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DYNAMIC ECONOMETRIC MODELS 
Vol. 9 – Nicolaus Copernicus University – Toruń – 2009 

Monika Kośko 
The University of Computer Science and Economics in Olsztyn 

Markov Switching Models with Application to Contagion 
Effect Analysis in the Capital Markets 

A b s t r a c t. This article presents the analysis of the contagion effect in the capital markets on 
the basis of the Markov switching models MS. The research is based on the return of the indexes. 
There is a distinction of two regimes with different volatility levels, the calm period and the crisis 
period. Then the analysis of the period’s occurrence was conducted, in reference to global finan-
cial crisis. Periods with a similar level of volatility occurrence in the same time. This analysis 
evidences the shocks transmission between financial markets, what confirms an occurrence of the 
contagion effect.  

K e y w o r d s: Markov switching model, contagion effect.  

1. Introduction 
 The aim of the article is an application of the Markov switching model MS 
to contagion effect analysis in the capital markets. There are different defini-
tions of contagion effect. The most popular definition affirms that shocks 
transmissions are caused by the herd behavior of investors and this is the most 
often assumed in the empirical research. There can be found three approaches in 
an application of the MS models to contagion effect analysis, such as: 
− univariate models with the switch in variance MSH  (Moore, Wang, 2007); 
− multivariate models with the switch in variance MSH-VAR or both in the 

variance and mean MSMH-VAR (Linne, 2001; Mandilaras, Bird, 2005); 
− the GARCH models with the Markov switching MS-GARCH (Edwards, 

Susmel, 2001). 

2. A Contagion Effect Definition  
 A contagion effect definition the most often concerns the financial markets, 
but the transmission processes envelope an economic connections too. The 



Monika Kośko 

 

74

Word Bank assumes three versions of the contagion effect definition1: broad, 
narrow and very narrow definition. According to broad definition the contagion 
effect is an international shocks transmission or wide-spread spillover effect. 
The transmission can refer to both good and bad periods and it’s not always 
identified as crisis. However in the crisis it can be more noticeable. In the nar-
row definition there is an assumption that contagion is a shocks transmission to 
other countries or the relations between economies except the fundamental con-
nections and common shocks. This definition the most often is reduced to very 
similar changes in the financial markets that are usually explained by the herd 
behavior. The very narrow definition assumes that a contagion effect occurs 
when in the crisis period the correlation between economies is stronger than in 
the calm period.         
 According to Fiszeder (2009) the narrow definition of contagion effect is 
the shocks transmission between countries that cannot be explained fundamen-
tally. These transmissions are real financial, economic and political connec-
tions. The most often cause of the contagion in narrow sense is the herd inves-
tors behavior. There can be noticed some specific group behavior of investors, 
what is more distinct in the crisis periods and causes crossing shocks over the 
financial markets. The understanding these behaviors could help to explain the 
transmission of the shocks. In the analysis of the contagion effect a transmission 
channels have the essential meaning. There can be found a three basic transmis-
sion channels:   
− real channel (international trade); 
− financial channel (global diversification of the investment portfolio); 
− herd behavior (a copy strategy in the investment); 
− international policy. 

3. The Markov Switching Model 
 The Markov switching model AR(p)-MSMH(r) for stochastic process ty  is 
given by:                                                       

( ) ( )( ) ( )( ) ( )( ) ,sy...sysysy tptptptttttt εμαμαμαμ +−+−+−+= −−−−−− 222111

 (1) 

( )( ),s,IID~ tt
20 εσε   

[ ],s,y|yEy ttttt 1−−=ε  

where ( )tSμ  is conditional expected value of ty  [ ]( )ttt syy ,| 1−Ε=μ  and 

( )ts2
εσ  is the variance of the disturbance term.  

                                                 
1 Contagion of Financial Crises, World Bank, http://www.worldbank.org.  



The Markov Switching Model with Application to Contagion Effect … 

 

75

In the Markov switching models the parameters ( ) ( ) jtt ,jS,jS πσμ ε == 2 2  are 
the unobserved variable tS  realizations, that has a Markov property3. The con-
ditional probabilities ( )tpij  create the transition probabilities matrix P with the 

rr ×  dimension that is given by: 

,

)t(p)t(p)t(p

)t(p)t(p)t(p
)t(p)t(p)t(p

rrrrrr

r

r

×

⎥
⎥
⎥
⎥

⎦

⎤

⎢
⎢
⎢
⎢

⎣

⎡

=

L

MOMM

L

L

21

22221

11211

P  (2)                                           

where r is the states (regimes) number of the tS  variable process. 

The P  matrix is the stochastic matrix, because its elements satisfy the follow-
ing conditions: 0≥)t(pij , 1=∑

j
ij )t(p . 

The homogeneous Markov chain probabilities ( )tpij  describing the one step 
change between states are constant and time independent.    
The Markov switching models MS can generate the skew distribution (when the 
third central moment significantly differs from zero) and the leptokurtic distri-
bution. In example, the model that can generate the skewness and the leptokur-
tosis of the distribution is the AR(0)-MSM(2). The AR(0)-MSH(2) model is 
given by: 

( ) ( )( ),sIsI,NID~,Y ttttYt 210 2
2

2
1 =+==− εε σσεεμ  (3) 

where 2
2

2
1 εε σσ ,  the variance of the disturbance term, in the following first and 

second state ( ) ( )21 == tt sI,sI  are dummies variables.  

The excess coefficient of the tY  process for the Markov switching AR(0)-
MSH(2) model can be written as: 

( )[ ]
( )[ ]

( )
( ) ,

Y

Y

Yt

Yt
22

22
2
11

22
2

2
121

22

4 33
εε

εε

σπσπ

σσππ

μΕ

μΕ

+

−
=−

−

−  (4)       

                                                 
2 Unconditional probabilities of the Markov chain (ergodic) for two states are received from 

equations: 
2211

22
1 2

1
pp

p
−−

−
=π ,

2211

11
2 2

1
pp

p
−−

−
=π . 

 3 The finite homogeneous Markov chain with the state space { }r,...,,21  is the stochastic 
process where for all { }rji ,...,2,1, ∈  the ( ) ==== − tpiSjS ijtt )|Pr( 1

( )tp)iS|jSPr( ijtt ==== −1  equality is fulfilled.  



Monika Kośko 

 

76

where ( )[ ]2
YtY μΕ −  is the second central moment and 21 ππ ,  are the ergodic 

probabilities in the following first and second states.  

 The (4) coefficient significantly differs from zero when 2
2

2
1 εε σσ ≠  and when

10 1 << π . Therefore the leptokurtosis is confirmed by the Markov structure 
which has heteroscedastic disturbance term and different from zero the excess 
coefficient of the distribution.  

4. The Financial Time Series Results 
 In the empirical analysis the weekly return rates of the main stock exchange  
indexes were used, such as: RTS (Russian), SAX (Slovakia), HIS (China), 
PX50 (Czech Republic), BUX (Hungary), CAC40 (France), DAX (Germany), 
FTSE100 (England). The analyzed series come from the period from September 
the 1th, 1995 to August 21th, 2009. The price series transformation into return 
series was achieved by calculating the week dynamics. The time series of the 
returns were multiplied by 100. Then the ADF test for unit root was applied and 
its results show that for all time series this test rejects the unit root hypothesis. 
In the next part of the empirical analysis the MSH models with the variance 
switch and 1, 2 or 3 states were estimated. The appropriate order of autocorrela-
tion and autoregression in these models were determined by the means of the 
Durbin and Watson test. For two states models one of the states is interpreted as 
low volatility periods and the second state as high volatility periods. For the 
MSH(3) models with three states the additional state is characterized as periods 
with the moderated volatility. The switching models were checking for the 
presence of the ARCH effect (Ljung and Box test for the squares of return rates) 
and for the normality of distribution (Jarque and Bery test). The tests results are 
presented in the Table 1. The distributions of the all residuals series are normal. 
In most models the ARCH effect doesn’t occur. The log-likelihood ratio analy-
sis indicates that the ratios are higher for all models with three states MSH(3) 
than ratios of the models with two states MSH(2). Moreover the log-likelihood 
test for the number of states4 was applied. The results of this test indicate the 
choice of the MSH(3) models. The estimation results of the MSH(2) and the 
MSH(3) models are shown in the Table 1 and in the Table 2 appropriately.  
 Models with the highest values of probabilities are presented in the Table 3. 
In MSH(2) models the variance of high volatility state is about two times higher 
than the variance of the low volatility state.   

                                                 
4 The log-likelihood test is constructed on the basis of the ( ) ( )( )AHLHLLR −−= 02  statistic 

that has a chi-square distribution ( )k2χ  and k is a number of additional parameters of the alterna-
tive hypothesis model.  



The Markov Switching Model with Application to Contagion Effect … 

 

77

Table 1. Estimation results of the MSH(2) models  

Series 11p  22p  1σ  2σ  μ  LR Jarque and Bery test 
(residual) 

WIG 0.9695 0.9801 0.0289 0.0626 0.0123 
[0.113] 1162.4 3.76 [0.152] 

RTS 0.9871 0.9866 0.1149 0.0655 0.0337 
[0.006] 795.1 3.18 [0.204] 

SAX 0.9435 0.9608 0.0567 0.0327 0.0016 
[0.797] 1242.8 6.30 [0.043] 

HSI 0.9949 0.9952 0.0290 0.0950 0.0143 
[0.032] 1137.9 2.42 [0.298] 

PX50 0.9846 0.9965 0.0775 0.0350 0.0111 
[0.007] 1249.3 0.57 [0.751] 

BUX 0.9557 0.9789 0.0835 0.0387 0.0215 
[0.006] 1060.5 4.52 [0.104]] 

CAC40 0.9953 0.9931 0.0488 0.0239 0.0134 
[0.007] 1291.1 1.49 [0.474] 

DAX 0.9778 0.9865 0.0653 0.0306 0.0146 
[0.022] 1215.0 1.98 [0.371] 

FTSE 0.9262 0.9864 0.0578 0.0249 0.0049 
[0.225] 1473.4 1.68  [0.43] 

Nikkei 0.9376 0.9961 0.0847 0.0370 -0.0006 
[0.915] 1253.9 1.07 [0.586] 

SP500 0.9870 0.9789 0.0232 0.0462 0.0091 
[0.029] 1430.2 1.49 [0.474] 

Note: p-values have been presented in brackets. 

Table 2. Estimation results of the MSH(3) models 

Series 11p  22p  33p  1σ  2σ  3σ  LR Jarque and Bery test 
(residual) 

WIG 0.6032 0.9685 0.9721 0.0113 0.0628 0.0288 1164.7 6.50 [0.039] 
HSI 0.9949 0.9812 0.9836 0.0287 0.0561 0.0834 1146.0 5.29 [0.071] 
BUX 0.9406 0.9827 0.9674 0.1034 0.0354 0.0566 1070.5 10.6 [0.005] 
DAX 0.9281 0.9940 0.9835 0.0962 0.0286 0.0462 1230.4 5.95 [0.051] 
FTSE 0.9913 0.9937 0.9081 0.0337 0.0177 0.1013 1513.3 7.89 [0.019] 
SP500 0.9930 0.9026 0.9642 0.0206 0.0600 0.0321 1441.9 5.24 [0.073] 

Note: p-values have been presented in brackets. 

The high volatility periods that were pointed out on the basis of MSH models 
are presented in the Table 3. These periods represent the high variance regime. 
The common markets (indexes) periods are following (marked in the Table 3):  
− 09.1998–12.2000 (the consequences of the Russian crisis); 
− 02.2001–12.2002 (the beginning of this crisis is seen in the DAX and SP500 

indexes,  then in the WIG, RTS and FTSE indexes); 
− 01.2003–12.2004 (the beginning of this crisis is seen in the SP500 and 

FTSE indexes, then in the WIG, DAX and RTS indexes); 
− 07.2008–08.2009 (the beginning of this crisis is seen in the SP500 index, 

then in the RTS, FTSE, WIG, PX500, DAX, SAX and BUX indexes). 



Monika Kośko 

 

78

Table 3. The high volatility periods pointed out on the basis of MSH models 

WIG RTS SAX HSI PX50 BUX CAC40 DAX FTSE Nikkei SP500 
11.95-
12.00 

10.95-
04.01 

01.96-
02.96 

04.97-
05.02 

06.98-
05.99 

01.96-
04.96 

01.97-
05.03 

07.97-
12.97 

09.97-
11.97 03.02 09.98-

06.00 
           

09.01-
02.02 

11.01-
01.02 

11.96-
02.97 

07.07-
08.09 

09.08-
08.09 

12.96-
03.97 

07.07-
08.09 

07.98-
03.00 

10.98-
02.99 

10.08-
05.09 

02.01-
12.01 

           
07.03-
01.04 

10.03-
12.03 

12.97-
01.99   10.97-

01.98  02.01-
05.01 

08.01-
11.01  05.02-

04.03 
           

04.06-
09.06  04.99-

01.00   05.98-
01.00  08.01-

05.03 
06.02-
09.00  01.08-

08.09 
           

03.07-
03.08  03.00-

12.00   09.00-
01.01  09.03-

11.03 
01.03-
05.03   

           
09.08-
08.09 

07.08-
08.09 

03.01-
07.01   11.05-

12-05  10.08-
08.09 

09.08-
05.09   

           
  05.02-

07.02   05.06-
09.06      

           
  11.02-

05.03   10.08-
08.09      

           
  01.04-

03.04         
           
  09.04-

04.05         
           
  10.08-

08.09         

 The first period, shaded in the Table 3, corresponds with the Russian crisis 
and its consequences can be notice in the 1998 year. This crisis has sunk into a 
memory of worldwide economic recession. Its beginnings were noticeable 
firstly in Czech Republic in March 1997 in the form of the banking crisis, and 
in markets of south-east Asia. The analysis of these periods shows that the Rus-
sian crisis periods coincide with the high volatility periods for the most index 
series. The next distinguished common markets periods of high volatility might 
be qualified as a derivative of economic crisis begun in the half of 2008 year in 
USA. These periods weren’t identified only in the case of the HIS and CAC40 
indexes. The contagion effect on financial markets in the crisis periods is seen 
more clearly. The high volatility periods occurrence usually at the beginning of 
the financial crisis. Then the shocks are transmitted between markets. The anal-
ysis evidences these transmissions between financial markets, what confirms an 
occurrence of the contagion effect.  



The Markov Switching Model with Application to Contagion Effect … 

 

79

5. Summary 
 The main aim of this article was an application of the Markov switching 
models to identification and analysis the contagion effect in the capital markets. 
The research allowed to distinguish two regimes with different volatility levels, 
the calm period (low volatility) and the crisis period (high volatility). Basing on 
this classification the identification of some interdependence pattern between 
markets was created. In the empirical part of the article the narrow contagion 
effect definition was assumed. According to this definition the cause of the 
shocks transmission are the herd behavior of investors mainly and the funda-
mental economic factors are not taken under attention. In these researches there 
were distinguished the high volatility periods (on the indicated capital markets) 
and the occurrence of these periods was analyzed. The conclusion is that in the 
case of the Russian crisis and the crisis which begun in the 2008 year, the pat-
terns of the high volatility periods were very similar. The capital markets are 
connected with themselves what is noticed clearly in investors behavior during 
the beginning of the crisis period. The high volatility periods usually occur with 
the price decrease and they coincide or come across on themselves. The more 
detailed analysis using the switching structure could be carried out on the basis 
of the multivariate Markov switching models for two series for example, what 
would allow appointing the periods of common volatility.     

References 
Dempster, A. P., Laird, N. M., Rubin, D. B. (1977), Maximum Likelihood from Incomplete Data 

via the EM Algorithm, Journal of the Royal Statistical Society, 39, 1–38. 
Doman, M., Doman, R. (2004), Ekonometryczne modelowanie dynamiki polskiego rynku finan-

sowego (An Econometric Modeling of Polish Financial Market Dynamic), Wydawnictwo 
Akademii Ekonomicznej, Poznań. 

Hamilton, J. D. (1989), A New Approach to the Economic Analysis of Nonstationary Time Series 
and the Business Cycle, Econometrics , 57, 357–384. 

Edwards, S., Susmel R. (2001), Volatility Dependence and Contagion in Emerging Equity Mar-
kets, Journal of Development Economics, 66, 505–532. 

Krolzig, H.-M. (1997), Markov-Switching Vector Autoregression. Modeling Statistical Inference 
and Application to Business Cycle Analysis, Springer Verlag Edition. 

Stawicki, J. (2004), Wykorzystanie łańcuchów Markowa w analizie rynków kapitałowych (The 
Markov Chains in Capital Markets Analysis), Wydawnictwo UMK, Toruń. 

Fiszeder, P. (2009), Modele klasy GARCH w empirycznych badaniach finansowych (The GARCH 
Models in the Empirical Financial Research), Wydawnictwo Naukowe UMK, Toruń. 

Moore, T., Wang, P. (2007), Volatility in Stock Returns for New EU Member States: Markov 
Regime Switching Model, International Review of Financial Analysis, 1, 282–292. 

Przełącznikowy model typu Markowa w badaniu efektu zarażania na 
rynkach kapitałowych 

Z a r y s  t r e ś c i. Artykuł stanowi próbę analizy efektu zarażania na rynkach kapitałowych 
z wykorzystaniem przełącznikowego modelu typu Markowa MS. Badanie przeprowadzono 



Monika Kośko 

 

80

w oparciu o indeksy giełdowe wybranych krajów. Wyznaczono stany o niskiej i wysokiej zmien-
ności dla poszczególnych szeregów oraz przeprowadzono analizę ich występowania w odniesie-
niu do globalnych kryzysów finansowych. Stwierdzono występowanie wspólnych okresów 
zmienności dla badanych indeksy. Okresy te wskazują na przenoszenie szoków pomiędzy rynka-
mi, potwierdzając występowanie efektu zarażania na tych rynkach.   

S ł o w a  k l u c z o w e: przełącznikowy model typu Markowa, efekt zarażania.  


