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© 2013 Nicolaus Copernicus University. All rights reserved.  
http://www.dem.umk.pl/dem 

D Y N A M I C  E C O N O M E T R I C  M O D E L S  
DOI: http://dx.doi.org/10.12775/DEM.2013.004  Vol. 13 (2013) 69−85 

Submitted October 10, 2013  ISSN 
Accepted December 30, 2013 1234-3862 

Milda Maria Burzała*  

Determination of the Time of Contagion  
in Capital Markets Based on the Switching Model 

A b s t r a c t. This article attempts to compare conclusions made about market contagion 
based on the periods indicated by using the Markov-switching model and based on a range for 
unconditional correlations as well as on arbitrary arrangements. DCC-model was used to 
control for correlation change over time. Determination of extremely high correlations by 
using a range for unconditional correlations and the MS(3) switching model yields similar 
results regarding conclusions about the occurrence of the process of contagion in a market. 
Conclusions about contagion are, however, made at a higher significance level in the case of 
the switching model. 

K e y w o r d s: switching model, DCC-GARCH model, contagion.  

J E L Classification: G01, G15, C24. 

Introduction  
 Current economical and financial crises in general have international –
character. Propagation mechanisms across countries and markets are called 
the transmissions for fundamental linkages. In literature contagion term is 
applied only to the financial markets, however it should not be identified 
only with the financial linkages – it can also concern the markets which are 
not significantly financially connected. Many authors claim that increase in 
financial integration intensifies contagion effects. On the subject of interde-

                                                 
* Correspondence to: Milda Burzala, Department of Econometrics, Faculty of Informatics 

and Electronic Economy, Poznan University of Economics, Towarowa 53, 61-896 Poznań, 
Poland,  e-mail: m.burzala@ue.poznan.pl. 



Milda Maria Burzała 

DYNAMIC ECONOMETRIC MODELS 13 (2013) 69–85 

70

pendence between markets, contagion effects and transmission channels 
treat, among others, the works of: Eichengreen et al. (1995, 1996), Goldstein 
(1998), Masson (1998), Kaminsky and Reinhart (2000, 2002), Forbes and 
Rigobon (2002), Pericoli and Sbracia (2003), Pesaran and Pick (2004), 
Dungey et al. (2005).      
 The most restrictive definition provided by the World Bank assumes that 
market contagion occurs when the correlation between markets in a time of 
crisis is significantly higher than during the period of tranquillity1. It is pos-
sible to control for correlation change over time by using, for example, 
a dynamic conditional correlation model. Researchers often also adopt an 
additional definition of contagion that would “suit the purpose of a given 
research method”. If volatility models are used, then contagion is identified 
with the spread of uncertainty across financial markets. 
 The assessment of the significance of a contagion process requires divid-
ing a sample into observations from the time of tranquillity and from the 
time of crisis in financial markets. A period of tranquillity is a benchmark 
period for determining connections between markets, which is a point of 
reference for changes observed during a crisis. The transition from a tran-
quillity period to the period of crisis is usually established based on events 
which may change the behaviour of certain indicators. The results of re-
search studies depend on the division which has been made and the time of 
crisis often covers both high and low correlations between researched mar-
kets. Establishing a potential time of market contagion by using the Markov-
switching model makes it possible to make an assumption about the differ-
ences in a stochastic process that determines correlations in particular  
regimes. The main hypothesis refers to the possibility of using the one-
dimensional Markov switching model to determine the time of contagion in 
capital markets. Results were compared with conclusions made about market 
contagion based on a range for unconditional correlations as well as on arbi-
trary arrangements. The consequences of adopting particular divisions are, in 
fact, important information for researchers.   
 Research results presented in this paper concern the assessment of the 
significance of contagion in certain capital markets in the years 2007–2009. 
Selected stock exchange indices represent the situation in securities mar-
kets2. In empirical studies that are described later, the concept of market 
                                                 

1 Contagion of Financial Crises, World Bank, http://www.worldbank.org/economic-
policy/managing%20volatility/contagion/definitions.htm (14 May 2012). This definition is 
cited based on Forbes and Rigobon’s paper (2002). 

2 Capital market crisis is identified with sharp decline in stock prices, maintaining for an 
extended period of time. Role of stock market indexes is broadly described by Jajuga (2006). 



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71

contagion means a contagion spreading from an index representing the U.S. 
market to an index representing the studied market3.  
 Section 1 presents the DCC-GARCH model and section 2 describes the 
Markov-switching model which has been used in the research. Section 3 
contains information on the tested stock exchange indices as well as the cri-
teria for an arbitrary division of the set of observations into those relating to 
the time of crisis and those relating to the period of tranquillity in securities 
markets. The obtained research results are presented in section 4. 

1. A dynamic Conditional Correlation Model 

 Let us assume that an n-dimensional vector of rates of return ts  
(t = 1,…,T) can be decomposed into the following form: 

,ttt εμs +=  (1) 

,2/1
tt ξHε =   (2)                                                    

where tμ  is the vector of conditional expected values of vector st based on 
model VAR(p). In empirical research it is usually assumed that p =14. The 
Student’s t-distribution was used because of an increased kurtosis for pro-
cess tξ . 
 The dynamic conditional correlation (DCC) model can be formulated as 
(Engle, 2002): 

,tttt DRDH =  (3) 

11, ,( ,..., ),t t NN tdiag h h=D  (4) 

                                                 
3 Causality tests are sometimes used for establishing the direction of contagion (Cheung, 

Ng, 1996; Coporale, Pittis and Spagnolo, 2002). Their usefulness, however, is limited. This is 
because these tests are based on Granger’s concept related to analysing correlations between 
studied processes and the consequences of events. It is often a researcher who decides 
whether to test a particular causal relationship exists based on his or her knowledge and ex-
perience (Osińska, 2006; Fiszeder, 2009).  

4 A vector autoregression model also controls for the mutual interdependence between 
markets through connections between the delayed values of endogenous variables. Empirical 
studies described in literature have found that linear relationship between stock returns are 
low significant. Some researchers suggest that it is better to resign from expected value model 
than include incorrectly specified model, especially in the case of total model for expected 
values and variances (Doman, Doman, 2009). 

 



Milda Maria Burzała 

DYNAMIC ECONOMETRIC MODELS 13 (2013) 69–85 

72

2 2
, , , , ,

1 1
( ( 0) ) ,

1,... ,

q p

i t i ij i t j ij i t j i t j ij i t j
j j

h c I h

i N

α ε γ ε ε β− − − −
= =

= + + < +

=

∑ ∑  (5) 

1/2 1/2( ( )) ( ( )) ,t t tdiag diag− −= tR Q Q Q  (6) 

'

1 1 1 1
1 ,

K L K L

t k l t k t k t k l t l
k l k l
α β α β− − −

= = = =

⎛ ⎞
= − − + +⎜ ⎟
⎝ ⎠

∑ ∑ ∑ ∑Q Q ξ ξ Q  (7) 

Matrix Rt is a positively defined symmetric matrix with ones along the main 
diagonal; vector 1

t t t
−=ξ D ε  in this case denotes the vector of standardised 

residuals from model VAR(1). Matrix Dt was estimated based on the one-
dimensional GJR-GARCH(1,1) model (Glosen, Jagannathan, Runkle, 1993). 
In equation (5) )(⋅I  is an indicator function and it assumes the value of 1 for

0, <− jtiε  and the value of 0 for 0, ≥− jtiε  ( Ni ,...,1= ). Positive values of 

parameter ijγ  which are significantly different from zero prove that the lev-
erage effect occurs5.  
 Covariance stationarity and thus a finite variance in equation (5) is en-
sured by satisfying these conditions:  

1 1
0, , 0 and ( / 2) 1.

q p

i ij ij ij ij ij
j j

c α β α γ β
= =

> ≥ + + <∑ ∑  (8) 

In equation (6) tQ  denotes a square matrix of unconditional covariances of 
the vector tξ  variables. In addition, it is required that , 0,k lα β ≥  

1 1
1

K L

k l
k l
α β

= =

+ <∑ ∑ . 

 The model’s parameters are estimated in two stages. The logarithmic 
likelihood function is the sum of likelihood functions for a volatility model 
and likelihood functions for the parameters of dynamic correlations (Engle, 
2002).  

                                                 
5 The leverage effect results from an asymmetric response of rates of return to positive 

and negative information reaching the financial market. It is a consequence of a negative 
correlation between securities prices and the volatility of rates of return. The higher the value 
of parametr ijγ , the stronger the leverage effect (the additional impact of negative infor-

mation). 



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2. The Markov-Switching Model 

 In Markov-switching models it is assumed that a switch between the 
behaviours of rates of return in regimes (periods), and thus the process of 
contagion in a market, depend on certain hidden factors which are not di-
rectly observable. One can only observe the external symptoms of regime 
change by observing, for example, the mutual correlations between rates of 
return. Theoretically, for a time series of dynamic conditional correlations 

tρ , a one-dimensional switching model can be used, in which switches oc-
cur as a result of changes in the expected value μ , variance σ2 or the ex-
pected value and variance of the studied correlations (Hamilton, 1989; 
Davidson, 2013).  
 If a switching model is only constructed for the purpose of classifying 
the already obtained theoretical values of correlations, it can be assumed 
that, in each regime, values are generated by independent processes with 
a different constant expected value and constant variance:   

,2,1,0),(,0(~)( 2 ==+== iirNir titittit σεεμρ  (9) 

where )( irt =μ , )(2 irt =σ  denote the expected value and variance of con-
ditional correlations, respectively, in the i-regime. Such an approach allows 
one to use a one-dimensional model, in which it is assumed that three re-
gimes will be analysed, i.e. tr  = i (i = 0, 1, 2), which are related to an ex-
tremely low, average and extremely high correlation between rates of return. 
The proposed sequential procedure entails estimating the model of dynamic 
conditional correlations and the switching model separately, which makes it 
possible to avoid many problems related to estimating multidimensional 
models6.   
The series of random variables rt in the subsequent moments in time t 
(t=1,..., T) has the Markov property, i.e. its value at the time moment t+1, 
i.e. rt+1, depends only on the regime at the t moment, rather than on all the 
preceding regimes, which is formally formulated as:  

                                                 
6 In practice, the use of multidimensional switching models is associated with many prob-

lems because of the number of estimated parameters which grows exponentially, as in the 
multidimensional VechGARCH model (Billio, Lo Duca, Pelizzon, 2005). If one assumes that 
only two states (of high and low volatility) can occur in each of two studied markets, than one 
already allows for the occurrence of four regimes, and the matrix of conditional probabilities 
has dimensions [4 x 4]. The final number of parameters depends on the assumptions regarding 
the differences between processes determining the behaviour of rates of return in particular 
regimes.  



Milda Maria Burzała 

DYNAMIC ECONOMETRIC MODELS 13 (2013) 69–85 

74

1 1 1( , ,...) ( ) ,

, 0,1,2.
t t t t t ijP r j r i r k P r j r i p

i j
+ − += = = = = = =

=
 (10) 

Probabilities pij denote the probability of transition of the dependence be-
tween rates of return from regime i to regime j.  
If at the t – 1 moment the process was under the rt-1 = i regime, then the con-
ditional density function of the explained variable tρ  can be represented as 

1( , )t t tf s r i I −= , where 1−tI  denotes the history of the process until the t – 1 
moment. Any suppositions on the i regime may be made by means of a con-
ditional probability: 

,
)(),(

)(),(
)( 2

0 11

11

∑ = −−

−−

=⋅=

=⋅=
==

j ttttt

ttttt
tt

IjrPIjrsf

IirPIirsf
IirP  (11)

  
where 

∑ = −−− === 2
0 111 ).()( j ttijtt IjrPpIirP  (12) 

The model’s parameters are estimated by using the maximum likelihood 
method (Davidson, 2013)7. 

3. The Statistical Material and an Arbitrary Division of the Sample 

 In the empirical research, daily continuously compounded rates of return 
on six indices representing the situation on stock exchanges during the pe-
riod from August 17, 2005 to July 31, 2009 were used (1022 observations 
for each stock exchange): 
  )).ln()(ln(100 1,, −−⋅= titiit PPs  

Two indices from strong EU economies – DAX and CAC, representing the 
situation on the German and French stock exchanges, as well as two indices 
from weaker economies from the “old” European Union – the Spanish IBEX 
and the Greek ATEX, and two indices from the countries of Central and 
Eastern Europe – the Hungarian BUX and the Polish WIG20 were selected 
for the purpose of the analysis (source: the Stooq database). The Dow Jones 
                                                 

7 The relevant likelihood function is presented as part of the description of the TSM 
(Time Series Modelling) program. It is not easy to estimate the model’s parameters. Numeri-
cal problems result from the occurrence of local extrema of the logarithmic likelihood func-
tion. This is why normally two, up to three, regimes under which a process may be are distin-
guished. 



Determination of the Time of Contagion in Capital Markets... 

DYNAMIC ECONOMETRIC MODELS 13 (2013) 69–85 

75

Industrial Average (DJIA) index represented the situation on the U.S. stock 
exchanges. Gaps in the data were filled by using the linear interpolation 
method. Due to the different quotation times, data were smoothed by using 
a two-period moving average (Dungey et al., 2007).   

 
Figure 1. The Dow Jones Industrial Average index during the period from August 

17, 2005 to July 31, 2009 (lower figure – daily rates of returns from DJIA 
index) 

 The behaviour of the Dow Jones Industrial index during the studied pe-
riod is shown in Figure 1. An arbitrary division of the observation set into 
two subsets related to the period of crisis (high volatility of rates of return) 
and the period of tranquillity (low volatility) in securities markets should be 
made in such a way that the time of tranquillity immediately precedes the 
time of crisis. This is because it constitutes a benchmark for comparisons. 
Quotations preceding, for example, the collapse of Lehman Brothers, hardly 
represented a time of tranquillity as the Dow Jones Industrial Average had 
already been declining for some time and the rate of return on the index had 
been characterised by increased volatility. Therefore, it was decided that the 
information about the difficulties related to evaluating assets which was 
announced by the French BNP Paribas would be used when dividing the set 
of observations. On 9 August 2007 this bank suspended payments from three 
funds investing in the market of bonds secured by subprime mortgages. The 
period of crisis was extended beyond the time when securities were trading 
at the lowest level because of the increased volatility of rates of return which 
persisted until the end of July 2009. There were 511 observations for the 
time of crisis determined in this way. In order to ensure comparability of 
results, 511 former observations were analysed for events contributing to 

6000

7000

8000

9000

10000

11000

12000

13000

14000

15000

20
05

-0
8-

17

20
05

-1
1-

17

20
06

-0
2-

17

20
06

-0
5-

17

20
06

-0
8-

17

20
06

-1
1-

17

20
07

-0
2-

17

20
07

-0
5-

17

20
07

-0
8-

17

20
07

-1
1-

17

20
08

-0
2-

17

20
08

-0
5-

17

20
08

-0
8-

17

20
08

-1
1-

17

20
09

-0
2-

17

20
09

-0
5-

17

09-08-2007 
BNP Paribas cannot 
determine security values

march-2009 
min on the stock

31-07-2009
end of the 
sample

15-09-2008 
bankruptcy of the bank
 Lehman Brothers

17-08-2005 
the beginning of 
the sample

-6

-4

-2

0

2

4

6

05 06 07 08 09



Milda Maria Burzała 

DYNAMIC ECONOMETRIC MODELS 13 (2013) 69–85 

76

financial turmoil. It was a time of rising asset prices, with minor adjust-
ments, which is why it has been assumed to be a period of tranquillity on the 
stock exchange. 

4. Research Results 

 The estimation of a model of dynamic conditional correlations should be 
preceded by a test justifying their use. The results of such tests depend on the 
adopted specification of volatility models. Therefore, two tests were used in 
the research, i.e. the Tse test (2000) as well as Engle and Sheppard test 
(2001) in two versions with delays p=5 and p=10. For all indices at least one 
test indicated the reasonableness of constructing a dynamic conditional cor-
relation model8.  
 A significant increase of correlations in the time of crisis confirms the 
occurrence of the process of market contagion. Forbes and Rigobon (2002) 
propose using Fisher’s transformation of correlation coefficients while test-
ing the significance of change in correlation between rates of return.  
After Fisher’s transformation, the sample correlation coefficient can be 
treated as the realization of a random variable with a normal distribution 

with the expected value of 
ˆ1 1( ) ln
ˆ2 1

E ρρ
ρ

+
=

−
, where ρ̂  is the estimated cor-

relation coefficient. The variance of this variable is 1( )
3

Var
T

ρ =
−

. The null 

hypothesis 0 : K SH ρ ρ≤  is tested against an alternative hypothesis, i.e. 

1 : K SH ρ ρ>  (index S means the tranquility period, K - the time of potential 
market contagion). 
The empirical statistic in the test for two expected values is as follows: 

ˆ ˆ1 10,5ln 0,5ln
ˆ ˆ1 1

.
1 1

3 3

K S

K S

K S

FR

T T

ρ ρ
ρ ρ

⎛ ⎞ ⎛ ⎞+ +
−⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠=
+

− −

 (13)          

The FR-statistic have a normal distribution N(0,1) and even if the sample is 
small it allows one to use the critical values of a standard normal distribu-
tion. 

                                                 
8 Calculations were carried out by using the OxMetrics 6.10 program. 
 



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 The parameters of the GJR-GARCH(1,1) model are presented in the 
upper part of Table 1. Let us remember that the models were estimated on 
the basis of residuals from model VAR(1). Therefore, a slightly different 
model for the DJIA index was connected with each index. In all the volatility 
models for the DJIA index, the alfa parameter which describes the impact of 
positive residual impulses was insignificant. In the models for the European 
indices, the alfa parameter was only significant for the Greek (ATH), the 
Hungarian (BUX) and the Polish (WIG20) indices. The parameters that were 
significant were beta and gamma which describe the impact of past variance 
as well as the leverage effect (an additional impact of negative information 
reaching the market). The model’s assumptions require that the alfa parame-
ter be significantly greater than zero. Thus, in order to standardise the re-
siduals from model VAR(1), it was finally the GARCH(1,1) models that 
were used9: 

,,...1,
1 1

,
2
,, Nihch

q

j

p

j
jtiiijjtiijitij =++= ∑ ∑

= =
−− βεα  (14)          

where 
1 1

1
q p

ij ij
j j
α β

= =

+ <∑ ∑ , .0,0,0 ≥≥> ijijic βα  This time the obtained pa-

rameter estimates meet the models’ assumptions; the arch effect is signifi-
cant in all of the models (the lower part of Table 1). This means that the 
impact of negative information on many markets was considerably stronger 
than the impact of positive information. Such markets could be identified by 
estimating the GJR-GARCH model at the first stage of the research.  

Estimates of parameter β  exceed the value of 0.8, which confirms 
the volatility clustering phenomenon. Both in the GARCH(1,1) model and in 
the conditional correlation model DCC-GARCH(1,1) the requirement of 
covariance stationarity is satisfied. Also the conditions for variance (non-
negative, significant model parameters) are met. The sum of parameters 
( )α β+  in the GARCH model is close to one, which means the occurrence 
of persistence (long-term dependencies) and suggest test of integrated model 
(IGARCH, FIGARCH) in further studies. The highest unconditional correla-

                                                 
9 The same volatility model was adopted for two time series due to program limitations. 

In the research GARCH (1,2), GARCH (2,1) and GARCH (2,2) models were also tested – 
only in GARCH (1,1) model all parameters were significant and was the lowest information 
criterion value (AIC). Conditional correlations from two models (DCC-GJR-GARCH (1,1) 
and DCC-GARCH) were slightly different. 

  



Milda Maria Burzała 

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78

tion with the DJI index was recorded for the CAC and the DAX indices and 
the lowest for the BUX and the WIG20 indices.   

Table 1. The DCC-GARCH model’s parameters 
 GJR–GARCH(1,1) 

Index for index i for index DJIA 
const(i) alfa(i) beta(i) gamma(i) const alfa beta gamma 

ATH 0.017*** 0.069** 0.842*** 0.146*** 0.006*** –0.003 0.894*** 0.211*** 
CAC 0.010*** 0.022 0.900*** 0.123*** 0.006*** 0.002 0.891*** 0.201*** 
DAX 0.013*** 0.032 0.868*** 0.162*** 0.006*** –0.005 0.892*** 0.214*** 
IBEX 0.015** 0.043 0.853*** 0.164*** 0.007*** 0.006 0.880*** 0.212*** 
BUX 0.037** 0.113*** 0.798*** 0.135*** 0.006*** 0.005 0.881*** 0.209*** 

WIG20 0.027** 0.059** 0.889*** 0.064** 0.006*** –0.003 0.889*** 0.221*** 
GARCH(1,1) 

Index for index i for index DJIA 
const(i) alfa(i) beta(i) const alfa beta 

ATH 0.016** 0.165*** 0.823*** 0.005** 0.111*** 0.882*** 
CAC 0.009** 0.107*** 0.883*** 0.005** 0.116*** 0.877*** 
DAX 0.011** 0.129*** 0.861*** 0.005** 0.112*** 0.881*** 
IBEX 0.015** 0.159*** 0.824*** 0.006** 0.118*** 0.875*** 
BUX 0.034** 0.172*** 0.806*** 0.004** 0.116*** 0.880*** 

WIG20 0.026* 0.098*** 0.883*** 0.005** 0.114*** 0.879*** 
DCC–GARCH(1,1) 

Index Alfa beta df unconditional 
correlations Log–likelihood 

ATH 0.022*** 0.958*** 16.960*** 0.384*** –2060.58 
CAC 0.016* 0.962*** 12.636*** 0.641*** –1820.90 
DAX 0.027** 0.951*** 10.960*** 0.649*** –1801.13 
IBEX 0.016** 0.971*** 11.360*** 0.590*** –1845.05 
BUX 0.075    0.829*** 16.020*** 0.319*** –2292.27 

WIG20 0.018** 0.967*** 16.971*** 0.378*** –2346.02 
Note: a parameter’s significance for α = 0.01 is marked with three asterisks, for α = 0.05 with two aster-
isks and for α = 0.1 with one asterisk. 

 The parameters of the MS(3) switching model are provided in Table 2. 
Regime 2 is related to an extremely high correlation (the shaded area in Fi-
gure 2).   
As for the two indices – CAC and DAX, regime 2 covered both significantly 
high and significantly low correlation between markets. This probably re-
sulted from a very high volatility of extreme correlations. A correct classifi-
cation was obtained by simplifying the process to a model in which only the 
expected value would change. 
 For the remaining four indices, the variance of extremely high and ex-
tremely low correlations was significantly higher than the variance of condi-



Determination of the Time of Contagion in Capital Markets... 

DYNAMIC ECONOMETRIC MODELS 13 (2013) 69–85 

79

tional correlations in the time of tranquillity, and the model made it possible 
to make a correct classification10. 

Table 2. The estimates of the MS(3) switching model’s parameters  
Switches occur as 

 a result of changes → the expected value and variance the expected value 

 Regime ATH IBEX BUX WIG20 CAC DAX 

Number of observa-
tions 

0 283 423 178 312 223 222 
1 379 387 405 336 614 461 
2 360 212 439 374 185 339 

Expected value 
0 0.264 0.511 0.103 0.255 0.567 0,520 
1 0.372 0.598 0.271 0.372 0.638 0,632 
2 0.482 0.672 0.440 0.468 0.691 0,710 

FR-statistic 1,820** 1.449* 2.811*** 1.545* 1.132 1.989** 

Variance 
0 0.060 0.031 0.085 0.047 x x 
1 0.027 0.017 0.052 0.029 x x 
2 0.049 0.029 0.078 0.038 x x 

H-statistic 601.6*** 589.2*** 854.1*** 596.2*** 492.4*** 650.4*** 

D-statistic 0 15.89*** 12.90*** 16.69*** 13.77*** 10.83*** 11.61*** 

1 24.53*** 23.10*** 29.23*** 23.78*** 19.97*** 22.72*** 

Probability  
of transition 

 
 
 
 
 

p_{0/0} 0.977 0.983 0.880 0.987 0.978 0,961 
p_{0/1} 0.023 0.017 0.120 0.013 0.000 0,039 
p_{0/2} 0.000 0.000 0.000 0.000 0.022 0,000 
p_{1/0} 0.017 0.018 0.052 0.012 0.008 0,018 
p_{1/1} 0.962 0.970 0.880 0.964 0.984 0,965 
p_{1/2} 0.021 0.012 0.068 0.024 0.008 0,017 
p_{2/0} 0.000 0.000 0.061 0.019 0.000 0,000 
p_{2/1} 0.022 0.021 0.000 0.000 0.046 0,021 
p_{2/2} 0.978 0.979 0.939 0.981 0.954 0,979 

Log-likelihood 1708,9 2257.9 1063.5 1824.5 2307.3 1797.5 
AIC –3,325 –4.399 –2.062 –3.551 –4.499 –3.502 

Note: the FR-statistic refers to the difference in correlation between regimes 2 and 1; the D-statistic refers 
to the difference distributions between regime 2 and regime 0 or 1.  

 The Jarque-Berra test rejects at conventional significance level the nor-
mality of correlation in three regimes (not reported).  It is the reason of the 
use of nonparametric variance analysis (Kruskal and Wallis-test) to evaluate 
the quality of classification (division of the sample into observations from 
the time of tranquillity and from the time of potential market contagion).  
In the first research stage, H-statistic indicates the diversification of distribu-
tion at least in two regimes. In the second stage, D-statistics indicates the 
                                                 

10 High variance in 0 regime (extremely low correlations) indicates, that differentiation of 
the sign of returns on two markets causes the increase of uncertainty among investors. 

 



Milda Maria Burzała 

DYNAMIC ECONOMETRIC MODELS 13 (2013) 69–85 

80

diversification of distribution in all regimes for all studied indices. Relevant 
statistics are provided in Table 2. Results confirms the legitimacy of the use 
of one-dimensional switching model. Simple model can give satisfactory 
results. 

Corr_ATH_DJIA Corr_IBEX_DJIA 

Corr_BUX_DJIA Corr_WIG20_DJIA

Corr_CAC_DJIA Corr_DAX_DJIA 

Figure 2.  Potential periods of market contagion as determined based on conditional 
correlations from the DCC-GARCH(1.1) model 

Note:  the shaded area: the time of extremely high correlations (regime 2); ── the average value of 
conditional correlations in the time of crisis (August 9, 2007 to July 31, 2009); - - -  the upper limit of the 
range: unconditional correlation + 2 error. 



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81

 The occurrence of extreme correlations under regime 2 only means the 
time of potential market contagion. Only the rejection of the null hypothesis 
in the test for two expected values means that the process of market conta-
gion has occurred. The expected value of correlation in regime 2 is signifi-
cantly higher than the expected value of correlation in regime 1 (the time of 
tranquillity in the market) in all securities markets except for the French 
market (CAC). The significance level that allows one to reject the null hy-
pothesis is, however, varied, which is highlighted in the table. The FR-
statistics assumes the highest value for the Hungarian market. It can be as-
sumed that the value of statistics reflects the effects of contagion. The higher 
the value of statistics, the more severe the effects of market contagion.  
 The probability of remaining under each of the regimes, which is pro-
vided in Table 2, is high, which means that the highlighted regimes are per-
sistent. The analysis of charts in Figure 2 allows one to compare the fre-
quency of an index being under regime 2 and the time of remaining there. 
The additional, horizontal and dashed line makes it possible to relate indi-
cated periods to the time of potential market contagion as identified based on 
a range for unconditional correlations. In this case, those correlations which 
fall outside the upper and the lower limits determined by a double estimation 
error are assumed to be extremely high and extremely low correlations, re-
spectively. The time of tranquillity in a market is represented by correlations 
from a range determined in this way. As for arbitrary arrangements, it should 
be remembered that the sample was only divided into two subsets.   
 Total duration times of the potential market contagion period are pro-
vided in Table 3. The longest time is for an arbitrary division and the short-
est for the range for unconditional correlations. The switching model indi-
cated that the period of potential contagion in the Hungarian market was the 
longest. 

Table 3. The number of observations during the potential period of market contagion 
Metod ATH IBEX BUX WIG20 CAC DAX 

Arbitrary arrangements 511 511 511 511 511 511 
Switching model – regime 2 360 212 439 374 185 339 

Range for unconditional correlations 161 89 243 101 57 81 

 A comparison of the results of tests of the significance of occurrence of 
contagion spreading from the U.S. market to a given market is presented in 
Table 4.   
  



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DYNAMIC ECONOMETRIC MODELS 13 (2013) 69–85 

82

Table 4. A comparison of the results of testing the significance of contagion in 
a market 

Method Index ATH IBEX BUX WIG20 CAC DAX 

Arbitrary 
arrangements 

Contagion 0.416 0.594 0.360 0.427 0.650 0.654 
Tranquility 0.349 0.562 0.267 0.315 0.614 0.613 
FR-statistic 1.309* 0.770 1.641* 2.072** 0.966 1.096 

Switching model 
Contagion 0.482 0.672 0.440 0.468 0.691 0.710 
Tranquility 0.372 0.598 0.271 0.372 0.638 0.632 
FR-statistic 1.820** 1.449* 2.811*** 1.545* 1.132 1.989** 

Range for 
unconditional correla-

tions 

Contagion 0.535 0.702 0.492 0.574 0.731 0.771 
Tranquility 0.384 0.584 0.321 0.405 0.639 0.653 
FR-statistic 2.019** 1.788** 2.647*** 1.572* 1.237 2.030** 

Note: “contagion” means the time of potential market contagion. 

For five indices the values of the FR-statistics recorded in the case of the 
switching model are lower than the corresponding statistics in the analysis of 
unconditional correlations, which has an effect on conclusions about conta-
gion. The occurrence of the contagion process is registered more often (for 
lower significance levels) in the analysis of a range for unconditional corre-
lation. The opposite is true only for the BUX index, which probably results 
from small differences between the duration times of market contagion that 
are determined by using different methods. The lowest values of the FR-
statistics are usually recorded for an arbitrary division. Detailed results and 
significance levels are provided in Table 4. 

Conclusions 
 It is relatively difficult to date a crisis in financial markets. During peri-
ods determined based on events that change the behaviour of rates of return, 
both high and low correlation between markets can be observed. 
 This paper proposes indicating the periods of potential market contagion 
on the basis of a one-dimensional switching model. Tests made confirm the 
legitimacy of the use of simple switching model to determine potential mar-
ket contagion periods. Further studies should be conducted – it is important 
to compare obtained results with the results from multidimensional model, 
where switches are determined based on the changes of expected value, vari-
ance, and covariance. In the paper such comparisons were not made because 
of the lack of appropriate software. Occurrence of persistence suggest, that 
further studies should also include inference based on the integrated model.  

Results confirm the conclusions made by the author on the subject of 
contagion on the basis of logit model for panel data (Burzała, 2012). Signifi-
cant contagion effects were observed on German market, less significant – 



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on Polish, and the lack of significant contagion effects were observed on 
French market.  

Determination of extremely high correlations by using a range for uncon-
ditional correlations and the MS(3) switching model yields similar results 
regarding conclusions about the occurrence of the process of contagion in 
a market. Conclusions about contagion are, however, made at a higher sig-
nificance level in the case of the switching model. It is worth emphasising 
that it is necessary that the appropriate tests be conducted which would con-
firm the significance of the increase of correlation between markets. Also, 
the time of potential market contagion determined on the basis of a regime 
with an extremely high correlation (the switching model) is longer. In further 
studies more attention should be paid to the issue of determining the direc-
tion of contagion as well as extremely low correlations which may be a har-
binger of a crisis.  

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Wyznaczanie czasu zarażania rynków kapitałowych na 
podstawie modelu przełącznikowego 

Z a r y s  t r e ś c i. W artykule podjęto próbę porównania wnioskowania o zarażaniu rynków 
na podstawie okresów wskazanych przez model przełącznikowy Markowa z wnioskowaniem 
opartym na przedziale dla korelacji bezwarunkowych i ustaleniach arbitralnych. W celu kon-
trolowania zmieniających się w czasie korelacji wykorzystano model DCC. Ustalenie ekstre-
malnie wysokich korelacji przy wykorzystaniu przedziału dla korelacji bezwarunkowych lub 
modelu przełącznikowego MS(3) prowadzi do podobnych rezultatów w zakresie wnioskowa-
nia o wystąpieniu procesu zarażania rynku. Wnioski o zarażaniu są jednak stawiane przy 
wyższym poziomie istotności w przypadku modelu przełącznikowego. 

S ł o w a  k l u c z o w e: model przełącznikowy, model DCC, zarażanie.  



 

 




