The Hiemstra-Jones Test Revisited Zhidong Bai School of Mathematics and Statistics, Northeast Normal University Yongchang Hui1 7 1 0 School of Mathematics and Statistics, Xi’an Jiaotong University 2 n Zhihui Lv a J 5 School of Mathematics and Statistics, Northeast Normal University 1 ] Wing-Keung Wong E M Department of Finance, Asia University, Taiwan . t a Department of Economics, Lingnan University, Hong Kong t s [ Zhen-Zhen Zhu 1 v 2 School of Mathematics and Statistics, Northeast Normal University 9 9 3 0 Abstract The famous Hiemstra-Jones (HJ) test developed by Hiemstra and Jones . 1 (1994) plays a significant role in studying nonlinear causality. Over the last two decades, 0 7 there have been numerous applications and theoretical extensions based on this pioneer- 1 : ing work. However, several works note that counterintuitive results are obtained from v i the HJ test, and some researchers find that the HJ test is seriously over-rejecting in sim- X r ulation studies. In this paper, we reinvestigate HJ’s creative 1994 work and find that a their proposed estimators of the probabilities over different time intervals were not consistent with the target ones proposed in their criterion. To test HJ’s novel hypothesis on Granger causality, we propose new estimators of the probabilities defined in their paper and reestablish the asymptotic properties to induce new tests similar to those of HJ. Some simulations will also be presented to support our findings. Keywords: Central limit theorem, Hiemstra-Jones test, Nonlinear Granger causality. 1 Yongchang Hui, School of Mathematics and Statistics, Xi’an Jiaotong University. No.28, Xianning West Road, Xi’an, Shaanxi, P.R. China. Tel: (86)-029-82663170,Email: huiyc180@xjtu.edu.cn . Conflict-of-interest disclosure statement 1. Zhidong Bai. I declare that there is no conflict of interest. 2. Yongchang Hui. I declare that there is no conflict of interest. 3. Zhihui Lv. I declare that there is no conflict of interest. 4. Wing-Keung Wong. I declare that there is no conflict of interest. 5. Zhen-Zhen Zhu. I declare that there is no conflict of interest. 1 Introduction After the pioneering work of Granger (1969), Granger causality tests have developed into a set of useful methods to detect causal relations between time series in economics and finance. Consider a strictly stationary bivariate time series (X ,Y ) , t Z. Intuitively, t t { } ∈ Y is a Granger cause of X if adding past observations of Y to the information set t t t { } { } increases knowledge about the distribution of current values of X . t Linear Granger causality tests within the linear autoregressive model class have been developed in many directions, e.g., Hurlin et al (2001) proposed a procedure for causality tests with panel data, Bai et al (2008) extend the traditional bivariate Granger causality test to multivariate situations, Ghysels et al (2016) test for Granger causality with mixed frequency data based on the multiple-horizon framework established by Dufour and Renault (1998) and Dufour et al (2006). Though linear tests of Granger causality have been investigated very deeply, they are limited in their capability to detect nonlinear causality. There is no need emphasize the importance of nonlinear structures between variables, since the real world is “almost certainly nonlinear,” as Granger (1989) notes. Modern developments in computer science and computing facilities motivate ever increasing interest in testing nonlinear Granger causality. Among the varioustests of nonlinear Granger causality, theHiemstra andJones (1994) test (hereafter, the HJ test) is the most cited by scholars and the most frequently applied by practitioners in economics and finance. There were over 1100 Google Scholar hits by September 2016, which illustrates its significance in the economics and finance literatures. However, some doubts about the efficacy of the HJ test arise from the many counterintuitive results. Diks and Panchenko (2005) find two serious problems with the HJ test. First, even if there is a strong evidence of linear Granger causality, the HJ test can fail to detect causality. Second, using simulation studies, they show that under the null hypothesis, the reject rate tends to 1 when sample size increases. In accordance with the evidence presented by Diks and Panchenko (2005, 2006), in this paper, we reinvestigate the HJ test (1994) and reveal some of the underlying reasons for this questionable performance. The remainder of this paper is organized as follows. In Section 2, we simply review the procedure of the HJ test. In Section 3, we describe the crux of the problem identified by Diks and Panchenko (2005) and revise it accordingly. Specifically, we re-estimate the probabilities defined by Hiemstra and Jones (1994) and 1 deduce the asymptotic distribution of the test statistics. Simulation results are presented in Section 4. Finally, we provide some concluding remarks in Section 5. 2 Hiemstra-Jones Nonlinear Causality Test Hiemstra and Jones (1994) consider a causality test between two strictly stationary and weakly dependent time series processes X and Y . The m-length lead vector of X , t t t { } { } L -length lag vector of X and L -length lag vector of Y are defined as x t y t Xm X ,X , ,X , m = 1,2, , t = 1,2, t ≡ t t+1 ··· t+m−1 ··· ··· XLx (cid:0)X ,X , ,X (cid:1) ,L = 1,2, ,t = L +1,L +2, t−Lx ≡ t−Lx t−Lx ··· t−1 x ··· x x ··· YLy (cid:0)Y ,Y , ,Y ,(cid:1)L = 1,2, ,t = L +1,L +2, . t−Ly ≡ t−Ly t−Ly ··· t−1 y ··· y y ··· (cid:0) (cid:1) Hiemstra and Jones (1994) define non-Granger causality from Y to X . t t { } { } Definition 2.1. For any given values of m, L , L > 1 and e > 0, series Y does x y t { } not strictly Granger cause X if t { } P Xm Xm < e XLx XLx < e, YLy YLy < e k t − s k | k t−Lx − s−Lx k k t−Ly − s−Ly k (cid:16) = P Xm Xm < e XLx XLx < e (cid:17), (1) k t − s k | k t−Lx − s−Lx k (cid:0) (cid:1) where Pr( ) denotes the conditional probability and denotes the maximum norm, ·|· k · k which is defined as X Y = max x y , x y , , x y for any two vectors 1 1 2 2 n n k − k | − | | − | ··· | − | X = x1, ,xn and Y = y1, ,(cid:0)yn . (cid:1) ··· ··· (cid:0) (cid:1) (cid:0) (cid:1) Using the notation C m+L ,L ,e Pr Xm+Lx Xm+Lx < e, YLy YLy < e , 1 x y ≡ k t−Lx − s−Lx k k t−Ly − s−Ly k (cid:16) (cid:17) C (cid:0)L ,L ,e P(cid:1)r XLx XLx < e, YLy YLy < e , 2 x y ≡ k t−Lx − s−Lx k k t−Ly − s−Ly k (cid:16) (cid:17) C (cid:0)m+L ,(cid:1)e Pr Xm+Lx Xm+Lx < e , and 3 x ≡ k t−Lx − s−Lx k C (cid:0)L ,e P(cid:1)r X(cid:0)Lx XLx < e , (cid:1) 4 x ≡ k t−Lx − s−Lx k (cid:0) (cid:1) (cid:0) (cid:1) 2 Hiemstra and Jones (1994) re-express Equation (1) as C m+L ,L ,e C m+L ,e 1 x y 3 x = . (2) C L ,L ,e C L ,e (cid:0)2 x y (cid:1) (cid:0)4 x (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) After this preparation, they propose the following nonlinear Granger causality test statistic C m+L ,L ,e,n C m+L ,e,n 1 x y 3 x √n , (3) C L ,L ,e,n − C L ,e,n (cid:0)2 x y (cid:1) (cid:0)4 x (cid:1)! where (cid:0) (cid:1) (cid:0) (cid:1) 2 C m+L ,L ,e,n I xm+Lx,xm+Lx,e I yLy ,yLy ,e , 1 x y ≡ n(n 1) t−Lx s−Lx · t−Ly s−Ly (cid:0) (cid:1) − Xt e I(x,y,e) = k − k .  1, if x y e  k − k ≤   They claimed that theC ( ,n)swere U-statistic estimators of their counterparts C ( ) j j ∗ ∗ and tried to show the limiting results for the test statistics (3). Although the estimators C ( ,n) looked like U-statistics, they were not because the expectations of the general j ∗ terms are not the same. Moreover, the C ( )s are related to the indices t and s (in fact, j ∗ to t s for strongly stationary processes). The C ( ,n)s were independent of t and s j | − | ∗ for summing up over them. Therefore, the C ( ,n) estimators are neither consistent nor j ∗ asymptotic normal estimators of their counterparts C ( ). Based on this analysis, one j ∗ sees that the center of statistic (3) should tend toward infinity; hence, the test must be over-rejecting when the sample size is large. 3 A new test of Hiemstra-Jones Nonlinear Causality It is worth reminding the reader that the pair (s,t) (in fact, t s for strongly stationary | − | processes) in Equation (1) of Definition 2.1 is a key parameter of the probabilities C ( ). j ∗ 3 In fact, Hiemstra and Jones (1994) note this, and there is no problem in Equation (1) of Definition 2.1. However, it seems that Hiemstra and Jones (1994) overlooked this fact in their proposed estimation of C ( ). The improper estimators C ( ,n) thus lead to an j j ∗ ∗ invalid asymptotic distribution of the test statistic. We now begin to state the procedure for our new test. For any given pair (s,t), we denote C m+L ,L ,e;t,s Pr Xm+Lx Xm+Lx < e, YLy YLy < e , 1 x y ≡ k t−Lx − s−Lx k k t−Ly − s−Ly k (cid:16) (cid:17) C (cid:0)L ,L ,e;t,s P(cid:1)r XLx XLx < e, YLy YLy < e , 2 x y ≡ k t−Lx − s−Lx k k t−Ly − s−Ly k (cid:16) (cid:17) C (cid:0)m+L ,e;t,(cid:1)s Pr Xm+Lx Xm+Lx < e , and 3 x ≡ k t−Lx − s−Lx k C (cid:0)L ,e;t,s P(cid:1)r X(cid:0)Lx XLx < e . (cid:1) 4 x ≡ k t−Lx − s−Lx k (cid:0) (cid:1) (cid:0) (cid:1) Furthermore, we have Pr Xm Xm < e XLx XLx < e, YLy YLy < e k t − s k k t−Lx − s−Lx k k t−Ly − s−Ly k (cid:16) C m+L ,L ,e(cid:12);t,s (cid:17) = 1 x y (cid:12) , C L ,L ,e;t,s (cid:0)2 x y (cid:1) (cid:0) (cid:1) and Pr Xm Xm < e XLx XLx < e k t − s k k t−Lx − s−Lx k (cid:0) C3 m+Lx,e;t,(cid:12)s (cid:1) = (cid:12) . C L ,e;t,s (cid:0)4 x (cid:1) (cid:0) (cid:1) Under the assumption of stationarity, for the given (t,s) with s t = l, we express − C m+L ,L ,e;t,s C m+L ,L ,e;t,l = C m+L ,L ,e;l , 1 x y 1 x y 1 x y ≡ C (cid:0)L ,L ,e;t,s C(cid:1) L ,L(cid:0) ,e;t,l = C L(cid:1),L ,e;(cid:0)l , (cid:1) 2 x y 2 x y 2 x y ≡ C (cid:0)m+L ,e;t,(cid:1)s C(cid:0) m+L ,e;t(cid:1),l = C(cid:0) m+L ,(cid:1)e;l , and 3 x 3 x 3 x ≡ C (cid:0)L ,e;t,s C(cid:1) L ,e(cid:0);t,l = C L(cid:1),e;l .(cid:0) (cid:1) 4 x 4 x 4 x ≡ (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) Thus, Y does not strictly Granger cause another series X nonlinearly, which t t { } { } meansthatforeachl > 0,C m+L ,L ,e;l /C L ,L ,e;l = C m+L ,e;l /C L ,e;l . 1 x y 2 x y 3 x 4 x (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) 4 If we now consider two sets of samples x ,x , ,x and y ,y , ,y , we can 1 2 T 1 2 T { ··· } { ··· } examine whether there is nonlinear Granger causality from Y to X . That is, we test t t { } { } the following hypothesis C m+L ,L ,e;l C m+L ,e;l 1 x y 3 x H : = . (4) 0 C L ,L ,e;l C L ,e;l (cid:0)2 x y (cid:1) (cid:0)4 x (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) We first provide the consistent estimators of C m+L ,L ,e;l , C L ,L ,e;l , 1 x y 2 x y C3 m+Lx,e;l and C4 Lx,e;l (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) T−l−m+1 1 Cˆ m+L ,L ,e;l I xm+Lx,xm+Lx ,e I yLy ,yLy ,e , 1 x y ≡ n t−Lx t+l−Lx · t−Ly t+l−Ly (cid:0) (cid:1) t=XLxy+1 (cid:0) (cid:1) (cid:16) (cid:17) T−l−m+1 1 Cˆ L ,L ,e;l I xLx ,xLx ,e I yLy ,yLy ,e , 2 x y ≡ n t−Lx t+l−Lx · t−Ly t+l−Ly (cid:0) (cid:1) t=XLxy+1 (cid:0) (cid:1) (cid:16) (cid:17) T−l−m+1 1 Cˆ m+L ,e;l I xm+Lx,xm+Lx ,e , 3 x ≡ n t−Lx t+l−Lx (cid:0) (cid:1) t=XLxy+1 (cid:0) (cid:1) T−l−m+1 1 Cˆ L ,e;l I xLx ,xLx ,e , 4 x ≡ n t−Lx t+l−Lx (cid:0) (cid:1) t=XLxy+1 (cid:0) (cid:1) 0, if x y > e where L = max(L ,L ), I(x,y,e) = k − k xy x y  1, if x y e  k − k ≤ and n = T L l m+1. − xy − −  The consistency of our proposed estimators can be shown straightforwardly and is omitted from this paper. We use a simple numerical study to show that our estimators iid are consistent whereas those of HJ are not. Let X = 2ε +ε , ε N(0,1), while Y t t−1 t t t ∼ { } could be any stationary sequence. Let l = 1, L = L = m = 1. We can calculate the x y exact values of C L ,e;l , which are 0.3169 and 0.4597, respectively, when e = 1 and 4 x e = 1.5. For simpl(cid:0)icity, w(cid:1)e denote the values of C4 Lx,e;l as C4 and the HJ estimate and our estimate Cˆ4HJ and Cˆ4, respectively, in Tab(cid:0)le 1. A(cid:1)dditionally, Table 1 provides the estimated values with their corresponding relative estimation errors in brackets when T = 1000,2000and4000. It is obvious that the HJ estimator is not consistent. 5 Table 1: C L ,e;l and its estimated values. 4 x e=1 (cid:0) (cid:1) e=1.5 T = C4 Cˆ4 Cˆ4HJ C4 Cˆ4 Cˆ4HJ 1000 0.3169 0.3056(3.56%) 0.2564(19.0%) 0.4597 0.4529(1.46%) 0.3755(18.2%) 2000 0.3169 0.3109(1.89%) 0.2531(20.1%) 0.4597 0.4629(0.69%) 0.3718(19.1%) 4000 0.3169 0.3128(1.29%) 0.2474(21.9%) 0.4597 0.4599(0.05%) 0.3636(20.9%) Note: The true value of C L ,e;l is denoted C , the HJ estimate and our estimate are 4 x 4 denoted CˆHJ and Cˆ , respectively. The relative estimation errors are in the accompanying 4 4 (cid:0) (cid:1) brackets. Now, we propose Cˆ m+L ,L ,e,l Cˆ m+L ,e,l 1 x y 3 x T = √n (5) n C(cid:0)ˆ2 Lx,Ly,e,l (cid:1) − C(cid:0)ˆ4 Lx,e,l (cid:1)! (cid:0) (cid:1) (cid:0) (cid:1) as the test statistic, and we establish the following asymptotic distribution of T for n statistical inference. Theorem 3.1. Stationary sequences x ,t = 1, ,T and y ,t = 1, ,T are both t t { ··· } { ··· } strong mixing, with mixing coefficients satisfying the conditions of Lemma 1 presented in Appendix, for given values of l,L ,L ,m and e > 0, under the null hypothesis that y x y t { } does not strictly Granger cause x , then the test statistic is defined in (3) t { } Cˆ m+L ,L ,e,l Cˆ m+L ,e,l √n 1 x y 3 x d N 0,σ2(m,L ,L ,e,l) . C(cid:0)ˆ2 Lx,Ly,e,l (cid:1) − C(cid:0)ˆ4 Lx,e,l (cid:1)! −→ x y (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) Theasymptoticvarianceσ2(m,L ,L ,e,l)withitsconsistentestimatorσˆ2(m,L ,L ,e,l) x y x y and the proof of theorem 3.1 are given in the Appendix. The hypothesis H defined in 0 (4) is rejected at α if T /σˆ2(m,L ,L ,e,l) > z , n x y α/2 (cid:12) (cid:12) (cid:12) (cid:12) where z is the up α/2 quantile of the standard normal distribution. In this situation, α/2 we will conclude that there exists nonlinear Granger causality from Y to X . t t { } { } Thereareseveralpossiblemethodstoestimatetheasymptoticcovarianceσ2(m,L ,L ,e,l). x y A model-based approach uses known laws of X and Y to calculate the expectations t t { } { } in the formula given in the Appendix and simply substitutes C ( ),j = 1,2,3,4with their j ∗ corresponding estimates. However, in practice, we can hardly avoid model misspecifica- 6 tion and may obtain improper laws of X and Y . We suggest the use of bootstrap t t { } { } methods as in the simulation studies we use to test hypothesis H . 0 4 Simulation In this section, we perform numerical studies using simulations to illustrate the applica- bility and superiority of the new nonlinear Granger causality test developed in Section 3. Let R be the times of rejecting the null hypothesis that Y does not strictly Granger t { } cause X nonlinearly in 10,000 replications at the α level, and thus, the empirical power t { } is R/10,000. In our simulation, the length of the testing sequences is 1000, and we chose the same lag length and lead length: L = L = m = 1. We set three situations of l and x y two situations of e: l = 1, l = 2, l = 3 and e = 1, e = 1.5. Consider the following two cases. 1 ρ iid Case 1: (X ,Y ) N(0,Σ), Σ = . ρ = 0,0.2, ,0.8. t t−1 • ∼  ρ 1  ···   Case 2: X = 1+0.4X2 ε , Y = ρX +η , where • t t−1 t t−1 t t iid iid εt ηt, εt N(p0,1), ηt N(0,1). ρ = 0,0.2, ,0.8. ⊥ ∼ ∼ ··· In neither case is there nonlinear Granger causality from Y to X when ρ = 0, and t t causality strengthens when ρ increases. From the results displayed in Figure 1 and Figure 2, we conclude first that there is no over-rejection problem in our new test. Second, our test possesses very appropriate power, as we see that empirical power sharply increases to 1 as ρ increases. Further, we find that different settings of e may influence detection, and we suggest that practitioners choose a couple of different values. 7 Figure 1: Test nonlinear Granger causality form Y to X : Case 1 t t l=1, e=1 l=1, e=1.5 1.0 1.0 0.8 0.8 Power 0.6 Power 0.6 Empirical 0.4 Empirical 0.4 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 rho rho l=2, e=1 l=2, e=1.5 1.0 1.0 0.8 0.8 Power 0.6 Power 0.6 Empirical 0.4 Empirical 0.4 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 rho rho l=3, e=1 l=3, e=1.5 1.0 1.0 0.8 0.8 Power 0.6 Power 0.6 Empirical 0.4 Empirical 0.4 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 rho rho Note: L = L = m = 1 in our test. Simulation is conducted with the test level α = 5%, and x y 10,000 replications. 8

The Hiemstra-Jones Test Revisited

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