by opening at 17:00 h, closing at 12:00 h. Hong Kong opens at 19:00 h, while closing at 01:30 h. These markets overlapped in terms of the trading day in real
time. Canada and the US market were both open simultaneously from 09:30 to 16:00 h. The problem arises when shocks to the US and Canadian markets
originate in the Asian markets. In calendar time this is measured as a contempora- neous shock, yet in real time this shock occurs with several hours of lag time.
Kayha 1997 shows that covariances between non-contemporaneous markets are somewhat upwardly biased. While noting this to be the case, any bias which exists
is exhibited in both the pre- and post-1987 crash period.
The data must be stationary to perform the appropriate analysis. Kwiatkowski et al. 1992 have shown that a test for stationarity adds to the robustness of
conclusions regarding unit root. This test is important due to the low power of Dickey – Fuller tests in rejecting the null hypothesis of unit root. These results are
reported in Table 1. Using the KPSS test, stationarity is rejected for the price series, while stationarity cannot be rejected for the return series.
3. Cross spectral analysis
Cross spectral analysis transforms time domain data into the frequency domain. This presents an alternative method to investigate pre- and post-crash correlations
among Pacific Rim equity markets. See Harvey 1994, pp. 235 – 240 for a detailed development of how the coherences and phase leads were estimated. The estimated
coherency spectrum between markets i and j for various frequencies is given by
w ˆ
ij
= h.
ij
v h
.
ii
vh .
jj
v
12
all v 1
where h
ii
and h
jj
i j are the autospectrum estimated from h
.
ii
v = 1
2p
N − 1 s = − N − 1
l
N
sR .
ii
s e
− ivs
. 2
Two tools from cross spectral analysis are used: coherence and phase lead. While coherence estimates the correlation in cross spectral analysis, the phase lead
indicates which variable leads lags. Due to Euler Relations and DeMoire’s Theorem the coherence can be written as
h .
ij
= cˆ
ij
+ iqˆ
ij
3 where the quantities c
ij
and q
ij
are the cospectrum and quadrature, respectively. Eq. 3 may also be written as,
h
ij
=
s = −
R
ij
scosvs − i sinvs 4
where the first term on the right hand side of Eq. 4 is the cospectrum and the second term is the quadrature.
The phase shift is given by
8 v = tan
− 1
c
ij
− iq
ij
. 5
The phase lead is measured in radians. A coefficient near zero indicates that the variables are in phase; they increase decrease simultaneously at a particular
frequency. A phase lead of 180° p results when the variables are moving inversely
Table 1 KPSS test
a
L = 10 L = 20
L = 0 Pre-crash prices
26.710 1.3170
US 2.4739
1.4204 UK
27.841 2.6196
19.584 Canada
1.8122 0.9659
26.456 Germany
1.2955 2.4386
1.7151 France
35.187 3.2365
2.7322 29.372
1.4630 Japan
Hong Kong 1.8468
37.700 3.4772
Australia 33.105
3.0599 1.6316
Pre-crash returns US
0.0382 0.0371
0.0406 UK
0.0257 0.0234
0.0247 Canada
0.0796 0.0954
0.0779 0.1556
0.1711 Germany
0.1861 0.0780
0.0667 0.0612
France Japan
0.0744 0.0754
0.0728 0.0536
0.0701 0.0578
Hong Kong 0.0426
Australia 0.0552
0.0463 Post-crash prices
7.194 US
0.3786 0.6915
0.8786 0.4764
9.348 UK
Canada 0.7716
15.437 1.4367
Germany 0.6624
13.401 1.2428
0.4561 0.8488
France 9.100
0.8937 Japan
18.269 1.6840
Hong Kong 29.361
2.5055 1.4573
1.8314 0.9778
19.590 Australia
Post-crash returns US
0.0269 0.0195
0.0258 0.0342
0.0402 0.0316
UK 0.0627
Canada 0.0421
0.0420 0.0559
0.0726 Germany
0.0738 0.0810
France 0.0975
0.0847 0.1019
Japan 0.1076
0.1053 0.0624
0.0662 Hong Kong
0.0650 0.0198
0.0513 0.0513
Australia
a
Test statistic is T
− 2
SS
t 2
S
2
L where S
t
= S
t
, t = 1, 2, . . . T. S
2
L = T
− 1
S
t 2
+ 2T
− 1
S1−SL+ 1S
tt−s
. The null hypothesis of stationarity is rejected if the test statistic is greater than the critical value. The critical values are 0.176 0.05 level and 0.216 0.01 level. The test includes a constant and trend.
at a particular frequency. Further, a positive phase value indicates that the input variable leads the output variable, while a negative phase value indicates that the
input variable lags. Even numbered figures plot the pre- and post-crash phase leads for each of the pairwise markets examined in this study. The input variable is listed
first in the title in each figure.
Greater coherence between the markets tends to reduce the benefits of interna- tional diversification. As mentioned above, both Meric and Meric 1989 and
Taylor and Tonks 1989 showed that diversification is greater in the short run high frequency, falling over longer periods low frequency. The coherence results
reported below are consistent with these conclusions and offer an alternative method to test the robustness of these results.
Phase leads can also indicate the degree of benefit of diversification. If markets are out of phase, the benefit of diversification is greater. Diversification is reduced
to the extent that markets are in phase markets are coincidental. Results reported below show that the markets tend to be out of phase at high frequencies, rising at
longer frequencies. Thus, benefits of diversification accrue over shorter periods, while falling over longer periods.
4. Results for north American and western pacific markets
