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

Tomasz Chruściński 
Nicolaus Copernicus University in Toruń 

The Study of Interdependence Between Capital 
and Currency Markets Using Multivariate 

GARCH Models 

A b s t r a c t. In the article an attempt was made to investigate the interaction among the various 
stock exchanges as well as various exchange rates and then to determine the direction 
of information flow between capital and currency markets. Tools used in this study are Multivari-
ate GARCH models. Presented results developed an earlier study of World Stock Exchange clas-
sification. These stock exchanges will be further analysed according to their interaction.  

K e y w o r d s: Multivariate GARCH Model, independence analysis, stock exchange, exchange 
rate. 

1. Introduction 
 The research on the interaction of capital and currency markets in the world 
has been going on for many years, but this topic has not lost its popularity. Es-
pecially in the last year, and when as a result of the rapid withdrawal 
of investors from the stock exchange, the inflow of speculative capital, and 
the bankruptcies of numerous well-known companies, discussions about inter-
penetrating influences of stock exchanges they have on one another were very 
hot. At the same time, there was considerable interest attracted into fluctuations 
in the currency market. The aim of this paper is to examine the relationship 
between stock exchanges of securities of eligible countries and exchange rates 
associated with them. The article presents a continuation of earlier studies by 
the author on the classification of stock exchanges in the world, which now will 
be analyzed in terms of the direction of information flow between the markets. 
Multivariate GARCH models are applied to determine the influence on  
volatility. 



Tomasz Chruściński 112 

2. Methodology 
 With a map-sharing of stock exchanges, resulting from the author’s previ-
ous studies1, an attempt was made to determine the direction in which informa-
tion flows between the dissociated groups, and thus to examine what their spill-
over effects are. The study used a multivariate model MGARCH, based on daily 
rates of return with the major indices and exchange rates. Since it would be 
difficult to examine changes in all indices, representatives of groups have been 
chosen. Therefore, for use of the specification of the model’s equations, loga-
rithms adopted rate of return (1) of the following indices were accepted: 
Nasdaq, Dow Jones, S & P500, DAX, CAC40, FTSE250, SSE Comp., WIG20, 
and the exchange rate representing the countries from which the selected indi-
ces. These are: EUR/USD, EUR/GBP, EUR/PLN, USD/GBP, USD/CNY, 
CNY/GBP, CNY/PLN, CNY/USD, GBP/PLN.  
 The starting point for analysis are the concepts of conditional expected val-
ues, conditional variance-covariance matrix and conditional distribution 
of standardized residuals of the model, presented in the equation: 

 ( ) .lnln100 1 tttttttt zhPPr +=+=−= − μεμ  (1) 

tP , 1−tP  - value of the index (exchange rate) at time t and t-1, tμ  - conditional 

expected value of rate of return rt at time t ( [ ]1| −Ε= ttt Irμ ), tε  - residual 

of the model, th  - conditional variance of returns at time t ( [ ]1|var −= ttt Irh ), 

tz  -  independent of the residuals of the model with zero mean and unit vari-
ance, It-1 - information available at time t.  
 More methodological information about the modelling of stochastic proc-
esses can be found in the literature (Osińska, 2006; Fiszeder, 2009). The paper 
focuses on key issues of MGARCH structure model. 
 Although it has been over 20 years since Engle and Bollerslev (1986) intro-
duced one-equation ARCH and GARCH models, their attitude to discrete-time 
variance of rates of return, is still valid and being developed. A natural exten-
sion of GARCH models for the analysis of financial markets was introduced 
by Bollerslev (1988) Multivariate GARCH model (MGARCH). In the empirical 
part of this paper this model was used to describe the mutual influence of stock 
markets and exchange rates in the world. 
 The general form of a multivariate model, which is equivalent to one-
equation model GARCH (1,1) defined in the work of Bollerslev, Engle, 
Wooldridge (1988) called VECH-GARCH, is given by equation: 

 ( ) ( ) ( ) ( ),11 1tτ HBAWΗ −−− +⋅+= vechvechvechvech t
T
t εε  (2) 

                                                 
1 Chruściński (2008). 



The Study of Interdependence Between Capital and Currency Markets ... 113

where ( )⋅vech  is an operator of symmetric vector. 

 Several conditions have to be met for an estimation of the model (2) to be 
properly conducted. A positive definiteness and stationary of matrix Ht for each 
time t shall be provided, what requires the positive definiteness of matrix A and 
B and is associated with introducing very complicated nonlinear constraints. 
The primary consequence of the full form of VECH equation is the need 
to estimate a large number of parameters, which even in the two-dimensional 
model amounts to 21. These problems resulted in little use the model has found 
in practice. Models with diagonal matrices A and B, that reduced the number of 
estimated parameters and eliminated the transmittance variance effect by condi-
tioning elements of matrix Ht from its past values and error products at time t  
( tjti ,, εε ), turned out to be a solution. The general form of the diagonal model 
VECH (DVECH) is: 

 ( ) ,11 tt HBAWH oo ++= −− t
T
t εε  (3) 

where: ( )( )AA diagivech= , ( )( )BB diagivech= , intersection YX o  is 
the Hadamard product, and ( )⋅ivech  is an inverse operator to ( )⋅vech . 
 After reducing matrices A and B to the diagonal form, we obtain the final 
form of the model DVECH, which is an extension of GARCH model (1.1)  
(see Yang, 2001).  
 A special variant of the VECH model is the BEKK model, which easily 
solves the problem of non-positive covariance matrix. However, since it is diffi-
cult to obtain stationarity of matrix Ht as well as a small number of estimated 
parameters (see Piontek, 2006), similarly applicable are diagonal matrices A 
and B, to obtain a model DBEKK (2) 

 .11 BHBAAWWH 1t
TTT

t −−− ++= tt εε  (4) 

 Diagonal forms can be used to estimate each equation of models (3) and (4) 
respectively, allowing to avoid numerous optimization problems of maximum 
likelihood method (ML) applied to several equations simultaneously. In the 
empirical example maximisation of the maximum likelihood function defined 
by the formula (5) was adopted: 

 [ ].ln
2
1

1

'∑
=

−+−=
T

t
ttLLF εε 1

tt HH  (5) 

3. Empirical Results 
 The daily rate of return on the main stock indices of the selected countries 
and the related exchange rates were used for the analysis. Time series were ad-
justed for comparability by removing rates of return for periods in which even 
only one variable was unknown. Finally, 1865 observations for the period from 



Tomasz Chruściński 114 

January 2001 to April 2009 were obtained. For such prepared data diagonal 
VECH models, diagonal BEKK models, and a  constant conditional correlation 
model CCC have been built. The comparison was conducted to assess which of 
them matches reality the best. After estimating several dozen models, a decision 
to use the conditional Student’s t-distribution was made. This has given better 
results of the parameters’ estimation than a normal distribution. 

Table 1. Comparison of estimation results for given models, according to the AIC crite-
rion 

Model 
AIC value Optimal 

model DVECH CCC DBEKK 

rCNY/GBP=f(rCAC40, rDAX); rDAX=f(rS&P500, rCAC40); 
rS&P500=f(rDJIA) -19.6769 -19.5990 -19.6505 DVECH 
rCNY/PLN=f(rDJIA, rCAC40); rCAC40=f(rS&P500, rDAX); 
rDAX=f(rNASDAQ) -20.1400 -20.0380 -20.1282 DVECH 
rCNY/EUR=f(rDJIA, rWIG20); rCAC40=f(rS&P500, rDAX); 
rWIG20=f(rFTSE250, rDAX) -18.8546 -18.7995 -18.8423 DVECH 
rGBP/PLN=f(rS&P500, rDAX); rDAX=f(rFTSE250, rCAC40); 
rWIG20=f(rFTSE250, rDAX) -18.5694 -18.5593 -18.5418 DVECH 
rEUR/PLN=f(rS&P500, rCAC40); rS&P500=f(rNASDAQ, rDJIA); 
rCAC40=f(rS&P500, rDAX) -20.0290 -20.0298 -19.9953 CCC 
rWIG20=f(rNASDAQ, rDAX); rDAX=f(rS&P500, rCAC40);  
r S&P500=f(rDJIA) -18.1299 -18.1305 -18.1000 CCC 
rWIG20=f(rS&P500, rDAX); rDAX=f(rS&P500, rCAC40); 
rS&P500=f(rDJIA) -18.1394 -18.1414 -18.1102 CCC 
rWIG20=f(rDJIA, rDAX); rDAX=f(rS&P500, rCAC40); rD-
JIA=f(rS&P500) -18.1705 -18.1764 -18.1469 CCC 
rWIG20=f(rNASDAQ, rFTSE250); rFTSE250=f(rS&P500, rCAC40); 
rCAC40=f(rS&P500, rDAX) -19.0869 -19.0830 -19.0638 DVECH 
rWIG20=f(rS&P500, rFTSE250); rFTSE250=f(rS&P500, rCAC40); 
rS&P500=f(rDJIA) -18.8132 -18.8329 -18.7793 CCC 
rWIG20=f(rDJIA, rFTSE250); rFTSE250=f(rS&P500, rCAC40); 
rDJIA=f(rS&P500) -18.8622 -18.8839 -18.8324 CCC 
rSSE=f(rFTSE250); rFTSE250=f(rS&P500);  
r S&P500=f(rNASDAQ, rDJIA) -18.5000 -18.5103 -18.4458 CCC 
rSSE=f( rFTSE250); rFTSE250=f(rS&P500, rCAC40); 
rWIG20=f(rS&P500, rDAX) -17.7720 -17.7780 -17.7281 CCC 

 The purpose of this research was to determine the direction of the impact 
of a stock exchange  representing one group on a representative of another stock 
exchange, the mutual influence of exchange rates on one another and, conse-
quently, to examine the relationship between stock market and currency market. 
Having available rates of return of 8 indices and 9 exchange rates, hundreds of 
multi-dimensional combinations have been analysed. Because the publication 
has limited size only examples of the results of estimated models are presented.  
The value of AIC has been chosen as the selection criterion. The results of the 
comparisons are shown in Table 1. 



The Study of Interdependence Between Capital and Currency Markets ... 115

 It is clearly visible while analysing the results of the estimation presented 
in Table 1 that the best models to depict the interaction between stock indices 
and exchange rates are the diagonal VECH model and the constant conditional 
correlation model CCC. The same structure of the optimal models occurs for all 
of the interdependence test results. An exemplary multivariate empirical 
DVECH model is shown below: 
 rEUR/PLN(t) = -0.000332 – 0.042826·rS&P500(t-1) + 0.024953·rCAC40(t-1) + εt 

 rS&P500(t) = 0.000576 + 0.016579·rNASDAQ(t-1) – 0.093922·rDJIA(t-1) + εt 
 rCAC40(t) = 0.0007 + 0.420494·rS&P500(t-1) – 0.20429·rDAX(t-1) + εt 

 

Figure 1. The reciprocal influence of stock exchanges, represented by the major indexes 
in the period from January 2001 to April 2009 

 The more interesting part of the calculations’ results is an economic inter-
pretation of the constructed multivariate models. Basing on correctly evaluated 
exogenous variables of individual equations the direction of the information 
flow from the day before between the individual stock exchanges is clearly 
visible. An illustration of the mutual  influence of stock exchanges used in the 
study is shown in Figure 1, while Figure 2 represents the results of studies using 
similar variables from the period from January 2001 to September 2008. 
A comparison of the two mentioned figures  provides us with a clear picture of 
changes in a correlation between stock exchanges over a 7-month period of the 
economic crisis. 



Tomasz Chruściński 116 

 

Figure 2. The reciprocal influence of stock exchanges, represented by the major indexes 
in the period from January 2001 to September 2008 

 Comparing the results of the analysis shown in Figures 1 and 2, there is 
a noticeable change in the direction of information flow between the stock ex-
changes studied, which occurred during the period from October 2008 to April 
2009. There is no longer a direct connection between DJIA and NASDAQ indi-
ces on the U.S. market, however the London FTSE250 index gained  a more 
direct influence on German DAX index on the European market. Both stock 
exchanges (British and German) have had an impact on Polish WIG20 index. 
French stock floor which had previously provided the Polish market with in-
formation contained in the S&P500 no longer fulfils this purpose and currently 
"yesterday's" changes of the American S&P500 directly affect the WIG20. 
The Warsaw Stock Exchange has lost its position, previously shared with the 
London Stock Exchange, and does not shape exchange rates in Shanghai any-
more. 

 

Figure 3. Mutual effect of exchange rates associated with the countries where 
the analysed stock exchanges come from 



The Study of Interdependence Between Capital and Currency Markets ... 117

 The knowledge of dependencies inside the markets is desirable to examine 
the mutual influence of stock markets and exchange rates. In this case, similarly 
as for stock exchanges connections, dependencies of rates of return on exchange 
rates associated with the countries where the investigated stock exchange come 
from were determined. These dependencies for individual exchange rates are 
presented in Figure 3. 
 The analysis allowed to determine the direction of the influence the selected 
stock exchanges and exchange rates have on one another. The results of these 
interactions are shown in Figure 4. 
 

 
Figure 4. Mutual effect of stock indexes and exchange rates associated with 

the countries where the analysed stock exchanges come from 

4. Summary 
 The study confirmed the intuitive assumption about the interdependence 
of stock markets and exchange rates. In majority of cases, exchange rates de-
pend on the stock exchanges indices changes, but not vice versa, what derives 
directly from investors’ tendency to generate profit. In the situation where stock 
floor is dominated by supply and shares prices of companies are dropping, play-
ers withdraw their capital, exacerbating the already declining trend, and invest 
in a foreign currency. Therefore, the usual changes of stock indices and ex-
change rates are negatively correlated with one another.  
 The S&P500 and the CAC40 indices have the biggest influence 
on exchange rates in a selected group of stock exchanges. Their combined im-
pact shapes, inter alia, EUR/PLN exchange rate. The CNY/PLN exchange rate 
depends in turn on the DJIA and the CAC40 index. This is due to the fact that 
the U.S. economy cooperates closely with the Chinese economy and 
in consequence has a strong influence on the Yuan. From the perspective 
of Polish stock market it is interesting that the WIG20 index has an impact 



Tomasz Chruściński 118 

on the CNY/USD exchange rate. This dependence may result from a shared 
impact of the non-distant, considering time, Warsaw stock exchange and the 
London stock exchange on indices of the Shanghai stock exchange. In such 
a case the connection with SSE exchange is very normal. 
 In this study, only EUR/GBP exchange rate turned out to be independent 
of any stock exchange. On the other hand, its value depends on USD/GBP ex-
change rate. As far as the interdependence of capital markets  and exchange 
rates is concerned, the analytical results presented in the paper are fragmentary. 
Models computation is a complex and time-consuming process, therefore mod-
els comparisons and analysis presented in this paper have an illustrative value 
only and provide an example of MGARCH models usage.  

References  
Bollerslev, T., Engle, R., Wooldridge, J. (1988), A Capital Asset Pricing Model with Time-

Varying Covariance, Journal of Political Economy, University of Chicago Press, 96(1). 
Bollerslev, T. (1986), Generalized Autoregressive Conditional Heteroskedasticity, Journal 

of Econometrics, 31, 307–327. 
Chruściński, T. (2008), Analiza wielowymiarowa giełd papierów wartościowych na świecie 

(Multivariate Analysis of Stock Exchanges in the World), Wiadomości statystyczne (Stati-
stical News), GUS, Warszawa, No. 9, 50–61. 

Chruściński, T. (2009), Wykorzystanie wielowymiarowych modeli klasy GARCH do badania 
wzajemnego wpływu rynków finansowych na świecie (Using multivariate GARCH Mo-
dels to Examine the Mutual Influence of Financial Markets in the World), Equilibrium, 
Wydawnictwo Naukowe UMK, Toruń, 1(2), 61–68. 

Engle, R. (1982), Autoregressive Conditional Heteroskedasticity with Estimates of the Variance 
of UK Inflation, Econometrica, 50, 987–1008. 

Fiszeder, P., (2009), Modele klasy GARCH w empirycznych badaniach finansowych (GARCH 
Models in Empirical Studies of Financial), Wydawnictwo UMK, Toruń. 

Osińska, M. (2006), Ekonometria finansowa (Financial Econometrics), PWE, Warszawa. 
Piontek, K. (2006), Wyzwania praktyczne w modelowaniu wielowymiarowych procesów 

GARCH (The Practical Challenges in the Modeling of Multivariate GARCH Processes), 
Taksonomia (Taxonomy), Akademia Ekonomiczna, Wrocław, No. 13. 

Yang, W. (2001), M-GARCH Hedge Ratios and Hedging Effectiveness in Australian Futures 
Markets, Edith Cowan University. 

Badanie współzależności rynków kapitałowych i walutowych 
z zastosowaniem modeli MGARCH 

Z a r y s  t r e ś c i. W artykule podjęto próbę zbadania wzajemnych relacji pomiędzy wybranymi 
indeksami giełdowymi, relacji pomiędzy kursami walut, a następnie określono kierunek przepły-
wu informacji pomiędzy rynkami kapitałowym i walutowym. Do oszacowania współzależności 
instrumentów inwestycyjnych posłużył wielowymiarowy model klasy GARCH. Praca jest konty-
nuacją poprzednich badań autora związanych z taksonometryczną klasyfikacją giełd na świecie. 

S ł o w a  k l u c z o w e: wielowymiarowy model GARCH, analiza współzależności, giełdy pa-
pierów wartościowych, waluty. 


