5.2 Some evidence on the variance ratio
What do we know about λj and λj' ? Based on the single factor model in (1),
a first, simple approach to obtain estimates of these variance ratios consists of
specifying a composite ‘global factor’, as the daily average return in a cross
section of stock markets.18 We estimate such global factor in different ways: we
first use the sample of the G7 countries, then our full sample excluding Hong
Kong; finally we adopt the ‘world stock market index’ produced by Thomson
Financial Datastream. After computing two-day rolling average of returns on
the global factor, we regress the two-day rolling average of Hong Kong’s returns
on it. The variance of the residuals from this regression gives an estimate of the
variance of the country specific shock, from which we obtain an estimate of λj∙.
The results from this procedure are shown in the first half of table 2. In our
sample, the order of magnitude of the variance ratio for Hong Kong is between
2 and 4: i.e. in the Hong Kong stock market, the variance of country-specific
shocks is between 2 and 4 times the variance of the global factor (multiplied by
the factor loading 7 ). Most interestingly, these ratios do not vary substantially
between the tranquil and the crisis period.
A second approach to estimating the variance ratio is based on principal
component analysis. First, we calculate the principal components for our full
sample of rolling averages of returns. We then regress the rolling average of
returns in country j on the principal components, using the residual from this
regression to estimate the variance of the country specific shocks. Results are
shown in the second half of table 2.
Our estimates of λj for the full sample are not too distant from what we
have obtained by using the composite global factor. The first principal com-
ponent gives an estimated variance ratio equal to 7.1. If we include the first
five components in the regression (so as to explain 76% of the variance in the
sample) λj is equal to 4.1. At the margin, the difference in the estimated value
of λj is only relevant in the case of Germany (for this country, λ = 4.4). Note
however that our test statistic is derived under the maintained assumption of a
single factor model, and thus is not directly applicable in a multi-factor world.
A key conclusion from these preliminary (and admittedly rough) estimates
based on a single factor model of returns, is that the variance ratio is well below
what is needed to justify a test based on unadjusted correlation coefficients (see
table 1). At the same time, however, the value of the Xs is bounded away from
zero. The strong results of interdependence reached by Boyer et al. (1999) and
Forbes and Rigobon (1999a) may not survive when the implicit bias in their
test is removed.
6 Conclusion
This paper has presented a general framework to approach tests of contagion
between stock markets in different countries based on correlation analysis. A
number of tests in the literature correct for potential bias due to changes in the
variance of global shocks driving returns. By analyzing these tests as special
cases of our framework, we show that these tests are conditional on arbitrary
18This approach is consistent with more general dynamic factor models, as shown in Forni
and Lippi (1997).
19