Empirical Modelling of Contagion: A Review of Methodologies

TL;DR: A review of the existing literature on contagion detection during financial market crises can be found in this paper, where a number of extensions are also suggested which allow for multivarite testing, endogeneity issues and structural breaks.

Abstract: The existing literature promotes a number of alternative methods to test for the presence of contagion during Þnancial market crises. This paper reviews those methods, and shows how they are related in a uniÞed framework. A number of extensions are also suggested which allow for multivarite testing, endogeneity issues and structural breaks.

Summary

1 Introduction

• There is now a reasonably large body of empirical work testing for the existence of contagion during Þnancial crises.
• Many of the methods proposed in these papers are adapted in some form to the current empirical literature on measuring contagion.
• For an overview of the literature see Pericoli and Sbracia (2003) and Dornbusch, Claessens and Park (2000).
• Interpreting the approaches in this way provides a natural ordering of models across the information spectrum with some models representing full information methods and others representing partial information methods.
• Tests are examined in Section 5 and additional methods are canvassed in Section 6.

2 A Model of Interdependence

• The model has its origins in the the factor models in Þnance based on Arbitrage Pricing Theory for example, where asset returns are determined by a set of common factors and a set of idiosyncratic factors representing non-diversiÞable risk (Sharpe (1964); Solnik (1974)).
• Let the returns of three asset markets during a non-crisis period be deÞned as {x1,t, x2,t, x3,t} .
• (1) All returns are assumed to have zero means.

3 Unanticipated Shock Models of Contagion

• The deÞnition of the term contagion varies widely across the literature.
• This deÞnition is consistent with that of Masson (1999a,b,c), who divides shocks to asset markets as either common, spillovers that result from some identiÞable channel, local or contagion, and as shown below that of other approaches, such as Forbes and Rigobon (2002) where contagion is represented by an increase in correlation during periods of crisis.
• The Þrst model discussed is based on the factor structure developed by Dungey, Fry, Gonzalez-Hermosillo and Martin (2002a,b) amongst others, where contagion is deÞned 2Of course, just two of the restrictions in (7) are sufficient for independence of asset markets.
• To distinguish between asset returns in a non-crisis and crisis period, yi,t represents the return during the crisis period and xi,t the return during the non-crisis period.
• The fundamental aim of all empirical models of contagion is to test the statistical signiÞcance of the parameter γ.3.

3.1 Bivariate Testing

• This is usually the situation observed in the data.
• Both situations are valid as both represent evidence of contagion via the impact of unanticipated shocks in (10).
• This assumption is discussed in Section 3.3. in the factor model (10).

3.2 Multivariate Testing

• It is possible to test for contagion in many directions provided that there are sufficient moment conditions to identify the unknown parameters.
• Γi,j, controlling the strength of contagion across all asset markets.
• By combining the empirical moments of the variance-covariance matrix during the crisis period, 6 moments, from the empirical moments from the variance-covariance matrix of the pre-crisis period, another 6 moments, gives 12 empirical moments in total which can be used to identify the 12 unknown parameters.
• A joint test of contagion using the factor model in (2) and (17), can be achieved by comparing the objective function from the unconstrained model, qu, with the value obtained from estimating the constrained model, qc, whereby the contagion parameters are set to zero.
• As before, the test of contagion can be interpreted as testing for changes in both variances and covariances.

3.3 Structural Breaks

• The model given by equations (2) and (18) is based on the assumption that the increase in volatility during the crisis period is solely generated by contagion, that is, γi,j 6= 0∀i, j. However, another scenario is that there is a general increase in volatility without any contagion; denoted as increased interdependence by Forbes and Rigobon (2002).
• This would arise if either the world loadings (λi) change, or idiosyncratic loadings (δi) change, or a combination of the two.
• The latter would arise from increases in the shocks of (some) individual asset markets which are entirely speciÞc to those markets and thus independent of other asset markets.
• Bekaert, Harvey and Ng (2003) adopt a different strategy for modelling structural breaks by specifying time varying factor loadings.
• For a trivariate model (N = 3) that allows for all potential structural breaks in world and idiosyncratic factors, no contagion channels can be tested as the model is just identiÞed.

3.4 Using Just Crisis Data

• IdentiÞcation of the unknown parameters in the factor model framework discussed above is based on using information from both non-crisis and crisis periods.
• There may be a problem for certain asset markets in using non-crisis data to obtain empirical moments to identify unknown parameters, such as for example in the move from Þxed to ßoating exchange rate regimes during the East Asian currency crisis.
• Specifying the factor model in (2) for N = 4 assets, means that there are 4 world parameters and 4 idiosyncratic parameters.
• This suggests that 2 contagious links can be speciÞed and identiÞed.
• 5In general, an allowance for both contagion and structural breaks results in identiÞcation problems if the number of structural breaks entertained is unrestricted.

3.5 Autoregressive and Hetroskedastic Dynamics

• The Þrst consists of including lagged values of the returns in the system.
• When the number of assets being studied is large, this approach can give rise to a large number of unknown parameters, thereby making estimation difficult.
• A third approach is to specify autoregressive representations for the idiosyncratic factors, ui,t.
• In particular, contagion has the effect of causing a structural shift during the crisis period in the conditional covariance by γδ1, and the conditional variance by γ2.
• The inclusion of a GARCH world factor into an N factor model of asset returns provides a parsimonious multivariate GARCH model.

4 Correlation and Covariance Analysis

• Forbes and Rigobon (2002) deÞne contagion as the increase in correlation between two variables during a crisis period.
• As crisis periods are typically characterised by an increase in volatility, a test based on the correlation is biased upwards resulting in evidence of spurious contagion (Forbes and Rigobon (2002), Boyer, Gibson and Loretan (1999), Loretan and English (2000), Corsetti, Pericoli and Sbracia (2003)).8.
• A feature of the correlation applications is that they are based on pair-wise comparisons and thus do not consider potential multivariate analogues of the test.
• To 7Problems in estimating multivariate GARCHmodels are noted by Malliaroupulos (1997), although research on this problem proceeds apace.
• Overcome this problem, a multivariate approach is proposed below based on simple regression equations augmented by dummy variables.

4.1 Bivariate Testing

• To demonstrate the Forbes and Rigobon (2002) approach, consider testing for contagion from country 1 to country 2.
• Ρx, giving the false appearance of contagion.
• To test that there is a signiÞcant change in correlation, the null hypothesis is H0 : νy = ρx, (25) against the alternative hypothesis of H1 : νy > ρx. (26) 9Forbes and Rigobon (2002) in their empirical application, compare the crisis period correlation with the correlation calculated over the total sample period (low volatility period).
• This alternative formulation is also discussed below.

4.2 Alternative Formulation

• In implementing the correlation test in (28), equation (24) shows that the conditional correlation needs to be scaled initially by a nonlinear function of the change in volatility in the asset return of the source country, country 1 in this case, over the pertinent sample periods.
• Another way to implement the Forbes and Rigobon test of contagion is to scale the asset returns and perform the contagion test within a regression framework.
• For the crisis returns the regression equation is given as follows, where the 11This tranformation is valid for small values of the correlation coefficients, ρx and vy.
• This alternative formulation suggests that another way to implement the ForbesRigobon adjusted correlation is to estimate (29) and (30) by OLS and test the equality of the regression slope parameters.

4.3 Relationship with Unanticipated Shock Models

• Interpreting the Forbes-Rigobon contagion test as a Chow test provides an important link connecting this approach with the unanticipated shocks model discussed in the previous section.
• That is, the low volatility period is deÞned as the total sample period and not the pre-crisis period.
• These variables are initially extracted from the asset returns data by regressing the returns on the chosen set of world factors and using the residuals form these regressions in the contagion tests given in (24) to (28).
• In conducting the contagion tests, the analysis is performed in pairs with the source country changing depending on the hypothesis being tested.
• An implication of the approach though is that it requires switching the exogeneity status of the variables, an issue that is discussed further below.

4.4 Multivariate Testing

• The regression framework developed above for implementing the Forbes and Rigobon test suggests that a multivariate analogue can be easily constructed as follows.
• Rigobon (2003b) suggests an alternative multivariate test of contagion.
• To highlight this point, consider the following bivariate factor model based on the Þrst two equations in (2) and (10).
• In implementing the DCC test, the covariance matrices employed tend to be conditional covariance matrices if dynamics arising from lagged variables and other exogenous variables are controlled for.
• The advantage of working with VAR residuals, as compared to structural residuals, is that the VAR represents an unconstrained reduced form, thereby circumventing problems of simultaneity bias.

4.5 Endogeneity Issues

• The potential simultaneity biases arising from the presence of endogenous variables are more evident when the Forbes and Rigobon test is case in a linear regression framework.
• The problem of simultaneity bias is the same whether the endogenous explanatory variables are scaled or not.
• To perform the Forbes and Rigobon contagion test while correcting for simultaneity bias, equations (40) and (41) need to be estimated initially using a simultaneous equations estimator and the tests of contagion performed on the simultaneous equation estimates of γi,j in (43).
• Corresponding to each sample period, in each VAR there are 4 parameters associated with the lagged variables which are used to identify the 4 structural parameters.
• This choice of instruments is an extension of the early suggestions of Wald (1940) and Durbin (1954).

5 Models of Asymmetries and Nonlinearities

• The motivation of these approaches is that the transmission processes across asset markets may be nonlinearly different during periods of extreme returns than during normal times.
• In these models contagion arises when signiÞcant relationships across asset markets are detected during periods of extreme movements.
• The underlying differences in the proposed approaches lie in the ways that extreme observations are modelled.
• Four models of asymmetries are now outlined.

5.1 Outliers

• Favero and Giavazzi (2002) use a VAR to control for the interdependence between asset returns, and use the heteroskedasticity and nonnormalities of the residuals from that VAR to identify unexpected shocks which may be transmitted across countries and hence considered contagion.
• The Favero and Giavazzi (2002) approach is very similar to the Forbes and Rigobon (2002) correlation test as both tests are based on testing the signiÞcance of dummy variables in an augmented model.
• Thus the Eichengreen et al (1995, 1996) approach can be viewed as focussing on the change in the strength of the correlation during crisis periods.
• The latter is somewhat similar to Bae, Karolyi and Stulz (2003) concept of co-exceedances involving contemporaneous extreme returns events in both countries.

6.1 Principal Components

• Principal components provide an alternative way to identify factors; examples include Calvo and Reinhart (1995) and Kaminsky and Reinhart (2001).
• The principal components are based on an eigen decomposition of either the variance-covariance matrix or the correlation matrix, with the principal components computed as the eigenvectors associated with the largest eigenvalues.
• Thus, each computed principal component represents a weighted average of individual asset returns.
• This assumption is unlikely to be appropriate when using high frequency asset returns data, especially estimated over a sample containing Þnancial crises where volatilities may change over time.
• One solution is to use a dynamic factor approach (Mody and Taylor (2003)), whilst a more general approach is to use the extended factor model discussed in Section 3.

6.2 Multiple Equilibria

• An important feature of theoretical models of contagion is that they yield multiple equilibria (Dornbusch, Park and Claessens (2000)).
• This suggests that the underlying distribution is multimodal in general where the modes correspond to stable equilibria and the antimodes correspond to the unstable equilibria.
• Jeanne and Masson (2000) adopt this strategy by employing Hamilton s Markovian switching model (Hamilton (1994)), which is equivalent to (70) with a time-varying weighting parameter, φt, based on a Markovian updating formula; see also Masson (1999c) for a discussion of the approach.
• 25 Pesaran and Pick (2003) show that the class of models that incorporate binary variables can generate multiple equilibria.
• They consider a model that is equivalent to the Favero and Giavazzi (2002) which uses (55) in the case of one outlier.

6.3 Spillovers

• In particular, transmissions through an identiÞed channel such as fundamental variables or Þnancial streams are more consistent with the concept of spillovers in the terminology of Masson (1998,1999a,b).
• In a similar vein van Rijikghem andWeder (2001) consider Þnancial ßows.
• A related channel of contagion is information ßows and investor preferences.
• Empirical work on this stream of research is limited to calibration and simulation experiments, due to the obvious lack of data.

6.4 Multiple Classes of Assets

• The majority of the existing literature on contagion considers transmissions across geographical borders for a particular asset market, although one important exception to this is the relatively large literature discussing joint banking and currency crises such as Kaminsky and Reinhart (1999) and Bordo and Eichengreen (1999).
• Þnd evidence of contagion from equity to currency markets in the East Asian crisis.
• Another approach for modelling the strength of interactions between markets is to determine if Þnancial assets are priced using the same stochastic discount factor.
• This could potentially increase the complexity of the modelling problem, and result in issues of dimension, these issues are discussed in an earlier version of this paper.

7 Conclusions

• This paper has overviewed a number of the important tests for the presence and characteristics of contagion in Þnancial markets in the current literature.
• Using an overarching framework of a latent factor model, similar to that proposed in the Þnance literature, the different test methodologies are shown to be related.
• The Þve tests of contagion speciÞcally considered in the paper were Þrst the latent factor framework developed by the current authors, and similar to that of Corsetti, Periocoli and Sbracia (2001) and Bekaert, Harvey and Ng (2003) in which testing for contagion is a test on the parameter γ.
• In a companion paper to this one the authors address the issues of time zones, data frequency, missing observations and endogenous deÞnitions of crisis periods.
• In addition, issues associated with the practical implementation of the tests described here are discussed.

Full text

WORKING PAPER NO 8
EMPIRICAL MODELLING
OF CONTAGION:
A REVIEW OF METHODLOGIES
MARDI DUNGEY, RENÉE FRY,
BRENDA GONZÁLEZ-HERMOSILLO,
VANCE L. MARTIN
The Working Paper is intended as a means whereby researchers’ thoughts and findings
may be communicated to interested readers for their comments. The paper should be
considered preliminary in nature and may require substantial revision. Accordingly, a
Working Paper should not be quoted nor the data referred to without the written consent
of the author. All rights reserved.
© 2003 Mardi Dungey, Renée Fry, Brenda González-Hermosillo and Vance L. Martin
Comments and suggestions would be welcomed by the authors:
e-mail: mardi.dungey@anu.edu.au

Em pirical M odelling of Con tagion: A Review of
M ethodologies
∗
Mardi Dungey
+%
, Renée Fry
+
,
Brenda González-Hermosillo
∗
and Vance L. Martin
#
+
Australian National Univ ersity
%
CERF, Cam bridge Universit y
∗
In ternational Monetary Fund
#
University of Melbourne
October 2003
Abstract
The existing literature promotes a n umber of alternative methods to test for
the presence of contagion during Þnancial market crises. This paper reviews those
methods, and shows how they are related in a uniÞed framework. A number of
extensions are also suggested which allow for multivarite testing, endogeneity
issues and structural breaks.
Key words: Contagion, Financial Crises.
JEL Classification: C15, F31
∗
This project was funded under AR C large gran t A00001350. We are grateful for commen ts from
David Cook, Jon Danielsson, Amil Dasgupta, Barry Eic hengreen, Charles Goodhart, P aul Masson,
Hashem Pesaran, Andreas Pick, Roberto Rigobon, Hyun Shin, Demosthenes Tambakis and partici-
pan ts at the Warwick Summer Workshop 2003. The paper was partly written while Mardi Dungey
w as a Visiting Fellow at CERF. The authors contact details are: mardi.dungey@an u.edu.au; re-
nee.fry@anu.edu.au; bgonzalez@imf.org; vance@unimelb.edu.au.
1

1Introduction
There is now a reasonably large body of empirical w ork testing for the existence of
contag ion during Þnancial crises. A range of different methodologies are in use, mak-
ing it difficu lt to assess the evidence for and against contagion, and particularly its
signiÞcance in transmitting crises bet ween count ries.
1
The origins of current empirical studies of con tag ion stem from Sharpe (1964) and
Grubel and Fadner (1971), and more recently from King and Wadhwani (1990), Engle,
Ito and Lin (1990) and Bekaert and Hodric k (1992). Man y of the m ethods pro posed in
these papers are adapted in some form to the curren t em pirical literature on measuring
con tagion.
The aim of the present paper is to provide a unifying framework to highlight the
key similarities an d differences between the various approac hes. For an overview of the
literature see Pericoli and Sbracia (2003) and Dornb u sch, Claessens and Park (2000).
The proposed framework is based on a latent factor structure whic h forms the basis of
the models of Dungey and Martin (2001), Corsetti, Pericoli and Sbracia (2001, 2003)
and Bekaert, Harvey and Ng (2003). This framew ork is used to compare directly the
correlation analysis appro ach pop u larised in this literature b y Forbes and Rigo bon
(2002), the VAR appro ach of Favero and Giava zzi (2002), the probability models of
Eic h engr een, Rose and Wyp losz (1995, 1996) and the co-exceedance approac h of Bae,
Karolyi and Stulz (2003).
An important outcome of this paper is that differenc es in the deÞnitions used to
test for contagion are minor and under certain conditions are even equivalent. In par-
ticular, all papers are interpreted as w or king from the same m odel, with the differences
stemming from the amo unt of information used in the data to detect contagion. In-
terpreting the approaches in this way pro vides a natural ordering of models across the
information spectrum with some models representing full information methods and
others represen ting partial information methods.
The paper proceeds as follow s. In Section 2 a framework draw n from basic rela-
tionship s between asset retu r n s is used to model returns in a non-crisis environment.
This framework is a ugmented in Sec t ion 3 to give a model which includes an avenue for
contagion during a crisis. Th e relationship between this model and the correlation tests
for con tagion are examined in Section 4 which includes a generalisation of the Forbes
and Rigobon bivariate test to a multivariate en viron m e nt. The remaining non-linear
1
The literature on Þnancial crises themselv es is m uch wider than that can vassed here and is reviewed
in Flood and Marion (1998) while more recent papers are represented by Allen and Gale (2000), Calvo
and Mendoza (2000), Kyle and Xiong (2001) and Kodres and Pritsker (2002).
2

tests are exam in e d in Section 5 and additional meth ods are canvassed in Section 6.
Each of the tests is show n to be a test of the signiÞcance of a slope dumm y. Section 7
concludes.
2 A M odel of I n terdependence
Before developing a model of con tagion, a model of in terdependence of asset mark ets
dur ing non -crisis periods is speciÞed as a latent factor model of asset returns. The
model has its origins in the the factor models in Þnance based on Arbitrage Pricing
Theory for example, where asset returns are determined by a set of common factors and
a set of idiosyncratic factors represen ting non-diversiÞable risk (Sharpe (1964); Solnik
(1974)). Similar latent factor models of contagion are used by Dungey and Martin
(2001), Dungey, Fry, Gonzalez-Hermos illo and Martin (2002a), Forbes and Rigobon
(2002) and Bekaert, Harvey and Ng (2003).
To simplify the analysis, the number of assets considered is three. Exte nd in g the
model to N assetsisstraightforwardwithanexamplegivenbelow.Letthereturnsof
three asset mark ets during a non-crisis period be deÞned as
{x
1,t
,x
2,t
,x
3,t
} . (1)
All returns are assumed to hav e zero means. The returns could be on currencies, or
national equity mark ets, or a combination of currency and equity returns in a partic-
ular coun try or across countries. The follow ing trivariate factor model is assumed to
summarise the dynamics of the three processes during a period of tranquility
x
i,t
= λ
i
w
t
+ δ
i
u
i,t
,i=1, 2, 3. (2)
The v ariable w
t
represents common shocks that impact upon all asset returns with
loadings λ
i
. These shocks could represen t Þnancial shoc ks arising from c h anges to the
risk aversion of international in vestors, or changes in wo rld endowments (Mahieu and
Sc h otm an (1994), Rigobon (2003b)). In general, w
t
represen ts market fundamentals
which determine the ave rage level of asset re t urns across internat i o nal markets during
“norm al”, that is, tranquil, times. Th is variable is commonly referred to as a world
factor, which may or ma y not be observed. For simplicit y, the world facto r is assumed
to be a latent stoc hastic process with zero mean and unit variance
w
t
∼ (0, 1) . (3)
The properties of this factor are extended below to capture ric her dynamics including
both autocorrelation and time-varying volatilit y. Th e terms u
i,t
in equation (2) are
3

idiosyncratic factors that are unique to a speciÞc asset mark et. The con tribution of
idiosyn cra tic shocks to the volatility of asset markets is determined by the loadings
δ
i
> 0. These factors are also assumed to be stochastic processes with zero mean and
unit variance
u
i,t
∼ (0, 1) . (4)
To complete the speciÞcation of the model, all factors are assumed to be independen t
E [u
i,t
,u
j,t
]=0, ∀i 6= j (5)
E [u
i,t
,w
t
]=0, ∀i. (6)
To highligh t the interrelationships amongst the three asset returns in (2) during a
non-crisis period, the covariances are given by
E [x
i,t
x
j,t
]=λ
i
λ
j
, ∀i 6= j, (7)
whilst the variances are
E
£
x
2
i,t
¤
= λ
2
i
+ δ
2
i
∀i. (8)
Expression (7) show s that any dependence bet ween asset returns is solely the result of
the inßuence of common shocks arising from w
t
, that simultaneously impact upon all
markets. Setting
λ
1
= λ
2
= λ
3
, (9)
results in independen t asset markets with all mo vements determined b y the idiosyn-
cratic shocks, u
i,t
.
2
The iden tifying assumption used by Mahieu and Schotman (1994)
in a similar problem is to set λ
i
λ
j
to a constant value, L,foralli 6= j.
3 Unanticipated Shoc k M odels of Con tagion
The deÞnition of the term contagio n varies widely across the literature. In this paper
conta gion is represented by the transm ission of unanticipated local shocks to another
country or market. This deÞnition is consistent with that of Masson (1999a,b,c), who
divides shocks to asset markets as either commo n, spillovers that result from some
identiÞable chann el, local or con tagion, and as sho wn below that of other approaches,
such as Forbes and Rigobon (2002) wh ere con tag ion is represented by an increase in
correlation during periods of crisis.
The Þrst model discussed is based on the factor structure dev eloped b y Dungey, Fry,
Gon zalez-Hermo sillo and Ma rtin (2002a,b) among st others, where con tagion is deÞned
2
Of course, just t wo of the restrictions in (7) are sufficient for independence of asset mark ets.
4

Frequently asked questions

Q1. What are the contributions in "Empirical modelling of contagion: a review of methodologies∗" ?

This paper reviews those methods, and shows how they are related in a uniÞed framework. A number of extensions are also suggested which allow for multivarite testing, endogeneity issues and structural breaks. 

Q2. What have the authors stated for future works in "Empirical modelling of contagion: a review of methodologies∗" ?

Finally, the extreme returns test of Bae, Karolyi and Stulz ( 2003 ) is a further reÞnement of the Eichengreen et al framework, and hence can be similarly cast in a latent factor model and expressed as a test on the parameter γ. Whilst the paper has drawn together many of the existing empirical methods to identify contagion there are many further questions to be addressed. 

Q3. What are the difficulties in modelling transmission across nancial assets?

Some of the difficulties in modelling transmission across Þnancial assets include controlling for different time zone issues, data frequency and volatility structures across both country and asset types. 

Q4. How does the Favero and Giavazzi test work?

In implementing the Favero and Giavazzi (2002) test, the structural model needs to be estimated using a simultaneous equation estimator to correct for simultaneity bias. 

Q5. What is the effect of contagion on the conditional variance and covariance of asset?

In particular, contagion has the effect of causing a structural shift during the crisis period in the conditional covariance by γδ1, and the conditional variance by γ2. 

Q6. What is the attraction of the Eichengreen et al approach?

One of the attractions of the Eichengreen et al (1995, 1996) approach is that it generates probability estimates (Pt) of the spread of Þnancial crises across countries. 

Q7. What is the identifying assumption used by Mahieu and Schotman in this paper?

The identifying assumption used by Mahieu and Schotman (1994) in a similar problem is to set λiλj to a constant value, L, for all i 6= j. 

Q8. What is the implication of the approach?

An implication of the approach though is that it requires switching the exogeneity status of the variables, an issue that is discussed further below. 

Q9. What is the way to capture the properties of two stable equilibria?

In the case of two stable equilibria, these properties can be captured by a mixture distributionf (yi,t) = φf1 (yi,t) + (1− φ) f2 (yi,t) , (70)where 0 < φ < 1 is a parameter which weights the individual densities fi () with means corresponding to the stable equilibria, to form the overall density. 

Q10. What is the determinant of the change in the covariance matrix?

This test is referred to as the determinant of the change in the covariance matrix (DCC) as it is based on comparing the covariance matrices across two samples and then taking the determinant to express the statistic as a scalar.