitutes contagion and how we should measure it. For example, Pericoli and Sbracia (2003) provide
an overview of the contagion literature presenting five different classifications of contagion (see also the World Bank’s
website for a comprehensive overview of the definitions of contagion). The major distinction when defining contagion
is between ‘‘fundamentals-based’’ and ‘‘pure’’ contagion (see Dornbusch et al., 2000; Kaminsky and Reinhart, 2000). The
definition of ‘‘fundamentals-based’’ contagion introduced by Calvo and Reinhart (1996) emphasizes the transmission of
shocks between countries and/or markets resulting from real linkages and financial market integration in both crisis and
non-crisis periods. These forms of co-movements reflect normal interdependence across markets and countries and are
often labelled as spillovers. The term ‘‘pure’’ contagion (see among others Eichengreen et al., 1996; Bae et al., 2003) refers to
the transmission of shocks from one country to another country in excess of what should be expected after controlling for
fundamental factors. This type of contagion is generally related to investors’ behavior such as herding, financial panic, loss of
∗ Tel.: +39 071 2207114; fax: +39 071 2207102.
E-mail address: .
0167-9473/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.csda.2010.11.003
, 3492 M. Gallegati / Computational Statistics and Data Analysis 56 (2012) 3491–3497
confidence, etc., and leads to excessive co-movements. Thus, during turbulent periods, some co-movements across markets
can merely be an implication of interdependence rather than the effect of contagion. A key issue in testing for contagion is
to draw a distinction between ‘‘excessive’’ and normal co-movements across financial markets.
Standard time-domain techniques can have problems in identifying fundamental contagion from other transmission
mechanisms because of the difficult task of finding good proxies for the impact of macroeconomic fundamentals using
current econometric techniques. The statistical problems in measuring financial contagion are well documented by the
variety of econometric techniques used for the analysis of co-movements between financial markets: they include testing
for changes in correlation coefficients (King and Wadhwani, 1990; Lee and Kim, 1993; Calvo and Reinhart, 1996), ARCH
and GARCH models (Chou et al., 1994; Hamao et al., 1990; Billio and Caporin, 2010), cointegrating relationships (Longin
and Solnik, 1995; Cashin et al., 1995), probit/logit models (Eichengreen et al., 1996; Kaminsky and Reinhart, 2000), regime
switching (Gallo and Otranto, 2008), the factor model (Corsetti et al., 2005), and the copula approach (Costinot et al., 2000;
Rodriguez, 2007).
A potentially useful different perspective on the empirical issue of discriminating between contagion and
interdependence can be provided by frequency domain analysis. Given its ability to partition each variable into components
of different frequencies, a frequency domain framework can provide a simple and intuitive way to distinguish between
contagion and interdependence. Examples of studies testing for contagion by associating contagion and interdependence
with distinct frequency ranges (high and low frequencies, respectively) are the recent papers by Bodart and Candelon (2009)
and Orlov (2009).
Wavelet analysis is a filtering method that provides an interesting alternative to time series and frequency domain
methods since it transforms the original data into different frequency components with a resolution matched to its scale.
This characteristic is particularly useful when dealing with signals that are non-stationary and exhibit changing frequencies
over time, as in the case of financial market data. The two main interesting features of wavelet analysis for our purposes are
(i) its ability to decompose macroeconomic time series, and data in general, into their time-scale components, and (ii) its
capacity to provide an alternative representation of the variability and association structure of certain stochastic processes
on a scale-by-scale basis. The multi-resolution decomposition property of the wavelet transform can be used to separately
identify contagion and interdependence by associating each to its corresponding frequency component. In addition, the
‘‘energy-preserving’’ property of the wavelet transform, allowing for a scale-based decomposition of the energy in a time
series, can provide the basis for a wavelet-based test for contagion. Thus, after identifying contagion and interdependence
with wavelet and scaling coefficients, respectively, we test for the occurrence of contagion by using a simple graphical test
based on non-overlapping confidence intervals of estimated wavelet correlation coefficients in crisis and non-crisis periods.
Two main findings emerge from the wavelet-based ‘correlation-breakdown’ test: (i) there is evidence for each country of
international financial contagion during the US subprime crisis, and (ii) these contagion effects are scale dependent, in the
sense that contagion does not display its effects uniformly across scales (exceptions are Brazil and Japan).
The paper is organized as follows. Section 2 introduces wavelet decomposition analysis and the notions of wavelet
variance, covariance, and correlation. In Section 3, we first describe the data used in this study and then present the results
of the wavelet-based test for contagion. Finally, Section 4 concludes.
2. Methodology
Since the use of wavelets is a well-established technique, in this section we only present the essential methods useful for
our purposes. For a more complete and insightful development of the theory and use of wavelets, see Percival and Walden
(2000) and Gençay et al. (2002).
2.1. Wavelet series expansion
The most important property that wavelets possess for the analysis of economic data is to decompose the time series into
components associated with different scales of resolution. Any function f (t ) in L2 (R) can be represented by the following
wavelet series expansion:
f (t ) = vJ ,k φJ ,k (t ) + wJ ,k ψJ ,k (t ) + · · · + wj,k ψj,k (t ) + · · · + w1,k ψ1,k (t ), (1)
k k k k
where the coefficients vJ ,k = k φJ ,k f (t ) and wj,k = k ψj,k f (t ) represent the underlying smooth behavior of the data at
the coarsest scale (the scaling coefficients) and the coarse-scale deviations from it (the wavelet coefficients), respectively,
and where φJ ,k , ψj,k are the so-called scaling and wavelet functions satisfying the following conditions:
φJ ,k (t )φJ ,k∗ (t )dt = δk,k∗ ,
ψj,k (t )ψj∗ ,k∗ (t )dt = δj,j∗ δk,k∗ ,
ψj,k (t )φJ ,k∗ (t )dt = 0, ∀j, k,
Contents lists available at SciVerse ScienceDirect
Computational Statistics and Data Analysis
journal homepage: www.elsevier.com/locate/csda
A wavelet-based approach to test for financial market contagion
Marco Gallegati ∗
Department of Economics, Faculty of Economics ‘‘G. Fuá’’, Universitá Politecnica delle Marche, Piazzale Martelli 8, 60121 Ancona, Italy
article info abstract
Article history: A wavelet-based approach to test whether contagion occurred during the US subprime
Received 20 January 2010 crisis of 2007 is proposed. After separately identifying contagion and interdependence
Received in revised form 22 October 2010 through wavelet decomposition of the original returns series, the presence of contagion
Accepted 3 November 2010
is assessed using a simple graphical test based on non-overlapping confidence intervals of
Available online 21 November 2010
estimated wavelet coefficients in crisis and non-crisis periods. The results indicate that all
stock markets have been affected by the US subprime crisis and that Brazil and Japan are
Keywords:
Wavelet decomposition
the only countries in which contagion is observed at all scales.
Contagion © 2010 Elsevier B.V. All rights reserved.
Interdependence
Wavelet correlation
1. Introduction
The repeated occurrence of turmoils in international financial markets has drawn a striking interest in the linkages
between financial markets during times of crisis. Indeed, in turbulent periods we observe an increase in cross-market
linkages due to the propagation of shocks from one country to other countries as a consequence of the high integration
of financial and goods markets. The increase in cross-market linkages from the pre-crisis period to the crisis period may
take the form of interdependence or contagion. The issue of discriminating between contagion and interdependence during
periods of high volatility in financial markets has important implications for the asset allocation strategy of risk managers
and for policymakers’ optimal policy response to a crisis. In effect, in turbulent periods the usefulness of hedging operations is
called into question, as correlations between financial time series in these periods may differ markedly from those in quiet
or normal periods. Such a changing correlation pattern, known as ‘‘correlation breakdown’’, suggests that the benefits of
international diversification for asset allocation and portfolio composition may be substantially reduced in stressful market
situations, just when these benefits are needed most.
Despite the large number of papers on financial market contagion, there is no agreement in the literature on the exact
definition of what constitutes contagion and how we should measure it. For example, Pericoli and Sbracia (2003) provide
an overview of the contagion literature presenting five different classifications of contagion (see also the World Bank’s
website for a comprehensive overview of the definitions of contagion). The major distinction when defining contagion
is between ‘‘fundamentals-based’’ and ‘‘pure’’ contagion (see Dornbusch et al., 2000; Kaminsky and Reinhart, 2000). The
definition of ‘‘fundamentals-based’’ contagion introduced by Calvo and Reinhart (1996) emphasizes the transmission of
shocks between countries and/or markets resulting from real linkages and financial market integration in both crisis and
non-crisis periods. These forms of co-movements reflect normal interdependence across markets and countries and are
often labelled as spillovers. The term ‘‘pure’’ contagion (see among others Eichengreen et al., 1996; Bae et al., 2003) refers to
the transmission of shocks from one country to another country in excess of what should be expected after controlling for
fundamental factors. This type of contagion is generally related to investors’ behavior such as herding, financial panic, loss of
∗ Tel.: +39 071 2207114; fax: +39 071 2207102.
E-mail address: .
0167-9473/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.csda.2010.11.003
, 3492 M. Gallegati / Computational Statistics and Data Analysis 56 (2012) 3491–3497
confidence, etc., and leads to excessive co-movements. Thus, during turbulent periods, some co-movements across markets
can merely be an implication of interdependence rather than the effect of contagion. A key issue in testing for contagion is
to draw a distinction between ‘‘excessive’’ and normal co-movements across financial markets.
Standard time-domain techniques can have problems in identifying fundamental contagion from other transmission
mechanisms because of the difficult task of finding good proxies for the impact of macroeconomic fundamentals using
current econometric techniques. The statistical problems in measuring financial contagion are well documented by the
variety of econometric techniques used for the analysis of co-movements between financial markets: they include testing
for changes in correlation coefficients (King and Wadhwani, 1990; Lee and Kim, 1993; Calvo and Reinhart, 1996), ARCH
and GARCH models (Chou et al., 1994; Hamao et al., 1990; Billio and Caporin, 2010), cointegrating relationships (Longin
and Solnik, 1995; Cashin et al., 1995), probit/logit models (Eichengreen et al., 1996; Kaminsky and Reinhart, 2000), regime
switching (Gallo and Otranto, 2008), the factor model (Corsetti et al., 2005), and the copula approach (Costinot et al., 2000;
Rodriguez, 2007).
A potentially useful different perspective on the empirical issue of discriminating between contagion and
interdependence can be provided by frequency domain analysis. Given its ability to partition each variable into components
of different frequencies, a frequency domain framework can provide a simple and intuitive way to distinguish between
contagion and interdependence. Examples of studies testing for contagion by associating contagion and interdependence
with distinct frequency ranges (high and low frequencies, respectively) are the recent papers by Bodart and Candelon (2009)
and Orlov (2009).
Wavelet analysis is a filtering method that provides an interesting alternative to time series and frequency domain
methods since it transforms the original data into different frequency components with a resolution matched to its scale.
This characteristic is particularly useful when dealing with signals that are non-stationary and exhibit changing frequencies
over time, as in the case of financial market data. The two main interesting features of wavelet analysis for our purposes are
(i) its ability to decompose macroeconomic time series, and data in general, into their time-scale components, and (ii) its
capacity to provide an alternative representation of the variability and association structure of certain stochastic processes
on a scale-by-scale basis. The multi-resolution decomposition property of the wavelet transform can be used to separately
identify contagion and interdependence by associating each to its corresponding frequency component. In addition, the
‘‘energy-preserving’’ property of the wavelet transform, allowing for a scale-based decomposition of the energy in a time
series, can provide the basis for a wavelet-based test for contagion. Thus, after identifying contagion and interdependence
with wavelet and scaling coefficients, respectively, we test for the occurrence of contagion by using a simple graphical test
based on non-overlapping confidence intervals of estimated wavelet correlation coefficients in crisis and non-crisis periods.
Two main findings emerge from the wavelet-based ‘correlation-breakdown’ test: (i) there is evidence for each country of
international financial contagion during the US subprime crisis, and (ii) these contagion effects are scale dependent, in the
sense that contagion does not display its effects uniformly across scales (exceptions are Brazil and Japan).
The paper is organized as follows. Section 2 introduces wavelet decomposition analysis and the notions of wavelet
variance, covariance, and correlation. In Section 3, we first describe the data used in this study and then present the results
of the wavelet-based test for contagion. Finally, Section 4 concludes.
2. Methodology
Since the use of wavelets is a well-established technique, in this section we only present the essential methods useful for
our purposes. For a more complete and insightful development of the theory and use of wavelets, see Percival and Walden
(2000) and Gençay et al. (2002).
2.1. Wavelet series expansion
The most important property that wavelets possess for the analysis of economic data is to decompose the time series into
components associated with different scales of resolution. Any function f (t ) in L2 (R) can be represented by the following
wavelet series expansion:
f (t ) = vJ ,k φJ ,k (t ) + wJ ,k ψJ ,k (t ) + · · · + wj,k ψj,k (t ) + · · · + w1,k ψ1,k (t ), (1)
k k k k
where the coefficients vJ ,k = k φJ ,k f (t ) and wj,k = k ψj,k f (t ) represent the underlying smooth behavior of the data at
the coarsest scale (the scaling coefficients) and the coarse-scale deviations from it (the wavelet coefficients), respectively,
and where φJ ,k , ψj,k are the so-called scaling and wavelet functions satisfying the following conditions:
φJ ,k (t )φJ ,k∗ (t )dt = δk,k∗ ,
ψj,k (t )ψj∗ ,k∗ (t )dt = δj,j∗ δk,k∗ ,
ψj,k (t )φJ ,k∗ (t )dt = 0, ∀j, k,
