Tests for contagion in financial returns using correlation analysis are seriously affected by the size of the “noncrisis” and “crisis” periods. Typically the crisis period contains relatively few observations, which seriously affects the power of the test.Financial crises ; Financial markets

Diana Zhumabekova

Mardi Dungey

English

Research Papers in Economics

Pacific Basin Working Paper Series 
 
 
 
 
 
 
 
 
 
TESTING FOR CONTAGION  
USING CORRELATIONS:  
SOME WORDS OF CAUTION 
 
 
 
Mardi Dungey 
Australian National University 
 
and 
 
Diana Zhumabekova 
Australian National University  
and 
Visiting Scholar 
Center for Pacific Basin Monetary and Economic Studies 
Federal Reserve Bank of San Francisco 
 
 
 
 
Working Paper No. PB01-09 
 
 
 
 
Center for Pacific Basin Monetary and Economic Studies  
Economic Research Department 
Federal Reserve Bank of San Francisco  
  WORKING PAPER PB01-09 
 
 
 
 
TESTING FOR CONTAGION USING CORRELATIONS: 
SOME WORDS OF CAUTION 
 
 
 
 
Mardi Dungey 
Australian National University 
 
and 
 
Diana Zhumabekova 
Australian National University  
and 
Visiting Scholar 
Center for Pacific Basin Monetary and Economic Studies 
Federal Reserve Bank of San Francisco 
 
 
 
 
September 2001 
 
 
Center for Pacific Basin Monetary and Economic Studies 
Economic Research Department 
Federal Reserve Bank of San Francisco 
101 Market Street 
San Francisco, CA  94105-1579 
Tel: (415) 974-3184 
Fax: (415) 974-2168 
http://www.frbsf.org 
 Testing for Contagion using Correlations: Some Words of Caution
* 
 
 
 
Mardi Dungey
1 and Diana Zhumabekova
2 
Australian National University 
 
 
 
 
 
Abstract 
Tests for contagion in financial returns using correlation analysis are seriously affected by the 
size of the ‘non-crisis’ and ‘crisis’ periods. Typically the ‘crisis’ period contains relatively few 
observations which seriously affects the power of the test. 
 
 
 
 
 
JEL Classification: G14 
 
Keywords: contagion, correlations, heteroskedasticity 
 
 
 
 
                                                            
1 RSPAS-Economics, Australian National University, Canberra, ACT 0200, Australia, ph. +61 2 6125 0304, fax. 
+61 2 6125 3700, email: mardi.dungey@anu.edu.au. 
2 email: diana.zhumabekova@anu.edu.au    1
1.  Introduction 
A popular and straightforward test for the presence of contagion in financial asset returns has 
been to examine changes in the correlations of those returns between ‘crisis’ and ‘non-crisis’ 
periods. A statistically significant change in the correlation is interpreted as evidence of contagion. 
This approach has been applied in the equity market literature, (for example Baig and Goldfajn 
(1998) and references cited in Forbes and Rigobon (1999)) and to some extent to yields and 
currency market data (Ellis and Lewis (2000)). Here we outline this basic test of contagion, 
including the heteroskedasticity correction developed by Forbes and Rigobon, and examine its 
properties. In particular we address the limitations of the power of this test. Using an application to 
equity market data during the US stock market crash of 1987 we show that the Forbes and Rigobon 
results consistently over-reject the hypothesis of contagion due in large part to the comparison of a 
large sample of non-crisis period data to a small sample of crisis period data. We suggest that as the 
relative size of the crisis sample is likely to be small in general, the correlation test is an unattractive 
approach to the problem. 
  
2.  Correlations as a measure of contagion 
The preferred test of contagion is to examine the relationship between returns in different 
markets using a simple linear model as follows. 
  t , i t , j t , i y y e + b =   (1) 
If there is a change in the relationship between the returns, yi,t and yj,t at some point, as given by 
a significant change in b, this is evidence of contagion. This simple test is complicated by the fact 
that in the move between a non-crisis and crisis period the volatility of the error term, ei,t, usually 
changes also, that is the assumption of homoskedasticity is violated. In order to avoid this problem   2
the test can be respecified in terms of testing for a statistical change in the correlation coefficient, ri, 
between the two periods. However, when examining a relationship such as (1) over a data sample 
involving a crisis, one may well infer that there are also changes in the variance structure of the data 
from the crisis period to the non-crisis period, and in general that the variance of the series will 
increase with the advent of a crisis. To allow for this Forbes and Rigobon introduce a simple 
correction: 
 
] ) ( 1 [ 1
2
i
i *
i
r - d +
r
= r   (2) 
where  1 - =
l
j
h
j
s
s
d is the relative increase in the variance of the equity returns from the low 
volatility non-crisis period, 
l
j s , to the high volatility crisis period, 
h
j s .  
Forbes and Rigobon compare two periods under the null hypothesis that the same model holds 
throughout (that is contagion is evidenced by a rejection of the null). Hence the 
l
j s  used in their test 
is drawn from the entire sample period, including the high volatility period, and the variance, 
h
j s  is 
drawn from a subperiod of the data. The general approach here is to first control for common effects 
between markets using a VAR. Practically, the process is implemented by running a VAR analysis 
between the two stock index returns for the total period as in (3) below.  
 
t , j l t , i t , j
t , i l t , j t , i
y ) L ( B y
y ) L ( B y
e + =
e + =
-
-
  (3) 
The correlations between the error terms ei,t and ej,t are then examined for contagion by 
calculating 
h *,
i r  for the high volatility period and 
l *,
i r  for the low volatility period. In practice Forbes 
and Rigobon draw the errors for the crisis period and the subsequent value 
h
i s  from the sub-period 
results of (3). The test for contagion is then:    3
 
l *,
i
h *,
i 1
l *,
i
h *,
i 0
: H
: H
r > r
r £ r
 
In fact the heteroskedasticity correction they propose is only necessary if one draws all the 
information from a system estimated across both crisis and non-crisis periods under the null of no 
contagion. A stronger test is to estimate crisis and non-crisis VARs separately and test the null of 
whether the unadjusted correlation coefficients are significantly different across the two samples. 
This has not generally been done due to limited degrees of freedom in a VAR for the crisis period. 
 
3.  Sample selection and the Fisher transformation 
To conduct the test of H0 versus H1, the Fisher transformation on the calculated correlation 
coefficients is required to achieve a distribution closer to the normal. The resulting mean and 
standard deviations are expressed as: 
  ￿ ￿
ł
￿
￿ ￿
Ł
￿
r -
r +
= r *
i
*
i *
i 1
1
ln
2
1
  (4) 
and 
 
3 n
1
s
i
i -
=   (5) 
where ni is the number of observations in the sample. The two-sample test on independent means 
is performed as: 
 
2 2
*, *,
l h
l
i
h
i
s s
t
+
-
=
r r
  (6) 
where sh and sl are the standard deviations of the correlation coefficients estimated during the 
subperiods of relative turmoil and stability respectively.    4
However, this transformation is an asymptotic result, and best suited for sample sizes of ni>50 
(see for example Kendall and Stuart (1979)). The literature applies this test to rather smaller 
subsamples, in particular tests are undertaken between non-turmoil periods of up to 500 
observations and turmoil periods of as few as 10 observations. A simple Monte Carlo experiment 
demonstrates the undesirable power properties of this application.  
The test of correlation coefficients is performed on errors from VARs of the form given in (3) 
where the errors are normally distributed. Hence in our experiment we simulate 2 populations of 
bivariate normal observations for a given correlation matrices P1 and P2.  
The correlations matrices are chosen arbitrarily so that the correlation coefficients r1 and r2 are 
statically different for 2 populations at 1% level of significance. These two population correlation 
coefficients then represent a ‘non-crisis’ and ‘crisis’ period for our purposes. If we randomly sample 
from each of the populations we can see that with reasonable size samples our ability to produce 
reliable estimates of the true correlation coefficients and most importantly their standard errors is 
good. However, as the sample size declines the estimates of the standard errors are seriously 
affected. Table 1 sets out these results, Experiments 5, 6 and 7 use some of the actual sample sizes 
tested in Forbes and Rigobon.  
Further, if we examine the t-test of the significance between the two correlation coefficients, we 
see that if one sample is large, and the other fairly small then the test has very little power, that is 
our ability to reject the null hypothesis can be seriously affected by the sample size. A chart of the 
dependence of the standard error of the correlation coefficient is given in Figure 1 – with rapidly 
increasing standard errors associated with decreasing sample size the chances of rejecting the null 
hypothesis become vanishingly small.   5
Table 1: Monte Carlo experiments on testing correlations 
  Sample 
size1 
Sample 
size 2  ?1  s1  ?2  s2  t-test 
on ?2-?1 
True values  2,000  2,000  0.31  0.022  0.50  0.022  7.37 
Simulated values for 
Experiment 1  1,000  1,000  0.31  0.032  0.50  0.032  5.22 
Experiment 2  500  500  0.31  0.045  0.50  0.044  3.70 
Experiment 3  500  300  0.31  0.045  0.50  0.058  3.21 
Experiment 4  500  100  0.31  0.045  0.50  0.101  2.12 
Experiment 5  467  22  0.31  0.046  0.49  0.229  1.03 
Experiment 6  509  10  0.31  0.044  0.49  0.377  0.70 
Experiment 7  466  35  0.31  0.046  0.50  0.176  1.30 
 
 
 
 
 
 
 
 
 
 
 
Figure 1: The effect of sample size on the calculated standard error 
 
4.  An example – contagion from the US stock market crash:  
In this section we demonstrate the consequences of this problem for an example drawn from 
Forbes and Rigobon. By increasing the sample size for the crisis period we find more evidence of 
contagion than did Forbes and Rigobon. This is a strong result given that the volatility increase is 
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 50 100 150 200 250 300 350 400 450
sample size
s
t
a
n
d
a
r
d
 
e
r
r
o
r
 
o
f
 
t
h
e
 
c
o
r
r
e
l
a
t
i
o
n
 
c
o
e
f
f
i
c
i
e
n
t  6
now smaller than in their sample as the crisis period contains some observations previously 
classified as ‘non-crisis’.  
Table 2 reports the results of correlation tests for contagion between the US stock market index 
and 9 other developed stock market indices. All data are daily returns taken from the Thomson 
Financial Datastream stock market indices. Forbes and Rigobon propose a non-crisis period from 1-
January-1986 to the 16-October-1987, a total of 466 observations. We also consider this as the 
stable period. The crisis or turmoil period in Forbes and Rigobon consists of 35 observations from 
17-October-1987 to 4-December-1987. To increase this sample size to one with more appropriate 
power properties we extend the turmoil period to 30-April-1988, a total of 142 observations. We 
estimate VAR model for the full period, and calculate the stable period correlation coefficient, and 
heteroskedasticity adjusted crisis period correlation coefficients for a short and long crisis periods. 
The correlation tests for contagion are conducted for the short and long crisis periods using 
heteroskedasticity-adjusted correlation coefficients. 
The results in Table 2 show that there is evidence of contagion in 6 of the 9 countries examined 
using the longer crisis period with adjusted correlation coefficients, whereas the short period with 
adjustments presents no evidence at all demonstrating the seriousness of the power problem 
discussed above. 
   7
Table 2: Correlation test for contagion in the US stock market crash:  
n1 is non crisis period, n2 is short crisis period, n3 long crisis period 
Correlation coefficient 
(std err)  Test of contagion 
Country  Non-crisis 
n1=466 
(s=0.046) 
 
Crisis 
n2=35 
(s=0.177) 
Crisis 
n3=142 
(s=0.085) 
t-test 
using 
rstab & r1adj 
Contag? 
t-test 
using 
rstab & r2adj 
Contag? 
  rstab  r1  r1adj  r2  r2adj         
Canada  0.610  0.800  0.253  0.778  0.388  0.48  No  -3.13  No 
France  0.122  0.658  0.227  0.500  0.319  -2.42  No  2.14  Yes 
Germany  0.076  0.564  0.217  0.345  0.265  0.64  No  2.01  Yes 
Netherlands  0.266  0.676  0.237  0.597  0.348  0.77  No  0.92  No 
Switzerland  0.072  0.750  0.239  0.524  0.334  -0.31  No  2.84  Yes 
U.K.  0.238  0.636  0.242  0.558  0.348  0.89  No  1.22  No 
Australia*  0.118  0.859  0.218  0.796  0.360  0.05  No  2.66  Yes 
Hong Kong*  0.052  0.209  0.168  0.296  0.253  0.65  No  2.14  Yes 
Japan*  0.144  0.855  0.214  0.697  0.345  0.39  No  2.21  Yes 
* Lagged returns were used for these countries to adjust for time period differences. 
 
5.  Conclusion 
The correlation test for contagion as proposed in the existing literature, and popularised in 
Forbes and Rigobon (1999) suffers from substantial power problems when applied to the typical 
situation of a large ‘non-crisis’ sample and a small ‘crisis’ sample. A simple example shows that 
when the crisis sample period is extended (to include what were previously defined as non-crisis 
observations) the correlation test finds statistically signficant evidence of contagion, where it did not 
with the shorter crisis period. For this reason we caution against using this method of identifying 
contagious episodes, as the biases in constructing the test statistics and choosing the sample period 
are not well defined.   8
References: 
Baig, T.  and I. Goldfajn, 1998, Financial Market Contagion in the Asian Crisis, IMF Working Paper 
WP/98/155. 
Ellis, L. and E. Lewis, 2000, The Response of Financial Markets in Australia and New Zealand to News 
about the Asian Crisis, BIS Conference on International Financial Markets and the Implications for 
Monetary and Financial Stability, Basle, 25-26 October, 1999, Vol. 8. 
Forbes, K. and R. Rigobon, 1999, No Contagion, only Interdepence : Measuring Stock Market Co-
movements, NBER Working Paper #7267. 
Kendall, M. and A. Stuart, 1979, The Advanced Theory of Statistics (Charles Griffin and Co., London). 
 
 
 

No Contagion, only Interdepence : Measuring Stock Market Comovements,

The Response of Financial Markets in Australia and New Zealand to News about the Asian Crisis,

Testing for contagion using correlations: some words of caution

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Testing for contagion using correlations: some words of caution

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