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
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
x
t
5 a 1
o
, x
i
5
1
a
i
x
t
2
i
1
o
, y
j
5
1
b
j
y
t
2
j
1 u
x ,t
4 y
t
5 b 1
o
, x
i
5
1
a
i
x
t
2
i
1
o
, y
j
5
1
b
j
y
t
2
j
1 u
y ,t
5 where x and y are stationary variables and ,x and ,y are the lag lengths of x and y,
respectively. The null hypothesis in the Granger causality test is that y does not cause x
, which is represented by H :b
1
5 ··· 5 b
, y
5 0, whereas the alternative hypothesis
is H
1
:b
j
? 0 for at least one j in Eq. 4. The test statistic has a standard F distribution
with ,x, T 2 ,x 2 ,y 2 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 y to x, the procedure consists of the following steps:
1. Treat x as a one-dimensional process as represented by Eq. 4 with b
j
5 ∀
j ,
→
I.A. M
oosa, P.
Silvapulle
International
Review of
Economics
and Finance
9 2000
11–30
19 Table 2
Estimated volatility equations mean and variance Series
Mean Variance
Full sample Df
t
1
3 t
Z
t
5 0.068 1 u
t
logh
t
5 0.007 1 0.899logh
t
2
1
1 0.039[
uu
t
2
1
uh
2
12 t
2
1
2 2p
12
] 4.27
1.62 17.20
3.42 Df
t
1
6 t
Z
t
5 0.070 1 u
t
logh
t
5 0.010 1 0.897logh
t
2
1
1 0.043[
uu
t
2
1
uh
2
12 t
2
1
2 2p
12
] 4.92
1.72 19.00
4.32 Dv
3 t
Z
t
5 2 0.001 1 u
t
h
t
, 5 0.0001 1 0.100u
2 t
2
1
1 0.074u
2 t
2
3
1 0.845h
t 2 1
0.79 12.72
11.15 6.70
13.29 Dv
6 t
Z
t
5 0.002 1 u
t
h
t
5 0.002 1 0.183u
2 t
2
1
1 0.072u
2 t
2
2
1 0.303u
2 t
2
3
1 0.0335h
t
2
1
6.72 16.72 31.74
33.71 51.62
33.33 First sub-sample
Df
t
1
3 t
Z
t
5 0.009 1 u
t
logh
t
5 0.005 1 0.419logh
t
2
1
1 0.021[
uu
t
2
1
uh
2
12 t
2
1
2 2p
12
] 3.28
1.68 10.51
4.28 Df
t
1
6 t
Z
t
5 2 0.015 1 u
t
h
t
5 0.032 1 0.442u
2 t
2
1
2.05 22.56 8.32
Dv
3 t
Z
t
5 0.005 1 u
t
h
t
5 0.001 1 0.231u
2 t
2
1
1 0.707h
t
2
1
1.98 4.55
12.41 16.00
Dv
6 t
Z
t
5 0.004 1 u
t
h
t
5 0.003 1 0.18u
2 t
2
1
1 0.702h
t
2
1
1.97 15.40 7.32
15.40 Second sub-sample
Df
t
1
3 t
Z
t
5 0.016 1 u
t
h
t
5 0.036 1 0.401u
2 t
2
1
2.62 18.17 8.30
Df
t
1
6 t
Z
t
5 2 0.009 1 u
t
h
t
5 0.028 1 0.131u
2 t
2
1
1 0.311h
t 2 1
1.68 4.57
3.99 4.39
Dv
3 t
Z
t
5 0.003 1 u
t
h
t
5 0.006 1 0.481u
2 t
2
1
2.96 7.91
3.89 Dv
6 t
Z
t
5 0.007 1 u
t
h
t
5 0.003 1 0.043u
2 t
2
1
1 0.845h
t
2
1
4.52 2.67
5.20 20.50
Figures in parentheses are the t statistics.
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. Treat x as 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 of y from 1 to L and determine ,y, which gives true minimum FPE, denoted
FPE
x
,x, ,y. 3. Compare FPE
x
,x, 0 with FPE
x
,x, ,y. If the former is greater than the latter, then it can be concluded that y causes x.
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
