I.A. Moosa, P. Silvapulle International Review of Economics and Finance 9 2000 11–30 15
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. f
t
1
3 t
and f
t
1
6 t
are the logarithms of the futures prices for maturities of three and six months, respectively, while v
3 t
and v
6 t
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.
The data sample used in this study covers a period between 1985 and 1996. Thus,
16 I.A. Moosa, P. Silvapulle International Review of Economics and Finance 9 2000 11–30
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 199091 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, Df
t
, and the absolute price change, |Df
t
|. 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
ARCH2 and ARCH3 specifications. The LM2 and LM3 test statistics have asymptotic x
2
2 and x
2
3 distributions, respectively. The one-sided score test statistic for ARCH2 has an asymptotic x¯
2
distribution, which is a weighted average of x
2
1 and x
2
2, while the one-sided test statistic for ARCH3 has an asymptotic x¯
2
distribu- tion, which is a weighted average of x
2
1, x
2
2, and x
2
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.
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
, 5
35 ,
5 45
, 5
35 ,
5 45
, 5
35 ,
5 45
Full sample f
t t
1
3
without trend 2
1.30 2
1.35 6.24
4.90 2
1.78 2
1.70 f
t t
1
3
with trend 2
1.61 2
1.74 1.07
0.85 2
1.58 2
1.51 f
t t
1
6
without trend 2
3.43 2
3.51 1.61
1.28 2
2.01 2
1.99 f
t t
1
6
with trend 2
3.10 2
3.19 0.80
0.64 2
2.81 2
2.48 v
t 3
without trend 2
2.27 2
2.32 5.83
4.62 2
2.48 2
1.90 v
t 3
with trend 2
2.91 2
2.98 0.92
0.74 2
1.90 2
2.05 v
t 6
without trend 2
2.88 2
2.16 0.61
0.49 2
2.23 2
2.13 v
t 6
with trend 2
2.96 2
2.58 0.28
0.25 2
1.92 2
1.51 First sub-samble
f
t t
1
3
without trend 2
1.69 2
1.58 3.35
2.30 2
1.67 2
1.52 f
t t
1
3
with trend 2
1.18 2
1.05 0.53
0.38 2
1.12 2
0.92 f
t t
1
6
without trend 2
2.00 2
1.06 2.08
1.48 2
1.98 2
1.02 f
t t
1
6
with trend 2
2.04 2
2.04 0.52
0.38 v
t 3
without trend 2
1.57 2
1.51 2.69
1.88 2
1.32 2
1.24 v
t 3
with trend 2
2.70 2
2.61 0.46
0.33 2
2.58 2
2.41 v
t 6
without trend 2
1.59 2
1.54 0.84
0.59 2
1.57 2
1.42 v
t 6
with trend 2
1.77 2
1.62 0.63
0.45 2
1.64 2
1.51 Second sub-sample
f
t t
1
3
without trend 2
2.60 2
2.52 0.59
0.46 2
2.25 2
2.04 f
t t
1
3
with trend 2
2.40 2
2.21 0.34
0.27 2
2.15 2
2.08 f
t t
1
6
without trend 2
2.41 2
2.30 0.90
0.69 2
2.21 2
2.18 f
t t
1
6
with trend 2
2.70 2
2.01 0.28
0.22 2
2.48 2
1.90 v
t 3
without trend 2
1.88 2
1.90 0.98
0.69 2
1.72 2
1.45 v
t 3
with trend 2
1.68 2
1.67 0.88
0.62 2
1.44 2
1.28 v
t 6
without trend 2
1.60 2
1.72 1.49
1.04 2
1.50 2
1.20 v
t 6
with trend 2
1.02 2
0.99 0.79
0.56 2
0.98 2
0.97 ,
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 are 22.86 and 0.431 without trend and 23.41 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 series Z
t
, denoted h
t
, is modelled as Z
t
5 m 1 u
t
1 where u
t
| φ
t
2
1
, N0, h
t
and h
t
5 h 1
o
q j
5
1
a
j
u
2 t
2
j
1
o
p k
5
1
b
k
h
t
2
k
2 where t 5 maxp, q, . . . , T. By imposing the restriction b
1
5 ··· 5 b
p
5 0 on Eq. 2,
we obtain ARCHq models. Although ARCHGARCH specifications have been
18 I.A. Moosa, P. Silvapulle International Review of Economics and Finance 9 2000 11–30
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
logh
t
5 h 1
o
p k
5
1
c
k
u
t
2
k
h
2
12 t
2
k
1
o
q j
5
1
a
j
logh
t
2
j
1 a [
uu
t
2
1
uh
2
12 t
2
1
2 2p
12
] 3
where t 5 maxp, 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 robust t statistics 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 EGARCH1, 1 is appropriate for Df
t
1
3 t
and Df
t
1
6 t
over the full sample period. The volume series Dv
3 t
and Dv
6 t
, on the other hand, follow GARCH1, 2 processes over the full sample period. Different model specifications are obtained
for the sub-samples.
4. Testing for linear causality
