1 Introduction

Over the past few decades, the directed influence flow between major financial markets, such as stock returns and exchange rate, stock returns and trading volume, has preoccupied the minds of policymakers, investors, and economists. These three variables are important in determining the development of a country. For policymakers, it is important to analyze the transmission channel between these markets to adopt proper policies to stimulate the real economy, especially during the crisis. For investors, the causal structure between these variables can be used to construct portfolio strategies. Therefore, objective to understanding the causal relations between stock returns, trading volumes, and exchange rates has become increasingly important for investors and policymakers.

Up to now, there are numerous studies about causal influences of those financial markets in the literature. In asset markets, although practitioners usually use the trading volume to analyze stock price, trading volume has long been playing second fiddle to the stock price in many theoretical studies. According to one viewpoint proposed by Fama that a market is weak-form efficient, past security prices can not be used to predict the future price changes and hence, technical analysis tools have no value (Fama, 1991). In contrast, technical analysts strongly believe that “It takes volume to make price move” (Karpoff, 1987). While in the other studies, several models have been developed to demonstrate that trading volume might act as an indicator of stock price owing to the dynamical (causal) relation between stock returns and trading volume (Blume et al., 1994; Hiemstra & Jones, 1994; Chordia & Swaminathan, 2000; Chuang et al., 2009a; Gebka & Wohar, 2013). Hence, the return-volume relation is more informative than the price-volume relation. For instance, Hiemstra and Jones (1994), Chordia and Swaminathan (2000), Chandrapala (2011) found daily returns of stocks with high trading volume lead daily returns of stocks with low trading volume, and concluded that trading volume plays a significant role in the dissemination of market-wide information. Another evidence to support this was provided by Chuang et al. (2009a) who used the quantile causal method to analyze the relation between daily stock return and trading volume data in the past seventeen years (Jan.2 or Jan.4, 1990–June 30, 2006) including NYSE, S&P 500 and FTSE 100, proving that trading volume has a positive effect on return volatility. Therefore, it is an interesting topic in finance to ascertain whether a market is week-form efficient.

In currency markets, as exchange rate plays a significant role in a country’s economic development, considerable attention has been paid to the causal relationship between exchange rates and stock returns, namely returns-exchange rates relation (Chow et al., 1997; Yang et al., 2014; Zhao, 2010; Chortareas et al., 2012; Liang et al., 2013). For foreign investors, their returns from market depend on not only the performance of the assets they invest but also the changes in exchange rates. There are two hypotheses of research in this sense. One hypothesis known as the ’goods market hypothesis’ demonstrates that changes in exchange rates affect the stock returns in the competitiveness of multinational firms. Conversely, the other hypothesis ’portfolio balance approach’ demonstrates that the fluctuations of stock returns can influence the movements of exchange rates. In general, it is found that the dynamic relationship between stock prices and foreign exchange rates is either unidirectional (Lin, 2012; Cuestas & Tang, 2015), or bidirectional (Granger et al., 2000; Pan et al., 2007; Rjoub, 2012). However, empirically, the causality of those variables still produces conflicting results (Lee & Rui, 2000; Aydemir & Demirhan, 2017).

As already mentioned above, why do most researches have produced conflicting results? There may be due to the indirect influences of other factors. In this paper, the great concern is to account for indirect causal relationships between stock returns, trading volume, and exchange rate, and eliminate spurious findings of causation to discover early warning signals of financial risk contagion. This concern is not only crucial for investment and risk management issues, but also the economic and financial stability.

Hence, this paper aims to propose a new test for possible causality that just detects the direct causal effects, and then use it to observe the various relationships between the stock and foreign exchange markets in Asia. To achieve this goal, a preliminary step is to develop a proper method to detect the causal relation among this three variables, or even more than three variables, to reconstruct the financial network.

Research over the last few years has shown that Granger causality (GC) is a key and popular technique to furnish this capability. The closed papers include studies of the relationship between exchange rates and stock returns, trading volume and stock returns based on bivariate Granger causality; see Chow et al. (1997), Yang et al. (2014), Zhao (2010), Chortareas et al. (2012), Liang et al. (2013), Lin (2012), Cuestas and Tang (2015), Granger et al. (2000, Pan et al. (2007), Rjoub (2012). None of above studies considers the integrity of prediction improvements using measures of the causality in quantiles.

On top of all this, the GC analysis, in general, is implemented in the conditional mean. However, in many econometric applications of interest, a tail area causal relation may be quite different from a causality based on the mean regression and the focus of a causality analysis on the mean might result in unclear news. For example, it may fail to detect a tail causal relation if the distributions of the variables involve fat tailed or nonlinear causal relationships. To overcome this barrier, an illustrating motivation comes from the well−known robustness properties of the conditional quantile and the set of conditional quantiles characterizes the entire distribution in more detail. Thus, the quantile regression may be a powerful alternative (Chuang et al., 2009a; Jeong et al., 2012; Lee & Yang, 2012, 2014). Yet another motivation comes from controlling and monitoring downside market risk and investigating large comovements between financial markets. Hong et al. (2009) point out that Granger causality in quantile analysis is important for risk management and portfolio diversification. Recently, a useful and flexible modeling strategy of implementing Granger causality based on parametric quantile models (GCQ) has been proposed at all quantile levels by Troster (2018).

Although GCQ can accurately identify the causal relation between a pair of time series, it fails to identify causal patterns for multivariate time series.

Generally speaking, the application of these GC methods might get in trouble, especially when the size of samples is small and the number of variables is large. To conquer such a problem, some composed methods are presented by integrating variable selection into GC method, such as conditional Granger causality (CGC). CGC is put forward to differentiate direct interactions from indirect ones (Lizier et al., 2011; Dhamala et al., 2008; Geweke, 1984). Yang et al. (2017) proposed a group lasso nonlinear conditional Granger causality (Glasso-NCGC) to handle nonlinearity and directionality of complex networked systems. Compared with CGC, Lasso-CGC and NCGC, all of the results in his paper demonstrate that the proposed method performs better in terms of higher are under precision-recall curve. Ding et al. (2006) and Barrett et al. (2010); Barnett and Seth (2014) define conditional Granger causality which has the ability to resolve whether the interaction between two time series is direct or is mediated by another time series. Dufour and Renault (1998) modified GC to detect hidden causal relations. Dufour and Taamouti (2010) make multi-horizon extensions of Geweke’s measures (Geweke 1984) in the context of a set of linear invertible process, and it allows for conditional causality with auxiliary variables. Meanwhile, some studies made interpretations on CGC analyses (Stokes & Purdon 2017, 2018; Barnett et al., 2018). For example, Stokes and Purdon (2017, 2018), pointed out their work was to characterize statistical properties of the traditional computation of CGC. They suggest analysts should pay more attention to underlying models, the dynamics they present, and the overall modeling process, all of which form the foundations for inferences on directed influences. Barnett et al. (2018) emphasized that CGC models statistical dependencies among observed responses and CGC derived from a single full regression. However, most of these CGC methods are confined to identify direct causal effect on mean.

In this paper, we propose a new method termed as conditional Granger causality test in quantiles (CGCQ), which is an extension of the GCQ for investigating direct causal effects between multivariate time series, as shown in the next section. It shows that CGCQ detects nonlinear causalities and tail causal relations, in which the causalities are all direct in multivariate variables case. To our knowledge, conditional Granger causality in quantiles by parametric methods in a flexible specification setting has not been introduced in the previous literature.

With the fast growth of Asian emerging economies and the increasing openness of the world economy, as further contributions, we employ our method to analyze the causal relationships between three Asian financial markets-China, Japan, and South Korea.Footnote 1 In the empirical section, to illustrate the applicability of our approach, we not only investigate the causality between stock returns, stock trading volume, and foreign exchange rate in the three domestic financial markets but also conduct a detailed investigation of the causality between stock returns for cross-country markets. Two main reasons can explain our motivation to propose CGCQ method to study the causal relations between foreign exchange and stock market in three Asian financial markets. For one thing, there are many existing indirect causal associations in the sense of pairwise causal analysis. For another, the results of preceding research clearly show that the dependence of stock returns on the changes of other asset classes is likely to vary across the conditional distribution of stock and other financial variables movements (Chuang et al., 2009a; Yang et al., 2014, among others).

The main marginal contributions of this paper lie in two ways. First, in multivariate cases, the GCQ test requires the correct specification of the parametric quantile auto-regressive model that not consider the information of other factors, it is suitable used in the cases of two variables but not multivariate variables. However, our test is based on marginal joint quantile regression (under the null of non-causality) that includes past measurements of the other factors, and then searches for rejections of the null hypothesis in multivariate variables. Second, using CGCQ, we examine the full causal relationships between stock returns, stock trading volume changes, and exchange rate changes in three Asian countries with domestic and cross-country markets. The results of the CGCQ test show that there is no causal relation from stock returns to stock trading volume changes in the South Korean market at high quantile. Besides, there is no evidence of Granger causality running from trading volume to exchange rates, and stock returns to exchange rates in China market at high quantile, but not for the low quantile. In contrast, the causal effects inferred by GC or GCQ method are different at the significance level of 5%, the mistaken identification of indirect influence as being direct(the results can been seen in Tables 6 and 8). In sum, the inclusion of additional variables is important both economically and statistically. We believe that the proposed approach is a useful statistical method to detect the real causal relation among multivariate economic variables.

The paper is organized as follows: Section 2 reviews related literature. In Sect. 3, we propose a test statistic based on the infinite number of unconditional moment restrictions for the null hypothesis of conditional Granger non-causality in parametric quantiles. We also derive the asymptotic distribution of the proposed test statistic under the null and the alternative hypothesis, and theoretically justify the validity of the sub-sampling approach. In Sect. 4, we present Monte Carlo simulation to assess the performance of our method for the case of finite sample. In Sect. 5, we apply our method to the empirical data observed from financial markets. In Sect. 6, we conclude the article.

2 Literature Review

This section briefly reviews the literature on the relationship between exchange rates and stock returns, stock returns and trading volume empirically in some countries. According to traditional theory, the change of exchange rate is influenced by international trade, and stock market returns can be predicted by financial and macroeconomic variables. The performance of domestic stock market over the few decades played important roles in economic development, indicating the significant impact of stock returns on exchanges rate movements. However, in empirical studies, the causal relationship between stock return, stock trading volume and exchange rate remains unresolved. Based on the causal relationship framework, the empirical results show some discrepancies among different markets.

The relationship between stock returns and trading volume has interested empirical financial economists for long. For example, Chuang et al. (2009b) found that the trading volume serves as a trigger to cause nonlinear dynamic cycle of stock returns in Taiwan, Hong Kong, Singapore, and Korea stock markets. Lin (2013) examined the dynamic stock return-volume relations for six emerging Asian markets in quantiles, and their finding is consistent with the theoretical models that trading volume contributes information to the distribution of stock return. Chuang et al. (2012) found the effect of volume on returns in only two out of ten Asian markets analyzed. Chuang et al. (2009a) examined the causal relations between stock return and trading volume based on quantile regressions. In the empirical studies, the results show that the causal effects from trading volume to stock return are usually heterogeneous across quantiles and those of return on volume are more stable. Chen (2012) reported for S&P 500 trading activity to affect subsequent returns only in bear markets but no volume-return causality at both market phases considered jointly. Gebka and Wohar (2013) further investigated the nature of the volume-return causality by using the quantile regression method to analyze the causality between past trading volume and stock returns in the Pacific Basin countries. However, Lee and Rui (2000) reported that trading volume does not predict the next day’s index returns on the Chinese A and B markets in Shanghai and Shenzhen. Lee and Rui (2002) find that trading volume does not Granger-cause stock market returns on each of three stock markets: New York, Tokyo, and London. Meanwhile, stock returns Granger-cause trading volume in the US and Japanese markets but not in the UK market. On the week form efficient, Chandrapala (2011) examined the relationship between trading volume and stock returns at the Colombo Stock Exchange(CSE) and found the trading volume change has predictive power on stock returns.

In the foreign exchange markets and stock markets, some studies have found evidence supporting the different connections between them (Kim, 2003; Aydemir & Demirhan, 2009; Chkili & Nguyen, 2014; Yang et al., 2014). Chkili and Nguyen (2014) suggest that exchange rate changes have no effects on stock returns, but stock returns have a significant impact on exchange rate changes in BRICS countries except for South Africa. As for the Asian markets, some other empirical studies are also available. For example, Pan et al. (2007) examined dynamic linkages between exchange rates and stock returns in seven East Asian countries, excluding China, and found that the relationship between stock and foreign exchange markets in Asian differs depending on countries and time. Granger et al. (2000) investigated the relationship between stock and foreign exchange markets of nine Asian countries. They found the causal effects of exchange rate on the stock returns in the Japanese stock market; the causal effect is bidirectional in South Korea. Yang et al. (2014) examined the dynamic linkages between stock prices and exchange rates of nine Asian markets over the period from 1997 to 2010 by Granger causality test in quantiles; their empirical results indicate that most stock and foreign exchange markets are negatively correlated, and they find there exist more bidirectional causal relations. Zhao (2010) studied the dynamic linkages between real effective exchange rates and stock returns in China with Vector AutoRegression (VAR) model and multivariate Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models. Chortareas et al. (2012) used the event-induced methodology to investigate the impact of float exchange rates on stock returns in three MENA countries. Their results show that the devaluation of currency has a positive effect on economic activity and stock returns. Besides, there exists a unidirectional causal effect from exchange rates to stock returns in China (Nieh & Yau, 2010). Liang et al. (2013) revisited the relationships between stock prices and exchange rates in ASEAN-5 economies using panel Granger causality and panel dynamic ordinary least squares (DOLS) methodologies. Their result supports the “stock-oriented” hypothesis of exchange rates proposed by Branson (1983) and Frankel et al. (1987).Footnote 2

To sum up, both theoretical models and empirical results mentioned above demonstrate that causal effects might spread from advanced financial markets to emerging markets (Chow et al., 2011; Coudert et al., 2011). The crisis happening in one country may also spread to its neighboring economies. Therefore, it is crucial to examine the causal relations not only for domestic stock returns, trading volume, and exchange rate but also for cross country stock markets. This paper uses a new approach to propose an explanation for the different empirical results of the relationship between stock and foreign exchange markets.

3 Econometric Modelling

3.1 Testing Conditional Granger Causality in Quantiles

We assume throughout the article that the time series process \((X_{t},Y_{t},Z_{t})_{t\in N}\) be a strictly stationary and ergodic time series process defined on the probability space \((\varOmega , {\mathcal {F}}, P)\). When we observe real-valued time-dependent variables \(X_{t}\),\(Y_{t}\),\(Z_{t}\) (such as stock returns, trading volume and exchange rates), we can construct a corresponding explanatory vector \({\mathbf {I}}_{t}=({\mathbf {I}}_{t}^{X},{\mathbf {I}}_{t}^{Y},{\mathbf {I}}_{t}^{Z})^{^{\prime }}\in {\mathbb {R}}^{d}\), \(d=s+q+h\), where \({\mathbf {I}}_{t}^{X}=(X_{t-1},\ldots ,X_{t-s})\in {\mathbb {R}}^{s}\), \({\mathbf {I}}_{t}^{Y}=(Y_{t-1},\ldots ,Y_{t-q})\in {\mathbb {R}}^{q}\) and \({\mathbf {I}}_{t}^{Z}=(Z_{t-1},\ldots ,Z_{t-h})\in {\mathbb {R}}^{h}\). We denote the conditional distribution functions of \(Y_{t}\) given \(\left( {\mathbf {I}}_{t}^{X},{\mathbf {I}}_{t}^{Y},{\mathbf {I}}_{t}^{Z}\right) \) and \(\left( {\mathbf {I}}_{t}^{Y},{\mathbf {I}}_{t}^{Z}\right) \) as \(F_{Y}(y|{\mathbf {I}}_{t}^{X},{\mathbf {I}}_{t}^{Y},{\mathbf {I}}_{t}^{Z})\) and \(F_{Y}(y|{\mathbf {I}}_{t}^{Y},{\mathbf {I}}_{t}^{Z})\) respectively, which are continuous for all \(y\in {\mathbb {R}}\). We denote T as the sample size throughout the manuscript.

According to Granger et al. (1969), Granger (1981, 1988), the null hypothesis of conditional Granger noncausality from \(X_{t}\) to \(Y_{t}\) as follows:

$$\begin{aligned} H_{0}^{X\nrightarrow Y|Z}:F_{Y}\left( y|{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{X},{\varvec{I}}_{t}^{Z}\right) =F_{Y}\left( y|{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{Z}\right) , \text {for all y }\in {\mathbb {R}}. \end{aligned}$$
(1)

Since the estimation of the conditional conditional distribution may be complicated in practice, and combined with historical achievements of predecessors, the conditional Granger noncausality in mean can defined as:

$$\begin{aligned} H_{0}^{X\nrightarrow Y|Z}:\mathrm {E}\left( Y_{t}|{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{X},{\varvec{I}}_{t}^{Z}\right) =\mathrm {E}\left( Y_{t}|{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{Z}\right) \text {a.s.,} \end{aligned}$$
(2)

where \(\mathrm {E}\left( Y_{t}|{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{X},{\varvec{I}}_{t}^{Z}\right) \) and \(\mathrm {E}\left( Y_{t}|{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{Z}\right) \) denote the mean of \(F_{Y}\left( \cdot |{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{X},{\varvec{I}}_{t}^{Z}\right) \) and \(F_{Y}\left( \cdot |{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{Z}\right) \), respectively.

Similarly, we can define noncausality in higher order moments, such as the noncausality in variance; see (Cheung & Ng,1996). The usual noncausality suffers a severe limitation since noncausality in mean (or in higher moments) cannot capture the dependence which may appear in the tails of the conditional distributions or other distribution characteristics. The quantile regression method provides a more detailed and flexible analysis of the whole distribution than the conditional mean−regression analysis. For example, Lee and Yang (2012) show that money−income Granger causality in mean is not significant for all data sets, but the money−income causality in quantile seems to be more significant in the tails. Previously, Troster (2018) proposed a method to test Granger noncausality in conditional quantiles. Therefore, in this study, we propose a conditional Granger causality test in quantiles as extension of the GCQ. The proposed approach can not only evaluate the linear or nonlinear direct causalities, but also can fully characterize the direct causality of the whole distribution when all quantiles are considered.

In order to better introduce the conditional Granger causality test, let \(Q_{\tau }^{Y|Z}\left( \cdot |{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{Z}\right) \) and \(Q_{\tau }^{Y,X|Z}\left( \cdot |{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{X},{\varvec{I}}_{t}^{Z}\right) \) denote the \(\tau \)-quantiles of \(F\left( \cdot |{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{Z}\right) \) and \(F\left( \cdot |{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{X},{\varvec{I}}_{t}^{Z}\right) \) respectively, \(\tau \in \varGamma \), \(\varGamma \subset [0,1]\). Equation (1) becomes equivalent to testing

$$\begin{aligned} H_{0}^{QC:X\nrightarrow Y|Z}:Q_{\tau }^{Y,X|Z}\left( Y_{t}|{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{X},{\varvec{I}}_{t}^{Z}\right) =Q_{\tau }^{Y|Z}\left( Y_{t}|{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{Z}\right) , \text {a.s. for all } \tau \in \varGamma , \end{aligned}$$
(3)

where \(\varGamma \) is a compact set such that \(\varGamma \subset \left[ 0,1\right] \) and the conditional \(\tau \)-quantiles of \(Y_{t}\) satisfy some restrictions as follows:

$$\begin{aligned}&\Pr \left\{ Y_{t}\le Q_{\tau }^{Y|Z}\left( Y_{t}|{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{Z}\right) |{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{Z}\right\} :=\tau , \text {a.s. for all } \tau \in \varGamma ,\nonumber \\&\Pr \left\{ Y_{t}\le Q_{\tau }^{Y,X|Z}\left( Y_{t}|{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{X},{\varvec{I}}_{t}^{Z}\right) |{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{X},{\varvec{I}}_{t}^{Z}\right\} :=\tau , \text {a.s. for all } \tau \in \varGamma . \end{aligned}$$
(4)

Given explanatory vector \({\mathbf {I}}_{t}\), \(\Pr \left\{ Y_{t}\le Q_{\tau }\left( Y_{t}|{\varvec{I}}_{t}\right) |{\varvec{I}}_{t}\right\} =\mathrm {E}\left\{ {\mathbb {I}}\left[ Y_{t}\le Q_{\tau }\left( Y_{t}|{\varvec{I}}_{t}\right) \right] |{\varvec{I}}_{t}\right\} ,\) holds, where \({\mathbb {I}}(a \le b)\) is an indicator function of the event that takes the value of unity when a is less or equal than b and zero otherwise. Then testing Eq. (3) becomes

$$\begin{aligned}&\mathrm {E}\left\{ {\mathbb {I}}\left[ Y_{t}\le Q_{\tau }^{Y,X|Z}\left( Y_{t}|{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{X},{\varvec{I}}_{t}^{Z}\right) \right] |{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{X},{\varvec{I}}_{t}^{Z}\right\} \nonumber \\&\quad =\mathrm {E}\left\{ {\mathbb {I}}\left[ Y_{t}\le Q_{\tau }^{Y|Z}\left( Y_{t}|{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{Z}\right) \right] |{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{X},{\varvec{I}}_{t}^{Z}\right\} , \text {a.s. for all } \tau \in \varGamma , \end{aligned}$$
(5)

where the left−hand side of Eq. (5) is equal to \(F_{Y}\left( \cdot |{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{X},{\varvec{I}}_{t}^{Z}\right) \). We postulate a parametric model to estimate the \(\tau \)-quantile of \(F_{Y}\left( \cdot |{\varvec{I}}_{t}\right) \). Let \({\mathcal {B}}\subset \mathcal {M=}\left\{ m(\cdot ,\theta (\tau ))|\theta (\cdot ):\tau \mapsto \theta (\tau ) \in \varTheta \subset {\mathbb {R}}^{p},\tau \in \varGamma \subset [0,1]\right\} \) be a family of uniformly bounded functions.

Then, under the null hypothesis in Eq. (3), the \(Q_{\tau }^{Y|Z}\left( \cdot |{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{Z}\right) \) is correctly specified by a parametric model \(m\left( {\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{Z},\theta _{0}\left( \tau \right) \right) \), for some \(\theta _{0}\in {\mathcal {B}}\). From the Eq. (5), the null and alternative hypotheses to be tested are

$$\begin{aligned}&H_{0}^{QC:X\nrightarrow Y|Z}:\mathrm {E}\left\{ {\mathbb {I}}\left[ Y_{t}\le m\left( {\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{Z},\theta _{0}\left( \tau \right) \right) \right] |{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{X},{\varvec{I}}_{t}^{Z}\right\} =\tau , \text {a.s. for all } \tau \in \varGamma , \end{aligned}$$
(6)
$$\begin{aligned}&H_{A}^{QC:X\nrightarrow Y|Z}:\mathrm {E}\left\{ {\mathbb {I}}\left[ Y_{t}\le m\left( {\varvec{I}}_{t}^{Y},I_{t}^{Z},\theta _{0}\left( \tau \right) \right) \right] |{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{X},{\varvec{I}}_{t}^{Z}\right\} \ne \tau , \text {for some } \tau \in \varGamma . \end{aligned}$$
(7)

Thus, we reject the null hypothesis that there is a directly causal influence from \(X_{t}\) to \(Y_{t}\) under \(\tau \) quantile.

3.2 Test Statistics

Our test is an application of the method proposed by Granger’s (1969, 1981), Gewek’s (1984) and Troster (2018) in the context of multivariate conditional dependencies and Granger causality in quantiles. The test statistic is a Cramér–von Mises(CvM)functional norm of quantile−market empirical processes.

In the family of functions \({\mathcal {M}}\), \(m\left( {\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{Z},\theta _{0}\left( \tau \right) \right) \) is the only element of \({\mathcal {M}}\) that correctly specifies the true conditional quantile \(Q_{\tau }^{Y|Z}\left( \cdot |{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{Z}\right) \), for all \(\tau \in \varGamma \). Thus, equation (6) can be rewritten to the following equation

$$\begin{aligned} H_{0}^{QC:X\nrightarrow Y|Z}:\mathrm {E}\left\{ \varPsi _{\tau ,t}\left( \theta _{0}\right) |{\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{X},{\varvec{I}}_{t}^{Z}\right\} =0, \text {a.s. for all } \tau \in \varGamma , \end{aligned}$$
(8)

where \(\varPsi _{\tau ,t}\left( \theta _{0}\right) \equiv \varPsi _{\tau }\left( Y_{t}-m\left( {\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{Z},\theta _{0}\left( \tau \right) \right) \right) \) is the check loss function.

Assume W is a compact set, that by appropriately choosing a parametric family of functions \(\left\{ g({\varvec{I}}_{t},{\varvec{w}}):{\varvec{w}}\in W\subset {\mathbb {R}}^{r}\right\} \). We further rewrite the hypotheses in term of unconditional moment restrictions to the so−called “density−weighted” version:

$$\begin{aligned} \mathrm {E}\left\{ \varPsi _{\tau ,t}\left( \theta _{0}\right) g\left( {\varvec{I}}_{t},{\varvec{w}}\right) \right\} =0, \text {for all } \tau \in \varGamma , \end{aligned}$$
(9)

where \(g\left( {\varvec{I}}_{t},{\varvec{w}}\right) \) is the weighting functions.

We use the parametric family of weighting function used in Troster (2018):

$$\begin{aligned} \mathrm {E}\left\{ \varPsi _{\tau ,t}\left( \theta _{0}\right) \exp \left( i{\varvec{w}}^{^{\mathrm {T}}}{\varvec{I}}_{t}\right) \right\} =0, \text {for all } {\varvec{w}}\in {\mathbb {R}}^{d} \text {and for all } \tau \in \varGamma , \end{aligned}$$
(10)

where \(r\le d\), \(i=\sqrt{-1}\) is the imaginary root.

Our test statistics is based on the sample analog of the Eq. (10)

$$\begin{aligned} v_{T}\left( {\varvec{w}},\tau \right) :=\frac{1}{\sqrt{T}}\sum \limits _{t=1}^{T}\varPsi _{\tau ,t}\left( \theta _{T}\right) \exp \left( i{\varvec{w}}^{^{\mathrm {T}}}{\varvec{I}}_{t}\right) ,\tau \in \varGamma , \end{aligned}$$
(11)

where \(\theta _{T}(\tau )\) is a \(\sqrt{\mathrm {T}}\) consistent estimator of \(\theta _{0}(\tau )\).

Given our sample \(\left\{ \left( Y_{t},X_{t},Z_{t}\right) \right\} _{t\in {\mathbb {Z}}}\), our proposed test statistic \(CvM_{T}\) in the framework defined as

$$\begin{aligned} CvM_{T}=\iint \left| v_{T}\left( {\varvec{w}},\tau \right) \right| ^{2}\mathrm {d}F_{{\varvec{w}}}\left( {\varvec{w}}\right) \mathrm {d}F_{\tau }\left( \tau \right) ,{\varvec{w}}\in {\mathbb {R}} ^{d},\tau \in \varGamma , \end{aligned}$$
(12)

where \(F_{{\varvec{w}}}(\cdot )\) is the CDF of a d-variate standard normal random vector. This choice not only satisfies the requirements of the weighting function (Kuan & Lee, 2004), but also simplifies the computations of \(CvM_{T}\). Hong (1999) shows that the power performance for similar tests to \(CvM_{T}\) is not too sensitive to the choice of \(F_{{\varvec{w}}}(\cdot )\). \(F_{\tau }(\cdot )\) a uniform discrete distribution over a grid of \(\varGamma \) in n equidistributed points, \(\varGamma _{n}=\left\{ \tau _{j}\right\} _{j=1}^{n}\). Then, our test statistic \(CvM_{T}\) has the form

$$\begin{aligned} Cvm_{T}=\frac{1}{Tn}\sum \limits _{t=1}^{T}\left| \varphi _{\cdot j}^{^{\prime }}{\varvec{W}}\varphi _{\cdot j}\right| . \end{aligned}$$
(13)

where \({\varvec{\varPsi }}\) is the \(T\times n\) matrix with elements \(\varphi _{i,j}=\varPsi _{\tau _{j}}\left[ Y_{i}-m\left( {\varvec{I}}_{i}^{Y},{\varvec{I}}_{i}^{Z},\theta _{T}\left( \tau \right) \right) \right] \), and \({\varvec{W}}\) is the \(T\times T\) matrix with elements \(w_{t,s}=\exp \left[ -0.5\left( {\varvec{I}}_{t}-{\varvec{I}}_{s}\right) \right] \); see details in Kuan and Lee (2004) and Escanciano and Velasco (2006).

Therefore, the computation of \(CvM_{T}\) becomes straightforward. Under the similar assumptions in Troster’s asymptotic theory (Troster, 2018), the test statistic \(CvM_{T}\) weakly converges to zero under the null hypothesis (8), and to a probability limit different than zero under the alternative.

The intuitive measuring of this definition is quite clear. When the causal influence from \(Y_{t}\) to \(X_{t}\) is entirely mediated by \(Z_{t}\), \({\mu _{i}^{*}(\tau )}\) is uniformly zero. Thus, we have \(E\left[ \varPsi _{\tau }(Y_{t}-m( {\mathbf {I}}_{t}^{Y},{\varvec{I}}_{t}^{Z},\theta _{0}(\tau )))g({\mathbf {I}}_{t},{\mathbf {w}})\right] =0\), measuring that no further improvement in the prediction of \(Y_{t}\) can be expected by including past measurement of \(X_{t}\). On the other hand, when there is still a direct component from \(X_{t}\) to \(Y_{t}\), the inclusion of past measurements of \(X_{t}\) in addition to that of \(Y_{t}\) and \(Z_{t}\) results in better predictions of \(Y_{t}\), leading \(E\left[ \varPsi _{\tau }(Y_{t}-m({\mathbf {I}}_{t}^{Y},{\mathbf {I}}_{t}^{Z},\theta _{0}(\tau )))g({\mathbf {I}}_{t},{\mathbf {w}})\right] \ne 0\).

Remark In Escanciano and Velasco (2010), their theory allows for \(n\rightarrow \infty \) as \(T\rightarrow \infty \) and \(\left\{ \tau _{j}\right\} _{j=1}^{n}\) are generated independently from a given distribution on \(\varGamma \). Yet in practice, for the ease of computation, they simply considered n as a finite number and \(\left\{ \tau _{j}\right\} _{j=1}^{n}\) generated deterministically. We will adopt such strategy throughout the article.

3.3 Asymptotic Theory

We consider the process \(v_{T}\left( {\varvec{w}},\tau \right) \) of Eq. (11) as the values in \({\mathcal {L}}^{\infty }\left( W\times \varGamma \right) \). That is the set of all complex−valued uniformly bounded functions defined with the supremum metric, say \(d_{\infty }\), and \({\mathcal {B}}_{d_{\infty }}\) is its Borel \(\sigma -\)algebra. In the whole paper, the parameter \(\theta _{0}\) takes values from the family \({\mathcal {B}}\), and \(\left\| \theta \right\| _{{\mathcal {B}}}=\sup _{\tau \in \varGamma }\left| \theta (\tau )\right| \). \({\mathcal {F}}_{t}=\sigma \left( {\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{Z},{\varvec{I}}_{t-1}^{Y},{\varvec{I}}_{t-1}^{Z},\cdots \right) \) be the \(\sigma -\)filed generated up to time t. \(N_{[\cdot ]}(\delta ,\varrho ,\left\| \cdot \right\| )\) be the \(\delta -\)bracketing number of a class of function \(\varrho \) with respect to a norm \(\left\| \cdot \right\| \), see Vaart and Wellner (1997) and Escanciano and Velasco (2006) for more details. For each \(t\in {\mathbb {X}}\), as \(\varepsilon _{t}\left( \tau \right) :=Y_{t}-Q_{t}\left( {\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{Z}\right) \) and the parametric quantile error as \(e_{t}\left( \theta \left( \tau \right) \right) :=Y_{t}-m\left( {\varvec{I}}_{t}^{Y},{\varvec{I}}_{t}^{Z},\theta \left( \tau \right) \right) \). In addition, \(f_{Z}\) denotes the density function of a conditional distribution function \(F_{Z}\). All limits are taken as \(T\rightarrow \infty \), where T is the sample size. Based on the relevant assumptions in Troster (2018), we deploy the following main corollaries to analyze the asymptotic behavior of our test statistic.

Corollary 1

Under the similar Assumptions A1–A5 in Troster (2018), extending other variables (such as the Z), we have the following conclusion,

  1. (1)

    Under the null hypothesis \(H_{0}^{QC:X\nrightarrow Y|Z}\) of Eq. (6),

    $$\begin{aligned} CvM_{T}\overset{d}{\longrightarrow }\iint \limits _{\varGamma \times {\mathcal {W}}}\left| S_{\infty }\left( {\mathbf {w}}, \tau \right) \right| ^{2}dF_{{\mathbf {w}}}({\mathbf {w}})dF_{\tau }(\tau ), \end{aligned}$$

    where \(S_{\infty }\left( {\mathbf {w}}, \tau \right) \) is a tight Gaussian process with zero-mean and covariance function:

    $$\begin{aligned} R_{S_{\infty }}\left( \tau _{1},\tau _{2}\right) =\lim _{T\rightarrow \infty }\frac{1}{T}\sum _{t=1}^{T}\sum _{s=1}^{T}E[\ell _{\tau _{1}}(Y_{t},{\mathbf {I}} _{t}^{Y,Z},\theta _{0}(\tau _{1})) \times \ell _{\tau _{2}}(Y_{s},{\mathbf {I}}_{s}^{Y,Z},\theta _{0}(\tau _{2}))], \end{aligned}$$

    where \(E[\ell _{\tau }(Y_{t},{\mathbf {I}}_{t}^{Y,Z},\theta _{0}(\tau ))\varPsi _{\tau , s}(\theta _{0})]=0\) if \(t\ne s,\) \(\ell _{\tau }(Y_{t},{\mathbf {I}}_{t}^{Y,Z},\theta _{0}(\tau ))\) is a process in \(\ell ^{\infty }(\varGamma ).\) Where \(S_{\infty }({\mathbf {w}},\tau ) \) is a tight mean zero Gaussian process.

  2. (2)

    Under the alternative hypothesis of Eq. (7), there exists \(\varepsilon >0\), such that

    $$\begin{aligned} \lim _{T\rightarrow \infty }\Pr \left( CvM_{T}>\varepsilon \right) =1. \end{aligned}$$

By Corollary 1, we reject the null hypothesis \(H_{0}^{QC:X\nrightarrow Y|Z}\) whenever \(CvM_{T}\) is significantly positive. In the rest of the subsection, we study the consistency of \(CvM_{T}\) and the asymptotic distribution of \(CvM_{T}\) against a sequence of Pitman’s local alternatives converging to the null hypothesis at the rate \(T^{-1/2}\). Under a sequence of local alternative \(H_{A,T}^{X\nrightarrow Y|Z}\), we have

$$\begin{aligned} H_{A,T}^{X\nrightarrow Y|Z}:E\left[ \varPsi _{\tau }\left( e_{t}\left( \theta _{0}\left( \tau \right) \right) \right) |{\mathcal {F}}_{t}\right] =\varGamma _{\tau }/\sqrt{T}, \text { ~~for~all~}\tau \in \varGamma , \text { for some }\theta _{0}\in {\mathcal {B}}, \end{aligned}$$
(14)

where the function \(\varGamma _{\tau }:{\mathbb {R}}^{d}\rightarrow {\mathbb {R}}\) satisfies the Assumption A6 in Troster (2018).

According to Pitman’s local alternatives, Corollary 2 demonstrates that the test statistics based on \(CvM_{T}\) is asymptotically strictly unbiased against the local alternative \(H_{A,T}^{X\nrightarrow Y|Z}\). Therefore, Corollary 2 follows from Theorem 4 of Escanciano and Velasco (2010) and Theorem 2 of Troster (2018). There are the following conclusions.

Corollary 2

Under the local alternatives \(H_{A,T}^{X\nrightarrow Y|Z}\), according to the Assumptions A1–A3, A6 and A4’ in Troster (2018), and extending other variables (such as Z), we have

$$\begin{aligned} CvM_{T}\overset{d}{\longrightarrow }\iint \limits _{\varGamma \times {\mathcal {W}}}\left| S_{\infty }\left( {\mathbf {w}},\tau \right) +\varDelta \left( {\mathbf {w}},\tau \right) \right| ^{2}dF_{{\mathbf {w}}}({\mathbf {w}} )dF_{\tau }(\tau ), \end{aligned}$$

where \(\varDelta \left( {\mathbf {w}},\tau \right) \) is non-trivial shift function, and the conditions see Theorem 4 of Escanciano and Velasco (2010).

3.4 Subsampling Approximation and Validity

As the proposed test statistic \(CvM_{T}\) is asymptotically nonpivotal and depends in a complex way on the data generation process (DGPs) under the null distribution, the critical values for the test statistics cannot be tabulated for general cases. Therefore, how to calculate the critical values for the test statistic is important in such a framework. In this section, we consider the subsampling method to approximate the critical values \(c_{T,b}(1-\alpha )\). Since the asymptotic properties of the subsampling tests depend on the choice of the subsample size b, in our framework, we follow the approach of Troster (2018) and choose a subsample of size \(b=\left[ kT^{2/5}\right] \), for different values of k, where \(\left[ \cdot \right] \) denotes the integer part of a number, while k and T are a constant parameter and the size of sample, respectively.

In this article, the algorithm for computing a subsampling realization of the test statistic \(CvM_{T}\) is conducted in two steps; see (Escanciano and Velasco 2010; Troster 2018) for more details. The asymptotic validity of the sampling critical values has the following conclusions.

Corollary 3

Under the similar Assumptions A1–A7 in Troster (2018), extending other variables (such as the Z), \(b/T\rightarrow 0\), \(b\rightarrow \infty \) as \(T\rightarrow \infty \), we have the following results.

  1. (1)

    Under the null hypothesis \(H_{0}^{X\nrightarrow Y|Z}, \,\, \lim _{T\rightarrow \infty }\Pr (CvM_{T}>c_{T,b}(1-\alpha ))=\alpha ,\)

  2. (2)

    Under any fixed alternative hypothesis \(H_{A}^{X\nrightarrow Y|Z},\,\,\lim _{T\rightarrow \infty }\Pr (CvM_{T}>c_{T,b}(1-\alpha ))=1,\)

  3. (3)

    Under the local alternative \(H_{A,T}^{X\nrightarrow Y|Z}\,\,\lim _{T\rightarrow \infty }\Pr (CvM_{T}>c_{T,b}(1-\alpha ))>\alpha ,\)

Corollary 3 implies that the proposed tests based on the subsampling critical value have a correct asymptotic level, are consistent and are able to detect alternatives tending to the null at the parametric rate \(\frac{1}{\sqrt{T}}\).

4 Numerical Studies

In this section we report the results of a Monte Carlo study to investigate the performance of our proposed test statistic. Throughout this section, we consider three stochastic processes \(X_{t},Y_{t}\) and \(Z_{t}\). The data are generated from two cases (linear and nonlinear DGPs) and the empirical rejection frequencies for \(5\%\) nominal level tests for different sample sizes T, subsample sizes b, conditional quantile parametric models \(m(\cdot ,\cdot )\), coupling strength parameter c and an equally spaced grid of 20 quantiles on the interval \(\varGamma =\left[ 0.10,0.90\right] \). In each experiment the number of iterations is fixed to 1000.

4.1 Monte Carlo Experiments

To validate our method, we first consider the DGP1 as presented in Ding et al. (2006) that the causal influence from \(Y_{t}\) to \(X_{t}\) is indirect and completely mediated by \(Z_{t}\) when the coupling strength \(c=0\) (we will change the value of c for the power analysis later).

$$\begin{aligned} DGP1: \left\{ \begin{array}{ll} X_{t}=0.8X_{t-1}+0.4Z_{t-1}+cY_{t-1}+\varepsilon _{t}, &{} \\ Y_{t}=0.9Y_{t-1}+\xi _{t}, &{} \\ Z_{t}=0.5Z_{t-1}+0.5Y_{t-1}+\eta _{t}.&{} \end{array} \right. \end{aligned}$$
(15)

Second, we consider the following nonlinear DGP2 to illustrate the effectiveness of our proposed method in identifying the nonlinear causality:

$$\begin{aligned} DGP2: \left\{ \begin{array}{ll} X_{t}=0.5X_{t-1}+0.4Z_{t-1}+\varepsilon _{t}, &{} \\ Y_{t}=0.9Y_{t-1}-0.3\exp (Z_{t-1})+\xi _{t}, &{} \\ Z_{t}=0.5Z_{t-1}+0.5Y_{t-1}+\eta _{t}. &{} \end{array} \right. \end{aligned}$$
(16)

where \(\varepsilon _{t},\xi _{t},\eta _{t}\) are independent Gaussian white noise processes with zero means and variances \(\sigma _{1}^{2}=0.3,\sigma _{2}^{2}=1,\sigma _{3}^{2}=0.2\). We note that the time unit here is arbitrary and has no financial meaning. DGP1-DGP2 are common linear and nonlinear models used in the time series literature.

For DGP1 above, we have under the null hypothesis \(c=0.00\), where the coefficient c captures the degree of causality from past \(Y_{t}\) to future \(X_{t}\), and hence a higher absolute value of c implies a stronger Granger causality. For DGP2, there exists a nonlinear from \(Z_{t}\) to \(Y_{t}\). All the coefficients are set so that the corresponding time series is stationary. In addition, we consider the sample sizes T from 100 to 1000 with an increment of 100. Subsample size \(b=\left[ kT^{2/5}\right] \) with \(k={3,4,5}\).

We set the estimate \(m(\cdot )\) (namely, QJR(p,h), the parametric quantile joint regressive specifications model) as followsFootnote 3:

$$\begin{aligned} {\widetilde{m}}\left( {\varvec{I}}_{t}^{X},{\varvec{I}}_{t}^{Z},\theta _{T}\left( \tau \right) \right) =\mu _{0}\left( \tau \right) +\sum \limits _{i=1}^{p}\mu _{i}\left( \tau \right) X_{t-i}+\sum \limits _{j=1}^{h}\mu _{j}^{*}\left( \tau \right) Z_{t-j}+\sigma _{t}\varPhi _{\varepsilon }^{-1}\left( \tau \right) , \end{aligned}$$
(17)

where \(\theta _{T}\left( \tau \right) =\left( \mu _{0}\left( \tau \right) ,\mu _{1}\left( \tau \right) ,\cdots ,\mu _{p}\left( \tau \right) ,\mu _{1}^{*}\left( \tau \right) ,\cdots ,\mu _{h}^{*}\left( \tau \right) ,\sigma _{t}\right) ^{^{\mathrm {T}}}\), \(\varPhi _{\varepsilon }^{-1}\left( \tau \right) \) is the \(\tau \)-quantile of the standard standard Gaussian error distribution.

When \(c=0.00\), there is no direct Granger causality between \(X_{t}\) and \(Y_{t}\); when \(c\ne 0.00\), there is a direct Granger causality from \(Y_{t}\) to \(X_{t}\).

4.2 Simulation Results

As a competitor to our test, we employ three other existing causal detection methods such as GC, GCQ, conditional GC (cGC)Footnote 4. The results of the linear case (DGP1) using the CGCQ, GCQ, GC and cGC methods at \(c=0.00\) are summarized in Tables 1 and 2. The numbers in parentheses in each direction in the first column represent the coupling strength corresponding to the causal relationship between the two variables. Each entry shows empirical rejection rate of the null hypothesis of non-Granger causality with different methods.

As noted in Troster (2018), the simulations show that the results of GCQ are insensitive to the choice of parameter k. We consider the parameter \(k=4\) of CGCQ in Tables 1 and 2 to save space, and the parametric model \(m(\cdot )\) of CGCQ is QJR(1,1) and QJR(2,2), while GCQ is QAR(1) and QAR(2), respectively. As is seen clearly, first, the obtained empirical rejection frequency using CGCQ test was mostly around 0.05, and with the increase of sample size T, the \(CvM_{T}\) has correct asymptotic size. Besides, the power is sensitive to the choice of T, that is, the larger T is, for the same coupling strength, the larger the power is. From a technical point of view, this makes sense, because the more data we have, the more evidence we can draw from to detect the “causality” effect. Secondly, the results of GCQ and GC in Tables 1 and 2 show an apparent unidirectional causal driving of \(X_{t}\) by \(Y_{t}\) that is in fact due to the indirect influence through \(Z_{t}\). This mistaken identification of an indirect influences as being direct one suggests the need for the conditional Granger causality measure, and it also implies that GCQ and GC cannot effectively distinguish from indirect and direct causal patterns between multi-variables. Although there is no evidence to support the causal relationship running from \(Y_{t}\) to \(X_{t}\) using CGCQ and cGC methods, the cGC test is that the critical values do not have the correct nominal size, as the empirical sizes are always smaller than the \(5\%\) nominal level of the test. This is a drawback of the cGC test; see the bold parts in Tables 1 and 2. Finally, the remain causalities are all detected by cGC and CGCQ tests, and it noted that the empirical rejection frequency of the causal influence from \(X_{t}\) to \(Y_{t}\) increases with the increase of T using GCQ method, which is not consistent with the asymptotical theory. The GC method cannot detect the direct causality between \(X_{t}\) and \(Z_{t}\).

The DGP1 case, although simple, thus plainly demonstrates that the pairwise measure of Granger causality by itself may be insufficient to reveal true relations. However, we have seen that conditional Granger causality analysis, including cGC and CGCQ measures, eliminated indirect causal influences that inadvertently resulted from application of the pairwise Granger causality measure. Knowing the system equations in DGP1 case allowed us to verify that the CGCQ measure yielded a true depiction of the system relations. Moreover, the test statistic of CGCQ has correct asymptotic size, is consistent against fixed alternatives, and has power against Pitman deviations from the null hypothesis.

Table 1 Empirical rejection frequencies of CGCQ, GCQ, GC, and cGC using QJR(1,1), QAR(1), and lag order 1 when \(c=0.00\) at nominal size 5% in DGPs1
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Table 2 Empirical rejection frequencies of CGCQ, GCQ, GC, and cGC using QJR(2,2), QAR(2), and lag order 1 when \(c=0.00\) at nominal size 5% in DGPs1
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We use different values of c to investigate the power of the test, such that the higher c is, the stronger the causality of \(Y_{t}\) on \(X_{t}\) is. Table 3 displays the results at \(T=100\) and 500 with different subsample size \(k=3,4,5\). As discussed before, the higher c is, the stronger the causality of \(Y_{t}\) on \(X_{t}\) is, which is confirmed by the larger and larger power values. In addition, the proposed method has the correct asymptotic power with the increase of T.

Table 3 For DGP1, empirical rejection frequencies for \(5\%\) subsampling CvM test–QJR(1,1) and QJR(2,2) model (17) with subsample sizes of \(b=\left[ kT^{\frac{2}{5}}\right] (k=3,4,5)\)
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The results for DGP2 are summarized in Table 4. In the first, there is no causality running from \(Y_{t}\) to \(X_{t}\). As for GCQ and GC, the results in the table indicate that there exist direct causal effects from \(Y_{t}\) to \(X_{t}\), but it is contradictory to the reality. On the contrary, the numbers in the column of CGCQ are all close to the size (5\(\%\)), and the column of cGC is hardly interpretable.

In the secondly, there exists a direct causality running from \(Z_{t}\) to \(Y_{t}\); as for GC, the figures in the table are hardly interpretable. It may reflect even sloppier size or it may be reflecting the weak power. In contrast, the power in the column of GCQ,CGCQ, and cGC grows when the sample size T gets large.

For the DGP2 examined, even using a small subsample size, the proposed \(CVM_{T}\) presents robust and reliable inference.

In sum, the numerical results demonstrate that the proposed test performs quite well for finite samples. It not only detects linear causal relations but also discovers the nonlinear causalities. Moreover, the new CGCQ method can capture more precise network connectivity patterns for multivariate time series than the other three GC methods.

5 Empirical Application

In this empirical study, we investigate the causality between stock returns, trading volume, and the foreign exchange rates in three domestic financial markets, as well as the causality between stock returns in cross-country markets. The data used in this study are daily stock market indexes, stock trading volume and nominal change rates for three Asia countries, including China, South Korea and Japan. The three stock indexes consist of Chinese Shanghai A−share Index (SHAI), Korea Composite Stock Price Index (KOSPI) and Japan’s Nikkei 225 Index (NIKKEI). The daily data from the beginning of January 4, 2002 to December 30, 2016 are taken from Wind Database, and there are 3635, 3520 and 3452 observations in each market, respectively. All exchange rates are expressed as the number of local currencies against US dollar. Similarly, the changes of foreign exchange rates are calculated as \(ex_{t}=\ln (P_{E,t})-\ln (P_{E,t-1})\), where \(P_{E,t}\) is the exchange rates indexes.

Returns are calculated as \(r_{t}=\ln (P_{r,t})-\ln (P_{r,t-1})\), where \(P_{r,t}\) is the index at time t. The variation in the trading volume is \(v_{t}=\ln (V_{r,t})-\ln (V_{r,t-1})\), where \(V_{t}\) is the closed trading volume at time t.

Fig. 1
figure 1

The series of log Trading volume, log Stock prices, log Exchange rates and volume changes, stock returns, exchange rate changes in China, Japan, South Korea, respectively

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Figure 1a, b display the daily log and log−difference series. The graphs of the log series indicate that the series are non-stationary and follow a common pattern. The differenced logarithmic data are stationary. Their summary statistics are collected in Table 5.

Table 4 For DGP2, empirical rejection frequencies for \(5\%\) subsampling CvM test–QJR(1,1) model of (17) with subsample sizes of \(b=\left[ kT^{\frac{2}{5}}\right] (k=3,4,5)\), lag order is 1
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It can be seen that the average daily returns of the stock market are much higher than the exchange rate change over the study period in three countries. Furthermore, the changes in the trading volume exhibit negative returns (0.02%) on a daily basis in South Korea. The unconditional volatility of the stock returns, measured by standard deviations, is more than twice the volatility of the exchange rate in Japanese stock market, and 15 times in Chinese stock market. However, the volatility of exchange rate is 27 times more than that of the stock returns in South Korea. The distributions of Stock returns and trading volume change are skew to the left for all countries, while exchange rate change is skew to the left for the China and South Korea and to the right for Japan. Also, all the market returns for the three countries exhibit excess that has fat tails. Skewness and kurtosis coefficients indicate that all series deviate from normally distributed. This departure from normality is formally confirmed by the JB statistics that rejects normality at the 1% level for all market time series.

To apply our proposed method, we estimate QJR(3) model as in Eq. (17) for each dependent variable on the \(CvM_{T}\) test with three different subsample sizes \(k=3,4,5\). Tables 5, 6, 7 and 8 report the subsampling p-values of CGCQ, GCQ, and p-values of GC.

Table 5 Summary statistics for stock returns \(r_t\), volume change \(v_t\) and exchange-rate change \(ex_t\)
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Table 6 Test of causal relationship among stock returns, trading volumes, and exchange rates in China, using CGCQ, GCQ, and GC, respectively: p-values
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If we take into account all the quantiles, the results of China suggest that trading volume change Granger-cause stock returns, and vice-versa, at the 5% significance level; the others are bi-directional Granger-causality at the 1% significance level. However, if we perform a high quantile-regression at \(\tau =0.9\), there is no evidence that the stock returns Granger-cause the exchange rate change at \(10\%\) significance level, and the change in trading volume does not cause large changes in the exchange rate at the 5% significance level. From the results of causality between trading volume change and stock returns, we find the quantile causal effects of trading volume change on stock returns are more significant at lower quantiles. These are somewhat similar to the results in Chuang et al. (2009a). Meanwhile, the two-way quantile causal relations between trading volume change and stock returns for China indicate that the volume and stock returns can help predict each other. Exchange rate changes exhibit correlation with stock returns, and it may be stronger in the tails than in the center. However, stock return does not exert a direct causal influence on the extreme high changes of the exchange rate. One might expect that the change of the nominal exchange rate has a significant impact on the Chinese stock market prices as the inflow in portfolio investment from foreign investors plunges.

In Table 7, the results demonstrate that there exists a bi-directional Granger-causality between the stock returns and foreign exchange at significance level of 1% when all quantiles are considered in Japanese market, and we often cannot reject the hypothesis that the change in trading volume and stock returns does not Granger-cause the change in exchange rates at the significance level of 10% when \(\tau =0.5\). Therefore, the observations imply that stock market change reflects extreme variation in exchange rates, stock returns Granger-cause trading volume change, and trading volume change can help predict returns in the Japanese markets.

Table 7 Test of causal relationship among stock returns, trading volumes, and exchange rates in Japan, using CGCQ, GCQ, and GC, respectively: p values
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The empirical results for South Korea (Table 8) indicate stock returns couldn’t lead the trading volume changes at high quantile(\(\tau =0.9\)). It means stock returns carry important information that is not contained in the extreme variation of the trading volume. In other words, there is no direct effect of stock returns on the high trading volume change, but there may be indirect effects. It shows that an increase in stock returns does not induce investors to big deals, and further shows that investors treat investment more rationally in South Korea. The causal relation between the stock returns and exchange rate is more pronounced when exchange rates are extremely high or low. This is in line with Tsai (2012) and Pan et al. (2007).

Table 8 Test of causal relationship among stock returns, trading volumes, and exchange rates in South Korea, using CGCQ, GCQ, and GC, respectively: p-values
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Table 9 Test of causal relationship among stock returns across country, using CGCQ, GCQ, and GC, respectively: p values
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To analyze the spillover effect of stock returns among different countries, we have examined empirical Granger-causality between stock returns for cross-country markets. Table 9 shows that there are significant causalities among the three Asian stock markets on the extreme tails of \(\tau =0.1\) and \(\tau =0.9\) at the 1% significance level. However, if we consider only \(\tau =0.5\), there is no evidence that the stock returns of the Japanese and the Korean markets Granger-cause the stock returns of the Chinese market at the 5% significance level. In sum, for the stock market of China, Japan, and Korea, the quantile causalities are active on the tails of the distribution of stock returns. These results are consistent with the conclusion in previous studies (Lee & Rui, 2002; Pan et al., 2007; Chuang et al., 2009a; Nieh & Yau, 2010; Chandrapala 2011; Tsai, 2012; Lee & Zhao, 2014).

In order to compare our approach with a standard test of Granger-causality, we perform Granger causality in mean between stock returns, trading volume change, and exchange rate change. We test the null hypothesis of non-causality using the Granger causality in the conditional mean with three different lag specifications(See the column of GC in Tables 5, 6, 6 and 8). The results show, in China, that no evidence supports the causality running from stock returns to exchange rates change, and causality running from exchange rates change to trading volume change, and vice-versa, at 1% significance level, with different lags. However, stock returns Granger-cause in mean trading volume changes at the 1% significance level. In Japan, there is no Granger-causality in mean from exchange rate change to trading volume change. Besides, we find no Granger-causality from variations in trading volume to stock returns. In addition, the trading volume changes to stock returns and to exchange rate changes are mostly no Granger-causality at 5% significance level in South Korea.

In the cross-country study, when we are using a pairwise GC analysis, the empirical evidence shows that there is a bidirectional causality between Korean stock markets and Japanese stock markets, while no evidence is found to support the causal relations between Korean stock markets and Chinese stock markets. Besides, the stock market of Japan and China does not exist causal correlation with each other. In sum, there is no evidence of a strong pattern of Granger-causality between the series using a conditional mean analysis.

In contrast to the GC analysis in mean and unreported simulations of Granger causality in quantiles with copula approach(\(GCQ-Copula\), Lee & Yang, 2014). Our proposed test detects rich causality among stock returns, trading volume change, and exchange rates change in each domestic market and causality between stock returns in cross-country. Therefore, the empirical results show that our causality measures present a more informative analysis of Granger causality than the conditional mean regression analysis. Our method can characterize the entire distribution in more detail, whereas the conditional mean method focuses only on a single part of the conditional distribution.

6 Conclusion

Many prominent financial and policy decisions can be made with the help of the Granger-causality method since GC can extract the causal relationship between multivariate economic time series. By extending the Troster (2018) idea to multivariate time series, we propose a consistent parametric test of conditional Granger-causality in quantiles. Different from pairwise Granger-causality in all conditional quantiles or conditional mean, our proposed test can investigate direct causal relations between two series excluding the influence by other time series. Its benefit is illustrated in three (China, Japan, and South Korea) domestic financial market applications where the causal relationships, among the exchange rates change, trading volume change and stock return, were shown to be different between a tail area and in the center of the distribution. In our study, as we find the trading volume has predictive power on stock returns, investors can make trading volume based strategies to make profits and theoretically this provides evidence of weak form inefficiency of these three markets.

We also illustrate that stock returns in cross-country markets have a significant bidirectional causality, especially on two sides of the stock returns distribution. These causal relationships suggest that contagion exists between the three Asian stock markets, and opportunities for international portfolio diversification in Asian stock markets still exist.

Furthermore, our results have broader theoretical implications. We find both the traditional approach and the portfolio approach exist at the same time in the Japanese and Korean markets. However, the empirical results show that the traditional approach is more prominent than the portfolio approach in Chinese markets. On the other hand, although the finding in China shows a unidirectional causal relationship running from exchange rate changes to stock returns at extremely high(0.9th quantile), this does not completely follow using the portfolio approach in the literature either. We believe that conditional Granger causality in quantiles offers a new way of looking at cooperative economics and finance, and it improves our ability to identify principal financial structures underlying the dynamics of economics and finance.

We close the paper with discussing the possible avenues of future work. Rejecting non-causality hypotheses in statistical tests implies that certain variables can help in forecasting others. Of course, statistical significance depends on the data and test power, and the outcomes of such tests do not represent the magnitude of causality. Therefore, a possible direction for future work is to extend this method by combining the strength of causality (Dufour & Taamouti, 2010; Zhang et al., 2016), thus capable of yielding a much more informative analysis of Granger causality than tests of non-causality. Another worthwhile extension would be to examine causality among volatility of stock price, exchange rates, and trading volume using the concept of second-order causality (Dufour & Zhang, 2015).