312 M.-S. Pan, Y.A. Liu International Review of Economics and Finance 8 1999 305–316
are given in Table 3. The variance ratio estimates, homoskedasticity- and heteroskedas- ticity-robust standard normal Z statistics, and the bootstrap percentiles are reported
for q 5 4, 8, 16, and 24. The bootstrapped percentiles show that at least one of the variance ratio estimates for each of the four periods are significantly different from
unity at the 5 level. Therefore, the deviations from the cointegrating relationship for all the four sample periods do not follow a martingale. Furthermore, the Wald
statistics for the joint test reported in Table 3 also reject the martingale null hypothesis.
It is worth noting that, for the periods of 1980–1984 and 1985–1992, the VR statistics about the individual horizons decrease when q increases, meaning that the deviations
from the cointegrating relationship follow a mean-reverting behavior. Given the frac- tional integration results reported in the previous section, it appears that the error
correction term for the 1980–1984 sample takes a much longer time to reverting back to the mean than the 1985–1992 period. That the error correction term is a long-
memory, mean-reverting process supports Baillie and Bollerslev’s 1994 contention that the seven exchange rates are fractionally cointegrated. This feature of long-
memory, mean-reversion, however, exists only in the 1980–1984 data.
5. Long memory, mean reversion and rescaled range analysis
To further examine the characteristics of long-memory and mean-reversion in the error correction term of the system of the seven exchange rates, we employ the
rescaled range RS test. The RS statistic is formed by measuring the range between the maximum and minimum distances that the cumulative sum of a stochastic random
variable has deviated from its mean and then dividing this by its standard deviation. An unusually small large RS statistic signifies mean-reversion mean-aversion.
Mandelbrot 1972 demonstrates that the RS statistic can uncover not only periodic dependence but also nonperiodic cycles. He further shows that the RS statistic is a
more general test of long-memory in time series than examining either autocorrelations i.e., the variance ratio test or spectral densities.
Lo 1991 points out that the original version of the RS analysis which may be termed as classical RS test has limitations in that it cannot distinguish between short-
and long-term dependence, nor is it robust to heteroskedasticity. Lo modifies the RS statistic by replacing the standard deviation with a weighted autocovariances up to a
finite number of lags:
V q 5
1
√
Ts ˆ q
5
Max 1 k T
o
k t
5
1
X
t
2 X 2 Min
1 k T
o
k t
5
1
X
t
2 X
6
, 6
where X
t
5 1 2 L Z
t
, sˆ
2
q 5 sˆ
2
1 2S
q i
5
1
[1 2 Iq 1 1]gˆI, and, respectively, sˆ
2
and gˆ denote the usual sample variance and autocovariance estimators. Thus, the modified RS statistic detects long memory by controlling for
M.-S. Pan, Y.A. Liu International Review of Economics and Finance 8 1999 305–316 313
Table 4 Rescaled range analysis of long memory
Wald Sample period
V V
4 V
8 V
16 V
24 statistic
1973–1979 1.378
1.353 1.311
1.230 1.182
8.791 p 5
.273 p 5
.326 p 5
.449 p 5
.461 q 5
.086 1980–1984
0.793 0.902
0.920 0.910
0.918 8.690
p 5 .107
p 5 .117
p 5 .097
p 5 .097
q 5 .096
1985–1992 0.501
0.723 0.807
0.856 0.887
16.476 p 5
.005 p 5
.043 p 5
.068 p 5
.079 q 5
.013 1973–1992
1.067 1.095
1.108 1.069
1.044 2.710
p 5 .327
p 5 .342
p 5 .290
p 5 .253
q 5 .535
Rescaled range RS analysis of the error correction term for long-term memory using the modified RS Vq and the classical RS V statistics. The sample period and the number of observations are as
noted in Table 1. The critical values of 5 significance level for left and right tails are 0.809 and 1.862, respectively. The p value reports the probability that the RS statistics from the bootstrap distribution
are less larger than the sample RS statistic if the sample value is less larger than the median of the bootstrap distribution in 1,000 iterations. The Wald statistic is computed by using the covariance matrix
that describes the dependencies among the four RS estimates as derived from the bootstrap distribution. The Wald statistic distributes as a x
2
variate with 4 degrees of freedom, with a critical value of 9.488. The q value reports the probability that the Wald statistic from the bootstrap distribution is larger than
the sample Wald statistic.
the possible short-term dependence in the time series and hence is more reliable in terms of providing evidence of long memory. Lo also derives the asymptotic distribu-
tion of the modified RS statistic under the hypothesis of no long-term dependence. As with the variance ratio test, the RS test that is based on an arbitrarily chosen
autocovariance adjustment tends to reject the no long-memory null hypothesis too often. For a joint test, we compute the Wald statistic in a similar way as in the variance
ratio test. Because the distribution of the RS statistics is right-skew and leptokurtotic, the hypothesis testing is properly done by comparing the sample Wald statistic with
the bootstrapped distribution of Wald statistics.
Both the classical and modified rescaled range analyses are performed, and the results are contained in Table 4. Except for the 1985–1992 period, the findings that
none of the modified RS Vq statistics are significant at the 5 level disclose no long-range dependence in the error correction term. Nevertheless, under a conservative
significance level such as 10, the two higher order V statistics for the 1980–1984 period are significant. Additionally, the joint tests based on the bootstrapped distribu-
tion of the Wald statistics suggest that the error correction term for the three sub- periods displays a long-run dependence at a 10 significance level. This long-run
dependence is associated with the observation that the deviations from cointegrating relationship follow a mean-reverting behavior. The evidence of significant classical
RS V in the 1980–1984 period and especially the 1985–1992 sample might simply partially reflect the presence of short-run dependence in the error correction term,
as echoed by the significant variance-ratio test results.
314 M.-S. Pan, Y.A. Liu International Review of Economics and Finance 8 1999 305–316
Although the RS test-based evidence of long-memory, mean-reverting characteris- tic in the error correction term is only marginal even for the 1980–1984 period, we
should caution the readers that the use of the RS statistic is, at best, the second best alternative, since it is developed for detecting general long-run memory. To estimate
a time series as the long-memory process i.e., the fractionally-integrated model, a semiparametric approach such as the GPH test is more preferable. Therefore, although
the evidence of long-memory from the RS test is weak, the GPH test of fractional integration does suggest the presence of long-memory for the 1980–1984 period.
6. Conclusions
