Correlation Analysis of Financial Contagion: What One Should Know Before Running a Test

A Appendix

This appendix derives the expression (2) of the coefficient of interdependence φ
in the general case. From the data generating process of ri, the unconditional
variance of the idiosyncratic shock ε₇ can be written as:

Var(εi) = Var(ri) — 7? ■ Var(f)

By the definition of λ_j and the data generating process of r_j∙, we can also get:

Var(f ) =

Var (r_j)
(1 + >)

Therefore, we find:

Var(ε_i )

7i ■ Var(f)

Var(r_i) _—₁ = 7∣ ^{(1 +} ʌ,^^)Var⁽r⁾

(A∙1)

7? ■ Var(f) 7jVar(r_j)

For convenience, we rewrite the expression of the correlation coefficient induced
by the process (1):

_ 1

^p [¹ + ⅛⅛⅛Γ'²-11 + >

(A.2)

Substituting (A.1) into (A.2), we obtain the unconditional correlation coefficient
as a function of the rates of return, the factor loadings and λ_j■:

1 7 V ar (ri ) A ¹A

^{1 +} λ₃ ∖Var(r_jУ

(A∙3)

We now turn to the crisis period. From the data generating process of the
rate of return of the stock market in country i, the variance of r during the
crisis is:

Var(r_i I C) = 7² ■ Var(f ∣ C) + Var(ε_i)

(A.4)

Note that by the data generating process (1) and by the definition of λj and
ʌɑ, it follows that:

^Var^(rj ⁱ^c⁾ = _{1 +}_δ = ^{1 +} ʌɑ ^Var⁽^fⁱ^c⁾

Var(r₁) 1 + λj Var(f )

(A.5)

By solving (A.5) for Var(f ∣ C) and substituting the resulting expression into
(A.4) we get:

Var(ri I C) = Var(ri) + ψ7²Var(f)

where ψ is defined as in follows

^^{(1 +} ʌj) ⁺ Q ^— > ⁾1 + >7

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Correlation Analysis of Financial Contagion: What One Should Know Before Running a Test

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