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Granger Causality in Expectiles: an M-vine copula test
{\bf Abstract.} {A model-free measure of Granger causality in expectiles is proposed, generalizing the traditional mean-based measure to arbitrary positions of the conditional distribution. Expectiles are the only law-invariant risk measures that are both coherent and elicitable, making them particularly well-suited for studying distributional Granger causality where risk quantification and forecast evaluation are both relevant. Based on this measure, a test is developed using M-vine copula models that accounts for multivariate Granger causality with $d+1$ series under non-linear and non-Gaussian dependence, without imposing parametric assumptions on the joint distribution. Strong consistency of the test statistic is established under some regularity conditions. In finite samples, simulations show accurate size control and power increasing with sample size. A key advantage is the joint testing capability: causal relationships invisible to pairwise tests can be detected, as demonstrated both theoretically and empirically. Two applications to international stock market indices at the global and Asian regional level illustrate the practical relevance of the proposed framework.}
The classical Granger causality framework (Granger, 1969) assesses whether lagged values of one stochastic process provide statistically significant information for forecasting another, conditional on the past of the latter. While widely applied, this approach is inherently limited to linear predictive structures in the conditional mean, rendering it inadequate in the presence of non-linear, asymmetric, or tail-dependent relationships that frequently characterize financial, macroeconomic, and environmental time series. \\
To overcome these limitations, recent methodological advances have extended Granger causality to notions that go beyond the conditional mean, offering a distributional perspective on causal dependence. Most of this literature has focused on the notion of Granger causality in quantiles. For instance, the parametric approach from troster2018, the non-parametric approaches from jeong2012 and balcilar2016, and the copula-based framework from lee2014, among others. Later, song2021 introduced model-free measures of Granger causality in quantiles, extending the approach of song2018 to different parts of the distribution. Based on this measure, jang2023 proposed a test for Granger causality in quantiles using vine copulas. More recently, bouezmarni2024 introduced the first parametric test for Granger non-causality in expectiles based on expectiles regressions. Despite this progress, no model-free or copula-based tests for Granger causality in expectiles have been proposed, particularly one that can accommodate multivariate causality under non-linear and non-Gaussian dependence.\\
Shifting the focus from Granger causality in quantiles to expectiles is motivated from the theory of risk measures. Quantiles have been widely used in financial and risk management literature under the name of Value-at-Risk, nonetheless, quantiles are not a coherent risk measure in the sense of artzner1999 as they lack sub-additivity. Expectiles were introduced by newey1987 as the minimizers of an asymmetric quadratic loss and they constitute the only class of law-invariant risk measures that are both coherent and elicitable bellini2015,ziegel2016. These two properties make expectiles particularly well-suited for distributional Granger causality analysis in contexts where risk quantification and forecast evaluation are both relevant.\\
Moreover, the incorporation of copula theory provides a flexible and model-agnostic tool for capturing non-linear and asymmetric dependence structures between variables. Copulas allow the joint distribution of time series to be decomposed into their marginal distributions and a dependence function, facilitating the explicit modeling of tail dependence and other complex interactions that traditional linear models fail to capture. For a comprehensive treatment of copula theory we refer to nelsen2007, joe2014, and durante2015. More recently, vine copulas, that are hierarchical constructions that decompose high-dimensional copulas into a cascade of bivariate building blocks, have been employed to model both cross-sectional and temporal dependence in stationary time series. Some examples of these structures are the M-vines beare2015, the COPAR-vines brechmannczado2015, and the D-vines smith2015. Building on their work, nagler2022 generalized these models introducing the S-vines, a broader class of all the vine structures that can represent stationary time series.\\
Integrating copula-based dependence modeling with expectile-based Granger causality constitutes a framework that jointly addresses distributional asymmetry, tail sensitivity, and non-linear dependence, yielding a more comprehensive characterization of causal mechanisms in complex stochastic systems. Such a framework has broad applicability, particularly in domains where extreme co-movements and heterogeneous dependence play a crucial role, such as risk management, financial contagion, and climate dynamics. \\
The contribution of the present paper is threefold. First, we define a model-free measure of Granger causality in expectiles that generalizes the mean-focused measure of song2018 to other positions of the conditional distribution. This measure inherits the desirable properties of its predecessors: being non-negative and equal to zero if and only if there is no Granger causality in the $\tau$-th expectile. Moreover, it reduces to the mean-based measure of song2018 when $\tau = 1/2$. Second, we propose a test for Granger causality in expectiles based on the M-vine copula model of beare2015, extending the bivariate vine copula testing framework of jang2022 and fuentesmartinez2025 to the expectile domain and, crucially, to the multivariate setting with $d+1$ series. Indeed, as we will illustrate both in simulations and with an empirical application, there are scenarios in which pairwise dependencies are present, but too small to be detected by pairwise tests, while the joint effect is sizable enough for the proposed multivariate test to catch it. In addition, we will also construct a theoretical case in which pairwise Granger causality in the mean is entirely absent, while joint Granger causality in the mean is present and, with simulations, we show that our test is able to detect it. Theoretical results supporting the consistency of the estimators are provided and, through a simulation study, we also demonstrate that the proposed test has good finite-sample size control and it exhibits a power that increases with sample size $T$ until reaching excellent values already for $T\geq 200$. Moreover, for $\tau=1/2$ (the case of Granger causality in the mean) it has been possible to compare the proposed method with the classical linear Granger causality in the mean F-test and the comparison has revealed a comparable size, but a strongly superior power in the presence of non-linear dependencies. Third, we provide two empirical applications of the proposed test to international stock markets: in particular, the joint testing reveals causal relationships that are invisible to pairwise tests. Indeed, in one of them, pairwise Granger causality is not strong enough to be significant for each series, but the joint test clearly rejects the null hypothesis of no Granger causality for expectiles located in the left tail of the distribution. \\
The remainder of this paper is organized as follows. Section (ref) defines Granger causality in expectiles and proposes the model-free measure. Section (ref) describes the M-vine copula test procedure. Section (ref) presents the simulation study. Section (ref) applies the test to global and Asian stock market data. Section (ref) concludes. Theoretical results on the consistency of the estimators are gathered in the Appendix.
Let $(X_t, \vZ_t)_{t\in\mathbb{Z}}$ be a (strictly) stationary stochastic process, with $X_t$ taking values in ${\mathbb R}$ and $\vZ_t=(Z_{1,t},\dots,Z_{d,t})$ taking values in ${\mathbb R}^{d}$, that is $$ [(X_t, \vZ_t),\dots, (X_{t+h}, \vZ_{t+h})] \stackrel{D}= [(X_s, \vZ_s),\dots, (X_{s+h}, \vZ_{s+h})]\qquad\forall s,\,t\in\mathbb{Z},\, h\in\mathbb{N} $$ where the symbol $\stackrel{D}=$ means equality in distribution. Suppose further that $(X_t, \vZ_t)_{t\in\mathbb{Z}}$ is a Markov process of order $k\in \mathbb{N}\setminus\{0\}$ (more briefly, a $k$-Markov process), that is $$ [X_{t},\vZ_t] \,|\, [(X_s,\vZ_s):\,s\leq t-1]\stackrel{D}= [X_{t},\vZ_t] \,|\, [(X_{t-1},\vZ_{t-1}),\dots, (X_{t-k},\vZ_{t-k})] \qquad\forall t\,. $$ Set $X=(X_t)_{t\in\mathbb{Z}}$ and $\vZ=(\vZ_t)_{t\in\mathbb{Z}}$. Moreover, denote by ${\mathcal I}_{X}(s)$ the $\sigma-$field $\sigma(X_u: u\leq s)$, that is the information brought by $X$ until time-step $s$ and by ${\mathcal I}_{X, \vZ}(s)$ the $\sigma-$field $\sigma((X_u, \vZ_u): u\leq s)$, that is the information brought by $X$ and $\vZ$ until time-step $s$. Assuming $X_t$ and $\vZ_t$ to be square integrable, we say that $\vZ$ causes $X$ (in the mean) in the sense of Granger, if
where
are the mean squared errors of the optimal prediction of $X_t$ given the information ${\mathcal I}_{X}(t-k)$ and given the information ${\mathcal I}_{X,\vZ}(t-k)$, respectively. song2018 proposed a model-free measure of Granger causality in the mean for the bivariate case, which can be easily adapted into our multivariate setting as
While most of the literature has focused on the notion of Granger causality in the mean, this can also be extended for other parts of the distribution. For instance, several tests have been proposed for testing Granger causality in quantiles jeong2012,lee2014,balcilar2016,troster2018,jang2023. Moreover, song2021 also propose a model-free measure of Granger causality in quantiles. Given that quantiles are often used as risk measures in financial risk management, testing for Granger causality in quantiles can be understood as testing if the past information of previous extreme events from other risk factors can improve the prediction of future extreme events of a given risk factor of interest. However, quantiles are not a coherent risk measure as they are not sub-additive. Hence, in order to properly measure Granger causality in risk, one could use other risk measures, such as expectiles. Expectiles are the only law invariant coherent and elicitable risk measure bellini2015,ziegel2016. They are defined as the minimizers of an asymmetric quadratic loss: formally, given $\tau\in (0,1)$, the {\em $\tau$-th expectile of $X$} is defined as $$ \mu_{\tau} (X)=\arg\min_{m\in\mathbb{R}} \mathbb{E}\left[\,\mathcal{R}_\tau(X-m)\,\right] $$ where
with $x_+=\max\{x,0\}$ and $x_-=\max\{-x,0\}$.\\
Using the definition of expectiles, we can provide a definition of Granger causality in expectiles.
In order to measure the degree of causality in the $\tau$-th expectile from $\vZ$ to $X$, one could use the difference between these two loss functions. This allows us to propose a model-free measure of Granger causality in expectiles.
The expectile causality measure $GC_{\tau}(\vZ \rightarrow X)$ is aligned with the definition of measure functions of dependence and feedback between time series in geweke1982, as well as with the ones introduced in song2018,song2021. In fact, it has the relevant properties that these measures include, such as being non-negative and canceling only when there is no Granger causality.
Let $(X,Z_1,\dots,Z_d)=(X_t,Z_{1,t},\dots,Z_{d,t})_t$ be a $k-$Markov square-integrable (strictly) stationary stochastic process and let $\{(x_t,z_{1,t},\dots,z_{d,t})\;:\;t=1,\dots,T\}$ be a sample of it. For simplicity, we assume that $k=1$, but everything can be naturally extended to any order $k\in\mathbb{N}\setminus\{0\}$. \\
The following test is an extension of the one introduced in fuentesmartinez2025 for Granger causality in the mean. \\ We start by describing part A (computation of the value of the test statistic). We use the sample version $GC_{\tau}(\vZ \rightarrow X)$ of the above proposed measure, that is
where the expectiles $\widehat{\mu}_\tau(X_t|\, X_{t-1}=x_{t-1})$ and $\widehat{\mu}_\tau(X_t|\, X_{t-1}=x_{t-1},\,\vZ_{t-1}=\vz_{t-1})$ are computed fitting an M-vine copula model to the observed sample and using it to generate i.i.d. observations of $X_t$ given $X_{t-1}=x_{t-1}$ and of $X_t$ given $X_{t-1}=x_{t-1}$ and $\vZ_{t-1}=\vz_{t-1}$, and using these generated samples to compute the corresponding expectiles empirically. More precisely, we proceed as follows:
Roughly speaking, $\widehat{GC}_{\tau}(\vZ \rightarrow X)$, measures the log-difference between the mean $\tau$-expectile loss related to the model with an M-vine copula structure fitted only on the information set from $X$, and the mean $\tau$-th expectile loss computed with the M-vine copula model fitted to the entire sample from both $X$ and $\vZ$, that is using the information set of all series. In line with the definition of Granger causality in $\tau$-th expectiles, if this estimated quantity is significantly higher than zero, we reject the null hypothesis of no Granger causality (in the $\tau$-th expectile) from $\vZ$ to $X$: indeed, under the null hypothesis, the measure $GC_{\tau}(\vZ \rightarrow X)$ is zero, and so, the higher is the value of the statistics $\widehat{GC}_{\tau}(\vZ \rightarrow X)$, the more statistically significant is the evidence of the presence of Granger causality (in the $\tau$-th expectile) running from $\vZ$ to $X$. Regarding $T_0\geq k+1$, we can choose it by taking into account the goodness of fit of the models to the data. \\
In order to test if $\widehat{GC}_{\tau}(\vZ \rightarrow X)$ is statistically greater than zero, we rely, as in jang2022, on a method, which takes inspiration from paparoditis2000 and also used in fuentesmartinez2025. This method relies on the M-vine copula model fitted on the entire sample $M_{x\vz}=\{(x_t,\vz_t):t=1,\dots,T\}$ in order to generate independent samples under the null hypothesis of no Granger causality (in the $\tau$-th expectile) that have the same dependence structure as the original sample, and use them for computing the $p-$value for the test. Previous literature has dealt with these methods in order to simulate exclusively bivariate series, hence, we propose an extension of it to the multivariate case in which we allow for $d+1$ series. Given the presence of more than two series, the methodology is not as straightforward as in the original case in which all the required copulas are present in the first tree of the M-vine structure. Therefore, the multivariate case requires to proceed sequentially using the first $d$ trees of the corresponding vine. More precisely, the part B (computation of the $p$-value) of the proposed procedure works as follows:
Therefore, we reject the null hypothesis of no Granger causality (in the $\tau$-th expectile) when $p<\alpha$, where $\alpha$ is a given significance level.
In order to analyze the finite sample properties of the proposed M-vine test for Granger causality in expectiles in terms of size and power, we perform a simulation study based on two size assessment models and four power assessment models. For the selected data generating processes (DGP's) we set $d=2$, i.e., we work with $d+1=3$ series. In order to ease both notation and exposition of the DGP's, we will denote the considered multivariate time series by $(X_t,Y_t,Z_t)_t$ and we will study the Granger causality in expectiles from $(Y,Z)\rightarrow X$. We simulate $S=500$ Monte Carlo replications for each model with sample sizes $T\in\{100,200,500\}$ and compute the empirical size and power of our test using a predefined significance level of $\alpha=0.05$. Furthermore we set $\tau\in\{0.1,0.5,0.9\}$, in order to study the statistical properties of our test both in the mean and at the tails of the distribution. In terms of the specification of our test, for Part A we perform $N=200$ predictions for each series and set $T_0=\frac{T}{2}$, whilst for part B, we work with $B=200$ generated samples under the null hypothesis. We begin by introducing the two size assessment models in our simulation study. \\
Size assessment models\\
where $(\eta_{x,t})_t, (\eta_{y,t})_t, (\eta_{z,t})_t$ are three independent white Gaussian noises.
where $(z_{x,t})_t, (z_{y,t})_t, (z_{z,t})_t$ are three independent sequences of i.i.d. random variables distributed as the standardized skewed Student's t distribution $\text{sstd}(0,1,\nu=5,\xi=-1.5)$.\\
Given that these models are employed to analyze the size of the test, they are specified such that the absence of Granger causality from $(Y,Z)\to X$ is established by construction. On the one hand, S1 represents the case of three independent stationary AR(1) processes with the same autoregressive coefficient and standard normal innovations. On the other hand, S2 has three independent GARCH(1,1) processes with the same parameter specification and independent innovations drawn from a standardized skewed Student's t distribution with 5 degrees of freedom and a skewness parameter of $-1.5$. Model S2 serves to emulate the behaviour of data from financial returns by inducing heavy tails and negative skewness in the innovations, which are well-known stylized facts in the literature related to financial returns cont2001.
Table (ref) presents the results of the empirical size assessment of the proposed test for both S1 and S2. Noticeably, for both DGP's the test manages to control the size around the predefined level of $\alpha=0.05$, meaning that the proposed M-vine test has a low probability of incorrectly rejecting the null hypothesis of no Granger causality in the $\tau$-expectile. Moreover, this conclusion holds for every considered value of $\tau$ and for every considered length $T$ of the series. For analyzing the power of the proposed test, we now introduce the power assessment models that we are going to employ.\\
Power assessment models\\
where $(\eta_{x,t})_t, (\eta_{y,t})_t, (\eta_{z,t})_t$ are three independent white Gaussian noises.
where $(\eta_{x,t})_t, (\eta_{y,t})_t, (\eta_{z,t})_t$ are three independent white Gaussian noises.
where $(\eta_{x,t})_t, (\eta_{y,t})_t, (\eta_{z,t})_t$ are three independent white Gaussian noises.
where $(z_{x,t})_t, (z_{y,t})_t, (z_{z,t})_t$ are three independent sequences of i.i.d. random variables distributed as the standardized skewed Student's t distribution $\text{sstd}(0,1,\nu=5,\xi=-1.5)$.\\
These four DGP's are built so that there is joint Granger causality $(Y,Z)\to X$ and with or without single Granger causality $Y\to X$ and $Z\to X$. Model P1 has a similar structure as S1, $Y_t$ and $Z_t$ are still independent AR(1) processes, whereas $X_t$ does not only depend linearly on its own lag, but also on the ones from $Y_t$ and $Z_t$. In P2, $Y_t$ and $Z_t$ are AR(1) processes with smaller autoregressive coefficient than in P1, but it represents a more complex type of dependence driven by the interaction term in $X_t$ that involves the lags of $Y_t$ and $Z_t$. Model P3 is relatively similar to P2 with the only difference that $Y_t$ and $Z_t$ are both purely Gaussian white noises. This subtle difference makes P3 a compelling model, as one can analytically show that there is no Granger causality in the mean from $Y\rightarrow X$ nor from $Z\rightarrow X$, however, there is joint Granger causality in the mean from $(Y,Z)\rightarrow X$ (for a formal proof of these facts, we refer to Appendix (ref). See also Appendix (ref)). Consequently, tests for pairwise Granger causality in the mean would correctly lead to the no-rejection of the null hypothesis of no Granger causality in the mean (e.g. Table (ref) in Appendix (ref)), but, differently from multivariate tests, they could not be applied to detect the presence of the joint Granger causality. Lastly, P4 is built to analyze the power of the test with simulated data that behaves as series of financial returns: specifically, it resembles S2 but in the case that $X$ depends on the lags of both $Y_t$ and $Z_t$ through a non-linear term. \\
The results for the empirical power across the four DGP's, three expectile levels, and three sample sizes are exhibited in Table (ref). The first noticeable pattern is that for each DGP and expectile level, the power is monotonically increasing as $T$ grows, explicitly illustrating the consistency of the M-vine test. In fact, for $T=500$ the test reaches the ideal power in most dependence scenarios and remarkably, even for $T=200$ the test has a considerably high power in some DGP's. The process for which our test has the lowest power in $T=500$ is P1. This result has two implications. On the one hand, it shows the sensitivity of the proposed test to the size of the autoregressive coefficients on the lags of the causing series. This has been well-documented for other tests for Granger causality, for instance in the simulation study of bouezmarni2024. On the other hand, this is an example of a scenario where the single autoregressive coefficients for each causing series might be small enough so that, in practice with a finite sample size, they lead to pairwise tests not rejecting the null hypothesis of no Granger causality. Therefore, the proposed test provides a tool that can address such settings where the pairwise dependencies are too weak to be detected by pairwise tests, but the joint influence is strong enough that our test can catch it and distinguish these cases from those in which there is indeed no Granger causality. To illustrate this, under the P1 specification, we compare the power of the proposed M-vine test for $(Y,Z)\to X$ with the ones of our pairwise M-vine Granger causality in expectiles tests and with the ones of the Granger causality in quantiles tests from balcilar2016\footnote{balcilar2016 introduces a Kernel-based Non-Parametric (KNP) test for Granger causality in quantiles that is able to test for individual causality between series. Using the R package ‘nonParQuantileCausality’, we implement this test for each direction: $Y\to X$ and $Z\to X$.}.
Table (ref) exhibits the results from the power comparison between the three tests. When $T=500$ and $\tau=0.5$ (which means Granger causality in the mean for the M-vine test and in the median for the quantile-based test), the proposed joint test has a notably higher power. In fact, our joint test reaches an almost ideal power, whereas the power of the pairwise tests are in the vicinity of $0.70$. When we move into the tails, the joint version of our test maintains power relatively closer to $0.90$ for $T=500$. The pairwise M-vine test has a power that increases with sample size also for extreme values of $\tau$, keeping the same pattern as in the case of the mean (although with lower values). In contrast, KNP test exhibits a power drastically close to or below $0.05$ (always below $0.10$) across all sample sizes, meaning that it is failing to detect a causal relationship that is present by construction.
In summary, these results highlight the two aforementioned merits of the proposed methodology. Firstly, the joint nature of the proposed test captures the combined predictive contribution of $Y$ and $Z$, whereas the pairwise tests yield considerably weaker evidence as the single dependencies are too small to be detected individually. Secondly, as shown from the comparison between the pairwise tests, the copula-based estimation tends to be more efficient than the kernel-based counterpart when working with finite samples, especially when focusing on the tails where copulas excel at modeling and capturing dependencies. \\
We repeat the simulation study of the previous section using the linear Granger causality in the mean F-test for $(Y,Z)\to X$, in order to compare its statistical properties with the ones of the M-vine test with $\tau=1/2$. Table (ref) reports the empirical size and the power of the above mentioned linear test for $(Y,Z)\to X$. Regarding the size, the linear test controls the size around the nominal level when $T\geq 200$, similarly to the proposed M-vine test (compare the “size part” of Table (ref) with Table (ref)). In terms of power, the results reveal evident differences between linear and non-linear settings. For the linear DGP P1, the F-test achieves an almost ideal power even with moderate sample sizes, showing a faster rate of convergence than the M-vine test with $\tau=1/2$. This is not a surprising result at all, given that the F-test is built on such linear models. However, for DGPs P2, P3, and P4, where the dependence structure is non-linear, the power of the F-test remains very low for every sample size, failing to increase with $T$. In contrast, the power of the M-vine test increases consistently with $T$ across all power assessment models, reaching near-ideal levels at $T=500$ (compare the “power part” of Table (ref) with Table (ref)).
These results confirm a very important aspect: a Granger causality test designed for linear dependence, such as the F-test, cannot detect non-linear causal relationships, whereas the proposed copula-based approach is able to capture them regardless of the underlying dependence structure (linear or non-linear).
In this section, we present two applications of the proposed test in real data. We employ the M-vine Granger causality in expectiles test for studying the relationships among major stock market indices at both the global and the Asian regional level. Regarding the specification of the test, we work with the same setting as in Section (ref), i.e., we set $N=200$, $T_0=\frac{T}{2}$, and $B=200$. We test for Granger causality at several positions of the distribution symmetrically around the mean by setting $\tau\in\{0.05,0.1,0.25,0.5,0.75,0.9,0.95\}$. For each application, we first perform the proposed test between all possible pairs of indices to assess individual Granger causality, and then we allow for pairs of indices to jointly cause an individual index, exploiting the multivariate feature of the tool. From the first application, we found that S&P 500 has a pivotal role in terms of transmitting causality to both the FTSE 100 and the Nikkei 225 across the entire distribution, which comes as an expected result given the international relevance of the U.S. market. Moreover, our expectile-based test reveals that the FTSE causes the S&P 500 in the sense of Granger only in the tails of the distribution, a pattern that would be invisible to a purely mean-based framework. Whereas in the second application, we find a more novel result: specifically, the test reveals that there is no Granger causality between pairs of the three Asian indices, but the Nikkei and the Shanghai Composite jointly Granger cause the Hang Seng Index in its left tail, empirically illustrating the advantage of the multivariate nature of the proposed test.
As first application, we study Granger causality among three globally relevant stock markets: the U.S., the U.K., and Japan. To this end, we work with data from their corresponding indices: the S&P 500 (SP), the FTSE 100 (FTSE), and the Nikkei 225 (NK). The data consists of daily observations between September 2012 and October 2014 retrieved from Yahoo Finance. Notice that the length of the sample is $T=500$ in order to be consistent with the simulation study in Section (ref). In order to make the data stationary, we transform stock prices into log-returns as $r_t=(\log(s_t)-\log(s_{t-1}))*100$, where $s_t$ is the stock price at time $t$.\\
From the first panel of Table (ref), the results show that the S&P500 causes both the FTSE and NK individually across the entire distribution. This reflects the fact that among the three markets, the U.S. stock market is the one with the highest relevance worldwide, and therefore, acts as a transmitter of causality to the others. The second panel illustrates that the FTSE also causes the NK in the sense of Granger across most of the distribution, but to a lower extent than the S&P500. Furthermore, it shows that the U.K. stock market has no predictive power in regular scenarios around the mean, however, extreme returns from the FTSE can help in predicting extreme movements in the S&P500, as reflected by this symmetric pattern of Granger causality in both tails of the distribution. The third panel shows that the Japanese stock market acts purely as a receiver of causality, as it does not cause any of the other two indices at any part of the distribution. This illustrates how the Japanese market lacks predictive power for the other two indices that are arguably more relevant at a global level. Finally, the last panel of Table (ref) exhibits the results for joint Granger causality. One can notice that the pairs (SP,NK) and (SP,FTSE) cause the FTSE and NK, respectively, having a statistically significant improvement in their prediction across all values of $\tau$. This comes as an expected result as the S&P500 causes each of these indices individually, hence, paired with another index the conclusion must hold. A similar argument applies to the pair (FTSE,NK) causing the S&P500. The U.K. stock market already has enough predictive power in the tails of the distribution of the U.S. market by itself, then the addition of information stemming from the Japanese market, whose prediction power was already shown to be negligible across most of the distribution, does not change the results.\\
Overall, these results present a coherent picture of global stock market interdependence in which the S&P500 occupies a pivotal transmitting role across the entire return distribution, while NK acts mostly as a receiver. The tail-specific nature of the causality flow from FTSE to S&P500, and the joint tail causality from (FTSE, NK) to S&P500, highlight the value of our expectile-based tool over traditional mean-based approaches in capturing the full distributional structure of international return spillovers.\\
For this second application, we shift the focus to three of the main Asian stock markets: mainland China, Hong Kong, and Japan. We examine causality between their respective indices: the Shanghai Composite (SSE), the Hang Seng Index (HSI), and the Nikkei 225 (NK). As in the previous application, the dataset consists of daily observations between September 2012 and October 2014 retrieved from Yahoo Finance, providing the same sample size of $T=500$. We also work with log-returns in order to achieve the stationarity of the three series.\\
Table (ref) reports the $p$-values for Granger causality in expectiles across all pairwise and joint directions among the Nikkei 225, Hang Seng Index, and Shanghai Composite. From the first three panels, there is evident absence of Granger causality at the pairwise level. None of the six pairwise combinations exhibit significant Granger causality across the entire distribution. Consequently, a standard pairwise analysis would therefore conclude that there are no spillover effects between these three Asian stock markets. However, the joint test provides a different conclusion. While (HSI, NK) $\rightarrow$ SSE and (HSI, SSE) $\rightarrow$ NK are still statistically insignificant for all $\tau$, the direction (NK, SSE) $\rightarrow$ HSI displays a strong level of joint causality in the left tail of the distribution. In fact, the null hypothesis of no Granger causality in this direction is rejected at the 1% significance level for $\tau=0.05$ and at the 5% level for $\tau=0.10$, whereas no evidence of causality is detected from the mean of the distribution until its right tail. This increase in predictability from (NK, SSE) $\rightarrow$ HSI as $\tau$ decreases, is suggestive of an increase in dependence under market stress. This pattern of no individual causality but significant joint causality constitutes a clear empirical instance where standard pairwise testing misses a genuine spillover mechanism at the tails. Furthermore, this provides evidence of a real situation in which the scenario from model P1 from Section (ref) occurs, highlighting the importance of being able to test for joint Granger causality. From an economic standpoint, this implies that neither the NK nor the SSE provide enough information to anticipate downturns of the HSI, despite the HSI being one of the most prominent indices of this region. Nevertheless, they do so jointly, suggesting that when both mainland China and Japan are simultaneously in distress they constitute a comprehensive Asian risk signal that carries predictive power for Hong Kong's downside risk.\\
Lastly, it is worth noting that NK $\rightarrow$ HSI exhibits a marginal significance at the 10% level for $\tau=0.05$ and $\tau=0.10$, which despite our interpretation of being not significant, it could be construed as evidence of a weak individual Granger causality from Japan to Hong Kong at these corresponding expectile levels. Even under this interpretation, the joint test together with SSE shifts the significance level with which the null is rejected from a borderline 10% to a clear 1%, indicating that the Shanghai Composite index contributes complementary information that can make this predictability become fully evident.
Traditional Granger causality analysis focuses on testing whether past values of one time series contain predictive information about another, typically within a linear conditional mean framework. However, many real-world systems — particularly in finance, economics, and environmental science — exhibit complex, asymmetric, and non-linear dependence structures that cannot be fully captured by standard linear models. To address these limitations, recent research has extended Granger causality concepts to the expectile domain, providing a flexible framework for investigating causality across different parts of a conditional distribution rather than merely the mean. In this paper, by generalizing the mean measure of song2018 we introduced a model-free measure of Granger causality in expectiles, and using this novel measure we proposed a testing procedure based on the M-vine copula models from beare2015 that can account for multivariate Granger causality under non-linear and non-Gaussian settings. Under some (standard) regularity conditions, we established the strong consistency of the proposed test statistic. Furthermore, by means of a simulation study, we showed that the proposed test controls the size around the nominal significance level while having a considerably high power for a wide array of data generating processes, even for medium sample sizes. Lastly, we employed the M-vine Granger causality in expectiles test in two empirical applications testing for Granger causality between global major stock market indices and Asian stock market indices.\\