Volume 39, Issue 3
Contemporaneous Causal Orderings of CSI300 and Futures Prices throughDirected Acyclic Graphs
Xiaojie Xu
North Carolina State University

Abstract
This paper examines contemporaneous causality among daily price series of the Chinese Stock Index 300 (CSI300),
nearby futures, and first distant futures for April 2010 ~ November 2014 through vector error correction modeling anddirected acyclic graphs. As non-Gaussian data are prominent in financial time series, the recently developed Linear
Non-Gaussian Acyclic Model (LiNGAM) algorithm is utilized to facilitate analysis. It refines results derived from thePC algorithm, which does not lead to the unique identification of a directed acyclic graph. The price series studied aretied together through cointegration and the nearby futures adjusts towards long-run relationships. Contemporaneous
price information is determined to be discovered in the nearby futures. The results suggest that a shock to the nearby
futures could have long-lasting effects on prices across the three series under consideration. Policy makers should pay
close attention to the nearby futures for financial stability.

Citation: Xiaojie Xu, (2019) ''Contemporaneous Causal Orderings of CSI300 and Futures Prices through Directed Acyclic Graphs'',
Economics Bulletin, Volume 39, Issue 3, pages 2052-2077
Contact: Xiaojie Xu - xxu6@ncsu.edu.
Submitted: March 06, 2019. Published: September 07, 2019.

1.

Introduction

The stock market in China was established in the early 1990s and has developed dramaticallysince then. However, there had been no index designed to reflect the overall performance ofstocks until China launched the Chinese Stock Index 300 (CSI300) on April 8, 2005, which
covers 300 stocks listed in the Shanghai Stock Exchange and Shenzhen Stock Exchange
and represents about 70% of total market capitalization of these two exchanges. To further
facilitate development of financial markets and risk management systems, the CSI300 futures
came into investors’ eyes on April 16, 2010. After its operating for several years, it is essential
to explore price information flows between the CSI300 and futures market, which are one ofinvestors and regulators’ central concerns.
The existing literature on price information flows has primarily concentrated on developed
economies. And previous studies generally indicate unidirectional leadership of a futures
against an associated spot market (Yang et al., 2012), though different empirical evidence
is presented in the literature. While this issue in emerging economies such as China has notreceived so much attention as compared to developed countries, it is becoming eye-catchingto economists and investors. High trading volumes of stocks and futures index contracts
that keep increasing in recent years provide strong supporting evidence of this point. In
fact, the CSI300 futures is one of the most actively traded contracts in the world. Further,unique market structures including high barriers for most domestic individual investors andqualified foreign institutional investors to participate in futures trading and dominance ofindividual over institutional investors in the stock market (Ng and Wu, 2007) suggest that
information flows between the CSI300 and futures are worthwhile avenues for research.
There have been two studies in the literature focusing on the issue of causal directions
between the CSI300 and futures market. Yang et al. (2012) investigate price dynamics between the CSI300 and nearby futures for the first three months since the launch of the futures
market and find that futures are overshadowed by spot prices in information leadership in
the long run. Utilizing data for each of March, May, July, September, and November of 2011
and January and March of 2012, Hou and Li (2013) show that futures lead spot prices in
the long run and short run except for March 2011 during which information feedback in the
short run is discovered. Therefore, the information importance of futures against spot prices
seems to be increasing.
Although there are extensive studies adopting time-series models to investigate lagged
causal relationships among prices, issues of contemporaneous causality, which is difficult toinfer in non-structural models, are to be confronted if one wants to understand contemporaneous consequences of shocks or interventions, whose occurrences are not rare in China due
to factors such as policy interventions and financial reform and development. Therefore, thepurpose of this article is to contribute to an understanding of contemporaneous causality
in the CSI300 and futures market. Directed acyclic graphs (DAGs) facilitate such analysis
because they identify structural models through data-determined orthogonalization of thecontemporaneous innovation covariance, which is critical in providing inference in innovation accounting (Swanson and Granger 1997). Applications in economics include Bessler
et al. (2003) and Wang (2010a). To the author’s knowledge, the only work dealing with
contemporaneous causality among spot and futures prices is by Chopra and Bessler (2005).
Continuing in this vein, two algorithms are considered in the current study for DAG

inference among the CSI300, nearby futures, and first distant futures, avoiding possible
spurious causality due to omission of important variables (Wang et al., 2007) in a bivariate
model only incorporating the CSI300 and nearby futures as in previous studies (e.g., Yanget al., 2012; Hou and Li, 2013). The PC algorithm, one of the earliest and most widely-usedapproaches, based on conditional independence is first applied to search for causal orderings.
For empirical analysis, however, there could exist observational equivalence and the DAG
will not be uniquely identified using this algorithm (Lai and Bessler, 2015; Moneta et al.,2013). The current study also encounters this problem. The recently developed Linear
Non-Gaussian Acyclic Model (LiNGAM) algorithm (Shimizu et al., 2006) is then adopted
because its requirement of non-Gaussian data is satisfied in this study and it suggests a
promising approach to identify a unique DAG. Based on the structural model derived from a
DAG, innovation accounting analysis, i.e. forecast error variance decompositions and impulse
responses, is conducted to provide insight into contemporaneous consequences of shocks orinterventions.
This study extends previous research to a recent period 2010 ∼ 2014 and to the author’s
knowledge, represents the first attempt to explore contemporaneous causality in the CSI300
and futures market. Results here could benefit market participants by providing them witha relatively new view of market interdependence and directions of causation within themarkets.
2.

Literature Review

To examine price relationships between spot and futures markets, lead-lag causality has
drawn economists’ attention1 . Meanwhile, because spot and futures prices of many financial
indexes are nonstationary, the notion of cointegration (Engle and Granger, 1987) is widelyadopted in the literature. These include Chan (1992), Ghosh (1993), Tse (1995), Kim et al.
(1999), Lin et al. (2002), Xu (2015, 2017c, 2019b), and Xu and Thurman (2015b).
Theoretically, because spot and futures prices adjust instantaneously to incorporate new
information under efficient markets where no profitable arbitrage opportunities are possible,
no lead-lag relationship is to be expected. Empirical results, however, are mixed on this issue.
Nonetheless, futures markets are found to be price leading sources more often as compared to
cash markets. This could be due to advantages such as low transaction costs, high leverageand low initial outlays, great transparency and liquidity and short selling opportunities infutures markets. Further, for most developed economies, it is widely perceived that the index
futures leads the spot index and plays the dominant role in price discovery (Hou and Li,2013).
1

The current study focuses on linear lead-lag causality. Another strand of the literature explores nonlinear relationships among time-series, which might be ignored by linear causality tests (Shu and Zhang,2012; Xu, 2014b, 2018c). For example, for the crude oil market, Silvapulle and Moosa (1999) find that
futures unidirectionally lead spot prices based on the linear causality test while the bidirectional leadershipis identified with the non-linear test. More recent applications of the non-linear causality test to the crude oilmarket include Bekiros and Diks (2008) and Lee and Zeng (2011). Xu (2018a) explores non-linear causality
between the Chinese Stock Index 300 and its futures. There also exist studies which investigate causality
in both time and frequency domains (e.g., Xu, 2018d) and differences between in-sample and out-of-sample
causality (e.g., Xu and Thurman, 2015a; Xu, 2018e).

There also exits empirical evidence for cash leading futures prices. For example, Moosa(1996) finds that crude oil market participants’ action is trigged by the spot price andthe futures adjusts subsequently. Rosenberg and Traub (2006) discover that the amountof information contained in currency spot prices is greater than that in futures, possibly
due to an increase in spot market transparency. Kawaller et al. (1988) state that spot andfutures prices are affected by their own histories, each other’s movements and current market
information. They point out that lead-lag patterns change dynamically with the arrival of
new information. And, at any time, one price might lead the other as market participants
filter information relevant to spot or futures positions. Tang et al. (1992) reveal bidirectionalcausality between the Hang Sang index and its futures in the post-crash period. Wahab andLashgari (1993) discover feedback between the cash and futures market for the S&P 500 and
FT-SE 100 index.
Although there are extensive studies adopting time-series models to investigate lead-lag
causal relationships, one must confront issues of contemporaneous causality if one wants
to understand contemporaneous consequences of shocks or interventions. Contemporaneous
causality is difficult to infer in non-structural models and DAGs have been utilized in theliterature to construct data-determined orthogonalization of the contemporaneous innovation
covariance, which is critical in providing inference in innovation accounting (Swanson andGranger, 1997)2 . The issue of contemporaneous causality has been explored for different
commodity markets, including farm and retail prices for pork and beef (Bessler and Akleman,1998), international wheat prices (Bessler et al., 2003), regional soybean prices (Haigh and
Bessler, 2004) and corn prices (Xu, 2014a, 2017a, 2019a), Indian black pepper prices (Chopra
and Bessler, 2005), and Chinese rice prices (Awokuse, 2007). It also has drawn researchers’
attention in areas of the US economy - relationships among the real GNP, real business
investment, GNP price deflator, M1 measure of money, unemployment, and Treasury-billrate (Awokuse and Bessler, 2003), international stock markets (Bessler and Yang, 2003;
Yang, 2003; Yang and Bessler, 2004), and Eurocurrency exchange rates (Wang et al., 2007).
The current study investigates contemporaneous causality among daily price series of the
CSI300, its nearby futures, and first distant futures.
As compared to the bivariate framework in previous studies (e.g., Yang et al., 2012; Houand Li, 2013), the multivariate framework in the current study allows for causal influencesof the CSI300 and two closely related futures, avoiding possible spurious causality due toomission of important variables (Wang et al., 2007)3 . While Yang et al. (2012) use only threemonth data and Hou and Li (2013) utilize data from a certain month a time among seven2

An improvement in innovation accounting analysis is to be expected over the traditional approaches that
impose causal structure restrictions based on human judgement and/or economic theories, especially when
limited prior knowledge exists on market interrelationships (Awokuse, 2007).
3
Xu (2017b) investigates the relationship between the CSI300 and its nearby futures and finds that these
two series adjust equally toward the long-run relationship. However, considering the non-trivial trading
volume of the first distant futures, it might also contribute to price dynamics of the three series. To measure
relative quantitative importance of futures as compared to spot prices, Hasbrouck’s (1995) information sharemodel is utilized (see Xu (2018b) for an introduction and a similar empirical application). The test basedon the CSI300 and its nearby futures shows that the former has an information share of 0.529 and the latter
0.471. Similarly, the test based on the CSI300 and its first distant futures shows that the former has an
information share of 0.514 and the latter 0.486. Therefore, the first distant futures should be an importantseries to be considered for analyzing price dynamics in the current study.

months they consider to approach the price dynamic problem, the sample examined here
covers a period of four years and a half, a much longer series that facilitates cointegration
modeling and analysis building on it.
3.

Data

Daily closing prices of the CSI300, nearby futures, and first distant futures are obtained
from Wind Information Co., Ltd. The futures is a financial contract for which the CSI300
serves as the underlying asset. The contract size is the index value of CSI300 multiplied by
RMB 300. The nearby futures refers to all most recent one month contracts concatenated.
It is constructed in a continuous way and reflects a dynamic concept. If today is April 11,
2019, the nearby futures is IF1904, where “IF” refers to the futures, “19” the year, and “04”the April contracts, and the first distant futures is IF1905. If the date exceeds April 15,
2019, the delivery date of IF1904, the nearby futures becomes IF1905 and the first distantfutures becomes IF19064 . The sample ranges from April 16, 2010, the launch date of the
CSI300 futures, to November 14, 2014, resulting in 1,112 observations. For the rest of thisstudy, prices are converted to their natural logarithms5 . Descriptive information of different
series is exhibited in Figure 1 and Table 2. To test for non-stationarity, two tests are usedthat set the null hypothesis of a unit root: the augmented Dickey-Fuller test (ADF; Dickey
and Fuller, 1981) and the Phillips-Perron test (PP; Phillips and Perron, 1988). Because
failure to reject the null of a unit root does not imply that a unit root exists, unit root tests
may not behave well in telling apart unit roots and weakly-stationary alternatives. Hence,
the Kwiatkowski-Phillips-Schmidt-Shin test (KPSS; Kwiatkowski et al., 1992), with the null
hypothesis of stationarity, also is applied. These three tests are implemented for both pricelevels and their first differences with results reported in Table 1. The results show that theprice series are stationary in differences but not in levels.
As one might expect, the series of the CSI300 and futures are close to each other, showing
a downward trend. The market, in general, is in contango, with spot prices less erratic than
the futures prices. The first differences of the spot prices, however, tend to be more erratic
than those of the futures. Normality is rejected for all series and their first differences at the
5% significance level.
4.
4.1.

Empirical Analysis

Cointegration and Vector Error Correction Modeling

Let a p × 1 (p = 3 in the current study) vector Xt be represented in a vector error correction
model (VECM):∆Xt = µ + ΠXt−1 +k−1XΓi ∆Xt−i + et for t = 1, ..., T ,

(1)

i=1

4

Readers are referred to Hou and Li (2013) and Yang et al. (2012) for more institutional backgrounds of
the CSI300 and futures.
5
Unless stated otherwise, we will refer to ”log prices” as ”prices” hereafter.

Table 1: Unit root tests on levels and first differences of price series of the CSI300, nearby
futures, and first distant futures
SeriesSpot
Nearby Futures
First Distant Futures
Spot
Nearby Futures
First Distant Futures

Without Trend
With Trend
ADF1
PP2
KPSS3
ADF1
PP2
KPSS3
Panel A: Test with Price Levels-1.740 -2.412 3.687-2.437 -2.807 0.303-1.701 -2.503 3.712-2.419 -2.952 0.301-1.695 -2.487 3.740-2.384 -2.949 0.299
Panel B: Test with First Differences-23.467 -33.653 0.116-23.459 -33.657 0.039-23.196 -34.900 0.116-23.188 -34.904 0.036-23.348 -35.168 0.116-23.341 -35.172 0.036

1

The numbers of lags are selected by the Bayesian information criterion (BIC). We arrive at the same decisionon the existence of a unit root if the Akaike information criterion (AIC) is used to select the numbers of lags.
The critical values of the ADF test with a constant but without a trend are -3.43, -2.86, and -2.57 at the 1%,5%, and 10% significance level, respectively. The critical values of the ADF test with a constant and a trend
are -3.96, -3.41, and -3.12 at the 1%, 5%, and 10% significance level, respectively.
2
The critical values of the PP test with a constant but without a trend are -3.439, -2.865 and -2.568 at the
1%, 5%, and 10% significance level, respectively. The critical values of the PP test with a constant and atrend are -3.971, -3.416 and -3.130 at the 1%, 5%, and 10% significance level, respectively.
3
The critical values of the KPSS test with a constant but without a trend are 0.739, 0.463, and 0.347 at the
1%, 5%, and 10% significance level, respectively. The critical values of the KPSS test with a constant and atrend are 0.216, 0.146, and 0.119 at the 1%, 5%, and 10% significance level, respectively.

Table 2: Summary statistics for price series of the CSI300, nearby futures, and first distantfutures
SeriesMeanMedianMinimum

MaximumSpot7.8538Nearby Futures
7.8550First Distant Futures 7.8580

7.8341
7.8352
7.83577.64357.63697.62698.17438.18818.2101

Standard
Deviation
0.12920.13130.1340Skewness

Kurtosis0.47840.4801
0.48272.22492.27782.3118

Spot
Prices in Natural Logarithm

8.3
8.2
8.1
8
7.9
7.8
7.7

7.6
03/03/2010

09/19/2010

04/07/2011

10/24/2011

05/11/2012
11/27/2012
Date 04/16/2010 ~ 11/14/2014 (n=1,112)

06/15/2013

01/01/2014

07/20/2014

02/05/2015

06/15/2013

01/01/2014

07/20/2014

02/05/2015

06/15/2013

01/01/2014

07/20/2014

02/05/2015

Nearby Futures
Prices in Natural Logarithm

8.3
8.2
8.1
8
7.9
7.8
7.7

7.6
03/03/2010

09/19/2010

04/07/2011

10/24/2011

05/11/2012
11/27/2012
Date 04/16/2010 ~ 11/14/2014 (n=1,112)
First Distant Futures

Prices in Natural Logarithm

8.4

8.2

8

7.8

7.6
03/03/2010

09/19/2010

04/07/2011

10/24/2011

05/11/2012
11/27/2012
Date 04/16/2010 ~ 11/14/2014 (n=1,112)

Figure 1: Price series of the CSI300, nearby futures, and first distant futures
where Xt = (St , Ftn , Ftf d ), St , Ftn , and Ftf d represent the CSI300, nearby futures, and first
distant futures, Π and Γi are p × p coefficient matrices, µ is a p × 1 deterministic term,
and k = 2 is selected based on the Bayesian information criterion. Trace and maximum
eigenvalue tests (Johansen, 1988, 1991) are adopted to assess cointegration. In particular,
twomodels are considered: (1) H1 (r): µ = µ0 (unrestricted constant), ∆Xt = µ0 +αβ ′ Xt−1 +Pk−1′
non-zero mean; (2)i=1 Γi ∆Xt−i + et , and the cointegrating relations β Xt may have a P′′∗′
H1 (r): µ = µ0 = αδ (restricted constant), ∆Xt = α(β Xt−1 + δ ) + k−1
i=1 Γi ∆Xt−i + et ,′6′and the cointegrating relations β Xt have a non-zero mean δ . Results in Table 3 indicate a
cointegration rank, r, of two. Possible structural breaks in long-run relationships among theseries are examined with Hansen and Johansen’s (1999) recursive method, which can reveal
the (in)stability of cointegration identified. Bessler et al. (2003) adopt this approach forthe same purpose when studying international wheat prices. Figure 2 shows the normalized
trace test statistics calculated at each data point between May 10, 2010 (point 16) andNovember 14, 2014 (point 1,112). The first 15 data points ranging from April 16, 2010 to
May 7, 2010 are used as the base period. As shown in Figure 2, the test statistics are scaledby the 5% critical values. Therefore, the null hypothesis at a data point can be rejected if
its corresponding entry in the figure is greater than one. It is obvious that the trivariate
model has two and almost never less than two cointegrating vectors. Therefore, the CSI300,
6Johansen’s trace statistic tests the nested hypotheses: null : r = r0 vs. alternative : r > r0 for r0 = 0, 1,
2, ..., p−1. Johansen (1992) proposes a sequential testing procedure to determine the number of cointegrating
vectors. Hypotheses are tested in the following order: H1∗ (0), H1 (0), H1∗ (1), H1 (1), ..., H1∗ (p − 1), H1 (p − 1).
For example, H1∗ (1) can only be rejected if also H1∗ (0) and H1 (0) are rejected, and H1 (1) can only be rejected
if also H1∗ (0), H1 (0), and H1∗ (1) are rejected. Testing is terminated and the corresponding hypothesis is not
rejected at the first failure to reject the null in the testing sequence. Johansen’s maximum eigenvalue statistic
tests the hypotheses: null : r = r0 vs. alternative : r = r0 + 1 for r0 = 0, 1, 2, ..., p − 1.

Table 3: Johansen’s trace and maximum eigenvalue tests for the cointegration rank of
the CSI300, nearby futures, and first distant futures
Null: Rank
r=0
r≤1
r≤2
Null: Rank
r=0
r=1
r=2

Panel A: Johansen’s Trace Test
λ − T race∗1 C(5%)∗2 Decision9
λ − T race3 C(5%)4 Decision9259.01934.910R
258.760
29.680R
41.78019.960R
41.52515.410
R
4.604
9.240
F
4.362
3.760
R10
Panel B: Johansen’s Maximum Eigenvalue Test
λ − max∗5 C(5%)∗6 Decision9
λ − max7 C(5%)8 Decision9217.23922.000R
217.235
20.970R
37.17615.670R
37.16314.070
R
4.604
9.240
F
4.362
3.760
R10

1

The trace statistic with a constant in the cointegration vector.
The critical value at the 5% significance level listed in Table 1* from Osterwald-Lenum (1992) for the
trace test with a constant in the cointegration vector.
3
The trace statistic with a constant outside the cointegration vector.
4
The critical value at the 5% significance level listed in Table 1 from Osterwald-Lenum (1992) for the
trace test with a constant outside the cointegration vector.
5
The maximum eigenvalue statistic with a constant in the cointegration vector.
6
The critical value at the 5% significance level listed in Table 1* from Osterwald-Lenum (1992) for the
maximum eigenvalue test with a constant in the cointegration vector.
7
The maximum eigenvalue statistic with a constant outside the cointegration vector.
8
The critical value at the 5% significance level listed in Table 1 from Osterwald-Lenum (1992) for the
maximum eigenvalue test with a constant outside the cointegration vector.
9”R” means ”Reject”. ”F” means ”Fail to Reject”.
10
The decision will be ”F” if the 1% significance level is used because the associated critical value is6.650.
2

nearby futures, and first distant futures are driven by one common stochastic trend, which isreferred to as the (unobservable) implicit efficient price in the literature (Baillie et al., 2002).
4.2.

Hypothesis Testing

In order to test economic implications such as: (a) whether a market is included in cointegration vectors, and (b) whether the price of a market responds to disturbances in long-run
equilibrium relationships, restrictions on β and/or α are imposed to perform corresponding
hypothesis tests on the cointegrating space: (a) a test of exclusion from cointegration vectors,and (b) a test of weak exogeneity. The null hypothesis of (a) can be formulated as:′

R1×p βp×r = O1×r ,(2)′

where R1×p = (0, ...0, 1, 0, ..., 0) and 1 is the i-th element (i = 1, ..., p), which mean thatthe i-th price series is not in cointegration vectors, i.e. the i-th column of matrix Π is zero.
Similarly, the null hypothesis of (b) can be formulated as:′

B1×p αp×r = O1×r ,(3)

Trace Test Statistics
8

X(t)

7
6
5
4
3
2
1
0
100

200

300

400

500

600

700

800

900

1000

1100

900

1000

1100

The test statistics are scaled by the 5% critical values
8

R1(t)

7
6
5
4
3
2
1
0
100

200

300

400

500
H(0)|H(3)600

700
H(1)|H(3)800
H(2)|H(3)Note: X(t) is associated with the case for which all parameters are estimated recursively and R(t) with thecase for which cointegrating relations are estimated recursively but short-run parameters are concentrated out(see Hansen and Johansen, 1999). H(0), H(1), and H(2) correspond to null hypotheses of the cointegration
rank being 0, 1, and 2, respectively. The black horizontal line represents the value, one.

Figure 2: Recursive cointegration analysis: plots of trace test statistics

Table 4: Hypothesis tests on the cointegrating space
Hypothesis1
Degrees of Freedom χ2 Test Statistics Decision2
Panel A: A Test of Exclusion from Cointegration Vectors
β11 = β12 = 0
2
192.365R
β21 = β22 = 0
2
191.661R
β31 = β32 = 0
2120.473
R
Panel B: A Test of Weak Exogeneity
α11 = α12 = 0
2
2.720
F
α21 = α22 = 0
2
7.206
R
α31 = α32 = 0
2
1.758
F
1

Subscripts of β and α correspond to series as follows for j = 1 and 2: β1,j andα1,j - Spot, β2,j and α2,j - Nearby Futures, and β3,j and α3,j - First Distant
Futures.
2
Decisions are made at the 5% significance level. ”R” means ”Reject”. ”F”
means ”Fail to Reject”.
′

where B1×p = (0, ...0, 1, 0, ..., 0) and 1 is the i-th element, which mean that the i-th priceseries does not respond to disturbances in long-run equilibrium relationships, i.e. the i-th
row of matrix Π is zero. Under the null hypothesis of (a) or (b), the test statistic has an
asymptotic chi-squared distribution. Results in Table 4 reveal that each of the CSI300,
nearby futures, and first distant futures is part of long-run equilibrium relationships andthe nearby futures responds and adjusts towards the equilibrium relationships. This finding
is consistent with Chopra and Bessler (2005), although different empirical applications are
pursued7 .
4.3.

Contemporaneous Causality

Let the innovation vector, et , from the estimated VECM in Equation (1) be written as′′′
Aet = vt , where A is a p × p matrix of structural parameters such that E(Aet et A ) = E(vt vt ),and vt is a p × 1 vector of orthogonal structural shocks, i.e. E(vi,t , vj,t ) = 0 for i 6= jcomponents of vt . Different algorithms of DAGs essentially search and place zeros on matrix
A. The identification condition of matrix A is given by Doan (1996): for all i 6= j andi, j = 1, 2, ..., p, there are no elements of matrix A such that both Aij and Aji 6= 0. When
conducting innovation accounting analysis, the VECM is converted into its equivalent levels
vector autoregressive model (VAR) to calculate impulse response functions and forecast error
variance decompositions. This VAR has cointegration constraints of the VECM imposed andyields consistent results on innovation accounting (Phillips, 1998). In particular, let the pvariate VAR representation of the estimated VECM be:
Xt = A−1 A1 Xt−1 + A−1 A2 Xt−2 + · · · + A−1 Ak Xt−k + A−1 vt for t = 1, ..., T ,(4)where Ai ’s are p × p matrices of coefficient parameters. The associated structural VAR is:
7

Chopra and Bessler (2005) investigate contemporaneous causality among the spot, nearby futures, andfirst distant futures for the black pepper market in Kerala, India.

AXt = A1 Xt−1 + A2 Xt−2 + · · · + Ak Xt−k + vt for t = 1, ..., T .

(5)

To reveal the relative effect of each variable, the response of Xt to the structural innovation
vt needs to be determined.
The PC algorithm, one of the earliest and most widely-used algorithms of DAGs thatis based on conditional independence, is first applied to search for causal orderings. An
introduction following Bessler and Akleman (1998), Bessler and Yang (2003), Bessler et al.
(2003), Wang (2010b), Wang et al. (2007), and Yang and Bessler (2004) can be found in the
Appendix. The estimated contemporaneous innovation correlation matrix of the VECM, V ,is used by this algorithm to determine DAGs. For the current study,
Spot Nearby Futures First Distant Futures
 Spot

1.000
.
V =
 Nearby Futures

0.945
1.000
First Distant Futures 0.947
0.989
1.000
Figure 3 (the left panel) shows the “Pattern” based on the algorithm at the 1% significancelevel because three edges connecting the CSI300, nearby futures, and first distant futures
cannot be directed8 .
Nearby Futures

Spot

Nearby Futures

First Distant Futures

Spot

First Distant Futures

Figure 3: The causal pattern on innovations on CSI300, nearby futures, and first distantfutures prices based on the PC algorithm (left) and the DAG based on the LiNGAM
algorithm (right)
Because observational equivalence exists when assigning causal flows based on the PC
algorithm, methods dealing with non-directed edges in Gaussian space are worth investigating. Due to non-normality of three residual series from the VECM, i.e. p-values of the
Jarque-Bera, Shapiro-Wilk, Kolmogorov-Smirnov, Cramer-von Mises, and Anderson-Darling
test being smaller than 0.01, the Linear Non-Gaussian Acyclic Model (LiNGAM) algorithm
suggests a promising approach that uses non-normality to assign causal flows. An introduc8

Spirtes et al. (2000) state: “In order for the method to converge to correct decisions with probability
1, the significance level used in making decisions should decrease as the sample size increases, and the useof higher significance levels (e.g., 0.2 at sample sizes less than 100, and 0.1 at sample sizes between 100 and300) may improve performance at small sample sizes.” Yang and Bessler (2004) compare the significancelevels at 1% and 0.1% for a system of 9 variables with a sample size of 1,800. Ramsey (2010) adopts thesignificance level at 0.1% for simulation exercises on various graphs, including one with 10 nodes, 20 edges,
and a sample size of 5,000. For the sample size of 1,316 in the current study, the 1% significance level seems
appropriate.

tion following Lai and Bessler (2015), Moneta et al. (2013), and Shimizu et al. (2006) can
be found in the Appendix. Figure 3 (the right panel) shows the uniquely identified DAG
based on the algorithm with the prune factor set to 19 . The associated matrix A can be
represented as:a11 a12 a13
A =  0 a22 0  .
0 a32 a33

Ninety day ahead forecast error variance decompositions and impulse responses are provided
in Table 5 and Figure 4. For the latter, results in different cells can be compared because
they are normalized by the standard deviation of the historical innovations for the associatedseries.
While the effects of a shock in one series on others vary in strength, they tend to bepersistent in the longer run. This is an indication of long-run relationship constraints (Ordenand Fisher, 1993). The nearby futures turns out to be the most exogenous series, the impulse
responses of the CSI300 and first distant futures to its shock are rather strong, and it accounts
for more than 96% of the variances across series even after ninety days. The CSI300 or first
distant futures does not have significant impacts on other series. These results indicatethat price information is discovered in the nearby futures. In a similar application to theblack pepper market in Kerala, India, Chopra and Bessler (2005) examine contemporaneous
causality among the cash, nearby futures, and first distant futures series and find that oneof the futures is the most exogenous depending on how causal flows are assigned between
the two futures series.
Our result of contemporaneous price information being discovered in the nearby futures
sheds light on the increasing informational importance of the CSI300 futures market ascompared to the spot with continuous development of the financial system. This suggests
the importance of reducing high barriers to participating in futures trading in China formany investors, such as qualified foreign institutional investors, to increase the information
content of the futures market. Greater openness of investment channels and policy incentives
to attract well-informed traders may further stimulate futures market development (Xu,
2017b). In terms of enhancing financial stability, our result suggests that policy makers
should pay close attention to the nearby futures for risk management. In particular, potential
risk associated with large shocks to the nearby futures should be controlled in a sound andeffective manner due to the long-lasting effects.
5.

Conclusion

This study investigates contemporaneous causality among price series of the CSI300 (Chinese
Stock Index 300), nearby futures, and first distant futures with vector error correction mod9
The prune factor is between 0 and 1. More edges would be pruned out as the factor increases. However,there have not been conclusive studies on the factor selection. Bizimana et al. (2015) use 0.5 for theirsample size of 158 based on communications with Dr. Joseph D. Ramsey from Carnegie Mellon University,
who is one of the administrators of the TETRAD software. Lai and Bessler (2015) use 1 for the numberof observations exceeding 200. For the sample size of 1,316 in the current study, the factor at 1 seems
appropriate.

Table 5: Variance decompositions of the CSI300, nearby
futures, and first distant futures series based on the DAG
derived from the LiNGAM algorithm

1

Day1

Spot

1
15
30
60
90

9.915
5.141
4.266
3.628
3.379

1
15
30
60
90

0.000
2.394
2.618
2.738
2.780

1
15
30
60
90

0.000
1.427
1.871
2.300
2.483

Nearby Futures
First Distant Futures
2
Spot
89.3570.728
94.4790.380
95.2830.452
95.8180.554
96.0200.601
2
Nearby Futures
100.000
0.000
97.0150.591
96.7320.651
96.5840.677
96.534
0.686
First Distant Futures2
97.7732.227
97.0871.486
96.913
1.216
96.7130.986
96.624
0.894

Day 1 is the contemporaneous period.
This subsection in the table shows how the variance of a particular series
is explained by price innovations from the three series listed in the firstrow. The numerical results are in percentage representations.
2

Innovation to
Nearby Futures

50

75

25

50

75

25

Nearby Futures
25

Spot

50

75

25

25

50

50

75

First Distant Futures

10
8
6
4
2
0
-2
-4
-6
-8

First Distant Futures

First Distant Futures
75

75

10
8
6
4
2
0
-2
-4
-6
-8

Nearby Futures

50

50

First Distant Futures

10
8
6
4
2
0
-2
-4
-6
-8

75

10
8
6
4
2
0
-2
-4
-6
-8
25

50

10
8
6
4
2
0
-2
-4
-6
-8

Nearby Futures

Nearby Futures

Nearby Futures

Spot
10
8
6
4
2
0
-2
-4
-6
-8
25

First Distant Futures

10
8
6
4
2
0
-2
-4
-6
-8

Spot

Spot
25

First Distant Futures

Response of

Spot

Spot
10
8
6
4
2
0
-2
-4
-6
-8

75

10
8
6
4
2
0
-2
-4
-6
-8
25

50

75

Figure 4: Impulse responses of one price to a one-time-only shock in innovations in anotherprice: based on the DAG derived from the LiNGAM algorithm

eling and directed acyclic graphs. These series are tied together through cointegration andthe nearby futures adjusts towards long-run relationships. Contemporaneous price information is found to be discovered in the nearby futures. The results provide market participants
with a relatively new view of market interdependence and directions of causation within themarkets, and shed light on the increasing informational importance of the CSI300 futures
market as compared to the spot with continuous development of the financial system. Meanwhile, the results suggest that a shock to the nearby futures could have long-lasting effects on
prices across the three series studied. Policy makers should pay close attention to the nearby
futures for financial stability. Future research incorporating macroeconomic variables, suchas the M1 measure of money, GDP deflator, and exchange rate, is a worthwhile avenue.
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Appendix: Directed Acyclic Graphs
Directed Acyclic Graphs (DAGs) facilitate the inference of causal relations with a nontime
sequence asymmetry. DAGs have been studied for decades with the recent development
documented in Spirtes et al. (2000) and Pearl (1995, 2000). Consider a causally sufficient
set constituting of three variables X, Y , and Z. A causal fork that X causes Y and Z can be
represented as: Y ← X → Z, suggesting that the unconditional association between Y and
Z is nonzero since both Y and Z have a common cause X, while the conditional association
between Y and Z, given knowledge of the common cause X, is zero since common causes
screen off associations between their joint effects. Another causal fork that both X and Z
cause Y can be expressed as: X → Y ← Z, suggesting that the unconditional association
between X and Z is zero, while the conditional association between X and Z, given thecommon effect Y , is nonzero since common effects do not screen off associations between
their joint causes. These “screening-off” phenomena have been built into an extensive DAG
literature. For more details about these screening-off asymmetries in causal relations, readers
can refer to Orcutt (1952), Papineau (1985), Reichenbach (1956), and Simon (1953).
Intuitively, a directed graph uses arrows and vertices to represent the causal relationship(or lack thereof) among a set of variables. Formally, a graph is an ordered triple (V, M, E),
where V is a nonempty set of vertices (variables), M is a nonempty set of marks (symbols
attached to the end of undirected edges), and E is a set of ordered pairs. Vertices are said tobe adjacent if they are connected by an edge. Given a set of vertices {A, B, C, D, E, F }, we
can consider the following four cases: (a) an undirected graph contains undirected edges (e.g.,A−B) only, which signify covariances that are given no particular causal interpretations; (b)a directed graph contains directed edges (e.g., B → C) only, which suggest that a variationin B, with all other variables held constant, causes a (linear) variation in C that is not
mediated by any other variables in the system; (c) an inducing path graph contains both
directed edges and bidirected edges (e.g., C ↔ D), the latter indicating the bidirectionalcausal interpretation between two variables; (d) a partially oriented inducing path graphcontains directed edges, bidirected edges, nondirected edges (e.g., D ◦ − ◦ E) and partiallydirected edges (e.g., E◦ → F ), the latter two with a small circle at the end of an edge tosignify the uncertainty as to whether an arrow should be contained or not. The lack of anedge between two variables indicates unconditional or conditional independence. A DAG
is a directed graph with no directed cyclic paths10 . Hence, an acyclic graph has no path
10

The number of possible labeled DAGs for n > 1 vertices is: Rn =Pn

k+1
k=1 (−1)



n
k



2k(n−k) Rn−k ,that leads away from a variable to return to it. In other words, an acyclic graph contains avariable no more than once in a path. The path such as A → B → C → A is cyclic since
the path leads away from A to B and returns to A via C. In this study, only acyclic graphs
are employed. A causal chain such as X → Y → X is not allowed in a final directed graph.
DAGs are designed to represent conditional independence as implied by the recursive
product decomposition:Pr(x1 , x2 , x3 , ..., xn ) = Πni=1 Pr(xi |pai ),(6)where Pr is the probability of variables x1 , x2 , ..., xn , pai is the realization of some subsets ofthe variables that precede (come before in a causal sense) xi in order (i = 1, 2, ..., n), and Π isthe product operation. Pearl (1986, 1995) has proposed d-separation (direction separation)
as a graphical characterization of conditional independence relations in Equation (6). If weformulate a DAG in which the variables corresponding to pai are represented as the parents(direct causes) of xi , then the independencies implied by Equation (6) can be read off thegraph using the notion of d-separation (Pearl, 1995):
Definition 1 (d-separation). Let X, Y , and Z be three disjoint subsets of vertices (variables) in a directed acyclic graph G, and let p be any path between a vertex (variable) in X
and a vertex (variable) in Y , where by “path” we mean any succession of edges, regardlessof their directions. Z is said to block p if there is a vertex w on p satisfying one of thefollowing: (a) w has converging arrows along p, and neither w nor any of its descendants areon Z; or, (b) w does not have converging arrows along p, and w is in Z. Further, Z is said tod-separate X from Y on graph G, written (X ⊥⊥ Y |Z)G , if and only if Z blocks every pathfrom a vertex (variable) in X to a vertex (variable) in Y .
Geiger et al. (1990) show that there is a one-to-one correspondence between the set of
conditional independencies X ⊥⊥ Y |Z implied by Equation (6) and the set of triples X, Y, Z
that satisfy the d-separation criterion in a graph G. Specifically, if G is a DAG with vertex(variable) set V, X and Y are in V, and Z is also in V, the implied linear correlation
between X and Y in G, conditional on Z, is zero if and only if X and Y are d-separated
given Z.
The notion of d-separation can be illustrated further following Pearl (2000). Consider
three vertices (variables), A, B, and C. A variable is a collider if arrows converge on it:
A → B ← C. The vertex B is called a collider, and A and C are d-separated, given the null
set. However, if we condition on B, the information flow between A and C is opened up, and
A and C are d-connected (directionally connected). By modifying the graph A → B ← C
to include variable D as a descendant of B, we have:
A → B ← C.
↓D
If we condition on D, the information flow between A and C is also opened up. This
where R0 = 1 (McKay et al., 2004).

illustrates part (a) of Definition 1.
If the information flow is characterized by diverging arrows as described in part (b)of Definition 1, the d-separation condition is different. We need to take two cases into
consideration. First, we consider three vertices (variables), K, L, and M , specified by
the graph K ← L → M , where L is a common cause of K and M . The unconditional
association (correlation) between K and M is nonzero since they have a common cause L,
and K and M are d-connected. However, if we condition on L (know the value of L), theassociation between K and M vanishes away (and is zero), and K and M are d-separated.
Hence, conditioning on common causes blocks the information flow between common effects.
Second, we consider a causal path, which is a causal chain, described by D → E → F , where
D causes E, and E causes F . The unconditional association between D and F is nonzero,and D and F (the end points) are not d-separated. However, if we condition on E (themiddle vertex or mediator), the association between D and F disappears (and is zero), and
D and F are d-separated.
In a word, two vertices, say X and Y , are said to be d-separated if the information flow
between them is blocked. This happens when: (a) X and Y have a common cause W withthe graph representation X ← W → Y , or X and Y are end points of a causal chain whose
middle vertex is U with the graph representation X → U → Y , and we condition on W or
U ; (b) X and Y have a common effect Z with the graph representation X → Z ← Y , andwe do not condition on Z or any of its descendants (descendants are not shown here).
The PC Algorithm
The PC algorithm is used for inference on DAGs based on observed data. Spirtes et al.
(2000) have incorporated into it the notion of d-separation for building DAGs using thenotion of sepset (defined later). This algorithm is an ordered set of commands. It starts
with a general unrestricted set of relations among variables and proceeds stepwise to remove
edges between variables and direct “causal flows”.
Briefly, on the vertex set V, we begin with a complete undirected graph G that has an
undirected edge between every variable of the system (every variable in V). Edges between
variables are removed sequentially based on zero-order correlations (unconditional correlations) or higher-order partial correlations (conditional correlations). First, the algorithm
removes edges from the complete undirected graph by checking for unconditional correlations between pairs of variables. Edges that connect variables with zero correlations are
removed. Second, the algorithm checks for first order partial correlations (correlations between pairs of variables conditional on a third variable) from the remaining edges. Edges thatconnect variables with zero first order partial correlations are removed. Similarly, provided
that we have N variables, the algorithm continues to check for partial correlations up to the(N − 2)-th order and removes edges that connect variables with zero partial correlations ofa corresponding order.
For efficiency of the PC algorithm, Bülmann and van de Geer (2011) have pointed outthat it is a clever iterative multiple testing approach for inferring zero partial correlations.
If a marginal correlation ρ(j, k) = 0, considerations of partial correlations ρ(j, k|C) of higher
orders with |C| ≥ 1 are not needed. Similarly, if a first order partial correlation ρ(j, k|m) = 0,
considerations of higher order partial correlations ρ(j, k|C) with m belonging to C and|C| ≥ 2 are not necessary. The same idea applies to partial correlations of other orders.
As a result, faithfulness allows a hierarchical testing process from marginal to first- and to′higher-order partial correlations. Shalizi (2013) has indicated that X ⊥⊥ Y |S , where all′
variables in S are adjacent to X or Y or both, if X ⊥⊥ Y |S for some sets of variables S. We
can consider a single long directed path from X to Y . If we condition on any of the variables
along the chain, X and Y become independent. Yet, we could always move the point where
we block the chain to either right next to X or right next to Y . Hence, when we are trying
to remove edges between X and Y to obtain independence, only conditioning on variables
which are still connected to X and Y (not those in totally different parts of the graph) isneeded. The PC algorithm tries to minimize the number of variables it conditions on, thus
avoiding many statistical tests and usually running fast. Also, Kalisch and Bülmann (2007)
have noted that the PC algorithm is computationally feasible for high-dimensional sparse
problems.
In applications, we use Fisher’s z statistic to test whether conditional correlations are
1
significantly different from zero. The formula for z is z[ρ(i, j|k), n] = 21 (n−|k|−3) 2 ×ln{[|1+ρ(i, j|k)|] × [|1 − ρ(i, j|k)|]−1 }, where n is the number of observations used to estimate thecorrelations, ρ(i, j|k) is the population correlation between series i and series j conditional onseries k (the influence of series k on series i and series j is removed), and |k| is the number ofvariables in k (|k| = 0 for unconditional correlation). Let r(i, j|k) be the sample correlation
between series i and series j conditional on series k. If series i, j, and k are all normally
distributed, z[ρ(i, j|k), n] − z[r(i, j|k), n] is standard normally distributed.
The edges that survive all the removals are directed by applying the notion of sepset:
Definition 2 (sepset). The conditioning variable(s) on removed edges between two variables
is (are) called the sepset of the variables whose edges have been removed (for vanishing zeroorder conditioning information, the sepset is the empty set).
Consider triples X − Y − Z, where X and Y , and Y and Z are adjacent, but not X and Z, asa simplified example. First, one directs edges between the triples X − Y − Z as X → Y ← Z
if Y is not in the sepset of X and Z. Second, if X → Y , Y and Z are adjacent, X and Z are
not adjacent, and there is no arrowhead at Y , one directs Y − Z as Y → Z. Third, if there
is a directed path from X to Y and an edge between them, one directs X − Y as X → Y .
Demiralp and Hoover (2003) and Spirtes et al. (2000) have studied the PC algorithmwith Monte Carlo simulations. With the sample size of 100, the PC algorithm may makemistakes on edge inclusions or exclusions (an edge that should be included is not included
or an edge that should not be included is included), and edge directions (an arrowhead thatshould be put at a specific vertex is not put there or an arrowhead that should not be put
at a specific vertex is put there). With extensive explorations of several versions of the PC
algorithm on simulated data with respect to errors on both edge inclusions or exclusions andedge directions, Spirtes et al. (2000) found that (a) there is little chance for the algorithm
to include an edge that is not in the “true” model, but with small sample sizes (say lessthan 200 observations), there is a considerable chance for the algorithm to omit an edgethat belongs to the model; (b) arrowhead commission errors (putting an arrowhead where it
does not belong) are more likely than edge commission errors (putting an edge where it does
not belong). Spirtes et al. (2000) stated: “in order for the method to converge to correct

decisions with probability 1, the significance level used in making decisions should decreaseas the sample size increases, and the use of higher significance levels (e.g. 0.2 at sample sizes
less than 100, and 0.1 at sample sizes between 100 and 300) may improve performance atsmall sample sizes.”
For the connection between directed graphs and Holland’s (1986) counterfactual randomvariable model (the random assignment experimental model), readers are referred to Spirtes
et al. (1999).
The LiNGAM Algorithm
The Gaussian data assumption made in the PC algorithm negates the need for informationfrom higher-order moment structures (Shimizu et al., 2006). It could result in a set ofindistinguishable causal patterns, which are equivalent in their (conditional) probability
structures. For example, when X, Y , and Z are normally distributed, two graphs, X ←Y → Z and X → Y → Z, are compatible with the same probability distribution and thus
are observationally equivalent (Pearl, 2000).
An important difference of the Linear Non-Gaussian Acyclic Model (LiNGAM) algorithm from most earlier work on the linear, causal sufficient, case is the assumption of
non-Gaussianity of variables (disturbances) (Shimizu et al., 2006), which is common in financial time series. When this assumption is valid, the complete causal structure could be
estimated without any prior information on the causal ordering of variables (Shimizu et al.,2006). The further the data are from normality, the more accurate the final causal structure
identified by the LiNGAM algorithm (Shimizu and Kano, 2008).
While the PC algorithm generally searches the causal pattern based on conditional independence, the LiNGAM algorithm discovers the causal directionality according to functional
composition (Pearl, 2000). In particular, independent component analysis (ICA) is utilized
by the LiNGAM algorithm here11 . ICA is only feasible on non-Gaussian data and, for Gaussian variables, it generally could not find the correct mixing matrix because many different
mixing matrices yield the exact same Gaussian joint density (Hyvärinen et al., 2001)12 .
Consider observed data generated from a process with properties as follows (Shimizu etal., 2006):
1. Variables xi , i ∈ {1, ..., m} could be arranged in a causal order, denoted by k(i), suchthat no later variable causes any earlier one, i.e., the generating process is recursive(Bollen, 1989), which could be represented graphically by a DAG (Pearl, 2000; Spirtes
et al., 2000).
11

An alternative approach known as DirectLiNGAM, which does not make use of ICA, is proposed by
Shimizu et al. (2011).
12
Based on the Central Limit Theory (CLT), any mixture of signals from independent sources usually
has a distribution closer to a normal distribution than any of the constituted original variable (Stone,2004). Under the assumption that one observes the mixtures, X = (x1 , x2 , ..., xn ), of independent signals,s = (s1 , s2 , ..., sn ), one has X = As with s representing mutually independent components. The CLT saysthat any of the s is less Gaussian than the mixture variables X. The independent components could be
rewrote inversely as the linear combination of the mixture variables. ICA aims at finding the demixing
matrix, W , which maximizes the sum of the non-Gaussianity of the mutually statistically independent
components of s̃, where s̃ = W̃ X and W̃ = A (Hyvärinen et al., 2001; Shimizu, Hyvärinenet al., 2006).

2. The value assigned to each variable xi is a linear function of values already assignedto
P earlier ones, plus a “disturbance” term ei and an optional constant term ci : xi =kj <ki bij xj + ei + ci .

3. Disturbances ei , i ∈ {1, ..., m} are all continuous-valued random variables with nonGaussian distributions
Q of non-zero variances and they are independent of each other,i.e., p(e1 , ..., em ) = i pi ei .
If each variable xi has a non-zero mean, we are left with the system of equations:
X = BX + e,

(7)

where B is the coefficient matrix of the model. Solving for X in Equation (7) results in:
X = Ae,(8)where A = (I − B)−1 . Equation (8) and the independence and non-Gaussianity of components of e form the standard linear ICA model (Hyvärinen et al., 2001; Shimizu et al., 2006).
Rewriting Equation (8), one obtains:e = (I − B)X.

(9)

In general, the LiNGAM algorithm first uses ICA to obtain an estimate of the mixing matrix
A and subsequently permutes and normalizes it appropriately before utilizing it to compute
B, which contains the sought connection strengths bij 13 . When the number of observed
variables xi is relatively small (e.g., less than eight), finding the best permutation is easy
with a simple exhaustive search. For higher dimensionalities, a more sophisticated approachis required14 .
After finding a causal ordering k(i), some estimated connection strengths might be exceedingly weak and are probably zero in the generating model. Under these circumstances,
the Wald test could be used to determine whether certain connections should be pruned.
Readers can refer to Kano and Shimizu (2003), Shimizu and Kano (2008), and Lai and
Bessler (2015) for an intuitive illustrative example of using higher order moments for determining the direction of causality for non-Gaussian variables. We closely follow these authors.
Consider two models:M1 : y = βx + εy ,(10)
M2 : x = ηy + εx ,(11)

where the explanatory variable is independent of the error in each model. Let xk and yk ,where k = 1, 2, ..., N , be observations on x and y with mean zero. The moment structure
can be defined asN
1 X i j
mij =
x y .
(12)
N k=1 k k
The first-order moment of observed data is not considered because E(x) = E(y) = 0.
1314Refer to Shimizu et al. (2006) for more details of the permutation and normalization problem.
Refer to Shimizu et al. (2006) for the detailed LiNGAM discovery algorithm.

The model-predicted second-order moment of M1 is

 


1 0 
m20
2
E
(x
)
= σ2 (τ̂2 )E  m11  =  β 0 E (ε2Y )
2
β 1
m02

(13)

where τ2 is the number of parameters and τ̂2 = (β, E (x2 ) , E (ε2Y )) in this case. We notethat the number of the distinct sample moments and that of τ2 are both 3. Meanwhile,the same second-order moment structures of M1 and M2 are the same. Therefore, M1 andM2 are equivalent, meaning that M1 could not be identified from M2 if one only considers
second-order moment structures. One could, however, apply higher-order moments of M1
and M2 to detect the causal direction if the relevant variables and disturbance terms are
non-normally distributed. For example, the third-order moment of M1 can be wrote as


 
m30
1 0 

 m21   β 0  E (x3 )




E
= σ3 (τ̂3 )(14)=
m12   β 2 0  E (ε3Y )β3 1m03
where τ̂3 = (β, E (x3 ) , E (ε3Y )). The fourth-order moment structure can be defined in a
iT
h
T

and σ(τ ) = σ2 (τ2 )T , σ3 (τ3 )T , σ4 (τ4 )T . There aresimilar way. Let m = mT2 , mT3 , mT4
twelve sample moments and seven parameters in this case and one could evaluate the model
fit. The null and alternative hypotheses to test the overall model fit can be expressed as
H0 : E(m) = σ(τ ) versus H1 : E(m) 6= σ(τ )(15)

The test statistic is based on the difference between m and σ(τ̂ ) by F (τ̂ ), T1 , and T2 , whereF (τ̂ ) = {m − σ(τ̂ )}T V̂ −1 {m − σ(τ̂ )},(16)

T1 = N × F (τ̂ ),

(17)

T1
,
1 + F (τ̂ )

(18)

T2 =

and V̂ is a weight matrix in generalized least squares estimation, which converges in probability to a certain positive definite matrix V .
Consider the case where M1 has a smaller chi-square value of the statistic T2 as comparedto M2 and does not reject H0 in Equation (15). This implies that M1 has better model-dataconsistency and one considers it the best-fitting model for this reason. Therefore, the correctcausal ordering between variables (x causes y) is reflected through M1 (Kano and Shimizu
2003; Shimizu and Kano 2008). The LiNGAM algorithm applies the above test statistics to
examine the overall model fit (Shimizu, Hoyer et al., 2006).