ON THE IDENTIFICATION OF INTERDEPENDENCE AND CONTAGION
OF FINANCIAL CRISES
EMANUELE BACCHIOCCHI
Working Paper n. 2015-12
LUGLIO 2015
FRANCESCO GUALA
Working Paper n. 2011-18
SETTEMBRE
2011
ARE PREFERENCES FOR REAL?
CHOICE THEORY, FOLK PSYCHOLOGY,
DIPARTIMENTO DI ECONOMIA, MANAGEMENT E METODI QUANTITATIVI
AND THE HARD CASE FOR COMMONSENSIBLE REALISM
Via Conservatorio 7
20122 Milano
tel. ++39 02 503 21501 (21522) - fax ++39 02 503 21450 (21505)
FRANCESCO GUALA
http://www.economia.unimi.it
E Mail: [email protected]
Working Paper n. 2011-18
On the Identification of Interdependence and Contagion of
Financial Crises
Emanuele Bacchiocchi⇤
July 2015
Abstract
In this paper we propose a new framework for modeling heteroskedastic structural
vector autoregressions. Although it is general enough to find potential applications in
many empirical economic fields, it reveals to be well suited in the distinction between
interdependence and contagion in the literature related to the transmission of financial
crises. The identification of the structural parameters is obtained by exploiting the
heteroskedasticity in the data, naturally arising during crisis periods. More precisely
we provide identification conditions when both heteroskedasticity and traditional restrictions on the parameters are jointly considered. Finally, this methodology is used
to investigate the relationships between sovereign bond yields for some highly indebted
EU countries.
Keywords: Heteroskedasticity, identification, interdependence, contagion, highly indebted
EU countries, financial crisis.
JEL codes: C01,C13,C30,C51.
⇤
DEMM, University of Milan, Via Conservatorio 7, 20122 Milan (Italy), tel +39 02 50321504, fax +39
02 50321450. Email: [email protected]
1
I
Introduction
The recent global crisis and the subsequent European debt sovereign crisis, in their drama,
o↵er economists and econometricians a rich laboratory to study the transmission of financial
shocks during periods of high turbulences. Since the seminal contribution by King and
Wadhwany (1990) many studies have been proposed to capture the rational and irrational
aspects of the spreads of financial crises. Contagion is the term which mainly represents this
stream of research, although over the years, other definitions were proposed according to the
di↵erent theoretical explanations causing this phenomenon. Masson (1999), for example,
distinguishes between monsoonal e↵ects, spillovers, and pure contagion. While the first
two are mainly related to macroeconomic fundamentals and external economic linkages, the
last one indicates all those cases when the co-movements can be considered as “excessive”.
The definition of contagion we have in mind for the aim of the present paper is to look
at whether shocks propagate di↵erently during normal or turbulent periods, indicated as
“shift contagion” by Caporin et al., 2013. From the empirical point of view, the literature is
huge. Di↵erent approaches have been proposed and many international crises investigated
(see Dungey et al, 2005, Bacchiocchi and Bevilacqua, 2009, and Caporin et al., 2013, for
surveys).
A consistent class of papers, in particular, proposes empirical tests for financial contagion by observing whether the correlations between di↵erent markets are significantly
higher during crisis periods. In this line of research Forbes and Rigobon (2002) point to
the importance of the distinction between the so called “interdependence”, indicating the
market co-movements occurring even in periods of stability, and the “pure contagion” effects, occurring only if cross-markets co-movements increase significantly after the shock.
Their approach is based on correcting the traditional tests for comparing cross-market correlations for the upward bias caused by heteroskedasticity, naturally arising during crises.
Once correcting for this bias, they find that for many crises occurred during the ’90s the
observed high cross-market correlations are mainly attributed to interdependence rather
than contagion. This approach, however, is not immune to the endogeneity problem, i.e.
when there are bi-directional simultaneous linkages between the two investigated countries
(the originating and the a↵ected countries).
Rigobon (2003a), instead, in studying the propagation of shocks proposes a test for parameter stability, taking into account the econometric problems that generally arise when one
investigates market relationships during high turbulent periods, i.e. simultaneous equations,
omitted variables and heteroskedasticity. The paper, however, does not make any distinction between the two possible transmission channels “interdependence” and “contagion”.
Ciccarelli and Rebucci’s (2007) work also represents an important contribution to the
analysis of transmission of financial shocks. They propose a time-varying coefficient model
to measure contagion and interdependence in a Bayesian framework, that jointly deals with
the presence of heteroskedasticity and omitted variables. The time-varying nature of the
coefficients, moreover, may be used without knowledge of the e↵ective timing of the crisis
prior to the empirical analysis. However, as recognized by the authors, they do not attempt
to address the possible simultaneity problems that might arise when modeling strictly linked
countries or markets1 .
1
The authors suggest to deal with the possible endogeneity problem by means of potential extensions of
the time-varying coefficients structural VAR models used by Cogley and Sargent (2005) and Ciccarelli and
Rebucci (2006)
2
In the present paper, instead, we concentrate on this last problem and propose a solution
based on a new specification of structural vector autoregressive (SVAR) models that explicitly accounts for possible heteroskedasticity of the endogenous variables which is exploited
to identify di↵erent volatility regimes. Such heteroskedasticity is thus used to understanding
whether the transmission of shocks is di↵erent across di↵erent volatility regimes. As an example, in periods of high instability of the financial markets, a shock hitting one particular
market might propagate in a di↵erent way than in relatively tranquil periods. The e↵ect of
the same shock, in di↵erent periods of time, might be completely di↵erent. The turbulences
of the market could either amplify the e↵ect of the shock in the same market in which it
originates, or allow for a propagation to other financial markets, or both the e↵ects. In
this context, the proposed model, although general enough to find potential applications in
many macro-economic and financial frameworks, reveals to be well suited for the distinction
between interdependence and contagion of financial markets as discussed above.
The particular structure of the model, in fact, helps in the distinction between these two
phenomena, while the endogeneity drawback highlighted by Forbes and Rigobon (2002) is
solved through the recent “identification through heteroskedasticity” proposed by Rigobon
(2003b). The intuition, originally introduced by Wright (1928), consists in using the second
moments to increase the number of relations mapping the parameters of the reduced and
structural forms. Lanne and Lütkepohl (2008) and Lanne et al. (2010) use the same idea
to identify the structural shocks in SVAR models. As will be shown in the next sections,
the specification introduced in the present paper reveals to be more general than those
proposed by Rigobon (2003b) and Lanne and Lütkepohl (2008). The paper also introduces
novel identification rules for heteroskedastic SVAR models that encompasses previous ones.
To the best of our knowledge, the most similar contributions to our approach are Dungey
et al (2010) and Dungey et al (2015). In the former, the authors use a multivariate identified
GARCH model and distinguish between tranquil periods, hypersensitivity and contagion
between Hong Kong, Indonesia, Korea and Thailand, over the period 1997-98. In the latter,
instead, the focus is on crises dating using smooth transition structural GARCH model, with
an application to US equity, bond and REIT returns for 2001-10. Similarly to our model,
contagion can be detected by looking at possible changes in the simultaneous relationships
among the investigated variables. Another related paper is Kosch and Caporin (2013), that
extends the Dynamic Conditional Correlation (DCC) multivariate GARCH by including a
threshold structure for the dynamics of the correlations. This model allows the authors to
show that, when considering a set of stock markets in 1994-2011, periods of market turmoils
are associated with higher market comovements. This paper, however, does not discriminate
between interdependence and contagion, but rather corrects the dynamic correlations from
the presence of heteroskedasticity as in the spirit of Forbes and Rigobon (2002).
Over the last years other authors have proposed approaches to obtain identification using
heteroskedasticity in the data. Klein and Vella (2010) use the heteroskedasticity of the residuals to identify the structural parameters in bivariate triangular systems. Prono (2013) also
discusses identification in linear bivariate triangular models where structural errors follow
a bivariate and diagonal GARCH(1, 1) process. Lewbel (2010), instead, considers bivariate
models with mismeasured or endogenous regressors. Identification in triangular and fully
simultaneous systems can be obtained by imposing restrictions on particular second moments involving regressors and heteroskedastic residuals. Bacchiocchi and Fanelli (2015)
propose an heteroskedastic SVAR model with structural breaks. Sentana and Fiorentini
(2001), in a context of conditionally heteroskedastic factor models, provide identification
3
conditions that can be applied in a large number of cases, like residuals following GARCH
specifications2 , regime switching processes3 or structural VAR models4 .
On one side, in our innovative specification for heteroskedastic SVAR models, the presence of di↵erent volatility regimes facilitates the identification of the interdependence among
financial markets when a priori restrictions on the direction of causality cannot be imposed.
More generally, and this is new to the best of our knowledge, identification conditions will
be provided when the heteroskedasticity in the data is mixed with the traditional approach
of imposing restrictions on the structural parameters of the model.
On the other side, the new transmission channels operating in high turbulent periods
provide evidence on the contagion e↵ects. A test for contagion, thus, simply becomes a test
on the parameters related to these further transmission channels.
The methodology developed in the paper is used to study the transmission of financial
shocks in some highly indebted EU countries during the recent global and debt crises: Spain,
Ireland, Portugal, Greece and Italy. Being part of a common market these countries are
highly connected one to the others. Moreover, during the recent turbulences all these countries have played a proactive role in the propagation of (negative) financial shocks, making
impossible to discriminate between originating and a↵ected countries. The assumption of
no endogeneity considered in many empirical contributions in the literature of contagion,
such as Forbes and Rigobon (2002) and Ciccarelli and Rebucci (2006), becomes seriously
questionable.5
While the empirical literature on contagion is huge, only few contributions explicitly
focus on the e↵ects of the recent financial and debt crises within the European Monetary
Union (EMU). One of these is the already mentioned work by Caporin et al (2013), that
using univariate quantile regression techniques find almost no presence of contagion over
the sample 2003-2013, for some European countries. Arezki et al (2011), using a VAR
with dummies, investigate the spillover e↵ects of rating news on some European credit and
financial markets over the period 2007-2010, excluding thus the more recent events of the
sovereign debt crisis. De Santis (2012) studies the factors a↵ecting the sovereign bond
yields. Using daily observations for the period September 2008-August 2011, he finds that
sovereign spreads are explained by an aggregate regional risk factor, by country-specific
credit risk and by spillover e↵ects from Greece. Giordani et al (2013), distinguish between
“wake-up-call” and pure (or shift) contagion and find that only the former occurred among
nine euro-area countries during the period 2000-2011.
The results obtained using our procedure, that accounts for possible endogeneity and
heteroskedasticity, show that these countries are characterized by strong and bidirectional
interdependence. During the recent financial and debt crises, the transmission of financial
shocks is completely changed and in some cases there is strong evidence of contagion.
The rest of the paper is organized as follows. In Section II we first present the statistical
model and then derive the conditions for identification of the structural parameters. Section
III uses this methodology to investigate the transmission of financial shocks among some
highly indebted EU countries over the last years. Section IV provides some concluding
remarks.
2
See Caporale, Cipollini and Demetriades (2005), Dungey and Martin (2001), King et al. (1994), Rigobon
(2002).
3
See Caporale, Cipollini and Spagnolo (2005) and Rigobon and Sack (2003, 2004).
4
See Normandin and Phaneuf (2004).
5
This issue has been discussed and addressed in Caporin et al (2013), too.
4
II
Modeling Interdependence and Contagion through a new
class of heteroskedastic structural VAR models
Periods of financial crises are characterized by clusters of higher volatility that, following
Forbes and Rigobon (2002), invalidate tests for contagion based on correlation coefficients.
Moreover, the two authors, using simulation exercises, highlight that endogeneity also biases
these kinds of tests for contagion. In the present section we propose a general econometric
framework that allows for both endogeneity and heteroskedasticity in the data.
II.1
A structural VAR model with heteroskedastic errors
In this section we present a new specification for structural vector autoregressions that
explicitly model the heteroskedasticity features of the data. Let the data generating process
(DGP) follows a VAR model of the form
yt = Ddt + A1 yt
1
+ · · · + Ap yt
p
+ ut
(1)
where yt is a g-dimensional vector of observable variables, dt is a set of deterministic components with related coefficients D, A1 , A2 , . . . , Ap are matrices of parameters, and ut a
vector of innovations characterized by s regimes of di↵erent covariance matrices ⌃i , i.e.
ut ⇠ L (0, ⌃i ) with i = 1 . . . s. The structural form of the model, connecting the observable
variables yt or, concentrating out the dynamics, the innovations ut , with the uncorrelated
heteroskedastic shocks "t ⇠ L (0, ⇤t ), can be written as
Aut = Bi (t 2 Ti ) "t
,
E "t "0t = ⇤i
t 2 Ti
for
(2)
where Ti collects all the time periods such that regime i is in place, and (·) is the indicator
function. Given the heteroskedasticity nature of the structural shocks, described by the ⇤i
diagonal matrices, an equivalent representation of the connections between the structural
and reduced forms of the model is given by ⌃i = A 1 Bi ⇤i Bi0 A0 1 , i = 1 . . . s.
The time-invariant A matrix contains all the parameters describing the simultaneous
relations among the observable variables. The time-varying Bi matrices, instead, contain
regime-specific structural parameters describing further transmission channels for the structural shocks, operating during each regime. Moreover, as already mentioned, such structural
shocks are allowed to have di↵erent (diagonal) covariance matrices ⇤i .
Remark 1. Rigobon (2003b) provides a sufficient rank condition for the identification of a bivariate simultaneous equation model characterized by s = 2 volatility regimes.
Lanne and Lütkepohl (2008), using a well known result from matrix algebra, generalize the
Rigobon’s result by considering heteroskedastic SVAR models for more than two observable
variables. In particular, their specification features the following relations
⌃1 = A
1
⇤1 A
10
and
⌃2 = A
1
⇤2 A
10
.
(3)
The generalization to s > 2 is straightforward and can be obtained by considering a constant
A matrix and regime-specific ⇤i covariance matrices for the structural shocks, with i =
1, . . . , s. This specification, as can be easily seen, can be obtained as a special case of
the SVAR model in Eq.s (1)-(2) when all Bi matrices are restricted to be unit diagonal,
5
i.e. Bi = Ig for i = 1, . . . , s. This di↵erence is substantial in that, in the Lanne and
Lütkepohl’s model, once re-scaled, the impulse response functions (IRFs) do not change
across the di↵erent volatility regimes, while in our specification, they can change according
to the di↵erent further transmission channels provided by the Bi matrices.⌅
Structural breaks in the dynamics of the VAR model
The DGP in Eq. (1) is characterized by constant dynamics and regime-specific covariance matrices ⌃i . It might be the case, as for instance in the empirical analysis in Section
III, that the VAR model is not stable over time, requiring regime-specific dynamics as well.
In this latter case all the reduced-form parameters will be regime-specific and, similarly
to the structural parameters, denoted by Di , A1i , . . . , Api , ⌃i , i = 1, . . . , s. This case, that
will be discussed in more details in the empirical analysis, does not alter the structural
characteristics of the model previously discussed. Under the assumption of known break
dates, the estimation of the reduced-form parameters can be performed separately in each
regime.
In the opposite situation, when there are no changes in both the dynamics and the
structural parameters of the model, i.e. Bi = B and ⇤i = Ig , the model reduces to
the traditional AB-SVAR model in the terminology of Amisano and Giannini (1997) and
Lütkepohl (2005).
Unit roots and Cointegration
If some of the variables appear to be non-stationary, it would be more convenient to
move to the Vector Error Correction (VECM) notation
yt = Ddt + ↵ 0 yt
1
+
yt
1
1
+ ··· +
p 1
yt
p+1
+ ut
(4)
where
is the di↵erencing operator, the full column rank g ⇥ r matrices ↵ and
are
the adjustment coefficients and the cointegration matrix, respectively, r is the number of
cointegrating relations, while j = (Aj+1 + . . . + Ap ), for j = 1, . . . , p 1.
The structural form of the model easily becomes
A yt = D ⇤ d t + ↵ ⇤ 0 yt
1
+
⇤
1
yt
1
+ ··· +
⇤
p 1
yt
p+1
+ Bi (t 2 Ti ) "t
(5)
where D⇤ = AD, ↵⇤ = A↵, ⇤i = A i and where E ("t "0t ) = ⇤i , for t 2 Ti . Furthermore, as
for the simple VAR model discussed before, if the structural breaks are not confined to the
second moments but involve the dynamics of the model as well, under the assumption of
known break dates, the reduced-form VECM can be studied separately in each regime.
Interdependence and Contagion
The specification discussed above, thus, allows to distinguish between relationships that
remain constant over the whole sample, and others that are regime-specific. Going back to
the core of the present paper, this is well suited to model interdependence and contagion
of financial crises. In particular, the interdependence relations among markets, that remain
constant both in tranquil and turbulent periods, can be captured through the A paramet-
6
ers, while the further relationships among the shocks, generally appearing during turmoil
periods, characterizing the pure contagion phenomenon, are described by the Bi matrices.
The transmission of shocks, thus, is much more complicated than in standard SVAR
models and is given by the combination of a) the natural linkages between two or more
markets (interdependence) and b) the e↵ect of contagion operating in each specific regime.
This point will be largely discussed in the empirical analysis in Section III.
II.2
Identification
As is known, all SVAR models su↵er from identification problems, which are generally solved
by imposing restrictions on the structural parameters. Our specification is not immune
to such a problem, although the heteroskedasticity in the data may solve or, at least,
alleviate it. In fact, as originally proposed by Rigobon (2003b) the di↵erent clusters of
volatility provides a useful source of information that can be used in the identification of
the parameters of the structural form of the model.
The approach we follow in the present paper is to mix the heteroskedasticity present in
the data with the traditional approach consisting of imposing restrictions on the parameters.
The A, B1 , . . . , Bs , ⇤1 , . . . , ⇤s parameters, thus, can be subjected to linear restrictions of the
form:
0
1
0
1
0
1
0
a
SA
sA
A
B b1 C
B
C
B
C
B
S
S
·
·
·
S
s
B11
B12
B1s
B
C
B
C
B B1 C
B B1
B b C
B
C
B
C
B s
SB22 · · · SB2s
B 2 C
B
C
B B2 C
B B2
B . C
B
C
B
C
B .
.
.
.
B .. C
B
C
B .. C
B ..
..
..
B
C
B
C
B
C
B
+ B
B
C = B
C
B
C
B bs C
B
C
B Bs C
B s Bs
SBss
B
C
B
C
B
C
B
B 1 C
B
C
B
C
B s⇤1
S
⇤
⇤
1
1
B
C
B
C
B
C
B
B .. C
B
C
B
C
B ..
.
..
@ . A
A
@ .. A
@ .
@
.
S⇤ s
s ⇤s
s
⇤s
✓
g 2 (2s+1)⇥1
=
S
+
g 2 (2s+1)⇥(pA +pB +p⇤ )
(pA +pB +p⇤ )⇥1
s
Assumption 1 The reduced form innovations are distributed as a multivariate normal
variable with time-varying covariance matrix
for
t 2 T Bi
(7)
where,
⌃i = A
1
Bi ⇤i Bi0 A0
1
6
,
i = 1, . . . , s.
(8)
This is the explicit notation for imposing restrictions on the parameters. An equivalent way is represented
by the implicit form R✓ = r where the p = pA + pB + p⇤ columns of the S matrix form a basis for the null
space of the rows of R, so that RS = [0].
7
C
C
C
C
C
C
C
C
C
C
C
C
C
A
g 2 (2s+1)⇥1
(6)
where a = vec A, bi = vec Bi and i = vec ⇤i , while pA , pB = pB1 +. . .+pBs and p⇤ = p⇤1 +
0 , 0 ,..., 0 , 0 ,...,
. . .+p⇤s are the number of free parameters to be estimated in = A
B1
Bs
⇤1
The restrictions in Eq. (6) allow for cross restrictions in the parameters contained in
the Bi matrices. This will be extremely useful in the empirical analysis in Section III, when
some parameters of the Bi matrices are imposed to not change among di↵erent regimes.
The following assumptions formalize some concepts previously introduced:
ut ⇠ N (0, ⌃i )
1
0 6
0
⇤s .
Assumption 2 The g ⇥ g matrices of parameters A, Bi and ⇤i , i = 1, . . . , s, are non
singular.
Assumption 1 describes the distributional aspects of the error terms of the VAR model,
which show a time-varying covariance matrix modeled as a function of the structural parameters characterizing the di↵erent volatility states7 . In the traditional literature on SVAR
models, Assumption 2 refers to the invertibility of the A and B matrices. In our framework,
instead, it is required that all Bi matrices are non-singular.
Throughout, use is made of the following notation: Kg is the g 2 ⇥g 2 commutation matrix
as defined in Magnus and Neudecker (2007), Ng = 1/2 Ig2 + Kg . The g ⇥ g 2 full-row rank
matrix Ug , instead, defined in Magnus (1988), is such that Ug0 w (M ) = vec (M ) with the g⇥1
vector w (M ) = (m11 , m22 , . . . , mss )0 , relating the vector w (M ) to the diagonal elements in
the diagonal g ⇥ g matrix M .
The following proposition presents the necessary and sufficient condition for identification of the structural parameters.
Proposition 1 Consider the SVAR model with s regimes of volatility described in Eq.s (1)(2). Under Assumptions 1-2, then (A, B1 , . . . , Bs , ⇤1 , . . . , ⇤s ) are locally identified if and
only if the following sg 2 ⇥ p matrix
0
2Ng B1⇤ 2Ng A⇤1
..
.
B
@
2Ng Bs⇤
A
..
.
1B
1
⌦A
1B
1
1
Ug
..
.
2Ng A⇤s
A
1B
s
A⇤1 , . . . , A⇤s , B1⇤ , . . . , Bs⇤
has full column rank. The non-singular matrices
lows
A⇤i = A 1 Bi ⇤1 ⌦ A
Bi⇤ = A 1 Bi ⇤i Bi0 A
⌦A
1B
s
Ug
C
AS
(9)
are defined as fol-
1
10
⌦A
1
.
A necessary condition for identification is that sg (g + 1) /2 p, where p = pA + pB + p⇤
represents the number of free parameters in the A, Bi , and ⇤i matrices, with i = 1, . . . , s.
Proof. The proof of Proposition 1 is discussed in the Appendix A.
Remark 2. When all Bi matrices are restricted to be unit diagonal, as already said, the
model in Eq.s (1)-(2) can be reconciled with the Lanne and Lütkepohl’s (2008) approach.
Using a well known result of matrix algebra in which two symmetric and positive definite
matrices can be simultaneously diagonalized by means of common squared matrices and
specific diagonal matrices, they show that simply exploiting the presence of two or more
levels of volatility is sufficient for the structural parameters (the A matrix) to be identified (without the need of parameters restrictions, as instead in the well-known Cholesky
triangularization). However, considering the necessary order condition in our Proposition
1, as well as the discussion in Lanne et al (2010, page 124), simultaneous diagonalization
discussed above is exact only for s = 2 volatility regimes, while for more than two regimes
7
Any other multivariate stochastic variable univocally defined by the first two moments will provide the
same results in terms of identification.
8
it is not necessarily possible. In other words, the model is exactly identified for s = 2,
but overidentified for s 2. If a hypothetical LR test for such overidentifying restrictions
would reject the null, the Lanne and Lütkepohl’s specification provides no alternatives.
The heteroskedastic SVAR model described in Eq. (1)-(2), introducing the Bi parameters,
fills this gap. The more complicated structure of the model, however, necessitates specific
identification rules, provided in the previous Proposition 1.⌅
In practical applications, the necessary and sufficient condition in Eq. (9) can be numerically checked, as suggested in Giannini (1992), using random numbers for the three
vectors A , B , and ⇤ such that the restrictions in Eq. (6) hold8 .
III
The European Debt Crisis
This empirical analysis aims to shed light on the relationships between sovereign bond
yields for some highly indebted EU countries. The data refer to 10-year bond maturity
yield spreads between the so called ‘PIIGS’ countries (Portugal, Italy, Ireland, Greece, and
Spain) versus Germany, used as benchmark, since German bonds have maintained their
benchmark status and have continued to display lower yields even during both the financial
and debt crises. We consider weekly observations over the period January 2005 - March
2014. All the series come from Datastream.
The investigated period covers many interesting events characterizing the recent European
and world-wide history, such as the global financial crises in 2007 and 2008, the 2008-2009
Spanish financial crisis, the great fear for the Greece default in 2010. All these events
have led instabilities and tensions on the financial markets, which raised the problem of
high public deficits and debt sustainability for the EU member states. Furthermore, given
the strong interconnections between the markets, financial shocks in one country are likely
propagated to other markets. Moreover, following Forbes and Rigobon (2002), Dungey et al
(2010,2015) and Caporin et al (2013), such mechanisms of propagation are di↵erent during
tranquil or turbulent periods. It becomes fundamental, thus, to distinguish between a) the
“natural” interconnections between financial markets, that we indicate as interdependence,
from b) the propagation of financial crises hitting one or more countries, that constitutes
the pure or shift contagion phenomenon.
As thoroughly discussed in the paper, such distinction, from an econometric point of
view, leads to two serious problems of identification: The distinction between contagion and
interdependence from one side, and the possible double causality between the two (or more)
markets investigated. If the trend of market A is important in explaining the trend of market
B, it could be possible that also the contrary holds. This clearly conflicts with the traditional
theory of identification in simultaneous equation systems and SVARs9 . The model we have
developed in this paper starts from the idea that the heteroskedasticity present in the data
can provide new information to solve the identification problem. On the other hand, as we
will discuss here below, the presence of di↵erent clusters of volatility cannot be excluded
given the financial pressures characterizing the markets in the investigated sample.
The model in Eq.s (1)-(2), and all the related results for identification, reveal to be well
8
A Gauss 13.0 package for checking for identification and estimating the unknown parameters of the
heteroskedastic SVAR model developed in this paper can be obtained from the author upon request.
9
The problem has been circumvented by Favero and Giavazzi (2002) by imposing restrictions on the
dynamic part of the model, leading the contemporaneous relationships unrestricted.
9
suited for distinguishing among interdependence and contagion, measured by the A and Bi
matrices respectively.
III.1
Stylized facts and historical events
In Figure 1 we show the interest rates (left panel) and spreads (right panel) series for the
sample period. From both graphs it emerges that the first years of the sample, at least up
to the first signals of the global financial crisis, i.e. the collapse of the U.S. housing bubble
and the consequent rise in interest rates in the second half of 2007, the interest rates for
all EU countries followed practically the same almost constant trend. Since that period,
and up to the almost overall recognized end of the global financial crisis in the late 2008,
the EU interest rates started to rise and highlighted positive spreads with respect to their
benchmark, the German Bund.
At the end of the global crisis, such di↵erentials are in the order of 3 percent for Greece
and Ireland, and 1.5 percent for Italy, Spain and Portugal. During the 2009 a moderate
realignment appeared, but the situation became critical since the beginning of 2010, when
the financial crisis turned into an even more dangerous debt crisis. Such debt crisis was
mostly centered on events in Greece, where there was concern about the rising cost of
financing government debt.
The global financial crisis, however, had contributed to transform other EU countries
into fertile ground for financial, economic and social instabilities to occur. Such weak economic and financial conditions acted as a trigger for the rise in the interest rates di↵erentials
realized in the markets since the beginning of 2010. Idiosyncratic policy interventions pursued by the National Governments, associated to wider rescue remedies proposed by the
EU and IMF, seem to have only moderate and transitory e↵ects on the situation of the
financial markets, that up to the end of the sample continue to register incredibly high
spreads with respect to Germany, leading to serious problems of sustainability of the public
debt for those countries.
The objectives of this section are twofold: First, we want to estimate the simultaneous
relationships between the interest rates of these countries measured by the A matrix in
our general model specification. The complicated economic and financial interconnections
between all these countries do not allow to follow any economic theoretical framework,
suggesting thus to estimate such matrix unrestrictedly. Second, the global and idiosyncratic
financial crises suggest to estimate di↵erent mechanisms of propagation of shocks in periods
of high volatility regimes as described by the Bi matrices in our formulation.
In Table 1 we report the di↵erent covariance and correlation matrices among the spreads
over di↵erent horizons in the sample. Among the di↵erent sub periods described above, very
di↵erent values for the variances and covariances among the spreads clearly emerge. Apart
for the first period, characterized by stable interest rates and spreads, for all the other
periods the correlations between the spreads are high and generally above 0.9.
10
III.2
Interdependence and contagion
We consider that the structural form of the model is given as follows
0
B
B
B
AB
B
@
spt
irt
ptt
grt
itt
1
get
get
get
get
get
0
C
B
C
B
C
B
C = c+ (L) B
C
B
A
@
spt
irt
ptt
grt
itt
get
get
get
get
get
1
C
C
C
C+⇥ (L)
C
A
V ixxt 1
Baat 1 Aaat
1
!
+Bi
0
B
B
B
(t 2 Ti ) B
B
@
(10)
where spt get , irt get , ptt get , grt get , and itt get are the interest rate spreads
for Spain, Ireland, Portugal, Greece, and Italy, respectively. Baat Aaat is the spread
between BAA and AAA corporate bonds and V ixxt measures market expectations of near
term volatility conveyed by stock index option prices. Both indicators represent exogenous
variables that could play a relevant role in the explanation of EU spreads. (L) and ⇥ (L)
are two matrix polynomials in the lag operator L, while c is a vector of constant terms.10
A and Bi , i = 1, . . . , s are the matrices of interest and measure the interdependence and
contagion relationships, respectively. The "’s are the idiosyncratic shocks, and are assumed
to be uncorrelated and with regime-specific variances given by ⇤i , i = 1, . . . , s. As before,
Ti collects all the time periods such that regime i is in place, and (·) is the indicator
function.
The reduced form of the model is trivially obtained by premultiplying both sides of Eq.
(10) by the invertible matrix A
0
B
B
B
B
B
@
spt
irt
ptt
grt
itt
get
get
get
get
get
1
C
C
C
C=A
C
A
1
c+A
1
0
B
B
B
(L) B
B
@
spt
irt
ptt
grt
itt
get
get
get
get
get
1
C
C
C
C+A
C
A
1
⇥ (L)
V ixxt 1
Baat 1 Aaat
1
!
+ ut
(11)
where the reduced-form residuals ut satisfy
0
B
B
B
AB
B
@
usp
t
uir
t
upt
t
ugr
t
uit
t
1
0
C
B
C
B
C
B
C = Bi (t 2 Ti ) B
C
B
A
@
"sp
t
"ir
t
"pt
t
"gr
t
"it
t
1
C
C
C
C
C
A
,
E "t "0t = ⇤i
for
t 2 Ti
(12)
that continue to share the same interdependence-contagion relationships as the original
model in Eq. (10). Without any constraint on the parameters of the predetermined variables, maximizing the likelihood for Eq. (10) is equivalent to maximizing the concentrated
likelihood in Eq. (12).
10
Originally, other explanatory variables, like US and EU stock price indices, Euro/US Dollar exchange
rate, EU and US short term interest rate, have been included as potential exogenous regressors. However,
the explanatory power was so poor that an F test strongly suggested to exclude all these variables from the
analysis.
11
"sp
t
"ir
t
"pt
t
"gr
t
"it
t
1
C
C
C
C
C
A
The reduced form and the structural breaks
The reduced form in Eq. (11) can be seen as a standard VAR model. The residuals ut
depend on the structural matrices A, Bi and ⇤i , but also on the volatility regime dates.
The first point, thus, concerns the determination of the regimes.
The recent events concerning the global financial crisis and the EU debt crisis provide
a natural framework to define the regimes. As mentioned before, these events have been
associated with large and persistent increases in volatility. Since June 2007 the five countries
experienced global and idiosyncratic shocks that allow us to distinguish, country by country,
tranquil from turbulent periods.
In Spain, the first and strong signals of instabilities appeared even before the ‘official’
start of the global financial crisis. During the second half of 2007, when the real estate
bubble burst, the crisis immediately overcame the whole banking system that, although
credited as one of the most solid and best equipped among all Western economies to cope
with the worldwide liquidity crisis, strongly relaxed his strict requirements from intending
borrowers during the housing bubble, o↵ering up to 50-year mortgages.
As for the Spanish case, the Irish crisis was triggered by the ‘terrible’ mix of a real
estate bubble from one side, and over-exposure of many large banks that financed the
property market, from the other side. The situation for the banking sector became critical
in September 2007, with the explosion of the global financial crisis.
The recent historical events were substantially di↵erent for Greek and Portugal. Given
the limited exposure of these countries with respect to international financial markets, they
were only marginally touched by the global financial crisis. The Portuguese financial crisis
was mainly an economic and political crisis and started during the first weeks of 2010.
In Greece, instead, in only few days the situation became completely out of control.
Facing the growing increase of the public debt, on April 23 the Greek government asks
an initial loan of 45 billion euro to the EU and IMF to cover its financial needs for the
remaining part of 2010. Only few days later Standard & Poor’s decided to relegate the
sovereign debt rating to ‘junk’.
The Italian case is completely di↵erent. As is well-known, Italy has the largest Debt/GDP
ratio amongst the major European countries, around 115% at the end of 2010. However,
Italy is not growing it faster than its neighbors. The government deficit is small, and the
country has a best in class primary balance. Nevertheless, unstable and weak political conditions, as well as the uncertainty due to the large amount of debt that has to be rolled over
each year, by August 2011 the spread between Italian ten-year bonds and their German
counterparts reached alarming levels, pushing ECB President and Bank of Italy Governor
to writing a joint letter to Italian Prime Minister calling for ‘immediate and bold’ measures
to promote growth.
The combination of these events indicates approximately s = 3 di↵erent volatility regimes: quiet until mid 2007, crises in Spain and Ireland until early 2010, and then crisis
everywhere. More precisely, we take as break dates a) June 2007, that is generally taken
as the first signal of the crisis in Spain and b) the end of January 2010, when the Portuguese government announces strong measures to reduce the budget deficit, raised up to
a worrisome 9.4%. These dates are clearly indicative, although warmly suggested by the
hystorical events and by the data reported in Figure 1. A robustness analysis, however, will
be conducted in order to confirm the main empirical findings.
Although the attention is mainly paid on the di↵erent volatility regimes, a special atten12
tion must be paid on the stability of the dynamics of the VAR. A set of statistical tests will
be thus performed in order to verify, firstly, whether the break dates discussed before are
e↵ectively structural breaks, and secondly whether such breaks are confined to the volatility
of the residuals or involve the dynamics too.
We first consider the VAR model without any break. A joint analysis of information
criteria and specification tests on the residuals suggests to include 6 lags for both the ⇥
and polynomials.
We then account for the structural breaks on the covariance matrices only. A (quasi)LR test for the null that there are no breaks against the alternative of two breaks on
the covariance matrices can be given by LR = 2 [372.51 (1219.81 + 668.51 213.45)] =
2604.71, which suggests to strongly reject the null with a p value = 0.000 (taken from
a 2(30) distribution). Moreover, when considering a completely unrestricted model, i.e.
di↵erent dynamics and di↵erent covariance matrices, and compare it with a model with
constant dynamics but di↵erent covariance matrices, the (quasi-)LR test becomes LR =
2 [1674.87 2806.32] = 2262.89 that continues to reject the null with a p value = 0.000
(taken from a 2(310) distribution). We thus find formal support that the two breaks involve
both the dynamics and the volatility of the VAR model.
The analysis of the structural form of the model, thus, will be focused on regime-specific
VAR models, each of which written as in Eq.s (1)-(2). In Figure 2 (middle panel) we report
the residuals of the estimated reduced forms, as well as the fitted and actual series of the
spreads11 (left panel). In the right panel, instead, we report the absolute values for the
reduced form residuals, which clearly highlight the presence of di↵erent volatility clusters.
The structural form
The structural form of the model in Eq.s (10)-(12) does not meet the identifying conditions reported in Proposition 1. More precisely, focusing on the concentrated version in
Eq. (12), the parameters to estimate are 20 for the A matrix (we impose unit values on the
main diagonal), 75 for the Bi matrices and 15 for the diagonal ⇤i matrices, i = 1, . . . , 3.
The number of empirical moments provided by the covariance matrices of the residuals ⌃i ,
i = 1 . . . 3, are instead 3g (g + 1) /2 = 45. The necessary condition in Proposition 1 states
that 20 + 75 + 15 45 = 65 further restrictions must be imposed.
First of all, given the nature of the first regime, characterized by the absence of significant turbulences on the financial markets, we suppose that the transmission of shocks takes
place only through the interdependence among the markets. The first set of restrictions,
thus, is B1 = I5 (25 restrictions). The other two regimes, instead, are characterized by
strong turmoils that, as largely discussed above, hit the financial markets with a well determined order; in sequence, Spain, Ireland, Portugal, Greece and, finally, Italy. This series
of historical events allows to restrict the B2 and B3 matrices to be lower triangular (with
unit values on the main diagonal), where the order of the variables respects the one just
mentioned (which corresponds to what indicated in the representations in Eq.s (10)-(12)).
11
As said above, the standard steps for the correct specification of the VAR models in each regime suggest
to include 6 lags for both the ⇥ and
polynomials. The included dynamics, however, reveals to be not
sufficient for cleaning the residuals in terms of departure from normality, especially for the excess of kurtosis.
Increasing the number of lags does not help to solve the problem. All estimates (and inference), thus,
should be referred to ‘quasi’-ML (and ‘quasi’-LR tests). As expected, the residuals show strong signals
of heteroskedasticity, that justifies the “identification through heteroskedasticity” strategy pursued in the
paper. All results are available upon request.
13
Moreover, we suppose that B2 and B3 do not change across the two regimes, i.e. B2 = B3 .
The diagonal covariance matrices ⇤1 , ⇤2 and ⇤3 , instead, are left free to vary across the
three volatility regimes.
This set of restrictions allows the model to meet both the order (necessary) and rank
(necessary and sufficient) conditions discussed in Proposition 1. More precisely, the model
is exactly identified, with 45 parameters to estimate, and 45 empirical moments.
Estimated results
The estimated parameters, with associated standard errors, are reported in Table 4.
Column (1) reports the estimates (and related standard errors) of A, Bi and ⇤i where some
highly insignificant coefficients are set to zero. Interestingly, this new procedure allows
to consistently estimate the parameters of the contemporaneous relationships among the
endogenous variables (the A matrix) without imposing exclusion restrictions as generally
required in the traditional approach for the simultaneous equation models and SVARs.
Di↵erently to standard SVARs, the proposed specification allows the identification and estimation of the elements in the Bi matrices, accounting for the propagation of the structural
shocks during turbulent periods. The (quasi-)LR test statistic, with associated p-value, for
such overidentifying restrictions, is reported in the last row of the table. The test suggests
to not reject the null hypotheses for all standard significant levels. Interestingly, the test
is not simply a test for the overidentifying restrictions, but represents a test for the entire
structure of the model, too. In other words, the test does not reject the lower-triangularity
and equivalence of B2 and B3 .
For a more comprehensible reading of the results we show, in the following equation,
the estimated A and Bi parameters in a matrix notation, as reported in Table 4:
0
B
B
B
B
B
@
1
1.294
1.141
1
0.226 0.102
0.109
0
0.585 0.084
0.331
0.215
1
0.345
0.558
0
B
B
B
+B
B
@
0
B
B
B
+B
B
@
0.046
0.025
0.454
1
0
1.760
0.949
0.441
1.206
1
10
CB
CB
CB
CB
CB
A@
usp
t
uir
t
upt
t
ugr
t
uit
t
1
0
0.380
1
0.119
0
0.156
0
0
0.763
0
0
0
0
1
0
0.959
1
0
0.260
0
0
0
0
1
1
0
0.380
1
0.119
0
0.156
0
0
0.763
0
0
0
0
1
0
0.959
1
0
0.260
0
0
0
0
1
1
C 
C
C
C = I5 (t 2 T1 ) +
C
A
1
C
C
C
C (t 2 T2 ) +
C
A
1
C
C
C
C (t 2 T3 )
C
A
0
B
B
B
B
B
@
(13)
"sp
t
"ir
t
"pt
t
"gr
t
"it
t
1
C
C
C
C
C
A
where the three covariance matrices of the structural shocks, together with those of the
residuals of the VAR model, are reported in Table 2.
The interpretation of the results is as follows. The first equation, for example, can be
14
read as an equation for the Spanish spread, that depends on the other contemporaneous
spreads and the idiosyncratic structural shock "sp
t , that with di↵erent levels of volatility,
hits the Spanish spread during the three regimes. The second equation, instead, is devoted
to explain the Irish spread and, other than depending from the other spreads and the
idiosyncratic structural shock "ir
t , during turbulent periods (second and third regimes), also
depends on the Spanish structural shock "sp
t , through the 0.380 coefficient. This evidence
highlights that during turmoils there is a further channel for the transmission of shocks
that, given the positive sign of the coefficient, can be interpreted as contagion (from Spain
to Ireland). The other equations can be interpreted accordingly.
The first comment highlights that there are bi-directional interdependent relations among
all the spreads, as indicated by the full structure of the A matrix. This result, allowed by
the heteroskedasticity present in the data, confirms that it would have been dangerous to
impose identifying restrictions on the A matrix. Unexpectedly, some of the coefficients
present a positive sign, indicating that the referring spread enters negatively in that specific
equation. In particular, two of these coefficients are surprisingly high (Irish spread in the
Spanish equation and Italian spread in the Irish equation). However, looking at the results
in Table 4, these coefficients are imprecisely estimated and, as shown in the robustness
checks in Section III.3, dramatically reduce when using the first di↵erences of the spreads
instead of the levels.
As can be seen from the coefficients shown in Eq. (13), during the second and third
regimes there are di↵erent episodes of contagion. However, one of the coefficient in the B2
and B3 parameters merits a special attention. Overall, the Greek spread depends, among
others, negatively from the Portuguese spread (interdependence). Furthermore, during
periods of crisis, an unexpected open of the Portuguese spread given by a positive "pt
t
shock, negatively transmits to the Greek spread through the 0.959 coefficient. During
turbulent periods, the negative impact of the Portuguese spread on the Greek one is even
reinforced. Due to identification issues, the imposed restrictions of a common transmission
structure during the global financial crisis and during the EU debt one (B2 = B3 ) is not
rejected by the data. In fact, when imposing some zero restrictions on A, B2 and B3 ,
obtaining thus degrees of freedom for possible LR or Wald tests, common parameters in
B2 and B3 are strongly supported by the data (see the LR test in the last row of Table 4,
columns (1)-(3), according to the di↵erent specification of the VAR model). This result is
in line with Caporin et al (2013), who find that when there is a change in the propagation
mechanism, this appended with the burst of the global financial crisis, and not when passing
from the global financial crisis to the debt one.
Although the transmission mechanism does not change during the global and debt crises,
the volatility of the shocks is much larger during the latter period, as shown by the extremely
di↵erent ⇤2 and ⇤3 matrices. This justifies the much larger volatility of the spreads during
the debt crisis with respect to the global financial one, as reported in Figure 1 and Figure
2.
Overall, the previous results highlight that the di↵erent regimes are characterized by
di↵erent transmission of shocks: through the A matrix during the quiet first regime, through
the combination of the A and the Bi coefficients during turbulent periods. This result,
obtained using the di↵erent volatility regimes featuring the data, is completely new in
the literature of SVAR models. Lanne and Lütkepohl (2008), in generalizing the original
Rigobon’s approach, consider di↵erent volatility regimes. However, as explained in the
Remark 1 in Section II.1, they explain the regimes as di↵erent variances of the uncorrelated
15
structural shocks. In our model, this is given by considering ⇤1 6= ⇤2 6= ⇤3 , but restricting
the Bi matrices as B1 = B2 = B3 = I5 . In other words, they consider the transmission of
shocks to remain the same across regimes. As said before, the novelty in our approach is
to consider possible di↵erences in the transmission of shocks. The Lanne and Lütkephol’s
specification is exactly identified for s = 2 volatility regimes, while becomes overidentified
for more than two regimes. In particular for s = 3 regimes, as in our empirical analysis,
we have g (g 1) /2 degrees of freedom for a formal (quasi-)LR test on the structure of the
model. Estimating our specification with the Lanne and Lütkephol’s restrictions we obtain
a log-likelihood value equal to 2704.35, that when compared to the log-likelihood of the
unrestricted model (with cointegration restrictions) gives LR = 2 [2704.35 2733.10] =
57.49 that is strongly rejected with a p value = 0.000 (taken from a 2(10) distribution).12
The transmission of shocks, in our empirical application is much more complicated than
what described by the Lanne and Lütkephol’s model.
III.3
Robustness
In this section we report a set of robustness checks. First, we consider alternative specifications of the model and di↵erent data. Second, we check the robustness of the results versus
the uncertainty associated to the break dates.
The main results presented in Eq. (13) and in Table 4 - column (1) - are obtained by
considering the levels of the spreads. In fact, if the VARs are correctly specified, Sims et
al. (1990) suggest that the parameters of the reduced form, and thus the residuals, can be
consistently estimated even in the case of non-stationary time series. In Table 4, column
(2), instead, we report the structural parameters for the model estimated with the first
di↵erences of the spreads, instead of the levels. The results are completely in line with
those presented for the main specification. A related point, concerns the presence of unit
roots and, eventually, cointegration among the investigated spreads. The investigation for
the presence of structural breaks has shown that the analysis should be restricted within
each regime. In Table 3 we report the results of the unit root and cointegration analysis
for each regime. The Johansen trace test provides evidence of non-stationarity and, more
precisely, suggests r = 1 cointegrating relation in the first and second regimes13 , and r = 2
in the third. The results of the estimated A and B matrices are reported in Table 4 - column
(3) - and show that accounting for unit roots and cointegration does not substantially alter
those obtained in the previous specifications, i.e. considering the levels of the spreads or
their growth rates. The last column (4) of Table 4, as already said, reports the estimated
coefficients when the Bi matrices are all set to the identity matrix, as in the Lanne and
Lütkepohl’s (2008) specification. Important, the results are strongly rejected by the LR
test (bottom part of the column) with a p-value practically equal to zero.
The previous results have been obtained by using interest rate spreads. The same
analysis has been repeated by using weekly bond returns. Once transformed to excess
12
The complete set of results can be obtained from the author upon request.
In both the first and third regimes the cointegration analysis has been performed by considering a
restricted constant as deterministic term. For the second regime, instead, given the increasing path of all
the spreads emerging from Figure 1 (right panel) a restricted trend seemed preferable. Furthermore, looking
at the trace test reported in Table 3, it seems to suggest r = 3, and as a consequence 2 = 5 3 unit roots
in the system. When imposing these restrictions, however, the system continues to be characterized by
two further roots practically equal to one. The most reasonable solution, thus, is to impose r = 1 (and
thus four unit roots) with a restricted deterministic trend. From here onward, we will include a (restricted)
deterministic trend in the VAR model for the second regime.
13
16
returns, the new series are almost proportional to the first di↵erences of the interest rate
spreads (e.g., the correlation between the excess return for Spanish bonds and the first
di↵erence of the interest rate spread (spt get ) is 0.98). The structural breaks for the
reduced form VAR model are strongly confirmed and the estimated A and Bi matrices are
practically identical.14
As already discussed in the previous sections, the way the breaks are detected might
have an influence on the estimation of the parameters. Di↵erent strategies could be used
in order to endogenously determine the breaks, e.g. through Markov switching models (see
Lanne et al, 2010, for an application to heteroskedastic SVARs) or the general framework
proposed by Qu and Perron (2007). However, given the consolidated sequence of events
characterizing the investigated period, we alternatively focus on whether the results are
robust to small or moderate variations of the break dates. We have thus performed a set
of simulations in which the break dates are randomly generated around the original breaks
(those used in the main empirical analysis). More precisely we use a uniform distribution
to generate breaks up to 4 or 8 weeks before or after the original breaks. For each new
break date we re-estimate the model and perform the LR tests both for the plausibility
of the break and for overidentifying restrictions, in order to compare the new results with
the main empirical findings. The structural form, for simplicity, refers to a VAR model in
levels, only. The results are reported in Table 5, for 1000 replications for each of the two
window widths. In column (1), for facilitating the comparison, we report the corresponding
results obtained in Table 4, column (3). In columns (2) and (3) we show the estimated
coefficients for the 8-16 weeks window width, respectively. Concerning the new break dates,
for both windows widths, the LR test for the constancy of the reduced form parameters
(both the dynamics and the covariance matrices) always rejects the null. Concerning the
overidentifying restrictions imposed in the main analysis, they are rejected only once and
five times for the two window widths, respectively. For each of the two simulation exercises
we report the median, the mean, the first decile and the ninth decile. The results are
clearly in line with those obtained in the main analysis. Overall, the simulation exercises
confirm the presence of the two breaks and, moreover, the results are robust to possible
misspecification of the break dates.
IV
Conclusion
This paper o↵ers two main contributions in the discussion of contagion: methodological and
empirical. Firstly, we have presented a theoretical framework for distinguishing between
interdependence and contagion in the transmission of financial shocks. In particular, we
have proposed a new specification of SVAR models that explicitly allows for di↵erent states
of volatility. The identification problem in the structural form of the model has been
solved by using the information coming from the di↵erent volatility regimes, under the
assumption that some of the structural parameters remain stable over time, in the spirit of
the “identification through heteroskedasticity” framework.
The specification adopted is well suited for distinguishing between interdependence and
contagion in an environment in which the investigated countries can behave alternatively
as originating or a↵ected actors. The endogeneity problem, thus, generally by-passed in the
literature (where a specific financial crisis naturally selects the originating country, leaving
14
The results, not reported here for saving space, are available from the author upon request.
17
all the others as simply a↵ected), is here solved by using the heteroskedastic structural VAR
model discussed in the paper.
Secondly, we have proposed an empirical analysis focusing on the transmission of financial shocks within five highly indebted EU countries; Spain, Ireland, Portugal, Greece
and Italy. The particular specification of the model provides a useful tool for modeling
the higher volatility of the interest rates on sovereign bonds observed during the recent
turbulences on the financial markets all over the world. The results highlight that a) there
are two structural breaks, a↵ecting both the reduced form parameters and the covariance
matrix of the residuals, b) the propagation of financial shocks is di↵erent between tranquil
and turbulent periods, although it does not change across global financial crisis and debt
crisis and c) the volatility of the shocks is much larger during the debt crisis, compared to
that of the global financial crisis.
These findings are robust to di↵erent specification of the VAR model, di↵erent data and
possible misspecification of the break dates.
18
References
Amisano, G. and Giannini, C. (1997), Topics in Structural VAR Econometrics, 2nd edn,
Springer, Berlin.
Arezki, R., Candelon, B. and Sy, A.N.R. (2011), Sovereign Rating News and Financial
Markets Spillovers: Evidence from the European Debt Crisis, IMF Working Paper
11/68.
Bacchiocchi, E. and Bevilacqua, M. (2009), International crises, instability periods and
contagion: the case of the ERM, International Review of Economics 56, 105-122.
Bacchiocchi, E. and Fanelli, L. (2015), Identification in Structural Vector Autoregressive
models with structural changes, with an application to U.S. monetary policy, Oxford
Bulletin of Economics and Statistics, forthcoming.
Caporale, G.M., Cipollini, A. and Demetriades, P. (2005), Monetary policy and the exchange rate during the Asian crisis: Identification through heteroskedasticity, Journal
of International Money and Finance 24, pages 39-53,.
Caporale, G.M., Cipollini, A. and Spagnolo, N. (2005), Testing for contagion: A conditional correlation analysis, Journal of Empirical Finance 12, 476-489.
Caporin, M., Pelizzon, L., Ravazzolo, F. and Rigobon, R. (2013), Measuring Sovereign
Contagion in Europe, NBER Working Paper Series 18741.
Ciccarelli, M. and Rebucci, A. (2007), Measuring Contagion and Interdependence with a
Bayesian Time-varying Coefficient Model: An Application to the Chilean FX Market
During the Argentine Crisis, Journal of Financial Econometrics 5, 285-320.
Ciccarelli, M. and Rebucci, A. (2006), Has the Transmission Mechanism of European
Monetary Policy Changed in the Run Up to EMU?, European Economic Review 50,
737-776.
Cogley, T. and Sargent, T.J. (2005), Drifts and Volatilities: Monetary Policies and Outcomes in the Post WWII U.S., Review of Economic Dynamics 8, 262-302.
De Santis, R.A. (2012), The Euro area sovereign debt crisis: safe haven, credit rating
agencies and the spread of the fever from Greece, Ireland and Portugal, ECB Working
Paper Series 1419.
Dungey, M. and Martin, V.L. (2001), Contagion across financial markets: An empirical
assessment, Australian National University mimeograph.
Dungey, M., Milunovich, G. and Thorp, S. (2010), Unobservable shocks as carriers of
contagion, Journal of Banking & Finance 34, 1008-1021.
Dungey, M., Milunovich, G. and Thorp, S. (2015), Endogenous crisis dating and contagion
using smooth transition structural GARCH, Journal of Banking & Finance 58, 71-79.
Favero, C.A. and Giavazzi, F. (2002), Is the International Propagation of Financial Shocks
Non-Linear? Evidence from the ERM, Journal of International Economics 57, 231246
19
Forbes, K. and Rigobon, R. (2002), No contagion, only interdependence: measuring stock
market co-movements, Journal of Finance 57, 2223-2261.
Giannini, C. (1992), Topics in structural VAR econometrics, Lecture Notes on Economics
and Mathematical Systems, Heidelberg and New York: Springer.
Giordano, R., Pericoli, M. and Tommasino P. (2013) “Pure or Wake-up-Call Contagion?
Another Look at the EMU Sovereign Debt Crisis”, International Finance 16, 131-160.
Harvey, A.C. (1990), The econometric analysis of time series, LSE Handbooks of Economics, Philip Allan, London.
King, M.A. and Wadhwani, S. (1990), Transmission of Volatility Between Stock Markets.
The Review of Financial Studies 3, 5-33
King, M., Sentana, E. and Wadhwani, S. (1994), Volatility and links between national
stock markets, Econometrica 62, 901-933.
Klein, R. and Vella, F. (2010), Estimating a Class of Triangular Simultaneous Equations
Models without Exclusion Restrictions, Journal of Econometrics 154, 154-164.
Kosch, M. and Caporin, M. (2013), Volatility Threshold Dynamic Conditional Correlations: An International Analysis, Journal of Financial Econometrics 11, 706-742.
Lanne, M. and Lütkepohl, H. (2008), Identifying monetary policy shocks via changes in
volatility, Journal of Money, Credit and Banking 40, 1131-1149.
Lanne, M., Lütkepohl, H. and Maciejowska, K. (2010), Structural vector autoregressions
with Markov switching, Journal of Economic Dynamics & Control 34, 121-131.
Lewbel, A. (2012), Using heteroskedasticity to identify and estimate mismeasured and
endogenous regressor models, Journal of Business and Economic Statistics 30, 67-80.
Lütkepohl, H. (2005) New Introduction to Multiple Time Series Analysis, Springer-Verlag,
Berlin Heidelberg.
Magnus, J.R. (1988), Linear structures, Griffin, London.
Magnus, J.R. and Neudecker, H. (2007), Matrix di↵erential calculus with applications in
statistics and econometrics, John Wiley and Sons.
Masson, P (1999), Contagion: Monsoonal E↵ects, Spillovers, and Jumps Between Multiple
Equilibria, In Agenor P.R., Miller M., Vines D. and Weber D. (eds), The Asian
Financial Crisis: Causes, Contagion and Consequences, Cambridge University Press,
Cambridge.
Normandin, M. and Phaneuf, B. (2004), Monetary policy shocks: Testing identification
conditions under time-varying conditional volatility, Journal of Monetary Economics
51, 1217-43.
Prono, T. (2013), The Role of Conditional Heteroskedasticity in Identifying and Estimating
Linear Triangular Systems with Applications to Asset Pricing Models that Include a
Mismeasured Factor, Journal of Applied Econometrics 29, 800-824.
20
Qu, Z. and Perron, P. (2007), Estimating and testing structural changes in multivariate
regressions, Econometrica 75, 459-502.
Rigobon, R. (2002), The curse of non-investment grade countries, Journal of Development
Economics 69, 423-449.
Rigobon, R. (2003a), On the measurement of the international propagation of shocks: is
the transmission stable?, Journal of International Economics 61, 261-283.
Rigobon, R. (2003b), Identification through heteroskedasticity, The Review of Economics
and Statistics 85, 777-792.
Rigobon, R. and Sack B. (2003), Measuring the reaction of monetary policy to the stock
market, Quarterly Journal of Economics 118, 639-669.
Rigobon, R. and Sack B. (2004), The impact of monetary policy on asset prices, Journal
of Monetary Economics 51, 1553-1575.
Sentana, E. and Fiorentini, G. (2001), Identification, Estimation and Testing of Conditionally Heteroskedastic Factor Models, Journal of Econometrics 102, 143-164.
Sims, C., Stock, J.H. and Watson, M.W. (1990), Inference in Linear Time Series Models
with Some Unit Roots, Econometrica 58, 113-144.
Wright, P.G. (1928), The tari↵ on animal and vegetable oils, New York, Macmillan.
21
A
Appendix: Proof of Proposition 1
The parameters of the reduced and structural form are connected through the following
relations:
⌃i = A 1 Bi ⇤i Bi0 A 10 ,
for i = 1, . . . , s.
(14)
These relations form a system of non-linear equations in A, Bi and ⇤i , whose solutions
provide a fundamental indication for the identifiability of the parameters. Di↵erentiating
Eq. (14), for i = 1, . . . , s, gives
A
1
dAA
1
Bi ⇤i Bi0 A
10
A
+A
1
1
dBi ⇤i Bi0 A
Bi ⇤i dBi0 A
10
10
+A
+A
1
1
Bi d⇤i Bi0 A
Bi ⇤i Bi0 A
10
10
dA0 A
+
10
= 0
(15)
Using the property vec (ABC) = (C 0 ⌦ A) vec B the system of equations can be written
✓
◆
✓
◆
✓
◆
1
0
10
1
1
1
1
1
A Bi ⇤i Bi A ⌦ A
vec dA + A Bi ⇤i ⌦ A
vec dBi + A Bi ⌦ A Bi vec d⇤i
✓
◆
✓
◆
1
1
1
1
0
10
+ A ⌦ A Bi ⇤i Kg vec dBi
A ⌦ A Bi ⇤i Bi A
Kg vec dA = 0
that, using the properties of the commutation matrix Kg , becomes
✓
2Ng A
1
Bi ⇤i Bi0 A 10
⌦A
1
◆
✓
1
1
◆
vec dA + 2Ng A Bi ⇤i ⌦ A
vec dBi +
✓
◆
A 1 Bi ⌦ A 1 Bi Ug dw (⇤i ) = 0 (16)
where Ug , already defined above, is such that Ug0 w (M ) = vec M with w (M ) = (m11 , m22 , . . . , mnn )0 .
The Jacobian and the necessary and sufficient condition in Eq. (9) immediately follow.
The necessary-only condition in Proposition 1, instead, refers to the number of free
parameters to be estimated (p) with respect to the number of empirical moments provided
by ⌃i , i = 1, . . . , s. This is easily proved by observing that the number of empirical moments
is given by sg (g + 1) /2. ⌅
22
B
Appendix: Estimation and Inference
In this appendix we turn to the problem of estimating the heteroskedastic SVAR model
in Eq.s (1)-(2), assuming that some sufficient condition for identification is satisfied. We
propose a Full-Information Maximum Likelihood (FIML) estimator that is based on the
maximization of the likelihood function of the structural form of the model. For the specific
case of three regimes used in the empirical analysis, the concentrated log-likelihood function
can be written as:
lT (✓) =
T1
log |A|2
2
T1
T2
T1
2
log |⇤1 |
log A 1 B2
log |⇤1 |
2
2
2
✓
◆
T3
T3
T1
2
10
1
1 ˆ
1
0
log A B3
log |⇤3 |
tr A B1 ⇤1 B1 A⌃1
2
2
2
✓
◆
✓
◆
T2
T3
10
1
1 ˆ
10
1
1 ˆ
0
0
tr A B2 ⇤2 B2 A⌃2
tr A B3 ⇤3 B3 A⌃1
2
2
(17)
ˆ i is the estimated covariance matrix of the VAR residuals in the i-th regime. Maxwhere ⌃
imizing the log-likelihood function in Eq. (17) provides ML estimators for the structural
parameters A, Bi and ⇤i , i = 1 . . . 3. However, if the actual distribution of the "t ’s is
non-normal, as it seems to be in our empirical application, the estimators are of course
quasi -ML, and the same holds for the related inference.
C
Appendix: Tables and Figures
Table 1: Covariances and Correlations across di↵erent sub periods
First regime
16/02/200530/05/2007
T1 = 126
Second regime
06/06/200720/01/2010
T2 = 138
Third regime
27/01/20105/3/14
T3 = 215
sp-ge
0.0006
0.0000
0.0004
-0.0003
-0.0002
0.0849
0.2119
0.1019
0.2229
0.1037
1.3517
0.8012
2.5103
7.0838
1.2422
ir-ge
0.0000
0.0048
0.0020
0.0024
0.0019
0.2119
0.6424
0.2667
0.6020
0.2517
0.8012
4.2626
4.0656
10.1390
0.7882
pt-ge
0.0004
0.0020
0.0050
0.0033
0.0028
0.1019
0.2667
0.1327
0.2755
0.1276
2.5103
4.0656
8.6490
23.3689
2.8558
gr-ge
-0.0003
0.0024
0.0033
0.0048
0.0035
0.2229
0.6020
0.2755
0.6617
0.2641
7.0838
10.1390
23.3689
78.2626
8.3973
23
it-ge
-0.0002
0.0019
0.0028
0.0035
0.0029
0.1037
0.2517
0.1276
0.2641
0.1363
1.2422
0.7882
2.8558
8.3973
1.3592
sp-ge
1
0.0255
0.2462
-0.1541
-0.1579
1
0.9077
0.9599
0.9406
0.9642
1
0.3338
0.7342
0.6887
0.9164
ir-ge
0.0255
1
0.4093
0.5047
0.5211
0.9077
1
0.9135
0.9233
0.8506
0.3338
1
0.6696
0.5551
0.3275
pt-ge
0.2462
0.4093
1
0.6654
0.7317
0.9599
0.9135
1
0.9296
0.9485
0.7342
0.6696
1
0.8982
0.8329
gr-ge
-0.1541
0.5047
0.6654
1
0.9267
0.9406
0.9233
0.9296
1
0.8794
0.6887
0.5551
0.8982
1
0.8142
it-ge
-0.1579
0.5211
0.7317
0.9267
1
0.9642
0.8506
0.9485
0.8794
1
0.9164
0.3275
0.8329
0.8142
1
Table 2: Estimated Covariance matrices: Reduced form (left panel)
and Structural form (right panel).
0.0002
0.0000
0.0000
0.0000
0.0000
0.0000
0.0004
0.0000
0.0000
0.0000
0.0013
0.0017
0.0014
0.0016
0.0013
0.0017
0.0069
0.0027
0.0028
0.0024
0.0461
0.0355
0.0329
0.0883
0.0366
0.0355
0.1044
0.0699
0.1062
0.0307
ˆ1
⌃
0.0000
0.0000
0.0001
0.0000
0.0000
ˆ2
⌃
0.0014
0.0027
0.0027
0.0030
0.0021
ˆ3
⌃
0.0329
0.0699
0.1667
0.1489
0.0318
0.0000
0.0000
0.0000
0.0002
0.0001
0.0000
0.0000
0.0000
0.0001
0.0001
0.0015
0
0
0
0
0
0.0009
0
0
0
0.0016
0.0028
0.0030
0.0074
0.0024
0.0013
0.0024
0.0021
0.0024
0.0026
0.0105
0
0
0
0
0
0.0075
0
0
0
0.0883
0.1062
0.1489
0.8668
0.0779
0.0366
0.0307
0.0318
0.0779
0.0445
0.1755
0
0
0
0
0
0.0743
0
0
0
ˆ1
⇤
0
0
0.0002
0
0
ˆ2
⇤
0
0
0.0011
0
0
ˆ3
⇤
0
0
0.2197
0
0
0
0
0
0.0002
0
0
0
0
0
0.0002
0
0
0
0.005
0
0
0
0
0
0.0035
0
0
0
0.6553
0
0
0
0
0
0.1259
Table 3: Cointegration analysis (standard errors in parentheses).
first regime
16/02/2005-30/05/2007
T1 = 120
Rank
Trace test p-value
0
83.035
0.015
1
38.575
0.548
2
20.857
0.673
3
11.604
0.493
4
4.205
0.394
second regime
06/06/2007-20/01/2010
T2 = 138
Rank Trace test p-value
0
180.490
0.000
1
110.220
0.000
2
57.664
0.001
3
20.731
0.194
4
6.069
0.462
1
es-ge
ir-ge
pt-ge
1
0.337
es-ge
0
ir-ge
(0.099)
0.225
pt-ge
(0.041)
gr-ge
0.570
gr-ge
(0.043)
it-ge
1
third regime
27/01/2010-05/03/2014
T3 = 215
Rank
Trace test p-value
0
111.270
0.000
1
60.654
0.010
2
27.023
0.292
3
12.827
0.386
4
1.923
0.788
1
es-ge
1
0
0.054
ir-ge
0
0
0.345
pt-ge
0
gr-ge
0
(0.028)
(0.098)
0.241
(0.027)
it-ge
2
1
0.560
1
( )
0.398
(0.048)
it-ge
1
0
(0.053)
const
0.073
trend
(0.021)
es-ge
ir-ge
pt-ge
gr-ge
it-ge
2
(1)
p-value
↵1
0.093
es-ge
0.539
ir-ge
(0.242)
(0.322)
0.561
(0.191)
0.325
(0.242)
0.434
0.001
const
(0.000)
↵1
1.236
es-ge
(0.184)
0.417
1.183
pt-ge
(0.260)
gr-ge
1.132
gr-ge
(0.430)
it-ge
0.763
(0.146)
(0.256)
0.202
0.653
exactly
identified
24
↵1
0.070
(0.044)
ir-ge
(0.416)
pt-ge
0.357
(0.466)
0.047
(0.066)
0.146
(0.084)
0.820
(0.191)
it-ge
2
(5)
p-value
0.010
(0.043)
9.402
0.094
4.280
(0.915)
↵2
0.021
(0.009)
0.003
(0.014)
0.047
(0.018)
0.128
(0.041)
0.026
(0.009)
Table 4: FIML estimates of A, Bi and ⇤i .
(1)
VAR levels
(2)
VAR first di↵.
(4)
RigobonLanne-Lütkepohl
param.
SE
-0.205
0.163
-0.229
0.075
0.027
0.071
0.371
0.070
-0.058
0.051
-0.035
0.039
-0.019
0.046
0.091
0.061
0.014
0.039
-0.284
0.060
0.168
0.130
0.195
0.067
-0.035
0.010
-0.014
0.025
-0.163
0.034
-0.121
0.030
-0.713
0.052
-0.545
0.165
-0.554
0.075
-0.914
0.131
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.018
0.001
0.021
0.001
0.013
0.001
0.013
0.001
0.010
0.001
0.028
0.002
0.065
0.004
0.028
0.002
0.074
0.005
0.069
0.007
0.121
0.006
0.259
0.014
0.367
0.019
0.898
0.048
0.299
0.028
SE
param.
SE
param.
SE
0.543
-0.668
0.269
-1.162
0.492
0.074
-0.296
0.081
-0.237
0.077
0.089
0.000
-0.087
0.083
0.108
0.637
0.122
0.634
0.118
0.881
0.450
0.202
1.402
0.837
0.066
0.140
0.070
0.127
0.065
0.000
0.000
0.083
0.000
0.063
0.079
0.261
-0.110
0.073
-0.362
0.252
0.065
-0.264
0.055
-0.196
0.066
0.159
0.000
0.359
0.138
0.149
0.444
0.117
0.519
0.148
0.040
-0.042
0.015
-0.057
0.047
0.031
-0.026
0.025
-0.039
0.030
0.115
-0.321
0.065
-0.426
0.090
0.000
0.000
0.668
-1.136
0.151
-1.816
0.647
0.594
0.474
0.259
1.190
0.594
0.148
-0.449
0.112
-0.474
0.125
0.166
-0.898
0.104
-1.150
0.141
0.215
0.544
0.103
0.351
0.194
0.078
0.279
0.098
0.140
0.082
0.126
0.000
0.061
0.091
0.000
0.000
0.099
0.083
0.000
0.000
0.000
0.180
0.599
0.137
0.849
0.166
0.128
-0.960
0.194
-0.916
0.158
0.000
0.000
0.080
0.218
0.059
0.255
0.075
0.016
0.021
0.003
0.039
0.016
0.008
0.022
0.002
0.030
0.008
0.001
0.014
0.001
0.014
0.001
0.001
0.014
0.001
0.014
0.001
0.002
-0.013
0.002
0.014
0.002
0.052
0.043
0.009
0.102
0.053
0.013
0.072
0.006
0.087
0.017
0.005
0.029
0.003
0.033
0.004
0.006
0.066
0.005
0.070
0.006
0.010
0.072
0.009
0.059
0.009
0.217
0.194
0.042
0.419
0.214
0.047
0.240
0.020
0.273
0.054
0.066
0.444
0.037
0.469
0.052
0.063
0.803
0.063
0.810
0.062
0.054
0.378
0.044
0.355
0.050
LR test for over-identifying restrictions
2
2
2
2
LR
1.255
3.839
1.257
57.485
6 =
6 =
6 =
10 =
(p-value)
(0.974)
(0.922)
(0.974)
(0.000)
Note: The generic ijj . parameter represents the squared root of the variance of the structural
shock in equation j for the i-th regime.
A21
A31
A41
A51
A12
A32
A42
A52
A13
A23
A43
A53
A14
A24
A34
A54
A15
A25
A35
A45
B21
B31
B41
B51
B32
B42
B52
B43
B53
B54
⇤111
⇤221
⇤331
⇤441
⇤551
⇤112
⇤222
⇤332
⇤442
⇤552
⇤113
⇤223
⇤333
⇤443
⇤553
param.
-1.141
-0.226
-0.109
0.585
1.294
0.102
0.000
0.084
-0.331
-0.215
0.345
0.558
-0.046
-0.025
-0.454
0.000
-1.760
0.949
-0.441
-1.206
0.380
0.119
0.156
0.000
0.000
0.000
0.763
-0.959
0.000
0.260
0.036
0.028
0.014
0.014
0.013
0.089
0.067
0.033
0.061
0.071
0.383
0.243
0.483
0.757
0.379
(3)
—
Cointegrated VAR
25
Table 5: Simulation results for the robustness towards the break dates.
(1)
(2)
median ninth
mean
first
median ninth
mean
decile
decile
decile
A21
-1.120
-0.995 -1.334 -2.284
-1.120
-0.977
-1.439
A31
-0.226
-0.208 -0.239 -0.281
-0.239
-0.213
-0.233
A41
-0.110
-0.083 -0.106 -0.155
-0.119
-0.084
-0.134
A51
0.607
0.631
0.609
0.588
0.612
0.665
0.614
A12
1.261
2.676
1.780
1.089
1.269
3.964
2.156
A32
0.108
0.117
0.107
0.100
0.115
0.132
0.125
A52
0.087
0.101
0.088
0.071
0.085
0.102
0.084
A13
-0.324
-0.266 -0.458 -1.062
-0.326
-0.229
-0.540
A23
-0.196
-0.175 -0.195 -0.216
-0.196
-0.158
-0.200
A43
0.343
0.356
0.338
0.293
0.355
0.528
0.376
A53
0.568
0.712
0.585
0.514
0.580
0.684
0.615
A14
-0.048
-0.031 -0.065 -0.185
-0.049
-0.030
-0.089
A24
-0.031
-0.023 -0.032 -0.041
-0.033
-0.026
-0.065
A34
-0.460
-0.379 -0.452 -0.584
-0.492
-0.392
-0.511
A15
-1.761
-1.676 -2.058 -3.180
-1.856
-1.674
-2.239
A25
0.928
1.782
1.253
0.843
0.972
2.508
1.426
A35
-0.450
-0.426 -0.453 -0.501
-0.434
-0.269
-0.385
A45
-1.211
-1.129 -1.199 -1.319
-1.223
-1.126
-1.200
B21
0.384
0.414
0.314
0.111
0.362
0.413
0.288
B31
0.106
0.184
0.113
0.056
0.111
0.187
0.118
B41
0.139
0.236
0.127
0.011
0.136
0.267
0.125
B52
0.770
0.923
0.778
0.549
0.769
0.942
0.770
B43
-0.963
-0.915 -0.956 -0.981
-0.954
-0.893
-0.937
B54
0.266
0.304
0.268
0.245
0.279
0.314
0.277
1
0.035
0.062
0.045
0.032
0.036
0.087
0.052
11
1
0.028
0.039
0.032
0.026
0.028
0.050
0.034
22
1
0.014
0.014
0.014
0.013
0.014
0.015
0.013
33
1
0.014
0.014
0.014
0.014
0.014
0.015
0.014
44
1
0.013
0.014
0.013
0.013
0.013
0.015
0.013
55
2
0.088
0.169
0.116
0.079
0.092
0.218
0.134
11
2
0.069
0.084
0.074
0.067
0.075
0.099
0.081
22
2
0.033
0.053
0.037
0.029
0.034
0.058
0.040
33
2
0.061
0.068
0.061
0.044
0.061
0.079
0.063
44
2
0.071
0.079
0.070
0.063
0.068
0.078
0.070
55
3
0.378
0.724
0.503
0.338
0.379
1.039
0.602
11
3
0.247
0.317
0.273
0.241
0.259
0.389
0.290
22
3
0.488
0.522
0.484
0.451
0.510
0.559
0.500
33
3
0.754
0.785
0.757
0.722
0.747
0.781
0.741
44
3
0.381
0.421
0.380
0.347
0.368
0.414
0.368
55
LR for common reduced form parameters
num. of rejection: 1000
num. of rejection: 1000
LR for overidentifying restrictions
num. of rejection: 1
num. of rejection: 5
Note: Results obtained through 1000 replications for each window width.
true
param.
-1.162
-0.237
-0.087
0.634
1.402
0.127
0.063
-0.362
-0.196
0.359
0.519
-0.057
-0.039
-0.426
-1.816
1.190
-0.474
-1.150
0.351
0.140
0.061
0.849
-0.916
0.255
0.039
0.030
0.014
0.014
0.014
0.102
0.087
0.033
0.070
0.059
0.419
0.273
0.469
0.810
0.355
first
decile
-1.735
-0.285
-0.134
0.591
1.089
0.094
0.074
-0.709
-0.227
0.280
0.512
-0.103
-0.040
-0.511
-2.640
0.830
-0.511
-1.269
0.191
0.067
0.041
0.667
-0.987
0.237
0.032
0.026
0.013
0.013
0.013
0.079
0.066
0.028
0.055
0.064
0.336
0.240
0.443
0.730
0.348
26
Figure 1: Interest rates (left panel) and spreads (right panel), January 2005 - March 2014.
45
ESBRYLD
PTBRYLD
ITBRYLD
45
40
IRBRYLD
GRBRYLD
BDBRYLD
40
es_ge
pt_ge
it_ge
ir_ge
gr_ge
35
35
30
30
25
25
20
20
15
15
10
10
5
5
0
2005 2006 2007 2008 2009 2010 2011 2012 2013 2014
2005 2006 2007 2008 2009 2010 2011 2012 2013 2014
Figure 2: Actual and fitted spreads (left panel), residuals (middle panel) and residuals in
absolute value (right panel), January 2005 - March 2014. Vertical bars: break dates.
5
es-ge
es-ge fitted
2010
ir_ge
ir-ge fitted
0
2010
pt-ge
pt-ge hat
2005
2010
gr-ge
gr-ge fitted
20
2010
it-ge
it-ge fitted
2.5
2005
2010
2010
gr-ge residuals
2005
2005
4
2010
abs pt-ge residuals
2010
abs gr-ge residuals
2
2010
it_ge residuals
2005
1.0
0
2010
abs ir-ge residuals
0.5
2005
1
2005
1.5
pt-ge residuals
2005
5
2010
0.5
2005
1
2005
1.5
ir-ge residuals
0
2005
5.0
2010
-1
0
40
1
abs es-ge residuals
0.25
2005
-1
2005
10
0.75
es-ge residuals
-0.5
0
2005
10
0.5
2010
abs it-ge residual
0.5
2010
27
2005
2010