Directory UMM :Data Elmu:jurnal:I:International Review of Economics And Finance:Vol9.Issue1.Jan2000:

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International Review of Economics and Finance 9 (2000) 11–30

The price–volume relationship in the crude oil

futures market

Some results based on linear and nonlinear

causality testing

Imad A. Moosa*, Param Silvapulle

Department of Economics and Finance, La Trobe University, Bundoora, Victoria 3083, Australia Received 9 March 1998; accepted 16 February 1999

Abstract

This article presents some evidence for the presence of a causal relationship between price and volume in the crude oil futures market. The results of linear causality testing reveal the presence of causality running from volume to price but not vice versa. While the results of testing for nonlinear causality are inconsistent, most of the evidence shows that causality runs in both directions. In general, there is evidence for the sequential information arrival hypothesis and the noise trading model, but not for market efficiency. There is also some evidence for the presence of a maturity or a liquidity effect. Finally, there is some variation in the results, depending on the sample period. 2000 Elsevier Science Inc. All rights reserved.

JEL classification:G14; C32

Keywords:Price-volume relationship; Nonlinear causality; Futures market

1. Introduction

This article examines the price–volume relationship in the crude oil futures market using linear and nonlinear causality testing. This relationship is important for a number

of reasons.1_{For example, Gallant et al. (1992) assert that more can be learned about}

the market by studying the joint dynamics of prices and trading volume than by focusing on the univariate dynamics of prices. Moreover, the trading volume is thought

*Corresponding author. fax: 03-9479-1654.

E-mail address: [email protected] (I.A. Moosa)

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12 I.A. Moosa, P. Silvapulle / International Review of Economics and Finance 9 (2000) 11–30

to reflect information about changes in investors’ expectations. Another reason for the interest in the presence or otherwise of a strong price–volume relationship is that it provides support for using technical analysis as opposed (or in addition) to

fundamental analysis.2

Apart from Hiemstra and Jones (1994), Fujihara and Mougoue (1997), and Abhy-ankar (1998), the available empirical evidence on the price–volume relationship is based on conventional (linear) Granger causality testing. Baek and Brock (1992) have shown that these tests have low power against the nonlinear alternatives. Hence, any test based on the assumption of linearity will fail to detect nonlinear dependence. Nonlinear dependence may be present if the price and volume are generated by nonlinear processes: this hypothesis is theoretically plausible and empirically

substanti-ated.3_{Recent work has revealed the existence of nonlinear structure in the process}

generating returns in financial and commodity markets. These nonlinearities are nor-mally attributed to nonlinear transaction cost functions, the role of noise traders, and market microstructure effects (Abhyankar, 1998). It is now widely accepted that the relationship between economic and financial time series are mainly nonlinear. As a result, testing for nonlinear causal relationships between two time series has received considerable attention in the recent literature. In this article, the nonlinear causality test proposed by Baek and Brock (1992) is used to study the price–volume relationship in the crude oil futures market.

Further motivation for using nonlinear models is provided by Savit (1988, pp. 271–272), who argued that financial and commodity markets are likely to be examples of dynamic systems manifesting nonlinearities. He disputes the argument that fluctua-tions in time series are random and argues instead that the fluctuafluctua-tions are generated by inherent nonlinearities. The distinction between linear and nonlinear adjustment to any deviation from the equilibrium price lies in whether or not the magnitude of adjustment is proportional to the deviation. A proportional adjustment implies a linear relationship, but this kind of adjustment cannot generate the randomness observed in financial and commodity markets. Savit argues that nonlinear adjustment can pro-duce this kind of behavior. Hsieh (1991) put forward a similar point by arguing that large moves in prices, which are greater than what is expected under a normal distribution, may be attributed to nonlinearities.

This article is organized as follows: We first present a discussion of the theory of the price–volume relationship, followed by a brief outline of the recent empirical evi-dence. We then present the data and examine the time series properties of the underlying variables. This is followed by an outline of the methodology used to test for linear causality and the results of the tests. The same follows for nonlinear causality testing.

2. The price–volume relationship: theory and empirical evidence 2.1. The theory

The rationale for postulating a positive relationship between volume and absolute price changes can be found in the basic supply and demand model. Starting from an

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initial equilibrium position, a change in demand induces a price change. While the adjustment takes place, transactions are executed in response to the change in demand until a new equilibrium price is reached. Hence, the volume of transactions rises as the price changes, regardless of the direction of this change. This line of reasoning, as put forward by Crouch (1970), Clark (1973), and Westerfield (1973), does not tell us anything about the direction of causality.

This theory was later refined by Copeland (1974), who used the volume as a proxy for the rate of information arrival. In general there are two competing hypotheses that explain information arrival in financial markets: the mixture of distributions hypothesis and the sequential information arrival hypothesis. According to the first hypothesis, information dissemination is contemporaneous (see for example Tauchen & Pittis, 1983; Harris, 1986, 1987). The sequential information arrival hypoth-esis allows for intermediate informational equilibria (see for example Jennings et al., 1981). Karpoff (1987) claimed that the mixture of distributions hypothesis is consistent with the empirical distribution of price changes. The sequential information arrival hypothesis implies a positive intertemporal causal relationship in both directions. Hiemstra and Jones (1994) argued that a sequential information flow results in lagged trading volume having predictive power for current absolute price changes, and lagged absolute price changes having predictive power for current volume. On the other hand, the mixture of distributions hypothesis implies only a positive contemporaneous causality from volume to absolute price changes with no intertemporal causality in either direction.

While the mixture of distributions hypothesis and the sequential information arrival hypothesis consider the relevant price variable to be the absolute price change, some economists argue that the relevant variable is the price change per se (the signed price change). Ying (1966) argues that if there is asymmetry in the behavior of the ratio of volume to price change, then what is important is the price change per se rather than the absolute price change. This postulation is supported by the observation that volume is relatively heavy in bull markets and light in bear markets. A theoretical

rationale for this proposition is provided by Epps (1975).4_{Thus, the signed}

contempora-neous relationship between trading volume and the price change reveals asymmetry of trading volume in up and down markets. Karpoff (1988) and Suominen (1996) argued that the observed positive correlation between volume and price change can be explained by the presence of differential costs in acquiring short and long positions. Because the costs of taking short and long positions in futures markets are identical, asymmetry should not be observed in these markets. On the other hand, an intertempo-ral relationship has implications for market efficiency. Specifically, intertempointertempo-ral cau-sality from volume to price changes implies informational inefficiency. If this is the case, then it may be possible to make abnormal profit, which explains interest in the price–volume relationship.

The price–volume relationship can also be explained in terms of the noise trading model of De Long et al. (1990). This model postulates that the activity of noise traders is not based on economic fundamentals and therefore results in a temporary mispricing. The price, however, moves toward its mean value in the long run in the absence of the

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14 I.A. Moosa, P. Silvapulle / International Review of Economics and Finance 9 (2000) 11–30

transitory component. Hence, the model predicts that the positive causal relationship running from price to volume is consistent with the positive-feedback trading strategy of noise traders who base their decisions on past price movements. Moreover, the model predicts that a positive causal relationship from volume to price is consistent with the hypothesis that price changes are caused by the actions of noise traders. Information asymmetry is also considered by Wang (1994) in his heterogeneous inves-tor model, which postulates that information asymmetry and invesinves-tor heterogeneity may affect the price–volume relationship. As a result, trading volume is always posi-tively correlated with absolute price change, and correlation increases with information asymmetry.

Stickel and Verrecchia (1994) explain the effect of volume on price by distinguishing between “informed” and “uninformed” trading. They present the hypothesis that price changes are more likely to reverse after weak volume support because price changes reflect demand and higher volume reflects a greater likelihood that the demand originates from informed rather than uninformed trading. Hence, as volume increases, the probability that the price change is information driven increases. They provided evidence indicating that large price changes on days with weak volume support tend to reverse, at least partially.

2.2. The empirical evidence

The empirical evidence on the price–volume relationship is vast and mixed. Since the older literature is surveyed by Karpoff (1987), we will concentrate on recent evidence, particularly on the studies dealing with futures markets. These studies also provide mixed evidence and often contradictory results. While most of them produced evidence for bi-directional causality, providing support for the sequential information arrival hypothesis, some evidence has been found for the mixture of distributions hypothesis. On the other hand, while most of the evidence points to market inefficiency, there is some evidence supporting efficiency, particularly in futures markets. The following is a brief description of some of the recent studies.

The evidence presented by Smirlock and Starks (1988) on stock markets and by McCarthy and Najand (1993) on currency futures markets supports the sequential information arrival hypothesis. Furthermore, McCarthy and Najand verified the effi-ciency of the currency futures markets. Kocagil and Shachmurove (1998) examined the same issue for 16 futures markets. The evidence on the contemporaneous relation-ship was consistent with the mixture of distributions hypothesis and the sequential information arrival hypothesis as well as Wang’s (1994) heterogeneous investors model. The results of intertemporal analysis, however, rejected the mixture of distributions hypothesis and provided support for the sequential information arrival hypothesis.

Hiemstra and Jones (1994) applied nonlinear causality testing to the U.S. stock market and found evidence for bi-directional causality. They also examined the propo-sition that causality from volume to price change can be explained by volume serving as a proxy for information flow in the stochastic process generating the price change variance. After correcting for volatility persistence in the returns, nonlinear causality from volume to price was still present.

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Foster (1995) found some contrasting results by examining the price–volume rela-tionship using data from the oil futures market. Based on GARCH and GMM models, Foster concluded that volume was not an adequate proxy for the rate of information flow but that volume and volatility were largely driven by the same factors (assumed to be information). Thus, his results support the mixture of distributions hypothesis. Moreover, he found evidence for intertemporal causality from volume to price, a result that he did not regard as being inconsistent with the mixture of distributions hypothesis but rather indicative of market inefficiency. Foster offered two possible explanations for inefficiency: (a) traders may condition their prices on previous vol-umes as a measure of market sentiment; and (b) it is a form of mimetic contagion, where agents set their prices with reference to the trading patterns of other agents.

Malliaris and Urrutia (1998) examined the price–volume relationship in six agricul-tural futures contracts using cointegration and error correction analysis. They found results in favor of bi-directional causality, such that the relationship was stronger from price to volume. These results are supported by those obtained by Fujihara and Mougoue (1997), who examined the relationship in three oil markets. While the results of linear causality testing were inconsistent, the results of nonlinear causality testing showed bi-directional causality.

3. Data and the time series properties of the variables 3.1. Data

The data sample used in this study consists of daily observations on futures prices and volumes of the West Texas Intermediate (WTI) crude oil covering the period between January 2, 1985, and July 11, 1996. Two futures contracts are considered:

the three-month and the six-month contracts.ft13

t and ftt16are the logarithms of the

futures prices for maturities of three and six months, respectively, whilev3

t andv6t are

the logarithms of the trading volumes of the two contracts. The data were obtained from the OPEC database as reported by the New York Mercantile Exchange.

At this point, we find it useful to say something about the reasons for using more than one maturity to test the price–volume relationship. The first reason is obvious: to find out if the results are consistent across maturities. If they are not, then we may conclude that there is a “maturity effect.” This may be the case if, for example, maturity reflects liquidity. Foster (1995) considered the maturity effect in the price–volume relationship and concluded that maturity had little effect other than that which could be attributed to liquidity. But liquidity is an important factor in this study, given that

the three-month contract is much more liquid than the six-month contract.5_{It would}

be interesting to find out whether or not a maturity or a liquidity effect is present. Moreover, it is a well-established finding in the literature that the volatility of futures prices depends on maturity (see for example Galloway & Kolb, 1996). Since volatility is a crucial element of the following analysis, another reason arises as to why more than one maturity is used.

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it encompasses two episodes of upheaval and turbulence in the oil market. The first of these episodes is the 1986 collapse of oil prices. The second is the 1990/91 rise in oil prices as a result of the invasion of Kuwait and the Gulf War. To find out if these events made any difference for the price–volume relationship, the tests are conducted over the full sample period and over two sub-sample periods. These sub-sample periods exclude the two periods of turbulence. The first sub-sample includes observations following the collapse of oil prices in 1986 and before the invasion of Kuwait (874 observations). The second sub-sample includes observations covering the more recent period after the effect of the Gulf War had subsided (1258 observations).

3.2. Time series properties

We first test for the staionarity of the time series using the Dickey-Fuller (1979) ADF test, the Kwiatkowski et al. (1992) KPSS test, and the Phillips-Perron (1988) PP test. The results, which are reported in Table 1, are mixed, but in general they indicate that all series are nonstationary. More importantly, the Phillips-Perron test confirms nonstationarity consistently. We consider the Phillips-Perron test as being the most reliable in this case, since the series exhibit ARCH effect as the results will show later. Hence, we conclude that all price and volume series are nonstationary.

The implication of this finding is that testing for causality between the price and volume should be based on unrestricted VARs in first differences or error correction

models, depending on whether or not the variables are cointegrated.6_{Specifying the}

price variable in first difference is also consistent with the theory of the price–volume relationship because this theory normally refers to price changes or returns. Because of the theoretical considerations mentioned earlier, two price variables are used: the

price change per se,Dft, and the absolute price change,|Dft|.

The next step, therefore, is to determine whether or not futures prices and volumes are cointegrated. The results of the cointegration tests are not reported but are available from the authors on request. They indicate the absence of cointegration in both cases. Thus, testing for causality will be based on unrestricted VARs, which will be specified in the following section.

To detect volatility in the series, we apply Engle’s (1982) LM test and the Silvapulle and Silvapulle (1995) one-sided score test to the first differences of all series for ARCH(2) and ARCH(3) specifications. The LM(2) and LM(3) test statistics have

asymptoticx2_{(2) and}_x2_{(3) distributions, respectively. The one-sided score test statistic}

for ARCH(2) has an asymptoticx¯2_{distribution, which is a weighted average of}_x2₍₁₎

andx2_{(2), while the one-sided test statistic for ARCH(3) has an asymptotic}_x_¯2

distribu-tion, which is a weighted average ofx2_(1),_x2_{(2), and}_x2_{(3). The results, which are not}

reported here, reveal that volatility is present in all of the series for both maturities and all sample periods.

It may be necessary to eliminate the effect caused by dynamic heteroskedasticity by fitting ARCH-type volatility models. This step may be necessary because it is possible that nonlinear causality could be caused by simple volatility effects associated with information flows, in which case the nonlinear causality test could be merely detecting spurious causality.

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I.A. Moosa, P. Silvapulle / International Review of Economics and Finance 9 (2000) 11–30 17 Table 1

Testing for stationarity

ADF KPSS PP

Series ,₅₃₅ ,₅₄₅ ,₅₃₅ ,₅₄₅ ,₅₃₅ ,₅₄₅

Full sample

ftt13(without trend) 21.30 21.35 6.24 4.90 21.78 21.70

ftt13(with trend) 21.61 21.74 1.07 0.85 21.58 21.51

ftt16(without trend) 23.43 23.51 1.61 1.28 22.01 21.99

ftt16(with trend) 23.10 23.19 0.80 0.64 22.81 22.48

vt3(without trend) 22.27 22.32 5.83 4.62 22.48 21.90

vt3(with trend) 22.91 22.98 0.92 0.74 21.90 22.05

vt6(without trend) 22.88 22.16 0.61 0.49 22.23 22.13

vt6(with trend) 22.96 22.58 0.28 0.25 21.92 21.51

First sub-samble

ftt13(without trend) 21.69 21.58 3.35 2.30 21.67 21.52

ftt13(with trend) 21.18 21.05 0.53 0.38 21.12 20.92

ftt16(without trend) 22.00 21.06 2.08 1.48 21.98 21.02

ftt16(with trend) 22.04 22.04 0.52 0.38

vt3(without trend) 21.57 21.51 2.69 1.88 21.32 21.24

vt3(with trend) 22.70 22.61 0.46 0.33 22.58 22.41

vt6(without trend) 21.59 21.54 0.84 0.59 21.57 21.42

vt6(with trend) 21.77 21.62 0.63 0.45 21.64 21.51

Second sub-sample

ftt13(without trend) 22.60 22.52 0.59 0.46 22.25 22.04

ftt13(with trend) 22.40 22.21 0.34 0.27 22.15 22.08

ftt16(without trend) 22.41 22.30 0.90 0.69 22.21 22.18

ftt16(with trend) 22.70 22.01 0.28 0.22 22.48 21.90

vt3(without trend) 21.88 21.90 0.98 0.69 21.72 21.45

vt3(with trend) 21.68 21.67 0.88 0.62 21.44 21.28

vt6(without trend) 21.60 21.72 1.49 1.04 21.50 21.20

vt6(with trend) 21.02 20.99 0.79 0.56 20.98 20.97

,_{in the ADF test indicates the number of augmentation terms included in the regression equation}

to whiten the noise term.,_{in the KPSS test and the PP test indicates the number of autocovariances}

included in the long-run variance. The 5% critical values of the ADF and KPSS test statistics are22.86 and 0.431 (without trend) and23.41 and 0.163 (with trend), respectively. The critical values of the PP test statistics are the same as those of the ADF statistic. A significant test statistic indicates the rejection of the null hypothesis, which is nonstationarity in the ADF and PP tests and stationarity in the KPSS test.

The conditional variance of the seriesZt, denoted ht, is modelled as

Zt5 m 1 ut (1)

whereut|φt21,N(0,ht) and

ht5h01

o

j51

aju2t2j1

o

k51

bkht2k (2)

wheret5max(p,q), . . . ,T.By imposing the restrictionb15···5bp50 on Eq. (2),

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found to be generally successful, there are some features of the data that these models fail to capture, the most interesting of which is the “leverage effect” (Nelson, 1991). Statistically, this effect implies that negative surprises to financial markets increase volatility more than positive surprises. To capture such effects, Nelson proposed the

exponential GARCH model. An EGARCH (p,q) model is given by

loght5h01

o

k51

ckut2kh2t21/2k 1

o

j51

ajloght2j1a[uut21uh2t21/212(2/p)1/2] (3)

wheret5max(p,q), . . .T.

Table 2 reports the results of fitting ARCH-type models, as represented by Eqs. (1)–(3), to the price and volume series. The table reports the mean and volatility

equations, including the robusttstatistics (in parentheses), which are computed using

the Newey-West (1987) procedure. The best model is selected on the basis of Akaike’s and the Schwartz Bayesian information criteria, which produced consistent results,

indicating that an EGARCH(1, 1) is appropriate for Dft13

t and Dftt16 over the full

sample period. The volume seriesDv3

t andDv6t, on the other hand, follow GARCH(1,

2) processes over the full sample period. Different model specifications are obtained for the sub-samples.

4. Testing for linear causality 4.1. Methodology

In this section, we illustrate Hsio’s (1981) linear causality test, which is based on a bivariate VAR representation of price and volume. Hsio’s (1981) sequential

proce-dure for linear Granger causality testing between two stationary series, x and y, is

based on the bivariate VAR representation

xt5 a01

o

i51

aixt2i1

o

j51

bjyt2j 1ux,t (4)

yt5 b01

o

i51

aixt2i1

o

j51

bjyt2j 1uy,t (5)

wherex and yare stationary variables and ,x and ,y are the lag lengths ofx and y,

respectively. The null hypothesis in the Granger causality test is thatydoes not cause

x, which is represented by H0:b15 ··· 5 b,y5 0, whereas the alternative hypothesis

isH1:bj?0 for at least onejin Eq. (4). The test statistic has a standardFdistribution

with (,_x,_T₂,_x₂,_y₂_{1) degrees of freedom, where}_T_{is the sample size. Obviously,}

the value of the test statistic depends on,_x _and ,_y_{, which makes it necessary to use}

various information criteria to choose the optimum lag lengths.

Hsio (1981) has suggested a sequential procedure for causality testing that combines Akaike’s final predictive error criterion (FPE) and the definition of Granger causality.

To test for causality from ytox, the procedure consists of the following steps:

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→

I.A.

oosa,

Silvapulle

International

Review

Economics

and

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11–30

Table 2

Estimated volatility equations (mean and variance)

Series Mean Variance

Full sample

Dft13

t Zt50.0681ut loght50.00710.899loght2110.039[uut21uh21/2t212(2/p)1/2]

(4.27) (1.62) (17.20) (3.42)

Dft16

t Zt50.0701ut loght50.01010.897loght2110.043[uut21uh21/2t212(2/p)1/2]

(4.92) (1.72) (19.00) (4.32)

Dv3

t Zt5 20.0011ut ht,50.000110.100ut22110.074u2t2310.845ht21

(0.79) (12.72) (11.15) (6.70) (13.29)

Dv6

t Zt50.0021ut ht50.00210.183u2t2110.072u2t2210.303u2t2310.0335ht21

(6.72) (16.72) (31.74) (33.71) (51.62) (33.33)

First sub-sample

Dft13

t Zt50.0091ut loght50.00510.419loght2110.021[uut21uh21/2t212(2/p)1/2]

(3.28) (1.68) (10.51) (4.28)

Dft16

t Zt5 20.0151ut ht50.03210.442u2t21

(2.05) (22.56) (8.32)

Dv3

t Zt50.0051ut ht50.00110.231u2t2110.707ht21

(1.98) (4.55) (12.41) (16.00)

Dv6

t Zt50.0041ut ht50.00310.18u2t2110.702ht21

(1.97) (15.40) (7.32) (15.40)

Second sub-sample

Dft13

t Zt50.0161ut ht50.03610.401u2t21

(2.62) (18.17) (8.30)

Dft16

t Zt5 20.0091ut ht50.02810.131u2t2110.311ht21

(1.68) (4.57) (3.99) (4.39)

Dv3

t Zt50.0031ut ht50.00610.481u2t21

(2.96) (7.91) (3.89)

Dv6

t Zt50.0071ut ht50.00310.043u2t2110.845ht21

(4.52) (2.67) (5.20) (20.50)

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20 I.A. Moosa, P. Silvapulle / International Review of Economics and Finance 9 (2000) 11–30

and compute its FPE with ,_x _{varying from 1 to} _{L, which is chosen arbitrarily.}

Choose the ,_x _{that gives the smallest FPE, denoted}_FPE_x₍,_{x, 0).}

2. Treatxas a controlled variable, with,_x_{as chosen in step 1 and}_y_{as a manipulated}

variable as in Eq. (4). Compute the FPE’s of Eq. (4) by varying the order of

lags ofyfrom 1 toLand determine,_{y, which gives true minimum FPE, denoted}

FPEx(,x, ,y).

3. CompareFPEx(,x, 0) withFPEx(,x,,y). If the former is greater than the latter,

then it can be concluded thaty causesx.7

4.2. Results

The Hsio procedure is used to test for causality between price and volume. The price variable is the first (log) difference (with sign and absolute) of the prices of the three-month and six-month contracts. The volume variable is the first (log) difference of the volume of trading of the three-month and six-month contracts. The results, which are reported in Table 3, are consistent. Regardless of the maturity of the contract and the definition of the price variable, there is unidirectional causality from volume to price. This finding provides support for market inefficiency, but not for the sequential information arrival hypothesis. Moreover, these results are more consistent than those obtained by Fujihara and Mougoue (1997, p. 399), who were led by the inconsistency of the results to conclude that they “imply that petroleum futures return series and volume series have no strong linear predictive power for one another.” We tend to believe that our results are more reliable because they are based on a more powerful test for linear causality than the conventional test used by Fujihara and Mougoue. For a discussion of why the test we used is more powerful, see Hsio (1981).

5. Testing for nonlinear causality 5.1. Methodology

We now turn to a discussion of the Baek-Brock (1992) nonparametric test as modified by Hiemstra and Jones (1994). This test is designed to detect nonlinear causal relations that cannot be detected by the conventional linear causality testing. The test is based on the concept of the correlation integral, which is an estimator of spatial probabilities across time.

Let {xt} and {yt} be two strictly stationary and weakly dependent time series,xmt be

them-length lead vector of {xt} defined as xmt 5{xt, xt11, . . . , xt1m21}. For m $1,

,_x _$_1, ,_y _$_{1, and} _e_. _0,_y _{does not strictly Granger cause} _x _if

Pr(uuxm

t 2xmsuu ,e

*

uux

,x t2,x2x

s2,xuu ,e,uuy

,y t2,y2y

s2,yuu ,e) 5Pr(uuxm

t 2xmsuu ,e

*

uux

t 2,x2x

s2,xuu ,e) (6)

where Pr(.) andk.kdenote the probability and the maximum norm, respectively. The

left hand side of Eq. (6) is the conditional probability that two arbitrarym-length lead

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I.A. Moosa, P. Silvapulle / International Review of Economics and Finance 9 (2000) 11–30 21 Table 3

Testing for linear causality between futures prices and volume

Controlled Manipulated FPE (,_x_,0) _{FPE (},_x_,,_y₎

Variable (x) ,_x _{Variable (}_y₎ ,_y ₍₃₁₀4₎ ₍₃₁₀4₎ _Result* Full sample

Dft13

t 27 Dv3t 16 1.632 1.620 C

Dv3

t 50 Dftt13 10 3.216 3.220 NC

uDft13

t u 43 Dv3t 3 1.068 1.016 C

Dv3

t 50 uDftt13u 7 3.216 3.219 NC

Dft16

t 45 Dv6t 25 2.087 2.041 C

Dv6

t 40 Dftt16 1 0.813 0.816 NC

uDft16

t u 23 Dv6t 7 1.359 1.330 C

Dv6

t 40 uDfttt16u 1 0.813 0.815 NC

First sub-sample

Dft13

t 3 Dv3t 5 0.323 0.319 C

Dv3

t 17 Dftt13 5 0.118 0.119 NC

uDft13

t u 23 Dv3t 3 0.229 0.205 C

Dv3

t 17 uDftt13u 1 0.118 0.119 NC

Dft16

t 20 Dv6t 1 0.390 0.372 C

Dv6

t 4 Dftt16 2 0.624 0.626 NC

uDft16

t u 27 Dv6t 1 0.254 0.250 C

Dv6

t 4 uDfttt16u 1 0.624 0.625 NC

Second sub-sample

Dft13

t 42 Dv3t 1 0.337 0.322 C

Dv3

t 25 Dftt13 1 0.587 0.589 NC

uDft13

t u 23 Dv3t 1 0.253 0.251 C

Dv3

t 25 uDftt13u 1 0.587 0.588 NC

Dft16

t 28 Dv6t 3 0.326 0.320 C

Dv6

t 3 Dftt16 1 0.342 0.348 NC

uDft16

t u 3 Dv6t 1 0.174 0.169 C

Dv6

t 3 uDftt16u 1 0.342 0.345 NC

* C indicates the presence of a causal relationship; NC indicates the absence of a causal relationship.

and,_y_{(lag vectors) are within}_e_{of each other. The probability on the right hand side}

of Eq. (6) is the conditional probability that two arbitrary m-length lead vectors of

{xt} are within a distancee of each other.

The definition of the nonparametric test statistic of Baek and Brock (1992) as modified by Hiemstra and Jones (1994) is as follows. Expressing the conditional probabilities in terms of the corresponding ratios of joint probabilities in Eq. (6), we obtain

C1(m1,_x,,_y,e)

C2(,_x,,_y,e) 5

C3(m1,_x,e)

C4(,_x,e) (7)

where the C’s are the correlation-integral estimators of the joint probabilities. If xt

does not causeyt, then form$ 1, ,x$ 1,,y$ 1, ande .0

√

1

C1(m 1,x,,y,e,n)

C2(,_x,,_y,e,n) 2

C3(m1,_x,e,n)

C4(,_x,e,n)

2

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22 I.A. Moosa, P. Silvapulle / International Review of Economics and Finance 9 (2000) 11–30

The nonlinear Granger causality test as represented by Eq. (8) is applied to the OLS residuals of Eqs. (4) and (5), which are free from linear predictive power. Baek and Brock (1992) argue that by removing linear predictive power with a linear VAR model, any remaining incremental predictive power of one residual series for another can be considered to be nonlinear predictive power.

5.2. Results

We now present the results of nonlinear causality using the procedure outlined

earlier. The appropriate values for the lead lag lengthm, the lag lengths ,_x _and ,_y,

and the scale parameter e must be chosen to apply this procedure. Hiemstra and

Jones (1994) recommend the following values:m51,,_x₅,_{y, and}_e₅_{1.5 with}_{s 5}

1. Tables 4 and 5 report the results of the nonlinear causality test as applied to the residuals of Eqs. (4) and (5) in which CS and TVAL denote the difference between two conditional probabilities as given by Eq. (8) and the standardized statistic, respectively.

Investigating the sensitivity of the results by varying the values ofe from 1.0 to 2.0

and sfrom 1.0 to 3.0 reveals only a marginal difference in the results.

The results reported in Table 4 provide evidence for the presence of bi-directional

nonlinear causality betweenDft13

t and Dv3t for all sample periods, except the second

sub-sample period, which reveals no causality from price to volume. These results hold for all common lag lengths from 1 to 21. The results reported in Table 5 show the same for the six-month maturity except for the first sub-sample period. For the

full sample period, the result thatDft16

t causesDv6t holds only for lags of 1–3. Similar

results are obtained if the price variable is the absolute price change. There is no readily available explanation for why the results differ for the second sub-sample period (for the three-month maturity) and for the first sub-sample period (for the six-month maturity). However, it is safe to conclude that the results provide evidence for bi-directional causality. Hence, nonlinear causality testing reveals what linear testing would not reveal. These results are consistent with those produced by Fujihara and Mougou (1997), who in this case used the same test we used. The results support the sequential information arrival hypothesis, and so they are in contrast with those produced by Foster (1995). The similarity to Foster’s results is that they indicate market inefficiency.

Hsieh (1991) finds that much of the nonlinear structure in daily stock prices is related to ARCH dependence, implying that the nonlinear test may only detect volatility dependence. Therefore, it may be useful and informative to apply the modi-fied Baek and Brock (1992) test to the volatility-filtered series (i.e., the series derived by removing the ARCH effect). These series are therefore adjusted for the linear and volatility effects that were estimated earlier.

The results of testing for nonlinear causality between the volatility-filtered price and volume series are reported in Tables 6 and 7. These results are again inconsistent across maturities or sample periods. For the three-month maturity the weight of the

evidence is for the presence of bi-directional nonlinear causality betweenDft13

t and Dv3

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unidirec-I.A. Moosa, P. Silvapulle / International Review of Economics and Finance 9 (2000) 11–30 23 Table 4

Testing for nonlinear causality between price and volume (three-month contract)

Dv3

t →Dftt13 Dftt13→Dv3t

,_x₅,_y _CS _TVAL _CS _TVAL

Full sample

1 0.181 19.792 0.115 11.481

3 0.221 19.363 0.171 9.273

5 0.293 18.924 0.158 8.457

7 0.279 18.354 0.142 7.859

9 0.399 16.281 0.132 6.999

11 0.398 15.120 0.128 5.968

13 0.366 12.613 0.099 5.821

15 0.354 8.124 0.084 4.720

17 0.329 6.180 0.071 4.521

19 0.303 5.124 0.058 3.141

21 0.287 4.920 0.031 3.918

First sub-sample

1 0.046 12.612 0.019 5.588

3 0.102 13.719 0.043 9.331

5 0.152 11.780 0.112 9.055

7 0.192 8.927 0.029 6.885

9 0.181 6.926 0.074 6.246

11 0.183 5.874 0.198 5.138

13 0.145 5.044 0.037 4.732

15 0.122 4.965 0.092 4.116

17 0.123 4.666 0.273 3.674

19 0.113 4.246 0.147 3.218

21 0.110 3.929 0.128 2.928

Second sub-sample

1 0.054 14.126 0.037 1.970

3 0.104 13.921 0.054 1.718

5 0.132 13.780 0.059 1.713

7 0.181 9.279 0.014 1.602

9 0.160 7.962 0.039 1.710

11 0.153 7.847 0.033 0.674

13 0.124 6.965 0.030 0.436

15 0.112 6.629 0.028 0.410

17 0.104 6.240 0.018 0.572

19 0.098 5.428 0.014 0.710

21 0.070 3.929 0.009 0.600

The results are based on the modified Baek and Brock nonlinear causality test. The test is applied to the VAR residuals. CS and TVAL denote the difference between two conditional probabilities in Eq. (7) and the standardized test statistic of Eq. (8), respectively. Under the null hypothesis of no causality, the test statistic has a standard normal distribution.

tional causality from Dft16

t to Dv6t (Table 7). The observed causality from Dftt16 to

Dv6

t appears to be due to volatility dependence. Similar results are obtained if the

price variable is the absolute price change. The results clearly show that there is a maturity effect, but we tend to agree with Foster (1995) that the maturity effect is in

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24 I.A. Moosa, P. Silvapulle / International Review of Economics and Finance 9 (2000) 11–30 Table 5

Testing for nonlinear causality between price and volume (six-month contract)

Dv6

t →Dftt16 Dftt16→Dv6t

,_x₅,_y _CS _TVAL _CS _TVAL

Full sample

1 0.099 11.360 0.010 2.418

3 0.124 12.240 0.014 2.109

5 0.098 12.521 0.012 2.216

7 0.083 12.641 0.009 1.892

9 0.075 11.659 0.010 1.428

11 0.067 10.765 0.009 1.329

13 0.065 9.981 0.009 1.210

15 0.112 7.999 0.007 1.208

17 0.098 5.812 0.007 1.202

19 0.089 3.520 0.007 1.112

21 0.074 2.992 0.005 1.100

First sub-sample

1 0.037 9.281 0.026 1.252

3 0.042 6.029 0.029 1.825

5 0.040 6.00 0.023 1.561

7 0.038 5.928 0.014 1.120

9 0.034 5.721 0.014 1.128

11 0.045 4.500 0.047 1.920

13 0.038 3.892 0.035 1.820

15 0.023 3.201 0.034 0.820

17 0.099 2.918 0.032 0.728

19 0.049 2.718 0.022 0.584

21 0.093 2.135 0.013 0.232

Second sub-sample

1 0.034 9.589 0.009 6.120

3 0.032 7.981 0.012 4.128

5 0.052 7.128 0.007 4.003

7 0.048 7.120 0.009 3.982

9 0.041 6.189 0.026 3.124

11 0.045 5.820 0.033 2.910

13 0.039 4.892 0.007 1.920

15 0.148 4.781 0.016 1.700

17 0.120 3.480 0.042 1.548

19 0.058 3.103 0.019 1.004

21 0.049 2.100 0.016 0.912

See footnote to Table 4.

essence a liquidity effect. It seems that volume serves better as a proxy for information arrival in liquid markets than otherwise.

6. Conclusion

We start these concluding remarks by recapitulating on the motivation for writing this article. Given the diversity of the conditions under which the price–volume

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rela-I.A. Moosa, P. Silvapulle / International Review of Economics and Finance 9 (2000) 11–30 25 Table 6

Testing for nonlinear causality between volatility-filtered series (three-month contract)

Dv3

t →Dftt13 Dftt13→Dv3t

,_x₅,_y _CS _TVAL _CS _TVAL

Full sample

1 0.132 9.997 0.095 6.142

3 0.181 9.736 0.115 5.731

5 0.208 8.999 0.102 3.743

7 0.249 8.374 0.098 2.952

9 0.218 7.821 0.082 1.818

11 0.253 7.528 0.073 1.868

13 0.300 6.316 0.071 1.821

15 0.259 5.421 0.060 1.728

17 0.228 3.581 0.055 1.652

19 0.203 2.912 0.028 1.241

21 0.199 2.001 0.017 0.819

First sub-sample

1 0.047 7.281 0.012 3.821

3 0.099 6.182 0.038 4.328

5 0.128 6.000 0.097 2.892

7 0.142 5.892 0.032 2.581

9 0.138 5.018 0.072 1.999

11 0.129 3.818 0.124 1.989

13 0.114 3.521 0.038 1.281

15 0.099 3.121 0.091 0.928

17 0.091 3.999 0.172 0.921

19 0.082 1.928 0.102 0.481

21 0.048 1.800 0.009 0.280

Second sub-sample*

1 0.052 10.112

3 0.081 8.480

5 0.106 6.129

7 0.109 5.892

9 0.121 4.182

11 0.098 4.003

13 0.091 3.891

15 0.072 2.918

17 0.080 1.281

19 0.052 1.015

21 0.033 0.928

The test is based on the linear and volatility-filtered prices and volume series. The values ofmand eare 1 and 1.5, respectively. The null hypothesis is no causality. * The results for causality from price to volume are not reported because they were insignificant even before volatility filtering.

tionship has been tested, it may be useful to state, ex ante, what we thought this study had to offer. We thought that since the available empirical evidence is mixed, a new piece of work could be potentially capable of contributing to the emergence of a consensus view on the underlying issue. The two studies of the oil market by Foster (1995) and by Fujihara and Mougoue (1997) produced contradictory results. Our study

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26 I.A. Moosa, P. Silvapulle / International Review of Economics and Finance 9 (2000) 11–30 Table 7

Testing for nonlinear causality between volatility-filtered series (six-month contract)

Dv6

t →Dftt16 Dftt16→Dv6t

,_x₅,_y _CS _TVAL _CS _TVAL

Full sample

1 0.085 2.631 0.004 1.859

3 0.097 2.820 0.003 1.109

5 0.058 3.652 0.003 1.216

7 0.053 3.401 0.002 1.016

9 0.075 2.965 0.002 1.011

11 0.091 2.975 0.002 1.009

13 0.099 2.798 0.002 1.003

15 0.112 2.777 0.001 0.991

17 0.072 2.128 0.001 0.912

19 0.060 1.920 0.001 0.512

21 0.041 1.912 0.001 0.300

First sub-sample*

1 0.043 4.218

3 0.042 3.185

5 0.038 2.018

7 0.037 1.992

9 0.028 1.721

11 0.037 1.528

13 0.028 1.321

15 0.091 1.120

17 0.028 0.998

19 0.022 0.921

21 0.018 0.900

Second sub-sample

1 0.038 5.958 0.010 1.990

3 0.024 4.912 0.019 1.928

5 0.041 4.112 0.005 1.418

7 0.032 3.818 0.009 1.381

9 0.039 3.101 0.018 1.222

11 0.040 2.412 0.021 1.200

13 0.034 1.928 0.009 0.981

15 0.100 1.721 0.015 0.920

17 0.050 1.528 0.038 0.810

19 0.047 1.008 0.018 0.618

21 0.042 0.926 0.010 0.412

See footnote to Table 6. * The results for causality from price to volume are not reported because they were insignificant even before volatility filtering.

aimed at combining some features of the two studies. Thus, we used the same technique used by Fujihara and Mougoue to test for nonlinear causality but a different test for linear causality. And, like Foster, we decided to consider the maturity effect. We also thought that it would be useful to examine the relationship by including and excluding periods of turmoil in the oil market.

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I.A. Moosa, P. Silvapulle / International Review of Economics and Finance 9 (2000) 11–30 27

We therefore proceeded to examine the relationship between prices and the trading volumes of the WTI crude oil futures contracts using daily data covering the period between January 2, 1985, and July 1, 1996. Two futures contracts were used for this purpose: the three-month and the six-month contracts. What has this study achieved, ex post?

As a preliminary step, we tested for unit root and the presence of volatility in the time series and found that they all have unit root and ARCH effect. We then tested for linear causality, using the Hsio procedure, and found that causality runs from volume to price regardless of the choice of the price variable, the maturity of the contract, and the time period. When we tested for nonlinear causality, using the Baek-Brock test, we obtained results mostly showing bi-directional causality for both contracts, albeit inconsistent across sample periods. However, the evidence suggested that the nonlinear causality running from price to volume in the case of the six-month contract was due to volatility dependence. Similar results were obtained when the price variable was the absolute price change.

The results presented in this study are broadly supportive of those obtained by Hiemstra and Jones (1994) and Fujihara and Mougoue (1997) in the sense of high-lighting the importance of nonlinear causality. There are, however, a few differences. These two studies produced weak evidence for linear causality. For example, Fujihara and Mougoue found no linear causality in the case of heating oil, unidirectional causality from price to volume in the case of unleaded gas, and unidirectional causality from volume to price in the case of crude oil. There is no readily available explanation for why different kinds of oil products produce different linear causality results. The results of this study are more consistent in revealing linear causality from volume to price regardless of the choice of the maturity, the price variable, and the time period. The nonlinear causality test results are slightly different because Fujihara and Mougoue found unidirectional causality from volume to price only in the case of crude oil. Although our results are not entirely consistent, they show bi-directional nonlinear causality in most cases. In agreement with Foster (1995), the results provide evidence for the presence of a liquidity or maturity effect and for market inefficiency. We find the explanations presented by Foster for inefficiency to be plausible.

In general, the results presented in this study are consistent with the predictions of the sequential information arrival hypothesis and the noise trading model. This conclusion would not have been reached on the basis of the results of linear causality testing alone. These results are useful for regulators, market participants, and the efficiency of the oil futures market. The results are useful for regulators when they consider such measures as limits on daily price movements and positions. For market participants, the results are useful since they imply that volume can be used to predict prices, lending support to technical analysis. Finally, the results imply market ineffi-ciency, which may be, as Foster (1995) puts it, due to mimetic contagion or the fact that traders may condition their prices on previous volumes.

A word of warning is warranted here. The test used in this paper is powerful in detecting nonlinear causal dependence, but it provides no guidance in relation to the source of this dependence. This explanation must be left to economic theory.

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28 I.A. Moosa, P. Silvapulle / International Review of Economics and Finance 9 (2000) 11–30

Acknowledgment

We are grateful to an anonymous referee for high-quality comments and suggestions. All remaining errors and omissions are entirely ours.

Notes

1. For a discussion, see Karpoff (1987) and Fujihara and Mougoue (1997). 2. For a general discussion of volume as a technical indicator see, for example,

Meyers (1994, pp. 139–146).

3. See, for example, Savit (1989), Hinich and Patterson (1985), Scheinkman and Le Baron (1989), Brock et al. (1991), Hsieh (1991), Hiemstra and Jones (1992, 1994), and Fujihara and Mougoue (1997).

4. For a good survey, see Karpoff (1987).

5. Over the sample period used in this study, the average daily trading of the three-month contract was 9,254.2, much higher than the corresponding figure of 1,320.5 for the six-month contract. This indicates that the former is much more liquid than the latter.

6. See Granger (1988).

7. Testing for causality from x to y can be carried out on the basis of Eq. (5)

using the same procedure. References

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30 I.A. Moosa, P. Silvapulle / International Review of Economics and Finance 9 (2000) 11–30 Smirlock, M., & Starks, L. (1988). An empirical analysis of the stock price-volume relationship.Journal

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Table 6

Testing for nonlinear causality between volatility-filtered series (three-month contract)

Dv3

t →Dftt13 Dftt13→Dv3t

,_x₅,_y _CS _TVAL _CS _TVAL

Full sample

1 0.132 9.997 0.095 6.142

3 0.181 9.736 0.115 5.731

5 0.208 8.999 0.102 3.743

7 0.249 8.374 0.098 2.952

9 0.218 7.821 0.082 1.818

11 0.253 7.528 0.073 1.868

13 0.300 6.316 0.071 1.821

15 0.259 5.421 0.060 1.728

17 0.228 3.581 0.055 1.652

19 0.203 2.912 0.028 1.241

21 0.199 2.001 0.017 0.819

First sub-sample

1 0.047 7.281 0.012 3.821

3 0.099 6.182 0.038 4.328

5 0.128 6.000 0.097 2.892

7 0.142 5.892 0.032 2.581

9 0.138 5.018 0.072 1.999

11 0.129 3.818 0.124 1.989

13 0.114 3.521 0.038 1.281

15 0.099 3.121 0.091 0.928

17 0.091 3.999 0.172 0.921

19 0.082 1.928 0.102 0.481

21 0.048 1.800 0.009 0.280

Second sub-sample*

1 0.052 10.112

3 0.081 8.480

5 0.106 6.129

7 0.109 5.892

9 0.121 4.182

11 0.098 4.003

13 0.091 3.891

15 0.072 2.918

17 0.080 1.281

19 0.052 1.015

21 0.033 0.928

The test is based on the linear and volatility-filtered prices and volume series. The values ofmand

eare 1 and 1.5, respectively. The null hypothesis is no causality. * The results for causality from price to volume are not reported because they were insignificant even before volatility filtering.

tionship has been tested, it may be useful to state, ex ante, what we thought this study

had to offer. We thought that since the available empirical evidence is mixed, a new

piece of work could be potentially capable of contributing to the emergence of a

consensus view on the underlying issue. The two studies of the oil market by Foster

(1995) and by Fujihara and Mougoue (1997) produced contradictory results. Our study

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26 I.A. Moosa, P. Silvapulle / International Review of Economics and Finance 9 (2000) 11–30

Table 7

Testing for nonlinear causality between volatility-filtered series (six-month contract)

Dv6

t →Dftt16 Dftt16→Dv6t

,_x₅,_y _CS _TVAL _CS _TVAL

Full sample

1 0.085 2.631 0.004 1.859

3 0.097 2.820 0.003 1.109

5 0.058 3.652 0.003 1.216

7 0.053 3.401 0.002 1.016

9 0.075 2.965 0.002 1.011

11 0.091 2.975 0.002 1.009

13 0.099 2.798 0.002 1.003

15 0.112 2.777 0.001 0.991

17 0.072 2.128 0.001 0.912

19 0.060 1.920 0.001 0.512

21 0.041 1.912 0.001 0.300

First sub-sample*

1 0.043 4.218

3 0.042 3.185

5 0.038 2.018

7 0.037 1.992

9 0.028 1.721

11 0.037 1.528

13 0.028 1.321

15 0.091 1.120

17 0.028 0.998

19 0.022 0.921

21 0.018 0.900

Second sub-sample

1 0.038 5.958 0.010 1.990

3 0.024 4.912 0.019 1.928

5 0.041 4.112 0.005 1.418

7 0.032 3.818 0.009 1.381

9 0.039 3.101 0.018 1.222

11 0.040 2.412 0.021 1.200

13 0.034 1.928 0.009 0.981

15 0.100 1.721 0.015 0.920

17 0.050 1.528 0.038 0.810

19 0.047 1.008 0.018 0.618

21 0.042 0.926 0.010 0.412

See footnote to Table 6. * The results for causality from price to volume are not reported because they were insignificant even before volatility filtering.

aimed at combining some features of the two studies. Thus, we used the same technique

used by Fujihara and Mougoue to test for nonlinear causality but a different test for

linear causality. And, like Foster, we decided to consider the maturity effect. We also

thought that it would be useful to examine the relationship by including and excluding

periods of turmoil in the oil market.

(3)

We therefore proceeded to examine the relationship between prices and the trading

volumes of the WTI crude oil futures contracts using daily data covering the period

between January 2, 1985, and July 1, 1996. Two futures contracts were used for this

purpose: the three-month and the six-month contracts. What has this study achieved,

ex post?

As a preliminary step, we tested for unit root and the presence of volatility in the

time series and found that they all have unit root and ARCH effect. We then tested

for linear causality, using the Hsio procedure, and found that causality runs from

volume to price regardless of the choice of the price variable, the maturity of the

contract, and the time period. When we tested for nonlinear causality, using the

Baek-Brock test, we obtained results mostly showing bi-directional causality for both

contracts, albeit inconsistent across sample periods. However, the evidence suggested

that the nonlinear causality running from price to volume in the case of the six-month

contract was due to volatility dependence. Similar results were obtained when the

price variable was the absolute price change.

The results presented in this study are broadly supportive of those obtained by

Hiemstra and Jones (1994) and Fujihara and Mougoue (1997) in the sense of

high-lighting the importance of nonlinear causality. There are, however, a few differences.

These two studies produced weak evidence for linear causality. For example, Fujihara

and Mougoue found no linear causality in the case of heating oil, unidirectional

causality from price to volume in the case of unleaded gas, and unidirectional causality

from volume to price in the case of crude oil. There is no readily available explanation

for why different kinds of oil products produce different linear causality results. The

results of this study are more consistent in revealing linear causality from volume to

price regardless of the choice of the maturity, the price variable, and the time period.

The nonlinear causality test results are slightly different because Fujihara and Mougoue

found unidirectional causality from volume to price only in the case of crude oil.

Although our results are not entirely consistent, they show bi-directional nonlinear

causality in most cases. In agreement with Foster (1995), the results provide evidence

for the presence of a liquidity or maturity effect and for market inefficiency. We find

the explanations presented by Foster for inefficiency to be plausible.

In general, the results presented in this study are consistent with the predictions

of the sequential information arrival hypothesis and the noise trading model. This

conclusion would not have been reached on the basis of the results of linear causality

testing alone. These results are useful for regulators, market participants, and the

efficiency of the oil futures market. The results are useful for regulators when they

consider such measures as limits on daily price movements and positions. For market

participants, the results are useful since they imply that volume can be used to predict

prices, lending support to technical analysis. Finally, the results imply market

ineffi-ciency, which may be, as Foster (1995) puts it, due to mimetic contagion or the fact

that traders may condition their prices on previous volumes.

A word of warning is warranted here. The test used in this paper is powerful in

detecting nonlinear causal dependence, but it provides no guidance in relation to the

source of this dependence. This explanation must be left to economic theory.

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28 I.A. Moosa, P. Silvapulle / International Review of Economics and Finance 9 (2000) 11–30

Acknowledgment

We are grateful to an anonymous referee for high-quality comments and suggestions.

All remaining errors and omissions are entirely ours.

Notes

1. For a discussion, see Karpoff (1987) and Fujihara and Mougoue (1997).

2. For a general discussion of volume as a technical indicator see, for example,

Meyers (1994, pp. 139–146).

3. See, for example, Savit (1989), Hinich and Patterson (1985), Scheinkman and

Le Baron (1989), Brock et al. (1991), Hsieh (1991), Hiemstra and Jones (1992,

1994), and Fujihara and Mougoue (1997).

4. For a good survey, see Karpoff (1987).

5. Over the sample period used in this study, the average daily trading of the

three-month contract was 9,254.2, much higher than the corresponding figure

of 1,320.5 for the six-month contract. This indicates that the former is much

more liquid than the latter.

6. See Granger (1988).

7. Testing for causality from

x

to

y

can be carried out on the basis of Eq. (5)

using the same procedure.

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