Granger Causality in Expectiles: an M-vine copula test

Granger Causalit y in Exp ectiles: an M-vine copula test Rob erto F uen tes-Martínez 1,2 and Irene Crimaldi 1 1 IMT Sc ho ol for A dv anced Studies Lucca 2 Univ ersidad de Alicante Abstract. A mo del-free measure of Granger causality in exp ectiles is proposed, gen- eralizing the traditional mean-based measure to arbitrary positions of the conditional distribution. Exp ectiles are the only law-in v arian t risk measures that are b oth coherent and elicitable, making them particularly w ell-suited for studying distributional Granger causalit y where risk quan tification and forecast ev aluation are b oth relev an t. Based on this measure, a test is developed using M-vine copula mo dels that accounts for multiv ariate Granger causalit y with d + 1 series under non-linear and non-Gaussian dependence, with- out imp osing parametric assumptions on the joint distribution. Strong consistency of the test statistic is established under some regularit y conditions. In finite samples, simulations sho w accurate size control and p o wer increasing with sample size. A k ey adv an tage is the join t testing capabilit y: causal relationships in visible to pairwise tests can b e detected, as demonstrated both theoretically and empirically . T wo applications to international sto c k mark et indices at the global and Asian regional lev el illustrate the practical relev ance of the proposed framew ork. 1 In tro duction The classical Granger causality framew ork (Granger, 1969) assesses whether lagged v al- ues of one sto c hastic process provide statistically significan t information for forecasting another, conditional on the past of the latter. While widely applied, this approach is inheren tly limited to linear predictiv e structures in the conditional mean, rendering it in- adequate in the presence of non-linear, asymmetric, or tail-dep enden t relationships that frequen tly c haracterize financial, macroeconomic, and en vironmen tal time series. T o ov ercome these limitations, recen t metho dological adv ances hav e extended Granger causalit y to notions that go b eyond the conditional mean, offering a distributional p erspec- tiv e on causal dep endence. Most of this literature has fo cused on the notion of Granger causalit y in quantiles. F or instance, the parametric approach from T roster (2018), the non- parametric approac hes from Jeong et al. (2012) and Balcilar et al. (2016), and the copula- based framework from Lee and Y ang (2014), among others. Later, Song and T aamouti (2021) introduced mo del-free measures of Granger causality in quantiles, extending the approac h of Song and T aamouti (2018) to differen t parts of the distribution. Based on 1 this measure, Jang et al. (2023) prop osed a test for Granger causality in quantiles using vine copulas. More recen tly , Bouezmarni et al. (2024) introduced the first parametric test for Granger non-causalit y in exp ectiles based on exp ectiles regressions. Despite this progress, no mo del-free or copula-based tests for Granger causalit y in exp ectiles ha ve b een prop osed, particularly one that can accommo date multiv ariate causality under non-linear and non-Gaussian dep endence. Shifting the focus from Granger causalit y in quan tiles to exp ectiles is motiv ated from the theory of risk measures. Quan tiles hav e b een widely used in financial and risk manage- men t literature under the name of V alue-at-Risk, nonetheless, quan tiles are not a coheren t risk measure in the sense of Artzner et al. (1999) as they lack sub-additivit y . Expectiles w ere introduced b y Newey and P ow ell (1987) as the minimizers of an asymmetric quadratic loss and they constitute the only class of la w-inv ariant risk measures that are both coher- en t and elicitable (Bellini and Bignozzi, 2015; Ziegel, 2016). These t wo prop erties make exp ectiles particularly w ell-suited for distributional Granger causality analysis in con texts where risk quantification and forecast ev aluation are b oth relev an t. Moreo ver, the incorp oration of copula theory provides a flexible and mo del-agnostic to ol for capturing non-linear and asymmetric dep endence structures b et ween v ariables. Copulas allo w the join t distribution of time series to b e decomp osed into their marginal distributions and a dep endence function, facilitating the explicit mo deling of tail dep en- dence and other complex interactions that traditional linear mo dels fail to capture. F or a comprehensive treatment of copula theory w e refer to Nelsen (2007), Joe (2014), and Duran te and Sempi (2015). More recently , vine copulas, that are hierarchical construc- tions that decompose high-dimensional copulas in to a cascade of biv ariate building blo c ks, ha ve b een emplo yed to mo del both cross-sectional and temp oral dependence in stationary time series. Some examples of these structures are the M-vines (Beare and Seo, 2015), the COP AR-vines (Brechmann and Czado, 2015), and the D-vines (Smith, 2015). Building on their w ork, Nagler et al. (2022) generalized these mo dels in tro ducing the S-vines, a broader class of all the vine structures that can represent stationary time series. In tegrating copula-based dep endence modeling with exp ectile-based Granger causality constitutes a framew ork that jointly addresses distributional asymmetry , tail sensitiv- it y , and non-linear dep endence, yielding a more comprehensiv e characterization of causal mec hanisms in complex sto chastic systems. Suc h a framework has broad applicability , particularly in domains where extreme co-mov emen ts and heterogeneous dep endence pla y a crucial role, suc h as risk management, financial contagion, and climate dynamics. The con tribution of the present pap er is threefold. First, w e define a mo del-free mea- sure of Granger causalit y in exp ectiles that generalizes the mean-fo cused measure of Song and T aamouti (2018) to other positions of the conditional distribution. This measure in- herits the desirable prop erties of its predecessors: b eing non-negative and equal to zero 2 if and only if there is no Granger causalit y in the τ -th expectile. Moreo ver, it reduces to the mean-based measure of Song and T aamouti (2018) when τ = 1 / 2 . Second, we prop ose a test for Granger causality in exp ectiles based on the M-vine copula mo del of Beare and Seo (2015), extending the biv ariate vine copula testing framew ork of Jang et al. (2022) and F uen tes-Martínez et al. (2025) to the exp ectile domain and, crucially , to the m ultiv ariate setting with d + 1 series. Indeed, as w e will illustrate b oth in simulations and with an empirical application, there are scenarios in which pairwise dep endencies are presen t, but too small to b e detected b y pairwise tests, while the join t effect is sizable enough for the prop osed m ultiv ariate test to catc h it. In addition, w e will also construct a theoretical case in which pairwise Granger causalit y in the mean is entirely absent, while join t Granger causality in the mean is presen t and, with sim ulations, w e show that our test is able to detect it. Theoretical results supp orting the consistency of the estimators are pro vided and, through a sim ulation study , we also demonstrate that the prop osed test has go od finite-sample size control and it exhibits a p o w er that increases with sample size T un til reaching excellen t v alues already for T ≥ 200 . Moreo ver, for τ = 1 / 2 (the case of Granger causality in the mean) it has b een p ossible to compare the prop osed metho d with the classical linear Granger causality in the mean F-test and the comparison has rev ealed a comparable size, but a strongly sup erior p o w er in the presence of non-linear dep endencies. Third, w e pro vide t wo empirical applications of the prop osed test to in ter- national sto c k mark ets: in particular, the joint testing rev eals causal relationships that are in visible to pairwise tests. Indeed, in one of them, pairwise Granger causalit y is not strong enough to b e significant for eac h series, but the joint test clearly rejects the null h yp othesis of no Granger causality for exp ectiles lo cated in the left tail of the distribution. The remainder of this pap er is organized as follows. Section 2 defines Granger causality in exp ectiles and prop oses the mo del-free measure. Section 3 describ es the M-vine copula test pro cedure. Section 4 presents the sim ulation study . Section 5 applies the test to global and Asian stock mark et data. Section 6 concludes. Theoretical results on the consistency of the estimators are gathered in the App endix. 2 Granger causalit y in exp ectiles Let ( X t , Z t ) t ∈ Z b e a (strictly) stationary sto c hastic pro cess, with X t taking v alues in R and Z t = ( Z 1, t , . . . , Z d , t ) taking v alues in R d , that is [ ( X t , Z t ) , . . . , ( X t + h , Z t + h ) ] D = [ ( X s , Z s ) , . . . , ( X s + h , Z s + h ) ] ∀ s , t ∈ Z , h ∈ N where the sym b ol D = means equality in distribution. Supp ose further that ( X t , Z t ) t ∈ Z is a Mark ov pro cess of order k ∈ N \ { 0 } (more briefly , a k -Marko v process), that is [ X t , Z t ] | [( X s , Z s ) : s ≤ t − 1 ] D = [ X t , Z t ] | [( X t − 1 , Z t − 1 ) , . . . , ( X t − k , Z t − k ) ] ∀ t . 3 Set X = ( X t ) t ∈ Z and Z = ( Z t ) t ∈ Z . Moreov er, denote by I X ( s ) the σ − field σ ( X u : u ≤ s ) , that is the information brought by X un til time-step s and b y I X , Z ( s ) the σ − field σ ( ( X u , Z u ) : u ≤ s ) , that is the information brought b y X and Z until time-step s . Assuming X t and Z t to be square in tegrable, we say that Z causes X (in the mean) in the sense of Granger, if σ 2 ( X t | I X , Z ( t − k ) ) < σ 2 ( X t | I X ( t − k ) ) where σ 2 ( X t | I X ( t − k ) ) = E h ( X t − E [ X t | I X ( t − k )] ) 2 i and σ 2 ( X t | I X , Z ( t − k ) ) = E h ( X t − E [ X t | I X , Z ( t − k )] ) 2 i are the mean squared errors of the optimal prediction of X t giv en the information I X ( t − k ) and giv en the information I X , Z ( t − k ) , resp ectiv ely . Song and T aamouti (2018) prop osed a mo del-free measure of Granger causality in the mean for the biv ariate case, which can b e easily adapted in to our multiv ariate setting as GC mean ( Z → X ) = log " σ 2 ( X t | I X ( t − k ) ) σ 2 ( X t | I X , Z ( t − k ) ) # . While most of the literature has fo cused on the notion of Granger causality in the mean, this can also b e extended for other parts of the distribution. F or instance, sev eral tests hav e b een prop osed for testing Granger causality in quantiles (Jeong et al., 2012; Lee and Y ang, 2014; Balcilar et al., 2016; T roster, 2018; Jang et al., 2023). Moreov er, Song and T aamouti (2021) also prop ose a mo del-free measure of Granger causalit y in quan tiles. Giv en that quantiles are often used as risk measures in financial risk managemen t, testing for Granger causalit y in quan tiles can b e understo o d as testing if the past information of previous extreme even ts from other risk factors can impro ve the prediction of future extreme even ts of a given risk factor of interest. How ever, quantiles are not a coherent risk measure as they are not sub-additiv e. Hence, in order to prop erly measure Granger causalit y in risk, one could use other risk measures, suc h as exp ectiles. Exp ectiles are the only la w in v ariant coherent and elicitable risk measure (Bellini and Bignozzi, 2015; Ziegel, 2016). They are defined as the minimizers of an asymmetric quadratic loss: formally , giv en τ ∈ ( 0, 1 ) , the τ -th exp e ctile of X is defined as µ τ ( X ) = arg min m ∈ R E [ R τ ( X − m ) ] where R τ ( X − m ) = | τ − I { ( X − m ) < 0 } | ( X − m ) 2 =  τ I { ( X − m ) ≥ 0 } + ( 1 − τ ) I { ( X − m ) < 0 }  ( X − m ) 2 = τ ( X − m ) 2 + + ( 1 − τ ) ( X − m ) 2 − with x + = max { x , 0 } and x − = max {− x , 0 } . 4 Using the definition of exp ectiles, we can pro vide a definition of Granger causality in exp ectiles. Definition 1. (Gr anger c ausality in the τ -th exp e ctile) A ssuming X t and Z t to b e squar e inte gr able and given τ ∈ ( 0, 1 ) , we say that Z Gr anger-c auses X thr ough its τ -th exp e ctile, if E [ R τ ( X t − µ τ ( X t | I X , Z ( t − k )) ] < E [ R τ ( X t − µ τ ( X t | I X ( t − k )) ] , wher e µ τ ( X | I ) = arg min m ∈ R E [ R τ ( X − m ) | I ] = arg min m ∈ R  τ E [( X − m ) 2 + | I ] + ( 1 − τ ) E [( X − m ) 2 − | I ]  . In order to measure the degree of causalit y in the τ -th exp ectile from Z to X , one could use the difference b et w een these tw o loss functions. This allows us to prop ose a mo del-free measure of Granger causality in exp ectiles. Definition 2. ( τ -th exp e ctile Gr anger c ausality me asur e) A ssuming X t and Z t to b e squar e inte gr able and given τ ∈ ( 0, 1 ) , we define the τ -th exp e ctile Gr anger c ausality me asur e fr om Z to X as GC τ ( Z → X ) = log " E [ R τ ( X t − µ τ ( X t | I X ( t − k )) ) ] E [ R τ ( X t − µ τ ( X t | I X , Z ( t − k )) ) ] # , pr ovide d the two me an values ar e strictly p ositive. The exp ectile causality measure GC τ ( Z → X ) is aligned with the definition of measure functions of dependence and feedback b etw een time series in Gew eke (1982), as well as with the ones introduced in Song and T aamouti (2018, 2021). In fact, it has the relev ant prop erties that these measures include, suc h as b eing non-negativ e and canceling only when there is no Granger causality . Remark 1. F or τ = 1 / 2 the GC τ ( Z → X ) is equal to GC mean ( Z → X ) b y Song and T aamouti (2018). This follows from the fact that when τ = 1 2 , the asymmetric quadratic loss R τ is equal to the mean squared error, and given that the minimizer of the mean squared error is the mean, it implies that µ 1 / 2 ( X ) = E [ X ] . Hence, GC 1 / 2 ( Z → X ) = GC mean ( Z → X ) . 3 M-vine test for Granger causality in exp ectiles Let ( X , Z 1 , . . . , Z d ) = ( X t , Z 1, t , . . . , Z d , t ) t b e a k − Marko v square-integrable (strictly) sta- tionary sto c hastic pro cess and let { ( x t , z 1, t , . . . , z d , t ) : t = 1, . . . , T } b e a sample of it. F or simplicit y , we assume that k = 1 , but everything can b e naturally extended to any order k ∈ N \ { 0 } . 5 The following test is an extension of the one in tro duced in F uen tes-Martínez et al. (2025) for Granger causality in the mean. W e start b y describing part A (computation of the v alue of the test statistic). W e use the sample version GC τ ( Z → X ) of the abov e prop osed measure, that is d GC τ ( Z → X ) = log " ( T − T 0 + 1 ) − 1 P T t = T 0 R τ ( x t − b µ τ ( X t | X t − 1 = x t − 1 ) ( T − T 0 + 1 ) − 1 P T t = T 0 R τ ( x t − b µ τ ( X t | X t − 1 = x t − 1 , Z t − 1 = z t − 1 ) # = log " P T t = T 0 R τ ( x t − b µ τ ( X t | X t − 1 = x t − 1 ) P T t = T 0 R τ ( x t − b µ τ ( X t | X t − 1 = x t − 1 , Z t − 1 = z t − 1 ) # where the exp ectiles b µ τ ( X t | X t − 1 = x t − 1 ) and b µ τ ( X t | X t − 1 = x t − 1 , Z t − 1 = z t − 1 ) are computed fitting an M-vine copula mo del to the observ ed sample and using it to generate i.i.d. observ ations of X t giv en X t − 1 = x t − 1 and of X t giv en X t − 1 = x t − 1 and Z t − 1 = z t − 1 , and using these generated samples to compute the corresp onding exp ectiles empirically . More precisely , we proceed as follo ws: Step A1) w e fit an M-vine copula mo del and estimate its resp ectiv e parameters for the obser- v ations from the series X , that is, for M x = ( x t ) t = 1 ... , T , by means of the selection and estimation pro cedure in tro duced and studied in Nagler et al. (2022); Step A2) using the mo del obtained from Step A1), for eac h t = T 0 , . . . , T , we generate N i.i.d. predictions { ˜ x M x i , t : i = 1, . . . , N } of X t giv en X t − 1 = x t − 1 , so that we can compute the empirical conditional τ -th expectile of X t giv en X t − 1 = x t − 1 b y the sample { ˜ x M x i , t } i = 1, ... , N , that is w e set b µ τ ( X t | X t − 1 = x t − 1 ) = b µ τ , N ( X t | X t − 1 = x t − 1 ) = arg min m N X i = 1 R τ ( ˜ x M x i , t − m ) ; Step A3) repeat Step A1) and Step A2) for the observ ations from the series ( X , Z ) , that is for M x z = ( x t , z t ) t = 1 ... , T , in order to obtain b µ τ ( X t | X t − 1 = x t − 1 , Z t − 1 = z t − 1 ) = b µ τ , N ( X t | X t − 1 = x t − 1 , Z t − 1 = z t − 1 ) = arg min m N X i = 1 R τ ( ˜ x M x z i , t − m ) . Remark 2. Note that Theorem 1 assures that the empirical quantit y b µ τ ( X t | X t − 1 = x t − 1 ) con verges a.s. (for N → + ∞ ) to ward the “pseudo-true” conditional τ -exp ectile of X t giv en X t − 1 = x t − 1 , that is to ward the conditional τ -exp ectile of X t giv en X t − 1 = x t − 1 with resp ect to the mo del obtained from Step A1. Indeed, it is enough to apply Theorem 1 taking X with distribution equal to the conditional distribution of X t giv en X t − 1 = x t − 1 determined by the model obtained from step A1), so that { ˜ x M x i , t : i = 1, . . . , N } is a realization of the random v ariables X 1 , . . . , X N in the theorem and b µ τ ( X t | X t − 1 = x t − 1 ) coincides with ˆ µ τ , N ( ˜ x M x 1, t , . . . , ˜ x M x N , t ) with ˆ µ τ , N ( x 1 , . . . , x N ) defined in the theorem. Similar 6 argumen ts hold true for b µ τ ( X t | X t − 1 = x t − 1 , Z t − 1 = z t − 1 ) . Hence, assuming that the fitted mo dels of step A1 are the true ones, by Corollary 1 and the contin uit y of the log ( · ) on ( 0, + ∞ ) , w e ha ve that, for eac h fixed T 0 and T , log " P T t = T 0 R τ ( X t − b µ τ , N ( X t | X t − 1 ) P T t = T 0 R τ ( X t − b µ τ , N ( X t | X t − 1 , Z t − 1 ) # a . s . − → log " P T t = T 0 R τ ( X t − µ τ ( X t | X t − 1 ) P T t = T 0 R τ ( X t − µ τ ( X t | X t − 1 , Z t − 1 ) # for N → + ∞ . Moreo ver, assuming that the fitted models are also ergo dic, b y Corollary 2 and again the con tinuit y of the log ( · ) on ( 0, + ∞ ) , the sample version d GC τ ( Z → X ) results (taking first T → + ∞ and then N → + ∞ ) a strongly consistent estimator of GC τ ( Z → X ) . Roughly speaking, d GC τ ( Z → X ) , measures the log-difference b et ween the mean τ - exp ectile loss related to the mo del with an M-vine copula structure fitted only on the information set from X , and the mean τ -th exp ectile loss computed with the M-vine cop- ula mo del fitted to the entire sample from b oth X and Z , that is using the information set of all series. In line with the definition of Granger causalit y in τ -th exp ectiles, if this estimated quantit y is significan tly higher than zero, we reject the null hypothesis of no Granger causality (in the τ -th exp ectile) from Z to X : indeed, under the n ull hypoth- esis, the measure GC τ ( Z → X ) is zero, and so, the higher is the v alue of the statistics d GC τ ( Z → X ) , the more statistically significant is the evidence of the presence of Granger causalit y (in the τ -th exp ectile) running from Z to X . Regarding T 0 ≥ k + 1 , we can c ho ose it by taking into account the go odness of fit of the models to the data. In order to test if d GC τ ( Z → X ) is statistically greater than zero, we rely , as in Jang et al. (2022), on a metho d, whic h tak es inspiration from Paparoditis and Politis (2000) and also used in F uentes-Martínez et al. (2025). This method relies on the M-vine copula mo del fitted on the entire sample M x z = { ( x t , z t ) : t = 1, . . . , T } in order to generate indep enden t samples under the null hypothesis of no Granger causality (in the τ -th exp ectile) that hav e the same dep endence structure as the original sample, and use them for computing the p − v alue for the test. Previous literature has dealt with these metho ds in order to sim ulate exclusively biv ariate series, hence, w e prop ose an extension of it to the multiv ariate case in whic h w e allo w for d + 1 series. Giv en the presence of more than t wo series, the metho dology is not as straightforw ard as in the original case in whic h all the required copulas are presen t in the first tree of the M-vine structure. Therefore, the m ultiv ariate case requires to proceed sequentially using the first d trees of the corresp onding vine. More precisely , the part B (computation of the p -v alue) of the prop osed pro cedure w orks as follows: Step B1) from the first tree of the obtained M-vine copula structure, extract, for eac h t = 1, . . . , T , b oth the copula b etw een X t and X t + 1 , sa y c X t , X t + 1 , related to the con- ditional distribution of X t + 1 giv en X t , and the copula b et ween X t and Z 1, t , sa y c X t , Z 1, t , related to the conditional distribution of Z 1, t giv en X t ; 7 Step B2) using the estimated marginal distribution b F X , generate x 0 t , and, conditional on this v alue, dra w x 0 t + 1 from c X t , X t + 1 ; Step B3) using also the estimated marginal distribution b F Z 1 , conditional on x 0 t , dra w z 0 1, t from c X t , Z 1, t ; Step B4) from T ree i , for i = 2, . . . , d , extract the conditional copula c X t , Z i , t ; Z 1, t , ... , Z i − 1, t ; then, using the estimated marginal distributions b F Z 2 , . . . , b F Z d , draw sequentially z 0 i , t from the conditional distribution of Z i , t giv en x 0 t , z 0 1, t , . . . , z 0 i − 1, t , for i = 1, . . . , d , yielding the full simulated vector z 0 t = ( z 0 1, t , . . . , z 0 d , t ) ; Step B5) using this generated sample { ( x 0 t , z 0 t ) : t = 1, . . . , T } , compute the quan tity d GC 0 τ ( Z → X ) , that gives a sim ulated v alue of the test statistic under the n ull h yp othesis; Step B6) repeat the ab ov e steps B times, so that w e get B indep enden t sim ulated v alues under the n ull h yp othesis: d GC 0 τ j ( Z → X ) for j = 1, . . . , B ; Step B7) compute the p -v alue for the test by the empirical mean p = 1 B B X j = 1 1 ( d GC 0 τ j ( Z → X ) ≥ d GC τ ( Z → X ) ) . Therefore, we reject the n ull h yp othesis of no Granger causality (in the τ -th exp ectile) when p < α , where α is a giv en significance lev el. 4 Sim ulation Study In order to analyze the finite sample prop erties of the prop osed M-vine test for Granger causalit y in exp ectiles in terms of size and p o wer, we p erform a simulation study based on tw o size assessment mo dels and four p o wer assessment models. F or the selected data generating pro cesses (DGP’s) we set d = 2 , i.e., we w ork with d + 1 = 3 series. In order to ease b oth notation and exp osition of the DGP’s, we will denote the considered multi- v ariate time series by ( X t , Y t , Z t ) t and we will study the Granger causalit y in exp ectiles from ( Y , Z ) → X . W e simulate S = 500 Monte Carlo replications for each mo del with sample sizes T ∈ { 100, 200, 500 } and compute the empirical size and p o w er of our test using a predefined significance level of α = 0.05 . F urthermore we set τ ∈ { 0.1, 0.5, 0.9 } , in order to study the statistical properties of our test b oth in the mean and at the tails of the distribution. In terms of the sp ecification of our test, for Part A we perform N = 200 predictions for eac h series and set T 0 = T 2 , whilst for part B, we w ork with B = 200 gener- ated samples under the null h yp othesis. W e begin b y introducing the t wo size assessment mo dels in our sim ulation study . Size assessment mo dels 8 S1 X t = 0.5 X t − 1 + η x , t , Y t = 0.5 Y t − 1 + η y , t , Z t = 0.5 Z t − 1 + η z , t , where ( η x , t ) t , ( η y , t ) t , ( η z , t ) t are three indep enden t white Gaussian noises. S2 X t = 0.05 + ε x , t , Y t = 0.05 + ε y , t , Z t = 0.05 + ε z , t , ε i , t = σ i , t z i , t , σ 2 i , t = 0.01 + 0.08 ε 2 i , t − 1 + 0.87 σ 2 i , t − 1 , i ∈ { x , y , z } , where ( z x , t ) t , ( z y , t ) t , ( z z , t ) t are three indep enden t sequences of i.i.d. random v ariables dis- tributed as the standardized sk ewed Student’s t distribution sstd ( 0, 1, ν = 5, ξ = − 1.5 ) . Giv en that these mo dels are emplo yed to analyze the size of the test, they are sp ecified suc h that the absence of Granger causalit y from ( Y , Z ) → X is established by construc- tion. On the one hand, S1 represents the case of three indep enden t stationary AR(1) pro cesses with the same autoregressive co efficien t and standard normal innov ations. On the other hand, S2 has three indep endent GARCH(1,1) pro cesses with the same parameter sp ecification and indep endent innov ations drawn from a standardized skew ed Student’s t distribution with 5 degrees of freedom and a sk ewness parameter of − 1.5 . Mo del S2 serv es to em ulate the b eha viour of data from financial returns b y inducing heavy tails and neg- ativ e skewness in the innov ations, which are w ell-known st ylized facts in the literature related to financial returns (Con t, 2001). DGP T = 100 T = 200 T = 500 τ = 0.1 S1 0.056 0.062 0.056 S2 0.056 0.040 0.046 τ = 0.5 S1 0.042 0.040 0.046 S2 0.056 0.042 0.052 τ = 0.9 S1 0.064 0.050 0.042 S2 0.060 0.042 0.044 T able 1: Empirical size of the prop osed M-vine Granger causality in exp ectiles test for ( Y , Z ) → X in each size assessmen t mo del. T able 1 presen ts the results of the empirical size assessment of the prop osed test for b oth S1 and S2. Noticeably , for b oth DGP’s the test manages to con trol the size around the predefined lev el of α = 0.05 , meaning that the prop osed M-vine test has a lo w proba- 9 bilit y of incorrectly rejecting the n ull hypothesis of no Granger causality in the τ -exp ectile. Moreo ver, this conclusion holds for every considered v alue of τ and for every considered length T of the series. F or analyzing the p ow er of the prop osed test, w e now in tro duce the pow er assessmen t models that w e are going to emplo y . P o wer assessmen t mo dels P1 X t = 0.5 X t − 1 + 0.2 Y t − 1 + 0.2 Z t − 1 + η x , t , Y t = 0.5 Y t − 1 + η y , t , Z t = 0.5 Z t − 1 + η z , t , where ( η x , t ) t , ( η y , t ) t , ( η z , t ) t are three indep enden t white Gaussian noises. P2 X t = 0.5 X t − 1 + 5 Y t − 1 Z t − 1 + η x , t , Y t = 0.25 Y t − 1 + η y , t , Z t = 0.25 Z t − 1 + η z , t , where ( η x , t ) t , ( η y , t ) t , ( η z , t ) t are three indep enden t white Gaussian noises. P3 X t = 0.5 X t − 1 + 5 Y t − 1 Z t − 1 + η x , t , Y t = η y , t , Z t = η z , t , where ( η x , t ) t , ( η y , t ) t , ( η z , t ) t are three indep enden t white Gaussian noises. P4 X t = 0.5 X t − 1 + 2.5 Y t − 1 Z t − 1 + ε x , t , Y t = ε y , t , Z t = ε z , t , ε i , t = σ i , t z i , t , σ 2 i , t = 0.01 + 0.08 ε 2 i , t − 1 + 0.87 σ 2 i , t − 1 , i ∈ { x , y , z } , where ( z x , t ) t , ( z y , t ) t , ( z z , t ) t are three indep enden t sequences of i.i.d. random v ariables dis- tributed as the standardized sk ewed Student’s t distribution sstd ( 0, 1, ν = 5, ξ = − 1.5 ) . These four DGP’s are built so that there is joint Granger causality ( Y , Z ) → X and with or without single Granger causality Y → X and Z → X . Mo del P1 has a similar structure as S1, Y t and Z t are still independent AR(1) pro cesses, whereas X t do es not only depend linearly on its o wn lag, but also on the ones from Y t and Z t . In P2, Y t and Z t are AR(1) pro cesses with smaller autoregressive coefficient than in P1, but it represen ts a more complex type of dep endence driven b y the interaction term in X t that inv olves the lags of Y t and Z t . Mo del P3 is relatively similar to P2 with the only difference that Y t and Z t are b oth purely Gaussian white noises. This subtle difference mak es P3 a com- p elling mo del, as one can analytically sho w that there is no Granger causality in the mean 10 from Y → X nor from Z → X , ho wev er, there is join t Granger causalit y in the mean from ( Y , Z ) → X (for a formal pro of of these facts, we refer to App endix A.2. See also App endix A.3). Consequen tly , tests for pairwise Granger causalit y in the mean w ould correctly lead to the no-rejection of the null h yp othesis of no Granger causality in the mean (e.g. T able 7 in App endix A.2), but, differen tly from multiv ariate tests, they could not b e applied to detect the presence of the joint Granger causalit y . Lastly , P4 is built to analyze the pow er of the test with simulated data that b eha ves as series of financial returns: sp ecifically , it resembles S2 but in the case that X dep ends on the lags of b oth Y t and Z t through a non-linear term. DGP T = 100 T = 200 T = 500 τ = 0.1 P1 0.420 0.586 0.882 P2 0.640 0.924 0.998 P3 0.676 0.934 1.000 P4 0.796 0.908 0.982 τ = 0.5 P1 0.516 0.724 0.978 P2 0.288 0.594 0.942 P3 0.296 0.610 0.972 P4 0.944 0.988 1.000 τ = 0.9 P1 0.412 0.624 0.886 P2 0.652 0.934 1.000 P3 0.672 0.958 1.000 P4 0.900 0.986 1.000 T able 2: Empirical p o wer of the prop osed M-vine Granger causality in exp ectiles test for ( Y , Z ) → X in each p o wer assessment mo del. The results for the empirical p ow er across the four DGP’s, three exp ectile levels, and three sample sizes are exhibited in T able 2. The first noticeable pattern is that for each DGP and exp ectile level, the p o w er is monotonically increasing as T grows, explicitly illustrating the consistency of the M-vine test. In fact, for T = 500 the test reac hes the ideal p o wer in most dependence scenarios and remarkably , even for T = 200 the test has a considerably high pow er in some DGP’s. The pro cess for which our test has the lo west p o w er in T = 500 is P1. This result has t w o implications. On the one hand, it sho ws the sensitivit y of the prop osed test to the size of the autoregressiv e co efficien ts on the lags of the causing series. This has b een w ell-do cumen ted for other tests for Granger causality , for instance in the simulation study of Bouezmarni et al. (2024). On the other hand, this is an example of a scenario where the single autoregressiv e coefficients for each causing series migh t be small enough so that, in practice with a finite sample size, they lead to pairwise tests not rejecting the n ull hypothesis of no Granger causality . Therefore, the prop osed test pro vides a tool th at can address such settings where the pairwise dep endencies are 11 to o weak to b e 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 causalit y . T o illustrate this, under the P1 sp ecification, w e compare the p o wer of the prop osed M-vine test for ( Y , Z ) → X with the ones of our pairwise M-vine Granger causalit y in exp ectiles tests and with the ones of the Granger causality in quan tiles tests from Balcilar et al. (2016) 1 . T est T = 100 T = 200 T = 500 τ = 0.1 M-vine ( Y , Z ) → X 0.420 0.586 0.882 M-vine Y → X 0.196 0.356 0.650 M-vine Z → X 0.280 0.388 0.530 KNP Y → X 0.041 0.048 0.084 KNP Z → X 0.033 0.040 0.094 τ = 0.5 M-vine ( Y , Z ) → X 0.516 0.724 0.978 M-vine Y → X 0.300 0.468 0.740 M-vine Z → X 0.344 0.464 0.720 KNP Y → X 0.154 0.340 0.709 KNP Z → X 0.180 0.344 0.705 τ = 0.9 M-vine ( Y , Z ) → X 0.412 0.624 0.886 M-vine Y → X 0.264 0.388 0.620 M-vine Z → X 0.268 0.352 0.560 KNP Y → X 0.000 0.000 0.082 KNP Z → X 0.000 0.014 0.052 T able 3: Empirical p o wer comparison for mo del P1: M-vine Granger causality in exp ectiles test for ( Y , Z ) → X versus the pairwise M-vine Granger causality in exp ectiles test and v ersus the Kernel-based Non-Parametric (KNP) tests for Granger causality in quantiles of Balcilar et al. (2016). Number of sim ulations: S = 500 . T able 3 exhibits the results from the p o wer comparison betw een the three tests. When T = 500 and τ = 0.5 (which means Granger causalit y in the mean for the M-vine test and in the median for the quan tile-based test), the prop osed joint test has a notably higher p o w er. In fact, our joint test reaches an almost ideal p o wer, whereas the p o wer of the pairwise tests are in the vicinit y of 0.70 . When we mo ve into the tails, the joint version of our test maintains p o w er relativ ely closer to 0.90 for T = 500 . The pairwise M-vine test has a p o wer that increases with sample size also for extreme v alues of τ , k eeping the same pattern as in the case of the mean (although with lo wer v alues). In contrast, KNP test exhibits a p o w er drastically close to or b elo w 0.05 (alwa ys b elo w 0.10 ) across all sample sizes, meaning that it is failing to detect a causal relationship that is present 1 Balcilar et al. (2016) introduces a Kernel-based Non-Parametric (KNP) test for Granger causality in quan tiles that is able to test for individual causalit y b et ween series. Using the R package ‘ nonParQuan- tileCausality ’, we implement this test for eac h direction: Y → X and Z → X . 12 b y construction. In summary , these results highlight the tw o aforementioned merits of the prop osed metho dology . Firstly , the joint nature of the prop osed test captures the com bined pre- dictiv e contribution of Y and Z , whereas the pairwise tests yield considerably weak er evidence as the single dependencies are to o small to b e detected individually . Secondly , as sho wn from the comparison b et ween the pairwise tests, the copula-based estimation tends to be more efficient than the kernel-based counterpart when working with finite samples, esp ecially when fo cusing on the tails where copulas excel at mo deling and capturing de- p endencies. 4.1 Granger causality in the mean (i.e. τ = 1 / 2 ) for ( Y , Z ) → X : comparison with the classic Granger causality F-test W e rep eat the sim ulation study of the previous section using the linear Granger causalit y in the mean F-test for ( Y , Z ) → X , in order to compare its statistical prop erties with the ones of the M-vine test with τ = 1 / 2 . T able 4 rep orts the empirical size and the p o wer of the ab ov e men tioned linear test for ( Y , Z ) → X . Regarding the size, the linear test con trols the size around the nominal lev el when T ≥ 200 , similarly to the prop osed M-vine test (compare the “size part” of T able 4 with T able 1). In terms of p o wer, the results reveal eviden t differences betw een linear and non-linear settings. F or the linear DGP P1, the F-test achiev es an almost ideal p o wer ev en with mo derate sample sizes, sho wing a faster rate of con vergence than the M-vine test with τ = 1 / 2 . This is not a surprising result at all, given that the F-test is built on such linear models. How ever, for DGPs P2, P3, and P4, where the dep endence structure is non-linear, the p o wer of the F-test remains very lo w for ev ery sample size, failing to increase with T . In contrast, the pow er of the M-vine test increases consisten tly with T across all p o we r assessmen t mo dels, reaching near-ideal lev els at T = 500 (compare the “p ow er part” of T able 4 with T able 2). DGP T = 100 T = 200 T = 500 Size S1 0.028 0.052 0.060 S2 0.042 0.062 0.060 P ow er P1 0.810 0.988 1.000 P2 0.454 0.438 0.448 P3 0.350 0.338 0.340 P4 0.388 0.410 0.500 T able 4: Empirical size and p o wer of the classical linear Granger causalit y in the mean F-test for ( Y , Z ) → X . These results confirm a very important aspect: a Granger causality test designed for linear dep endence, suc h as the F-test, cannot detect non-linear causal relationships, whereas the prop osed copula-based approach is able to capture them regardless of the 13 underlying dependence structure (linear or non-linear). 5 Empirical Applications In this section, we presen t tw o applications of the prop osed test in real data. W e employ the M-vine Granger causalit y in exp ectiles test for studying the relationships among ma jor sto c k mark et indices at b oth the global and the Asian regional lev el. Regarding the sp ecification of the test, we work with the same setting as in Section 4, i.e., we set N = 200 , T 0 = T 2 , and B = 200 . W e test for Granger causality at several p ositions of the distribution symmetrically around the mean by setting τ ∈ { 0.05, 0.1, 0.25, 0.5, 0.75, 0.9, 0.95 } . F or eac h application, we first p erform the prop osed test b et ween all p ossible pairs of indices to assess individual Granger causalit y , and then w e allo w for pairs of indices to jointly cause an individual index, exploiting the multiv ariate feature of the to ol. F rom the first application, we found that S&P 500 has a pivotal role in terms of transmitting causality to b oth the FTSE 100 and the Nikkei 225 across the entire distribution, which comes as an exp ected result given the international relev ance of the U.S. market. Moreo v er, our exp ectile-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 b e in visible to a purely mean-based framew ork. Whereas in the second application, w e find a more nov el result: sp ecifically , the test reveals that there is no Granger causalit y betw een pairs of the three Asian indices, but the Nikkei and the Shanghai Comp osite jointly Granger cause the Hang Seng Index in its left tail, empirically illustrating the adv antage of the multiv ariate nature of the prop osed test. 5.1 Causalit y among Global Sto c k Mark ets As first application, w e study Granger causalit y among three globally relev an t sto ck mar- k ets: the U.S., the U.K., and Japan. T o this end, w e work with data from their corre- sp onding indices: the S&P 500 (SP), the FTSE 100 (FTSE), and the Nikkei 225 (NK). The data consists of daily observ ations b et ween September 2012 and Octob er 2014 retriev ed from Y aho o Finance. Notice that the length of the sample is T = 500 in order to b e consisten t with the simulation study in Section 4. In order to make the data stationary , w e transform sto c k 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 . 14 τ Direction 0.05 0.10 0.25 0.50 0.75 0.90 0.95 SP → FTSE 0.000 ∗∗∗ 0.000 ∗∗∗ 0.003 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ SP → NK 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ FTSE → SP 0.030 ∗∗ 0.003 ∗∗∗ 0.150 0.290 0.500 0.035 ∗∗ 0.025 ∗∗ FTSE → NK 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ NK → SP 0.340 0.453 0.470 0.525 0.248 0.575 0.233 NK → FTSE 0.260 0.472 0.165 0.145 0.295 0.470 0.123 (FTSE, NK) → SP 0.008 ∗∗∗ 0.001 ∗∗∗ 0.135 0.340 0.128 0.030 ∗∗ 0.022 ∗∗ (SP , NK) → FTSE 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ (SP , FTSE) → NK 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ 0.000 ∗∗∗ T able 5: p -v alues for the Granger causalit y in exp ectiles test betw een FTSE, NK and SP . ∗ p < 0.10 , ∗∗ p < 0.05 , ∗∗∗ p < 0.01 . F rom the first panel of T able 5, the results sho w that the S&P500 causes b oth the FTSE and NK individually across the en tire distribution. This reflects the fact that among the three mark ets, the U.S. stock market is the one with the highest relev ance w orldwide, and therefore, acts as a transmitter of causalit y 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 lo wer exten t than the S&P500. F urthermore, it sho ws that the U.K. stock mark et has no predictive p o w er in regular scenarios around the mean, how ever, extreme returns from the FTSE can help in predicting extreme mov emen ts in the S&P500, as reflected by this symmetric pattern of Granger caus alit y in b oth tails of the distribution. The third panel shows that the Japanese sto c k market acts purely as a receiver of causalit y , as it do es not cause an y of the other tw o indices at an y part of the distribution. This illus- trates ho w the Japanese market lac ks predictive pow er for the other t wo indices that are arguably more relev ant at a global level. Finally , the last panel of T able 5 exhibits the results for joint Granger causalit y . One can notice that the pairs (SP ,NK) and (SP ,FTSE) cause the FTSE and NK, respectively , having a statistically significan t impro v ement in their prediction across all v alues of τ . This comes as an exp ected result as the S&P500 causes eac h of these indices individually , hence, paired with another index the conclusion m ust hold. A similar argument applies to the pair (FTSE,NK) causing the S&P500. The U.K. sto c k market already has enough predictive p o wer in the tails of the distribution of the U.S. market b y itself, then the addition of information stemming from the Japanese mark et, whose prediction p o wer w as already shown to b e negligible across most of the distribution, does not c hange the results. Ov erall, these results present a coherent picture of global sto ck market interdepen- dence in which the S&P500 o ccupies a piv otal transmitting role across the en tire return distribution, while NK acts mostly as a receiv er. The tail-sp ecific nature of the causality flo w from FTSE to S&P500, and the join t tail causalit y from (FTSE, NK) to S&P500, highligh t the v alue of our expectile-based to ol o v er traditional mean-based approac hes in capturing the full distributional structure of international return spillo vers. 15 5.2 Causalit y among Asian Sto c k Mark ets F or this second application, we shift the fo cus to three of the main Asian sto c k markets: mainland China, Hong K ong, and Japan. W e examine causalit y b etw een their resp ectiv e indices: the Shanghai Comp osite (SSE), the Hang Seng Index (HSI), and the Nikkei 225 (NK). As in the previous application, the dataset consists of daily observ ations b et ween Septem b er 2012 and Octob er 2014 retriev ed from Y aho o Finance, providing the same sam- ple size of T = 500 . W e also work with log-returns in order to ac hieve the stationarity of the three series. τ Direction 0.05 0.10 0.25 0.50 0.75 0.90 0.95 NK → HSI 0.080 ∗ 0.080 ∗ 0.145 0.470 0.365 0.305 0.370 NK → SSE 0.410 0.520 0.325 0.345 0.285 0.340 0.425 HSI → NK 0.440 0.275 0.330 0.365 0.310 0.305 0.360 HSI → SSE 0.150 0.370 0.255 0.240 0.235 0.515 0.195 SSE → NK 0.445 0.370 0.325 0.410 0.365 0.305 0.360 SSE → HSI 0.340 0.155 0.260 0.535 0.450 0.325 0.275 (NK, SSE) → HSI 0.005 ∗∗∗ 0.025 ∗∗ 0.080 ∗ 0.545 0.340 0.235 0.195 (HSI, NK) → SSE 0.225 0.395 0.305 0.220 0.305 0.385 0.575 (HSI, SSE) → NK 0.325 0.200 0.210 0.230 0.335 0.270 0.185 T able 6: p -v alues for the Granger causalit y in exp ectiles test b et ween HSI, NK and SSE. ∗ p < 0.10 , ∗∗ p < 0.05 , ∗∗∗ p < 0.01 . T able 6 rep orts the p -v alues for Granger causalit y in exp ectiles across all pairwise and join t directions among the Nikkei 225, Hang Seng Index, and Shanghai Composite. F rom the first three panels, there is eviden t absence of Granger causality at the pairwise level. None of the six pairwise com binations exhibit significan t Granger causalit y across the en- tire distribution. Consequen tly , a standard pairwise analysis would therefore conclude that there are no spillo ver effects b et ween these three Asian sto c k markets. Ho wev er, the joint test pro vides a differen t conclusion. While (HSI, NK) → SSE and (HSI, SSE) → NK are still statistically insignifican t for all τ , the direction (NK, SSE) → HSI displays a strong lev el of join t causalit y in the left tail of the distribution. In fact, the null h yp othesis of no Granger causalit y in this direction is rejected at the 1% significance level for τ = 0.05 and at the 5% level for τ = 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) → HSI as τ decreases, is suggestive of an increase in dep endence under market stress. This pattern of no individual causality but significant join t causality constitutes a clear empir- ical instance where standard pairwise testing misses a gen uine spillo ver mec hanism at the tails. F urthermore, this provides evidence of a real situation in which the scenario from mo del P1 from Section 4 o ccurs, highligh ting the imp ortance of b eing able to test for joint 16 Granger causality . F rom an economic standp oint, this implies that neither the NK nor the SSE provide enough information to an ticipate do wnturns of the HSI, despite the HSI b eing one of the most prominent indices of this region. Nev ertheless, they do so jointly , suggesting that when b oth mainland China an d Japan are simultaneously in distress they constitute a comprehensiv e Asian risk signal that carries predictive p ow er for Hong K ong’s do wnside risk. Lastly , it is worth noting that NK → HSI exhibits a marginal significance at the 10% lev el for τ = 0.05 and τ = 0.10 , whic h despite our in terpretation of b eing not significan t, it could b e construed as evidence of a weak individual Granger causality from Japan to Hong Kong at these corresp onding exp ectile levels. Ev en under this interpretation, the join t test together with SSE shifts the significance level with which the null is rejected from a b orderline 10% to a clear 1%, indicating that the Shanghai Comp osite index con tributes complemen tary information that can make this predictabilit y become fully eviden t. 6 Conclusion T raditional Granger causality analysis fo cuses on testing whether past v alues of one time series contain predictiv e information ab out another, t ypically within a linear conditional mean framework. How ev er, many real-world systems — particularly in finance, economics, and en vironmental science — exhibit complex, asymmetric, and non-linear dep endence structures that cannot b e fully captured by standard linear mo dels. T o address these limitations, recent researc h has extended Granger causality concepts to the expectile do- main, providing a flexible framew ork for in v estigating causality across differen t parts of a conditional distribution rather than merely the mean. In this pap er, b y generalizing the mean measure of Song and T aamouti (2018) w e in tro duced a mo del-free measure of Granger causality in expectiles, and using this no vel measure w e proposed a testing proce- dure based on the M-vine copula mo dels from Beare and Seo (2015) that can account for m ultiv ariate Granger causality under non-linear and non-Gaussian settings. Under some (standard) regularit y conditions, we established the strong consistency of the prop osed test statistic. 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T esting for granger-causalit y in quan tiles. Ec onometric R eviews 37 (8), 850–866. v an der V aart, A. W. (1998). A symptotic Statistics , V olume 3 of Cambridge Series in Statistic al and Pr ob abilistic Mathematics . Cambridge: Cambridge Universit y Press. Ziegel, J. F. (2016). Coherence and elicitabilit y . Mathematic al Financ e 26 (4), 901–918. 19 A App endix A.1 Theoretical results By standard arguments (e.g. (v an der V aart, 1998, Sec. 5.7)), w e can pro ve the follo wing theorem. Theorem 1 (Strong consistency of the empirical τ -exp ectile) . L et X b e a r e al r andom variable with E [ X 2 ] < ∞ and let X 1 , X 2 , . . . b e i.i.d. r e al r andom variables with the same distribution of X and fix τ ∈ ( 0, 1 ) . Denote by µ τ the τ -exp e ctiles of X , i.e. the unique minimizer (over m ) of Q τ ( m ) = E  R τ ( X − m )  . Denote by b µ τ , N the empir ic al τ -exp e ctile, i.e. the r andom variable b µ τ , N ( X 1 , . . . , X N ) , wher e b µ τ , N ( x 1 , . . . , x N ) is the unique minimizer (over m ) of Q τ , N ( x 1 , . . . , x N , m ) = 1 N N X i = 1 R τ ( x i − m ) . Then we have b µ τ , N a . s . − − − − → N →∞ µ τ . Pr o of. W e split the pro of in some steps. The first tw o steps are devoted to verify that µ τ and ˆ µ τ , N are well-defined; while the other steps pro ve the strong consistency of ˆ µ τ , N . W e b egin by showing the differentiabilit y of m 7→ Q τ , N ( x 1 , . . . , x N , m ) and m 7→ Q τ ( m ) . F or eac h x ∈ R , the function m 7→ R τ ( x − m ) = | τ − I { ( x − m ) < 0 } | ( x − m ) 2 =    τ ( x − m ) 2 on m ≤ x ( 1 − τ ) ( x − m ) 2 on m > x is differen tiable ev erywhere in R and, for each m ∈ R , w e ha ve ∂ ∂ m R τ ( x − m ) = ψ τ ( x , m ) = 2  τ 1 { ( x − m ) ≥ 0 } + ( 1 − τ ) 1 { ( x − m ) < 0 }  ( m − x ) =    2 τ ( m − x ) if m ≤ x 2 ( 1 − τ ) ( m − x ) if m > x . (1) (Note that piecewise defined functions are often not differen tiable in the p oin ts where the definition changes, but in the case of m 7→ R τ ( x − m ) , w e hav e the term ( x − m ) 2 that mak es the function differentiable also in m = x .) It immediately follows that, for all x 1 , . . . , x n , the real function m 7→ Q τ , N is differen tiable with deriv ativ e Q ′ τ , N ( m ) = z τ , N ( x 1 , . . . , x N , m ) = 1 N N X i = 1 ψ τ ( x i , m ) . (2) 20 Moreo ver, setting D h ( x , m ) = R τ ( x − ( m + h ) ) − R τ ( x − m ) h , b y the mean v alue theorem, we ha ve D h ( x , m ) = ψ τ  x , m + y h  for some y = y ( x , h ) ∈ ( 0, 1 ) and so, for | h | ≤ 1 , w e ha ve | D h ( x , m ) | =    ψ τ  x , m + y h     ≤ 2  | m + y h | + | x |  ≤ 2  | m | + 1 + | x |  . Therefore, since | D h ( X , m ) | ≤ 2 ( | m | + 1 + | X | ) for | h | ≤ 1 and E [ | X | ] < + ∞ , w e can apply the dominated con vergence theorem and obtain, for each m , Q ′ τ ( m ) = lim h → 0 Q τ ( m + h ) − Q τ ( m ) h = lim h → 0 E [ D h ( X , m )] = E  lim h → 0 D h ( X , m )  = E [ ψ τ ( X , m )] . W e no w turn to the existence, uniqueness and c haracterization of the minimizers. The real functions m 7→ Q τ , n ( x 1 , . . . , x N , m ) and m 7→ Q τ ( m ) are d ifferen tiable (as shown ab o v e) and strictly con vex (since empirical av erage/exp ectation of strictly con vex func- tions). Therefore, they ha ve unique minimizers ˆ µ τ , N ( x 1 , . . . , x N ) ∈ R and µ τ ∈ R , which are c haracterized b y z τ , N ( x 1 , . . . , x N , ˆ µ τ , N ( x 1 , . . . , x N ) ) = 0 and z τ ( µ τ ) = 0 where z τ , N ( x 1 , . . . , x N , m ) is defined in (2) and z τ ( m ) = Q ′ τ ( m ) = E [ ψ τ ( X , m )] and b oth of them are con tinuous and strictly increasing functions of m . W e then set Z τ , N ( m ) = z τ , N ( X 1 , . . . , X N , m ) = 1 N N X i = 1 ψ τ ( X i , m ) and ˆ µ τ , N = ˆ µ τ , N ( X 1 , . . . , X N ) . (Note that, since m 7→ z τ , N ( x 1 , . . . , x N ) is contin uous and strictly increasing and, for each fixed m , the function x 7→ ψ τ ( x , m ) is con tinuous (and strictly decreasing), w e hav e that ˆ µ τ , N ( x 1 , . . . , x N ) is Borel-measurable and so ˆ µ τ , N is a w ell-defined real random v ariable.) A k ey asp ect for the up coming conv ergence argumen t is the follo wing Lipschitz prop- ert y of m 7→ z τ , N ( x 1 , . . . , x N , m ) and m 7→ z τ ( m ) . F or all m , ˜ m and x , w e hav e | ψ τ ( x , m ) − ψ τ ( x , ˜ m ) | ≤ 2 | m − ˜ m | and so, for all m , ˜ m and x 1 , . . . , x N , w e ha v e | z τ , N ( x 1 , . . . , x N , m ) − z τ , N ( x 1 , . . . , x N , ˜ m ) | ≤ 2 | m − ˜ m | , | z τ ( m ) − z τ ( ˜ m ) | ≤ 2 | m − ˜ m | . W e can now deriv e the uniform strong law of large num b ers on compacts for Z τ , N . Fix an y M > 0 . F or all m ∈ [ − M , M ] and all x ∈ R , w e hav e | ψ τ ( x , m ) | ≤ 2 ( | m | + | x | ) ≤ 2 ( M + | x | ) . 21 Hence, since E [ | X | ] < + ∞ , the strong law of large num b ers applies p oint-wise in [ − M , M ] , that is, for eac h fixed m ∈ [ − M , M ] , w e ha ve Z τ , N ( m ) a . s . − → z τ ( m ) as N → + ∞ . (3) In addition, the ab o ve limit holds true also uniformly on [ − M , M ] . Indeed, tak e an arbi- trary δ > 0 . Since [ − M , M ] is compact, there exist finitely man y grid p oin ts m 1 , . . . , m K suc h that, for every m ∈ [ − M , M ] , there exists k that satisfies | m − m k | ≤ δ . Since the grid is finite and the strong law of large n umbers holds true at each grid p oin t, w e get max 1 ≤ k ≤ K | Z τ , N ( m k ) − z τ ( m k ) | a . s . − − − − → N →∞ 0. (4) Moreo ver, we can write | Z τ , N ( m ) − z τ ( m ) | ≤ | Z τ , N ( m ) − z τ , N ( m k ) | + | Z τ , N ( m k ) − z τ ( m k ) | + | z τ ( m k ) − z τ ( m ) | . Hence, using the Lipschitz prop ert y established ab ov e (with ˜ m = m k suc h that | m − m k | ≤ δ ), w e find | Z τ , N ( m ) − z τ ( m ) | ≤ | Z τ , N ( m k ) − z τ ( m k ) | + 4 δ , whic h implies sup m ∈ [ − M , M ] | Z τ , N ( m ) − z τ ( m ) | ≤ max 1 ≤ k ≤ K | Z τ , N ( m k ) − z τ ( m k ) | + 4 δ . T aking lim sup N →∞ and using (4), we get lim sup N →∞ sup m ∈ [ − M , M ] | Z τ , N ( m ) − z τ ( m ) | ≤ 4 δ a.s. Since δ > 0 is arbitrary , we conclude that sup m ∈ [ − M , M ] | Z τ , N ( m ) − z τ ( m ) | a . s . − − − − → N →∞ 0. (5) Lastly , we combine the uniform conv ergence and the characterization of the minimizers to obtain the desired result. Cho ose M > 0 such that | µ τ | < M . T aking an arbitrary ϵ > 0 such that µ τ ± ϵ ∈ ( − M , M ) , w e hav e z τ ( µ τ − ϵ ) < 0 and z τ ( µ τ + ϵ ) > 0 (since z τ is strictly increasing with z τ ( µ τ ) = 0 ). Moreo ver, since Z τ , N ( µ τ − ϵ ) ≤ | Z τ , N ( µ τ − ϵ ) − z τ ( µ τ − ϵ ) | + z τ ( µ τ − ϵ ) ≤ sup m ∈ [ − M , M ] | Z τ , N ( m ) − z τ ( m ) | + z τ ( µ τ − ϵ ) Z τ , N ( µ τ + ϵ ) ≥ z τ ( µ τ + ϵ ) − | Z τ , N ( µ τ + ϵ ) − z τ ( µ τ + ϵ ) | ≥ z τ ( µ τ + ϵ ) − sup m ∈ [ − M , M ] | Z τ , N ( m ) − z τ ( m ) | , b y means of (5), w e get that almost surely Z τ , N ( µ τ − ϵ ) < 0 and Z τ , N ( µ τ + ϵ ) > 0 for sufficien tly large N . 22 (Note that it w ould b e enough to use the p oin t-wise strong law of large n umbers (3) at the tw o p oin ts µ τ ± ϵ ∈ [ − M , M ] , but w e decided to state and apply the more elegan t v ersion (5).) Since ˆ µ τ , N is the unique zero of m 7→ Z τ , N ( m ) which is contin uous and strictly increasing, it follo ws that, almost surely , the (unique) zero-p oin t ˆ µ τ , N b elongs to ( µ τ − ϵ , µ τ + ϵ ) for sufficien tly large N , that is almost surely | b µ τ , N − µ τ | < ϵ for sufficiently large N . Since ϵ > 0 is arbitrarily sufficiently small, we can conclude that b µ τ , N a . s . − → µ τ . Corollary 1. If ( Y t ) t is a Markov squar e-inte gr able stationary sto chastic pr o c ess, then, for e ach fixe d T , we have 1 T T X t = 0 R τ ( Y t − µ τ , N ( Y t | Y t − 1 ) ) a . s . − → 1 T T X t = 0 R τ ( Y t − µ τ ( Y t | Y t − 1 ) ) for N → + ∞ . wher e µ τ ( Y t | Y t − 1 ) and µ τ , N ( Y t | Y t − 1 ) ar e define d as in the pr evious the or em taking as the distribution of X the c onditional distribution of Y t given Y t − 1 . Pr o of. It is an immediate consequence of the ab o v e theorem and the con tinuit y of the function m 7→ R τ ( x − m ) . Corollary 2. If ( Y t ) t is a Markov squar e-inte gr able er go dic stationary pr o c ess, then, taking first T → + ∞ and then N → + ∞ , we have 1 T T X t = 0 R τ ( Y t − µ τ , N ( Y t | Y t − 1 ) ) a . s . − → E [ R τ ( Y t − µ τ ( Y t | Y t − 1 ) ) ] . wher e µ τ ( Y t | Y t − 1 ) and µ τ , N ( Y t | Y t − 1 ) ar e define d as in the pr evious the or em taking as the distribution of X the c onditional distribution of Y t given Y t − 1 . Pr o of. By the strong la w of large num b ers for ergodic stationary processes, w e ha ve 1 T T X t = 0 R τ ( Y t − µ τ ( Y t | Y t − 1 ) ) a . s . − → E [ R τ ( Y t − µ τ ( Y t | Y t − 1 ) ) ] for T → + ∞ . Hence, with probability one, for eac h ϵ > 0 , there exists T ∗ (dep ending on ω and ϵ ) large enough suc h that      1 T T X t = 0 R τ ( Y t − µ τ , N ( Y t | Y t − 1 ) ) − E [ R τ ( Y t − µ τ ( Y t | Y t − 1 ) ) ]      ≤      1 T T X t = 0 R τ ( Y t − µ τ ( Y t | Y t − 1 ) ) − E [ R τ ( Y t − µ τ ( Y t | Y t − 1 ) ) ]      +      1 T T X t = 0 R τ ( Y t − µ τ , N ( Y t | Y t − 1 ) ) − 1 T T X t = 0 R τ ( Y t − µ τ ( Y t | Y t − 1 ) )      ≤ ϵ 2 +      1 T T X t = 0 R τ ( Y t − µ τ , N ( Y t | Y t − 1 ) ) − 1 T T X t = 0 R τ ( Y t − µ τ ( Y t | Y t − 1 ) )      ∀ T ≥ T ∗ . 23 Then, by the previous corollary , for eac h fixed T , there exists N ∗ sufficien tly large (de- p ending on T ) suc h that      1 T T X t = 0 R τ ( Y t − µ τ , N ( Y t | Y t − 1 ) ) − 1 T T X t = 0 R τ ( Y t − µ τ ( Y t | Y t − 1 ) )      ≤ ϵ 2 ∀ N ≥ N ∗ . Summing up, with probability one, for each ϵ > 0 , there exists T ∗ (dep ending on ω and ϵ ) suc h that, for each T ≥ T ∗ , there exists N ∗ (dep ending on T ) suc h that      1 T T X t = 0 R τ ( Y t − µ τ , N ( Y t | Y t − 1 ) ) − E [ R τ ( Y t − µ τ ( Y t | Y t − 1 ) ) ]      ≤ ϵ . A.2 P o wer-assessmen t mo del P3: non-pairwise but join t Granger causal- it y in the mean Recall that mo del P3 is P3 X t = 0.5 X t − 1 + 5 Y t − 1 Z t − 1 + η x , t , Y t = η y , t , Z t = η z , t , where ( η x , t ) t , ( η y , t ) t , ( η z , t ) t are three indep enden t white Gaussian noises. W e will show analytically that there is no individual Granger causalit y in the mean from Y → X nor from Z → X , but there is join t Granger causality in the mean from ( Y , Z ) → X . T o this end, we start with Y → X . T aking conditional exp ectations of X t giv en X t − 1 yields E [ X t | X t − 1 ] = 0.5 X t − 1 + 5 E [ Y t − 1 Z t − 1 | X t − 1 ] + E [ η x , t | X t − 1 ] , b y the independence of [ Y t − 1 , Z t − 1 ] from X t − 1 , and the fact that b oth hav e zero mean as w ell as η x , t , w e get E [ X t | X t − 1 ] = 0.5 X t − 1 . Similarly , w e no w tak e conditional expectations of X t giv en X t − 1 and Y t − 1 yielding E [ X t | X t − 1 , Y t − 1 ] = 0.5 X t − 1 + 5 Y t − 1 E [ Z t − 1 | X t − 1 , Y t − 1 ] + E [ η x , t | X t − 1 , Y t − 1 ] . Giv en that Z t − 1 and η x , t are indep endent of [ X t − 1 , Y t − 1 ] , and b oth ha ve mean zero, we ha ve E [ X t | X t − 1 , Y t − 1 ] = 0.5 X t − 1 = E [ X t | X t − 1 ] . Since both conditional expectations are identical, the mean squared prediction error using the lags of b oth X t − 1 and Y t − 1 is the same as the one using only X t − 1 , implying that there is no Granger causality in the mean from Y to X . The exact same argumen t holds for the case of Z → X . 24 No w, we will show that there is Granger causality in the mean from ( Y , Z ) → X . W e take conditional expectation of X t giv en [ X t − 1 , Y t − 1 , Z t − 1 ] , considering that η x , t is a Gaussian white noise, indep enden t of ( η y , t − 1 , η z , t − 1 ) , w e get E [ X t | X t − 1 , Y t − 1 , Z t − 1 ] = 0.5 X t − 1 + 5 Y t − 1 Z t − 1 Since this conditional exp ectation differs from E [ X t | X t − 1 ] = 0.5 X t − 1 , w e now compute the mean squared prediction errors explicitly to confirm the improv emen t. The mean squared prediction error using only X t − 1 is E h ( X t − E [ X t | X t − 1 ] ) 2 i = E h ( 5 Y t − 1 Z t − 1 + η x , t ) 2 i . Expanding and using the indep endence of Y t − 1 , Z t − 1 , and η x , t , as w ell as E [ Y 2 t − 1 ] = E [ Z 2 t − 1 ] = E [ η 2 x , t ] = 1 , we obtain E h ( X t − E [ X t | X t − 1 ] ) 2 i = 25 E [ Y 2 t − 1 ] E [ Z 2 t − 1 ] + E [ η 2 x , t ] = 25 + 1 = 26. The mean squared prediction error using X t − 1 , Y t − 1 , and Z t − 1 is E h ( X t − E [ X t | X t − 1 , Y t − 1 , Z t − 1 ] ) 2 i = E h η 2 x , t i = 1. Since 1 < 26 , the mean squared prediction error strictly decreases when b oth Y t − 1 and Z t − 1 are included alongside the past of X . Therefore, there is Granger causality in the mean from ( Y , Z ) to X . W e also provide in T able 7 the results of a simulation study , where we ha ve applied the pairwise classic linear Granger causality F-test based on restricted and unrestricted autoregressiv e mo dels as w ell as the pairwise version of our M-vine test with τ = 1 / 2 . T est T = 500 M-vine test Y → X 0.072 M-vine test Z → X 0.048 Linear F-test Y → X 0.244 Linear F-test Z → X 0.246 T able 7: Mo del P3: empirical rejection rates for the pairwise classic linear F-test for Granger causalit y in the conditional mean and for the pairwise M-vine test in the mean ( τ = 1 / 2 ). W e hav e tak en α = 0.05 and S = 500 simulations. The results in T able 7 show that the M-vine test has a rejection rate close to the nominal significance level of α = 0.05 for b oth directions, demonstrating the absence of individual Granger causality from both Y → X and Z → X . In contrast, the linear F-test exhibits clear size distortions stemming from the fact that this test is built under the assumption of a linear mo del, whic h is evidently missp ecified given that P3 features an 25 in teraction term. These results not only confirm the analytical findings from this section, but also further illustrate the adv antage of our copula-based approach in settings with non-linear dependence. A.3 P o wer-assessmen t mo del P3: pairwise and joint Granger causality in τ -exp ectiles, with τ  = 1 / 2 W e can deep en the analysis of model P3, showing that, for ev ery τ ∈ ( 0, 1 ) , there is join t Granger causality in the τ -exp ectile from ( Y , Z ) → X ; while there is only pairwise Granger causality in the τ -expectile from Y → X (and from Z → X ) if and only if τ  = 1 / 2 . By the mo del assumptions and the prop erties of the exp ectiles (Bellini et al., 2014), we can write µ τ ( X t | X t − 1 ) = 0.5 X t − 1 + c τ , where c τ = µ τ ( W t ) is a real constan t dep ending on the distribution of W t = 5 Y t − 1 Z t − 1 + η x , t and suc h that c 1 / 2 = 0 . W e no w compute the conditional τ -exp ectile of X t giv en X t − 1 and Y t − 1 . W e ha v e X t | ( X t − 1 , Y t − 1 ) D = 0.5 X t − 1 + U q 25 Y 2 t − 1 + 1 where U D = N ( 0, 1 ) . Hence, b y the prop erties of the expectiles, w e get µ τ ( X t | X t − 1 , Y t − 1 ) = 0.5 X t − 1 + k τ q 25 Y 2 t − 1 + 1, where k τ = µ τ ( U ) . In the case of τ  = 1 / 2 , we hav e k τ  = 0 , so that k τ q 25 Y 2 t − 1 + 1 is a non-trivial random v ariable, and consequen tly , µ τ ( X t | X t − 1 , Y t − 1 ) differs almost surely from µ τ ( X t | X t − 1 ) . Hence, in this case, there is individual Granger causalit y in the τ -exp ectile from Y → X . The same argumen t follows for the case of Z → X given the symmetry of P3. Lastly , w e show the join t Granger causality in the τ -exp ectile from ( Y , Z ) to X for ev ery τ ∈ ( 0, 1 ) . Conditioning on ( X t − 1 , Y t − 1 , Z t − 1 ) , the only remaining randomness in X t comes from η x , t D = N ( 0, 1 ) , so we obtain µ τ ( X t | X t − 1 , Y t − 1 , Z t − 1 ) = 0.5 X t − 1 + 5 Y t − 1 Z t − 1 + k τ . Since Y t − 1 Z t − 1 is a non-trivial random v ariable, the ab o v e quan tity differs almost surely from µ τ ( X t | X t − 1 ) for ev ery τ . Therefore, for all v alues of τ , there is joint Granger causalit y in τ -exp ectile from ( Y , Z ) → X . 26