is provided by the following two-factor model:
ri = ai + yi ■ f + βi ■ η + εi
rj = a1 + yj ■ f + η1 + ηj)
where βj has been normalized to 1. If interdependence, βi = 0, so that the
process is equivalent to the data generating process (1) by setting εj = η + η.∙.
Contagion occurs when the country specific shock η becomes a global factor,
i.e. when βi = 0. As shown below, our measure of interdependence is derived
under the null hypothesis βi = 0. Thus, it will be unaffected by a change in
the specification of the process for the rates of return, which uses the above
expressions instead of the process (1).
These definitions provide a general framework for the empirical test discussed
below.
3.2 Conditional correlation analysis
How can one derive a theoretical measure of correlation suitable to discrimi-
nate between contagion and interdependence according to the model presented
above? Suppose that we can identify the ‘origin’ of an international financial
crisis in some country j (e.g. Mexico at the end of 1994, Thailand in July 1997,
Hong Kong in October 1997). Let δ denote the proportional change in the vari-
ance of the stock market return r7 relative to the pre-crisis period. Then, we
can write
Var(rj I C) = (1 + δ)Var(rj)
where C denotes the event ‘crisis in country j’. Note that the observed change in
the variance of r7 does not necessarily coincide with an increase in the variance
of the global factor, as the variance of the country-specific component may also
change during the crisis.
In order to test whether changes in the correlation between r and r7 during
a crisis in j are consistent with the data generating process (1), we must specify
a measure of interdependence under the assumption that yi, yj∙, Var(εi) and
Cov(εi,εj) do not vary with the crisis in country j. In the Appendix we show
that, under such an assumption, the correlation coefficient between r⅛ and rj
can be written as the following function φ:
1/2
(2)
≠(λj∙,λ
'j ,
δ,p) = p
1+-∖i
1+λ7
1] (1 + ∖)
where λj (ʌɑ) denotes the ratio between the variance of the idiosyncratic shock
ε.j and the variance of the global factor f, scaled by the factor loading yj∙, during
the tranquil (crisis) period:
λ. = Var(^1 d λc = Var(^ i c)
3 ■ Var(f) 1 ■ Var(f ∣ C) .
In what follows, we will refer to φ as a theoretical measure of interdependence.
The correlation coefficient between r and r7 observed during the crisis, denoted