contributions.
Assume that the rates of return of the stock markets in country i and country
j are generated by the process
ri = ai + yi ■ f + εi (1)
rj = a1 + Tj ■ f + εj
where a,ι and a.j are constant numbers, yi and yj are market-specific factor
loadings, f is a global factor, ε⅛ and ε7 denote idiosyncratic risks, and where
f, ε-i and εj are mutually independent random variables with finite and strictly
positive variance.7
For simplicity, let both yi and yj be strictly positive. From the process
above, the correlation coefficient between r⅛ and rj∙ can be written as:8
о = Corr(r- r 1 = c°<TT3ï
ρ = corr[r∙l,r1 ) ------- -----
ʌ/vɑr(r) ■ Var(rj■)
_ 1
lɪ + Var(εt) 1 ɪ/2 ι [ι + Var(εj) 11/2
[1+ fiVar( f)∖ [1+7j2M∕)J
For given factor loadings yi and yj∙, a rise in correlation must correspond to
shocks increasing the variance of the global factor f relative to the variance of the
idiosyncratic noise ε-ι and/or εj∙. Given the variances of the global factor and the
idiosyncratic components, however, a rise in correlation could also correspond to
an increase in the magnitude of the factor loadings yi and yj, or to an increase
in the correlation between the idiosyncratic risks.
This distinction is at the root of recent empirical studies contrasting conta-
gion to interdependence. Consider a financial crisis in country j. The increase
in the variance of the stock market return in such a country may be due to
an increase in the variance of either the global factor f, or the country specific
component εj∙, or both. It is apparent that, if the change in the variance of the
global factor f is large enough relative to the change in the variance of the coun-
try specific component εj, cross-market correlation must increase during a crisis
in country j. This change in correlation is interdependence, in the sense that,
conditional on the occurrence of a financial crisis in country j, it is consistent
with the data generating process (1). Contagion, as opposed to interdependence,
occurs if the increase in correlation turns out to be ‘too strong’ relative to what
is implied by the process (1); i.e. it is too strong to be explained by the behavior
of the global factor and the country specific component. In other words, con-
tagion occurs when, conditional on a crisis, correlations are stronger because of
some structural change in the international economy — affecting the link across
markets.
In a related definition, contagion occurs when a country-specific shock be-
comes ‘regional’ or ‘global’. This means that there is some factor η for which
factor loadings are zero in all countries but one during tranquil periods, and be-
come positive during crisis periods. An illustration of this concept of contagion
7Allowing for some covariance across country-specific terms does not substantially modify
the main result of our analysis, on the need to adjust correlation coefficients for the variance
of country-specific shocks.
8We denote with Var the variance operator, Cov the covariance operator and Corr the
linear correlation operator.