,1 lj .11 1 „ , , , r ,. jl j . jl j1 1 1 1 τC r
case, the result will be a threshold function, that gives the threshold λ for any
positive λj∙.
Tests of equality between two correlation coefficients can be performed using
the Fisher z-transformation
ɪl 1+ p
z(p) = 2lnr—p
where p is the estimated correlation coefficient. Under the assumption that two
samples are drawn from two independent bivariate normal distributions with
the same correlation coefficient, Stuart and Ord (1991, 1994) show that the
difference between estimated z (p) in the two samples converges to a normal
distribution with mean and variance specified below:
N (0.—
∖ nι
3 n2
з)
where nɪ and n2 denote the size of the two samples.
We proceed as follows. We estimate the correlation coefficients during the
tranquil period, p, and during the crisis period, pP, as well as the increase in the
variance in the Hong Kong stock market, δ. By substituting p and δ into (2),
we obtain an estimation of our measure of interdependence as a function of λj
and λj', that is p(ʌj,ʌɑ). Given z(pP) and z(p(ʌj,ʌɑ)), we derive threshold
1 Г \ 1 ∖ C
from:
values of λj and λj
z(pc') — z(p(λ, λc)) = 1.645σz
(9)
where σz = —Ц +
n—3
rac1-3, with n and nɑ denoting the sample size of the tranquil
and the crisis period.
A problem in the above procedure is that the assumption of independent
samples is violated, since δ depends on both the tranquil and the crisis period
samples: the significance level of the test (9) is not the standard 5 per cent.
To assess the significance level of our test, we have resorted to Montecarlo sim-
ulation experiments. We have run 1, 000, 000 replications for different country
pairs, varying the parameter values and sample size. In all our simulations, the
significance level of the statistic (9) is comprised between 7 and 9 per cent. For
instance, setting n = 208, nc = 30 and δ = 8.72, as in our benchmark estima-
tion, and p = 0.219, pɑ = 0,661, which are the observed correlation coefficients
between the markets of Hong Kong and the Philippines, the significance level
of the test corresponding to (9) is 8.1 per cent.
5.1.1 The case of a constant variance ratio
Consider first the case λj = λj'. The threshold level of the variance ratio, λ,
can be easily found by inverting equation (9). This yields
-λ = {[:
- ʌ . 1 -∣2
ʌtp +1
p-—7
ω — 1
(1+δ)—1} ⅛— 1
(10)
where ω = exp 2 ^z(jjP) — 1.645σz) ], and (as defined above) p and pɑ are the
sample correlation coefficients. Consistently with the logic of our test, if one
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