University of Groningen Faculty of Economics & Business
MSc Finance
Volatility spillovers between the Benelux’ stock market indices
January 5, 2019
Author: Jelmer Potkamp Student number: s2363003 Supervisor: Steffen Eriksen
Abstract: This paper aims to investigate volatility spillovers between the Benelux’ stock market indices. This is relevant to investors operating in the Benelux, because it can help to create benefits from stock selection and portfolio diversification. Returns of the AEX, BEL 20 and LuxX Index were used to capture volatility spillovers between the three stock markets. Daily close-to-close log returns were collected from January 2, 2004 until July 29, 2016, resulting in 3222 observations. The model used to capture the volatility spillover effects is a multivariate GARCH BEKK (1, 1) model. One trivariate model and three bivariate models were estimated. The results show volatility spillovers from The Netherlands to Luxembourg, from Belgium to Luxembourg and from Luxembourg to Belgium. Shocks occurring in all stock markets impact the volatility of the other stock markets, except from Luxembourg to The Netherlands and from The Netherlands to Belgium. Thus, volatility spillovers do exist between the Benelux’ stock market indices and the three countries are therefore partially interdependent in terms of their stock market’s volatility. This finding can be used by investors and portfolio managers to optimize their investments.
1. Introduction
Within the current era of financial globalization, financial markets are becoming increasingly integrated (Raghavan and Dark, 2008). In order to optimize investments, it is crucial for investors and portfolio managers to both understand the direction of this increasing integration and to have knowledge of the characteristics of the financial instruments in the market they operate in. An important aspect of a stock is its volatility, which is measured by the standard deviation or variance of a stock’s return. A stock’s volatility displays uncertainty and risk about the value of the stock (Singh, Kumar and Pandey, 2010). If financial markets become integrated, the volatility of one financial market can spill over to other financial markets. Events in one market can then explain the volatility of another market (Ghini and Saidi, 2017). This phenomenon is known as volatility spillovers.
Much research has investigated volatility spillover effects between stock markets, with a large focus on the U.S. or Asia and to a lesser degree focusing on stock markets within the Eurozone1.
A gap in the literature exists regarding the investigation of volatility spillovers between the Benelux’ stock market indices. In this paper, the integration of the stock markets of The Netherlands, Belgium and Luxembourg will be investigated, by assessing the volatility spillovers between these countries for the first time. This leads to the following research question: Do volatility spillovers between the Dutch, Belgian and Luxembourgish stock market
indices exist? The answer to this research question is useful for investors and companies that
invest and operate in the Benelux because benefits from portfolio diversification may be reduced if the three stock markets are integrated in terms of their volatility.
First, it is important to fully understand the concept of volatility and volatility spillovers, the stock market dynamics within the Benelux, and the level of integration between the three nations. This is necessary for making expectations of volatility spillovers between the three stock market indices. The next section of this paper will elaborate on these concepts. Previous studies that are relevant to this paper will also be outlined in that section.
The third section describes the used data and methodology. The final dataset contains 3222 observations of daily close-to-close log returns from January 2, 2004 until July 29, 2016. This time frame covers the financial crisis and some preceding and following years. The choice for this dataset is relevant for the potential outcomes of this study, and it is worthwhile to assess the influence of the financial crisis on the volatility of financial markets in The Netherlands, Belgium and Luxembourg. In addition, the employed methodology is elaborated on in section three. Details of different econometric methods to model volatility and volatility spillovers as well as the multivariate GARCH BEKK (1, 1) model (which is applied in this paper), will be
explained. Four models are applied in this paper. First, all three indices will be simultaneously analyzed in a trivariate application, after which the indices will be analyzed in three bivariate cases.
The fourth section contains the results obtained from the four estimated models and explains how these results can be used to draw relevant implications for investors, companies and portfolio managers that invest and operate in the Benelux. The results from these four models are also compared to assess their robustness. Regarding the results, volatility spillovers are found from The Netherlands to Luxembourg, from Belgium to Luxembourg and from Luxembourg to Belgium. The three stock markets are therefore interdependent in terms of their volatility, which may result in a reduction in portfolio diversification benefits when investing in these stock markets simultaneously. The results of this paper are in line with other studies that focused on stock markets within the Eurozone. Billio and Pelizzon (2003), Connor and Golubovskaja (2012) and Alexakis and Vasila (2013) all found significant volatility spillovers between stock markets within the Eurozone. This paper contributes to the current available literature by supporting the presence of volatility spillovers within the Eurozone and extending this to the stock markets in the Benelux, which are not incorporated in any previous paper2.
Lastly, the conclusions of this paper are presented in the fifth section. The paper and the empirical results are summarized first, followed by the implications, concluding remarks and potential avenues for future research.
2. Literature
The following section will provide a brief overview of the most important literature found relating to volatility spillovers. Section 2.1 will describe volatility within stock markets. After that, the concept of volatility spillovers will be described in section 2.2. Section 2.3 will elaborate on the dynamics within the Benelux. Lastly, section 2.4 will end with the research question.
2.1 Volatility
The purpose of this paper is to investigate dynamics within stock markets. Both a stock’s return and its volatility can explain those dynamics (Peng and Ng, 2012). However, Peng and Ng (2012, pp 50) also argued that “the rate of change of market volatility is much higher than the rate of change of market return”. Therefore, a stock’s volatility explains more of the market dynamics than a stock’s return does (Badshah, 2018). For those reasons, this paper will use a stock market index’ volatility to measure interdependence between financial markets, rather than a stock market index’ return.
Volatility is a measure of risk and uncertainty about a stock’s value (Singh, Kumar and Pandey, 2010; Brooks, 2014) and is measured by the standard deviation or variance of a stock’s return. In periods of high volatility, the price of the stock fluctuates strongly, which increases perceived uncertainty. Subsequently, high levels of volatility result in both an increased upside potential and an increased downside risk for the investor. Low levels of volatility are more desirable for investors and portfolio managers, as it relates to a reduced uncertainty of a stock’s return.
In order to attempt to reduce volatility, understanding the different sources of volatility is critical. Hull (2015) and Ozenbas, Pagano and Schwartz (2010) argued that volatility is caused by new information reaching the market. As soon as new information is available on the market, investors’ opinions regarding the value of the stock might change. This will cause a change in the price and the return of the stock, resulting in higher volatility. Volatility thus reflects the sentiment of an investor with regard to the value of the stock. However, earlier research by Roll (1984) contradicts this proposed mechanism. This study investigated volatility in the futures market for orange juice and attempted to explain volatility with the most important news elements for orange juice futures, namely weather announcements. Results showed that the weather only explained a very small part of the volatility in orange juice prices. This may suggest that new information is not the most important factor explaining volatility. Fleming, Kirby and Ostdiek (2006) argued that different sources of volatility can also be disentangled by discriminating between trading and non-trading periods. Fama (1965) argued that it is reasonable to assume that weekend or holiday returns display higher variances than trading-period returns. Fama (1965) is the first author that succeeded in identifying this phenomenon that has been replicated often since then. The difference between tranding and non-trading periods was, however not as high as expected. Fama (1965) hypothesized that the variance between the close of trading on Friday and the close of trading on Monday would be three times higher than the variance between the close of trading between two trading days. However, the weekend variance was only 22% higher than the within-week variance. Roll (1984) and French and Roll (1986) found similar results. The weekend variance in their studies was respectively 54% and 10.7% higher than the within-week variance. In sum, discriminating between trading and non-trading periods does not seem to be the most appropriate way to disentangle different sources of volatility since weekend variances and within-week variances do not differ much.
volatility. Attanasio (1990) described how asymmetric information can lead to higher levels of volatility. In case of asymmetric information, stock prices may not fully reveal all information that is available, because of the presence of exogenous noise. “If this is the case it is possible that such noise gets ‘amplified’ by the behavior of the uniformed traders; this will make prices more volatile than in a situation of symmetric information” (Attanasio, 1990, pp 159). In line with this, Du and Wei (2003) argued that stock market volatility increases with the number of informed agents. Importantly, both Attanasio (1990) and Du and Wei (2003) assumed in their research that new information influences the volatility of a stock’s return, which contradicts the results of Roll (1984). Following the results of Attanasio (1990) and Du and Wei (2003), information asymmetries lead to an increase in a stock’s volatility. Next to that, Dellas and Hess (2005) showed that liquidity reduces stock volatility, because sufficient liquidity allows investors to smooth their trades. Thus, a qualitatively well performing banking system in terms of the availability of liquidity will decrease the stock market’s volatility. Lastly, the results of Green, Maggioni and Murinde (2000) showed a significant increase in the stock market’s volatility as transaction costs increase. Transaction costs in this case include “dealer’s margins, broker’s commissions and stamp duty” (Green, Maggioni and Murinde, 2000, pp 586). These findings may contribute to the investor’s ability to reduce the volatility of their stock’s returns. Volatility can also be discerned in terms of its degree. Ozenbas, Pagano and Schwartz (2010) provided some information about the patterns of intraday volatility. They argued that the pattern of a stock’s volatility follows a U-shape. The opening of trading is a stressful period because share values are determined based on new information that occurred overnight. The end of the trading is also a stressful period since traders are seeking to execute their last orders just before closing. Thus, at the beginning and at the end of a trading day, the volatility of a stock is usually higher than the period in between. This information is useful when choosing the data to use for analyzing and comparing stock market features. The volatility should be measured during the same time of the day, to make a good comparison.
We have seen that volatility is an important part of the characteristics of a stock. It reveals much about the sentiment of investors and their opinion regarding the value of the stock. Studies show that the within-week variance is usually a little lower than the weekend variance, but the difference is not proportional to the difference in days. Several factors can explain the origin of volatility. The most important one is new information that comes to the market, despite contradicting results by Roll (1984).
2.2 Volatility spillover
part of their portfolio in that stock market and invest in some other stock market. When the investor moves to other stock markets, the shocks will subsequently have an impact on that stock market as well (Bagchi, Dandapat and Chatterjee, 2016). Bagchi, Dandapat and Chatterjee (2016) argued that volatility spillovers can also take place between a stock market and some other financial market. Investors move their funds away from the stock market to other monetary markets. These types of volatility spillovers will not be incorporated in this study since the data consist of only stock market returns.
Central to the discipline of volatility spillovers is the concept of capital market integration within the European Union. Financial markets get more and more integrated due to the current era of financial globalization (Raghavan and Dark, 2008). Alexakis and Vasila (2013) also stressed that trade barriers are gradually removed by the European Union, to fulfill its vision of an integrated European stock market. Detailed examination of the growing stock market integration process by Alexakis and Vasila (2013) showed that the Markets in Financial Instruments Directive (MiFID) opened a new era in the integration process. The MiFID is a regulation, with the purpose of an increase in financial markets’ transparency across the European Union and liberalization within the European Union. Alexakis and Vasila (2013) also pointed out that economic and monetary convergence, new electronic platforms and a single currency all contributed to a reduction in transaction costs, an improvement in efficiency, an increase in the quality of public information and integration of European cross-border trade. Next, Marelli and Signorelli (2017) explained that economic integration is to be achieved by a Customs Union3, a common agricultural policy, free movements of persons and services,
monetary policy coordination among the national central banks, and a competition policy. These are established in the Treaty of Rome, signed in 1957 and this still holds for the countries based in Europe. Fratzscher (2002) explained that the degree of integration has increased even more since the introduction of the Euro, which in turn requires more coordination among the national central banks. The still increasing integration and interdependence of the European stock market is something important to consider, because it might result in more volatility spillover effects taking place.
Mentioned previously, volatility is measured by the variance or standard deviation of a stock’s return. Longin and Solnik (1995) argued that periods of high volatility go hand in hand with a high level of correlation between the different stocks. We can naturally derive this statement from the formula for correlations and variances4. If variances and covariances of some national
and foreign stocks are high, the corresponding correlations will be high as well. Majewska and Olbryś (2017, pp 46) explained that “a relatively high degree of financial integration is usually
3 Adopting a Customs Union involves the introduction of common external tariffs as well as the abolition of
internal tariffs (Marelli and Signorelli, 2017).
4 Define the covariance between national stock X and foreign stock Y by 𝜎
"#= 𝜌"#𝜎"𝜎#, with 𝜌"# being the
coupled with high cross-market correlations and therefore it might produce a substantial drop in cross-border portfolio diversification benefits”. We have seen that the European stock markets become increasingly integrated. Benefits from international portfolio diversification might be reduced due to this financial integration.
2.3 Benelux
A number of studies have examined volatility spillovers between stock market indices (see e.g., Cheng and Yip, 2017; Fleming, Kirby and Ostdiek, 1998; Harris and Pisedtasalasai, 2006; Ng, 2000; Nikmanesh, Nor, Sarmidi and Janor, 2014; Wang, Pan and Wu, 2017). All these papers focused on the stock markets within the U.S. or Asia. A limited number of papers that focused on stock markets within the Eurozone were found and most of those focused on the bigger countries within the Eurozone, based on the corresponding GDP per country. Germany and France were incorporated by most of the studies focusing on volatility spillovers between stock market indices and also Italy and Spain were investigated by a few authors. A gap was found in the literature regarding volatility spillovers between stock market indices within the Eurozone. The Netherlands was incorporated in only one previous study (see Connor and Golubovskaja, 2012). This country belongs to the Benelux, next to Belgium and Luxembourg which both have not been investigated in any prior research with respect to volatility spillovers. I aim to fill the current gap in the literature by investigating volatility spillovers between the Dutch (AEX), Belgian (BEL 20) and Luxembourgish (LuxX Index) indices. Most interesting for this study is the degree of integration between the three countries, which may tell us something about the extent to which volatility spillovers take place between the three countries’ indices.
Drucker (2015, pp 66) described that “all three Benelux nations operate in fairly diversified economies based on services and industry”. Beine, Cosma and Vermeulen (2010) explained that countries will not react in the same way to a certain shock if the considered countries operate in a different economy and they use direct trade as a measure for integration. With regard to the strength of direct trade, The Netherlands is the biggest supplier for imported goods for Belgium. For Luxembourg, Belgium is the biggest supplier for imported goods, and for The Netherlands, Belgium is the second biggest supplier of imported goods5. Also, Ghini and Saidi
(2017) stated that the financial crisis has revealed high levels of interdependence between financial markets. Last, it is described in section 2.2 that countries within Europe are financially and economically integrated and that the current era of financial globalization is still increasing the integration of the capital market within Europe.
5 The Trade Map database of the International Trade Centre provides data regarding import and export between
2.4 Research question
We can conclude that The Netherlands, Belgium and Luxembourg are integrated in a sense that the strength of direct trade is high since the Benelux’ nations are important trade partners for each other. The capital markets and economies of the three countries are integrated with each other because they all belong to the same economic and monetary union. However, we will expect that the three countries will not react in the same way to an occurring shock since the countries do not operate in the same economy in terms of service or industry. If a shock happens in one of the three Benelux’ countries, this might trigger an investor to withdraw from this market and invest in one of the other Benelux’ countries. To summarize, it is expected that volatility spillovers do exist between the AEX, BEL 20 and LuxX Index. In other words, it is expected that the Benelux’ financial markets exhibit high levels of interdependence. To test if this is indeed the case, the following research question is to be answered:
Do volatility spillovers between the Dutch, Belgian and Luxembourgish stock market indices exist?
3. Data and methodology
The next section will elaborate on the data and methodology used for this paper. Section 3.1 will describe the collection of the sample data. After that, section 3.2 will explain the most important models that can be used to capture volatility and volatility spillovers. That section will also describe the MGARCH BEKK (1,1) models that will be used in this paper.
3.1 Sample
Volatility spillovers will be investigated between the Dutch, Belgium and Luxembourgish stock market indices. The corresponding stock market indices are respectively the AEX, BEL 20 and LuxX Index. The AEX consists of 25 stocks and all those stocks are traded in Amsterdam. The BEL 20 consists of 20 stocks. 16 of those stocks are traded in Brussels, while three stocks are traded in Amsterdam and one in Paris6. The LuxX Index consists of nine stocks
traded in Luxembourg City.
Data will be collected from the database Eikon over the period January 2, 2004 until July 29, 2016. All three indices are price indices. Data will be converted to log returns, as log returns are easier to compare and because they function time-additive (Brooks, 2014). Close prices are collected for all indices, so the final dataset will consist of daily close-to-close log returns7.
The daily close-to-close log return ri,t, of index i at time t can be calculated as
6 Companies that are not listed on Euronext Brussels are also allowed to enter the BEL 20 index, if they meet
some requirements. The Index Rule Book from Euronext (2015) provides the full list of requirements.
7 It is described in the literature section that a stock’s volatility follows a U-shape. It is therefore important to be
𝑟',) = 100% × ln ( 23,4
23,456), (1)
with pi,t being the close price of index i at time t. Fig. 1, fig. 2 and fig. 3 show the dynamics of
all close-to-close log return series.
Initially, some data points were missing for the Luxembourgish stock market index. This is due to different public holidays in Luxembourg. The models to be estimated require a continuous dataset without any missing values. In order to establish a continuous dataset, missing returns are replaced by the previous known return. This strategy follows the methodology of the paper written by Mohammadi and Tan (2015). The result is a dataset consisting of 3222 observations for the AEX, BEL 20 and LuxX Index.
Fig. 1
Daily close-to-close log returns of the AEX index in the period 02/01/04 – 29/07/16.
Fig. 2
Daily close-to-close log returns of the BEL 20 index in the period 02/01/04 – 29/07/16.
Fig. 3
Daily close-to-close log returns of the LuxX index in the period 02/01/04 – 29/07/16.
It can already be seen that all return series follow approximately the same path. The returns did not fluctuate strongly from January 2004 until January 2008. In 2008, returns started to fluctuate a lot, indicating the start of the financial crisis. In the period after 2008, we see that returns evaporated but are still less stable than before the financial crisis. This feature is known as volatility clustering. According to Franq and Zakoïan (2010), it is expected that small returns are followed by small returns and that large returns are followed by large returns. Brooks (2014) also argued that a possible explanation could be that information arrivals are not evenly spaced over time but occur in bunches.
3.2 Methodology
This section will describe the methodology that will be used in this paper. Various models were developed that capture volatility. Historical volatility is the simplest and most widely used way to model volatility. The volatility in terms of variance or standard deviation is calculated over some historical period, and these variances are used to predict future volatilities. However, the use of historical volatility may not result in the most precise estimations, since recent research has shown that more complex models for volatility result in more accurate volatility forecasts (Brooks, 2014). Further, a model is needed that accounts for volatility clustering because the series display this property. Univariate- and multivariate models are examples of such models and those will be discussed throughout this section.
3.2.1 Univariate models for volatility
The most well-known univariate model for volatility is the generalized autoregressive conditionally heteroscedastic (GARCH) model. The GARCH (p, q) model was first developed by Bollerslev (1986) and is defined as
with conditional variance of the error term 𝜎)8 at time t, constant 𝛼
:, squared error 𝑢)>'8 at time t-i and variance 𝜎)>C8 at time t-j. This means that the conditional variance of the error term
depends on q squared errors of the previous period and p lagged variances of the previous period. The conditional variance must strictly be positive, and a negative variance should be avoided (Francq and Zakoïan, 2010). Brooks (2014) argued that GARCH models allow for the variance of the error term to fluctuate over time and that they allow for volatility clustering. Those are the two most important reasons to use GARCH models for volatility modelling. GARCH type of models enforce the volatility to have a symmetric response to both negative and positive shocks (Francq and Zakoïan, 2010). Meanwhile, Xekalaki and Degiannakis (2010) provided us with evidence that a negative shock has more impact on the volatility than a positive shock. Both Gloster, Jagannathan and Runkle (1993) and Nelson (1991) built a model that allows for asymmetric responses. The model of Nelson (1991) also ensures that the estimation will be positive. Both models (the GJR model and the EGARCH model) can be found in Appendix A.
So far, the EGARCH model proposed by Nelson (1991) seems to be the best model to capture volatility. It is a parsimonious model, ensuring non-negative outcomes of the conditional variances and it allows negative and positive shocks to have a different impact on the volatility. However, in this paper volatility spillovers between indices will be modelled. The volatility of an index should therefore not only depend on its own lagged variance and its own lagged error, but also on the covariance with the other indices. The EGARCH model is not capable of capturing such spillovers. Nevertheless, multivariate GARCH models can be used to capture volatility depending on the variances and covariances of all indices used.
3.2.2 Multivariate GARCH models
Different multivariate GARCH models exist and Brooks (2014) gives a brief overview of the different models. The VECH model, the BEKK model, the CCC model and the DCC model are the most widely used multivariate GARCH models.
The constant conditional correlation (CCC) model constructs the correlations between series directly. However, in this model the correlation between series is assumed to be constant over time. This assumption may not always hold and is therefore an unrealistic assumption according to Francq and Zakoïan (2010). The dynamic conditional correlation model (DCC) relates to the CCC model but does allow correlations to vary over time. The DCC model was first proposed by Engle (2002) and is given by
with conditional variance-covariance matrix 𝐻) at time t, conditional standard deviation matrix
𝐷) and conditional correlation matrix 𝑅). The DCC model requires a two-step estimation approach (Buriev, Dewandaru, Zainal and Masih, 2018). First, the standard deviation matrix is obtained by estimating a GARCH (1, 1) model. After that, the conditional variance-covariance matrix has to be estimated using 𝐷) and 𝑅). Most multivariate GARCH models require a one-step estimation approach. Using the DCC model may therefore not be the most efficient and approachable way to capture volatility spillovers.
Another multivariate GARCH model is the vector-half operator GARCH model (VECH model). This model is developed by Bollerslev, Engle and Wooldridge (1988). Brooks (2014, pp 469) argued that for a VECH model “the conditional variances and conditional covariances depend on the lagged values of all of the conditional variances of, and conditional covariances between, all of the asset returns in the series, as well as the lagged squared errors and the error cross-products”. We can therefore assume that applying a VECH model will result in a highly accurate prediction of the parameters. However, using a VECH model for three assets (three indices will be used in this study) will result in 78 parameters to be estimated8. This is
impractical. Another disadvantage of using a VECH model, is that the model does not provide a guarantee for a positive definite covariance matrix9, a result that can naturally be obtained
from the VECH representation. Next, a positive semi-definite covariance matrix would also not be enough for this study, since this type of covariance matrix allows for a variance of zero or higher and the variance is not equal to zero for all analyzed series. Thus, a guarantee of a positive definite covariance matrix is important in this study from both a financial and econometric point of view. For this reason and because of the high number of parameters to estimate, a VECH model will not be the most appropriate model for this study.
A good alternative that resolves these obstacles is the Baba-Engle-Kraft-Kroner (BEKK) model. The BEKK (1, 1) model is first developed by Engle and Kroner (1995) 10. A positive
definite covariance matrix is guaranteed and the number of parameters to be estimated is smaller than in the VECH model. In the VECH model, 78 parameters need to be estimated for
8 Three variables need to be estimated. C is an 𝑁(𝑁 + 1)/2 column vector, and A and B are square parameter
matrices of order 𝑁(𝑁 + 1)/2. For 𝑁 = 3, this will result in 78 (6+36+36) parameters to be estimated (see Brooks, 2014).
9 Let the random variable Y be defined by 𝑌 = 𝑎𝑋 and 𝑣𝑎𝑟(𝑌) = 𝑎∑𝑎′, with ∑ being the covariance matrix.
The variance must implicitly be a positive number, so ∑ must also be positive semi-definite. If the covariance matrix is not positive definite, the variance obtained from the covariance matrix will not be bigger than zero, which is meaningless (Holton, 2003).
10 A decision should be made upon the number of lags p and q to incorporate in the BEKK (p, q) model. Qiao,
three indices. The BEKK (1, 1) model only requires an estimation of 24 parameters when three indices are included11. The full model is defined as
𝐻',) = 𝐶𝐶′ + 𝐵𝐻',)>A𝐵′ + 𝐴Ξ',)>AΞT
',)>A𝐴′ (4)
with Ξ',)|𝜓',)>A ~ 𝑁(0, 𝐻',)),
and conditional variance-covariance matrix Hi,t of index i at time t, innovation column vector
Ξi,t,and information set ψi,t-1 at t-1 so that Ξi,t can be seen as a random error at time t capturing
shocks occurring in index i’s stock market, given all available information at time t-1 (Jayasinghe, Tsui and Zhang, 2014). C is a lower triangular matrix of parameters. Silvennoinen and Teräsvirta (2008, pp 4) described that “the decomposition of the constant term into a product of two triangular matrices is to ensure positive definiteness of Hi,t”. A is the 𝑁 × 𝑁
ARCH parameter matrix describing the relation between the conditional variance and historical innovations, and B is the 𝑁 × 𝑁 GARCH parameter matrix describing the relation between the conditional variance and its first order lag (see Brooks, 2014; Chen and Weng, 2018). In case of three indices (trivariate case), the parameters to be estimated are
𝐴 = X 𝑎YY 𝑎YZ 𝑎Y[ 𝑎ZY 𝑎ZZ 𝑎Z[ 𝑎[Y 𝑎[Z 𝑎[[\, 𝐵 = X 𝑏YY 𝑏YZ 𝑏Y[ 𝑏ZY 𝑏ZZ 𝑏Z[ 𝑏[Y 𝑏[Z 𝑏[[\, 𝐶 = X 𝑐YY 0 0 𝑐ZY 𝑐ZZ 0 𝑐[Y 𝑐[Z 𝑐[[\, (5) with subscript x, b and l corresponding to respectively the AEX, BEL 20 and LuxX Index. The following trivariate model needs to be estimated to capture volatility spillovers:
𝐻) = _
ℎYY,) ℎYZ,) ℎY[,) ℎZY,) ℎZZ,) ℎZ[,) ℎ[Y,) ℎ[Z,) ℎ[[,) a = b 𝑐YY 0 0 𝑐ZY 𝑐ZZ 0 𝑐[Y 𝑐[Z 𝑐[[c b 𝑐YY 𝑐ZY 𝑐[Y 0 𝑐ZZ 𝑐[Z 0 0 𝑐[[c + b 𝑏YY 𝑏YZ 𝑏Y[ 𝑏ZY 𝑏ZZ 𝑏Z[ 𝑏[Y 𝑏[Z 𝑏[[c _
ℎYY,)>A ℎYZ,)>A ℎY[,)>A ℎZY,)>A ℎZZ,)>A ℎZ[,)>A ℎ[Y,)>A ℎ[Z,)>A ℎ[[,)>A
a b 𝑏YY 𝑏ZY 𝑏[Y 𝑏YZ 𝑏ZZ 𝑏[Z 𝑏Y[ 𝑏Z[ 𝑏[[c + b 𝑎YY 𝑎YZ 𝑎Y[ 𝑎ZY 𝑎ZZ 𝑎Z[ 𝑎[Y 𝑎[Z 𝑎[[c _ ΞY,)>A ΞZ,)>A Ξ[,)>Aa [ΞY,)>A ΞZ,)>A Ξ[,)>A] b 𝑎YY 𝑎ZY 𝑎[Y 𝑎YZ 𝑎ZZ 𝑎[Z 𝑎Y[ 𝑎Z[ 𝑎[[c. (6)
11 The number of parameters to be estimated is given by 𝑁8+ 𝑁8+ [f(fgA)
Some authors (see e.g., Brooks, 2014; Ghini and Saidi, 2017; de Oliveira, Maia, de Jesus and Besarria, 2018; Lötkepohl, 2005) argued that a BEKK (1, 1) model may be difficult to estimate as the number of parameters increases, which can have a negative impact on the estimation results. To overcome this problem, three bivariate models including two indices instead of three indices can be estimated12. The three bivariate models will then be used to estimate the
conditional variances of and covariances between respectively the AEX and the BEL 20 (1), the AEX and the LuxX Index (2) and the BEL 20 and LuxX Index (3). In this paper, both the trivariate model and the three bivariate models will be estimated. This will be done because estimating a MGARCH BEKK (1, 1) model becomes more difficult as the number of indices (k) increases. An increase in k usually results in a decrease in the number of significant parameters (Tsay, 2014). Also, it is worthwhile to compare the results of the four different estimations and to discover differences and similarities.
For the bivariate model, the following 11 parameters need to be estimated:
𝐴 = h𝑎𝑎AA 𝑎A8 8A 𝑎88i, 𝐵 = j 𝑏AA 𝑏A8 𝑏8A 𝑏88k, 𝐶 = j 𝑐AA 0 𝑐8A 𝑐88k, (7)
Subscript 1 corresponds to the AEX and subscript 2 corresponds to BEL 20 for the first bivariate estimation. For the second model, subscript 1 corresponds to the AEX and subscript
2 to the LuxX Index. For the last model, subscript 1 and 2 respectively signify the BEL 20 and
LuxX Index.
Those parameters are used to compute the following variance and covariance equations, capturing volatility spillovers:
𝐻) = lℎℎAA,) ℎA8,) 8A,) ℎ88,)m = l𝑐𝑐AA 0 8A 𝑐88m n 𝑐AA 𝑐8A 0 𝑐88o + l𝑏AA 𝑏A8 𝑏8A 𝑏88m l ℎAA,)>A ℎA8,)>A ℎ8A,)>A ℎ88,)>Am l 𝑏AA 𝑏8A 𝑏A8 𝑏88m + n𝑎𝑎AA 𝑎A8 8A 𝑎88o l ΞA,)>A
Ξ8,)>Am [ΞA,)>A Ξ8,)>A] n
𝑎AA 𝑎8A
𝑎A8 𝑎88o. (8)
12 For the bivariate case (𝑁 = 2), the number of parameters to be estimated is given by 𝑁8+ 𝑁8+ [f(fgA) 8 ], this
4. Results
The results will be described in the following section. Section 4.1 contains some descriptive statistics about the return series and an analysis regarding the data. Section 4.2 shows the results of tests for autocorrelation and ARCH-effects. Next, section 4.3 contains the estimation results and interpretation of the MGARCH BEKK (1,1) models, as well as some diagnostic testing.
4.1 Preliminary analysis
Table 1 provides summary statistics and some test results of the returns of the AEX, BEL 20 and LuxX Index. The mean return is highest for the Luxembourgish index while it is lowest for the Dutch index. The minimum and maximum values and the standard deviations are quite similar for all indices. The distribution of the three indices is negatively skewed indicating that most of the data are clustered on the right and that the distribution has a left-hand tail. The kurtosis for all three indices is higher than 3, indicating a leptokurtic distribution of the return series. The Jarque-Bera statistic rejects the null hypotheses for normality for all return series, supporting the leptokurtic distribution and skewness of all series. Lastly, the Augmented-Dickey-Fuller test is performed to test for unit root13. In all three cases, the null hypothesis is
rejected at the 1% significance level, indicating that all three series have no unit roots14. To
conclude, the null hypotheses of non-stationarity and normality are rejected, for all return series.
Table 1
Summary statistics of the AEX, BEL 20 and LuxX Index.
Statistics are provided over the period January 2, 2004 until July 29, 2016, resulting in 3222 observations. P-values smaller than 0.01, 0.05 and 0.10 are indicated by ***, ** and * respectively. ADF=Augmented-Dickey-Fuller.
Returns of stock indices
Coefficient AEX BEL20 LuxX Index
January 2, 2004 until July 29, 2016
Mean 0.0084 0.0131 0.01750 Median 0.0555 0.0488 0.0475 Maximum 10.0280 9.2212 9.1043 Minimum -9.5903 -8.3193 -11.1586 Std. Dev. 1.3288 1.2440 1.3336 Skewness -0.2200 -0.2297 -0.3302 Kurtosis 11.3310 9.3320 9.41970 Jarque-Bera statistic 9345.0850*** 5411.0130*** 5591.2630*** ADF t-statistic -56.9469*** -54.7219*** -54.3344***
13 Palma (2016) argued that we can only draw statistical conclusions from time series if the time series show
strict or weak forms of stationarity. It is therefore important to test whether the indices have unit root or not.
4.2 Autocorrelation and ARCH effects
As described in section 3.2.1, the most important reasons to model volatility using a certain type of (multivariate generalized) autoregressive conditionally heteroskedasticity model is because the variances of the error terms are not constant over time and because the returns show volatility clustering. In other words: the variance of the error term is heteroskedastic, and the volatility of the return series is autocorrelated (Brooks, 2014). It is important to test whether the data used for this paper show autocorrelation and heteroskedasticity. If that is the case, a GARCH type of model is appropriate to capture the volatility of the indices15.
A Ljung-Box Q-statistic test is performed to test for autocorrelation and an ARCH-test is performed to test for heteroskedasticity. Both tests are performed twice. First five lags were included and in the second case 30 lags were included. This is to incapsulate the effect of all trading days, because daily data are used and to see whether results are the same in both cases. Table 2 shows the results of these tests. The null hypotheses of the heteroskedasticity ARCH-test (representing no ARCH-effects) are rejected at any conventional significance level for all series, in both cases, indicating potential heteroskedasticity. Following the results of the Ljung-Box Q-statistic, the null hypotheses (representing no autocorrelation) can be rejected for all series, signifying potential autocorrelations. However, the Q(5)-statistic is only significant at the 5%- and 10% significance level for respectively the BEL 20 and LuxX Index. The Ljung-Box Q(30)-statistic is nevertheless significant at any conventional significance level for all series. It can be concluded that the AEX, BEL 20 and LuxX Index show autocorrelation and ARCH-effects. Thus, the variance of the error term is not constant for all indices and all indices show volatility clustering. Therefore, the most appropriate model to capture volatility of the stock market indices is a GARCH type of model. In that case, the conditional variance depends on both the squared errors and the lagged variances of the previous periods. Since the aim of this paper is to capture spillovers in volatility, a multivariate type of GARCH model is required so that the conditional variance also depends on the covariance with another index.
Table 2
Autocorrelation and heteroskedasticity tests for the AEX, BEL 20 and LuxX Index.
The first and second Ljung-Box follow respectively a 𝜒8(5)- and 𝜒8(30)-distribution since the tests are performed with five and 30 lags. The null hypotheses of no autocorrelation are rejected at the 1% significance level for all series in the case with 30 lags. When including five lags, the null hypotheses of no autocorrelation are rejected at the 1%-, 5%- and 10% significance level for respectively the AEX, BEL 20 and LuxX Index. The ARCH-test follows an F-distribution. Five lags are included in the first case and 30 lags in the second case. In both cases, the null hypotheses of no ARCH-effects are rejected at the 1% significance level for all series. P-values smaller than 0.01, 0.05 and 0.10 are indicated by ***, ** and * respectively.
Ljung-Box Heteroskedasticity ARCH-test
Index Q(5)-statistic Q(30)-statistic F(5)-statistic F(30)-statistic
AEX 23.0960*** 87.4220*** 219.2030*** 53.8097***
BEL 20 12.8510** 64.9650*** 160.5440*** 34.6220***
LuxX Index 9.9445* 58.8810*** 125.2950*** 37.4964***
15 According to the Gauss-Markov assumptions, a linear regression would fit the data if the data did not show
4.3 Estimation results
The MGARCH BEKK (1, 1) models are estimated using R. The “MTS” Package by Tsay and Wood (2018) contains an estimation method for a BEKK (1, 1) model. The estimated parameters are stated in table 3 for the trivariate case and in table 4 for the bivariate cases. The coefficients obtained from the estimation can be used to model the conditional variances of, and covariances between, the AEX, BEL 20 and LuxX Index. Fig. A1 – A6 in Appendix B.1 show the conditional variances and covariances obtained from the trivariate case and fig. A7 – A15 in Appendix B.2, B.3 and B.4 show the conditional variances and covariances of the three bivariate cases.
Table 3
Estimation results of the trivariate (k=3) MGARCH BEKK (1, 1) model.
Subscript x corresponds to the AEX, subscript b to the BEL 20 and subscript l to the LuxX Index. The C parameters are given in the first column, the A parameters are given in the third column and the B parameters are given in the fifth column. The corresponding estimation results are provided in the second, fourth and sixth columns. The standard errors are reported in parentheses. P-values associated with the coefficients smaller than 0.01, 0.05 and 0.10 are indicated by ***, ** and * respectively.
AEX – BEL 20 – LuxX Index
Parameter Coefficient Parameter Coefficient Parameter Coefficient
Table 4
Estimation results of the three bivariate (k=2) MGARCH BEKK (1, 1) models.
Subscript x corresponds to the AEX, subscript b to the BEL 20 and subscript l to the LuxX Index. The parameters are given in the first, third and fifth columns and the corresponding estimation results are provided in respectively the second, fourth and sixth columns. The standard errors are reported in parentheses. P-values associated with the coefficients smaller than 0.01, 0.05 and 0.10 are indicated by ***, ** and * respectively.
AEX – BEL 20 AEX – LuxX Index BEL 20 – LuxX Index
Parameter Coefficient Parameter Coefficient Parameter Coefficient
cxx 0.2658*** cxx 0.2658*** cbb 0.2488*** (-0.0205) (0.0245) (0.0299) cbx 0.2774*** clx 0.1823*** clb 0.2845*** (0.0269) (0.0300) (0.0431) cbb 0.1133*** cll 0.1947*** cll 0.4528*** (0.0131) (0.0274) (0.0785) axx 0.3592*** axx 0.3247*** abb 0.2022*** (0.0486) (0.0276) (0.0299) abx 0.1666*** alx -0.0016 alb -0.3491*** (0.0519) (0.0335) (0.0728) axb 0.0180 axl 0.0788*** abl 0.1671*** (0.0536) (0.0257) (0.0270) abb 0.2686*** all 0.3409*** all 0.5835*** (0.0533) (0.0297) (0.0463) bxx 0.9189*** bxx 0.9298*** bbb 1.000*** (0.0271) (0.0140) (0.0154) bbx -0.0363 blx 0.0199 blb 0.2445*** (0.0260) (0.0173) (0.0552) bxb -0.0184 bxl -0.0447*** bbl -0.1266*** (0.0324) (0.0145) (0.0210) bbb 0.9032*** bll 0.9045*** bll 0.6455*** (0.0302) (0.0163) (0.0749)
4.3.1 Shocks and past variance
To give an interpretation of the estimation outputs, it is important to understand what A and B indicate. Jayasinghe, Tsui and Zhang (2014) describe that the diagonal elements of A capture the relationship between the conditional variances (𝐻',)) in the current period and the error terms in the previous period (Ξ',)>A). The diagonal elements of B capture the relationship between the current conditional variances (𝐻',)) and past variances (𝐻',)>A).
For the trivariate model, the coefficients of the parameters axx (≈ 0.35), abb (≈ 0.18), and all (≈
0.37) are all significant at the 1% significance level. Those coefficients tell us that the conditional variances of the AEX, BEL 20 and LuxX Index are dependent on shocks occurring in that specific market, since they capture the impact of the past error terms. All coefficients
bxx (≈ 1.00), bbb (≈ 0.77), and bll (≈ 0.89) are also significant at the 1% significance level. From
example, has a 100% (bxx ≈ 1.00) impact on the variance of the AEX index in the current period.
Results like these are not surprising when compared with previous research (see e.g., Chen and Weng, 2018; Ghini and Saidi, 2017; Mohammadi and Tan, 2015; Tsay, 2014). In these papers, the impact of the past variance on the current variance is also close to 1.00. Ultimately, it can be said that the impact of the past variances on the current variances is higher than the impact of past shocks on the current variances, since the B estimates are all close to 1.00 and the A estimates closer to zero.
Regarding the three bivariate models, all coefficients axx, abb, all, bxx, bbb and bll are estimated
twice. Again, all coefficients are significant at the 1% significance level, as it is the case in the trivariate model. Table A1 in Appendix C gives a brief overview of all diagonal elements of matrices A and B which are estimated using the four models. For all diagonal elements, the results of the trivariate estimation are close to the results of the two bivariate estimations. Therefore, the results regarding the diagonal elements of matrices A and B of the bivariate models confirm the robustness of the results regarding the diagonal elements of matrices A and
B of the trivariate model. 4.3.2 Spillover effects
As the diagonal elements of the matrices A and B respectively capture the impact of the past error terms of index i and the past conditional variance of index i, on the conditional variance of that same index, the off-diagonal elements of the matrices A and B capture the spillover effects between the three different indices. Regarding the trivariate model, the coefficients abx,
axl and abl are all significant at the 1% significance level while coefficient alb is only significant
at the 10% significance level. The coefficients alx and axb are both not significant since their
p-values are 0.9568 and 0.3264 respectively. For the bivariate models, alx and axb are again not
significant at any conventional significance level. All other off-diagonal elements of matrix A are significant at the 1% significance level, for the bivariate models. To summarize, the significant elements of the off-diagonal elements of the parameter matrix A (abx, axl, abl and alb)
show that (unexpected) shocks occurring in the stock market of Belgium spillover to the stock markets of The Netherlands and Luxembourg, that (unexpected) shocks occurring in the stock market of The Netherlands spillover to the stock market of Luxembourg and that (unexpected) shocks occurring in Luxembourg spillover to the stock market of Belgium. As with the diagonal elements, the results regarding the off-diagonal elements of matrix A of the bivariate models confirm the robustness of the results regarding the off-diagonal elements of matrix A of the trivariate model.
The off-diagonal elements of the parameter matrix B describe the volatility spillovers from one stock market to the other. In the trivariate estimation, the coefficients bbx, blx, blb, bxl, and bbl
are all significant at the 1% significance level while the coefficient bxb is only significant at the
conventional significance level. Combining those results, it can be derived that changes in the volatility of the Dutch stock market spillover to the stock market of Luxembourg, that changes in the volatility of the Belgium stock market spillover to the stock market of Luxembourg and that changes in the volatility of the Luxembourgish stock market spillover to the stock markets of Belgium. In contrary to the results of the diagonal elements of matrices A and B and the off-diagonal elements of matrix A, the results of the off-off-diagonal elements of matrix B of the bivariate models are not completely in line with those of the trivariate model.
As previously described, volatility spillover effects are defined as interdependence between two or more different markets in terms of their volatility, so that the volatility in one market can be explained by events in the other market (Ghine and Saidi, 2017). The specification of the BEKK (1, 1) model allows for the differentiation of two different spillover effects: a volatility-to-volatility spillover effect and a shock-to-volatility spillover effect. The results indicate that volatility spillovers occur from The Netherlands to Luxembourg, from Belgium to Luxembourg and from Luxembourg to Belgium. Next, shocks occurring in all stock markets impact the volatility of the other stock markets, except from Luxembourg to The Netherlands and from The Netherlands to Belgium.
4.4 Implications
It is described in the literature section of this paper that volatility spillovers capture interdependence between stock markets in terms of their volatility. A high degree of such an interdependence might reduce the benefits from international portfolio diversification. The spillover effects obtained in this paper might therefore result in important implications to be taken into account by investors and portfolio managers. It is described that the empirical results have shown volatility spillovers from the AEX to the LuxX Index, from the BEL 20 to the LuxX Index and from the LuxX Index to the BEL 20. These spillovers imply that investors and portfolio managers could benefit from portfolio diversification by not investing in the BEL 20 and LuxX Index simultaneously or by not investing in the AEX and LuxX Index simultaneously. In times of high levels of volatility, the volatility of that specific market might spill over to one of the other stock markets. This in turn results in high levels of perceived risk and uncertainty. However, since no volatility spillover effect takes place between the AEX and BEL 20, a portfolio of stocks could be diversified by investing in those two markets simultaneously because those two markets are not interdependent in terms of their volatility.
4.5 Diagnostic tests
indeed the case, a test for autocorrelation was performed. The Ljung-Box test was performed on the squared residuals. Ghini and Saidi (2017) argued that the Q-statistic should be insignificant if the variance equation is correctly specified and thereby adequate to model the conditional variance of the time series. Table 5 contains the Ljung-Box Q-statistic for the four obtained models. As in section 4.2, the tests are performed with both five lags and 30 lags. The Q(5)-statistic and Q(30)-statistic reject the null hypothesis of no autocorrelation for the trivariate model, indicating that there is still autocorrelation. This might suggest that still some information is missing in the volatility that is modelled and that the specification of the model is therefore wrong. Therefore, the trivariate model may not adequately capture the conditional variances and covariances of the three stock market indices. However, the Q(30)-statistics of the three bivariate models fail to reject the null hypotheses of no autocorrelation, indicating an absence of autocorrelation. The Q(5)-statistic does reject the null hypothesis of no autocorrelation for the AEX – LuxX Index model, but does not reject the null hypotheses of no autocorrelation for the AEX – BEL 20 model and the BEL 20 – LuxX Index model. This suggests that autocorrelation slowly dies away in the AEX – LuxX Index model, and we therefore adopt the results of the Q(30)-statistic. The three bivariate models can therefore be accepted as adequate models to capture the conditional variances and conditional covariances of the AEX, BEL 20 and LuxX Index, if we follow the Q(30)-statistic.
Table 5
Results of tests for autocorrelation for the obtained trivariate model and the three bivariate models.
The Ljung-Box test is performed with five lags and 30 lags and they follow respectively a χ2(5)-distribution and a χ2 (30)-distribution. The null hypotheses of no autocorrelation are rejected at the 1% significance level for the trivariate model, for both tests. The null hypotheses of no autocorrelation are not rejected when the test is performed with 30 lags for the bivariate models since all p-values are above any conventional significance level. When the test is performed with 5 lags, the null hypothesis (representing no autocorrelation) is only rejected for the bivariate case modelling the AEX and LuxX Index. P-values smaller than 0.01, 0.05 and 0.10 are indicated by ***, ** and * respectively.
Diagnostic tests
Model Q(5)-statistic Q(30)-statistic
Trivariate model
AEX – BEL 20 – LuxX Index 110.7365*** 223.8210***
Bivariate models
AEX – BEL 20 0.1876 32.0010
AEX – LuxX Index 20.9559*** 39.7210
BEL 20 – LuxX Index 0.4605 31.1620
The estimations of the bivariate models are repeated to see if the results hold or differ, as a last robustness check. The dataset is divided in two, resulting in two subsets. The first subset contains the first until the 1611st observation (January 2, 2004 until July 8, 2010) and the second
subset contains the 1612nd observation until the 3222nd observation (July 9, 2010 until July 29,
The first subset contains the start of the financial crisis. This period displays higher levels of volatility than the second subset, as shown in fig. 1 – 3 in section 3.1 and in fig. A1 – A15 in Appendix B. When comparing the off-diagonal elements of matrices A and B, it can be seen that 11 out of the 12 parameters were significant (only parameter axb was not significant) in
the first subset and that only five out of the 12 parameters were significant in the second subset16. When estimating the total dataset (Table 4 in section 4.3), seven out of the 12
parameters were significant. This result implies that more volatility spillovers took place from January 2, 2004 until July 8, 2010 than from July 9, 2010 until July 29, 2016. We can derive from this result that more spillovers take place when volatility is high, since the returns fluctuated more in the first period.
5. Conclusion
The aim of this paper was to fill a gap in the existing literature regarding volatility spillovers within the stock markets of the Eurozone. Numerous papers were found investigating volatility spillovers. However, the stock markets of Belgium and Luxembourg have never been incorporated in such a study, and The Netherlands only once. This paper therefore investigated volatility spillover effects between the AEX, BEL 20 and LuxX Index, the three main stock market indices within the Benelux. In order to examine volatility spillovers between the three indices, four MGARCH BEKK (1, 1) models were applied to the three indices. The first estimation included all the indices and the last three estimations only incorporated two out of the three indices.
The data that were used in this paper were collected during the period January 2, 2004 until July 29, 2016. The return series of the three indices follow approximately the same path and they all reveal volatility clustering. Ljung-Box Q-statistics confirm volatility clustering for all series and the ARCH-heteroskedasticity test confirms that the variance of the error term is not constant over time for all series. Because of the volatility clustering and the ARCH-effects, we conclude that applying a GARCH model is the appropriate way to model the volatility for the AEX, BEL 20 and LuxX Index.
The empirical estimation results of the diagonal elements of the trivariate- and bivariate models reveal that the conditional variances of the AEX, BEL 20 and LuxX Index depend on their own past variances as well as shocks that occur in that specific market. Subsequently, domestic shocks and past variances can help to predict the future volatility of the stock market in The Netherlands, Belgium and Luxembourg. The results were significant at any conventional significance level for both the trivariate- and bivariate models. The empirical estimation results of the off-diagonal elements of the trivariate- and bivariate models explain spillover effects.
16 Only the off-diagonal elements of matrices A and B are compared since those elements capture spillover
The results of the trivariate estimation showed that shocks occurring in Belgium spill over to The Netherlands and Luxembourg, that shocks occurring in The Netherlands spill over to Luxembourg and that shocks occurring in Luxembourg spill over to Belgium. Therefore, a shock taking place in one country’s stock market can affect the volatility of the other country’s stock market. The results of the trivariate estimation also reveal that volatility spills over between all three stock market indices. In other words, the volatility in the stock market of The Netherlands can be explained by the volatility of the Belgian and Luxembourgish stock market, and vice versa for Belgium and Luxembourg. However, the performed diagnostic tests indicate that the trivariate model still shows autocorrelation, suggesting that information is still missing in the volatility that is modelled. The results of the trivariate model are therefore unreliable. Nevertheless, the bivariate estimations show no autocorrelation and those models can therefore be adopted, if we follow the Q(30)-statistics. The bivariate estimations with regard to the shock spillovers display the same results as in the trivariate case. The results regarding the volatility spillover effects are, however, somewhat different. When adopting the results of the bivariate estimations, the conclusion is that volatility spillovers take place from The Netherlands to Luxembourg, from Belgium to Luxembourg and from Luxembourg to Belgium. Moreover, the last diagnostic tests have shown that more spillover effects took place from January 2, 2004 until July 8, 2010, than from July 9, 2010 until July 29, 2016.
The research question of this paper was, ‘Do volatility spillovers between the Dutch, Belgian
and Luxembourgish stock market indices exist?’ The expectation was that volatility spillovers
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Appendix A: The GJR and EGARCH models
The GJR model (Glosten, Jagannathan and Runkle, 1993) is defined as
𝜎)8 = 𝛼:+ 𝛼A𝑢)>A8 + 𝛽𝜎)>A8 + 𝛾𝑢)>A8 𝐼)>A, (9)
with constant 𝛼:, the second term being the ARCH term, the third term being the GARCH term and the last term accounting for possible asymmetric responses.
The EGARCH model (Nelson, 1991) is defined as ln(𝜎)8) = 𝜔 + 𝛽 ln(𝜎 )>A8 ) + 𝛾 t456 uv456w + 𝛼 _ |t456| uv456w − u 8 ya, (10)
Appendix B: conditional variances and covariances B.1 Trivariate case AEX – BEL 20 – LuxX Index
The data composing the graphs in this sub appendix is obtained by estimating a trivariate MGARCH BEKK (1,1) model with the AEX, BEL 20 and LuxX Index.
Fig. A1
Conditional variance of the AEX in the period 02/01/04 – 29/07/16.
Fig. A2
Conditional variance of the BEL 20 in the period 02/01/04 – 29/07/16.
Fig. A3
Conditional variance of the LuxX Index in the period 02/01/04 – 29/07/16.
Fig. A4
Conditional covariance between the AEX and BEL 20 in the period 02/01/04 – 29/07/16.
Fig. A5
Conditional covariance between the AEX and LuxX Index in the period 02/01/04 – 29/07/16.
Fig. A6
Conditional covariance between the BEL 20 and LuxX Index in the period 02/01/04 – 29/07/16.
0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 01-04 01-05 01-06 01-07 01-08 01-09 01-10 01-11 01-12 01-13 01-14 01-15 01-16 Covariance Date (mm/yy) AEX - BEL 20 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 01-04 01-05 01-06 01-07 01-08 01-09 01-10 01-11 01-12 01-13 01-14 01-15 01-16 Covariance Date (mm/yy) AEX - LuxX Index
B.2 Bivariate case AEX – BEL 20
The data composing the graphs in this sub appendix is obtained by estimating a bivariate MGARCH BEKK (1, 1) model with the AEX and BEL 20.
Fig. A7
Conditional variance of the AEX in the period 02/01/04 – 29/07/16.
Fig. A8
Conditional variance of the BEL 20 in the period 02/01/04 – 29/07/16.
Fig. A9
Conditional covariance between the AEX and BEL 20 in the period 02/01/04 – 29/07/16.
B.3 Bivariate case AEX – LuxX Index
The data composing the graphs in this sub appendix is obtained by estimating a bivariate MGARCH BEKK (1,1) model with the AEX and LuxX Index.
Fig. A10
Conditional variance of the AEX in the period 02/01/04 – 29/07/16.
Fig. A11
Conditional variance of the LuxX Index in the period 02/01/04 – 29/07/16.
Fig. A12
Conditional covariance between the AEX and LuxX Index in the period 02/01/04 – 29/07/16.
