The Research on the Interactions between the Emerging and Developed Markets: From Region and Structural Break Perspectives

Abstract

This study utilizes a bivariate BEKK-EGARCH model with the setting of a structural break to investigate whether the interactions between stock indices in emerging and developed markets are different in terms of region, emerging stock indices, and subperiod. Then, this study investigated how the results of interactions vary with geographical location, emerging stock indices, and subperiod. Empirical results show that the interactions between emerging and developed markets indeed vary with geographical location and emerging stock indices, but are almost the same in the two subperiods. For example, for the paired stock indices in ‘Asia-America’ and ‘Asia-Europe’, or related to ‘XU100’, ‘SSE’ and ‘BSE’, the developed market mainly spills into the emerging market in terms of return and volatility. The findings from these empirical results can help investors and fund managers undertake different investment strategies in different regions and subperiods, and make effective investments.

1. Introduction

Owing to globalization and financial liberalization, the capital of enterprise can flow freely all over the world. Enterprises in countries with developed markets attempt to find cheap labor and low-priced factories to make their products. Emerging market can provide the above cheap production resources, thus the businesses in countries with developed markets invest their capital into regions or countries in the emerging market to obtain more profit. The above reasoning can be proved by the trend of GDP and the growth rate of GDP for countries in the E7 and G7 during the period from 2000 to 2019, as reported on the website of stlouisfed. For example, the Group of Seven (G7) still have positive growth rates of gross domestic product (GDP), which range from 0.7621% (France) to 1.4817% (Canada), except for Italy, even if the countries in the G7 have high GDP values ranging from USD 36,055 (Italy) to USD 49,506 (USA). On the other hand, the people in the region of the emerging market obtain more wealth because they make more money due to obtaining more job opportunities. This reasoning can be proved by the high GDP growth rates, from 0.5739% (Mexico) to 8.4597% (China), for the countries of the Emerging Seven (E7). Notably, we find that the values of GDP ranging from USD 36,055 (Italy) to USD 49,506 (USA) for the G7 are far greater than those of the E7, which range from USD 1375 (India) to USD 11,370 (Turkey). We also find that the GDP growth rates of the E7, which range from 0.5739% (Mexico) to 8.4597% (China), are far greater than those of the G7, which range from −0.0279% (Italy) to 1.4817% (Canada). In addition, we also find that in 2008 and 2009, most of the countries in the E7 and G7 have negative values of GDP growth rate, indicating that the GFC in 2008 caused a significant impact on the economy in both E7 and G7 countries. The above phenomena bring economic growth to countries of developed and emerging markets, thus the stock markets in emerging and developed markets may interact with each other, and some structural breaks may have occurred in the above markets around 2008. The interactions between the emerging and developed markets include two short-term connections (return and volatility spillovers) and one long-term relation (correlation). From the past literature on spillover issues, the research method scholars have used include the multivariate GARCH model, Diebold–Yilmaz’s [1] spillover index, Hong’s [2] causality-in-mean and causality-in-variance tests, and the wavelet coherence approach. In this study, I utilize a bivariate BEKK-EGARCH model with the setting of a structural break to investigate the following two questions. First, whether the interactions between the stock indices in emerging and developed markets are different in terms of geographical locations, emerging stock indices, and subperiods. Second, how the geographical locations, emerging stock indices, and subperiods (or structural break) affect the results of the interactions between the stock indices in emerging and developed markets. To the best of my knowledge, this study is the first to explore the spillover issues based on simultaneously considering geographical locations, emerging stock indices, and structural breaks. The sample in this study includes the stock indices in G7 and E7 countries, and the study covers the period from 16 October 2000 to 28 July 2020, which covers the global financial crisis (GFC) that occurred in 2008 and the quantitative easing (QE) implemented from November 2008 to September 2012. This approach belongs to the multivariate GARCH model. Notably, this approach can find the results of four financial features for two subperiods with only an estimated process, unlike any of the other approaches. Hence, it is the best. The four financial features include the correlation, return and volatility spillovers between two stock indices and the leverage effect in a stock index.

Subsequently, this study applies the variation in significant situations within five geographical locations (or seven emerging stock indices) to test the four hypotheses H1–H4. Then, the obtained results are used to explore whether the interactions between the stock indices in emerging and developed markets or the leverage effect in a market are different in terms of geographical location and emerging stock indices. Then, I used the variation in significant situations for the two subperiods to test the four hypotheses H5–H8. The results are used to explore, regarding five regions or seven emerging stock indices, whether the interactions between the stock indices in emerging and developed markets or the leverage effect of a market are different in two subperiods. The test results of hypotheses H1–H8 are used to answer the first question of this study.

Hypothesis 1 (H1):

The status of return spillover of paired markets is different within five regions or seven emerging stock indices.

Hypothesis 2 (H2):

The status of volatility spillover of paired markets is different within five regions or seven emerging stock indices.

Hypothesis 3 (H3):

The status of the correlation of paired markets is different within five regions or seven emerging stock indices.

Hypothesis 4 (H4):

The status of the leverage effect of a market is different within five regions or seven emerging stock indices.

Hypothesis 5 (H5):

The status of return spillover of paired markets is different during the two subperiods.

Hypothesis 6 (H6):

The status of volatility spillover of paired markets is different during the two subperiods.

Hypothesis 7 (H7):

The status of the correlation of paired markets is different during the two subperiods.

Hypothesis 8 (H8):

The status of the leverage effect of a market is different during the two subperiods.

Thus, this study uses three model-fitting ability tests to inspect the accuracy of the dates of structural breaks of return series of the stock indices in the G7 and E7 countries, and to verify the appropriateness of the set of two subperiods on the B-EGARCH-SB model. In addition, this study utilizes the unit root test and the cointegration test to ascertain the suitability of the VAR-type mean equation. Further, I find the eigenvalues of the variance–covariance matrix to ensure the semidefinite positivity of this matrix, and further to ascertain the suitability of the BEKK-EGARCH type of the variance–covariance equation. Subsequently, this study employs the B-EGARCH-SB model to estimate the conditional variance and covariance of 46 pairs of returns data. Empirical results show that the stock indices in the G7 and E7 countries have different investment attributes at the pre- and post-SB periods. Then, the return series of them has a structural break, also confirmed by three model-fitting ability tests of a bivariate model used in this study. Moreover, I find that the short- and long-term interactions between the emerging and developed markets vary with geographical location and emerging stock indices. For example, first, for the paired stock indices in ‘Asia-America’ and ‘Asia-Europe’ or related to ‘XU100’, ‘SSE’ and ‘BSE’, the developed market mainly spills into the emerging market in terms of return and volatility. The above results are the same as those found from the entire sample of paired stock indices. Second, for the paired stock indices in ‘Asia’, ‘America’ and ‘America-Europe’ or related to ‘MXX’ and ‘BVSP’, the emerging market primarily spills into the developed market in terms of return and volatility. Third, the leverage effect significantly exists in the emerging and developed markets, especially for the developed market. Fourth, the correlation is positive for all the paired stock indices, especially for the paired stock indices in ‘America’ and ‘America-Europe’ or related to ‘MXX’ and ‘BVSP’, indicating that the emerging stock market in America and MXX and BVSP are closely interrelated to the entire developed stock market. However, the above results are the average phenomena for the entire study period. Thus, they may be changed by the structural break and are expressed as follows. First, regarding the entire sample of paired stock indices, the volatility spillover is affected by the structural break the most significantly, followed by the correlation and leverage effect, whereas the return spillover is nearly unaffected by the structural break. Moreover, the impact of both the correlation and leverage effect increases, whereas that of volatility spillover decreases, after the structural break. However, the impact of return spillover is nearly unaffected by the structural break. Second, the paired stock indices in ‘Asia’ or related to ‘JKSE’ are significantly affected by the structural break. On the contrary, the paired stock indices in ‘America’ are nearly unaffected by the structural break. Via the findings from the empirical results, this study proposes some policy implications to help investors and fund managers to undertake different investment strategies in different regions and subperiods, then to make an effective investment.

The remainder of this paper Is organized as follows. Section 2 reviews the past literature about the spillover issue and then highlights the contributions of this study. Section 3 describes the empirical model utilized in this study, the B-EGARCH-SB model. Section 4 states the basic statistical features of the return series for the stock indices in the E7 and G7 countries during the overall period and two subperiods, the pre- and post-SB periods. Section 5 analyzes the results of the empirical model and further explores the issues addressed in this study. In Section 6, I give some reasons for the obtained results. Finally, Section 7 concludes the findings in Section 4 and Section 5 and proposes some policy implications for the investors and fund managers.

2. Literature Review

Owing to globalization and financial liberalization, the capital of enterprise can flow freely all over the world. Moreover, the stock market is the most popular financial market. This indicates that stock markets all over the world, especially in the G7 and E7 countries, may interact with each other. Hence, many works of literature on the spillover issues in the stock market have explored the interrelationship between the United States (US) and Japan with countries in Asia, and they found that the stock markets in the US and Japan, especially the US, can affect those of countries in Asia [3,4,5,6,7,8,9]. For example, Wei et al. [3] found that the United States (US) market has more influence than the Japanese market on the Taiwan and Hong Kong markets. Moreover, Miyakoshi [5] found that only the influence of the US is important for Asian market returns, whereas the volatility of the Asian market is influenced more by the Japanese market than by the US one (Asian market in Miyakoshi [5] includes Korea, Taiwan, Singapore, Thailand, Malaysia, Hong Kong, and Indonesia). Ng [4] found that there are significant spillovers from Japan to many of the Pacific Basin countries (Pacific Basin countries in Ng [4] denote Korea, Taiwan, Singapore, Thailand, Malaysia, and Hong Kong). In addition, there are several works of literature focusing on the spillover issues between China or BRIC with the other countries in the world [10,11,12,13,14,15,16,17] (BRIC represents Brazil, Russia, India, and China, four countries in the Emerging Seven (E7)). For instance, Do et al. [17] found that China’s equities are more related to their neighboring countries such as Singapore, Japan, Australia, and ASEAN-5 than to the US, Germany, and the UK (ASEAN-5 in Do et al. [17] represents Singapore, Thailand, Malaysia, Philippines, and Indonesia, five countries in Asia). From the above discussion, the US and Japan, representing the developed market, and China or BRIC, denoting the emerging market, play a crucial role in the spillover issues of the stock market. However, the sample stock indices representing the developed and emerging markets seem to be not enough. Thus, this study selects the stock indices in the G7 and E7 countries to represent the developed and emerging markets, respectively, in order to investigate two short-term and one long-term interactions between the emerging and developed markets. This is the first contribution of this study, because the stock indices in the G7 and E7 countries are the most representative stock market for the developed and emerging markets, respectively.

In addition, most of the past literature has examined the spillover issues for the entire sample period [3,4,5,6,10,14,15,16,18,19,20,21,22]. The results should vary with the study period or its subperiods because some structural breaks may exist due to some special incidents happening during the study period. Hence, some studies have investigated the spillover issues for several subperiods of the study period [8,9,12,13,17,23]. For instance, Gilenko and Fedorova [13] examined the return and volatility spillovers for three subperiods such as pre-crisis period, crisis period, and recovery period. The crisis is the US subprime financial crisis period or global financial crisis (GFC). Conversely, Do et al. [17] explored both the volatility spillover and dynamic conditional correlation for the four subperiods, the pre-GFC period, GFC period, extended-crisis period, and post-crisis period. According to the suggestion of Green [24], Allen et al. [12] divided the study period into four subperiods and found that there is some evidence of volatility spillovers across these markets in the pre-GFC period, but there is little evidence of spillover effects from China to the related markets during the GFC (the markets in Allen et al. [12] cover both the Shenzhen and Shanghai Chinese markets and several other markets around the Pacific Basin Area, including Australia, Hong Kong, Japan, Taiwan, and Singapore. On the other hand, four subperiods are 27 August 1991–30 June 1992, 1 July 1992–31 December 1996, 1 January 1997–29 December 2006, and 1 January 2007–17 November 2010. The first three subperiods are the pre-GFC period whereas the last subperiod is the GFC period). Vo and Ellis [9] explored the returns and volatility spillover for three subperiods: pre-, during, and post-GFC, and found that Vietnamese stock markets are influenced by the leading world stock markets. Moreover, the above results are stronger during and after the GFC. The subperiods in the above literature are divided by some special events such as the GFC. In reality, when some special event happens, the behavior of some markets may not change immediately but could change before or change after. Hence, the obtained results mentioned above may not be credible. Thus, this study uses the maximum likelihood-ratio test of Eizaguirre et al. [25] to accurately find the date of the structural break (SB), then divides the study period into the two subperiods: pre-SB and post-SB periods. This is the second contribution of this study. The findings should be more credible than those in [8,9,12,13,17].

Regarding the empirical models, all literature used the vector autoregressive (VAR) model to estimate the return spillover. To estimate the volatility spillover, the univariate generalized autoregressive conditional heteroscedasticity (GARCH) model is used at the early stage. Subsequently, the bivariate GARCH or exponential GARCH (EGARCH), the bivariate DCC-EGARCH model, and the bivariate BEKK-GARCH model are popular methods utilized recently (‘DCC’ is the abbreviation of ‘dynamic conditional correlation’ and ‘BEKK’ is named after Baba, Engle, Kraft, and Kroner [26]). Among these models, the BEKK-GARCH model proposed by Engle and Kroner [27] guarantees positive semidefiniteness by working with quadratic forms. However, it is not easy to estimate the parameters and it is very hard to clearly explain their meaning. Turning to the other models, they cannot estimate all the parameters of the model with only one procedure when the mean return and variance–covariance equations for several subperiods are estimated together. Thus, this study uses Su’s [28] diagonal bivariate BEKK-EGARCH model with the setting of a structural break (hereafter, B-EGARCH-SB) to explore two short-term and one long-term interaction between the emerging and developed markets for two subperiods (Su [28] adopted the suggestion of Moschini and Myers [29] to simplify the BEKK model, and then proposed a positive definite type of bivariate BEKK-GARCH model in diagonal representation to let the parameters’ estimates be parsimonious. Hence, this diagonal bivariate variance–covariance specification owns two properties: first, the positive definite in the variance–covariance matrix, and second, the parsimony in the parameter estimation and thus ease of parameter explanation. Please refer to Su [28] for more details). This is the third contribution of this study because this model has the following three merits. First is the positive definite in the variance–covariance matrix. Second is the parsimony in the parameter estimation and thus ease of parameter explanation. Third, the mean equation and variance–covariance equation for the two subperiods can be estimated together with only one procedure, owing to the setting of time dummy variables on the above two equations. (Conversely, Vo and Ellis [9] used the bivariate VAR model and diagonal BEKK-GARCH model to examine the returns and volatility spillovers, respectively, between Vietnam and advanced countries (US, Japan, and Hong Kong) for three subperiods. Corresponding to three subperiods of pre-crisis, during crisis, and post-crisis, they needed to estimate the parameters of both the VAR model and diagonal BEKK-GARCH model three times. At the same inference, Allen et al. [12], Li and Giles [8], and Do et al. [17] divided the study period into four, two, and four subperiods, respectively, thus, they needed to estimate all parameters of the model for four, two, and four times, respectively). Finally, this study analyzes the results of interactions or spillover issues between the emerging and developed stock markets from the viewpoints of geographical location and the emerging stock indices. This is the fourth contribution of this study because it can help to check whether the results of interactions or spillover issues vary with geographical location and the emerging stock indices. This is a problem that past studies have seldom discussed. Hence, in order to close the gap in the literature, this study utilizes a bivariate BEKK-EGARCH model with the setting of a structural break to investigate whether the interactions between stock indices in emerging and developed markets are different in terms of geographical location, emerging stock indices, and subperiods. Furthermore, this study investigates how geographical locations, emerging stock indices, and subperiods (or structural break) affect the results of the interactions between the stock indices in emerging and developed markets.

3. Methodology

To obtain a credible result regarding the interactions between emerging and developed markets, the selected empirical model should not only capture the explored issues such as the short-term and long-term interactions but should also seize the common features of financial assets. The short-term interactions are the return and volatility spillovers whereas the long-term interaction is the correlation. The common features of financial assets include the volatility clustering and leverage effect, usually existing during the volatility of financial asset returns series [30,31,32] (volatility clustering means that large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes [30]. Conversely, the leverage effect is the extra increase in volatility caused by the bad news [31]. Notably, the volatility clustering and leverage effect significantly appear in financial assets, especially in the stock market [32]). In addition, the study period covers the periods of the global financial crisis (GFC) and the quantitative easing (QE), indicating that the return series of stock indices around the world may have a structural break. Thus, this study utilizes a diagonal bivariate BEKK-EGARCH model with the setting of a structural break, named B-EGARCH-SB, to seize the above financial features and the explore issues for the pre- and post-SB periods. The diagonal bivariate BEKK-EGARCH model without the setting of structural break, named B-EGARCH-NSB, is composed of a two-dimensional mean equation ($R_t$) and two-dimensional variance–covariance equation ($H_t$) with bivariate Student’s t distribution.

The two-dimensional mean equation is expressed in the form of a bivariate vector autoregressive with lag one period (hereafter, VAR (1)) to seize the return spillover between two markets, and is shown below.

$$\begin{gathered} {\mathrm{r}}_{1,\mathrm{t}}={\mathsf{\varphi }}_{10}+{\mathsf{\varphi }}_{11}{\mathrm{r}}_{1,\mathrm{t}-1}+{\mathsf{\varphi }}_{12}{\mathrm{r}}_{2,\mathrm{t}-1}+{\mathsf{\epsilon }}_{1,\mathrm{t}} \\ {\mathrm{r}}_{2,\mathrm{t}}={\mathsf{\varphi }}_{20}+{\mathsf{\varphi }}_{21}{\mathrm{r}}_{1,\mathrm{t}-1}+{\mathsf{\varphi }}_{22}{\mathrm{r}}_{2,\mathrm{t}-1}+{\mathsf{\epsilon }}_{2,\mathrm{t}} \end{gathered}$$

(1)

where ${\mathrm{r}}_{1,\mathrm{t}}$ and ${\mathrm{r}}_{2,\mathrm{t}}$ represent the returns of the two stock indices in the emerging and developed markets, respectively, where ${\mathrm{r}}_{\mathrm{i},\mathrm{t}}=\left({\mathrm{lnP}}_{\mathrm{i},\mathrm{t}}-{\mathrm{lnP}}_{\mathrm{i},\mathrm{t}-1}\right)\times 100$ for i = 1,2 and ${\mathrm{P}}_{\mathrm{i},\mathrm{t}}$ is the close price of the $i^{\mathrm{th}}$ stock index at time t. $R_t={\left({\mathrm{r}}_{1,\mathrm{t}},{\mathrm{r}}_{2,\mathrm{t}}\right)}^{\prime }$ is a column vector of returns. ${\mathsf{\varphi }}_{10}$ and ${\mathsf{\varphi }}_{20}$, two constants, denote the mean return in the first and second stock markets, respectively. Notably, parameters ${\mathsf{\varphi }}_{12}$ and ${\mathsf{\varphi }}_{21}$ can be used to explore the return spillover between two different types of stock markets. For example, if parameter ${\mathsf{\varphi }}_{12}$ is significant then there exists a return spillover from the second stock index in the developed market to the first stock index in the emerging market, and vice versa (the subscripts ‘1’ and ‘2’ in the mean equation and variance–covariance equation denote the stock indices in the emerging and developed markets, respectively. For instance, if parameter ${\mathsf{\varphi }}_{21}$ is significant, then there exists a return spillover from the first stock index in the emerging market to the second stock index in the developed market. Because the trading hours of stock markets in the E7 and G7 countries are different, the term ‘${\mathsf{\varphi }}_{12}{\mathrm{r}}_{2,\mathrm{t}-1}$’ in the first mean equation and the term ‘${\mathsf{\varphi }}_{21}{\mathrm{r}}_{1,\mathrm{t}-1}$’ in the second mean equation must undergo some change to solve the non-synchronous trading problem. For example, if the trading hours of the second stock index are before those of the first stock index, then the term ‘${\mathsf{\varphi }}_{12}{\mathrm{r}}_{2,\mathrm{t}-1}$’ in the first mean equation is replaced with the term ‘${\mathsf{\varphi }}_{12}{\mathrm{r}}_{2,\mathrm{t}}$’. Conversely, if the trading hours of the first stock index are before those of the second stock index, then the term ‘${\mathsf{\varphi }}_{21}{\mathrm{r}}_{1,\mathrm{t}-1}$’ in the second mean equation is replaced with the term ‘${\mathsf{\varphi }}_{21}{\mathrm{r}}_{1,\mathrm{t}}$’. The trading hours of all stock indices must be converted into the Greenwich Mean Time [18]). ${\mathsf{\epsilon }}_t={\left({\mathsf{\epsilon }}_{1,\mathrm{t}},{\mathsf{\epsilon }}_{2,\mathrm{t}}\right)}^{\prime }$ is a column vector of error terms, and its conditional distribution is assumed to follow the bivariate Student’s t distribution with $\mathrm{E}\left({\mathsf{\epsilon }}_t\right)=0$ and $\mathrm{E}\left({\mathsf{\epsilon }}_t·{\mathsf{\epsilon }}_t^{\prime }\right)=H_t$. That is, ${\mathsf{\epsilon }}_t\left|{\Omega }_{t-1}~\mathrm{t}\left(0,H_t,\mathrm{n}\right)\right.$. Subsequently, two-dimensional variance–covariance equation, $H_t$, is expressed in the form of diagonal bivariate BEKK-EGARCH (1,1)-X model, to capture not only both the correlation and volatility spillover between two markets but also the leverage effect in each of two markets, and is expressed as follows (the diagonal bivariate BEKK-EGARCH(1,1)-X model can capture the leverage effect in each of the markets just like the univariate EGARCH model does. It also can seize the volatility spillover between two markets via involving the variance of another asset as the exogenous variable. In general, this model can measure the correlation between two markets. Hence, the variance–covariance equation of this diagonal bivariate BEKK-EGARCH(1,1)-X model can be used to explore the following three issues. The first two issues are the volatility spillover and the correlation relation between two different markets. The third issue is the leverage effect in each of the two markets).

$${\mathrm{h}}_{\mathrm{t}}=\mathrm{vech}\left(H_t\right)={\left[{\mathrm{h}}_{11,\mathrm{t}},{\mathrm{h}}_{12,\mathrm{t}},{\mathrm{h}}_{22,\mathrm{t}}\right]}^{\prime }$$

(2)

$$\begin{gathered} \ln \left({\mathrm{h}}_{11,\mathrm{t}}\right)={\mathsf{\omega }}_1+{\mathsf{\beta }}_1\ln \left({\mathrm{h}}_{11,\mathrm{t}-1}\right)+{\mathsf{\gamma }}_1\frac{{\mathsf{\epsilon }}_{1,\mathrm{t}-1}}{\sqrt{{\mathrm{h}}_{11,\mathrm{t}-1}}}+{\mathsf{\alpha }}_1\left[\frac{\left|{\mathsf{\epsilon }}_{1,\mathrm{t}-1}\right|}{\sqrt{{\mathrm{h}}_{11,\mathrm{t}-1}}}-\mathrm{E}\left(\frac{\left|{\mathsf{\epsilon }}_{1,\mathrm{t}-1}\right|}{\sqrt{{\mathrm{h}}_{11,\mathrm{t}-1}}}\right)\right] \\ +{\mathsf{\nu }}_{12}\ln \left({\mathrm{h}}_{22,\mathrm{t}-1}\right) \end{gathered}$$

(3)

$${\mathrm{h}}_{12,\mathrm{t}}={\mathsf{\omega }}_{12}+{\mathsf{\alpha }}_{12}{\mathsf{\epsilon }}_{1,\mathrm{t}-1}{\mathsf{\epsilon }}_{2,\mathrm{t}-1}+{\mathsf{\beta }}_{12}{\mathrm{h}}_{12,\mathrm{t}-1}$$

(4)

$$\begin{gathered} \ln \left({\mathrm{h}}_{22,\mathrm{t}}\right)={\mathsf{\omega }}_2+{\mathsf{\beta }}_2\ln \left({\mathrm{h}}_{22,\mathrm{t}-1}\right)+{\mathsf{\gamma }}_2\frac{{\mathsf{\epsilon }}_{2,\mathrm{t}-1}}{\sqrt{{\mathrm{h}}_{22,\mathrm{t}-1}}}+{\mathsf{\alpha }}_2\left[\frac{\left|{\mathsf{\epsilon }}_{2,\mathrm{t}-1}\right|}{\sqrt{{\mathrm{h}}_{22,\mathrm{t}-1}}}-\mathrm{E}\left(\frac{\left|{\mathsf{\epsilon }}_{2,\mathrm{t}-1}\right|}{\sqrt{{\mathrm{h}}_{22,\mathrm{t}-1}}}\right)\right] \\ +{\mathsf{\nu }}_{21}\ln \left({\mathrm{h}}_{11,\mathrm{t}-1}\right) \end{gathered}$$

(5)

where $\mathrm{E}\left(\frac{\left|{\mathsf{\epsilon }}_{1,\mathrm{t}-1}\right|}{\sqrt{{\mathrm{h}}_{11,\mathrm{t}-1}}}\right)=\mathrm{E}\left(\frac{\left|{\mathsf{\epsilon }}_{2,\mathrm{t}-1}\right|}{\sqrt{{\mathrm{h}}_{22,\mathrm{t}-1}}}\right)=\mathrm{E}\left(\left|{\mathrm{z}}_{\mathrm{t}-1}\right|\right)=\frac{2}{\sqrt{\mathsf{\pi }}}\frac{\mathsf{\Gamma }\left(0.5\left(\mathrm{n}+1\right)\right)}{\mathsf{\Gamma }\left(0.5\mathrm{n}\right)}\frac{\sqrt{\mathrm{n}-2}}{\mathrm{n}-1}$ and vech($H_t$) denotes the vech operator that stacks the ‘upper triangular’ portion of a two-dimensional matrix $H_t$ into a vector with a single column. If parameters ${\mathsf{\gamma }}_1$ and ${\mathsf{\gamma }}_2$ are significantly negative, then the leverage effect of volatility exists in the first and second stock indices. Notably, parameters ${\mathsf{\nu }}_{12}$ and ${\mathsf{\nu }}_{21}$ can be used to explore the volatility spillover between two different types of stock markets. For example, if parameter ${\mathsf{\nu }}_{12}$ is significant, then there exists a volatility spillover from the second stock index in the developed market to the first stock index in the emerging market, and vice versa (if parameter ${\mathsf{\nu }}_{21}$ is significant then there exists a volatility spillover from the first stock index in the emerging market to the second stock index in the developed market. With the same inference process in the mean equation, the term ‘${\mathsf{\nu }}_{12}\ln \left({\mathrm{h}}_{22,\mathrm{t}-1}\right)$’ in Equation (3) and the term ‘${\mathsf{\nu }}_{21}\ln \left({\mathrm{h}}_{11,\mathrm{t}-1}\right)$’ in Equation (5) are replaced with the terms ‘${\mathsf{\nu }}_{12}\ln \left({\mathrm{h}}_{22,\mathrm{t}}\right)$’ and ‘${\mathsf{\nu }}_{21}\ln \left({\mathrm{h}}_{11,\mathrm{t}}\right)$’, respectively, to solve the non-synchronous trading problem appearing at the stock markets in the E7 and G7 countries). If parameter ${\mathsf{\omega }}_{12}$ is significant, then for the estimate period there exists a relationship between two stock indices in the emerging and developed markets in terms of price trends. Notably, the parameters of this bivariate asymmetric GARCH model are estimated by maximum likelihood (ML), numerically optimizing the bivariate Student’s t log-likelihood function by using Rat’s 6.0 software (regarding the bivariate Student’s t log-likelihood function, please refer to Braione and Scholtes [33] for more details). Hence, the log-likelihood function of the B-EGARCH-NSB model can be written as follows:

$$\begin{gathered} \mathrm{L}\left(\Psi \right)={\displaystyle \sum }_{\mathrm{t}=1}^{\mathrm{T}}\ln \left\{\mathrm{f}\left(\left.R_t\right|{\Omega }_{t-1};\Psi \right)\right\}=-0.5{\displaystyle \sum }_{\mathrm{t}=1}^{\mathrm{T}}\left[\ln \left|H_t\right|+\left(\mathrm{m}+\mathrm{n}\right)\ln \left(1+\frac{{\mathsf{\epsilon }}_t^{\prime }{H}_t^{-1}{\mathsf{\epsilon }}_t}{\mathrm{n}-2}\right)\right] \\ +\mathrm{T}\left[\ln \mathsf{\Gamma }\left(\frac{\mathrm{n}+\mathrm{m}}{2}\right)-\ln \mathsf{\Gamma }\left(\frac{\mathrm{n}}{2}\right)-\frac{\mathrm{m}}{2}\ln \left(\mathsf{\pi }\right)-\frac{\mathrm{m}}{2}\ln \left(\mathrm{n}-2\right)\right] \end{gathered}$$

(6)

where T denotes the sample size of the estimate period, $\mathrm{f}\left(·\right)$ denotes the bivariate Student’s t density with the shape parameter $\mathrm{n}$ and ${\Omega }_{t-1}$ denotes the information set of all observed returns up to time t − 1. m represents the dimension of this bivariate GARCH model, and thus equals two for this case. $\Psi =\left[{\mathsf{\varphi }}_{10},{\mathsf{\varphi }}_{11},{\mathsf{\varphi }}_{12},{\mathsf{\varphi }}_{20},{\mathsf{\varphi }}_{21},{\mathsf{\varphi }}_{22}\right.,{ \mathsf{\omega }}_1,{\mathsf{\alpha }}_1,{\mathsf{\beta }}_1,{ \mathsf{\omega }}_{12},$ ${\mathsf{\alpha }}_{12},{\mathsf{\beta }}_{12},{ \mathsf{\omega }}_2,{\mathsf{\alpha }}_2,{\mathsf{\beta }}_2,{\mathsf{\nu }}_{12},{\mathsf{\nu }}_{21},{ \mathsf{\gamma }}_1,\left.{\mathsf{\gamma }}_2,\mathrm{n}\right]$ is the vector of parameters to be estimated. $R_t$,$H_t$ and ${\mathsf{\epsilon }}_t$ are defined in Equations (1)–(5). Notably, both the B-EGARCH-NSB and B-EGARCH-SB models have the same mean equation and variance–covariance equation. Only the difference between these two models is the settings of parameters ${\mathsf{\varphi }}_{12},{\mathsf{\varphi }}_{21},{ \mathsf{\omega }}_{12},{\mathsf{\nu }}_{12},{\mathsf{\nu }}_{21},{ \mathsf{\gamma }}_1$ and ${\mathsf{\gamma }}_2$ on the B-EGARCH-SB model. The above parameters are associated with the explored issues and some specific financial features mentioned above, and they are expressed including three time dummy variables and shown below (for example, two pairs of parameters ‘${\mathsf{\varphi }}_{12}$ and ${\mathsf{\varphi }}_{21}$’ and ‘${\mathsf{\nu }}_{12}$ and ${\mathsf{\nu }}_{21}$’ can capture the return spillover and the volatility spillover for paired stock markets in the emerging and developed markets, respectively. Parameter ${\mathsf{\omega }}_{12}$ can seize the correlation between two stock indices in the emerging and developed markets. Parameters ${\mathsf{\gamma }}_1$ and ${\mathsf{\gamma }}_2$ can capture the leverage effect of stock indices in the emerging and developed markets, respectively).

$$\begin{gathered} {\mathsf{\varphi }}_{12}={\mathsf{\varphi }}_{12}^{\mathrm{B}}·{\mathrm{d}}_{\mathrm{t}}^{\mathrm{B}}+{\mathsf{\varphi }}_{12}^{\mathrm{D}}·{\mathrm{d}}_{\mathrm{t}}^{\mathrm{D}}+{\mathsf{\varphi }}_{12}^{\mathrm{A}}·{\mathrm{d}}_{\mathrm{t}}^{\mathrm{A}},{\mathsf{\varphi }}_{21}={\mathsf{\varphi }}_{21}^{\mathrm{B}}·{\mathrm{d}}_{\mathrm{t}}^{\mathrm{B}}+{\mathsf{\varphi }}_{21}^{\mathrm{D}}·{\mathrm{d}}_{\mathrm{t}}^{\mathrm{D}}+{\mathsf{\varphi }}_{21}^{\mathrm{A}}·{\mathrm{d}}_{\mathrm{t}}^{\mathrm{A}}; \\ {\mathsf{\nu }}_{12}={\mathsf{\nu }}_{12}^{\mathrm{B}}·{\mathrm{d}}_{\mathrm{t}}^{\mathrm{B}}+{\mathsf{\nu }}_{12}^{\mathrm{D}}·{\mathrm{d}}_{\mathrm{t}}^{\mathrm{D}}+{\mathsf{\nu }}_{12}^{\mathrm{A}}·{\mathrm{d}}_{\mathrm{t}}^{\mathrm{A}},{ \mathsf{\nu }}_{21}={\mathsf{\nu }}_{21}^{\mathrm{B}}·{\mathrm{d}}_{\mathrm{t}}^{\mathrm{B}}+{\mathsf{\nu }}_{21}^{\mathrm{D}}·{\mathrm{d}}_{\mathrm{t}}^{\mathrm{D}}+{\mathsf{\nu }}_{21}^{\mathrm{A}}·{\mathrm{d}}_{\mathrm{t}}^{\mathrm{A}}; \\ {\mathsf{\omega }}_{12}={\mathsf{\omega }}_{12}^{\mathrm{B}}·{\mathrm{d}}_{\mathrm{t}}^{\mathrm{B}}+{\mathsf{\omega }}_{12}^{\mathrm{D}}·{\mathrm{d}}_{\mathrm{t}}^{\mathrm{D}}+{\mathsf{\omega }}_{12}^{\mathrm{A}}·{\mathrm{d}}_{\mathrm{t}}^{\mathrm{A}}; \\ {\mathsf{\gamma }}_1={\mathsf{\gamma }}_1^{\mathrm{B}}·{\mathrm{d}}_{\mathrm{t}}^{\mathrm{B}}+{\mathsf{\gamma }}_1^{\mathrm{D}}·{\mathrm{d}}_{\mathrm{t}}^{\mathrm{D}}+{\mathsf{\gamma }}_1^{\mathrm{A}}·{\mathrm{d}}_{\mathrm{t}}^{\mathrm{A}}; {\mathsf{\gamma }}_2={\mathsf{\gamma }}_2^{\mathrm{B}}·{\mathrm{d}}_{\mathrm{t}}^{\mathrm{B}}+{\mathsf{\gamma }}_2^{\mathrm{D}}·{\mathrm{d}}_{\mathrm{t}}^{\mathrm{D}}+{\mathsf{\gamma }}_2^{\mathrm{A}}·{\mathrm{d}}_{\mathrm{t}}^{\mathrm{A}} \end{gathered}$$

(7)

where ${\mathrm{d}}_{\mathrm{t}}^{\mathrm{B}}$ and ${\mathrm{d}}_{\mathrm{t}}^{\mathrm{A}}$ are two time dummy variables that take a value of 1 if the time is before the date of the first structural break and after the date of second structural break until the end date of the study sample, respectively. That is, ${\mathrm{d}}_{\mathrm{t}}^{\mathrm{B}}=1$ if ${\mathrm{date}}_{\mathrm{start}}\le \mathrm{t}<{\mathrm{date}}_{\mathrm{SB}1}$, and ${\mathrm{d}}_{\mathrm{t}}^{\mathrm{B}}=$ 0 otherwise; ${\mathrm{d}}_{\mathrm{t}}^{\mathrm{A}}=1$ if ${\mathrm{date}}_{\mathrm{SB}2}\le \mathrm{t}\le {\mathrm{date}}_{\mathrm{end}}$, and ${\mathrm{d}}_{\mathrm{t}}^{\mathrm{A}}=$ 0 otherwise. On the other hand, ${\mathrm{d}}_{\mathrm{t}}^{\mathrm{D}}$ is a time dummy variable that takes the value 1 if the time is between the dates of first and second structural breaks (i.e., ${\mathrm{date}}_{\mathrm{SB}1}\le \mathrm{t}<{\mathrm{date}}_{\mathrm{SB}2}$ or the transition period). Because 14 stock indices have different dates of structural breaks during the study period, there is an additional time dummy variable, ${\mathrm{d}}_{\mathrm{t}}^{\mathrm{D}}$, to express the transition period between two different dates of the structural break for a pair of stock indices. The date of the structural break is obtained by using the maximum likelihood-ratio test of Eizaguirre et al. [25]. Hence, the sample log-likelihood function of the B-EGARCH-SB model can be written in the same way as Equation (6), only the vector of parameters is expressed as $\Psi =[{\mathsf{\varphi }}_{10},{\mathsf{\varphi }}_{11}, {\mathsf{\varphi }}_{12}^{\mathrm{B}},{\mathsf{\varphi }}_{12}^{\mathrm{D}}, {\mathsf{\varphi }}_{12}^{\mathrm{A}}, {\mathsf{\varphi }}_{20},{\mathsf{\varphi }}_{21}^{\mathrm{B}},{\mathsf{\varphi }}_{21}^{\mathrm{D}}, {\mathsf{\varphi }}_{21}^{\mathrm{A}},{\mathsf{\varphi }}_{22},{\mathsf{\omega }}_1,{\mathsf{\alpha }}_1,{\mathsf{\beta }}_1,{ \mathsf{\omega }}_{12}^{\mathrm{B}},{\mathsf{\omega }}_{12}^{\mathrm{D}},{\mathsf{\omega }}_{12}^{\mathrm{A}},{\mathsf{\alpha }}_{12},{\mathsf{\beta }}_{12},{ \mathsf{\omega }}_2,{\mathsf{\alpha }}_2,{\mathsf{\beta }}_2,{\mathsf{\nu }}_{12}^{\mathrm{B}}$$,{\mathsf{\nu }}_{12}^{\mathrm{D}},{\mathsf{\nu }}_{12}^{\mathrm{A}},{\mathsf{\nu }}_{21}^{\mathrm{B}},{\mathsf{\nu }}_{21}^{\mathrm{D}},{\mathsf{\nu }}_{21}^{\mathrm{A}},{\mathsf{\gamma }}_1^{\mathrm{B}},{\mathsf{\gamma }}_1^{\mathrm{D}},{\mathsf{\gamma }}_1^{\mathrm{A}},{\mathsf{\gamma }}_2^{\mathrm{B}},{\mathsf{\gamma }}_2^{\mathrm{D}},{\mathsf{\gamma }}_2^{\mathrm{A}},\mathrm{n}]$. Notably, parameters with the superscript ‘B’ (respectively, ‘A’) can seize the financial feature related to that parameter during the pre-SB (respectively, post-SB) period. For instance, parameters ${\mathsf{\varphi }}_{12}^{\mathrm{B}}$ and ${\mathsf{\varphi }}_{12}^{\mathrm{A}}$ can capture the return spillover from the second stock index in the developed market to the first stock index in the emerging market during the pre-SB and post-SB periods, respectively. Moreover, parameters ${\mathsf{\gamma }}_1^{\mathrm{B}}$ and ${\mathsf{\gamma }}_1^{\mathrm{A}}$ can seize the leverage effect on the volatility in the first stock index of the emerging market during the pre-SB and post-SB periods, respectively.

4. Data and Descriptive Statistics

This study mainly explores the interactions between the emerging and developed markets from a structural break perspective. Then, the stock indices in the E7 and G7 countries are used to represent the emerging and developed markets, respectively. The stock indices in the E7 countries are XU100, RTSI, JKSE, SSE, BSE, MXX, and BVSP, corresponding to Turkey (Western Asia), Russia (North Asia), Indonesia (Southeast Asia), China (East Asia), India (South Asia), Mexico (North America), and Brazil (South America), respectively. Conversely, the stock indices in the G7 countries include FTSE, CAC40, DAX, MIB, DowJones (DJ), TSX, and N225, corresponding to the United Kingdom (Northwestern Europe), France (Western Europe), Germany (Central Europe), Italy (Southern Europe), the United States (North America), Canada (North America), and Japan (East Asia), respectively (the expression inside the bracket beside the country denotes the name of the continent the country is located. In addition, there are six continents in the world. The six continents are Asia, America, Europe, Africa, Oceania, and Antarctica). Hence, the study data include 14 stock indices in the E7 and G7 countries, that are mainly distributed in Asia, America, and Europe. Moreover, all the daily close price data of the 14 stock indices cover the period from 16 October 2000 to 28 July 2020, and were obtained from the Yahoo finance website. Figure 1 and Figure 2 depict the trends in the price level and return for the stock indices in the E7 and G7 countries, respectively. From Figure 1 and Figure 2, we can see that in 2008, all stock indices experienced a serious decline in price and an abrupt increase in volatility because of the global financial crisis (GFC) that occurred in 2008. Hence, owing to the global financial crisis (GFC) occurring in 2008 and the quantitative easing (QE) implemented from November 2008 to September 2012, the return series of the stock indices may have a structural break within the study period. Subsequently, the study period is divided into the pre-SB and post-SB periods according to the dates of structural breaks found by the maximum likelihood-ratio test of Eizaguirre et al. [25]. The dates of structural breaks for DAX, BVSP, N225, DJ, MIB, RTSI, TSX, BSE, MXX, XU100, CAC40, JKSE, FTSE, and SSE are 15 October 2003 (32.722), 21 January 2004 (46.576), 12 May 2004 (25.722), 27 February 2007 (24.329), 9 October 2007 (35.798), 23 January 2008 (47.984), 28 May 2008 (25.560), 15 September 2009 (32.410), 11 November 2009 (51.201), 13 January 2010 (32.117), 1 September 2010 (23.349), 12 January 2011 (53.596), 15 May 2013 (17.668), and 1 March 2017 (30.139), respectively. The number inside the bracket beside the dates of structural break denotes the maximum value of a series of likelihood-ratio statistics (i.e., Max. LRS) produced by the procedure proposed by Eizaguirre et al. [25]. Regarding the procedure of finding the Max. LRS, please see Su and Hung [34] for more details. Due to limited space, the information about a series of likelihood-ratio statistics is omitted here and is available upon request. In addition, I assume that all the parameters in the univariate EGARCH-t model will change at a structural breakpoint to generate a series of likelihood-ratio statistics. The univariate EGARCH-t model is expressed as follows. ${\mathrm{r}}_{\mathrm{t}}={\mathsf{\varphi }}_0+{\mathsf{\varphi }}_1{\mathrm{r}}_{\mathrm{t}-1}+{\mathsf{\epsilon }}_{\mathrm{t}}$, ${\mathsf{\epsilon }}_{\mathbf{t}}\left|{\mathsf{\Omega }}_{\mathrm{t}-1}~\mathrm{t}\left(0,{\mathrm{h}}_{\mathrm{t}},\mathrm{n}\right)\right.$, $\ln \left({\mathrm{h}}_{\mathrm{t}}\right)=\mathsf{\omega }+\mathsf{\beta }\ln \left({\mathrm{h}}_{\mathrm{t}-1}\right)+\mathsf{\gamma }\left({\mathsf{\epsilon }}_{\mathbf{t}-1}/\sqrt{{\mathrm{h}}_{\mathrm{t}-1}}\right)+\mathsf{\alpha }\left[\left(\left|{\mathsf{\epsilon }}_{\mathbf{t}-1}\right|/\sqrt{{\mathrm{h}}_{\mathrm{t}-1}}\right)-\mathrm{E}\left(\left|{\mathsf{\epsilon }}_{\mathbf{t}-1}\right|/\sqrt{{\mathrm{h}}_{\mathrm{t}-1}}\right)\right]$. Hence, the total number of parameters being changed during the process of finding the structural breakpoint, q, is equal to seven. Regarding q = 7, the critical values of the sup LR test for $l$ = 0 at the 90%, 95% and 99% levels are 21.23, 23.50 and 28.01, respectively. Variable $l$ denotes the total number of structural breakpoints. Notably, the values of the sup LR test (i.e., Max. LRS) are above the critical values at the 90% level for most of the cases. Hence, the null hypothesis of zero structural breakpoints is rejected for most of the stock indices at the 90% level, indicating that there exists one structural breakpoint in the return series of 14 stock indices in the G7 and E7 countries. As for the critical values of the sup LR test, please see Table 2 of Bai and Perron [35] for more details. Notably, most of the dates of structural breaks are within the period of the GFC happening and the QE policy being implemented.

Table 1. Descriptive statistics of daily return for the E7 and G7 stock indices.

	Mean	SD	Max.	Min.	SK	KUR	J-B	${\mathit{Q}}^2\left(24\right)$	Obs.
Panel A. The overall period
XU100	0.0626	2.3730	17.764	−20.330	−0.3468 c	9.712 c	13,993 c	1074 c	3542
RTSI	0.0624	2.4110	17.764	−20.779	−0.4155 c	9.625 c	13,776 c	1164 c	3542
JKSE	0.0713	1.6112	13.624	−12.925	−0.3621 c	11.093 c	18,239 c	932.9 c	3542
SSE	0.0277	1.9904	45.160	−12.763	3.1167 c	78.069 c	905,235 c	8.67	3542
BSE	0.0654	1.7302	14.412	−17.183	−0.5450 c	11.082 c	18,302 c	776.6 c	3542
MXX	0.0526	1.4561	10.153	−16.277	−0.4461 c	9.871 c	14,499 c	1306 c	3542
BVSP	0.0540	2.1034	13.022	−18.749	−0.4944 c	7.284 c	7976 c	2028 c	3542
FTSE	−0.0004	1.3964	10.870	−11.512	−0.3793 c	8.767 c	11,428 c	2985 c	3542
CAC40	−0.0057	1.6802	11.285	−13.098	−0.3133 c	6.508 c	6309 c	2383 c	3542
DAX	0.0185	1.7380	11.588	−13.054	−0.3638 c	6.003 c	5396 c	2335 c	3542
MIB	−0.0224	1.8261	13.761	−18.541	−0.8237 c	10.102 c	15,463 c	1144 c	3542
DJ	0.0270	1.3578	10.764	−12.577	−0.6376 c	11.107 c	18,449 c	4064 c	3542
TSX	0.0126	1.2871	11.294	−16.998	−1.4318 c	23.325 c	81,506 c	2154 c	3542
N225	0.0111	1.7027	11.805	−12.715	−0.3873 c	6.035 c	5464 c	1444 c	3542
Panel B. The pre-SB period
XU100	0.0855	2.9423	17.764	−20.330	−0.2487 c	7.4985 c	3943.89 c	410.3 c	1676
RTSI	0.1027	3.0401	17.764	−18.992	−0.1873 c	6.8078 c	2591.69 c	544.0 c	1338
JKSE	0.1174	1.8733	13.624	−12.925	−0.2982 c	9.5211 c	6984.90 c	390.7 c	1842
SSE	0.0180	1.9350	12.950	−12.763	−0.1973 c	5.1914 c	3299.20 c	771.6 c	2921
BSE	0.0910	2.0984	14.412	−17.183	−0.5462 c	8.0961 c	4527.30 c	213.7 c	1628
MXX	0.1012	1.7349	10.153	−16.277	−0.3997 c	8.9867 c	5596.33 c	499.0 c	1650
BVSP	0.0680	2.3039	12.933	−12.225	0.0110	3.0426 c	235.31 c	8.542	610
FTSE	0.0033	1.4760	10.870	−10.327	−0.1260 b	6.8439 c	4395.25 c	2491 c	2249
CAC40	−0.0287	1.8192	11.285	−11.476	0.0139	5.2222 c	2037.51 c	1656 c	1793
DAX	−0.1111	2.4495	8.005	−9.790	−0.0921	1.3016 c	40.39 c	226.6 c	561
MIB	−0.0058	1.4282	10.498	−12.900	−0.3739 c	8.9981 c	4385.43 c	548.8 c	1291
DJ	0.0154	1.2201	6.154	−7.396	−0.0752	3.6927 c	667.59 c	580.4 c	1173
TSX	0.0252	1.1025	4.709	−10.482	−1.0141 c	8.3560 c	4300.71 c	203.2 c	1396
N225	−0.0479	1.8132	7.883	−7.020	0.0742	0.9694 c	26.57 c	16.03	663
Panel C. The post-SB period
XU100	0.0419	1.7075	6.919	−11.063	−0.697 c	4.069 c	1438.6 c	134.9 c	1866
RTSI	0.0379	1.9320	9.727	−20.779	−0.911 c	9.806 c	9136.8 c	69.7 c	2204
JKSE	0.0214	1.2661	9.704	−9.635	−0.650 c	9.138 c	6035.4 c	740.8 c	1700
SSE	0.0732	2.2340	45.160	−6.712	13.213 c	268.109 c	1,878,04 c	0.682	621
BSE	0.0436	1.3399	11.573	−11.327	−0.459 c	12.927 c	13394 c	724.1 c	1914
MXX	0.0102	1.1583	5.618	−7.336	−0.625 c	4.760 c	1910.2 c	898.9 c	1892
BVSP	0.0511	2.0597	13.022	−18.749	−0.640 c	8.561 c	9154.5 c	2203 c	2932
FTSE	−0.0071	1.2464	8.666	−11.512	−1.1052 c	14.179 c	11,095 c	932.2 c	1293
CAC40	0.0177	1.5248	8.056	−13.098	−0.8544 c	8.409 c	5366 c	882.3 c	1749
DAX	0.0429	1.5678	11.588	−13.054	−0.4558 c	8.356 c	8776 c	1764 c	2981
MIB	−0.0319	2.0195	13.761	−18.541	−0.8759 c	9.065 c	7996 c	672.3 c	2251
DJ	0.0328	1.4212	10.764	−12.577	−0.8113 c	12.803 c	16,440 c	2930 c	2369
TSX	0.0044	1.3943	11.294	−16.998	−1.5275 c	25.682 c	59,810 c	1449 c	2146
N225	0.0247	1.6763	11.805	−12.715	−0.5169 c	7.611 c	7078 c	1377 c	2879

Notes: 1. The superscripts a, b and c denote significance at the 10%, 5%, and 1% levels, respectively. 2. SK and KUR denote the skewness and excess kurtosis, respectively. 3. J-B statistics are based on Jarque and Bera [36] and are asymptotically chi-squared-distributed with 2 degrees of freedom. 4. $Q^2\left(24\right)$ statistics are asymptotically chi-squared-distributed with 24 degrees of freedom. 5. Obs. denotes the number of observations. 6. The bold and italic fonts in column ‘Mean’ denote the largest and smallest values of mean value, respectively, when the mean values for the overall, pre-SB, and post-SB periods are compared to each other. Conversely, when the values of standard deviation for the overall, pre-SB, and post-SB periods are compared to each other, the bold and italic fonts in column ‘SD’ denote the largest and smallest values of standard deviation, respectively. 7. The dates of structural breaks for the XU100, RTSI, JKSE, SSE, BSE, MXX, and BVSP are 2010/01/13, 2008/01/23, 2011/01/12, 2017/03/01, 2009/09/15, 2009/11/11 and 2004/01/21, respectively. Conversely, the dates of structural breaks for FTSE, CAC40, DAX, MIB, DJ, TSX, and N225 are 2013/05/15, 2010/09/01, 2003/10/15, 2007/10/09, 2007/02/27, 2008/05/28 and 2004/05/12, respectively. 8. The overall period is from 16 October 2000 to 27 July 2020.

reports basic descriptive statistics of the daily return of stock indices in the E7 and G7 countries during the overall period and two subperiods. The descriptive statistics in panel A illustrate the average behavior of said descriptive statistics for the overall period. On the other hand, the descriptive statistics in panel B and panel C demonstrate the behavior of said descriptive statistics during the pre-SB and post-SB periods, respectively. We can see some interesting phenomena in the columns ‘Mean’ and ‘SD’ in Table 1 when the values of mean or standard deviation for the overall, pre-SB, and post-SB periods are compared to each other. First, irrespectively of the G7 or E7 countries, all the greatest and smallest values of mean or standard deviation are distributed in the pre-SB period or the post-SB period. This result indicates that the values of the mean (or standard deviation) for the overall period are analogously the average values of the mean (or standard deviation) for the pre-SB and post-SB periods. Second, regarding the E7 countries, irrespectively of ‘mean’ or ‘SD’, all the greatest and smallest values are distributed in the pre-SB and post-SB periods, respectively, except for the SSE. This result indicates that the stock indices in the E7 countries have a higher return and higher risk in the pre-SB period but a lower return and lower risk in the post-SB period. Third, regarding the G7 countries, irrespectively of ‘mean’ or ‘SD’, all the greatest and smallest values are distributed in the pre-SB and post-SB periods randomly. More precisely, if the greatest mean values and the smallest SD values are distributed in the pre-SB period then the smallest mean values and the greatest SD values are distributed in the post-SB period, and vice versa. The only two exceptions are FTSE and DJ. This result indicates that the stock indices in the G7 have higher return and lower risk in the pre-SB (respectively, post-SB) period, but lower return and higher risk in the post-SB (respectively, pre-SB) period. To sum up, the stock indices in the G7 and E7 countries have different investment attributes in the two subperiods, and then there exists a structural break on the return series of them. As for the other descriptive statistics, they have the same features as those for most of the financial return series. For example, the distribution of returns is left-skewed and has a larger and thicker tail than the normal distribution, indicating that the return series is not normally distributed. The above results are found by the coefficient of skewness, excess kurtosis, and the J-B normality test statistics [36]. In addition, the return series exhibit linear dependence and strong ARCH effect as shown by the Ljung-Box ${\mathrm{Q}}^2\left(24\right)$ statistics for the squared returns. From the above findings, a GARCH family model is very suitable to seize the fat tails and time-varying volatility found in these series of asset returns.

5. Empirical Results

This study mainly utilizes the empirical results of the B-EGARCH-SB model to explore the issue of the interactions between the emerging and developed markets from a structural break perspective. To ensure the findings are credible, the results of three model-fitting ability tests for 46 paired stock indices are used to inspect the accuracy of the dates of SB found in Section 4 and further confirm the appropriateness of the subperiods settings on the B-EGARCH-SB model. The three model-fitting ability tests include the likelihood ratio test (LR), the mean absolute error (MAE), and the parameters’ equality test. The first two tests are related to the data-fitting-ability comparison of two models with SB and without SB (i.e., the B-EGARCH-SB and the B-EGARCH-NSB models). Conversely, the third test is used to inspect the equality of two parameters, corresponding to the pre- and post-SB periods on some parameters of the B-EGARCH-SB model, which are associated with the explored issues. In addition, this study utilizes the unit root test and the cointegration test to ascertain the suitability of the VAR type mean equation. Further, the eigenvalues of the variance–covariance matrix are found to ensure the semidefinite positivity of this matrix, and further to ascertain the suitability of the BEKK-EGARCH type of the variance–covariance equation.

5.1. The Unit Root Test and the Cointegration Test

The empirical model of this study is a diagonal bivariate BEKK-EGARCH model with the setting of a structural break. The mean equation of this model is expressed in the form of a bivariate vector autoregressive with lag one period (VAR (1)). If the price levels of a pair of stock indices are cointegrated, then the VAR type of the mean equation should be replaced with the vector error correction model (VECM). Hence, in order to check that the model of this study is suitable, one should perform a Johansen cointegration test to check whether there exists a relationship between a pair of stock indices in terms of the price level. Before the above test is executed, we first perform the Augmented Dickey–Fuller (ADF) and Phillips–Perron (PP) unit root tests to check whether the price level of the stock index is stationary or not. The ADF test consists of estimating the following regression.

$${\Delta \mathrm{y}}_{\mathrm{t}}={\mathsf{\psi }\mathrm{y}}_{\mathrm{t}-1}+{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{p}}{\mathsf{\beta }}_{\mathrm{i}}{\Delta \mathrm{y}}_{\mathrm{t}-\mathrm{i}}+{\mathsf{\alpha }}_0+{\mathsf{\alpha }}_1\mathrm{t}+{\mathrm{u}}_{\mathrm{t}}$$

(8)

where ${\mathrm{u}}_{\mathrm{t}}$ is a pure white noise error term and ${\Delta \mathrm{y}}_{\mathrm{t}}={\mathrm{y}}_{\mathrm{t}}-{\mathrm{y}}_{\mathrm{t}-1}$. The number of lagged difference terms being included is to ensure the error term in the equation is serially uncorrelated. Notably, ${\mathsf{\alpha }}_0\ne 0$ and ${\mathsf{\alpha }}_1=0$ for the case of constant but no trend, whereas ${\mathsf{\alpha }}_0\ne 0$ and ${\mathsf{\alpha }}_1\ne 0$ for the case of constant and trend. The null hypothesis of ADF test is ${\mathrm{H}}_0: \mathsf{\psi }=0$. This indicates that if the null hypothesis is not rejected, then the series contains a unit root or this series is not stationary. Hence, regarding the above two cases, the test statistics for the ADF test are defined as follows.

$$\mathrm{Test} \mathrm{statistic}=\frac{\hat{\mathsf{\psi }}}{\widehat{\mathrm{SE}}\left(\hat{\mathsf{\psi }}\right)}$$

(9)

As for the critical values for this test, please see Fuller [37] for more details.

Table 2. The results of ADF and PP unit roots tests in the price level and first difference.

	ADF				PP
	Price Level		First Difference		Price Level		First Difference
	No Trend	Trend	No Trend	Trend	No Trend	Trend	No Trend	Trend
XU100	−1.19	−2.68	−18.40 c	−18.40 c	−1.11	−2.57	−58.18 c	−58.18 c
RTSI	−1.14	−2.61	−15.21 c	−15.21 c	−1.11	−2.58	−58.52 c	−58.52 c
JKSE	−1.56	−1.28	−11.56 c	−11.62 c	−1.71	−0.85	−57.50 c	−57.52 c
SSE	−1.34	−2.38	−13.71 c	−13.74 c	−1.01	−1.97	−59.48 c	−59.48 c
BSE	−1.11	−1.99	−23.29 c	−23.30 c	−1.06	−1.99	−58.90 c	−58.89 c
MXX	−1.82	−0.70	−16.57 c	−16.66 c	−2.09	−0.68	−55.56 c	−55.62 c
BVSP	−1.22	−2.09	−16.49 c	−16.49 c	−1.09	−1.91	−59.97 c	−59.96 c
FTSE	−1.96	−3.39 a	−14.54 c	−14.54 c	−2.08	−3.42 b	−60.88 c	−60.87 c
CAC40	−2.47	−2.92	−29.40 c	−29.42 c	−2.59 a	−3.01	−59.63 c	−59.64 c
DAX	−0.83	−3.60 b	−20.28 c	−20.30 c	−0.79	−3.41 b	−58.63 c	−58.63 c
MIB	−2.53	−2.61	−16.54 c	−16.56 c	−2.36	−2.51	−59.52 c	−59.52 c
DJ	−0.30	−2.55	−13.58 c	−13.61 c	−0.39	−2.61	−63.22 c	−63.23 c
TSX	−1.26	−2.85	−15.30 c	−15.30 c	−1.17	−3.25 a	−59.20 c	−59.19 c
N225	−1.21	−2.49	−60.25 c	−60.26 c	−1.15	−2.46	−60.27c	−60.29 c

Note: 1. The superscripts a, b and c denote significance at the 10%, 5%, and 1% levels, respectively. 2. The numbers in the column ‘No trend’ denote the values of the test statistic for the case of constant but no trend, whereas the numbers in the column ‘Trend’ denote the values of the test statistic for the case of constant and trend. 3. The critical values of the ADF unit root test, including the constant term but no trend, are −3.43, −2.86, and −2.57 at the 1%, 5%, and 10% levels, respectively. 4. The critical values of the ADF unit root test, including constant and trend terms, are −3.96, −3.41, and −3.12 at the 1%, 5%, and 10% levels, respectively. 5. As for the above critical values, please see Fuller [37] for more details.

lists the results of ADF and PP tests for the stock indices in E7 and G7 countries From Table 2, we can see that, irrespectively of the case of constant but no trend or the case of constant and trend, the values of the test statistic for the price level are greater than the critical value at the 10%. This result indicates that the series of price levels of the stock index contains a unit root, or this series is not stationary. On the contrary, the values of the test statistic for the first difference in price level are smaller than the critical value at the 1% level. This result indicates that the series of the first difference in price level for the stock index is stationary.

If the price levels are considered to be non-stationary, the next step is to check whether the price levels for a pair of stock indices are cointegrated. When the price levels for a pair of stock indices are cointegrated it means that the price levels for a pair of stock indices have a long-term relationship between them. Subsequently, this study uses the Johansen lambda tests to check whether there exists a relationship for a pair of stock indices in terms of the price level. Suppose that a set of variables ($\mathrm{k}\ge 2$) under consideration are I(1) and are thought to be potentially cointegrated. A vector autoregressive (VAR) with p lags containing these variables could be set up:

$$Y_{\mathrm{t}}={\mathsf{\beta }}_1{Y}_{\mathrm{t}-1}+{\mathsf{\beta }}_2{Y}_{\mathrm{t}-2}+\dots +{\mathsf{\beta }}_p{Y}_{\mathrm{t}-\mathrm{p}}+u_{\mathrm{t}}$$

(10)

where $Y_{\mathrm{t}}$, $Y_{\mathrm{t}-1}$, … $Y_{\mathrm{t}-\mathrm{p}}$ are a $\mathrm{k}$ vector of non-stationary I(1) variable, ${\mathsf{\beta }}_1$, ${\mathsf{\beta }}_2$,… ${\mathsf{\beta }}_p$ are $\mathsf{\kappa }\times \mathrm{k}$ matrices, and $u_{\mathrm{t}}$ is a vector of innovations. In order to use the Johansen test, the above VAR needs to be turned into a vector error correction model (VECM) of the following form.

$$\Delta Y_{\mathrm{t}}=\Pi Y_{\mathrm{t}-\mathrm{p}}+{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{p}-1}{\Gamma }_i\Delta Y_{\mathrm{t}-\mathrm{i}}+u_{\mathrm{t}}$$

(11)

where ${\Gamma }_i=\left({{\displaystyle \sum }}_{\mathrm{j}=1}^{\mathrm{i}}{\beta }_j\right)-I_k$, $\Pi =\left({{\displaystyle \sum }}_{\mathrm{j}=1}^{\mathrm{p}}{\beta }_j\right)-I_k$ is called as a long-run coefficient matrix, since in equilibrium, all the $\Delta Y_{t-i}$ will be zero, and setting the error term, $u_{\mathrm{t}}$, to their expected value of zero will mean $\Pi Y_{\mathrm{t}-\mathrm{p}}=0$. Hence, the test for cointegration between the $\mathrm{k}$ variables in $Y_{\mathrm{t}}$ is calculated by looking at the rank of the $\Pi$ matrix via its eigenvalues. The rank of a matrix is equal to the number of its characteristic roots (eigenvalues) that are different from zero. The eigenvalues, denoted ${\mathsf{\lambda }}_{\mathrm{i}}$, are put in ascending order: ${\mathsf{\lambda }}_1\ge {\mathsf{\lambda }}_2\ge \dots \ge {\mathsf{\lambda }}_{\mathrm{k}}$. Notably, the above ${\mathsf{\lambda }}_{\mathrm{i}}, \mathrm{i}=1,\dots \mathrm{k}$ must be less than 1 in absolute value and positive, and ${\mathsf{\lambda }}_1$ will be the largest (i.e., the closet to one), whereas ${\mathsf{\lambda }}_{\mathrm{k}}$ will be the smallest (i.e., the closest to zero). If the $\mathrm{k}$ variables are not cointegrated, the rank of $\Pi$ will not be significantly different from zero, so ${\mathsf{\lambda }}_{\mathrm{i}}\approx 0$ for $\mathrm{i}=1,2,\dots ,\mathrm{k}$. There are two test statistics for cointegration under the Johansen approach, which are formulated as

$${\mathsf{\lambda }}_{\mathrm{trace}}\left(\mathrm{r}\right)=-\mathrm{T}{{\displaystyle \sum }}_{\mathrm{i}=\mathrm{r}+1}^{\mathrm{k}}\ln \left(1-{\hat{\mathsf{\lambda }}}_{\mathrm{i}}\right)$$

(12)

$${\mathsf{\lambda }}_{\max }\left(\mathrm{r},\mathrm{r}+1\right)=-\mathrm{Tln}\left(1-{\hat{\mathsf{\lambda }}}_{\mathrm{r}+1}\right)$$

(13)

${\mathsf{\lambda }}_{\mathrm{trace}}$ is a joint test where the null is that the number of cointegrating vectors is less than or equal to $\mathrm{r}$ against the alternative that there is more than $\mathrm{r}$. ${\mathsf{\lambda }}_{\mathrm{trace}}=0$ when all the ${\mathsf{\lambda }}_{\mathrm{i}}=0$, for $\mathrm{i}=1,\dots ,\mathrm{k}$. ${\mathsf{\lambda }}_{\max }$ conducts separate tests on each eigenvalue and has as its null hypothesis that the number of cointegrating vectors is $\mathrm{r}$ against an alternative $\mathrm{r}+1$. Notably, Osterwald-Lenum [38] provided the critical values for these two Johansen lambda tests.

Table 3. The results of Johansen lambda tests for the cointegrating rank.

	${\mathbf{\lambda }}_{\mathbf{m}\mathbf{a}\mathbf{x}}$		${\mathbf{\lambda }}_{\mathbf{T}\mathbf{r}\mathbf{a}\mathbf{c}\mathbf{e}}$			${\mathbf{\lambda }}_{\mathbf{m}\mathbf{a}\mathbf{x}}$		${\mathbf{\lambda }}_{\mathbf{T}\mathbf{r}\mathbf{a}\mathbf{c}\mathbf{e}}$
	${\mathbf{H}}_0: \mathbf{r} \le 0^{ }$	${\mathbf{H}}_0: \mathbf{r} \le 1$	${\mathbf{H}}_0: \mathbf{r} = 0^{ }$	${\mathbf{H}}_0: \mathbf{r} = 1$		${\mathbf{H}}_0: \mathbf{r} \le 0$	${\mathbf{H}}_0: \mathbf{r} \le 1$	${\mathbf{H}}_0: \mathbf{r} = 0$	${\mathbf{H}}_0: \mathbf{r} = 1$
Xu-ft	12.126	0.425	12.551	0.425	rt-ft	19.102 b	0.502	19.604 a	0.502
Xu-ca	10.621	1.013	11.634	1.013	rt-ca	11.781	0.287	12.068	0.287
Xu-da	14.374 a	0.416	14.791	0.416	rt-da	21.360 c	0.017	21.377 b	0.017
Xu-mi	7.333	1.616	8.949	1.616	rt-mi	6.693	0.653	7.345	0.653
Xu-dj	7.084	0.134	7.218	0.134	rt-dj	12.363	0.048	12.411	0.048
Xu-ts	12.943	0.444	13.387	0.444	rt-ts	21.851 c	0.258	22.109 b	0.258
Xu-n2	7.654	0.614	8.268	0.614	rt-n2	8.973	0.028	9.001	0.028
jk-ft	15.289 a	1.052	16.341	1.052	ss-ft	6.765	2.359	9.124	2.359
jk-ca	10.004	1.985	11.988	1.985	ss-ca	12.592	2.451	15.043	2.451
jk-da	9.408	1.452	10.860	1.452	ss-da	6.614	0.587	7.201	0.587
jk-mi	7.404	2.037	9.441	2.037	ss-mi	10.215	1.506	11.721	1.506
jk-dj	3.885	3.189	7.074	3.189	ss-dj	5.121	0.199	5.320	0.199
jk-ts	9.374	1.079	10.453	1.079	ss-ts	7.366	1.488	8.854	1.488
jk-n2	5.066	2.471	7.538	2.471	ss-n2	8.225	2.160	10.385	2.160
bs-ft	14.098 a	0.130	14.228	0.130	mx-ft	13.717	3.169	16.886	3.169
bs-ca	11.837	0.107	11.944	0.107	mx-ca	9.721	4.552	14.272	4.552
bs-da	16.218 b	0.032	16.250	0.032	mx-da	6.352	4.054	10.406	4.054
bs-mi	7.205	0.187	7.392	0.187	mx-mi	7.615	3.890	11.504	3.890
bs-dj	7.482	0.001	7.484	0.001	mx-dj	7.689	1.754	9.443	1.754
bs-ts	12.895	0.111	13.007	0.111	mx-ts	9.562	2.920	12.482	2.920
bs-n2	7.390	0.042	7.432	0.042	mx-n2	5.437	3.950	9.387	3.950
bv-ft	7.063	0.529	7.592	0.529
bv-ca	10.541	0.659	11.200	0.659
bv-da	4.408	0.299	4.706	0.299
bv-mi	8.275	0.646	8.921	0.646
bv-dj	3.251	0.027	3.278	0.027
bv-ts	5.654	0.630	6.284	0.630
bv-n2	4.673	0.314	4.987	0.314

Notes: 1. a, b and c denote significance at the 10%, 5%, and 1% levels, respectively. 2. Regarding ${\mathrm{H}}_0:\mathrm{r}=0$, the critical values of the ${\mathsf{\lambda }}_{\max }$$\left({\mathsf{\lambda }}_{\mathrm{Trace}}\right)$ test statistics at the 10%, 5%, and 1% significance levels are 13.75 (17.85), 15.67 (19.96) and 20.20 (24.60), respectively. 3. Regarding ${\mathrm{H}}_0:\mathrm{r}=1$, the critical values of the ${\mathsf{\lambda }}_{\max }$$\left({\mathsf{\lambda }}_{\mathrm{Trace}}\right)$ test statistics at the 10%, 5%, and 1% significance levels are 7.52 (7.52), 9.24 (9.24), and 12.97 (12.97), respectively. 4. As for the critical values of the ${\mathsf{\lambda }}_{\max }$ and ${\mathsf{\lambda }}_{\mathrm{Trace}}$ test statistics, please see Osterwald-Lenum [38] for more details.

lists the results of two Johansen lambda tests for all 49 pairs of stock indices. From Table 3, we can see that the values of ${\mathsf{\lambda }}_{\mathrm{trace}}$ (${\mathsf{\lambda }}_{\max })$ for the null hypothesis ${\mathrm{H}}_0:\mathrm{r}\le 0$ (${\mathrm{H}}_0:\mathrm{r}=0$) are less than the critical values for most cases, indicating that the null hypothesis ${\mathrm{H}}_0:\mathrm{r}\le 0$ (${\mathrm{H}}_0:\mathrm{r}=0$) is not rejected. It would be concluded that there is no cointegration between the two stock indices for each of the 49 pairs of data. Hence, there is no long-run equilibrium relationship between these two stock indices, indicating that the setting of VAR type of the mean equation in this study is suitable.

5.2. Confirm the Appropriateness of the Subperiods’ Settings on the B-EGARCH-SB Model

Table 4. The results of three types of model-fitting ability tests for the B-EGARCH-SB model.

	XU100-FTSE	XU100-CAC40	XU100-DAX	XU100-MIB	XU100-DJ	XU100-TSX	XU100-N225	Sum
Pre-SB	2000/10/16~ 2010/01/13	-	2000/10/16~ 2003/10/15	2000/10/16~ 2007/10/09	2000/10/16~ 2007/02/27	2000/10/16~ 2008/05/28	2000/10/16~ 2004/05/12
Post-SB	2013/05/15~ 2020/07/28	-	2010/01/13~ 2020/07/28	2010/01/13~ 2020/07/28	2010/01/13~ 2020/07/28	2010/01/13~ 2020/07/28	2010/01/13~ 2020/07/28
LL	−12,319.22 (−12,307.01)	−	−13,202.77 (−13,181.63)	−13,373.04 (−13,341.68)	−12,269.78 (−12,245.46)	−11,934.13 (−11,909.50)	−13,600.69 (−13,557.08)
LR	23.98 b	-	42.28 c	62.72 c	48.64 c	49.26 c	87.22 c	6
MAE	2.7053 (2.7001)	-	3.1172(3.0914)	3.2339(3.2123)	2.4365(2.4398)	2.4387(2.4466)	2.7952(2.7858)	2(4)
RMSE	7.7184 (7.7165)	-	8.5565(8.5490)	9.9569(9.9417)	7.4856(7.4852)	8.3094(8.2929)	8.0160(8.0231)	1(5)
${\mathrm{H}}_0^1$(${\mathrm{H}}_0^2$)	1.9570 (4.2445)	-	8.7291 b(1.4409)	3.1123(2.0789)	0.0136 c(2.0214)	5.9070 c(5.7799 a)	1.1849(4.2360)	3(1)
${\mathrm{H}}_0^3$(${\mathrm{H}}_0^4$)	10.793 c (5.2046a)	-	48.013c(8.0897b)	62.611 c(30.595 c)	12.158 c(7.9315 b)	5.3624 a (43.397 c)	144.58 c(2.1254)	6(5)
${\mathrm{H}}_0^5$	7.3077 b	-	1.3172	90.295 c	16.323 c	9.1210 b	25.413 c	5
${\mathrm{H}}_0^6$(${\mathrm{H}}_0^7$)	3.4005 (5.2454 a)	-	0.5926(2.2128)	3.1377(13.655 c)	0.7000(2.2049)	1.5030(7.9256 b)	5.3099 a (12.373 c)	1(4)
${\lambda }_1$(${\lambda }_2$)	2.593 (1.961)	-	2.638(1.758)	3.245(1.709)	2.164(1.340)	2.103(0.445)	2.300(1.377)
	RTSI-FTSE	RTSI-CAC40	RTSI-DAX	RTSI-MIB	RTSI-DJ	RTSI-TSX	RTSI-N225	Sum
Pre-SB	2000/10/16~ 2008/01/23	2000/10/16~ 2008/01/23	2000/10/16~ 2003/10/15	2000/10/16~ 2007/10/09	2000/10/16~ 2007/02/27	2000/10/16~ 2008/01/23	2000/10/16~ 2004/05/12
Post-SB	2013/05/15~ 2020/07/28	2010/09/01~ 2020/07/28	2008/01/23~ 2020/07/28	2008/01/23~ 2020/07/28	2008/01/23~ 2020/07/28	2008/05/28~ 2020/07/28	2008/01/23~ 2020/07/28
LL	−12,692.4 (−12,677.8)	−13,421.9 (−13,404.8)	−13,555.5 (−13,541.4)	−13,712.1 (−13,693.3)	−12,480.5 (−12,472.5)	−12,162.5 (−12,149.9)	−13,784.6 (−13,745.1)
LR	29.14 b	34.3 c	28.12 b	37.46 c	16.06	25.24 b	78.98 c	6
MAE	2.0570(2.0579)	2.3111(2.3291)	2.3750(2.4002)	2.4634 (2.4646)	2.0762(2.0898)	2.0460(2.0568)	2.3690(2.3584)	6(1)
RMSE	5.5180(5.5205)	6.2396(6.2402)	6.5154(6.5095)	6.8188 (6.8214)	5.7304(5.7346)	5.9208(5.9163)	6.0970(6.0756)	4(3)
${\mathrm{H}}_0^1$(${\mathrm{H}}_0^2$)	2.6214(0.9844)	1.9834(2.0542)	0.1055(1.3241)	9.3509 c(2.7657)	0.7063(4.7977 a)	1.4306(1.2099)	14.752 c(0.1911)	2(1)
${\mathrm{H}}_0^3$(${\mathrm{H}}_0^4$)	3.1115(22.913 c)	46.630 c(4.7351 a)	32.234 c(10.450 c)	22.535 c(18.628 c)	5.2858 a(5.7721 a)	6.2926 b(6.0103 b)	95.933 c(5.6198 a)	6(7)
${\mathrm{H}}_0^5$	1.1522	1.4935	0.9593	0.7305	2.1496	2.4627	5.1074 a	1
${\mathrm{H}}_0^6$(${\mathrm{H}}_0^7$)	1.7919(2.3733)	2.5016(0.08104)	1.4701(0.6377)	1.2448(1.6234)	1.7917(3.3834)	5.2013 a(11.794 c)	0.5440(10.109 c)	1(2)
${\lambda }_1$(${\lambda }_2$)	2.701(1.934)	2.826(2.124)	2.459(2.259)	3.096(2.386)	2.794(1.450)	2.556(0.473)	2.726(1.305)
	JKSE-FTSE	JKSE-CAC40	JKSE-DAX	JKSE-MIB	JKSE-DJ	JKSE-TSX	JKSE-N225	Sum
Pre-SB	2000/10/16~ 2011/01/12	2000/10/16~ 2010/09/01	2000/10/16~ 2003/10/15	2000/10/16~ 2007/10/09	2000/10/16~ 2007/02/27	2000/10/16~ 2008/05/28	2000/10/16~ 2004/05/12
Post-SB	2013/05/15~ 2020/07/28	2011/01/12~ 2020/07/28	2011/01/12~ 2020/07/28	2011/01/12~ 2020/07/28	2011/01/12~ 2020/07/28	2011/01/12~ 2020/07/28	2011/01/12~ 2020/07/28
LL	−10,986.02 (−10,971.85)	−11,722.61 (−11,709.63)	−11,864.14 (−11,839.05)	−12,049.52 (−12,026.59)	−10,794.98 (−10,773.99)	−10,441.79 (−10,419.10)	−11,972.22 (−11,952.79)
LR	28.34 b	25.96 b	50.18 c	45.86 c	41.98 c	45.38 c	38.86 c	7
MAE	1.5747(1.5803)	1.8910(1.8937)	1.8987(1.9058)	2.0232(2.0218)	1.4000(1.3921)	1.4367(1.4153)	2.0883(2.0894)	4(3)
RMSE	5.3189(5.3199)	5.9638(5.9642)	5.6038(5.6118)	7.0081(6.9783)	4.7924(4.7716)	5.6947(5.6623)	6.5592(6.5867)	4(3)
${\mathrm{H}}_0^1$(${\mathrm{H}}_0^2$)	11.008 c(2.8001)	2.3778(5.9817 a)	17.194 c(4.5393)	2.2015(1.5129)	14.618 c(4.9716 a)	0.3605(0.5264)	3.6580(0.7792)	3(2)
${\mathrm{H}}_0^3$(${\mathrm{H}}_0^4$)	4.8726 a(6.0281 b)	20.548 c(0.7417)	17.652 c(11.467c)	29.806 c(11.818 c)	0.5074(8.4995 b)	10.072 c(30.524 c)	32.478 c(10.928 c)	6(6)
${\mathrm{H}}_0^5$	0.4100	0.7582	1.0028	9.3453 c	0.0782	6.7104 b	8.9595 b	3
${\mathrm{H}}_0^6$(${\mathrm{H}}_0^7$)	4.5370(1.9570)	7.9884 b(0.8104)	7.2664 b(3.6869)	13.025 c(1.8152)	24.246 c(1.0319)	7.3548 b(17.984 c)	10.881 c(18.010 c)	6(2)
${\lambda }_1$(${\lambda }_2$)	2.082(0.766)	2.747(0.671)	2.226(0.682)	3.114(0.674)	1.534 (0.885)	0.799 (0.355)	1.819(0.611)

Note: 1. ‘Pre-SB’ denotes the period before the first structural breakpoint whereas ‘Post-SB’ represents the period after the second structural breakpoint. 2. The superscripts a, b and c denote significance at the 10%, 5%, and 1% levels, respectively. 3. LL indicates the log-likelihood value of a model with no structural break (i.e., the B-EGARCH-NSB model or the restricted model), $L{L}_r$. Conversely, the number in the bracket underneath the preceding number denotes the log-likelihood value of the same model with a structural break (i.e., the B-EGARCH-SB model or the unrestricted model), $L{L}_u$. 4. LR denotes the statistics of likelihood ratio test, the null hypothesis of this test is that both the restricted model and unrestricted model are not significantly different and the alternative hypothesis is the unrestricted model has the better fitting performance. $\mathrm{LR}=-2\left(L{L}_r-L{L}_u\right)~{\chi }^2\left(m\right)$, m is the total number of constraints of a restricted model, and is equal to 14 (i.e., ${\mathsf{\varphi }}_{12}^{\mathrm{B}}={\mathsf{\varphi }}_{12}^{\mathrm{D}}$, ${\mathsf{\varphi }}_{12}^{\mathrm{D}}={\mathsf{\varphi }}_{12}^{\mathrm{A}}$; ${\mathsf{\varphi }}_{21}^{\mathrm{B}}={\mathsf{\varphi }}_{21}^{\mathrm{D}}$, ${\mathsf{\varphi }}_{21}^{\mathrm{D}}={\mathsf{\varphi }}_{21}^{\mathrm{A}}$; ${\mathsf{\nu }}_{12}^B$=${\mathsf{\nu }}_{12}^D$, ${\mathsf{\nu }}_{12}^D={\mathsf{\nu }}_{12}^A$; ${\mathsf{\nu }}_{21}^B={\mathsf{\nu }}_{21}^D$, ${\mathsf{\nu }}_{21}^D={\mathsf{\nu }}_{21}^A$; ${\mathsf{\omega }}_{12}^B$ = ${\mathsf{\omega }}_{12}^D$, ${\mathsf{\omega }}_{12}^D={\mathsf{\omega }}_{12}^A$; ${\mathsf{\gamma }}_1^B={\mathsf{\gamma }}_1^D$, ${\mathsf{\gamma }}_1^D={\mathsf{\gamma }}_1^A$; and ${\mathsf{\gamma }}_2^B={\mathsf{\gamma }}_2^D$, ${\mathsf{\gamma }}_2^D={\mathsf{\gamma }}_2^A$). The critical value of the ${\chi }^2\left(14\right)$ at the 10%, 5%, and 1% significance levels is 21.064, 23.685, and 29.141, respectively. 5. MAE denotes the value of the mean absolute error of a model with no structural break (i.e., the B-EGARCH-NSB model). Conversely, the number in the bracket underneath the preceding number denotes the value of MAE for the same model with a structural break (i.e., the B-EGARCH-SB model). 6. RMSE denotes the value of the root-mean-square error of a model with no structural break (i.e., the B-EGARCH-NSB model). Conversely, the number in the bracket underneath the preceding number denotes the value of RMSE for the same model with a structure break (i.e., the B-EGARCH-SB model). 7. The bold font in the row MAE (RMSE) denotes the lower value of MAE (RMSE) when the predictive accuracies of two models (i.e., the B-EGARCH-NSB and the B-EGARCH-NSB) are compared each other. 8. All the parameter equality tests for the unrestricted model—a model with structure break (i.e., the B-EGARCH-SB model)—follow the chi-squared-distributed with 2 degrees of freedom, ${\chi }^2\left(2\right)$, because there are two constraints for all seven parameter equality tests. For example, there are two constraints in the first parameter equality test, as shown by the null hypothesis of this test, ${\mathrm{H}}_0^1:{\mathsf{\varphi }}_{12}^{\mathrm{B}}={\mathsf{\varphi }}_{12}^{\mathrm{D}}={\mathsf{\varphi }}_{12}^{\mathrm{A}}$(i.e., ${\mathrm{H}}_0:{\mathsf{\varphi }}_{12}^{\mathrm{B}}={\mathsf{\varphi }}_{12}^{\mathrm{D}}$ and ${\mathrm{H}}_0:{\mathsf{\varphi }}_{12}^{\mathrm{D}}={\mathsf{\varphi }}_{12}^{\mathrm{A}}$). Additionally, ${\mathrm{H}}_0^2:{\mathsf{\varphi }}_{21}^{\mathrm{B}}={\mathsf{\varphi }}_{21}^{\mathrm{D}}={\mathsf{\varphi }}_{21}^{\mathrm{A}}$, ${\mathrm{H}}_0^3:{\mathsf{\nu }}_{12}^{\mathrm{B}}$=${\mathsf{\nu }}_{12}^{\mathrm{D}}={\mathsf{\nu }}_{12}^{\mathrm{A}}$, ${\mathrm{H}}_0^4:{\mathsf{\nu }}_{21}^{\mathrm{B}}={\mathsf{\nu }}_{21}^{\mathrm{D}}={\mathsf{\nu }}_{21}^{\mathrm{A}}$, ${\mathrm{H}}_0^5:{\mathsf{\omega }}_{12}^{\mathrm{B}}$=${\mathsf{\omega }}_{12}^{\mathrm{D}}={\mathsf{\omega }}_{12}^{\mathrm{A}}$, ${\mathrm{H}}_0^6:{\mathsf{\gamma }}_1^{\mathrm{B}}={\mathsf{\gamma }}_1^{\mathrm{D}}={\mathsf{\gamma }}_1^{\mathrm{A}}$, and ${\mathrm{H}}_0^7:{\mathsf{\gamma }}_2^{\mathrm{B}}={\mathsf{\gamma }}_2^D={\mathsf{\gamma }}_2^{\mathrm{A}}$ are the null hypotheses of the others six parameter equality tests. The critical values of the ${\chi }^2\left(2\right)$ at the 10%, 5%, and 1% significance levels are 4.605, 5.991 and 9.210, respectively. 9. ${\lambda }_1$ and ${\lambda }_2$ denote the eigenvalues of the matrix of variance–covariance $H_t$ and the value of ${\lambda }_1$ is greater than the value of ${\lambda }_2$. They are determined by the approach of Johnson [39] via using the ‘eigen’ function in Rat 6.0. 10. The numbers in column Sum corresponding to the rows ${\mathrm{H}}_0^1~{\mathrm{H}}_0^7$ (respectively, LR) denote the total number of paired data that a specific parameter equality tests (respectively, likelihood ratio test), being significantly for each panel. 11. When the predictive accuracies of the restricted and unrestricted models are compared with each other, the number in column Sum corresponding to the row MAE (RMSE) denotes the total number of paired data of the restricted model with better volatility forecast performance, whereas the number in the bracket underneath the preceding number denotes the total number of paired data in the unrestricted model with better volatility forecast performance. The restricted and unrestricted models are the B-EGARCH-NSB and the B-EGARCH-SB models, respectively. 12. The empirical results of XU100-CAC40, MXX-CAC40, and MXX-DJ paired data are omitted here because I cannot obtain the convergent results for the above three paired data.

lists the empirical results of three model-fitting ability tests and two eigenvalues of the variance–covariance matrix of the B-EGARCH-SB model for the 46 paired stock indices data. In Table 4, seven panels list the results of three model-fitting tests of 49 paired stock indices. The 49 paired stock indices are composed of one index from the E7 countries and another index from the G7 countries. Each panel includes seven paired stock indices corresponding to a specific emerging stock index in the E7 countries. For example, the first panel includes seven paired stock indices related to XU100, such as XU100-FTSE, XU100-CAC40, XU100-DAX, XU100-MIB, XU100-DJ, XU100-TSX, and XU100-N225. The seven paired stock indices are composed of XU100 and one stock index from the G7 countries. At the same inference, the second, third, fourth, fifth, sixth, and seventh panels include seven paired stock indices related to RTSI, JKSE, SSE, BSE, MXX, and BVSP, respectively. Notably, I cannot obtain the converging results for the XU100-CAC40, MXX-CAC40 and MXX-DJ paired stock indices. Hence, the results of the above three paired stock indices are omitted in Table 4, indicating that the total number of paired stock indices in Table 4 is 46. Due to limited space, only the results of the first three panels are listed in Table 4. Subsequently, I use the empirical results of three model-fitting ability tests in Table 4 to inspect the appropriateness of two subperiods’ settings of the B-EGARCH-SB model. The two subperiods are the pre-SB period and the post-SB period. As reported in Section 3, the dates of SB for the 14 stock indices in the G7 and E7 countries are all different. Hence, the study period is divided into three subperiods according to the dates of the first and second structural breakpoints. Taking an example of the XU100-FTSE paired stock indices data, the ending date of the pre-SB period is 13 January 2010, the date of SB for the XU100. On the contrary, the starting date of the post-SB period is 15 May 2013, the date of SB for FTSE. In addition, the transition period is between 13 January 2010 and 15 May 2013. As for the others paired stock indices data, the setting of two subperiods is defined at the rows ‘Pre-SB’ and ‘Post-SB’ of each panel in Table 4. As shown by the data listed in the row ‘LR’ of the first panel in Table 4, the LR tests for six paired stock indices related to XU100 are all significant. Then, at the first panel in Table 4, the result ‘6’ is recorded at the column ‘Sum’ corresponding to the row ‘LR’. At panel A of

Table 5. The summary results of three types of model-fitting ability tests for the B-EGARCH-SB model.

	XU100(6)	RTSI(7)	JKSE(7)	SSE(7)	BSE(7)	MXX(5)	BVSP(7)	Sum(46)
Panel A. The likelihood ratio test
LR	6	6	7	6	7	5	7	44
Panel B. Volatility forecasting performance comparison
MAE	2(4)	6(1)	4(3)	1(6)	2(5)	4(1)	3(4)	22(24)
RMSE	1(5)	4(3)	4(3)	2(5)	1(6)	1(4)	7(0)	20(26)
Panel C. The parameters equality tests
${\mathrm{H}}_0^1$	3	2	3	3	4	1	2	18
${\mathrm{H}}_0^2$	1	1	2	1	0	1	2	8
${\mathrm{H}}_0^3$	6	6	6	6	7	5	7	43
${\mathrm{H}}_0^4$	5	7	6	5	4	5	5	37
${\mathrm{H}}_0^5$	5	1	3	3	5	3	5	25
${\mathrm{H}}_0^6$	1	1	6	7	5	1	2	23
${\mathrm{H}}_0^7$	4	2	2	2	5	3	3	21

Note: 1. The numbers in this table are summarized from the numbers in the column ‘Sum’ of Table 4. For example, the number in the column ‘XU100’ and row ‘LR’ in panel A of this table is the number in the column ‘Sum’ and row ‘LR’ of Table 4 corresponding to the panel of all XU100-based paired data. On the other hand, the numbers in the column ‘XU100’ and row ‘MAE’ (respectively, ‘RMSE’) in panel B of this table are the corresponding numbers in the column ‘Sum’ and row ‘MAE’ (respectively, ‘RMSE’) of Table 4, corresponding to the panel of all XU100-based paired data. In addition, the number in column ‘XU100’ and row ‘${\mathrm{H}}_0^1$’ in panel C of this table is the number in the column ‘Sum’ and row ‘${\mathrm{H}}_0^1$’ of Table 4, corresponding to the panel of all XU100-based paired data; so are the numbers in column ‘XU100’ and the rows ${\mathrm{H}}_0^2$, ${\mathrm{H}}_0^3$, ${\mathrm{H}}_0^4$, ${\mathrm{H}}_0^5$, ${\mathrm{H}}_0^6$ and ${\mathrm{H}}_0^7$. 2. The numbers in column ‘Sum’ denote the summation of seven corresponding numbers in the columns ‘XU100’, ‘RTSI’, ‘MXX’, ‘BVSP’, ‘JKSE’, ‘SSE’, and ‘BSE’. 3. The numbers in the brackets beside the alternative column titles corresponding to alternative emerging stock indices denote the total number of paired data that the convergent empirical results obtained. 4. The bold font in the column ‘Sum’ and row ‘MAE’ (‘RMSE’) denotes the greater value of the total number of paired data when the number in the column ‘Sum’ and row ‘MAE’ (‘RMSE’) and the number in the bracket beside the preceding number are compared to each other.

, the above result ‘6’ is also recorded at the column ‘XU100’ corresponding to the row ‘LR’. From the data listed in the row ‘MAE’ of the first panel in Table 4, I observe that for the XU100-DJ and XU100-TSX paired stock indices, the B-EGARCH-NSB model has lower MAE values than the B-EGARCH-SB model. Thus, the total number of paired data that the B-EGARCH-NSB model has for better volatility forecast performance is two. Conversely, for the XU100-FTSE, XU100-DAX, XU100-MIB, and XU100-N225 paired stock indices, the B-EGARCH-SB model bears lower MAE values than the B-EGARCH-NSB model. Thus, the total number of paired data that the B-EGARCH-SB model has for better volatility forecast performance is four Hence, in the first panel in Table 4, the results ‘2(4)’ are recorded at the column ‘Sum’ corresponding to the row ‘MAE’. The above results, ‘2(4)’, are also recorded in the column ‘XU100’ corresponding to row ‘MAE’ in panel B of Table 5. With the same inference process, the result of the data listed in row ‘RMSE’ of the first panel in Table 4 is recorded. Thus, in the first panel in Table 4, the results ‘1(5)’ are recorded at the column ‘Sum’ corresponding to the row ‘RMSE’. The above results, ‘1(5)’, are also recorded in column ‘XU100’ corresponding to row ‘RMAE’ in panel B of Table 5. From the data listed in row ‘${\mathrm{H}}_0^1$’ of the first panel in Table 4, I discover that the first parameter equality test is significant for the XU100-DAX, XU100-DJ, and XU100-TSX pairs of stock indices. Then, in the first panel in Table 4, the result ‘3’ is recorded in the column ‘Sum’ corresponding to the row ‘${\mathrm{H}}_0^1$’. The above result, ‘3’, is also recorded in the column ‘XU100’ corresponding to the row ‘${\mathrm{H}}_0^1$’ at panel C of Table 5. With the same inference process, the results of the data listed in rows ‘${\mathrm{H}}_0^2~{\mathrm{H}}_0^7$’ in the first panel of Table 4 are recorded into Table 5, and so are the results of the data listed at the others panels of Table 4. Finally, the values of two eigenvalues (${\mathsf{\lambda }}_1$ and ${\mathsf{\lambda }}_2$) are all greater than zero in Table 4. This result indicates that the variance–covariance matrix expressed in a diagonal bivariate BEKK-EGARCH (1,1)-X model in this study is positive definite.

Table 5 lists the summary results of three model-fitting ability tests of the B-EGARCH-SB model for the 46 paired stock indices data. For example, the data listed in the column ‘XU100’ of Table 5 report the summary results of three model-fitting ability tests for all pairs of stock indices related to ‘XU100’. With the same inference process, the data listed in the column ‘Sum’ of Table 5 report the summary results of three model-fitting ability tests for all 46 paired stock indices related to all seven emerging stock indices. For example, the number ‘44’ listed in the column ‘Sum’ corresponding to row ‘LR’ means that there are 44 paired stock indices passing the LR test. In other words, the summation of all numbers in row ‘LR’ of panel A is equal to 44. This result indicates that the B-EGARCH-SB model has a better data fitting performance than the B-EGARCH-NSB model because the passing rate of the LR test is 95.65% (=44/46). As shown by the numbers ‘22(24)’ and ‘20(26)’ listed in the column ‘Sum’, respectively, corresponding to the rows ‘MAE’ and ‘RMSE’, I find that the B-EGARCH-SB model has a better volatility forecast performance than the B-EGARCH-NSB model because the B-EGARCH-SB model has lower MAE and RMSE values for most of the paired stock indices. For example, the B-EGARCH-SB model has a lower MAE (RMSE) value than the B-EGARCH-NSB model for 24 (26) of 46 pairs of stock indices. The numbers 24 and 26 are greater than 23, which is half of 46. From the above results of the LR test, MAE and RMSE, I can infer that the B-EGARCH-SB model fits the 46 paired stock indices data well. Hence, the settings of two subperiods on the B-EGARCH-SB model are suitable and the dates of a structural break on the stock indices in the G7 and E7 countries are accurate enough. Subsequently, this study investigates which parameters of the model or which financial features are significantly affected by the structural break. Thus, this study uses the empirical results of seven parameter equality tests of the B-EGARCH-SB model for all 46 paired stock indices data to explore which parameters of this model are significantly different for the pre- and post-SB periods. The seven parameter equality tests can be divided into four groups of tests to explore the following issue. The issue is whether the behavior of four financial features (i.e., the return spillover, volatility spillover, correlation, and leverage effect) in the emerging and developed stock markets is different for the pre-SB and post-SB periods. The first group of the test includes the first and second parameter equality tests (i.e.,${\mathrm{H}}_0^1:{\mathsf{\varphi }}_{12}^{\mathrm{B}}={\mathsf{\varphi }}_{12}^{\mathrm{D}}={\mathsf{\varphi }}_{12}^{\mathrm{A}}{ \mathrm{and} \mathrm{H}}_0^2:{\mathsf{\varphi }}_{21}^{\mathrm{B}}={\mathsf{\varphi }}_{21}^{\mathrm{D}}={\mathsf{\varphi }}_{21}^{\mathrm{A}}$) to discuss the issue about the return spillover between two markets, whereas the second group of tests includes the third and fourth parameter equality tests ($\mathrm{i}.\mathrm{e}.,{\mathrm{H}}_0^3:{\mathsf{\nu }}_{12}^{\mathrm{B}}$=${\mathsf{\nu }}_{12}^{\mathrm{D}}={\mathsf{\nu }}_{12}^{\mathrm{A}}$ and ${\mathrm{H}}_0^4:{\mathsf{\nu }}_{21}^{\mathrm{B}}={\mathsf{\nu }}_{21}^{\mathrm{D}}={\mathsf{\nu }}_{21}^{\mathrm{A}}$) to talk about the issue of volatility spillover between two markets. The third group of test only includes the fifth parameter equality test ($\mathrm{i}.\mathrm{e}.,{ \mathrm{H}}_0^5:{\mathsf{\omega }}_{12}^{\mathrm{B}}$=${\mathsf{\omega }}_{12}^{\mathrm{D}}={\mathsf{\omega }}_{12}^{\mathrm{A}}$) to talk about the issue of correlation between two markets, whereas the fourth group of the tests includes the sixth and seventh parameter equality tests (i.e.,${ \mathrm{H}}_0^6:{\mathsf{\gamma }}_1^{\mathrm{B}}={\mathsf{\gamma }}_1^{\mathrm{D}}={\mathsf{\gamma }}_1^{\mathrm{A}}$, and ${\mathrm{H}}_0^7:{\mathsf{\gamma }}_2^{\mathrm{B}}={\mathsf{\gamma }}_2^{\mathrm{D}}={\mathsf{\gamma }}_2^{\mathrm{A}}$) to discuss the issue of leverage effect in each of the two markets. The above two markets denote the emerging and developed stock markets. Notably, the seven parameter equality tests are mainly used to inspect whether a specific financial feature is significantly different in three subperiods (i.e., the pre-SB, transition, and post-SB periods). For example, the first and second tests (${\mathrm{H}}_0^1:{\mathsf{\varphi }}_{12}^{\mathrm{B}}={\mathsf{\varphi }}_{12}^{\mathrm{D}}={\mathsf{\varphi }}_{12}^{\mathrm{A}}{ \mathrm{and} \mathrm{H}}_0^2:{\mathsf{\varphi }}_{21}^{\mathrm{B}}={\mathsf{\varphi }}_{21}^{\mathrm{D}}={\mathsf{\varphi }}_{21}^{\mathrm{A}}$) are used to test whether the return spillover between the emerging and developed stock markets is significantly different at three subperiods (i.e., the pre-SB, transition, and post-SB). As shown by the numbers listed in the column ‘Sum’ corresponding to the rows ‘${\mathrm{H}}_0^1~{\mathrm{H}}_0^7$’, there are 43, 37, and 25 paired stock indices passing the third, fourth and fifth parameter equality tests, respectively. Notably, the parameters corresponding to the third, fourth and fifth parameter equality tests are ${\mathsf{\nu }}_{12}$, ${\mathsf{\nu }}_{21}$ and ${\mathsf{\omega }}_{12}$, respectively. In other words, the volatility spillover and correlation are significantly by the structural break because more than half of the 46 paired stock indices pass the third, fourth, and fifth parameter equality tests related to the above two financial features. In addition, there are 18 and 8 paired stock indices passing the first and second parameters equality tests, respectively. This result indicates that the return spillover is not affected by the structural break because fewer than half of the 46 paired stock indices pass the first and second parameter equality tests related to the return spillover.

5.3. The Interactions between the Emerging and Developed Markets

This study utilizes the stock indices in the E7 and G7 countries to represent the emerging and developed markets, respectively, then explores the interactions between these markets. However, the E7 and G7 countries are mainly distributed over Asia, America, and Europe. Thus, the 46 paired stock indices can be divided into five groups according to the continents in which the two-component stock markets are located. The five groups are Asia, America, Asia and America, Asia and Europe, and America and Europe. Then, the empirical results of the B-EGARCH-SB model of 46 paired stock indices are listed in

Table 6. The results of the B-EGARCH-SB model for alternative paired stock indices in Asia.

	XU100-N225	RTSI-N225	JKSE-N225	SSE-N225	BSE-N225	Sum
Panel A. Return spillover
${\mathsf{\varphi }}_{12}^{\mathrm{B}}$	0.0011 (0.0216)	0.0292 (0.0625)	−0.0295 (0.0260)	0.0053 (0.0236)	0.0151 (0.0258)	0
${\mathsf{\varphi }}_{12}^{\mathrm{A}}$	−0.0257 (0.0214)	0.0098 (0.0192)	−0.0235 (0.0152)	0.0573 (0.0275) b	−0.0138 (0.0113)	1(+)
${\mathsf{\varphi }}_{21}^{\mathrm{B}}$	0.0498 (0.0176) c	0.0111 (0.0166)	−0.0098 (0.0353)	−0.0467 (0.0352)	0.0904 (0.0321) c	2(+)
${\mathsf{\varphi }}_{21}^{\mathrm{A}}$	0.0975 (0.0162) c	0.0101 (0.0139)	0.0274 (0.0247)	−0.0166 (0.0149)	0.1238 (0.0212) c	2(+)
Panel B. Volatility spillover
${\mathsf{\nu }}_{12}^{\mathrm{B}}$	0.1101 (0.0226) c	0.1191 (0.0274) c	0.0453 (0.0120) c	−0.0057 (0.0044)	0.0224 (0.0086) c	4(+)
${\mathsf{\nu }}_{12}^{\mathrm{A}}$	−0.0003 (0.0100)	0.0371 (0.0120) c	0.0143 (0.0108)	−0.0175 (0.0074)b	−0.0005 (0.0077)	2(1+;1−)
${\mathsf{\nu }}_{21}^{\mathrm{B}}$	0.0195 (0.0082) b	0.0354 (0.0106) c	0.0424 (0.0109) c	0.0381 (0.0121)c	0.0391 (0.0118) c	5(+)
${\mathsf{\nu }}_{21}^{\mathrm{A}}$	0.0131 (0.0138)	0.0438 (0.0162) c	0.0142 (0.0089)	−0.0168 (0.0103)	0.0137 (0.0096)	1(+)
Panel C. Correlation
${\mathsf{\omega }}_{12}^{\mathrm{B}}$	0.0384 (0.0137) c	−0.3274 (0.2686)	0.0397 (0.0126) c	−0.0018 (0.0021)	0.0627 (0.0159) c	3(+)
${\mathsf{\omega }}_{12}^{\mathrm{A}}$	0.0166 (0.0051) c	0.2284 (0.1075) b	0.0236 (0.0056) c	0.0045 (0.0023) b	0.0395 (0.0064) c	5(+)
Panel D. Leverage effect
${\mathsf{\gamma }}_1^{\mathrm{B}}$	−0.0644 (0.0229) c	−0.0789 (0.0281) c	0.0057 (0.0228)	−0.0767 (0.0187) c	−0.0308 (0.0194)	3(−)
${\mathsf{\gamma }}_1^{\mathrm{A}}$	−0.1157 (0.0190) c	−0.0980 (0.0201) c	−0.0976 (0.0202) c	−0.0675 (0.0179) c	−0.1008 (0.0162) c	5(−)
${\mathsf{\gamma }}_2^{\mathrm{B}}$	−0.0467 (0.0281) a	−0.0515 (0.0316)	−0.0301 (0.0254)	−0.0711 (0.0285) b	−0.0280 (0.0269)	2(−)
${\mathsf{\gamma }}_2^{\mathrm{A}}$	−0.1190 (0.0163) c	−0.1593 (0.0190) c	−0.1096 (0.0165) c	−0.1173 (0.0257) c	−0.0913 (0.0158) c	5(−)

Note: 1. The superscripts a, b and c denote significance at the 10%, 5%, and 1% levels, respectively. 2. Numbers in parentheses in the rows of parameters are standard errors. 3. The numbers in column ‘Sum’ denote the total number of a specific parameter that is significant and the sign ‘+’ (respectively, ‘−’) in the bracket besides this number denotes that this parameter is significantly positive (respectively, negative).

,

Table 7. The results of the B-EGARCH-SB model for alternative paired stock indices in America.

	MXX-TSX	BVSP-DJ	BVSP-TSX	Sum
Panel A. Return spillover
${\mathsf{\varphi }}_{12}^{\mathrm{B}}$	0.0412(0.0240) a	0.0389(0.0417)	−0.0343(0.0514)	1(+)
${\mathsf{\varphi }}_{12}^{\mathrm{A}}$	0.0012(0.0181)	−0.0069(0.0178)	0.1170(0.0252) c	1(+)
${\mathsf{\varphi }}_{21}^{\mathrm{B}}$	0.0318(0.0126) b	0.0142(0.0177)	0.0298(0.0138) b	2(+)
${\mathsf{\varphi }}_{21}^{\mathrm{A}}$	0.0259(0.0121) b	0.0084(0.0049) a	0.0197(0.0080) b	3(+)
Panel B. Volatility spillover
${\mathsf{\nu }}_{12}^{\mathrm{B}}$	−0.0017(0.0045)	0.0110(0.0050) b	−0.0053(0.0050)	1(+)
${\mathsf{\nu }}_{12}^{\mathrm{A}}$	0.0105(0.0037)c	0.0147(0.0032) c	0.0039(0.0033)	2(+)
${\mathsf{\nu }}_{21}^{\mathrm{B}}$	0.0151(0.0045)c	0.0313(0.0069) c	0.0202(0.0057) c	3(+)
${\mathsf{\nu }}_{21}^{\mathrm{A}}$	0.0116(0.0044)c	0.0283(0.0058) c	0.0196(0.0050) c	3(+)
Panel C. Correlation
${\mathsf{\omega }}_{12}^{\mathrm{B}}$	0.0081(0.0015) c	0.0102(0.0042) b	0.0150(0.0040) c	3(+)
${\mathsf{\omega }}_{12}^{\mathrm{A}}$	0.0054(0.0010) c	0.0070(0.0016) c	0.0119(0.0020) c	3(+)
Panel D. Leverage effect
${\mathsf{\gamma }}_1^{\mathrm{B}}$	−0.0447(0.0084) c	−0.0294(0.0123) b	−0.0415(0.0114) c	3(−)
${\mathsf{\gamma }}_1^{\mathrm{A}}$	−0.0260(0.0095) c	−0.0284(0.0045) c	−0.0418(0.0069) c	3(−)
${\mathsf{\gamma }}_2^{\mathrm{B}}$	−0.0161(0.0089) a	−0.0850(0.0163) c	−0.0578(0.0152) c	3(−)
${\mathsf{\gamma }}_2^{\mathrm{A}}$	−0.0721(0.0077) c	−0.0528(0.0063) c	−0.0854(0.0072) c	3(−)

Note: 1. See notes 1–3 of Table 6. 2. The empirical results of MXX-DJ paired data are omitted here because I cannot obtain the convergent results for these paired data.

,

Table 8. The results of the B-EGARCH-SB model for alternative paired stock indices in Asia and America.

	XU100-DJ	XU100-TSX	RTSI-DJ	RTSI-TSX	JKSE-DJ	JKSE-TSX	Sum
Panel A. Return spillover
${\mathsf{\varphi }}_{12}^{\mathrm{B}}$	0.4035(0.0562) c	0.3328(0.0487) c	−0.0306(0.0548)	0.0072(0.0532)	0.2160(0.0297)c	0.2512(0.0287) c	4(+)
${\mathsf{\varphi }}_{12}^{\mathrm{A}}$	0.1176(0.0287) c	0.1465(0.0343) c	0.0155(0.0271)	0.0124(0.0307)	0.2289(0.0210)c	0.2398(0.0285) c	4(+)
${\mathsf{\varphi }}_{21}^{\mathrm{B}}$	−0.0091(0.0084)	0.0110(0.0076)	0.0057(0.0098)	0.0056(0.0084)	−0.0594(0.0159)c	−0.0117(0.0141)	1(−)
${\mathsf{\varphi }}_{21}^{\mathrm{A}}$	0.0109(0.0098)	−0.0002(0.0091)	−0.0035(0.0094)	−0.0048(0.0063)	−0.0129(0.0164)	−0.0009(0.0135)	0
Panel B. Volatility spillover
${\mathsf{\nu }}_{12}^{\mathrm{B}}$	0.0203(0.0088) b	0.0136(0.0095)	0.0237(0.0106) b	0.0248(0.0084) c	0.0315(0.0131) b	0.0086(0.0126)	4(+)
${\mathsf{\nu }}_{12}^{\mathrm{A}}$	−0.0017(0.0044)	−0.0015(0.0037)	0.0085(0.0053)	0.0098(0.0038) b	0.0346(0.0122) c	0.0536(0.0072) c	3(+)
${\mathsf{\nu }}_{21}^{\mathrm{B}}$	0.0123(0.0057) b	0.0072(0.0047)	0.0179(0.0064) c	0.0066(0.0058)	0.0261(0.0069) c	0.0392(0.0047) c	4(+)
${\mathsf{\nu }}_{21}^{\mathrm{A}}$	0.0110(0.0079)	−0.0018(0.0065)	0.0239(0.0088) c	0.0032(0.0072)	0.0092(0.0058)	0.0190(0.0042) c	2(+)
Panel C. Correlation
${\mathsf{\omega }}_{12}^{\mathrm{B}}$	0.0076(0.0050)	0.0209(0.0054) c	0.0046(0.0078)	0.1653(0.0923) a	0.0033(0.0022)	0.0056(0.0019) c	3(+)
${\mathsf{\omega }}_{12}^{\mathrm{A}}$	0.0215(0.0050) c	0.0191(0.0042) c	0.0006(0.0014)	0.0510(0.0572)	0.0030(0.0014) b	0.0026(0.0010) b	4(+)
Panel D. Leverage effect
${\mathsf{\gamma }}_1^{\mathrm{B}}$	−0.0272(0.0141) a	−0.0232(0.0134) a	−0.0456(0.0176) c	−0.0371(0.0165) b	−0.0080(0.0196)	−0.0439(0.0191) b	5(−)
${\mathsf{\gamma }}_1^{\mathrm{A}}$	−0.0413(0.0142) c	−0.0468(0.0142) c	−0.0647(0.0173) c	−0.0634(0.0152) c	−0.0979(0.0256) c	−0.0986(0.0267) c	6(−)
${\mathsf{\gamma }}_2^{\mathrm{B}}$	−0.1433(0.0235) c	−0.0720(0.0179) c	−0.1556(0.0258) c	−0.0616(0.0189)c	−0.1288(0.0242) c	−0.0659(0.0172) c	6(−)
${\mathsf{\gamma }}_2^{\mathrm{A}}$	−0.1186(0.0132) c	−0.1218(0.0150) c	−0.1654(0.0152) c	−0.1370(0.0173) c	−0.1470(0.0160) c	−0.1529(0.0144) c	6(−)
	SSE−DJ	SSE−TSX	BSE−DJ	BSE−TSX	MXX−N225	BVSP−N225	Sum
Panel A. Return spillover
${\mathsf{\varphi }}_{12}^{\mathrm{B}}$	0.0380(0.0208) a	0.0485(0.0294) a	0.1730(0.0289) c	0.2340(0.0314) c	−0.0170(0.0268)	0.0132(0.0378)	4(+)
${\mathsf{\varphi }}_{12}^{\mathrm{A}}$	0.1337(0.0284) c	0.1298(0.0364) c	0.2095(0.0214)c	0.2539(0.0244) c	−0.0385(0.0140)c	0.0029(0.0197)	5(4+;1−)
${\mathsf{\varphi }}_{21}^{\mathrm{B}}$	0.0241(0.0134) a	0.0082(0.0114)	−0.0288(0.0148) a	−0.0071(0.0126)	0.2720(0.0391) c	0.1247(0.0269) c	4(3+;1−)
${\mathsf{\varphi }}_{21}^{\mathrm{A}}$	−0.0295(0.0201)	−0.0144(0.0088)	−0.0035(0.0143)	−0.0100(0.0131)	0.2878(0.0231) c	0.1970(0.0117) c	2(+)
Panel B. Volatility spillover
${\mathsf{\nu }}_{12}^{\mathrm{B}}$	−0.0149(0.0042) c	−0.0046(0.0053)	−0.0241(0.0069) c	−0.0201(0.0074) c	0.0178(0.0038) c	0.0261(0.0099) c	5(2+;3−)
${\mathsf{\nu }}_{12}^{\mathrm{A}}$	−0.0002(0.0036)	−0.0022(0.0046)	0.0113(0.0058) a	0.0118(0.0071) a	−0.0087(0.0041) b	0.0167(0.0092) a	4(3+;1−)
${\mathsf{\nu }}_{21}^{\mathrm{B}}$	0.0015(0.0054)	0.0165(0.0043) c	−0.0024(0.0058)	0.0177(0.0054) c	0.0392(0.0095) c	0.0519(0.0161) c	4(+)
${\mathsf{\nu }}_{21}^{\mathrm{A}}$	−0.0006(0.0061)	−0.0092(0.0074)	0.0039(0.0065)	0.0111(0.0064) a	−0.0001(0.0100)	0.0349(0.0142) b	2(+)
Panel C. Correlation
${\mathsf{\omega }}_{12}^{\mathrm{B}}$	0.0453(0.0550)	0.1021(0.0683)	0.0069(0.0033) b	0.0130(0.0036) c	0.6043(0.1398) c	0.0296(0.0120) b	4(+)
${\mathsf{\omega }}_{12}^{\mathrm{A}}$	0.1049(0.0375) c	0.1033(0.0648)	0.0122(0.0032) c	0.0094(0.0027) c	0.3993(0.0470) c	0.0105(0.0043) b	5(+)
Panel D. Leverage effect
${\mathsf{\gamma }}_1^{\mathrm{B}}$	−0.0419(0.0114) c	−0.0291(0.0122) b	−0.0761(0.0171) c	−0.0635(0.0147) c	−0.0835(0.0233) c	−0.0620(0.0247) b	6(−)
${\mathsf{\gamma }}_1^{\mathrm{A}}$	0.0015(0.0186)	−0.0796(0.0209) c	−0.0953(0.0169) c	−0.0867(0.0169) c	−0.0962(0.0171) c	−0.0789(0.0142) c	5(−)
${\mathsf{\gamma }}_2^{\mathrm{B}}$	−0.1277(0.0194) c	−0.0879(0.0188) c	−0.1403(0.0215) c	−0.0646(0.0156) c	−0.0349(0.0312)	−0.0323(0.0296)	4(−)
${\mathsf{\gamma }}_2^{\mathrm{A}}$	−0.1085(0.0145) c	−0.1562(0.0218) c	−0.1251(0.0133) c	−0.1264(0.0149) c	−0.1148(0.0146) c	−0.1064(0.0169) c	6(−)

Note: 1. See notes 1–3 of Table 6.

,

Table 9. The results of the B-EGARCH-SB model for alternative paired stock indices in Asia and Europe.

	XU100-FTSE	XU100-DAX	XU100-MIB	RTSI-FTSE	RTSI-CAC40	RTSI-DAX	RTSI-MIB	Sum
Panel A. Return spillover
${\mathsf{\varphi }}_{12}^{\mathrm{B}}$	0.0399(0.0365)	0.0084(0.0365)	0.0901(0.0425) b	0.1002(0.0495) b	0.0275(0.0396)	0.0297(0.0458)	0.0135(0.0462)	2(+)
${\mathsf{\varphi }}_{12}^{\mathrm{A}}$	0.0729(0.0325) b	0.0195(0.0216)	0.0100(0.0169)	0.0000(0.0377)	−0.0113(0.0255)	0.0131(0.0223)	0.0051(0.0163)	1(+)
${\mathsf{\varphi }}_{21}^{\mathrm{B}}$	0.0068(0.0079)	−0.0011(0.0095)	0.0103(0.0097)	0.0144(0.0089)	0.0286(0.0113) b	0.0261(0.0192)	0.0238(0.0102) b	2(+)
${\mathsf{\varphi }}_{21}^{\mathrm{A}}$	0.0329(0.0136) b	0.0145(0.0134)	0.0280(0.0177)	0.0007(0.0139)	0.0035(0.0137)	0.0058(0.0125)	−0.0000(0.0153)	1(+)
Panel B. Volatility spillover
${\mathsf{\nu }}_{12}^{\mathrm{B}}$	0.0013(0.0028)	0.0230(0.0066) c	0.0131(0.0059) b	0.0175(0.0075) b	0.0436(0.0116) c	0.0479(0.0112) c	0.0156(0.0067) b	6(+)
${\mathsf{\nu }}_{12}^{\mathrm{A}}$	0.0018(0.0037)	0.0071(0.0040) a	0.0188(0.0039) c	0.0050(0.0066)	−0.0039(0.0055)	0.0132(0.0066) b	−0.0067(0.0032) b	4(3+;1−)
${\mathsf{\nu }}_{21}^{\mathrm{B}}$	−0.0011(0.0027)	0.0015(0.0039)	−0.0064(0.0032) b	−0.0023(0.0040)	0.0006(0.0046)	0.0076(0.0058)	−0.0046(0.0029)	1(−)
${\mathsf{\nu }}_{21}^{\mathrm{A}}$	−0.0049(0.0039)	−0.0048(0.0058)	0.0034(0.0054)	−0.0104(0.0055) a	−0.0029(0.0066)	−0.0013(0.0073)	0.0066(0.0052)	1(−)
Panel C. Correlation
${\mathsf{\omega }}_{12}^{\mathrm{B}}$	0.0158(0.0041) c	0.0415(0.0204) b	0.0257(0.0061) c	0.0015(0.0013)	0.0015(0.0015)	0.3916(0.3676)	0.0186(0.0258)	3(+)
${\mathsf{\omega }}_{12}^{\mathrm{A}}$	0.0098(0.0030) c	0.0311(0.0065) c	0.0332(0.0064) c	−0.0001(0.0013)	−0.0000(0.0008)	0.0612(0.0788)	0.0083(0.0230)	3(+)
Panel D. Leverage effect
${\mathsf{\gamma }}_1^{\mathrm{B}}$	−0.0042(0.0083)	−0.0271(0.0152) a	−0.0205(0.0122) a	−0.0482(0.0157) c	−0.0594(0.0162) c	−0.0557(0.0226) b	−0.0462(0.0149) c	6(−)
${\mathsf{\gamma }}_1^{\mathrm{A}}$	−0.0310(0.0125) b	−0.0414(0.0121) c	−0.0490(0.0121) c	−0.0778(0.0212) c	−0.0973(0.0199) c	−0.0912(0.0178) c	−0.0660(0.0142) c	7(−)
${\mathsf{\gamma }}_2^{\mathrm{B}}$	−0.0932(0.0096) c	−0.1217(0.0228) c	−0.1043(0.0158) c	−0.1737(0.0212) c	−0.1652(0.0220) c	−0.1312(0.0299) c	−0.1436(0.0206) c	7(−)
${\mathsf{\gamma }}_2^{\mathrm{A}}$	−0.1151(0.0162) c	−0.0935(0.0116) c	−0.0866(0.0116) c	−0.1597(0.0183) c	−0.1605(0.0162) c	−0.1583(0.0167) c	−0.1169(0.0120) c	7(−)
	JKSE−FTSE	JKSE−CAC40	JKSE−DAX	JKSE−MIB	SSE−FTSE	SSE−CAC40	SSE−DAX	Sum
Panel A. Return spillover
${\mathsf{\varphi }}_{12}^{\mathrm{B}}$	0.1290(0.0212) c	0.1030(0.0181) c	0.0678(0.0227) c	0.0937(0.0245) c	0.0892(0.0206) c	0.0452(0.0019) c	0.0300(0.0184)	6(+)
${\mathsf{\varphi }}_{12}^{\mathrm{A}}$	0.0795(0.0231) c	0.1015(0.0165) c	0.1045(0.0138) c	0.0782(0.0128) c	0.0982(0.0348) c	0.0877(0.0318) c	0.0811(0.0263) c	7(+)
${\mathsf{\varphi }}_{21}^{\mathrm{B}}$	−0.0277(0.0125) b	−0.0481(0.0157) c	−0.0467(0.0425)	−0.0427(0.0174) b	0.0009(0.0084)	0.0004(0.0118)	0.0198(0.0490)	3(−)
${\mathsf{\varphi }}_{21}^{\mathrm{A}}$	0.0113(0.0207)	0.0067(0.0221)	0.0173(0.0224)	−0.0036(0.0254)	−0.0307(0.0134) b	−0.0198(0.0160)	−0.0260(0.0184)	1(−)
Panel B. Volatility spillover
${\mathsf{\nu }}_{12}^{\mathrm{B}}$	0.0072(0.0061)	0.0160(0.0068) b	0.0185(0.0057) c	0.0075(0.0068)	0.0010(0.0030)	−0.0000(0.0029)	−0.0074(0.0036) b	3(2+;1−)
${\mathsf{\nu }}_{12}^{\mathrm{A}}$	0.0039(0.0079)	−0.0082(0.0062)	−0.0080(0.0066)	−0.0151(0.0050) c	−0.0063(0.0071)	−0.0157(0.0059) c	−0.0228(0.0085) c	3(−)
${\mathsf{\nu }}_{21}^{\mathrm{B}}$	0.0138(0.0040) c	0.0067(0.0040) a	0.0276(0.0088) c	−0.0154(0.0044) c	0.0112(0.0028) c	0.0083(0.0032) b	0.0270(0.0097) c	7(6+;1−)
${\mathsf{\nu }}_{21}^{\mathrm{A}}$	0.0033(0.0054)	0.0025(0.0049)	0.0057(0.0055)	0.0024(0.0043)	0.0060(0.0057)	0.0024(0.0060)	−0.0017(0.0083)	0
Panel C. Correlation
${\mathsf{\omega }}_{12}^{\mathrm{B}}$	0.0088(0.0028) c	0.0128(0.0046) c	0.0143(0.0089)	0.0088(0.0037) b	0.0012(0.0009)	0.0008(0.0015)	0.0216(0.0557)	3(+)
${\mathsf{\omega }}_{12}^{\mathrm{A}}$	0.0080(0.0028) c	0.0093(0.0037) b	0.0135(0.0047) c	0.0097(0.0041) b	0.0023(0.0012) a	0.0026(0.0024)	0.0914(0.0854)	5(+)
Panel D. Leverage effect
${\mathsf{\gamma }}_1^{\mathrm{B}}$	−0.0338(0.0131) c	−0.0322(0.0132) b	0.0041(0.0260)	−0.0119(0.0160)	−0.0276(0.0102) c	−0.0269(0.0095) c	−0.1203(0.0245) c	5(−)
${\mathsf{\gamma }}_1^{\mathrm{A}}$	−0.0650(0.0221) c	−0.0779(0.0202) c	−0.0964(0.0217) c	−0.0920(0.0199) c	−0.0632(0.0181) c	−0.0602(0.0168) c	−0.0767(0.0189) c	7(−)
${\mathsf{\gamma }}_2^{\mathrm{B}}$	−0.1223(0.0139) c	−0.1158(0.0144) c	−0.1289(0.0251) c	−0.1104(0.0169) c	−0.1537(0.0130) c	−0.1707(0.0188) c	−0.1888(0.0347) c	7(−)
${\mathsf{\gamma }}_2^{\mathrm{A}}$	−0.1338(0.0186) c	−0.1351(0.0155) c	−0.1307(0.0164) c	−0.1032(0.0138) c	−0.1165(0.0251) c	−0.1306(0.0213) c	−0.1211(0.0254) c	7(−)
	SSE−MIB	BSE−FTSE	BSE−CAC40	BSE−DAX	BSE−MIB			Sum
Panel A. Return spillover
${\mathsf{\varphi }}_{12}^{\mathrm{B}}$	0.0292(0.0165) a	0.1295(0.0237) c	0.1249(0.0204) c	0.0911(0.0148) c	0.1154(0.0240) c			5(+)
${\mathsf{\varphi }}_{12}^{\mathrm{A}}$	0.0712(0.0281) b	0.0683(0.0233) c	0.0731(0.0160) c	0.0757(0.0166) c	0.0716(0.0110) c			5(+)
${\mathsf{\varphi }}_{21}^{\mathrm{B}}$	−0.0004(0.0144)	0.0075(0.0118)	−0.0010(0.0138)	0.0649(0.0250) c	0.0143(0.0157)			1(+)
${\mathsf{\varphi }}_{21}^{\mathrm{A}}$	−0.0270(0.0181)	0.0069(0.0195)	−0.0209(0.0176)	−0.0145(0.0194)	−0.0170(0.0081) b			1(−)
Panel B. Volatility spillover
${\mathsf{\nu }}_{12}^{\mathrm{B}}$	−0.0092(0.0036) b	−0.0002(0.0030)	0.0020(0.0025)	0.0001(0.0039)	−0.0070(0.0044)			1(−)
${\mathsf{\nu }}_{12}^{\mathrm{A}}$	−0.0217(0.0061) c	−0.0046(0.0039)	−0.0085(0.0025) c	−0.0082(0.0043) a	−0.0152(0.0037) c			4(−)
${\mathsf{\nu }}_{21}^{\mathrm{B}}$	−0.0124(0.0044) c	0.0027(0.0022)	0.0016(0.0019)	0.0065(0.0058)	−0.0183(0.0039) c			2(−)
${\mathsf{\nu }}_{21}^{\mathrm{A}}$	−0.0047(0.0072)	0.0000(0.0036)	0.0029(0.0030)	0.0050(0.0036)	−0.0015(0.0041)			0
Panel C. Correlation
${\mathsf{\omega }}_{12}^{\mathrm{B}}$	−0.0001(0.0015)	0.0093(0.0032) c	0.0132(0.0028)c	0.0071(0.0065)	0.0154(0.0056) c			3(+)
${\mathsf{\omega }}_{12}^{\mathrm{A}}$	0.0116(0.0087)	0.0115(0.0032) c	0.0144(0.0019)c	0.0161(0.0038) c	0.0294(0.0072) c			4(+)
Panel D. Leverage effect
${\mathsf{\gamma }}_1^{\mathrm{B}}$	−0.0304(0.0116) c	−0.0190(0.0104) a	−0.0215(0.0101) b	−0.0421(0.0203) b	−0.0338(0.0143) b			5(−)
${\mathsf{\gamma }}_1^{\mathrm{A}}$	−0.0830(0.0192) c	−0.0747(0.0166) c	−0.0616(0.0089) c	−0.0762(0.0142) c	−0.0939(0.0148) c			5(−)
${\mathsf{\gamma }}_2^{\mathrm{B}}$	−0.1607(0.0224) c	−0.1089(0.0119) c	−0.1023(0.0128) c	−0.1525(0.0224) c	−0.1134(0.0156)c			5(−)
${\mathsf{\gamma }}_2^{\mathrm{A}}$	−0.1487(0.0249) c	−0.1100(0.0156) c	−0.1086(0.0119) c	−0.0999(0.0134) c	−0.0873(0.0114) c			5(−)

Note: 1. See notes 1–3 of Table 6. 2. The empirical results of XU100-CAC40 paired data are omitted here because I cannot obtain the convergent result for these paired data.

and

Table 10. The results of the B-EGARCH-SB model for alternative paired stock indices in America and Europe.

	MXX-FTSE	MXX-DAX	MXX-MIB	BVSP-FTSE	BVSP-CAC40	BVSP-DAX	BVSP-MIB	Sum
Panel A. Return spillover
${\mathsf{\varphi }}_{12}^{\mathrm{B}}$	−0.0038(0.0130)	0.0321(0.0046) c	−0.0031(0.0217)	0.0265(0.0487)	−0.0339(0.0329)	0.0363(0.0329)	0.0209(0.0095) b	2(+)
${\mathsf{\varphi }}_{12}^{\mathrm{A}}$	−0.0547(0.0200) c	−0.0255(0.0145) a	−0.0114(0.0111)	0.0038(0.0369)	0.0218(0.0219)	0.0751(0.0204) c	0.0362(0.0172) b	4(2+;2−)
${\mathsf{\varphi }}_{21}^{\mathrm{B}}$	0.0900(0.0105) c	0.0389(0.0437)	0.0898(0.0176) c	0.0335(0.0177) a	0.0698(0.0181) c	0.0026(0.0025)	0.0994(0.0178) c	5(+)
${\mathsf{\varphi }}_{21}^{\mathrm{A}}$	0.0669(0.0196) c	0.0976(0.0226) c	0.0960(0.0255) c	0.0487(0.0125) c	0.0793(0.0117) c	0.0651(0.0108) c	0.0726(0.0151) c	7(+)
Panel B. Volatility spillover
${\mathsf{\nu }}_{12}^{\mathrm{B}}$	0.0019(0.0026)	−0.0009(0.0032)	−0.0043(0.0034)	0.0034(0.0049)	0.0056(0.0036)	0.0091(0.0041) b	0.0144(0.0064) b	2(+)
${\mathsf{\nu }}_{12}^{\mathrm{A}}$	−0.0008(0.0029)	−0.0101(0.0039) c	−0.0128(0.0031) c	0.0035(0.0045)	−0.0025(0.0030)	0.0016(0.0042)	−0.0012(0.0037)	2(−)
${\mathsf{\nu }}_{21}^{\mathrm{B}}$	0.0087(0.0026) c	0.0147(0.0062) b	−0.0152(0.0046) c	0.0029(0.0045)	0.0152(0.0050) c	0.0304(0.0069) c	0.0160(0.0084) a	6(5+;1−)
${\mathsf{\nu }}_{21}^{\mathrm{A}}$	0.0050(0.0032)	0.0050(0.0047)	−0.0069(0.0044)	0.0031(0.0041)	0.0135(0.0042) c	0.0198(0.0048) c	0.0251(0.0099) b	3(+)
Panel C. Correlation
${\mathsf{\omega }}_{12}^{\mathrm{B}}$	0.0153(0.0030) c	0.0473(0.0104) c	0.0283(0.0058) c	0.0189(0.0072) c	0.0199(0.0064) c	0.0882(0.0201) c	0.0519(0.0162) c	7(+)
${\mathsf{\omega }}_{12}^{\mathrm{A}}$	0.0130(0.0027) c	0.0230(0.0047) c	0.0257(0.0054) c	0.0133(0.0038) c	0.0138(0.0028) c	0.0317(0.0064) c	0.0465(0.0103) c	7(+)
Panel D. Leverage effect
${\mathsf{\gamma }}_1^{\mathrm{B}}$	−0.0287(0.0075) c	−0.0328(0.0203)	−0.0237(0.0119) b	−0.0150(0.0145)	−0.0395(0.0135) c	−0.0389(0.0182) b	−0.0432(0.0200) b	5(−)
${\mathsf{\gamma }}_1^{\mathrm{A}}$	−0.0351(0.0151) b	−0.0520(0.0126) c	−0.0685(0.0130) c	−0.0358(0.0128) c	−0.0326(0.0085) c	−0.0510(0.0093) c	−0.0482(0.0081) c	7(−)
${\mathsf{\gamma }}_2^{\mathrm{B}}$	−0.1013(0.0094) c	−0.0679(0.0195) c	−0.0813(0.0143) c	−0.1488(0.0217) c	−0.1035(0.0189) c	−0.0833(0.0201) c	−0.1139(0.0218) c	7(−)
${\mathsf{\gamma }}_2^{\mathrm{A}}$	−0.0997(0.0133) c	−0.0910(0.0115) c	−0.0747(0.0102) c	−0.1092(0.0132) c	−0.1094(0.0088) c	−0.0934(0.0105) c	−0.0818(0.0095) c	7(−)

Note: 1. See notes 1–3 of Table 6. 2. The empirical results of MXX-CAC40 paired data are omitted here because I cannot obtain the convergent results for these paired data.

according to the above five groups. Further, the results listed in Table 6, Table 7, Table 8, Table 9 and Table 10 are used to explore how the explored issues vary with the five geographical locations of Asia, America, Asia and America, Asia and Europe, and America and Europe. For example, Table 6 lists the results of the B-EGARCH-SB model for five paired stock indices in Asia, which are XU100-N225, RTSI-N225, JKSE-N225, SSE-N225, and BSE-N225. The five paired stock indices are composed of one stock index from the emerging market in Asia (XU100, RTSI, JKSE, SSE, and BSE) and another stock index from the developed market in Asia also (N225). At the same inference process, Table 7 reports the results of the B-EGARCH-SB model for three paired stock indices in America, which are MXX-TSX, BVSP-DJ, and BVSP-TSX. Table 8 lists the results of the B-EGARCH-SB model for twelve paired stock indices in Asia and America, namely XU100-DJ, XU100-TSX, RTSI-DJ, RTSI-TSX, JKSE-DJ, JKSE-TSX, SSE-DJ, SSE-TSX, BSE-DJ, BSE-TSX, MXX-N225, and BVSP-N225. The twelve paired stock indices are composed of one stock index from the emerging market in Asia (XU100, RTSI, JKSE, SSE, and BSE) and another stock index from the developed market in America (DJ and TSX). Or, they are composed of one stock index from the emerging market in America (MXX and BVSP) and another stock index from the developed market in Asia (N225). With the same inference process, Table 9 reports the results of the B-EGARCH-SB model for 19 paired stock indices in Asia and Europe, which are XU100-FTSE, XU100-DAX, XU100-MIB, RTSI-FTSE, RTSI-CAC40, RTSI-DAX, RTSI-MIB, JKSE-FTSE, JKSE-CAC40, JKSE-DAX, JKSE-MIB, SSE-FTSE, SSE-CAC40, SSE-DAX, SSE-MIB, BSE-FTSE, BSE-CAC40, BSE-DAX, and BSE-MIB. Table 10 lists the results of the B-EGARCH-SB model for seven paired stock indices in America and Europe, namely MXX-FTSE, MXX-DAX, MXX-MIB, BVSP-FTSE, BVSP-CAC40, BVSP-DAX, and BVSP-MIB. The empirical results of MXX-DJ, XU100-CAC40, and MXX-CAC40 paired stock indices data are omitted in Table 7, Table 9 and Table 10, respectively, because I cannot obtain the convergent result for these three paired stock indices data.

To easily explore the interactions between the emerging and developed markets, and how geographical locations, emerging stock indices, and structural breaks affect the above interactions, I must perform one calculation for some specific parameters in Table 4, Table 5, Table 6, Table 7 and Table 8. These parameters are related to the following four financial features: the leverage effect (${\mathsf{\gamma }}_1^{\mathrm{B}}$, ${\mathsf{\gamma }}_1^{\mathrm{A}}$, ${\mathsf{\gamma }}_2^{\mathrm{B}}$, and ${\mathsf{\gamma }}_2^{\mathrm{A}}$) in a single market, and the correlation (${\mathsf{\omega }}_{12}^{\mathrm{B}}$ and ${\mathsf{\omega }}_{12}^{\mathrm{A}}$), return spillover (${\mathsf{\varphi }}_{12}^{\mathrm{B}}$, ${\mathsf{\varphi }}_{12}^{\mathrm{A}}$, ${\mathsf{\varphi }}_{21}^{\mathrm{B}}$, and ${\mathsf{\varphi }}_{21}^{\mathrm{A}}$) and volatility spillover (${\mathsf{\nu }}_{12}^{\mathrm{B}}$, ${\mathsf{\nu }}_{12}^{\mathrm{A}}$, ${\mathsf{\nu }}_{21}^{\mathrm{B}}$, and ${\mathsf{\nu }}_{21}^{\mathrm{A}}$) for paired stock markets. The parameters inside the bracket beside a specific financial feature denote the parameters related to this specific financial feature. This work is to calculate the total number of paired stock indices with a significant value for specific parameters such as ${\mathsf{\nu }}_{12}^{\mathrm{B}}$ and ${\mathsf{\nu }}_{12}^{\mathrm{A}}$, two general examples in Table 6. For example, regarding panel B of Table 6, parameter ${\mathsf{\nu }}_{12}^{\mathrm{B}}$ is significantly positive for four pairs of data such as the XU100-N225, RTSI-N225, JKSE-N225, and BSE-N225. In panel B of Table 6, the result ‘4(+)’ is thus recorded at the column ‘Sum’ corresponding to the row ‘${\mathsf{\nu }}_{12}^{\mathrm{B}}$’. In panel B of

Table 11. The summary results of the B-EGARCH-SB model for alternative paired stock indices in three continents of the world.

	Geographical Location							Emerging Stock Indices
	Asia (5)	America (3)	Asia–America (12)	Asia–Europe (19)	America–Europe (7)	${\Delta }^G$	SUM (46)	XU100 (6)	RTSI (7)	MXX (5)	BVSP (7)	JKSE (7)	SSE (7)	BSE (7)	${\Delta }^E$
Panel A. Return spillover
${\mathsf{\varphi }}_{12}^{\mathrm{B}}$	0	1(+)	8(+)	13(+)	2(+)		24(+)	3(+)	1(+)	2(+)	1(+)	6(+)	5(+)	6(+)
${\mathsf{\varphi }}_{12}^{\mathrm{A}}$	1(+)	1(+)	9(8+;1−)	13(+)	4(2+;2−)		28(25+;3−)	3(+)	0	3(−)	3(+)	6(+)	7(+)	6(+)
${\mathsf{\varphi }}_{21}^{\mathrm{B}}$	2(+)	2(+)	5(3+;2−)	6(3+;3−)	5(+)		20(15+;5−)	1(+)	2(+)	4(+)	5(+)	4(−)	1(+)	3(2+;1−)
${\mathsf{\varphi }}_{21}^{\mathrm{A}}$	2(+)	3(+)	2(+)	3(1+;2−)	7(+)		17(15+;2−)	2(+)	0	5(+)	7(+)	0	1(−)	2(1+;1−)
${\mathsf{\Sigma }}_{12}^{\mathrm{R}}$	1 (10%)	2 (33.3%)	17 (70.8%)	26 (68.4%)	6 (42.9%)	60.8%	52 (56.5%)	6 (50%)	1 (7.1%)	5 (50%)	4 (28.6%)	12 (85.7%)	12 (85.7%)	12 (85.7%)	78.6%
${\mathsf{\Sigma }}_{21}^{\mathrm{R}}$	4 (40%)	5 (83.3%)	7 (29.2%)	9 (23.7%)	12 (85.7%)	62%	37 (40.2%)	3 (25%)	2 (14.3%)	9 (90%)	12 (85.7%)	4 (28.6%)	2 (14.3%)	5 (35.7%)	75.7%
${\mathsf{\Sigma }}_{\mathrm{B}}^{\mathrm{R}}$	2 (20%)	3 (50%)	13 (54.2%)	19 (50%)	7 (50%)		44 (47.8%)	4 (33.3%)	3 (21.4%)	6 (60%)	6 (42.9%)	10 (71.4%)	6 (42.9%)	9 (64.3%)
${\mathsf{\Sigma }}_{\mathrm{A}}^{\mathrm{R}}$	3 (30%)	4 (66.7%)	11 (45.8%)	16 (42.1%)	11 (78.6%)		45 (48.9%)	5 (41.7%)	0 (0%)	8 (80%)	10 (71.4%)	6 (42.9%)	8 (57.1%)	8 (57.1%)
${\Delta }^R$	10%	16.7%	−8.4%	−7.9%	28.6%		1.1%	8.4%	−21.4%	20%	28.5%	−28.5%	14.2%	−7.2%
Panel B. Volatility spillover
${\mathsf{\nu }}_{12}^{\mathrm{B}}$	4(+)	1(+)	9(6+;3−)	10(8+;2−)	2(+)		26(21+;5−)	4(+)	7(+)	1(+)	4(+)	4(+)	3(−)	3(1+;2−)
${\mathsf{\nu }}_{12}^{\mathrm{A}}$	2(1+;1−)	2(+)	7(6+;1−)	11(3+;8−)	2(−)		24(12+;12−)	2(+)	4(3+;1−)	4(1+;3−)	2(+)	3(2+;1−)	4(−)	5(2+;3−)
${\mathsf{\nu }}_{21}^{\mathrm{B}}$	5(+)	3(+)	8(+)	10(6+;4−)	6(5+;1−)		32(27+;5−)	3(2+;1−)	2(+)	5(4+;1−)	6(+)	7(6+;1−)	6(5+;1−)	3(2+;1−)
${\mathsf{\nu }}_{21}^{\mathrm{A}}$	1(+)	3(+)	4(+)	1(−)	3(+)		12(11+;1−)	0	3(2+;1−)	1(+)	6(+)	1(+)	0	1(+)
${\mathsf{\Sigma }}_{12}^{\mathrm{V}}$	6 (60%)	3 (50%)	16 (66.7%)	21 (55.3%)	4 (28.6%)	38.1%	50 (54.3%)	6 (50%)	11 (78.6%)	5 (50%)	6 (42.9%)	7 (50%)	7 (50%)	8 (57.1%)	35.7%
${\mathsf{\Sigma }}_{21}^{\mathrm{V}}$	6 (60%)	6 (100%)	12 (50%)	11 (28.9%)	9 (64.3%)	71.1%	44 (47.8%)	3 (25%)	5 (35.7%)	6 (60%)	12 (85.7%)	8 (57.1%)	6 (42.9%)	4 (28.6%)	60.7%
${\mathsf{\Sigma }}_{\mathrm{B}}^{\mathrm{V}}$	9 (90%)	4 (66.7%)	17 (70.8%)	20 (52.6%)	8 (57.1%)		58 (63%)	7 (58.3%)	9 (64.3%)	6 (60%)	10 (71.4%)	11 (78.6%)	9 (64.3%)	6 (42.9%)
${\mathsf{\Sigma }}_{\mathrm{A}}^{\mathrm{V}}$	3 (30%)	5 (83.3%)	11 (45.8%)	12 (31.6%)	5 (35.7%)		36 (39.1%)	2 (16.7%)	7 (50%)	5 (50%)	8 (57.1%)	4 (28.6%)	4 (28.6%)	6 (42.9%)
${\Delta }^V$	−60%	16.6%	−25%	−21%	−21.4%		−23.9%	−41.6%	−14.3%	−10%	−14.3%	−50%	−35.7%	0%
Panel C. Correlative relation
${\mathsf{\omega }}_{12}^{\mathrm{B}}$	3(+)(60%)	3(+)(100%)	7(+)(58.3%)	9(+)(47.4%)	7(+)(100%)		29(+)(63%)	5(+)(83.3%)	1(+)(14.3%)	5(+)(100%)	7(+)(100%)	5(+)(71.4%)	0(0%)	6(+)(85.7%)
${\mathsf{\omega }}_{12}^{\mathrm{A}}$	5(+)(100%)	3(+)(100%)	9(+)(75%)	12(+)(63.2%)	7(+)(100%)		36(+)(78.2%)	6(+)(100%)	1(+)(14.3%)	5(+)(100%)	7(+)(100%)	7(+)(100%)	3(+)(42.9%)	7(+)(100%)
${\mathsf{\Sigma }}_{12}^{\mathsf{\omega }}$	8 (80%)	6 (100%)	16 (66.7%)	21 (55.3%)	14 (100%)	44.7%	65(70.7%)	11 (91.7%)	2 (14.3%)	10 (100%)	14 (100%)	12 (85.7%)	3 (21.4%)	13 (92.9%)	85.7%
${\Delta }^C$	40%	0%	16.7%	15.8%	0%		15.2%	16.7%	0%	0%	0%	28.6%	42.9%	14.3%
Panel D. Leverage effect
${\mathsf{\gamma }}_1^{\mathrm{B}}$	3(−)	3(−)	11(−)	16(−)	5(−)		38(−)	5(−)	7(−)	4(−)	6(−)	3(−)	7(−)	6(−)
${\mathsf{\gamma }}_1^{\mathrm{A}}$	5(−)	3(−)	11(−)	19(−)	7(−)		45(−)	6(−)	7(−)	5(−)	7(−)	7(−)	6(−)	7(−)
${\mathsf{\gamma }}_2^{\mathrm{B}}$	2(−)	3(−)	10(−)	19(−)	7(−)		41(−)	6(−)	6(−)	4(−)	6(−)	6(−)	7(−)	6(−)
${\mathsf{\gamma }}_2^{\mathrm{A}}$	5(−)	3(−)	12(−)	19(−)	7(−)		46(−)	6(−)	7(−)	5(−)	7(−)	7(−)	7(−)	7(−)
${\mathsf{\Sigma }}_1^{\mathrm{L}}$	8 (80%)	6 (100%)	22 (91.7%)	35 (92.1%)	12 (85.7%)	20%	83 (90.2%)	11 (91.7%)	14 (100%)	9 (90%)	13 (92.9%)	10 (71.4%)	13 (92.9%)	13 (92.9%)	28.6%
${\mathsf{\Sigma }}_2^{\mathrm{L}}$	7 (70%)	6 (100%)	22 (91.7%)	38 (100%)	14 (100%)	30%	87 (94.6%)	12 (100%)	13 (92.9%)	9 (90%)	13 (92.9%)	13 (92.9%)	14 (100%)	13 (92.9%)	10%
${\mathsf{\Sigma }}_{\mathrm{B}}^{\mathrm{L}}$	5 (50%)	6 (100%)	21 (87.5%)	35 (92.1%)	12 (85.7%)		79 (85.9%)	11 (91.7%)	13 (92.9%)	8 (80%)	12 (85.7%)	9 (64.3%)	14 (100%)	12 (85.7%)
${\mathsf{\Sigma }}_{\mathrm{A}}^{\mathrm{L}}$	10 (100%)	6 (100%)	23 (95.8%)	38 (100%)	14 (100%)		91 (98.9%)	12 (100%)	14 (100%)	10 (100%)	14 (100%)	14 (100%)	13 (92.9%)	14 (100%)
${\Delta }^L$	50%	0%	8.3%	7.9%	14.3%		13%	8.3%	7.1%	20%	14.3%	35.7%	−7.1%	14.3%

Note: 1. The numbers in the left panel of this table are summarized from the numbers in the columns ‘Sum’ in Table 6, Table 7, Table 8, Table 9 and Table 10. For example, the numbers at the column ‘Asia’ (respectively, ‘America’) underneath ‘Geographical locations’ are the corresponding numbers in the column ‘Sum’ in Table 6 (respectively, Table 7). On the other hand, the numbers in the column ‘Asia-America’ (respectively, ‘Asia-Europe’) underneath ‘Geographical locations’ are the corresponding numbers in the column ‘Sum’ in Table 8 (respectively, Table 9). In addition, the numbers in the column ‘America-Europe’ underneath ‘Geographical locations’ are the corresponding numbers in the column ‘Sum’ in Table 10. 2. The numbers in each column corresponding to a specific emerging stock index in the right panel of this table denote the total number of a specific parameter that is significant for all paired data related to that specific emerging stock index. Notably, the sign ‘+’ (respectively, ‘-’) in the bracket beside the preceding number denotes that this specific parameter is significantly positive (respectively, negative). 3. The number in the bracket besides the alternative column titles corresponding to alternative geographical locations (respectively, emerging stock indices) denotes the sample size or the total number of paired data that the convergent empirical results obtained in that geographical location (respectively, emerging stock index). 4. The numbers in the column ‘SUM’ denote the summation of five corresponding numbers in the columns ‘Asia’, ‘America’, ‘Asia-America’, ‘Asia-Europe’ and ‘America-Europe’ underneath ‘Geographical locations’. The preceding numbers in the column ‘SUM’ are also equal to the total sum of seven corresponding numbers in the columns ‘XU100’, ‘RTSI’, ‘MXX’, ‘BVSP’, ‘JKSE’, ‘SSE’ and ‘BSE’ underneath ‘Emerging stock indices’. 5. The number in row ‘${\mathsf{\Sigma }}_{12}^{\mathrm{R}}$’ in each column denotes the sum of the number corresponding to ${\mathsf{\varphi }}_{12}^{\mathrm{B}}$ and the number corresponding to ${\mathsf{\varphi }}_{12}^{\mathrm{A}}$ in the same column. On the other hand, the number in row ‘${\mathsf{\Sigma }}_{21}^{\mathrm{R}}$’ in each column denotes the sum of the number corresponding to ${\mathsf{\varphi }}_{21}^{\mathrm{B}}$ and the number corresponding to ${\mathsf{\varphi }}_{21}^{\mathrm{A}}$ in the same column. 6. The number in row ‘${\mathsf{\Sigma }}_{12}^{\mathrm{V}}$’ in each column denotes the sum of the number corresponding to ${\mathsf{\nu }}_{12}^{\mathrm{B}}$ and the number corresponding to ${\mathsf{\nu }}_{12}^{\mathrm{A}}$ in the same column. On the other hand, the number in row ‘${\mathsf{\Sigma }}_{21}^{\mathrm{V}}$’ in each column denotes the sum of the number corresponding to ${\mathsf{\nu }}_{21}^{\mathrm{B}}$ and the number corresponding to ${\mathsf{\nu }}_{21}^{\mathrm{A}}$ in the same column. 7. The number in row ‘${\mathsf{\Sigma }}_{12}^{\mathsf{\omega }}$’ in each column denotes the sum of the number corresponding to ${\mathsf{\omega }}_{12}^{\mathrm{B}}$ and the number corresponding to ${\mathsf{\omega }}_{12}^{\mathrm{A}}$ in the same column. 8. The number in row ‘${\mathsf{\Sigma }}_1^{\mathrm{L}}$’ in each column denotes the sum of the number corresponding to ${\gamma }_1^{\mathrm{B}}$ and the number corresponding to ${\gamma }_1^{\mathrm{A}}$ in the same column. On the other hand, the number in row ‘${\mathsf{\Sigma }}_2^{\mathrm{L}}$’ in each column denotes the sum of the number corresponding to ${\gamma }_2^{\mathrm{B}}$ and the number corresponding to ${\gamma }_2^{\mathrm{A}}$ in the same column. 9. The number in row ‘${\mathsf{\Sigma }}_{\mathrm{B}}^{\mathrm{R}}$’ in each column denotes the sum of the number corresponding to ${\mathsf{\varphi }}_{12}^{\mathrm{B}}$ and the number corresponding to ${\mathsf{\varphi }}_{21}^{\mathrm{B}}$ in the same column. On the other hand, the number in row ‘${\mathsf{\Sigma }}_{\mathrm{A}}^{\mathrm{R}}$’ in each column denotes the sum of the number corresponding to ${\mathsf{\varphi }}_{12}^{\mathrm{A}}$ and the number corresponding to ${\mathsf{\varphi }}_{21}^{\mathrm{A}}$ in the same column. 10. The number in row ‘${\mathsf{\Sigma }}_{\mathrm{B}}^{\mathrm{V}}$’ in each column denotes the sum of the number corresponding to ${\mathsf{\nu }}_{12}^{\mathrm{B}}$ and the number corresponding to ${\mathsf{\nu }}_{21}^B$ in the same column. On the other hand, the number in row ‘${\mathsf{\Sigma }}_{\mathrm{A}}^{\mathrm{V}}$’ in each column denotes the sum of the number corresponding to ${\mathsf{\nu }}_{12}^A$ and the number corresponding to ${\mathsf{\nu }}_{21}^A$ in the same column. 11. The number in the row ‘${\mathsf{\Sigma }}_{\mathrm{B}}^{\mathrm{L}}$’ in each column denotes the sum of the number corresponding to ${\gamma }_1^{\mathrm{B}}$ and the number corresponding to ${\gamma }_2^{\mathrm{B}}$ in the same column. On the other hand, the number in row ‘${\mathsf{\Sigma }}_{\mathrm{A}}^{\mathrm{L}}$’ in each column denotes the sum of the number corresponding to ${\gamma }_1^{\mathrm{A}}$ and the number corresponding to ${\gamma }_2^{\mathrm{A}}$ in the same column. 12. The numbers in the bracket in rows ‘${\mathsf{\Sigma }}_{12}^{\mathrm{R}}$’, ‘${\mathsf{\Sigma }}_{21}^{\mathrm{R}}$’, ‘${\mathsf{\Sigma }}_{12}^{\mathrm{V}}$’, ‘${\mathsf{\Sigma }}_{21}^{\mathrm{V}}$’, ‘${\mathsf{\Sigma }}_{12}^{\mathsf{\omega }}$’, ‘${\mathsf{\Sigma }}_1^{\mathrm{L}}$’, ‘${\mathsf{\Sigma }}_2^{\mathrm{L}}$’, ‘${\mathsf{\Sigma }}_{\mathrm{B}}^{\mathrm{R}}$’, ‘${\mathsf{\Sigma }}_{\mathrm{A}}^{\mathrm{R}}$’, ‘${\mathsf{\Sigma }}_{\mathrm{B}}^{\mathrm{V}}$’, ‘${\mathsf{\Sigma }}_{\mathrm{A}}^{\mathrm{V}}$’, ‘${\mathsf{\Sigma }}_{\mathrm{B}}^{\mathrm{L}}$’, and ‘${\mathsf{\Sigma }}_{\mathrm{A}}^{\mathrm{L}}$’ denote the corresponding significant ratio for the numbers beside them. 13. The number in the row ‘${\Delta }^{\mathrm{R}}$’ in each column denotes the difference between two significant ratios corresponding to ${\mathsf{\Sigma }}_{\mathrm{B}}^{\mathrm{R}}$ and ${\mathsf{\Sigma }}_{\mathrm{A}}^{\mathrm{R}}$, whereas the number in row ‘${\Delta }^{\mathrm{V}}$’ in each column denotes the difference between two significant ratios corresponding to ${\mathsf{\Sigma }}_{\mathrm{B}}^{\mathrm{V}}$ and ${\mathsf{\Sigma }}_{\mathrm{A}}^{\mathrm{V}}$. On the other hand, the number in row ‘${\Delta }^{\mathrm{C}}$’ in each column denotes the difference between two significant ratios corresponding to ${\mathsf{\omega }}_{12}^{\mathrm{B}}$ and ${\mathsf{\omega }}_{12}^{\mathrm{A}}$, whereas the number in row ‘${\Delta }^{\mathrm{L}}$’ in each column denotes the difference between two significant ratios corresponding to ${\mathsf{\Sigma }}_{\mathrm{B}}^{\mathrm{L}}$ and ${\mathsf{\Sigma }}_{\mathrm{A}}^{\mathrm{L}}$. 14. The bold font in rows ‘${\mathsf{\Sigma }}_{12}^{\mathrm{R}}$’, ‘${\mathsf{\Sigma }}_{21}^{\mathrm{R}}$’, ‘${\mathsf{\Sigma }}_{12}^{\mathrm{V}}$’, ‘${\mathsf{\Sigma }}_{21}^{\mathrm{V}}$’, ‘${\mathsf{\Sigma }}_1^{\mathrm{L}}$’ and ‘${\mathsf{\Sigma }}_2^{\mathrm{L}}$’ denotes the larger number when two numbers in the rows corresponding to ‘${\mathsf{\Sigma }}_{12}^{\mathrm{R}}$’ and ‘${\mathsf{\Sigma }}_{21}^{\mathrm{R}}$’ (or, ‘${\mathsf{\Sigma }}_{12}^{\mathrm{V}}$’ and ‘${\mathsf{\Sigma }}_{21}^{\mathrm{V}}$’) are compared to each other, so ‘${\mathsf{\Sigma }}_1^{\mathrm{L}}$’ and ‘${\mathsf{\Sigma }}_2^{\mathrm{L}}$’. Conversely, the bold font in rows ‘${\mathsf{\Sigma }}_{\mathrm{B}}^{\mathrm{R}}$’, ‘${\mathsf{\Sigma }}_{\mathrm{A}}^{\mathrm{R}}$’, ‘${\mathsf{\Sigma }}_{\mathrm{B}}^{\mathrm{V}}$’, ‘${\mathsf{\Sigma }}_{\mathrm{A}}^{\mathrm{V}}$’, ‘${\mathsf{\Sigma }}_{\mathrm{B}}^{\mathrm{L}}$’ and ‘${\mathsf{\Sigma }}_{\mathrm{A}}^{\mathrm{L}}$’ denotes the larger number when two numbers in the rows corresponding to ‘${\mathsf{\Sigma }}_{\mathrm{B}}^{\mathrm{R}}$’ and ‘${\mathsf{\Sigma }}_{\mathrm{A}}^{\mathrm{R}}$’ (or, ‘${\mathsf{\Sigma }}_{\mathrm{B}}^{\mathrm{V}}$’ and ‘${\mathsf{\Sigma }}_{\mathrm{A}}^{\mathrm{V}}$’) are compared to each other, so ‘${\mathsf{\Sigma }}_{\mathrm{B}}^{\mathrm{L}}$’ and ‘${\mathsf{\Sigma }}_{\mathrm{A}}^{\mathrm{L}}$’. 15. The bold (respectively, italic) font in row ${\Delta }^{\mathrm{R}}$ at the left-hand side of this table denotes the greatest (respectively, smallest) number among five numbers corresponding to five geographical locations in the same row. On the other hand, the bold (respectively, italic) font in row ${\Delta }^{\mathrm{R}}$ at the right-hand side of this table denotes the greatest (respectively, smallest) number among seven numbers corresponding to seven emerging stock indices in the same row. With the same inference process, I can obtain the meaning of the bold and italic fonts in rows ${\Delta }^{\mathrm{V}}$, ${\Delta }^{\mathrm{C}}$, and ${\Delta }^{\mathrm{L}}$. 16. The numbers in column ‘${\Delta }^{\mathrm{G}}$’ in each of the rows ‘${\mathsf{\Sigma }}_{12}^{\mathrm{R}}$’, ‘${\mathsf{\Sigma }}_{21}^{\mathrm{R}}$’, ‘${\mathsf{\Sigma }}_{12}^{\mathrm{V}}$’, ‘${\mathsf{\Sigma }}_{21}^{\mathrm{V}}$’, ‘${\mathsf{\Sigma }}_{12}^{\mathsf{\omega }}$’, ‘${\mathsf{\Sigma }}_1^{\mathrm{L}}$’, and ‘${\mathsf{\Sigma }}_2^{\mathrm{L}}$’ denote the difference between two significant ratios corresponding to the greatest and smallest numbers among five numbers corresponding to five geographical locations in the same row. Conversely, the numbers in column ‘${\Delta }^{\mathrm{E}}$’ in each of the rows ‘${\mathsf{\Sigma }}_{12}^{\mathrm{R}}$’, ‘${\mathsf{\Sigma }}_{21}^{\mathrm{R}}$’, ‘${\mathsf{\Sigma }}_{12}^{\mathrm{V}}$’, ‘${\mathsf{\Sigma }}_{21}^{\mathrm{V}}$’, ‘${\mathsf{\Sigma }}_{12}^{\mathsf{\omega }}$’, ‘${\mathsf{\Sigma }}_1^{\mathrm{L}}$’, and ‘${\mathsf{\Sigma }}_2^{\mathrm{L}}$’ denote the difference between two significant ratios corresponding to the greatest and smallest number among seven numbers corresponding to seven emerging stock indices in the same row.

, the above result ‘4(+)’ is also recorded at row ‘${\mathsf{\nu }}_{12}^{\mathrm{B}}$’ and column ‘Asia’ underneath ‘Geographical location’. On the other hand, parameter ${\mathsf{\nu }}_{12}^{\mathrm{A}}$ is significantly positive and negative only for the RTSI-N225 and SSE-N225 pairs of data, respectively. Thus, irrespectively of the significant cases being positive or negative, the total number of paired stock indices with a significant value is two for parameter ${\mathsf{\nu }}_{12}^{\mathrm{A}}$. In panel B of Table 6, the result ‘2(1+;1−)’ is thus recorded at the column ‘Sum’ corresponding to row ‘${\mathsf{\nu }}_{12}^{\mathrm{A}}$’. In panel B of Table 11, the above result ‘2(1+;1−)’ is also recorded in row ‘${\mathsf{\nu }}_{12}^{\mathrm{A}}$’ and column ‘Asia’ underneath ‘Geographical location’. Hence, Table 11 reports the summary results of the B-EGARCH-SB model for 46 paired stock indices in Table 4, Table 5, Table 6, Table 7 and Table 8. Notably, the left-hand side of Table 11 reports the summary results of the B-EGARCH-SB model for 46 paired stock indices based on five different combinations of geographical locations such as Asia, America, Asia and America, Asia and Europe, and America and Europe. On the other hand, the right-hand side of Table 11 lists the summary results of the B-EGARCH-SB model for 46 paired stock indices based on seven emerging stock indices (i.e., XU100, RTSI, MXX, BVSP, JKSE, SSE, and BSE). For example, the numbers in the column ‘XU100’ underneath ‘Emerging stock index’ in Table 11 denote the summary results of the B-EGARCH-SB model for six paired stock indices related to ‘XU100’. The six paired stock indices are XU100-FTSE, XU100-DAX, XU100-MIB, XU100-DJ, XU100-TSX, and XU100-N225. XU100-CAC40 is omitted here because I cannot obtain the convergent result for it. The 46 paired stock indices distribute over five different combinations of geographical locations or seven emerging stock indices. For example, in panel A of Table 11, the data ‘24(+)’ listed in the column ‘SUM’ corresponding to the row ‘${\mathsf{\varphi }}_{12}^{\mathrm{B}}$’ mean that parameter ‘${\mathsf{\varphi }}_{12}^{\mathrm{B}}$’ is significantly positive for 24 of 46 paired stock indices. In other words, irrespectively of the left-hand side or right-hand side of Table 11, the summation of all numbers in row ‘${\mathsf{\varphi }}_{12}^{\mathrm{B}}$’ of panel A is equal to 24. Subsequently, for the entire sample of paired stock indices and from the viewpoints of geographical locations and emerging stock indices, I will explore the interactions between the emerging stock market and the developed stock market.

As shown by the data listed in the column ‘SUM’ in Table 11, we can see the following phenomena regarding the issues explored in this study for the entire sample of paired stock indices. First, the return spillover from the developed market to the emerging market is more significant than the return spillover from the emerging market to the developed market because the significant ratio for ${\mathsf{\Sigma }}_{12}^{\mathrm{R}}$ (56.5%) is greater than that for ${\mathsf{\Sigma }}_{21}^{\mathrm{R}}$. (40.2%). The significant ratio for ${\mathsf{\Sigma }}_{12}^R$ (56.5%) equals ‘52’ divided by ‘92’, where ‘92’ equals twice of the total number of all sample paired stock indices (46). The number ‘52’ equals the sum of the number corresponding to ${\mathsf{\varphi }}_{12}^{\mathrm{B}}$ (24) and the number corresponding to ${\mathsf{\varphi }}_{12}^A$ (28). On the other hand, the significant ratio for ${\mathsf{\Sigma }}_{21}^R$ (40.2%) equals ‘37’ divided by ‘92’, where ‘92’ equals twice of total number of all sample paired stock indices (46). The number ‘37’ equals the sum of the number corresponding to ${\mathsf{\varphi }}_{21}^B$ (20) and the number corresponding to ${\mathsf{\varphi }}_{21}^A$ (17). The significant ratio for ${\mathsf{\Sigma }}_{12}^R$ can measure the intensity of return spillover from the developed market to the emerging market. Conversely, the significant ratio for ${\mathsf{\Sigma }}_{21}^R$ can measure the intensity of return spillover from the emerging market to the developed market. With the same inference process, the significant ratios for ${\mathsf{\Sigma }}_{12}^V$, ${\mathsf{\Sigma }}_{21}^V$, ${\mathsf{\Sigma }}_1^L$, ${\mathsf{\Sigma }}_2^L$ and ${\mathsf{\Sigma }}_{12}^{\mathsf{\omega }}$ are obtained. Thus, the developed market mainly spills into the emerging market from the viewpoint of return, or the developed market plays the dominant position on the return spillover. Second, the volatility spillover from the developed market to the emerging market is more significant than the volatility spillover from the emerging market to the developed market because the significant ratio for ${\mathsf{\Sigma }}_{12}^{\mathrm{V}}$ (54.3%) is greater than that for ${\mathsf{\Sigma }}_{21}^{\mathrm{V}}$ (47.8%). The significant ratio for ${\mathsf{\Sigma }}_{12}^V$ (54.3%) equals ‘50’ divided by ‘92’, where ‘92’ equals twice of the total number of all sample paired stock indices (46). The number ‘50’ equals the sum of the number corresponding to ${\mathsf{\nu }}_{12}^B$ (26) and the number corresponding to ${\mathsf{\nu }}_{12}^A$ (24). The significant ratio for ${\mathsf{\Sigma }}_{12}^V$ can measure the intensity of volatility spillover from the developed market to the emerging market. In addition, the significant ratio for ${\mathsf{\Sigma }}_{21}^V$ can measure the intensity of volatility spillover from the emerging market to the developed market. Hence, the developed market primarily spills into the emerging market from the viewpoint of volatility, or the developed market plays the dominant position on the volatility spillover. Third, the rankings of significant ratios from the largest to the smallest in sequence are 94.6% (${\mathsf{\Sigma }}_2^{\mathrm{L}}$), 90.2% (${\mathsf{\Sigma }}_1^{\mathrm{L}}$), 70.7% (${\mathsf{\Sigma }}_{12}^{\mathsf{\omega }}$), 56.5% (${\mathsf{\Sigma }}_{12}^{\mathrm{R}}$), 54.3% (${\mathsf{\Sigma }}_{12}^{\mathrm{V}}$), 47.8% (${\mathsf{\Sigma }}_{21}^{\mathrm{V}}$), and 40.2% (${\mathsf{\Sigma }}_{21}^{\mathrm{R}}$). This result indicates that in the stock market, the leverage effect is the most important. Moreover, the leverage effect significantly exists in the emerging and developed stock markets, especially in the developed market, because the significant ratio for ${\mathsf{\Sigma }}_2^{\mathrm{L}}$ (94.6%) is greater than that for ${\mathsf{\Sigma }}_1^L$. (90.2%) and these two significant ratios are large. The significant ratio for ${\mathsf{\Sigma }}_1^L$ (90.2%) equals ‘83’ divided by ‘92’, where ‘92’ equals twice of the total number of all sample paired stock indices (46). The number ‘83’ equals the sum of the number corresponding to ${\gamma }_1^B$ (38) and the number corresponding to ${\gamma }_1^A$ (45). The significant ratio for ${\mathsf{\Sigma }}_1^L$ (respectively, ${\mathsf{\Sigma }}_2^L$) can measure the intensity of leverage effect in the emerging (respectively, developed) market. Hence, the leverage effect cannot be neglected in this study. In addition, a long-term interaction (${\mathsf{\Sigma }}_{12}^{\mathsf{\omega }}$) is more significant than two short-term interactions (${\mathsf{\Sigma }}_{12}^{\mathrm{R}}$, ${\mathsf{\Sigma }}_{12}^{\mathrm{V}}$, ${\mathsf{\Sigma }}_{21}^{\mathrm{V}}$, and ${\mathsf{\Sigma }}_{21}^{\mathrm{R}}$). Finally, from the empirical results of parameters related to return spillover (${\mathsf{\varphi }}_{12}^{\mathrm{B}}$, ${\mathsf{\varphi }}_{12}^{\mathrm{A}}$,${\mathsf{\varphi }}_{21}^{\mathrm{B}},$ and ${\mathsf{\varphi }}_{21}^{\mathrm{A}}$), volatility spillover (${\mathsf{\nu }}_{12}^{\mathrm{B}}$, ${\mathsf{\nu }}_{12}^{\mathrm{A}}$, ${\mathsf{\nu }}_{21}^{\mathrm{B}}$, and ${\mathsf{\nu }}_{21}^{\mathrm{A}}$) and correlation (${\mathsf{\omega }}_{12}^{\mathrm{B}}$ and ${\mathsf{\omega }}_{12}^{\mathrm{A}}$), most of the significant cases are positive, especially for the correlation. This phenomenon may be attributed to the same type of commodities being explored in this study. Hence, both the emerging stock market and developed stock market positively affect each other in terms of the three types of interactions.

With the same inference process as in the previous paragraph, I use the data listed in the columns ‘Asia’, ‘America’, ‘Asia-America’, ‘Asia-Europe’, and ‘America-Europe’ at the left-hand side of Table 11 to discuss the two research questions of this study. The first question was whether the interactions between the stock indices in emerging and developed markets are different in terms of geographical location. As shown by the numbers listed in column ‘${\Delta }^G$’ and rows ‘${\mathsf{\Sigma }}_{12}^{\mathrm{R}}$’ (60.8%), ‘${\mathsf{\Sigma }}_{21}^{\mathrm{R}}$’ (62%), ‘${\mathsf{\Sigma }}_{12}^{\mathrm{V}}$’ (38.1%), ‘${\mathsf{\Sigma }}_{21}^{\mathrm{V}}$’ (71.1%), ‘${\mathsf{\Sigma }}_{12}^{\mathsf{\omega }}$’ (44.7%), ‘${\mathsf{\Sigma }}_1^{\mathrm{L}}$’ (20%), and ‘${\mathsf{\Sigma }}_2^{\mathrm{L}}$’ (30%), the interactions (correlation, return and volatility spillovers) between the stock indices in emerging and developed markets are significantly different in terms of geographical location because the differences in significant ratios among the five geographical locations for ‘${\mathsf{\Sigma }}_{12}^{\mathrm{R}}$’ (60.8%), ‘${\mathsf{\Sigma }}_{21}^{\mathrm{R}}$’ (62%), ‘${\mathsf{\Sigma }}_{12}^{\mathrm{V}}$’ (38.1%), ‘${\mathsf{\Sigma }}_{21}^{\mathrm{V}}$’ (71.1%), and ‘${\mathsf{\Sigma }}_{12}^{\mathsf{\omega }}$’ (44.7%) all are greater than 30%. The numbers in column ‘${\Delta }^G$’ in each of the rows ‘${\mathsf{\Sigma }}_{12}^{\mathrm{R}}$’, ‘${\mathsf{\Sigma }}_{21}^{\mathrm{R}}$’, ‘${\mathsf{\Sigma }}_{12}^{\mathrm{V}}$’, ‘${\mathsf{\Sigma }}_{21}^{\mathrm{V}}$’, ‘${\mathsf{\Sigma }}_{12}^{\mathsf{\omega }}$’, ‘${\mathsf{\Sigma }}_1^{\mathrm{L}}$’, and ‘${\mathsf{\Sigma }}_2^{\mathrm{L}}$’ denote the difference between two significant ratios corresponding to the greatest and smallest numbers corresponding to five geographical locations in the same row. This result indicates that Hypotheses 1–3 are not rejected. On the other hand, the leverage effect of a market is nearly the same within five geographical locations because the differences in significant ratios among the five geographical locations, ‘${\mathsf{\Sigma }}_1^{\mathrm{L}}$’ (20%) and ‘${\mathsf{\Sigma }}_2^{\mathrm{L}}$’ (30%), are less than (or equal) 30%. This result indicates that Hypothesis 4 is rejected. Subsequently, the second question was how the geographical locations affect the results of the interactions between the stock indices in emerging and developed markets. I found the following results about this question. First, in ‘Asia-America’ and ‘Asia-Europe’, the developed market mainly spills into the emerging market from the viewpoint of both return and volatility. In other words, the developed market plays the dominant position for both the return and volatility spillovers. This is because the significant ratios for ${\mathsf{\Sigma }}_{12}^{\mathrm{R}}$ and ${\mathsf{\Sigma }}_{12}^{\mathrm{V}}$ are greater than those for ${\mathsf{\Sigma }}_{21}^{\mathrm{R}}$.and ${\mathsf{\Sigma }}_{21}^{\mathrm{V}}$, respectively. Taking an example of ‘Asia-Europe’, the significant ratio for ${\mathsf{\Sigma }}_{12}^{\mathrm{R}}$ (68.4%) is greater than that for ${\mathsf{\Sigma }}_{21}^{\mathrm{R}}$ (23.7%). Moreover, the significant ratio for ${\mathsf{\Sigma }}_{12}^{\mathrm{V}}$ (55.3%) is also greater than that for ${\mathsf{\Sigma }}_{21}^{\mathrm{V}}$(28.9%). The above results are the same as those found from the entire sample of paired stock indices. Second, in ‘Asia’, ‘America’ and ‘America-Europe’, the emerging market primarily spills into the developed market from the viewpoint of both return and volatility. That is, the emerging market plays the dominant position for both the return and volatility spillovers. This is because the significant ratios for ${\mathsf{\Sigma }}_{21}^{\mathrm{R}}$ and ${\mathsf{\Sigma }}_{21}^{\mathrm{V}}$ are greater than those for ${\mathsf{\Sigma }}_{12}^{\mathrm{R}}$ and ${\mathsf{\Sigma }}_{12}^{\mathrm{V}}$, respectively. Taking an example of ‘America-Europe’, the significant ratio for ${\mathsf{\Sigma }}_{21}^{\mathrm{R}}$ (85.7%) is greater than that for ${\mathsf{\Sigma }}_{12}^{\mathrm{R}}$ (42.9%). Furthermore, the significant ratio for ${\mathsf{\Sigma }}_{21}^{\mathrm{V}}$ (64.3%) is greater than that for ${\mathsf{\Sigma }}_{12}^{\mathrm{V}}$.(28.6%). The above results are different from those found from the entire sample of paired stock indices. Third, the correlation is positive for all the paired stock indices. Particularly, in ‘America’ and ‘America-Europe’, there exists a positive correlation between the emerging and developed markets for all cases because the significant ratios for ${\mathsf{\Sigma }}_{12}^{\mathsf{\omega }}$. equal 100% for the above two geographical locations. The above result indicates that the emerging stock markets in America, MXX and BVSP, are closely interrelated to the developed stock markets in America and Europe, such as DJ, TSX, FTSE, CAC40, DAX, and MIB. Finally, the leverage effect exists in the emerging and developed stock markets for all paired stock indices because all the significant ratios for ${\mathsf{\Sigma }}_1^{\mathrm{L}}$ and ${\mathsf{\Sigma }}_2^{\mathrm{L}}$ are greater than 70%. Particularly, in ‘America’, the leverage effect exists in the emerging markets (MXX and BVSP) and developed markets (DJ and TSX) for all cases because the significant ratios for ${\mathsf{\Sigma }}_1^{\mathrm{L}}$ and ${\mathsf{\Sigma }}_2^{\mathrm{L}}$ equal 100%. Moreover, the leverage effect exists at the developed stock markets in ‘Asia-Europe’ and ‘America-Europe’ such as DJ, TSX, FTSE, CAC40, DAX, and MIB, but not for the N225, because the significant ratio for ${\mathsf{\Sigma }}_2^{\mathrm{L}}$ equals 100%. The above results indicate that the leverage effect significantly exists in the emerging and developed stock markets, especially for the developed market.

With the same inference process as in the previous two paragraphs, I use the data listed in columns ‘XU100’, ‘RTSI’, ‘MXX’, ‘BVSP’, ‘JKSE’, ‘SSE’, and ‘BSE’ at the right-hand side of Table 11 to discuss the two research questions of this study. The first question was whether the interactions between the stock indices in emerging and developed markets are different in terms of the emerging stock indices. As shown by the numbers listed in column ‘${\Delta }^E$’ and rows ‘${\mathsf{\Sigma }}_{12}^{\mathrm{R}}$’ (78.6%), ‘${\mathsf{\Sigma }}_{21}^{\mathrm{R}}$’ (75.7%), ‘${\mathsf{\Sigma }}_{12}^{\mathrm{V}}$’ (35.7%), ‘${\mathsf{\Sigma }}_{21}^{\mathrm{V}}$’ (60.7%), ‘${\mathsf{\Sigma }}_{12}^{\mathsf{\omega }}$’ (85.7%), ‘${\mathsf{\Sigma }}_1^{\mathrm{L}}$’ (28.6%), and ‘${\mathsf{\Sigma }}_2^{\mathrm{L}}$’ (10%), the interactions (correlation, return and volatility spillovers) between the stock indices in emerging and developed markets are significantly different in terms of emerging stock indices because the differences in significant ratios among the seven emerging stock indices for ‘${\mathsf{\Sigma }}_{12}^{\mathrm{R}}$’ (78.6%), ‘${\mathsf{\Sigma }}_{21}^{\mathrm{R}}$’ (75.7%), ‘${\mathsf{\Sigma }}_{12}^{\mathrm{V}}$’ (35.7%), ‘${\mathsf{\Sigma }}_{21}^{\mathrm{V}}$’ (60.7%), and ‘${\mathsf{\Sigma }}_{12}^{\mathsf{\omega }}$’ (85.7%) all are greater than 30%. The numbers in column ‘${\Delta }^E$’ in each of the rows ‘${\mathsf{\Sigma }}_{12}^{\mathrm{R}}$’, ‘${\mathsf{\Sigma }}_{21}^{\mathrm{R}}$’, ‘${\mathsf{\Sigma }}_{12}^{\mathrm{V}}$’, ‘${\mathsf{\Sigma }}_{21}^{\mathrm{V}}$’, ‘${\mathsf{\Sigma }}_{12}^{\mathsf{\omega }}$’, ‘${\mathsf{\Sigma }}_1^{\mathrm{L}}$’, and ‘${\mathsf{\Sigma }}_2^{\mathrm{L}}$’ denote the difference between two significant ratios corresponding to the greatest and smallest number among seven numbers corresponding to seven emerging stock indices in the same row. This result indicates that Hypotheses 1–3 are not rejected. On the other hand, the leverage effect of a market is nearly the same within the seven emerging stock indices because the differences in significant ratios among ‘${\mathsf{\Sigma }}_1^{\mathrm{L}}$’ (28.6%) and ‘${\mathsf{\Sigma }}_2^{\mathrm{L}}$’ (10%) are less than 30%. This result indicates that Hypothesis 4 is rejected. Subsequently, the second question is how the emerging stock indices affect the results of the interactions between the stock indices in emerging and developed markets. I found the following results regarding this question. First, regarding the paired stock indices related to ‘XU100’, ‘SSE’, and ‘BSE’, the developed market mainly spills into the emerging market from the viewpoint of both return and volatility. In other words, the developed market plays the dominant role for both the return and volatility spillovers. Taking an example of the paired stock indices related to ‘XU100’, the significant ratio for ${\mathsf{\Sigma }}_{12}^R$ (50%) is greater than that for ${\mathsf{\Sigma }}_{21}^R$ (25%). Moreover, the significant ratio for ${\mathsf{\Sigma }}_{12}^V$ (50%) is also greater than that for ${\mathsf{\Sigma }}_{21}^V$ (25%). The above results are the same as those found from the entire sample of paired stock indices. Second, regarding the paired stock indices related to ‘MXX’ and ‘BVSP’, the emerging market primarily spills into the developed market from the viewpoint of both return and volatility. That is, the emerging market plays the dominant role for both the return and volatility spillovers. Taking an example of the paired stock indices related to ‘MXX’, the significant ratio for ${\mathsf{\Sigma }}_{21}^R$ (90%) is greater than that for ${\mathsf{\Sigma }}_{12}^R$ (50%). Moreover, the significant ratio for ${\mathsf{\Sigma }}_{21}^V$ (60%) is also greater than that for ${\mathsf{\Sigma }}_{12}^V$ (50%). The above results are different from those found from the entire sample of paired stock indices. Third, regarding the paired stock indices related to ‘RTSI’ and ‘JKSE’, the results of both return and volatility spillovers are different from those found from ‘XU100’, ‘SSE’, ‘BSE’, ‘MXX’, and ‘BVSP’. For example, regarding the paired stock indices related to ‘RTSI’, the emerging market (i.e., RTSI) and developed market play the dominant roles in the return and volatility spillovers, respectively. On the contrary, regarding the paired stock indices related to ‘JKSE’, the developed market and emerging market (i.e., JKSE) play the dominant roles in the return and volatility spillovers, respectively. Fourth, the correlation is positive for all the paired stock indices. Particularly, regarding the paired stock indices related to ‘MXX’ and ‘BVSP’, there exists a positive correlation between the emerging and developed markets for all cases because the significant ratios for ${\mathsf{\Sigma }}_{12}^{\mathsf{\omega }}$. equal 100%. The above result indicates that the emerging stock markets, MXX and BVSP, are closely interrelated to all the developed stock markets such as DJ, TSX, FTSE, CAC40, DAX, MIB, and N225. The results found from the paired stock indices related to the ‘MXX’ and ‘BVSP’ are similar to those found from the paired stock indices in ‘America’ and ‘America-Europe’. Finally, the leverage effect exists in the emerging and developed stock markets for all the paired stock indices because all the significant ratios for ${\mathsf{\Sigma }}_1^{\mathrm{L}}$ and ${\mathsf{\Sigma }}_2^{\mathrm{L}}$ are greater than 71.4%. Particularly, regarding the paired stock indices related to the ‘RTSI’, the significant ratio for ${\mathsf{\Sigma }}_1^{\mathrm{L}}$ equals 100%, thus, the leverage effect exists in emerging stock market such as RTSI. Moreover, regarding the paired stock indices related to ‘XU100’ and ‘SSE’, the significant ratios for ${\mathsf{\Sigma }}_2^{\mathrm{L}}$ equal 100%, thus, the leverage effect exists in all the developed stock markets such as FTSE, CAC40, DAX, MIB, DJ, TSX, and N225. The above results found from the paired stock indices related to ‘XU100’ and ‘SSE’ are similar to those found from the paired stock indices in ‘America’, ‘Asia-Europe’, and ‘America-Europe’. The above results also indicate that the leverage effect exists significantly in the emerging and developed stock markets, especially in the developed market.

5.4. The Impact of a Structural Break on the Interactions between the Emerging and Developed Markets

In this subsection, I will explore the interactions between the emerging and developed markets from a structural break perspective, or how the structural break changes the results of the interactions between the emerging stock market and the developed stock market, which are illustrated in the previous subsection. From the data listed in column ‘SUM’ in Table 11, I find that, when I take the entire sample of paired stock indices as a whole, the volatility spillover is affected by the structural break the most significantly, followed by the correlation and leverage effect. However, the return spillover is nearly unaffected by the structural break, because the differences in significant ratios for ${\Delta }^V$, ${\Delta }^C,$ and ${\Delta }^L$ are ‘−23.9%’, ‘15.2%’ and ‘13%’, respectively, and they are not high in absolute value. The differences in significant ratios for ${\Delta }^V$ (−23.9%) equal the significant ratios for ${\mathsf{\Sigma }}_A^V$ (39.1%) minus the significant ratios for ${\mathsf{\Sigma }}_B^V$ (63%). The significant ratio for ${\mathsf{\Sigma }}_A^V$ (39.1%) equals ‘36’ divided by ‘92’, where ‘92’ equals twice of the total number of all the sampled paired stock indices (46). The number ‘36’ equals the sum of the number corresponding to ${\mathsf{\nu }}_{12}^A$ (24) and the number corresponding to ${\mathsf{\nu }}_{21}^A$ (12). The significant ratios for ${\mathsf{\Sigma }}_A^V$ can measure the intensity of bidirectional volatility spillover for the post-SB period. On the other hand, the significant ratio for ${\mathsf{\Sigma }}_B^V$ (63%) equals ‘58’ divided by ‘92’, where ‘92’ equals twice the total number of all sampled paired stock indices (46). The number ‘58’ equals the sum of the number corresponding to ${\mathsf{\nu }}_{12}^B$ (26) and the number corresponding to ${\mathsf{\nu }}_{21}^B$ (32). The significant ratios for ${\mathsf{\Sigma }}_B^V$ can measure the intensity of bidirectional volatility spillover for the pre-SB period. Hence, the differences in significant ratios for ${\Delta }^V$ can measure the impact of a structural break on the volatility spillover. Subsequently, the differences in significant ratios for ${\Delta }^C$ (15.2%), the differences in significant ratios for ${\Delta }^L$ (13%) and the differences in significant ratios for ${\Delta }^R$ (1.1%) can measure the impact of a structural break on the correlation, leverage, and return spillover, respectively, and they are obtained with the same inference process. However, the difference in significant ratios for ${\Delta }^R$ is ‘1.1%’, and it nearly approaches zero. The above results are consistent with those found in the parameters’ equality tests in Section 5.2. Notably, after the structural break, the impact of the volatility spillover decreases, whereas that of the correlation and leverage effect increases. Subsequently, with the same inference process as in the previous subsection, I use the data listed in the columns ‘Asia’, ‘America’, ‘Asia-America’, ‘Asia-Europe’, and ‘America-Europe’ in the left-hand side of of Table 11 to discuss the research questions of this study. Regarding the question of whether the interactions between the stock indices in emerging and developed markets are different in terms of the two subperiods, as shown by the numbers listed in rows ‘${\Delta }^R$’, ‘${\Delta }^V$’, ‘${\Delta }^C$’, and ‘${\Delta }^L$’, only the value of ‘${\Delta }^V$’ for Asia (−60%), the value of ‘${\Delta }^C$’ for Asia (40%), and the value of ‘${\Delta }^L$’ for Asia (50%) are greater than 30%. The numbers in row ‘${\Delta }^R$’ in each column denote the difference between two significant ratios corresponding to ${\mathsf{\Sigma }}_{\mathrm{B}}^{\mathrm{R}}$ and ${\mathsf{\Sigma }}_{\mathrm{A}}^{\mathrm{R}},$ whereas the numbers in row ‘${\Delta }^V$’ in each column denote the difference between two significant ratios corresponding to ${\mathsf{\Sigma }}_{\mathrm{B}}^{\mathrm{V}}$ and ${\mathsf{\Sigma }}_{\mathrm{A}}^{\mathrm{V}}$. On the other hand, the numbers in row ‘${\Delta }^C$’ in each column denote the difference between two significant ratios corresponding to ${\mathsf{\omega }}_{12}^{\mathrm{B}}$ and ${\mathsf{\omega }}_{12}^{\mathrm{A}}$, whereas the numbers in row ‘${\Delta }^L$’ in each column denote the difference between two significant ratios corresponding to ${\mathsf{\Sigma }}_{\mathrm{B}}^{\mathrm{L}}$ and ${\mathsf{\Sigma }}_{\mathrm{A}}^{\mathrm{L}}$. These results indicate that the Hypotheses 5–8 are rejected for most cases except for Asia. In other words, regarding the five geographical locations, the impact of a structural break on the interactions between the emerging and developed markets is very small. Notably, in Asia, the correlation and volatility spillovers between the stock indices in emerging and developed markets and the leverage effect in a market are significantly different in the two subperiods. Hence, financial features in Asia are affected by the structural break. The second question is how the structural break affects the results of the interactions between the stock indices in emerging and developed markets. I omit discussion about this because the impact of a structural break on the interactions between the emerging and developed markets is very small. That is, in Asia, the impact of the volatility spillover decreases, whereas that of the correlation and leverage effect increases, after the structural break.

With the same inference process, I use the data listed in columns ‘XU100’, ‘RTSI’, ‘MXX’, ‘BVSP’, ‘JKSE’, ‘SSE’, and ‘BSE’ in the right-hand side of Table 11 to discuss the research questions of this study. Regarding the question of whether the interactions between the stock indices in emerging and developed markets are different in the two subperiods, as shown by the numbers listed in rows ‘${\Delta }^R$’, ‘${\Delta }^V$’, ‘${\Delta }^C$’, and ‘${\Delta }^L$’, only the values of ‘${\Delta }^V$’ for XU100 (−41.6%), JKSE (−50%), and SSE (−35.7%), the value of ‘${\Delta }^C$’ for SSE (42.9%), and the value of ‘${\Delta }^L$’ for JKSE (35.7%) are greater than 30%. This result indicates that Hypotheses 5–8 are rejected for most cases except for JKSE. In other words, regarding the seven emerging stock indices, the impact of a structural break on the interactions between the emerging and developed markets is very small. Regarding the second question, how the structural break affects the results of the interactions between the stock indices in emerging and developed markets, I omit discussion about it because the impact of a structural break on the interactions between the emerging and developed markets is very small. Notably, in JKSE, the correlation between the stock indices in emerging and developed markets and the leverage effect in a market are significantly different in the two subperiods. Hence, the financial features in JKSE are affected by the structural break. That is, in JKSE, the impact of the volatility spillover decreases, whereas that of the leverage effect increases, after the structural break.

6. Discussion

This study utilizes a bivariate BEKK-EGARCH model with the setting of a structural break to investigate whether the interactions between stock indices in emerging and developed markets are different in terms of geographical location, emerging stock indices, and subperiods. Then, it investigates how the results of interaction vary with geographical location, emerging stock indices, and subperiods. From the empirical results, I found that the short- and long-term interactions between the emerging and developed markets significantly vary with geographical location and emerging stock indices, but are almost the same for the two subperiods. Subsequently, I listed the results for the entire sample of paired stock indices and will now give some discussions of these results.

First, the developed market mainly spills into the emerging market in terms of return and volatility. This result is consistent with that found in Wei et al. [3], Miyakoshi [5], and Ng [4], because the stock markets in the US and Japan, representing the developed market, can affect the stock markets in Korea, Taiwan, Singapore, Thailand, Malaysia, Hong Kong, and Indonesia in Asia, representing the emerging market in terms of return and volatility. This phenomenon may be attributed to the following two reasons. First, owing to advanced technology owned by the developed countries, developed countries such as those in the G7 take on the research and design of products. On the other hand, emerging countries such as those in the E7 and other emerging countries are in charge of the manufacture of products assigned by the developed countries. That is, the economic growth in emerging countries is determined by the production orders provided by developed countries. Second, owing to the high economic capacity, such as the high GDP owned by the developed countries, capitalists in the developed countries directly invest their capital into the enterprise or capital market of the emerging countries.

Second, as reported by the rankings of significant ratios for the leverage effect and short- and long-term interactions, the leverage effect is the most important, followed by the long-term interaction and two short-term interactions. Notably, the leverage effect significantly exists in the emerging and developed stock markets, especially for the developed market. This result is consistent with that found in Su [32] and Su [40], because Su [40] found the leverage effect exists in the stock market and is also the most crucial effect in volatility forecasts in the stock market compared with the other two financial features of price level and distribution effect. In other words, the volatility of the stock market is easily increased with bad news from the market. This phenomenon may be attributed to the following reason. The leverage effect belongs to the financial features of an individual market. On the contrary, long- and short-term interactions belong to the financial features of several markets. Hence, when there is an incident happening in a market, the financial feature of an individual market is the first to respond, and then the financial feature of several markets reacts. Thus, the leverage effect is more important than the long-term interaction and two short-term interactions. In addition, the stock market is the most popular capital market for most investors and fund managers. Thus, the stock market is very sensitive to the variation, especially in terms of bad news. This is why the leverage effect significantly exists in the stock market.

Third, both the emerging stock market and developed stock market positively affect each other with three types of interactions, especially correlation. This result is consistent with Ng [4]. This phenomenon may be attributed to the same type of commodities being explored in this study. Another reason is attributed to globalization and financial liberalization, which mean that the capital of enterprise can flow freely all over the world. The rise in the price of the stock market in developed markets such as the US one can drive the rise in the price of the stock market in the emerging market.

However, the above results for the entire sample of paired stock indices may vary with geographical location, emerging stock indices, and subperiods. This phenomenon may be attributed to the fact that economic capacity (GDP), financial liberalization (or financial liquidity), and industrial development and its distribution are different within each country. Thus, the above country factors in a group of countries in a region may be different from those in another region. Hence, the above results for the entire sample of paired stock indices may vary with geographical locations and emerging stock indices. In addition, the trends in the behavior of a financial feature may be affected by an incident, such as the GFC and QE in this study. Thus, the above results for the entire sample of paired stock indices may vary with subperiod. In other words, this study explores the interactions between the stock indices in emerging and developed markets from the viewpoints of time domain, region domain, and individual entity. Or, the above issues may be affected by the time factor, region factor, and entity factor. Thus, to the best of my knowledge, this study is the first to explore the interaction issues (the correlation and return and volatility spillovers) based on simultaneously considering geographical locations, emerging stock indices, and structural breaks. However, the results of this study still need to be compared with results obtained from a new crisis, the COVID-19 pandemic, for a robust check to be performed.

7. Conclusions

This study utilizes a bivariate BEKK-EGARCH model with the setting of a structural break to investigate the interactions between emerging and developed markets from a structural break perspective. The interactions include one long-term interaction, the correlation, and two short-term interactions, the return and volatility spillovers. Moreover, the stock indices in E7 and G7 countries are used to represent the emerging and developed markets, respectively. In addition, the study period covers the global financial crisis (GFC) and quantitative easing (QE). Hence, a structural break (SB) exists in the return series of stock indices detected by the maximum likelihood-ratio test of Eizaguirre et al. [23]. The accuracy of dates of SB is proved by the stock indices in the G7 and E7 countries having different investment attributes in the pre- and post-SB periods, and is also confirmed by three model-fitting ability tests on a bivariate model used in this study. The two subperiods are partitioned from the study period using the above dates of SB.

From the empirical results, I found that the short- and long-term interactions between the emerging and developed markets vary significantly with geographical location and emerging stock indices, but are almost the same in the two subperiods. Subsequently, I illustrate the above conclusions for the entire sample of paired stock indices and from the viewpoints of both geographical location and emerging stock indices. First, for the entire sample of paired stock indices, I find that the developed market mainly spills into the emerging market in terms of return and volatility. This phenomenon may be attributed to the developed countries (G7) taking on the research and design of products whereas the emerging countries (E7) are in charge of the manufacture of products assigned by the developed countries. Moreover, as reported by the rankings of significant ratios for the leverage effect and short- and long-term interactions, the leverage effect is the most important followed by the long-term interaction and two short-term interactions. Notably, the leverage effect significantly exists in the emerging and developed stock markets, especially for the developed market. In addition, both the emerging stock market and developed stock market positively affect each other in terms of the three types of interactions, especially for the correlation. Second, from the viewpoint of geographical location, I found that the interactions (correlation, return and volatility spillovers) between the stock indices in emerging and developed markets are significantly different within the five geographical locations because Hypotheses 1–3 are not rejected. However, the leverage effect of a market is nearly the same within the five geographical locations since Hypothesis 4 is rejected. More precisely, I find that the developed market mainly spills into the emerging market in terms of return and volatility in ‘Asia-America’ and ‘Asia-Europe’, whereas the emerging market primarily spills into the developed market in terms of return and volatility in ‘Asia’, ‘America’ and ‘America-Europe’. Moreover, the correlation is positive for all the paired stock indices, especially in ‘America’ and ‘America-Europe’, indicating that the emerging stock market in America, MXX and BVSP, are closely interrelated to the developed stock markets in America and Europe such as DJ, TSX, FTSE, CAC40, DAX, and MIB. In addition, the leverage effect significantly exists in the emerging and developed markets, especially for the developed market. Third, from the viewpoint of the emerging stock indices, I found that the interactions (correlation, return and volatility spillovers) between the stock indices in emerging and developed markets are significantly different within the seven emerging stock indices because Hypotheses 1–3 are not rejected. However, the leverage effect of a market is nearly the same within the seven emerging stock indices since Hypothesis 4 is rejected. More precisely, I found that, regarding the paired stock indices related to ‘XU100’, ‘SSE’ and ‘BSE’, the developed market mainly spills into the emerging market in terms of return and volatility. Conversely, regarding the paired stock indices related to ‘MXX’ and ‘BVSP’, the emerging market primarily spills into the developed market in terms of return and volatility. Notably, regarding the paired stock indices related to ‘RTSI’ and ‘JKSE’, the results of both return and volatility spillovers are different from those found for ‘XU100’, ‘SSE’, ‘BSE’, ‘MXX’, and ‘BVSP’. In addition, the correlation is positive for all the paired stock indices, especially for the paired stock indices related to ‘MXX’ and ‘BVSP’, indicating that the emerging stock markets, MXX, and BVSP, are closely interrelated to all the developed stock markets such as DJ, TSX, FTSE, CAC40, DAX, MIB, and N225. The above results are similar to those found for the paired stock indices in ‘America’ and ‘America-Europe’. Moreover, the leverage effect significantly exists in the emerging and developed stock markets, especially for the developed market. Finally, I demonstrate the impact of a structural break on the interactions between emerging and developed markets for the entire sample of paired stock indices and from the viewpoints of both geographical location and emerging stock indices. First, for the entire sample of paired stock indices, I found that the volatility spillover is affected by the structural break the most significantly, followed by the correlation and leverage effects, whereas the return spillover is nearly unaffected by the structural break. However, the above phenomena are not significant because the values of the significant ratio for them are not high. Second, from the viewpoint of geographical location, I found that the impact of a structural break on the interactions between emerging and developed markets is very small because Hypotheses 5–8 are rejected for most cases except in Asia. Regarding the paired stock indices within ‘Asia’, the correlation and volatility spillovers between the stock indices in emerging and developed markets and the leverage effect in a market are significantly different in the two subperiods. More precisely, the impact of the volatility spillover decreases, whereas that of the correlation and leverage effect increases, after the structural break. Third, from the viewpoint of the emerging stock indices, I found that the impact of a structural break on the interactions between the emerging and developed markets is very small because Hypotheses 5-8 are rejected for most cases except JKSE. Regarding the paired stock indices related to ‘JKSE’, the correlation between the stock indices in emerging and developed markets and the leverage effect in a market are significantly different in the two subperiods. More precisely, the impact of the volatility spillover decreases, whereas that of the leverage effect increases, after the structural break.