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
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 {x
t
} and {y
t
} be two strictly stationary and weakly dependent time series, x
m t
be the m-length lead vector of {x
t
} defined as x
m t
5 {x
t
, x
t
1
1
, . . . , x
t
1
m
2
1
}. For m 1, ,
x 1, ,y 1, and e . 0, y does not strictly Granger cause x if
Pr uux
m t
2 x
m s
uu , e uux
, x
t
2
, x
2 x
, x
s
2
, x
uu , e,uuy
, y
t
2
, y
2 y
, y
s
2
, y
uu , e 5
Pr uux
m t
2 x
m s
uu , e uux
, x
t
2
, x
2 x
, x
s
2
, x
uu , e 6
where Pr. and k.k denote the probability and the maximum norm, respectively. The left hand side of Eq. 6 is the conditional probability that two arbitrary m-length lead
vectors of {x
t
} are within a small distance e of each other, given that the corresponding ,x
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
310
4
310
4
Result Full sample
Df
t
1
3 t
27 Dv
3 t
16 1.632
1.620 C
Dv
3 t
50 Df
t
1
3 t
10 3.216
3.220 NC
uDf
t
1
3 t
u 43
Dv
3 t
3 1.068
1.016 C
Dv
3 t
50 uDf
t
1
3 t
u 7
3.216 3.219
NC Df
t
1
6 t
45 Dv
6 t
25 2.087
2.041 C
Dv
6 t
40 Df
t
1
6 t
1 0.813
0.816 NC
uDf
t
1
6 t
u 23
Dv
6 t
7 1.359
1.330 C
Dv
6 t
40 uDf
t t
1
6 t
u 1
0.813 0.815
NC First sub-sample
Df
t
1
3 t
3 Dv
3 t
5 0.323
0.319 C
Dv
3 t
17 Df
t
1
3 t
5 0.118
0.119 NC
uDf
t
1
3 t
u 23
Dv
3 t
3 0.229
0.205 C
Dv
3 t
17 uDf
t
1
3 t
u 1
0.118 0.119
NC Df
t
1
6 t
20 Dv
6 t
1 0.390
0.372 C
Dv
6 t
4 Df
t
1
6 t
2 0.624
0.626 NC
uDf
t
1
6 t
u 27
Dv
6 t
1 0.254
0.250 C
Dv
6 t
4 uDf
t t
1
6 t
u 1
0.624 0.625
NC Second sub-sample
Df
t
1
3 t
42 Dv
3 t
1 0.337
0.322 C
Dv
3 t
25 Df
t
1
3 t
1 0.587
0.589 NC
uDf
t
1
3 t
u 23
Dv
3 t
1 0.253
0.251 C
Dv
3 t
25 uDf
t
1
3 t
u 1
0.587 0.588
NC Df
t
1
6 t
28 Dv
6 t
3 0.326
0.320 C
Dv
6 t
3 Df
t
1
6 t
1 0.342
0.348 NC
uDf
t
1
6 t
u 3
Dv
6 t
1 0.174
0.169 C
Dv
6 t
3 uDf
t t
1
6
u 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
{x
t
} are within a distance e 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
C 1m 1 ,x,,y,e
C 2,x,,y,e
5 C
3m 1 ,x,e C
4,x,e 7
where the C’s are the correlation-integral estimators of the joint probabilities. If x
t
does not cause y
t
, then for m 1, ,x 1, ,y 1, and e . 0
√
n
1
C 1m 1 ,x,,y,e,n
C 2,x,,y,e,n
2 C
3m 1 ,x,e,n C
4,x,e,n
2
| AN
0,s
2
m,,x,,y,e 8
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 length m, 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: m 5 1, ,x 5 ,y, and e 5 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 of e from 1.0 to 2.0 and s from 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 between Df
t
1
3 t
and Dv
3 t
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 that Df
t
1
6 t
causes Dv
6 t
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 between Df
t
1
3 t
and Dv
3 t
, which is not explained by volatility Table 6. There is, however, only a 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
Dv
3 t
→ Df
t
1
3 t
Df
t
1
3 t
→ Dv
3 t
, x 5 ,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 Df
t
1
6 t
to Dv
6 t
Table 7. The observed causality from Df
t
1
6 t
to Dv
6 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
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
Dv
6 t
→ Df
t
1
6 t
Df
t
1
6 t
→ Dv
6 t
, x 5 ,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
