### Data sample and explanation

We collected the data for our estimation from the stock index of ASEAN countries by Thomson Reuters in the period from January 2001 to December 2017. We used the equity indices VN Index, SET Index, FTSE Straits times Index, PSEi Index, FTSE Bursa Malaysia KLCI Index, and Jakarta SE Composite Index, which are representative of the stock markets of Vietnam, Thailand, Singapore, the Philippines, Malaysia, and Indonesia, respectively. The main reason for our choice of these countries was to ensure the availability of datasets from between 2001 and 2017. Additionally, these six stock markets occupy 80% the ASEAN regions, while the remaining ones are small and newly established, for example, Cambodia, Laos, Myanmar, etc. In addition, Do et al. (2016) found that most studies examining the capital market in the ASEAN region also use these 6 countries, such as Balli et al. (2014). Therefore, our results can be interpreted to relate to the ASEAN area. After streamlining our data by eliminating the missing data from the various holidays in these stock markets, we calculate the log-return as theorized by Fama and Miller (1972).^{Footnote 1} After calculating the ASEAN stock returns, we summarize the data based on some basic criteria such as mean, standard deviation, skewness, and kurtosis. By doing this, we can note some features from each economy in the ASEAN region.

As a result in Table 1 and Figure 1, we determined that the average return in Indonesia was notably higher than the other stock markets in the ASEAN area. The Indonesia index increased 15 times during the period from 2001 to 2017. This explains why many investors choose to earn their return from this market. Furthermore, the Indonesian Government has gradually developed its policies to build infrastructure as well as to attract foreign direct investment, which fosters this equity market to become one of the most rapidly growing emerging stock markets. In contrast, Singapore experiences the lowest average return in the ASEAN region, although Singaporean market capitalization is the largest. However, because Singapore has reached a saturation point, investors might not take opportunities to earn profits. Thus, the volatility in Singapore is the lowest, which presents stable movements as well as neutralizes volatile risks. When it comes to other risk factors such as skewness and kurtosis, Thailand has suffered from a higher absolute value with − 1.4041 and 14.0140, respectively. It can be interpreted that Thailand has negative skewness in the left tail. The standardized kurtosis, then, is over four, which means that the heavy tail is also a risk factor in this country. The monetary crisis was catalyzed by Thailand in 1997. Additionally, not long ago, Thailand’s SET index decreased by 108.41 points, equivalent to the loss of 816 Baht (nearly 23 billion USD) (Sutheebanjard and Premchaiswadi 2010). The main reason for these statistics in Thailand is the failure to control the depreciation of Thai Baht, and the presence of risks associated with the country.

### Copulas approaches

Huynh et al. (2018), Huynh and Burggraf (2020) assert that the intent of the copulas approach comes from Sklar (1959)‘s theorem of generating joint multivariate probability-distribution functions. In our research, we refer to studies from Huynh et al. (2018), specifically the fundamental concept of joint density function *H* in Eq. (2) with *C* representing the copulas:

$$ \exists C:{\left[0,1\right]}^d\to \left[0,1\right] $$

(1)

which satisfies the main condition for x = (x_{1}, x_{2}, …, x_{d})

$$ H(x)=C\left\{{F}_1\left({x}_1\right),\dots, {F}_d\left({x}_d\right)\right\}\ x\in {R}^d $$

(2)

The function ‘C’ here is called ‘traditional copulas’ (or vanilla copulas), with two main characteristics: (i) dividing into many sides and (ii) marginal distribution following a standard normal distribution.^{Footnote 2} This copula has a disadvantage in distribution. Hence, we would like to refer the other copulas’ functions as follows. Among the variety of copula families, we mainly focus on t-DCC copulas (t-student’s distribution of Dynamic Conditional Correlation copulas), Gaussian DCC copulas (Gaussian Dynamic Conditional Correlation copulas), tv-copulas (time-varying copulas) and tv-SJC copulas (time-varying and static bivariate symmetrized Joe-Clayton copulas).

Nevertheless, Charfeddine and Benlagha (2016) conclude that t-student copulas allow us to estimate fat-tail shape. Additionally, it may increase the joint probability that results in the same events happening. Therefore, this study also indicates that t-student copulas will more applicable for dynamic data, whereas the Gaussian faces limitation. Thus, the t-student copulas (with *u* and *v* as uniform random variables obtained from the cumulative distribution function) are written as:

$$ C\left(u,\mathrm{v}|\rho, \nu \right)={\int}_{-\infty}^{t_{\nu}^{-1}(u)}{\int}_{-\infty}^{t_{\nu}^{-1}\left(\mathrm{v}\right)}\frac{1}{2\pi {\left(1-{\rho}^2\right)}^{\frac{1}{2}}}\ {\left\{1+\frac{x^2-2\rho xy+{y}^2}{\nu \left(1-{\rho}^2\right)}\right\}}^{-\frac{\nu +2}{2}} dsdt $$

(3)

Its components include parameter *ρ*, which represents a parameter of estimated copulas, and *ν*, which is the degree of freedom of t-student’s distribution. Meanwhile, \( {t}_{\nu}^{-1} \) is the inverse of the standard univariate (Charfeddine and Benlagha 2016). To be more specific, SJC copulas present both upper and lower tail with two parameters *τ*_{U}, *τ*_{L}, respectively. Clearly, the other copulas require symmetrical dependence of random variables for estimation. Thus, Static Bivariate Symmetrized Joe-Clayton copulas have an advantage over symmetrized data, and offers high precision in relation to the tail, calculating exactly the expected coefficient for interpretation. In particular, the Joe-Clayton copula is written as a function:

$$ {C}_{JC}\left(u,v|{\tau}_U,{\tau}_L\right)=1-{\left({\left\{{\left[1-{\left(1-u\right)}^k\right]}^{-\gamma }+{\left[1-{\left(1-v\right)}^k\right]}^{-\gamma }-1\right\}}^{-\frac{1}{\gamma }}\right)}^{\frac{1}{k}} $$

(4)

$$ k=1/{\log}_2\left(2-{\tau}_U\right) $$

(5)

$$ where\ \gamma =-1/{\log}_2\left({\tau}_L\right) $$

$$ {\tau}_U\epsilon \left(0,1\right),{\tau}_L\epsilon \left(0,1\right) $$

Hotta et al. (2006) expands the approach mentioned earlier by symmetric *τ*_{U} = *τ*_{L} by this formula:

$$ {C}_{SJC}\left(u,v|{\tau}_U,{\tau}_L\right)=0.5{C}_{JC}\left(u,v|{\tau}_U,{\tau}_L\right)+0.5{C}_{JC}\left(1-u,1-v|{\tau}_U,{\tau}_L\right)+u+v-1 $$

(6)

We develop this model by adding the time-varying factor then standardize it into tv-SJC copulas with the research of Charfeddine and Benlagha (2016) regarding dependence parameters:

$$ \hat{de{p}_t}={c}_j+{u}_t,t={T}_{j-1}+1,{T}_{j-1}+2,\dots, {T}_j $$

(7)

where *j* = 1, 2, …, *m* + 1; *T*_{0} = 0. *T*_{m + 1} = *T* and *c*_{j} is the conditional mean of estimated dependency parameters for each regime.

This is done under the assumption that the dependence parameter is calculated by past information and follows an ARMA (1,k). Hence, the Gaussian coefficient according to Nguyen and Bhatti (2012) is:

$$ {\rho}_t=\Lambda \left({\beta}_{\rho }{\rho}_{t-1}+{\omega}_{\rho }+{\gamma}_{\rho}\frac{1}{k}{\sum}_{i=1}^k\left|{u}_{t-i}-{v}_{t-i}\right|\right) $$

(8)

This means that parameter *k* is very wide and historical information is needed to choose the most appropriate one. We also propose the Gumbel and Clayton copulas here with the same assumption of ARMA (1,k)^{Footnote 3}:

$$ {\delta}_t={\beta}_U{\delta}_{t-1}+{\omega}_U+{\gamma}_U\frac{1}{k}{\sum}_{i=1}^k\left|{u}_{t-i}-{v}_{t-i}\right| $$

(9)

$$ {\theta}_t={\beta}_L{\theta}_{t-1}+{\omega}_L+{\gamma}_L\frac{1}{k}{\sum}_{i=1}^k\left|{u}_{t-i}-{v}_{t-i}\right| $$

(10)

After defining how to establish the tv-SJC copulas above, in the following equation by Nguyen and Bhatti (2012) is indicated the dynamics of the upper- and lower-tail dependences, respectively.

$$ {\tau}^U=\Pi \left({\beta}_U^{SJC}{\tau}_{t-1}^U+{\omega}_U^{SJC}+{\gamma}_U^{SJC}\frac{1}{k}{\sum}_{i=1}^k\left|{u}_{t-i}-{v}_{t-i}\right|\right) $$

(11)

$$ {\tau}^L=\Pi \left({\beta}_L^{SJC}{\tau}_{t-1}^L+{\omega}_L^{SJC}+{\gamma}_L^{SJC}\frac{1}{k}{\sum}_{i=1}^k\left|{u}_{t-i}-{v}_{t-i}\right|\right) $$

(12)

In order to interpret the level of dependency of the data structure, we utilized the literature as well as the empirical evidence of Meneguzzo and Vecchiato (2004), Mashal and Zeevi (2002), Breymann et al. (2003), and Galiani (2003).

The main method to estimate copulas parameters, therefore, is Inference-function-for-margins (IFM), which extracts the exact maximum likelihood (EML). Thus, these parameters are quite important to define the level of dependency.

$$ {\hat{\theta}}_{it}= argmax{\sum}_{t=1}^T\ln {f}_{it}\left({z}_{i,t}|{\Omega}_{t-1},{\theta}_{it}\right) $$

(13)

$$ {\hat{\theta}}_{ct}= argmax{\sum}_{t=1}^T\ln {c}_t\left({F}_{1t}\left({z}_{i,t}|{\Omega}_{t-1}\right),{F}_{2t}\left({z}_{2,t}|{\Omega}_{t-1}\right),\dots {F}_{nt}\left({z}_{n,t}|{\Omega}_{t-1}\right),{\hat{\theta}}_{it},{\hat{\theta}}_{ct}\right) $$

(14)

This approach is derived from the study of Joe and Xu (1996), Joe (1997), and Joe (2005), in which *F* and *c*_{t} are the functions for unknown marginal parameter vectors and unknown copula parameter, respectively. This value is used under the assumption that the margins are correctly specified, and sample variables are unobservable, independent, and identically distributed random variables (i.i.d.).

### Non-parametric approach

We used the study by Nguyen et al. (2016) and Huynh et al. (2020a) for plotting each point. We plotted on graphs of a wide area (*λ*_{i}, *χ*_{i}) for the movement of both variables (*X*_{i}, *Y*_{i}) with *i* = 1, 2, …, *n*. In order to draw this pair (*X*_{i}, *Y*_{i}), the calculation was as follows:

$$ {\mathrm{X}}_{\mathrm{i}}=\frac{{\mathrm{H}}_{\mathrm{i}}-{\mathrm{F}}_{\mathrm{i}}{\mathrm{G}}_{\mathrm{i}}}{\sqrt{{\mathrm{F}}_{\mathrm{i}}\left(1-{\mathrm{F}}_{\mathrm{i}}\right){\mathrm{G}}_{\mathrm{i}}\left(1-{\mathrm{G}}_{\mathrm{i}}\right)}} $$

(15)

$$ {\uplambda}_{\mathrm{i}}=4{\mathrm{S}}_{\mathrm{i}}\max \left\{{\left({\mathrm{F}}_{\mathrm{i}}-\frac{1}{2}\right)}^2,{\left({\mathrm{G}}_{\mathrm{i}}-\frac{1}{2}\right)}^2\right\} $$

(16)

Here, \( {S}_i=\mathit{\operatorname{sign}}\left\{\left({F}_i-\frac{1}{2}\right)\left({G}_i-\frac{1}{2}\right)\right\} \). The confidence interval lies in \( \pm {c}_p/\sqrt{n} \) (approximately at *C*_{p} at the significance level 95%, which is nearly 1.78).

Quantile-Quantile-plot (QQ-plot) was used, and the value of *H*_{i} is defined as follows:

$$ {\mathrm{K}}_0\left(\upomega \right)=\mathrm{P}\left(\mathrm{U}\mathrm{V}\le \upomega =\mathrm{P}\left(\mathrm{U}\le \frac{\upomega}{\upvartheta}\right)\mathrm{d}\upvartheta \right)=\mathrm{Id}\upvartheta +\frac{\upomega}{\upvartheta}\mathrm{d}\upvartheta =\upomega -\upomega \mathrm{log}\left(\upomega \right) $$

(17)

$$ {\mathrm{W}}_{\mathrm{i}}:\mathrm{n}=\upomega {\mathrm{k}}_0\left(\upomega \right){\left\{{\mathrm{K}}_0\left(\upomega \right)\right\}}^{\mathrm{i}-1}{\left\{1-{\mathrm{K}}_0\left(\upomega \right)\right\}}^{\mathrm{n}-\mathrm{i}}\mathrm{d}\upomega $$

(18)

Therefore, k_{0} is the relative density. This is the main approach of the K-plot (or Kendall-plot). Furthermore, we also refer to the studies by Dastgir et al. (2019a, b) and other performance measures from Saito (2019) and Eom et al. (2019) for recent literature incorporating new research methodologies of copula Causality.