instructive to use our model in order to highlight the conditions under which
a simple test on correlation is consistent with our measure of interdependence.
Looking at equation (3), note that φ is identically equal to p only when
ʌɑ = ʌj = ι∕p2 - 1 ∙ (4)
For this particular value of the variance ratio,11 interdependence implies that
the correlation coefficient should not respond to a crisis in country j. Thus, we
can perform a test of contagion just verifying whether the simple correlation
coefficient has changed significantly during a crisis.
Interestingly, the implicit assumption in condition (4) is a negative rela-
tionship between the correlation coefficient during tranquil period p and the
variance ratio λj■: the higher the correlation between r-ι and rj, the higher the
importance of the global factor and, in turn, the lower T1∙. This is not an unrea-
sonable assumption in general. However, unless λj happens to be exactly equal
(or close) to the inverse of the squared correlation coefficient minus one, tests of
contagion based on comparing simple correlation will be biased — it could be
interesting to explore the loss of accuracy of the test in the region around that
value of the variance ratio.
4.2 Tests based on adjusted correlation coefficient with
A = 0
Consider the approach championed by Forbes and Rigobon (1999a,b). The key
to these contributions is the (implicit) assumption that the rate of return of the
stock market in country j coincides with a global factor. In terms of our factor
model, this is equivalent to assuming that the data generating process of the
rates of return is:
r = ai + 7i ■ f + εi (5)
rι = aι + Tj ■f
so that
T ai A Ti
ri = T--+--- r j + Si
∖ Tj J Tj
corresponding to the linear equation at the root of Forbes and Rigobon’ esti-
mates:12
r = βo + βι ■ rj + εi
(6)
Thus, there is no country-specific shock affecting rj. In terms of our framework,
Var(ε1) = O implies λ1' = λj = O.
11A similar but more cumbersome expression could be derived for the general case in which
λf ≠ A
12Forbes and Rigobon (1999a) filter their data estimating a VΛT( model with domestic and
international interests rates and lagged returns. Then, they analyze the correlation between
the residuals of their estimates with the model (6). Note that in the theoretical part of the
paper Forbes and Rigobon (1999a) actually write a symmetric model, where rι and rj are
interdependent. However, the symmetric model is not estimated.
13