A Copula Nonlinear Granger Causality

Journal Pre-proof A Copula Nonlinear Granger Causality Jong-Min Kim, Namgil Lee, Sun Young Hwang PII:

S0264-9993(18)31661-4

DOI:

https://doi.org/10.1016/j.econmod.2019.09.052

Reference:

ECMODE 5019

To appear in:

Economic Modelling

Received Date: 16 November 2018 Revised Date:

11 July 2019

Accepted Date: 29 September 2019

Please cite this article as: Kim, J.-M., Lee, N., Hwang, S.Y., A Copula Nonlinear Granger Causality, Economic Modelling (2019), doi: https://doi.org/10.1016/j.econmod.2019.09.052. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.

A Copula Nonlinear Granger Causality Jong-Min Kim1 , Namgil Lee2 , and Sun Young Hwang3,* 1

Statistics Discipline, Division of Science and Mathematics, University of Minnesota-Morris, Morris, MN, 56267, USA 2 Department of Information Statistics, Kangwon National University, Chuncheon, Gangwon, South Korea 3 Department of Statistics, Sookmyung Women’s University, Seoul, South Korea * Corresponding author: Sun Young Hwang, [email protected]

Abstract We propose a new copula nonlinear Granger causality test that is more robust than the current available linear and nonlinear Granger causality tests when there exists an asymmetric and nonlinear directional dependence. To perform the statistical test of the copula nonlinear causality, the Gaussian Copula Marginal Regression (GCMR) model and copula directional dependence (Kim and Hwang, 2017) are employed in this paper. By using GCMR and twosample permutation test with rank sum statistic for the copula nonlinear Granger causality, we can confirm that the result of the proposed copula nonlinear Granger causality test is a reliable test through the simulated data and real data both for small and large sample sizes. Keywords: Copula, Granger Causality, Directional Dependence, Permutation Test.

1

Introduction

The causality test by Granger (1969, 1980) considers a fundamental concept for detecting time-varying causal relationships between time series in economics and finance. This fundamental concept is defined in terms of predictability at horizon one of a variable from its own past and the past of another variable of auxiliary variables. The traditional linear Granger causality tests within the linear autoregressive model class have been developed in many directions to detect causal relations between time series but they are limited in their capability to detect nonlinear causality. To study the bivariate nonlinear causal relationship 1

between these two time series, Hiemstra and Jones (1994) proposed the Hiemstra-Jones (HJ) nonlinear causality test which is an improved version based on a nonlinear Granger causality test by Baek and Brock (1992). The HJ nonlinear Granger test is the most cited by scholars and the most frequently applied by practitioners in economics and finance. There were 1,448 Google Scholar hits by February 26th 2019, which illustrates its significance in the economics and finance literatures. To mitigate the severe over-rejection rates that characterise the HJ test under the null hypothesis of Granger non-causality between variables, Diks and Panchenko (2005, 2006) proposed the bivariate Diks and Panchenko (DP) nonparametric Granger causality test which is a modified version of the widely used HJ test of Hiemstra and Jones (1994). Diks and Wolski (2016) extend its bivariate nonparametric Granger causality test to a multivariate case. Jeong, et al. (2012) proposed a nonparametric test for causality in quantile. To consider new causality test without assuming the gaussian distribution to explore a causal relationship between two financial time series, Lee and Yang (2014) proposed copula nonparmetric approach to Granger causality in conditional quantile which considers a statistic based on the Kullback-Leibler (1951) information criterion. Using a copula-based approach has several attractive properties. Copula models have been widely used to model dependence between macroeconomic or financial time series (Cherubini et al. (2011)). So we also propose a new copula nonlinear Granger test which uses the copula directional dependence with beta marginal distribution (Kim and Hwang, 2017) for testing the causality hypothesis. Our proposed copula nonlinear Granger causality test is a simple and robust nonlinear method which is free from normal assumption. The remainder of this paper is organized as follows: Section 2 describes linear and nonlinear causality framework. Section 3 describes copula concepts, directional dependence by copula and copula nonlinear causality test. To verify our Granger causality test, Sections 4 and 5 illustrates the proposed test on simulated and real data, respectively. Finally, conclusions are presented in Section 6.

2

Linear and nonlinear causality framework

In this section, we briefly define linear Granger causality (Granger, 1969) in a VAR system to explore informational linkages between pairs of markets. Given any two stationary data pair, say Xt and Yt , variable Xt Granger-causes Yt linearly provided that lags of Xt offer a significant information for explaining current values of Yt . The bivariate Granger causality is specified in a VAR system as follows: Xt = ϕ1 + and Yt = ϕ2 +

k X

a1i Xt−i +

k X

i=1

j=1

k X

k X

a2i Xt−i +

i=1

j=1

2

b1j Yt−j + ν1t

(1)

b2j Yt−j + ν2t

(2)

where ϕ1 and ϕ2 are the constant terms of the system of equation; a and b denote estimated coefficients; k is the optimal lag length based on the Akaike Information Criterion (AIC); and ν1t and ν2t represent residuals from the VAR model. Hiemstra and Jones (1994) uncovered nonlinear causal relations which modified Baek and Brock’s (1992) nonparametric method. Diks and Panchenko (2006) stated that the HJ-test under the null hypothesis of no Granger causality suffers from lack of power and over-rejection problems. To implement the HJ-test, Diks and Panchenko (2006) proposed a nonparametric non-linear Granger causality test. Diks and Panchenko (2006) considered the causality between two variables X and Y using q and p lags of those variables, respectively. Consider the vectors Xtq = (Xt−q+1 , . . . , Xt ) and Ytp = (Yt−p+1 , . . . , Yt ), with q, p > 1. The null hypothesis that Xtq does not contain any additional information about Yt+1 is expressed by H0 : Yt+1 |(Xtq ; Ytp ) ∼ Yt+1 |Ytp .

(3)

where we let ∼ denote equivalence in distribution. This null hypothesis is a statement about the invariant distribution of the vector of random variables Wt = (Xtq , Ytp , Zt ), where Zt = Yt + 1. After dropping the time indexes, the joint probability density function fX,Y,Z (x, y, z) and its marginals must satisfy the following relationship: fX,Y (x, y) fY,Z (y, z) fX,Y,Z (x, y, z) = × , fY (y) fY (y) fY (y)

(4)

for each vector (x, y, z) in the support of (X, Y, Z). Diks and Panchenko (2006) show that, for a proper choice of weight function, g(x, y, z) = fY2 (y), the corresponding functional is denoted as q: q = E[fX,Y,Z (X, Y, Z)fY (Y ) − fX,Y (X, Y )fY,Z (Y, Z)]. (5) Diks and Panchenko (2006) proposed the following estimator for q: Tn () =

(n − 1) X ˆ (fX,Y,Z (Xi , Yi , Zi )fˆY (Yi ) − fˆX,Y (Xi , Yi )fˆY,Z (Yi , Zi )) n(n − 2) i

(6)

where n is the sample size, and fˆW is a local density estimator of a dW -variate random vector W . In the case of bivariate causality, the test is consistent if the bandwidth  is given by n = Cn−β , for any positive constant C and β ∈ ( 14 , 13 ). The test statistic is asymptotically normally distributed in the absence of dependence between the vectors Wi . Since Diks and Panchenko (2006) showed that in samples smaller than 500 observations their test may under-reject, it would be wise to make further investigations in case the test fails to reject the null hypothesis. It is one of our motivations to propose a copula non-linear Granger causality test which has small bias accuracy with small sample size.

3

3

Copula Nonlinear Granger Causality

3.1

Copula and Directional Dependence

A copula is a multivariate uniform distribution representing a way of trying to extract the dependence structure of random variables from the joint distribution function. It is a useful approach for understanding and modelling dependent random variables. A copula is a multivariate distribution function defined on the unit [0, 1]r , with uniformly distributed marginals. Sklar (1959) shows that any bivariate distribution function, FXY (x, y), can be represented as a function of its marginal distribution of X and Y , FX (x) and FY (y), by using a two-dimensional copula C(·, ·). More specifically, the copula may be written as FXY (x, y) = C(FX (x), FY (y)) = C(u, v), where u and v are the continuous empirical marginal distribution functions FX (x) and FY (y), respectively. Note that u and v have uniform distribution U (0, 1). Therefore, the copula function represents how the function FXY (x, y) is coupled with its marginal distribution functions, FX (x) and FY (y). It also describes the dependent mechanism between two random variables by eliminating the influence of the marginals or any monotone transformation of the marginals. Definition 1. A r-dimensional copula is a function C : [0, 1]r → [0, 1] with the following properties: 1. For all (u1 , . . . , ur ) ∈ [0, 1]r , C(u1 , . . . , ur ) = 0 if at least one coordinate of (u1 , . . . , ur ) is 0; 2. C(1, . . . , 1, ui , 1, . . . , 1) = ui , for all ui ∈ [0, 1], (i = 1, . . . , r); 3. C is r-increasing, (see Nelsen (2013)). From Definition 1, any bivariate distribution function of X and Y , FX,Y , can be represented as a function of its marginals, FX and FY , through a copula C. So, any monotone strictly increasing transformation of X and Y will leave the joint dependence between them unchanged, and thus the copula associated with a pair (X, Y ) will be the same as the copula with transformed variables (U, V ). Therefore, we can describe the dependence structure between X and Y by considering only the pairs of ranks of X and Y . Given n pairs of data points {(xi , yi ); i = 1, . . . , n} from a pair of continuous random variables (X, Y ), we can approximate the corresponding couple (ui , vi ) = (FX (xi ), FY (yi )) using the pairs of ranks {(Ri , Si ); i = 1, . . . , n} where Ri is the rank of xi among x1 , . . . , xn and Si is the rank of yi Si Ri and vi = n+1 . Using these normalized ranks, one can transform among y1 , . . . , yn : ui = n+1 X and Y into U and V . Since the marginal distributions of U and V are uniform on [0, 1] and thus parameter-free, this rank-based approach allows us to compute joint probabilities without knowing marginal distributions. So we used empirical cumulative distribution function method for the transformation for U = F (X) and V = F (Y ), 4

Let Yt be a response variable bounded on the unit interval (0, 1), t = 1, . . . , n, and let xt be a vector of p concomitant covariates. According to Paolino (2001), Ferrari and Cribari-Neto (2004), Cribari-Neto and Zeileis (2010), Schmid, et al. (2013), and Guolo and Varin (2014), beta regression assumes that Yt given xt follows a beta distribution Beta(µt , κt ) parametrized in terms of the mean parameter 0 < µt < 1 and the precision parameter κt > 0. It follows that var(Yt ) = µt (1 − µt )/(1 + κt ) and the density function of Yt is f (yt ; µt , κt ) =

Γ(κt ) ytµt κt −1 (1 − yt )(1−µt )κt −1 , Γ(µt κt )Γ((1 − µt )κt )

where Γ(·) denotes the Gamma function and subscript t emphasizes the time dependence of the beta density through µt and κt . Dependence of the response Yt on the covariates xt is obtained by assuming a logit model for the mean parameter, logit(µt ) = xTt βx , where βx is a p-dimensional vector of coefficients. Guolo and Varin (2014) developed a marginal extension of the beta regression model for time series analysis and the cumulative distribution function of a normal variable is denoted by Φ(·). The Guolo and Varin (2014) marginal beta regression model exploited the probability integral transformation to relate response Yt to covariates xt and to a standard normal error t , Yt = Ft−1 {Φ(t ); β} where Ft (·; β) is the cumulative distribution function associated with density. The probability integral transformation implies that Yt is marginally beta distributed, Yt ∼ Beta(µt , κt ). By using the GCMR by Masarotto and Varin (2012), we estimate the parameter of βx for the logit(µt ) = xTt βx . The inference is performed through a likelihood approach. Computation of the exact likelihood is possible only for continuous responses. Otherwise the likelihood function is approximated by importance sampling. Details of likelihood computations are discussed in Guolo and Varin (2014). So Kim and Hwang (2017) assume that Ut given Vt = vt follows a beta distribution Beta(µt , κt ) parametrized in terms of the mean parameter 0 < µt < 1 and the precision parameter κt > 0, and we denote by F (Ut ; θ) the cumulative distribution function of a beta random variable of mean µUt = E(Ut |vt ). Dependence of the response Ut on the covariate vt is obtained by assuming a logit model for the mean parameter, logit(µUt ) = xTt βx , where βx is a 2-dimensional vector of coefficients. Data are generated from the marginal regression model Ut = F −1 {Φ(t )}, and  µ Ut = β0 + β1 vt , where t = 1, . . . , n, logit(µUt ) = log 1 − µ Ut 

5

(7)

exp(β0 +β1 vt ) so that µUt = E(Ut |vt ) = 1+exp(β and κUt = 1 + exp(β0 + β1 vt ) with the correlation 0 +β1 vt ) matrix of the errors corresponding to the white noise process. The directional dependency from Vt to Ut is measured by

ρ2Vt →Ut =

V ar(E(Ut |vt )) = 12V ar(µUt ) = 12σU2 , V ar(Ut )

(8)

where Vt is a uniform random variable in [0, 1] and we denote σU2 = V ar(µUt ). The variance, σU2 , can be calculated from the given whole financial data of µUt by the simple variance. Kim and Hwang (2017) also proposed multivariate copula direction dependence by using the Guolo and Varin (2014) marginal extension of the beta regression model for time series analysis and the cumulative distribution function of a normal variable. Kim and Hwang (2017) assume that Ut given on the covariates vt = (v1t , v2t , . . . , vkt ) follows a beta distribution Beta(µt , κt ) and obtain the directional dependence of the response Ut on the covariates vt = (v1t , v2t , . . . , vkt ) by taking a logit model for the mean parameter, logit(µUt ) = xTt βx , where βx = (β0 , β1 , . . . , βk ) is a k + 1-dimensional vector of coefficients,  k X µ Ut = β0 + βi vit , where t = 1, . . . , n, logit(µUt ) = log 1 − µ Ut i=1 

(9)

P P exp(β + k βi vit ) so that µUt = E(Ut |vt ) = 1+exp(β0 +Pi=1 and κUt = 1 + exp(β0 + ki=1 βi vit ) with the k β v ) 0 i=1 i it correlation matrix of the errors corresponding to the white noise process. The directional dependency can be obtained by

ρ2(V1t ,V2t ,...,Vkt )→Ut =

V ar(E(Ut |vt )) V ar(Ut )

(10)

where Ut , V1t , V2t , . . . and Vkt are in [0, 1]. To compute the estimation of ρ2(V1t ,V2t ,...,Vkt )→Ut , Kim and Hwang (2017) used the “GCMR” R package (Guido and Varin, 2017) and chose beta marginal distribution to find the estimates of βx from Gaussian marginal regression. With these estimates of βx and the covariates vt , we compute E(Ut |vt ) and then calculate Var(E(Ut |vt )) and Var(Ut ). With these computed values, we obtain the estimation ρˆ2(V1t ,V2t ,...,Vkt )→Ut in the equation (10).

3.2

Copula Nonlinear Granger Causality

We suppose that {Xt } is the preceding variable and {Yt } is the trailing variable and {(Xt , Yt )} is a stationary bivariate time series. Let FX (·) and FY (·) denote the stationary cumulative distribution functions of {Xt } and {Yt } respectively. Let Ut = FX (Xt ) and Vt = FY (Yt ) be the cumulative probability distributions of Xt and Yt . Based on the information of {Yt−1 , Yt−2 , . . . , Yt−k }, we denote   Zt = FY (Yt−1 ), FY (Yt−2 ), . . . , FY (Yt−k )   = Vt−1 , Vt−2 , . . . , Vt−k 6

and consider Kt based on the information {Xt−1 , Xt−2 , . . . , Xt−k } defined by   Kt = FX (Xt−1 ), FX (Xt−2 ), . . . , FX (Xt−k )   = Ut−1 , Ut−2 , . . . , Ut−k where k is the predetermined time lag. Assume that Vt given on the covariates (zt , kt ) = (vt−1 , vt−2 , . . . , vt−k , ut−1 , ut−2 , . . . , ut−k ) follows a beta distribution Beta(µVt , κVt ), then we have a model based on Kim and Hwang (2017) who obtained the dependence of the response Vt on the covariates (zt , kt ) by taking a logit model for the mean parameter as follows:  k k X X µ Vt = α0 + αi vt−i + βj ut−j , logit(µVt ) = log 1 − µVt i=1 j=1 

where µVt = E(Vt |(vt−1 , vt−2 , . . . , vt−k , ut−1 , ut−2 , . . . , ut−k )) =

(11)

P P exp(α0 + ki=1 αi vt−i + kj=1 βj ut−j ) P Pk 1+exp(α0 + i=1 αi vt−i + kj=1 βj ut−j )

with the correlation matrix of the errors corresponding to the white noise process. The copula directional dependence is given by ρ2(Vt−1 ,Vt−2 ,...,Vt−k ,Ut−1 ,Ut−2 ,...,Ut−k )→Ut =

Var(E(Vt |(vt−1 ,vt−2 ,...,vt−k ,ut−1 ,ut−2 ,...,ut−k ))) , Var(Vt )

(12)

where Vt , vt−1 , vt−2 , . . . , vt−k , ut−1 , ut−2 , . . . , and ut−k ∈ [0, 1]. From now on, the lag k is predetermined (fixed). In the data analysis, k can be chosen according to the minimum AIC in the context of vector autoregression (VAR). Definition 2. (Copula Nonlinear Granger Causality; CNGC) {Ut } does not CNGC {Vt } if and only if βi = 0 for all i = 1, 2, . . . , k in Equation (11). {Ut } does CNGC {Vt } if and only if βi 6= 0 for some i = 1, 2, . . . , k in Equation (11). That is, H0 : βi = 0 for all i = 1, 2, . . . , k (Non-CNGC) and Ha : βi 6= 0 for some i = 1, 2, . . . , k. (CNGC) To conduct the test for testing H0 (Non-CNGC) against Ha (CNGC), we are willing to compare the two directional dependency measures ρ2(Vt−1 ,Vt−2 ,...,Vt−k )→Vt and ρ2(Vt−1 ,Vt−2 ,...,Vt−k ,Ut−1 ,Ut−2 ,...,Ut−k )→Vt to see if there is a significant difference between the two. Definition 3. {Ut } does not CNGC {Vt } if and only if ρ2(Vt−1 ,Vt−2 ,...,Vt−k )→Vt = ρ2(Vt−1 ,Vt−2 ,...,Vt−k ,Ut−1 ,Ut−2 ,...,Ut−k )→Vt .

7

(13)

{Ut } does CNGC {Vt } if and only if ρ2(Vt−1 ,Vt−2 ,...,Vt−k )→Vt 6= ρ2(Vt−1 ,Vt−2 ,...,Vt−k ,Ut−1 ,Ut−2 ,...,Ut−k )→Vt . To test the hypothesis, we will use two-sample permutation test with rank sum statistic (explained in the next subsection) to show whether there exist a difference between the two directional dependencies of the model under H0 (Non-CNGC) and the model under Ha (CNGC), where the estimated difference is given by ρˆ2(Vt−1 ,Vt−2 ,...,Vt−k )→Vt − ρˆ2(Vt−1 ,Vt−2 ,...,Vt−k ,Ut−1 ,Ut−2 ,...,Ut−k )→Vt . where ρˆ2 is an estimator of ρ2 based on the sample.

3.3

Two-Sample Permutation Test with Rank Sum Statistic

A permutation test (also called a randomization test, or an exact test) is a type of statistical significance test in which the distribution of the test statistic under the null hypothesis is obtained by calculating all possible values of the test statistic under rearrangements of the labels on the observed data points. The idea of the permutation test was introduced by R.A. Fisher in the 1930’s. For the two-sample permutation test, we set M = n + n = 2n observations such that X1 , X2 , . . . , Xn are independent and identical distributed random sample under H0 (Non-CNGC) and Y1 , Y2 , . . . , Yn are independent and identical distributed random sample under Ha (CNGC). We want to make inferences about the difference in distributions. Before performing two sample permutation test with rank sum statistic, we employ simple random sampling of the sample size (n) from each population under the models of H0 (Non-CNGC) and Ha (CNGC). In this paper, we take different sample sizes from large to small to verify the justification of our proposed test. So we perform simple random sampling 1000 times for computing copula directional dependencies of ρ2Vt →Ut under the models of H0 (Non-CNGC) and Ha (CNGC). With these 1,000 difference values for each copula directional dependence of ρ2Vt →Ut under the models of H0 (Non-CNGC) and Ha (CNGC), we take different sample sizes of the total observations and calculate p-values from two sample permutation test for rank sum statistic which is based on permutation Achieved Significance Level (ASL). You can see the detailed explanation slide and R code from Helwig (2017) (Dr. Nathaniel E. Helwig’s Permutation Tests) as follows. That is, we let F1 and F2 denote the distributions of H0 (Non-CNGC) and Ha (CNGC). The null hypothesis is the same distribution, that is, H0 : F1 (a) = F2 (a) for all a and the alternative hypothesis is not the same distribution which is specified by Ha : F1 (a) 6= F2 (a) for some a. Let p = (p1 , p2 , . . . , pM ) denote the permutation vector denoting which observation belongs to which group. We denote pi to be a group membership of
ai where ai is the i-th observation for combined sample of M observations. There are M different n possible p vectors.

8


n!n! Lemma 1. Under H0 : F1 (a) = F2 (a) for all a, the vector p has probability 1/ M = M! n
M M! of equaling each of the n = n!n! different possible outcomes. The permutation Achieved ˆ Significance Level (ASL) is the permutation probability that θˆ∗ exceeds θ.

ASLperm =

#{|θˆb∗ |

  M ˆ ≥ |θ|}/ n

(14)

(Mn ) where {θˆb∗ }b=1 is the set of all possible test statistics under H0 . Note that the above is for

possible p the two-sided alternative Ha : θ 6= θ0 . When M is large, forming θˆb∗ for all M n n vectors is computationally expensive so that Helwig (2017) uses a Monte Carlo approach as follows. Procedure for approximating ASLperm using Monte Carlo approach is summarized as follows. First, we randomly sample B = 10, 000 permutation vectors p∗1 , . . . , p∗B . Second, evaluate the permutation replication θˆb∗ = s(p∗b , a) where a = (a1 , . . . , aM ) is the observed vector of combined data. Third, approximate ASLperm using ˆ b perm = #{|θˆb∗ | ≥ |θ|}/B. ASL

(15)

This assumes that the statistic θˆb∗ = s(p∗b , a) is designed such that larger absolute values provide more evidence against H0 . To test the equation (13) in this paper, we use θˆ = PM n(M +1) where Ri = rank(|ai − θ0 |) and 1{pi =1} is the indicator function. In i=1 Ri 1{pi =1} − 2 the same way, we perform the same procedure for ρ2Ut →Vt under the models of H0 and Ha .

4

Simulation Study

Nelsen (2013) describes the asymmetric bivariate copulas, and Liebscher (2008) proposes the construction of asymmetric multivariate copulas by the multiplication of two symmetric archimedean copulas. Let A and B be copulas. Then Cα,β (U, V ) = A(U α , V β ) · B(U 1−α , V 1−β ), defines a family of copulas Cα,β , with parameters α, β ∈ [0, 1]. In particular, if α = β = 1, then C1,1 = A, and, if α = β = 0, then C0,0 = B. For α 6= β, the Cα,β is, in general, asymmetric, that is C(U, V ) 6= C(V, U ) for some U and V ∈ [0, 1]. Plackett (1965) has defined the copula p [1 + (θ − 1)(U + V )] − [1 + (θ − 1)(U + V )]2 − 4U V θ(θ − 1) Cθ (U, V ) = , 2(θ − 1) for θ > 0. Independence corresponds to θ = 1. An asymmetric copula by the composition of two symmetric copulas is useful for sophisticated structures of dependence between variables. For verifying our CNGC test, we generate two highly correlated (Pearson correlation r = 0.686, r2 = 0.47 and N = 1, 000) simulation data (U, V ) with the composition of two 9

Figure 1: Scatterplot of two highly correlated (Pearson correlation r = 0.686, r2 = 0.47 and N = 1, 000) simulated data (U, V ) with the composition of two Plackett copula (θ = 500) × Plackett copula (θ = 5) with (α = 0.50, β = 0.37).

Plackett copula (θ = 500) × Plackett copula (θ = 5) with (α = 0.50, β = 0.37). So Figure 1 shows the scatterplot of asymmetric dependence between two highly correlated (Pearson correlation r = 0.686, r2 = 0.47 and N = 1, 000) simulation data (U, V ). For the Granger causality tests with simulated data, we compute the log-returns in percentage, that is, Xt = 100× log returns of U and Yt = 100×log returns of V so that total observation of (Xt , Yt ) is N = 999. In Table 1, we performed the linear Granger causality test and a nonlinear DP test when  = 1 and  = 1.5 with the simulated data (Xt , Yt ). The results are that the traditional linear Granger causality shows that {Yt } does not Granger cause {Xt } and {Xt } does not Granger cause {Yt } when the lags k = 1, 2, 3. But the nonlinear DP tests when  = 1 and  = 1.5 show that {Yt } does Granger cause {Xt } when the lags k = 1, 2, 3 since p-values are smaller than the significance level α = 0.05 and {Xt } does not Granger cause {Yt } when the lags k = 1, 2, 3 since p-values are greater than the significance level α = 0.05. When the lag k = 1 with uniform transformed random variables Ut = FX (Xt ) and Vt = FY (Yt ) in Table 2, the GCMR output of our proposed copula nonlinear causality test shows that there is statistical significance for intercept, Ut−1 and Vt−1 of the direction (Vt → Ut ) under the Ha assumption of CNGC but there is no statistical significance for Ut−1 of the direction (Ut → Vt ) under the Ha assumption of CNGC. Therefore, we test whether there exists a difference in Eq.(13) by taking different sample sizes n = 300, 400, 500, 600, 700, 800 of the total observations N = 999 with simple random sam10

pling 1000 times for computing copula directional dependence and then computing 1,000 difference values of directional dependencies of Vt → Ut and Ut → Vt . With these 1,000 difference values, we perform two sample permutation test for rank sum statistic with generating 10,000 replicates to compute p-values of the differences of ρ2Vt →Ut under the models of H0 and Ha , and ρ2Ut →Vt under the models of H0 and Ha . It turns out that there is a statistically significant difference for directional dependence of ρ2Vt →Ut under the models of H0 and Ha because p-values are smaller than the significant level α = 0.05 at the different sample sizes n = 300, 400, 500, 600, 700, 800. The total observations of size N = 999 is used to discover nonlinear Granger causality relationships: {Vt } does Granger cause {Ut }. See Tables 1–4. Since the subsamples of size n are subsamples from the total number of observations N = 999, we can assume that they have been generated under Ha : {Vt } does Granger cause {Ut }. There is, however, no statistically significant difference in directional dependencies for ρ2Ut →Vt under the models of H0 and Ha because p-values are larger than the significant level α = 0.05 at the different sample sizes n = 300, 400, 500, 600, 700, 800 from Table 5. When the lags k = 2, 3 with uniform transformed random variables Ut = FX (Xt ) and Vt = FY (Yt ) in Table 3 and Table 4, the GCMR outputs of our proposed copula nonlinear causality test show that there is no statistical significance for Vt−2 and Vt−3 of the direction (Vt → Ut ) under the Ha assumption of CNGC and there is also no statistical significance for Ut−2 and Ut−3 of the direction (Ut → Vt ) under the Ha assumption of CNGC. But we want to confirm the results of Tables 3 and 4 whether there exists a difference in Eq.(13) by taking different sample sizes n = 300, 400, 500, 600, 700, 800 of the total observations with simple random sampling 1000 times for computing copula directional dependence and then computing 1,000 difference values of directional dependencies of of Vt → Ut and Ut → Vt . With these 1,000 difference values, we perform two sample permutation test for rank sum statistic with generating 10,000 replicates to compute p-values of the differences of ρ2Ut →Vt under the models of H0 and Ha . It turns out that there is a statistically significant difference for directional dependencies of ρ2Ut →Vt under the models of H0 and Ha because p-values are smaller than the significant level α = 0.05 at the different sample sizes n = 300, 400, 500, 600, 700, 800 from Table 5. The results of our proposed test when the lags k = 2, 3 are different from that for the lag k = 1. The reason is that, the lag k = 1, Ut−1 and Vt−1 are included in the model so that these Ut−1 and Vt−1 variables influenced on the statistical significance for directional dependencies of ρ2Vt →Ut and ρ2Ut →Vt . From these findings from Table 1 and Table 2, our copula nonlinear causality confirms that {Vt } does CNGC {Ut } and {Ut } does Non-CNGC {Vt } when the lag k = 1 which is the same result as the nonlinear DP tests when  = 1 and  = 1.5 even though the traditional linear traditional Granger causality shows that there are no Granger causality between {Xt } and {Yt }. Furthermore, we considered the copula nonlinear Granger causality test with the sample sizes which are smaller than 500. By using two-sample permutation test with rank sum statistic, we show that the results of our proposed test performed with the small samples are the same as the result of our proposed test performed with the large samples which are more than 500 sample sizes in Table 5.

11

The power of a hypothesis test is the probability of making a correct decision to reject the null hypothesis (H0 ) when an alternative hypothesis (Ha ) is true. And the size of a test refers to the probability of making an incorrect decision to reject H0 when H0 is true. While the significance level α is the nominal size, the actual size of the test can be different from the nominal size due to small sample sizes. We performed Monte Carlo simulations to see the power and the actual size of the CNGC test defined in Definition 2. First, we generated bivariate time series {(Ut , Vt )} of length n by Eq. (11) as logit(µUt ) = 1 + 0.5ut−1 + β1 vt−1 ,

logit(µVt ) = 1 + 0.5vt−1 .

(16)

That is, {Vt } does CNGC {Ut } if and only if β1 6= 0. The time series were further transformed by Xt = F −1 (Ut ) and Yt = F −1 (Vt ), where F (·) is a cumulative distribution function of one of the four distributions: the standard normal (N (µ = 0, σ 2 = 1)), Student’s t (tν=4 ), lognormal (Lognormal(µ = 0, σ 2 = 1)), or beta (Beta(µ = 2/7, κ = 7)) distributions. The four distributions were selected due to their diverse skewness and kurtosis values: the N (0, 1) and t4 have a skewness of 0, and Lognormal(0, 1) and Beta(2/7, 7) have skewnesses of 6.185 and −0.596, respectively. The kurtoses of the N (0, 1), t4 , Lognormal(0, 1), and Beta(2/7, 7) are 3, ∞, 113.936, and 2.88, respectively. We changed the value of β1 as β1 = 0, −0.25, −0.5, and increased n from 20 to 5000. We repeated the experiment 1000 times independently and estimated the power and the actual size of the CNGC test with α = 0, 0.01, . . . , 0.15. Table 6 shows the observed rejection rates, i.e., the actual size (β1 = 0) and the power (β1 = −0.25, −0.50) over the length n from 20 to 5000. We can see that the power increases to 1 and the actual size converges to the nominal size 0.05 as n increases for all cases. It is noteworthy that there is almost no difference between the four cases of the marginal distributions, F (·). It is because the copula Granger causality is not influenced by any monotone transformation of the marginals. Figure 2 shows the size-size plot (Davidson and MacKinnon, 1998; Diks and Panchenko, 2006) of the CNGC test for time series lengths n = 20, 50, 100, 200 and the standard normal distribution F (·) ∼ N (0, 1). We can see that actual size converges to nominal size as n increases so that the size-size curve with a larger n is closer to the diagonal line. Therefore, we can say that our proposed method is a valid nonlinear Granger causality test which is robust to relatively small sample sizes. We did not include the plots for the other cases of the distributions, i.e., t4 , Lognormal(0, 1), and Beta(2/7, 7), because they are almost similar to the case of N (0, 1). Next, we performed another Monte Carlo simulations based on bivariate autoregressive (AR) process generated by Xt = 1 + 0.5Xt−1 + β1 Yt−1 + ν1t ,

Yt = 1 + 0.5Yt−1 + ν2t ,

(17)

where the residual νit is independently and identlcally distributed from one of the four distributions: N (0, 1), t4 , Lognormal(0, 1), and Beta(2/7, 7). We can check that Yt does Granger-causes Xt if and only if β1 6= 0. As in the previous simulations, we changed the value of β1 as β1 = 0, −0.25, −0.5, and increased n from 20 to 5000. We repeated the experiment 1000 times independently and estimated the power and the actual size of the CNGC test with α = 0, 0.01, . . . , 0.15. 12

Figure 2: Size-size plot of the CNGC test for time series lengths n = 20 (solid line), n = 50 (dashed line), n = 100 (dotted line), and n = 200 (dashdotted line). The actual size is estimated from the Monte Carlo simulations for bivariate process (16) generated under H0 : β1 = 0, and the standard normal distribution, N (0, 1), as marginal distribution, F (·). ● ●

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Table 7 summarizes the observed rejection rates over the length n from 20 to 5000. We can check that the power increases to 1 as n increases for all cases, and the power for β1 = −0.50 is higher than 0.99 when n ≥ 100. We can also see that the size converges to the nominal size 0.05 as n increases, and it lies between 0.03 and 0.07 when n ≥ 100. Figure 3 shows the size-size plot of the CNGC test for time series lengths n = 20, 50, 100, 200 and distributions ν1t , ν2t ∼ N (0, 1), t4 , Lognormal(0, 1), and Beta(2/7, 7). We can see that actual size converges to nominal size as n increases for all cases. For the heavy tailed distribution (t4 ) and the skewed distributions (Lognormal(0, 1) and Beta(2/7, 7)), the rejection rate is relatively high with the small sample size n = 20 compared to the standard normal distribution N (0, 1). With the larger sample sizes n ≥ 50, the rejection rate of t4 converges fast to the nominal size, while the rejection rates of Lognormal(0, 1) and Beta(2/7, 7) are similar to that of the N (0, 1) distribution. This implies that the proposed CNGC test is robust to the outliers which are due to heavy tail and high kurtosis of the residual distribution. In addition, we can see that the CNGC can incorporate the linear Granger causality, which implies that the CNGC does not sensitively depend on the choice of the link function such as the logit function in Eq. (11).

5

Real Data Analysis

In this section, we consider monthly Arabica and Robusta coffee price data. Robusta is cheaper than Arabica on the commodity market because it is generally accepted that Arabica beans are of higher quality. Figure 4 shows that the price of Arabica coffee is higher than the price of Robusta coffee. Coffee as a tradable commodity is popular because of its increasing volatility. Coffee commodity is traded in US dollars so that the exchange rate largely influences the price of coffee. The data were obtained from the following website, (DATAZEN). Time series span from the January 1960 to the March 2011. He and Maekawa (1999) noted that the typical Granger-causality test can lead to spurious causality when one or both time series are non-stationary. With real data, we performed an augmented Dickey Fuller (ADF) and Kwiatkowski Phillips Schmidt Shin (KPSS) tests that are used for testing a null hypothesis that an observable time series is stationary around a deterministic trend against the alternative of a unit root. Table 8 shows that two Arabica and Robusta coffee price time series are nonstationary. Therefore, we did the log-returns of these two coffee price time series data denoted by (Xt =Log-returns of Arabica, Yt =Log-returns of Robusta, in percentage) to make a stationary data (the total number of each data observation is 614). The ADF and KPSS tests in Table 8 show that Xt and Yt are stationary time series data. The results of these two log-returns (in percentage) of Arabica and Robusta coffee price time series data (Xt , Yt ) can be found in Figure 5. In Table 9, we performed the linear Granger causality test and the nonlinear DP test when  = 1 and  = 1.5 with the real data (Xt , Yt ). The results in Table 9 are that the linear traditional Granger causality shows that {Xt } does not Granger cause {Yt } and {Yt } does not Granger cause {Xt } when the lags k = 1, 2, 3 but the nonlinear DP tests when  = 1 and  = 1.5 show that {Yt } does Granger cause {Xt } and {Xt } does Granger cause {Yt } when 15

Figure 4: Price Time Plot of Arabica and Robusta.

the lags k = 1, 2, 3 since p-values are smaller than the significance level α = 0.05. When the lag k = 1 with uniform transformed random variables Ut = FX (Xt ) and Vt = FY (Yt ) in Table 10, the GCMR output of our proposed copula nonlinear causality test shows that there is statistical significance for intercept, Ut−1 and Vt−1 of the direction (Ut → Vt ) under the Ha assumption of CNGC but there is not statistically significant for Vt−1 of the direction (Vt → Ut ) under the Ha assumption of CNGC from Table 10. Therefore, we test whether there exists a difference in Eq.(13) by taking different samples sizes of 40%, 50%, 60%, 70%, 80%, 90% of the total observation with simple random sampling 1000 times for computing copula directional dependence and then computing 1,000 difference values of directional dependencies of Vt → Ut and Ut → Vt . With these 1,000 difference values, we perform two sample permutation test for rank sum statistic with generating 10,000 replicates to compute p-values of the differences of ρ2Ut →Vt under the models of H0 (Non-CNGC) and Ha (CNGC) and ρ2Vt →Ut under the models of H0 and Ha . For the lag k = 1 in Table 13, it turns out that there is a statistically significant difference for directional dependence of ρ2Ut →Vt under the models of H0 (Non-CNGC) and Ha (CNGC) because p-values are smaller than the significant level α = 0.05 at the different samples sizes of 40%, 50%, 60%, 70%, 80%, 90% of the total observation. But it turns out that there is no statistically significant difference for directional dependence of ρ2Vt →Ut under the models of H0 (Non-CNGC) and Ha (CNGC) because p-values are greater than the significant level α = 0.05 at the different samples sizes of 40%, 50%, 60%, 70%, 80%, 90% of the total observation. Consequently, with k = 1, the conclusions of our test are in between the linear Granger causality test and the nonlinear DP test, since the linear Granger causality shows 16

Figure 5: Price Time Plot of Log-returns (in percentage) for Arabica and Robusta.

that there is no causality between {Xt } and {Yt } while bidirectional causality exists between {Xt } and {Yt } according to the nonlinear DP test. When the lags k = 2, 3 with uniform transformed random variables Ut = FX (Xt ) and Vt = FY (Yt ) in Table 11 and Table 12, the GCMR outputs show that there is no statistical significance for Ut−2 and Ut−3 of the direction (Ut → Vt ) while Ut−1 is significant under the Ha assumption of CNGC. For the direction (Vt → Ut ), Vt−1 and Vt−2 are not statistically significant but Vt−3 is significant in Table 12. Further, we want to confirm the result whether there exists a difference in Eq.(13) under the models of H0 (Non-CNGC) and Ha (CNGC). Regarding ρ2Ut →Vt , the same conclusion (as for k = 1) continues to be valid for k = 2 or 3, that is, there is a statistically significant difference for ρ2Ut →Vt . However, for k = 2, 3, contrary to k = 1, we observe that there is a statistically significant difference for directional dependencies for ρ2Vt →Ut under the models of H0 and Ha . Thus, for k = 2, 3, we conclude bidirectional causality which is similar to that of DP test.

17

6

Conclusion

This research considered the copula nonlinear Granger causality test which is easier to interpret the causality result and more robust (to distributional assumptions and to relatively small sample sizes) than the traditional Granger causality test and the nonlinear DP test when there exists a small sample asymmetric and nonlinear directional dependence. In a future study, we will extend our copula nonlinear Granger causality test to the multivariate setup with the theoretical justification for relevant asymptotic limit distribution.

Acknowledgment We thank the two Reviewers and AE for careful reading and constructive comments which led to substantial improvements in the revised version. This work was supported by a grant from the National Research Foundation of Korea (NRF-2018R1A2B2004157).

References 1. Baek, E. and Brock, W. (1992). A general test for nonlinear Granger causality: Bivariate model. Working paper, Iowa State University and University of Wisconsin, Madison. 2. Cherubini U., Mulinacci, S., Gobbi, F., and Romagnoli, S. (2011). Dynamic Copula Methods in Finance, first ed. Wiley. 3. Cribari-Neto, F. and Zeileis, A. (2010). Beta Regression in R. Journal of Statistical Software, 34(2), 1-24. 4. Davidson, R. and MacKinnon, J. G. (1998). Graphical methods for investigating the size and power of hypothesis tests. The Manchester School, 66(1), 1-26. 5. Diks, C. and Panchenko, V., 2005. A Note on the Hiemstra-Jones test for Granger Noncausality. Studies in Nonlinear Dynamics and Econometrics, 9(2). 1-9 6. Diks, C. and Panchenko, V., 2006. A New Statistic and Practical Guidelines for Nonparametric Granger Causality Testing. Journal of Economic Dynamics and Control, 30, 1647-1669. 7. Diks, C., Wolski, M., 2016. Nonlinear Granger causality: Guidelines for multivariate analysis. Journal of Applied Econometrics, 31, 1333-1351. 8. Ferrari, S. L. P. and Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31, 799-815.

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9. Granger, C. W. J. (1980). Testing for Causality. Journal of Economic Dynamics and Control, 2, 329-352. 10. Granger, C. W. J., 1969. Investigating causal relations by econometric models and cross-sectional methods. Econometrica, 37(3), 424-438. 11. Guido, M. and Varin, C. (2017). Gaussian Copula Regression in R. Journal of Statistical Software, 77(8), 1-26. 12. He, Z. and Maekawa, K. (1999). On spurious Granger causality. Economic letters, 73(3), 307-313. 13. Helwig, N. E. (2017). Permutation Tests. http://users.stat.umn.edu/ helwig/notes/permNotes.pdf. 14. Hiemstra, C. and Jones, J. D. (1994). Testing for linear and nonlinear Granger causality in the stock price-volume relation. Journal of Finance, 49(5), 1639-1664. 15. Jeong, K., Hardle, W. K. and Song, S. (2017). A Consistent Nonparametric Test for Causality in Quantile. Econometric Theory, 28(4), 861-887. 16. Kim, J.-M. and Hwang, S. (2017). Directional Dependence via Gaussian Copula Beta Regression Model with Asymmetric GARCH Marginals. Communications in Statistics: Simulation and Computation, 46(10), 7639-7653. 17. Lee, T.-H. and Yang, W. (2017). Granger-Causality in Quantiles between Financial markets: Using Copula Approach. International Review of Financial Analysis, 33, 70-78. 18. Nelsen, R. B. (2013). An Introduction to Copulas. Springer Science & Business Media, 2nd. 19. Paolino, P. (2001). Maximum likelihood estimation of models with beta-distributed dependent variables. Political Analysis, 9, 325-346. 20. Schmid, M., Wickler, F., Maloney, K. O., Mitchell, R., Fenske, N. and Mayr, A. (2013). Boosted Beta Regression. PLOS ONE, 8(4), e61623. 21. Sklar, A. (1959). Fonctions de repartition a´ n dimensions et leurs marges. Publ.Inst. Statist. Univ. Paris, 8: 229-231. 22. Toda H. Y. and Yamamoto T. (1995). Statistical inference in vector autoregressions with possibly integrated processes. Journal of Econometrics, 66, 225-250.

19

Table 1: Granger Causality with Simulated Data (Xt , Yt ): p-values. Method

Direction

k=1

k=2

k=3

Linear Granger Test

Xt → Yt

0.37

0.41

0.25

Yt → Xt

0.18

0.49

0.30

Xt → Yt

0.32

0.75

0.46

Yt → Xt

0.003

0.001

0.004

Xt → Yt

0.23

0.28

0.26

Yt → Xt

0.024 0.0003 0.0003

Nonlinear DP Test ( = 1)

Nonlinear DP Test ( = 1.5)

20

Table 2: The GCMR for the lag k = 1 with Simulated Data (Ut = FX (Xt ), Vt = FY (Yt )). Ut → Vt Ha : Copula Causality

Estimate Std. Error

z value

P-value

Intercept

0.981

0.069

14.262

0.000

Vt−1

-2.099

0.147

-14.290

0.000

Ut−1

0.088

0.144

0.608

0.543

dispersion

2.707

0.107

25.188

0.000

z value

P-value

H0 : No Copula Causality

Estimate Std. Error

Intercept

0.997

0.063

15.720

0.000

Vt−1

-2.044

0.115

-17.800

0.000

dispersion

2.706

0.107

25.170

0.000

z value

P-value

Vt → Ut Ha : Copula Causality

Estimate Std. Error

Intercept

0.952

0.070

13.564

0.000

Ut−1

-1.610

0.148

-10.878

0.000

Vt−1

-0.333

0.147

-2.272

0.023

dispersion

2.557

0.101

25.368

0.000

z value

P-value

H0 : No Copula Causality

Estimate Std. Error

Intercept

0.889

0.064

13.820

0.000

Ut−1

-1.821

0.116

-15.690

0.000

dispersion

2.544

0.100

25.400

0.000

21

Table 3: The GCMR for the lag k = 2 with Simulated Data (Ut = FX (Xt ), Vt = FY (Yt )). Ut → Vt Ha : Copula Causality

Estimate Std. Error

z value

P-value

Intercept

1.888

0.130

14.531

0.000

Vt−1

-2.587

0.165

-15.716

0.000

Ut−1

-0.067

0.154

-0.434

0.664

Vt−2

-0.941

0.165

-5.721

0.000

Ut−2

-0.237

0.156

-1.519

0.129

dispersion

2.916

0.117

24.889

0.000

z value

P-value

H0 : No Copula Causality

Estimate Std. Error

Intercept

1.834

0.120

15.274

0.000

Vt−1

-2.633

0.135

-19.555

0.000

Vt−2

-1.093

0.132

-8.282

0.000

dispersion

2.908

0.117

24.856

0.000

z value

P-value

Vt → Ut Ha : Copula Causality

Estimate Std. Error

Intercept

1.784

0.130

13.693

0.000

Ut−1

-2.072

0.158

-13.134

0.000

Vt−1

-0.433

0.162

-2.678

0.007

Ut−2

-1.103

0.161

-6.861

0.000

Vt−2

-0.009

0.166

-0.054

0.957

dispersion

2.768

0.111

25.047

0.000

z value

P-value

H0 : No Copula Causality

Estimate Std. Error

Intercept

1.677

0.114

14.773

0.000

Ut−1

-2.326

0.130

-17.955

0.000

Ut−2

-1.079

0.127

-8.469

0.000

dispersion

2.740

0.109

25.124

0.000

22

Table 4: The GCMR for the lag k = 3 with Simulated Data (Ut = FX (Xt ), Vt = FY (Yt )). Ut → Vt Ha : Copula Causality

Estimate Std. Error

z value

P-value

Intercept

2.6113

0.193

13.533

0.000

Vt−1

-2.8208

0.168

-16.803

0.000

Ut−1

-0.0627

0.156

-0.401

0.688

Vt−2

-1.5091

0.190

-7.951

0.000

Ut−2

-0.1735

0.173

-1.003

0.316

Vt−3

-0.9146

0.164

-5.576

0.000

Ut−3

0.1948

0.158

1.236

0.217

dispersion

3.0345

0.123

24.730

0.000

z value

P-value

H0 : No Copula Causality

Estimate Std. Error

Intercept

2.6

0.176

14.751

0.000

Vt−1

-2.8679

0.140

-20.482

0.000

Vt−2

-1.6127

0.158

-10.204

0.000

Vt−3

-0.7835

0.131

-5.968

0.000

dispersion

3.0198

0.122

24.753

0.000

z value

P-value

Vt → Ut Ha : Copula Causality

Estimate Std. Error

Intercept

2.31885

0.194

11.934

0.000

Ut−1

-2.19606

0.162

-13.550

0.000

Vt−1

-0.48465

0.166

-2.921

0.003

Ut−2

-1.34689

0.179

-7.518

0.000

Vt−2

-0.10961

0.192

-0.571

0.568

Ut−3

-0.47065

0.161

-2.927

0.003

Vt−3

-0.07986

0.167

-0.478

0.633

dispersion

2.81549

0.113

24.992

0.000

z value

P-value

H0 : No Copula Causality

Estimate Std. Error

Intercept

2.1515

0.166

12.951

0.000

Ut−1

-2.4742

0.135

-18.354

0.000

Ut−2

-1.3731

0.148

-9.293

0.000

Ut−3

-0.5079

0.129

-3.930

0.000

dispersion

2.7845

0.111

25.077

0.000

23

Table 5: P-value of Two-Sample Permutation Test for Copula Granger Causality with Different Sample Sizes from Simulated Data (Ut = FX (Xt ), Vt = FY (Yt ), Number of total observations N = 999). Lag=k

k=1

k=2

k=3

Directional dependence

ρ2Ut →Vt

ρ2Vt →Ut

ρ2Ut →Vt

ρ2Vt →Ut

ρ2Ut →Vt

ρ2Vt →Ut

H0 : No Copula Causality

0.24

0.19

0.28

0.24

0.33

0.28

Ha : Copula Causality

0.24

0.20

0.28

0.25

0.33

0.29

Lag=k

k=1

k=2

k=3

P-value

Ut → Vt

Vt → Ut

Ut → Vt

Vt → Ut

Ut → Vt

Vt → Ut

P-value with n = 800

0.3659

0.000

0.000

0.000

0.000

0.000

P-value with n = 700

0.4051

0.000

0.003

0.000

0.000

0.000

P-value with n = 600

0.4167

0.000

0.007

0.000

0.000

0.000

P-value with n = 500

0.3419

0.000

0.010

0.000

0.000

0.000

P-value with n = 400

0.3890

0.001

0.017

0.000

0.000

0.000

P-value with n = 300

0.2655

0.001

0.025

0.000

0.000

0.000

24

Table 6: Observed rejection rates (size and power) of the CNGC test for bivariate process (16) as a function of the time series length n (nominal size 0.05). F (·) ∼ N (0, 1) n

20

Size 0.092 Power (β1 = −0.25) 0.077 Power (β1 = −0.50) 0.097

50

100

200

500

1000

2000

5000

0.054 0.065 0.111

0.053 0.082 0.144

0.042 0.098 0.217

0.052 0.157 0.510

0.043 0.279 0.807

0.054 0.526 0.975

0.057 0.863 1.000

F (·) ∼ t4 n

20

Size 0.092 Power (β1 = −0.25) 0.078 Power (β1 = −0.50) 0.096

50

100

200

500

1000

2000

5000

0.055 0.065 0.110

0.053 0.082 0.144

0.042 0.098 0.217

0.052 0.157 0.511

0.043 0.279 0.807

0.054 0.526 0.975

0.057 0.863 1.000

F (·) ∼ Lognormal(0, 1) n

20

Size 0.092 Power (β1 = −0.25) 0.074 Power (β1 = −0.50) 0.096

50

100

200

500

1000

2000

5000

0.055 0.066 0.113

0.053 0.082 0.144

0.043 0.100 0.217

0.052 0.157 0.511

0.043 0.280 0.807

0.055 0.526 0.975

0.057 0.863 1.000

F (·) ∼ Beta(2/7, 7) n

20

Size 0.091 Power (β1 = −0.25) 0.077 Power (β1 = −0.50) 0.099

50

100

200

500

1000

2000

5000

0.055 0.066 0.111

0.053 0.082 0.144

0.043 0.100 0.217

0.052 0.157 0.510

0.043 0.279 0.807

0.055 0.526 0.975

0.057 0.863 1.000

25

Table 7: Observed rejection rates (size and power) of the CNGC test for bivariate AR process (17) as a function of the time series length n (nominal size 0.05). ν1t , ν2t ∼ N (0, 1) n

20

Size 0.089 Power (β1 = −0.25) 0.279 Power (β1 = −0.50) 0.603

50

100

200

500

1000

2000

5000

0.051 0.503 0.946

0.063 0.759 1.000

0.056 0.967 1.000

0.061 1.000 1.000

0.066 1.000 1.000

0.041 1.000 1.000

0.055 1.000 1.000

ν1t , ν2t ∼ t4 n

20

Size 0.103 Power (β1 = −0.25) 0.275 Power (β1 = −0.50) 0.598

50

100

200

500

1000

2000

5000

0.085 0.485 0.914

0.055 0.736 0.995

0.062 0.951 1.000

0.051 1.000 1.000

0.042 1.000 1.000

0.044 1.000 1.000

0.046 1.000 1.000

ν1t , ν2t ∼ Lognormal(0, 1) n

20

Size 0.113 Power (β1 = −0.25) 0.401 Power (β1 = −0.50) 0.674

50

100

200

500

1000

2000

5000

0.076 0.650 0.927

0.068 0.883 0.998

0.058 0.992 1.000

0.061 1.000 1.000

0.061 1.000 1.000

0.046 1.000 1.000

0.053 1.000 1.000

ν1t , ν2t ∼ Beta(2/7, 7) n

20

Size 0.134 Power (β1 = −0.25) 0.551 Power (β1 = −0.50) 0.776

50

100

200

500

1000

2000

5000

0.075 0.690 0.971

0.058 0.895 1.000

0.056 0.987 1.000

0.057 1.000 1.000

0.053 1.000 1.000

0.047 1.000 1.000

0.051 1.000 1.000

Table 8: Stationary Test by ADF and KPSS with Real Data (Xt =Log-returns of Arabica and Yt =Log-returns of Robusta, in percentage). P-value of ADF Test P-value of KPSS Test Arabica

0.2937

0.01

Robusta

0.2715

0.01

Log-returns of Arabica

0.01

0.1

Log-returns of Robusta

0.01

0.1

26

Table 9: Granger Causality with Real Data (Xt =Log-returns of Arabica and Yt =Log-returns of Robusta, in percentage): p-values. Method

Direction

Linear Granger Test

Xt → Yt

0.1416 0.1829 0.3346

Yt → Xt

0.4568 0.5784 0.3269

Xt → Yt

0.0004

Yt → Xt

0.0074 0.0072 0.0083

Xt → Yt

0.0017 0.0026 0.0045

Yt → Xt

0.0044 0.0076 0.0033

Nonlinear DP Test ( = 1)

Nonlinear DP Test ( = 1.5)

27

k=1

k=2

0.001

k=3

0.0135

Table 10: The GCMR for the lag k = 1 with Real Data (Ut = FX (Xt ), Vt = FY (Yt )). Ut → Vt Ha : Copula Causality

Estimate Std. Error

z value P-value

Intercept

-0.6675

0.0938

-7.117

0.000

Vt−1

0.8351

0.2037

4.100

0.000

Ut−1

0.4975

0.2043

2.436

0.015

dispersion

2.2385

0.1100

20.347

0.000

H0 : No Copula Causality

Estimate Std. Error

z value P-value

Intercept

-0.5857

0.0876

-6.683

0.000

Vt−1

1.1688

0.1516

7.709

0.000

dispersion

2.2163

0.1086

20.406

0.000

Vt → Ut Ha : Copula Causality

Estimate Std. Error

z value P-value

Intercept

-0.5278

0.094

-5.613

0.000

Ut−1

1.1077

0.2044

5.420

0.000

Vt−1

-0.0529

0.2033

-0.260

0.795

dispersion

2.1830

0.1066

20.480

0.000

H0 : No Copula Causality Estimate Std. Error

z value P-value

Intercept

-0.5365

0.0879

-6.107

0.000

Ut−1

1.0722

0.1524

7.038

0.000

dispersion

2.1828

0.1066

20.469

0.000

28

Table 11: The GCMR for the lag k = 2 with Real Data (Ut = FX (Xt ), Vt = FY (Yt )). Ut → Vt Ha : Copula Causality

Estimate Std. Error

z value P-value

Intercept

-0.6819

0.1135

-6.009

0.000

Vt−1

0.8338

0.2103

3.966

0.000

Ut−1

0.4848

0.2091

2.319

0.0204

Vt−2

-0.0274

0.2088

-0.131

0.8957

Ut−2

0.0702

0.2069

0.339

0.7343

dispersion

2.2389

0.1099

20.379

0.000

H0 : No Copula Causality

Estimate Std. Error

z value P-value

Intercept

-0.5835

0.1049

-5.560

0.000

Vt−1

1.1705

0.1574

7.434

0.000

Vt−2

-0.0061

0.1570

-0.039

0.969

dispersion

2.2164

0.1087

20.394

0.000

Vt → Ut Ha : Copula Causality

Estimate Std. Error

z value P-value

Intercept

-0.6212

0.1134

-5.476

0.000

Ut−1

-0.1226

0.2093

-0.586

0.558

Vt−1

0.0487

0.2096

0.232

0.816

Ut−2

0.002

0.209

0.009

0.993

Vt−2

0.1992

0.2108

0.945

0.345

dispersion

2.1914

0.1070

20.472

0.000

H0 : No Copula Causality

Estimate Std. Error

z value P-value

Intercept

-0.6036

0.1060

-5.697

0.000

Ut−1

1.0304

0.1567

6.575

0.000

Ut−2

0.1757

0.1546

1.137

0.000

dispersion

2.1876

0.1068

20.474

0.000

29

Table 12: The GCMR for the lag k = 3 with Real Data (Ut = FX (Xt ), Vt = FY (Yt )). Ut → Vt Ha : Copula Causality

Estimate Std. Error

z value P-value

Intercept

-0.7664

0.1316

-5.825

0.000

Vt−1

0.8516

0.2104

4.048

0.000

Ut−1

0.47137

0.2089

2.256

0.0241

Vt−2

-0.0863

0.2147

-0.402

0.6877

Ut−2

0.0771

0.2109

0.366

0.7147

Vt−3

0.1694

0.2062

0.822

0.4114

Ut−3

0.0459

0.2065

0.222

0.8240

dispersion

2.2458

0.1102

20.382

0.000

H0 : No Copula Causality

Estimate Std. Error

z value P-value

Intercept

-0.6662

0.1220

-5.462

0.000

Vt−1

1.1792

0.1577

7.478

0.000

Vt−2

-0.0594

0.1618

-0.367

0.714

Vt−3

0.2089

0.1557

1.342

0.180

dispersion

2.2236

0.1091

20.379

0.000

Vt → Ut Ha : Copula Causality

Estimate Std. Error

z value P-value

Intercept

-0.6984

0.1308

-5.341

0.000

Ut−1

1.0965

0.2084

5.261

0.000

Vt−1

-0.1043

0.2093

-0.498

0.6183

Ut−2

0.1245

0.2129

0.585

0.5586

Vt−2

0.0865

0.2173

0.398

0.6907

Ut−3

-0.2771

0.2075

-1.336

0.1817

Vt−3

0.4695

0.2067

2.271

0.0232

dispersion

2.2103

0.1083

20.405

0.000

H0 : No Copula Causality

Estimate Std. Error

z value P-value

Intercept

-0.6227

0.1223

-5.093

0.000

Ut−1

1.0304

0.1568

6.572

0.000

Ut−2

0.1648

0.1585

1.039

0.299

Ut−3

0.0490

0.1565

0.313

0.754

dispersion

2.1880

0.1069

20.470

0.000

30

Table 13: P-value of Two-Sample Permutation Test for Copula Granger Causality with Different Sample Sizes from Real Data (Ut = FX (Xt ), Vt = FY (Yt ), Number of total observations N = 614). Lag=k

k=1

k=2

k=3

Directional dependence

ρ2Ut →Vt

ρ2Vt →Ut

ρ2Ut →Vt

ρ2Vt →Ut

ρ2Ut →Vt

ρ2Vt →Ut

H0 : No Copula Causality

0.0744

0.0675

0.0746

0.0692

0.0787

0.0703

Ha : Copula Causality

0.0855

0.0678

0.086

0.072

0.0884

0.0796

Lag=k

k=1

k=2

k=3

P-value

Ut → Vt

Vt → Ut

Ut → Vt

Vt → Ut

Ut → Vt

Vt → Ut

P-value with 90 % of n = 550

0.000

0.4794

0.000

0.000

0.000

0.000

P-value with 80 % of n = 491

0.000

0.3722

0.000

0.000

0.000

0.000

P-value with 70 % of n = 430

0.000

0.3187

0.000

0.000

0.000

0.000

P-value with 60 % of n = 368

0.000

0.2529

0.000

0.000

0.000

0.000

P-value with 50 % of n = 307

0.000

0.1718

0.010

0.000

0.000

0.000

P-value with 40 % of n = 246

0.000

0.0751

0.000

0.000

0.000

0.000

31

Highlights  We propose a new copula nonlinear Granger causality test.  Both the Gaussian Copula Marginal Regression model and the concept of copula directional dependence are employed to construct the test.  The permutation test with rank sum statistic is used to confirm the proposed test.  Monte Carlo simulation results demonstrate robustness to distributional assumptions and relatively small sample sizes.  A real data analysis is conducted to illustrate our test.