This document presents a new test for financial contagion based on changes in co-kurtosis and co-volatility between a non-crisis period and crisis period. The author applies this test to analyze contagion in the US banking sector during the 2008-2009 financial crisis. Specifically, the author 1) develops new statistics to measure extremal dependence through co-kurtosis and co-volatility, 2) applies these statistics to test for contagion from the US banking sector to other markets, and 3) finds significant contagion effects spreading from the US through these channels to global equity markets and banking sectors.

1. A New Test of Financial Contagion with Application to
the US Banking Sector
Cody Yu-Ling Hsiao
Centre For Applied Macroeconomic Analysis
Australian National University
November 2012
Cody Hsiao () Contagion Tests 11/27 1 / 18

2. Outline
Motivations
Research questions
Contribution
Statistics of co-kurtosis and co-volatility
Contagion tests (testing for smile e¤ect)
Application to the US banking sectors

3. Motivations
Smile e¤ect through co-kurtosis channel
Smile e¤ect is represented as the co-movements between return and
return skewness (E (ri1 , rj3 ) and E (ri3 , rj1 )) turning from a negative
relation to a positive relation in the crisis period
Theoretically speculators invest in securities with a positive asset return
and negative skewness in normal periods (Brunnermeier and Pedersen,
2008)
Funding constraints lead to higher kurtosis and co-kurtosis risk

4. Motivations
Smile e¤ect through co-volatility channel
Smile e¤ect is represented as the co-movements between return
volatility and return volatility turning from a negative relation to a
positive relation in the crisis period
Co-volatility (E (ri2 , rj2 )) is described as the relationship between return
volatility and return volatility

5. Research questions
How to test for …nancial contagion (smile e¤ect) through co-kurtosis
or co-volatility channels?
Does …nancial contagion (smile e¤ect) through co-kurtosis or
co-volatility channels really exist in the …nancial markets during the
global …nancial crisis of 2008-2009?

6. Contributions
A new class of tests of …nancial contagion based on increases in
co-kurtosis or co-volatility is proposed
De…nition
Contagion is de…ned as a signi…cant increase in the fourth order co-moments
of two markets between a non-crisis and a crisis period
This new approach is applied to test for …nancial contagion (smile
e¤ect) in equity markets and banking sectors following the global
…nancial crisis of 2008-2009

7. Statistics of co-kurtosis
A non-normal multivariate returns distribution is speci…ed
The bivariate generalized exponential family of the distribution with
the …rst form of co-kurtosis (Cokurtosis13 ), that is
"
1 1 r1,t µ1 2 r2,t µ2 2
f (r1,t , r2,t ) = exp 2
+
2 1 ρ σ1 σ2
r1,t µ1 r2,t µ2
2ρ ,
σ1 σ2
#
1 3
r1,t µ1 r2,t µ2
+θ 7 η
σ1 σ2
The Lagrangian multiplier statistic for co-kurtosis (LM1 ) is used to
test for extremal dependence with restriction of θ 7 = 0
T 1 3 2
1 r1,t µ1b r2,t µ2b
LM1 =
T (18b2 +6 )
ρ
∑ b
σ1 b
σ2 T (3b)
ρ
t =1
Statistics of co-volatility is derived based on the LM test either

8. Contagion Tests
Extremal dependence tests
Co-kurtosis and co-volatility are used for measuring extremal dependence
The …rst type of statistic CK13 is to detect the shocks emanating
from the asset returns of a source market i to the cubed returns of
asset in a recipient market j.
0 12
B by (r 1 ,r 3 )
ξ b (r 1 ,r 3 ) C
ξx i j
CK13 (i ! j; ri1 , rj3 ) = @ s i 2j A
18b
v +6
y jx i 18b2 +6
ρ
Ty + Txx
The second type of statistic CK31 is to measure the shocks
transmitting from the cubed returns of asset in a source market i to
the returns of asset in a recipient market j
0 12
B by (r 3 ,r 1 )
ξ b (r 3 ,r 1 ) C
ξx i j
CK31 (i ! j; ri3 , rj1 ) = @ s i 2j A
18b
v +6
y jx i 18b2 +6
ρ
Ty + Txx

9. Contagion Tests
Extremal dependence tests
Ty m n
b
yi ,t µyi b
yj ,t µyj
b rm, rn =
ξy i j 1
∑ 3by jxi
v
Ty b
σyi b
σyj
t =1
Tx m b
xj ,t µxj n
b rm, rn = 1 b
xi ,t µxi
ξx i j Tx ∑ b
σxi b
σxj (3bx ) ,
ρ
t =1
and
by
ρ
b
vy jx = s
2 2
.
i sy ,i sx ,i
1+ 2 (1 b2 )
ρy
sx ,i
b
vy jx represents the adjusted correlation coe¢ cient proposed by Forbes
i
and Rigobon (2002).
They consider that estimation of cross-market correlation coe¢ cients
is biased due to heteroscedasticity in market returns.

10. Contagion Tests
Extremal dependence tests
The third type of statistic CV22 is to detect the shocks transmitting
from the volatility of asset returns in a source market i to the
volatility of asset returns in a recipient market j.
0 12
B b (
ξy ri2 ,rj2 ) b r 2 ,r 2
(
ξx i j ) C
CV22 (i ! j; ri2 , rj2 ) = @ s A
4b4 +16b2 +4
v v
y jx i y jx i 4b4 +16b2 +4
ρ ρ
Ty + x Tx x
where
Ty 2 2
b
yi ,t µyi b
yj ,t µyj
b (r 2 , r 2 ) =
ξy i j 1
∑ v2
1 + 2by jxi
Ty b
σyi b
σyj
t =1
Tx 2 b
xj ,t µxj 2
b (r 2 , r 2 ) b
xi ,t µxi
ξx i j = 1
Tx ∑ b
σxi b
σxj 1 + 2b2
ρx
t =1

11. Contagion Tests
Extremal dependence tests
To test that there is a signi…cant change in co-kurtosis or co-volatility
between the non-crisis period and the crisis period, the null and
alternative hypotheses are
Ho : ξ y (rim , rjn ) ξ x (rim , rjn )
H1 : ξ y (rim , rjn ) > ξ x (rim , rjn )
Under the null hypothesis of no contagion, tests of contagion based on
changes in co-kurtosis or co-volatility are asymptotically distributed as
d
CK13 , CK31 , CV22 (i ! j ) ! χ2 .
1

12. Application to the US banking sector
The …nancial crisis of 2008-2009 is escalated into a global
phenomenon
The tests of contagion are applied to identify transmission channels
through changes in extremal dependences during the global …nancial
crisis of 2008-2009
Data
The daily banking equity indices and equity indices are collected for
four regions
Asian region (Hong Kong and Korea), European region (France,
Germany, the UK), Latin American region (Chile and Mexico), North
American region (the US)
The non-crisis period is chosen from April 1, 2005 to June 29, 2007
and the crisis period is from March 3, 2008 to August 31, 2009
A Vector Autoregressive (VAR) model is estimated in order to control
for market fundamentals and handle the problems of serial correlation

13. Application to the US banking sector
Contagion channel through extremal dependence
Testing for contagion based on changes in extremal dependence during the
global …nancial crisis of 2008-2009. Source market is the US banking
sector
(a) CK 13 : co-kurtosis contagion test with co-kurtosis measured in terms of
1
E (r US , r 3 ), (b) CK 31 : co-kurtosis contagion test with co-kurtosis measured in
J
3
terms of E (r US , r 1 ), (c) CV 22 : co-volatility contagion test with co-volatility
J
2
measured in terms of E (r US , r 2 )
J

14. Conclusion
A new test of …nancial contagion based on changes in co-kurtosis and
co-volatility is proposed
This new approach is applied to test for …nancial contagion in equity
markets and banking sectors following the global …nancial crisis of
2008-2009
The results of the tests show that signi…cant contagion e¤ects (smile
e¤ects) are widespread from the US banking sector to global equity
markets and global banking sectors through one of the extremal
dependence channels

15. Any questions?

16. Further studies
A joint test of contagion through the co-skewness channel
A joint test of contagion through the co-kurtosis and co-volatility
channel
Selection of the period of crisis period could a¤ect the results of
contagion
Sensitivity tests given the di¤erent periods of the crisis period
Markov switching model in contagion analysis (the crisis period are
endogenously by the MS model)

17. Finite sample properties
Monte Carlo simulations are performed to calculate the critical values
due to relative large sample period of the non-crisis period but
relative short sample period of the crisis.
To generate the asset returns in the simulation, the parameters are
chosen for the following non-crisis and crisis variance-covariance
matrices of returns in two equity market i and j as
0.557 0.143 29.195 14.350
Vx = , Vy = .
0.143 2.660 14.350 16.430

18. Finite sample properties
The size of non-crisis period is Tx = 585 and the size of crisis period is
Ty = 391. α is signi…cant level. Based on 10,000 replications.
Test statistics α = 0.025 α = 0.05 α = 0.1
CS 12 5.40 4.02 2.73
CS 21 4.91 3.81 2.70
CK 13 9.33 7.04 4.79
CK 31 9.28 7.00 4.81
CV 22 5.03 3.82 2.73
χ2
1 5.02 3.84 2.71
The statistics for contagion based on co-skewness and co-volatility present a
good approximation of the …nite sample distribution
The test statistics for contagion based on co-kurtosis tend to be biased
The results of contagion tests seem quite robust given relative large sample
period of the non-crisis period but relative short sample period of the crisis