THE MULTIVARIATE DCC-GARCH MODEL WITH INTERDEPENDENCE AMONG MARKETS IN CONDITIONAL VARIANCES’ EQUATIONS

The article seeks to investigate the issue of interdependence that during crisis periods in the capital markets is of particular importance due to the likelihood of causing a crisis in the real economy. The research objective of the article is to identify this interdependence in volatility. Therefore, first we propose our own modification of the DCC-GARCH model which is so designed as to test for interdependence in conditional variance. Then, the DCC-GARCH-In model was used to study interdependence in volatility of selected stock market indices. The results of the research confirmed the presence of interdependence among the selected markets.


INTRODUCTION
The development of the modern global world economy caused the occurrence of interdependence between individual economies, in fact, in every aspect of their economic activities.This interdependence is a natural mechanism that ensures a proper functioning of capital markets.Another property of the global nature of capital markets is the transmission of shock phenomena that easily transfer from one market to another.The existence of interdependence between markets allows mitigating shocks of various kinds.The observed transmission of shocks causes disruption in the functioning of markets, but in the long run the situation regains its balance.In the global economy there are also situations where transmissions of shocks may not be mitigated by the existing interdependence between markets.In a specific crisis period, subsequent transmissions of shocks, in addition to creating distortions in the functioning of markets, may also raise significantly the level of interdependence between markets.An increase in interdependence may, in turn, strengthen unpredictably a crisis situation and lead to upsetting the functioning of both financial markets and real economies.It means that considering interdependence between markets is significant for a better understanding of the functioning of stock markets (see Bekaert, Wu, 2000;Pritsker, 2001;Forbes, Rigobon, 2002;Baur, 2003;Pericoli, Sbracia, 2003;Corsetti, Pericoli, Sbracia, 2005;Fiszeder, 2009;Billio, Caporin, 2010;Doman, Doman, 2014;Burzała, 2014;Burzała, 2015).
Under the process of on-going globalization one can observe a steady increase in interdependence between markets resulting from fundamental linkages.In addition, during a crisis period fundamental linkages are disrupted by shock transmissions.The implication is that there is a change in fundamental linkages.The result is that under crisis periods it is hardly possible to distinguish explicitly the impact of fundamental linkages on the increase in interdependence between markets from the impact of other factors.Under a tranquillity period we probably deal with the transmission of shocks stemming mostly from fundamental linkages.The existing interdependence between markets allows the elimination of shocks and regaining the state of balance by markets.Under a crisis period it is possible to generate shock transmissions caused mostly by factors not related to the existing fundamental linkages and evoke a change in fundamental linkages resulting from these factors.An unpredictable change in fundamental linkages in conjunction with the growth of interdependence between markets may contribute to the development of a crisis even to a larger degree than non-fundamental factors.
Research into the phenomenon of interdependence between markets is very important, since in today's economy a crisis situation on capital markets, or, more broadly, on financial markets, can shift very quickly to the real sphere and lead to an economic crisis.In addition, the 2008 world financial crisis pointed to the fact that collapses of large financial entities with international connections or even of countries' financial systems are quite likely to happen.Moreover, an increase in the level of interdependence among markets resulting from the on-going globalization, on the one hand, allows a better neutralization of subsequent shock transmissions in a crisis situation; on the other hand, however, under contagious crisis periods it may contribute to the strengthening and extension of this situation.Therefore, identification of interdependence between markets seems to be an essential element in the evaluation of risk related to the functioning of capital markets, and, in consequence, in the determination of the tools and actions for such situations.
The objective of the present article is to identify interdependence between markets for volatility.The article proposes the DCC-GARCH-In (In for interdependence) model whose design takes into account the impact of the volatility of other markets.This model is an extended specification of the DCC-GARCH model, and this extension can be regarded as the determination of the level of interdependence between markets in volatility.The DCC-GARCH model was applied in this paper due to its advantages, i.e., the relatively easy parameters estimation and simple interpretation of results.The model specification proposed by the authors will be used for the purpose of analysis of selected capital markets regarding interdependence in volatility.The research performed allowed the identification of a significant increase in interdependence for conditional variance during selected periods.Furthermore, the estimation of the proposed DCC-GARCH-In model parameters showed that interdependence between markets largely determines the volatility of individual capital markets and should be taken into account when modelling conditional variance.

THE DCC-GARCH-IN MODEL SPECIFICATION
The introduction of the GARCH class models allowed the modelling of the conditional variance for individual assets or indices.However, it was pointed very quickly to the need to account for interdependence between the studied markets.In 2002 Engle proposed a DCC-GARCH3 model whose construction made it possible to analyse interdependence between markets by estimating the time-varying conditional correlation.The DCC-GARCH model can be written as (Engle, 2009) , ( 5) where Y t -the multivariate process of returns, μ t -the vector of conditional means of returns, H i,t -the conditional variance for i-th returns, where i = 1,…,N, V t-1 (η t ) -the conditional covariance matrix of the residuals η t based on the past information set, R t -the time-varying conditional correlation matrix, Q t -a positive-definite quasi correlation matrix, ω i , α i , β i -the parameters of the conditional variance equation, where i = 1,…,N, α, β -the parameters of the conditional correlation equation, -the unconditional correlation matrix of the epsilons and can be estimated as .
In the DCC-GARCH model the conditional variance depends on the lagged conditional variance and on squared returns with the consideration of conditional means.However, in the case of the conditional correlation equation, the variables describing are the standardized residuals ε t and lagged conditional correlations Q t -1 .In addition, to guarantee H t to be positive definite the parameters α, β must satisfy the conditions, α ≥ 0, β ≥ 0 and α + β < 1.
The authors propose to extend the DCC-GARCH model by taking into account the volatility of other markets in the conditional variance equation.This procedure will allow one to capture the interdependence between markets in volatility.The DCC-GARCH-In model4 specification is as follows: , ( 8) , where w i,j = 0 for i = j, (10) , ( 11) -the weighted mean of the conditional variances based on other where j = 1,…, N and j ≠ i, γ i -the parameter of the , V t-1 (η t ) -the conditional covariance matrix of the residuals η t based on the past information set, -the conditional variance for j = 1,…, N, w i,j -the weights specifying the share of conditional variance, R t -the time-varying conditional correlation matrix, Q t -a positive-definite quasi correlation matrix, -the unconditional correlation matrix of the epsilons and can be estimated as .
In this paper the normal distribution is assumed as the error distribution.It is possible to assume the fat-tailed distribution, e.g., t-distribution or skew t-distribution which may lead to a better fit of the model to the data.
The factor exhibits the weighted mean of conditional variances from remaining equations of the model.The introduction of the factor allows taking into account the N -1 conditional variances in each equation.
A separate issue is the determination of the weights matrix.Due to the possibility of transferring capital between stock markets, it is assumed that all of the stock markets under study are considered to be neighbours to each other.Therefore, it is necessary to determine the values of the elements of the weights matrix that will reflect the share of j-th conditional variances in formula ( 10).The weights matrix can be expressed by the following formula: It is assumed that the weight values w i,j for i = j are equal to 0, since in the i-conditional variance equation the variable is already present.In this paper it is assumed that weights are given prior to the estimation.One may wish to estimate weights, but this leads to a high number of additional parameters to estimate in high dimensional cases, which could be not feasible.The method of determining the weights can be based on information describing individual markets or on the determination of arbitrary weights drawing on the researcher's knowledge and experience.The weights can be determined on the basis of stock market capitalization, GDP per capita, or on the basis of the share of stock market capitalization in GDP.In the article the authors took capitalization of stock markets as the criterion.Therefore, the values of the weight matrix can be determined in the following way: (16) where the sum of the weights for each row is 1.
The estim ation of the DCC-GARCH-In model parameters can be carried out using the maximum likelihood method.Similar to the case of the DCC-GARCH model, one may use the two-step estimation method described by Engle (2002Engle ( , 2009)), where the following logarithm of the likelihood function is used: The two-step method is a simplification consisting in estimating separately the parameters of the conditional variance equations and means in the first step, and in estimating the parameters of the conditional correlation equations in the second step.
It is worth noting that in the first step all of the parameters of the conditional variance equations and means are estimated simultaneously utilizing the first component of log likelihood function L 1 (θ 1 ).
The reason behind this is that the i-th conditional variance depends on all remaining conditional variances.The second step is about estimating the parameters of the conditional correlation equation.The second component of the logarithm of the likelihood function can be expressed using the following formula: .( 19)

EMPIRICAL RESEARCH
The empirical research used time series of the selected stock market indices -WIG, BUX, DAX, S&P 500, FTSE 100, CAC 40, KOSPI, BOVESPA, SSE, HSI, RTS, SSMI, NIKKEI 225, ATG, MIBTEL5 and adequate capitalization of stock markets6 .For this purpose we took daily observations covering the period from 3 January 2000 to 3 January 2012, which gave us a total of T = 3000 observations.The assumption of such a long time period allowed the consideration of two crisis periods (the 2000-2002 dot-com bubble and the 2007-2009 global financial crisis).The study used logarithmic returns . For the estimation purpose, we applied the maximum likelihood method with a conditional normal distribution for both the DCC-GARCH and the DCC-GARCH-In models.
Table 1 shows the weights matrix determined by the capitalization of the stock markets in accordance with formul a (16).As can be seen, the diagonal elements of the weights matrix equal zero, which corresponds to the previous assumptions of the model.For instance, the conditional variance equation of the DCC-GARCH-In model for the WIG index take the following form: (20) Tables 2 and table 3 contain the results of the estimation of th e DCC-GARCH and DCC-GARCH-In models parameters.In the case of the stock market indices under study, the constant was not taken into account in the conditional mean equation, since for each equation the constant was found to be statistically insignificant at the significance level of 5%.The parameter γ i was found to be statistically significant for the following indices: WIG, BUX, DAX, S&P 500, FTSE 100, CAC 40, BOVESPA, RTS, SSMI, NIKKEI 225, MIBTEL, which indicates the presence of interdependence in volatility among the markets.In the case of the ATG, HSI, SSE and KOSPI indices, the parameter γ i was found to be statistically insignificant at the significance level of 5%.The highest parameter estimate at the level of 0.0615 was obtained for the conditional variance equation of the BOVESPA.The lowest parameter estimate at the level of 0.0046 was obtained for the parameter γ i in the equation for the KOSPI index.Statistically significant parameters γ i indicate the presence of a stronger linkage in volatility between the selected index and remaining markets.Additionally, we compared the DCC-GARCH model with the DCC-GARCH-In model using standard Lagrange Ratio test, but applying it only to the conditional variance equations, i.e., utilizing .The obtained test statistic (LR = 106.15with p-value lower than 0.01) means that the DCC-GARCH-In fits the data better than the standard DCC-GARCH model.Conditional correlations for both models are usually very similar.This is due to the fact that essentially the conditional variance does not impact the conditional correlation.The only linkage that exists refers to the standardized residuals ε t .The results obtained for three indices (WIG, DAX, S&P500) will be interpreted to provide an example.The values of the factor are similar for all markets, too.The factor allows taking into account the weighted mean of volatility of other markets.In the case of the WIG and the DAX the values of this factor are at a similar level.However, the values of the factor for the S&P500 index achieve a higher level (see figure 4).
We can see that the volatility of the factor is much higher under turmoil periods in the capital markets, than in periods of tranquillity, due to the design of this factor.Next, figure 5 shows the values of the factor multiplied by the parameter estimate of γ i for the S&P500, WIG and DAX indices.For all of the three indices the parameter γ i was found to be statistically significant.The greatest value of the parameter estimate γ i was obtained for the DAX index, and the lowest for the WIG index.The parameter estimate γ i for the DAX index is one and half times greater than the estimate for the S&P500 index, and five times higher than the estimate for the WIG index.Due to the expression , the DCC-GARCH-In model specification allows for the consideration of the interdependence between selected markets in volatility.The entire interdependence is partially composed of the interdependence of fundamental linkages and partially of contagion stemming from crisis periods.It is not possible to clearly separate and determine the structures of these two parts.Based on the parameter estimates of the DCC-GARCH-In model, it can be asserted, however, that there occurs interdependence within the volatility of the studied markets.The entire study period was highly heterogeneous and contained as many as two crisis periods.In subsequent crisis periods, i.e., during the 2000-2002 dot-com bubble and the 2007-2009 global financial crisis, the interdependence of market volatility and its structure could change.It seems reasonable to attempt in further research to estimate parameters for the two crisis periods as well as for the tranquillity period.A considerable difference in the parameter estimates γ i for the tranquillity period and for the crisis period would indicate a considerable change in the interdependence between markets in volatility.) results in either an increase or a decrease in the value of the conditional variance in relation to the value resulting from the DCC-GARCH model.It should be emphasized that the differences in the conditional variance for both models reach 40%, which indicates that the factor within the process of explaining the conditional variance is considerable.As a result of the consideration of the information about the interdependence of the markets, we received different levels of volatility, compared with the levels of volatility obtained exclusively based on information on the domestic market (DCC-GARCH).The article seeks to investigate the issue of interdependence that during crisis periods in the capital markets is of particular importance due to the likelihood of causing a crisis in the real economy.The research objective of the article is to identify this interdependence in volatility.Therefore, first we propose our own modification of the DCC-GARCH model which is so designed as to test for interdependence in conditional variance.Then, the DCC-GARCH-In model was used to study interdependence in volatility of selected stock market indices.The results of the research confirmed the presence of interdependence among the selected markets.

Figure 1 .
Figure 1.The conditional correlation between the WIG and the DAX indices Source: own elaboration.

Figure 2 .
Figure 2. The conditional correlation between the WIG and the S&P500 indices Source: own elaboration.

Figure 3 .
Figure 3.The conditional correlation between the DAX and S&P500 indices Source: own elaboration.
Figures 1 to 3 show the values of conditional correlations obtained from the DCC-GARCH and the DCC-GARCH-In for pairs of the WIG-DAX, WIG-S&P500, DAX-S&P500.

Figure
Figure 4.The variable for indices DAX, S&P500 and WIG Source: own elaboration.

Figure 5 .
Figure 5.The values of for indices DAX, S&P500 and WIG Source: own elaboration.

Figu re 6 .
Figu re 6.The conditional variance obtained from the DCC-GARCH and DCC-GARCH-In models for the WIG index Source: own elaboration.

Figure 7 .
Figure 7.The conditional variance obtained from the DCC-GARCH and DCC-GARCH-In models for the DAX index Source: own elaboration.

Figure 8 .
Figure 8.The conditional variance obtained from the DCC-GARCH and DCC-GARCH-In models for the S&P500 index Source: own elaboration.

Table 1 .
The assumed weight matrix based on the capitalization of stock markets Variable w i1 w i2 w i3 w i4 w i5 w i6 w i7 w i8 w i9 w i10 w i11 w i12 w i13 w i14 w i15

Table 2 .
The results of the estimation of the multivariate DCC-GARCH model parameters

Table 3 .
The results of the estimation of the multivariate DCC-GARCH-In model parameters