We examine the size properties of tests for causality in variance in thepresence of structural breaks in volatility. Extensive Monte Carlo simulationsdemonstrate that these tests suffer from severe size distortions when suchbreaks are not taken into account. Pre-testing the series for structuralchanges in volatility is shown to largely remedy the problem.structural change;causality tests;volatitilty

Dijk, D.J.C. van

Osborn, D.R.

Sensier, M.

English

Research Papers in Economics

Testing for causality in variance in the presence of
breaks
Dick van Dijky
Econometric Insitute
Erasmus University Rotterdam
Denise R. Osbornz
Centre for Growth and Business Cycle Research
Economic Studies
University of Manchester
Marianne Sensierx
Centre for Growth and Business Cycle Research
Economic Studies
University of Manchester
Econometric Institute Report EI 2004-48
November 2004
Abstract
We examine the size properties of tests for causality in variance in the presence of
structural breaks in volatility. Extensive Monte Carlo simulations demonstrate that
these tests suer from severe size distortions when such breaks are not taken into
account. Pre-testing the series for structural changes in volatility is shown to largely
remedy the problem.
Keywords: volatility; causality tests; structural change.
JEL Classication Codes: C12; C22.
The second and third authors acknowledge support from the Economic and Social Research Council
(UK) under grant number L138251030.
yEconometric Institute, Erasmus University Rotterdam, P.O. Box 1738, NL-3000 DR Rotterdam, The
Netherlands, e-mail: djvandijk@few.eur.nl (corresponding author)
zCentre for Growth and Business Cycle Research, Economic Studies, University of Manchester, Manch-
ester M13 9PL, United Kingdom, e-mail: denise.osborn@manchester.ac.uk
xCentre for Growth and Business Cycle Research, Economic Studies, University of Manchester, Manch-
ester M13 9PL, United Kingdom, e-mail: marianne.sensier@manchester.ac.uk1 Introduction
During the last two decades, a great deal of attention has been paid to modelling the dy-
namic properties of volatility. An important strand of this research has analysed volatility
spillovers, particularly across nancial series. Morgenstern (1959) rst investigated if -
nancial market crises spill over to other countries. The modelling of spillover eects has
subsequently been developed primarily through correlation analysis based on GARCH
models, including King and Wadhwani (1990), Lin, Engle and Ito (1994), Susmel and En-
gle (1994), Ng (2000) and Billio and Pelizzon (2003). These papers estimate parametric
models to examine specic formulations for the spillover eects, while Cheung and Ng
(1996) and Hong (2001) develop general causality-in-variance tests within this framework.
Through a distinct strand of literature, there has recently been increasing awareness
that many time series experience occasional structural breaks in (unconditional) volatility;
see Andreou and Ghysels (2002), Lamoureux and Lastrapes (1990), McConnell and Perez-
Quiros (2000), Stock and Watson (2003) and Sensier and van Dijk (2004), among others.
In the light of the evidence for such structural breaks, it is important to investigate whether
they aect the appearance of volatility spillover eects. The present paper examines this
question by analyzing the impact of volatility breaks on the causality-in-variance tests of
Cheung and Ng (1996) and Hong (2001).
Based on an extensive set of Monte Carlo simulations, we demonstrate that these tests
suer from severe size distortions when such breaks are neglected. However, we also show
that pre-testing for volatility breaks provides an eective solution to the size problem.
2 Tests for causality in variance
Let yt = (y1t;y2t)0 be a bivariate series of interest. In Cheung and Ng (1996), y2t is said
to cause y1t in variance if
E[(y1t   1t)
2j
t 1] 6= E[(y1t   1t)
2j
1t 1] (1)
1where 
it is the information set dened by 
it = fyit j;j  0g, i = 1;2, 
t = 
1t [ 
2t,
and it is the mean of yit conditional on 
it. Let "it = yit   it; i = 1;2, and assume
that "it = it
p
hit, where hit is a positive, time-varying function measurable with respect
to 
it 1, and it is an innovation process with E[itj
it 1] = 0 and E[2
itj
it 1] = 1. Hence
hit is the (univariate) conditional variance of "it by construction, that is hit = V["itj
it 1].
The null hypothesis that y2t does not cause y1t in variance can now be formulated as
H0 : V[1tj
t 1] = V[1tj
1t 1]: (2)
Dene the squared standardized residuals
^ ut = (y1t   ^ 1t)
2=^ h1t and ^ vt = (y2t   ^ 2t)
2=^ h2t (3)
where hats indicate suitable estimates of the corresponding quantities, and the sample
cross-correlation at lag k,
ruv(k) = cuv(k)=
p
cuu(0)cvv(0); (4)
where cuv(k) is the sample cross-covariance
cuv(k) =
(
1
T
PT
t=k+1(ut   u)(vt k   v); if k  0
1
T
PT k
t=1 (ut   u)(vt k   v); if k < 0;
with T denoting sample size, u and v the sample means of ut and vt, respectively, and
cxx(0) = 1=T
PT
t=1(xt   x)2 for x = u;v.
Cheung and Ng (1996) suggest testing H0 using the statistic
S = T
K X
k=1
r
2
uv(k); (5)
which has an asymptotic chi-square distribution with K degrees of freedom. The reverse
hypothesis that y1t does not cause y2t in variance can be tested analogously, summing the
squared cross-correlations r2
uv(k) from k =  K to  1.
Hong (2001) modies the Cheung-Ng test in two ways. First, it, i = 1;2, is dened
to be the mean of yit conditional on the complete information set 
t 1. Using such a
2\bivariate" conditional mean denition ensures that any causality-in-mean is ltered out
when testing for causality-in-variance.1 Pantelidis and Pittis (2004) show that neglected
causality-in-mean leads to severe (positive) size distortions for the Cheung-Ng test, but
this is easily remedied by obtaining ^ it from a bivariate conditional mean specication. In
this paper, we abstract from conditional mean issues by using a data generating process
with no causality in mean, so that we can use the corresponding sample mean for ^ it.
Second, Hong (2001) suggests weighting the cross-correlations to obtain more powerful
tests. In particular, he denes the statistic
Q =
T
PT 1
k=1 w2(k;K)r2
uv(k)   C(w)
p
2D(w)
asy
 N(0;1); (6)
where w(k;K) is a weight function, and
C(w) =
T 1 X
k=1
(1   k=T)w
2(k;K);
D(w) =
T 1 X
k=1
(1   k=T)(1   (k + 1)=T)w
4(k;K);
are approximately the mean and variance, respectively, of T
PT 1
k=1 w2(k=K)r2
uv(k). There
is considerable freedom in the choice of weight function w(k;K), see Hong (2001) for
examples. Here we use the Bartlett kernel2
w(k;K) =
(
1   jk=(K + 1)j if k=(K + 1)  1
0 otherwise:
:
3 Monte Carlo design and size results
We examine the size of causality-in-variance tests for a data generating process with
yt = (y1t;y2t)0  NID(0;t), where t =

2
1t 1t2t
1t2t 2
2t

. The eects of neglected
changes in volatility are examined through the following ve experiments:3
1See also Hong (2001, footnote 5) for discussion.
2Results for other weight functions are qualitatively similar and are available upon request.
3Analogously to experiments D and E, we also examined increases in volatility in 2
1t and 2
2t from 1 to
2. Results for these are the mirror images of the corresponding cases shown. These results are available
on request.
3A. Shift in volatility of y2t only: 2
1t = 1 for all t, while 2
2t = 2
b for t  T=2 and
2
2t = 2
a for t > T=2;.
B. Simultaneous and identical changes in volatility: 2
1t = 2
2t = 2
b for t  T=2 and
2
1t = 2
2t = 2
a for t > T=2;
C. Simultaneous but opposite changes in volatility: 2
1t = 2
a;2
2t = 2
b for t  T=2; and
2
1t = 2
b;2
2t = 2
a for t > T=2;
D. Simultaneous decline in volatility: 2
2t and 2
1t change from 1 to 0.5 at t = T;
E. Declines at dierent times: 2
1t changes at t = T=2 while 2
2t changes at t = T,
both from 1 to 0.5.
Experiments A to C use 2
b = 1 and 2
a such that
2
a
2
b = 0:1;0:2;:::0:9;1:0; 1
0:9;:::; 1
0:1. Ex-
periments D and E use  = 0:10;0:15;:::;0:90 with  = 0:45;0:46;:::;0:55 also examined
for E).
We consider sample sizes of T = 160 and 480, corresponding to 40 years of quarterly
and monthly data, respectively; 10000 replications are used throughout. The test statistics
S in (5) and Q in (6) for testing causality-in-variance from y2t to y1t and vice versa are
computed for K = 1;2;:::;10. The tests are applied to yt, replacing ^ it and ^ hit in (3)
by the sample mean and variance, respectively. Rejection frequencies at the nominal 5%
signicance level4 are shown graphically, with empirical rejection frequencies for sample
sizes of 160 and 480 shown in panels (a) and (b) respectively of Figures 1-5. The results
for the Cheung-Ng and Hong test statistics are very similar; hence, to save space, only
results for the former test are shown.
Figure 1 shows that neglecting structural breaks in volatility has only minor eects
when just one of the series experiences a volatility change, although tests using jKj > 1
are slightly oversized when the volatility shift is relatively large. In contrast, simultaneous
4Results for other nominal signicance levels are qualitatively similar and are available upon request.
4changes in volatility lead to substantially larger size distortions, irrespective of whether
the volatility change is identical or opposite (Figures 2 and 3). In addition, these plots
demonstrate that the size distortion is symmetric in the ratio
2
a
2
b , and becomes larger as
the sample size T increases. Figure 4 makes clear that the timing of the volatility change
matters: the size distortions are largest when the simultaneous volatility decline occurs
at one-third of the sample period. Finally, from Figure 5 it appears that simultaneous
volatility changes lead to the largest size distortions, with the distortion declining as the
time interval between breaks increases.
Since volatility changes have been shown to occur across a wide range of observed eco-
nomic and nancial time series, the severity of the size distortions revealed in our results
appears to indicate that these tests may, in practical applications, provide unreliable in-
ference about (the non-existence of) causality in variance. Size problems arise particularly
when both series exhibit volatility changes in close temporal proximity, in which case the
tests frequently and incorrectly attribute this occurrence to an underlying causality.
4 Solving the problem: pre-testing for volatility breaks
The Monte Carlo results in the previous section convincingly demonstrate that the tests
for causality-in-variance suer from considerable size distortions when breaks in volatility
are neglected. In this section we explore the usefulness of pre-testing for structural changes
in volatility to remedy this problem.
As in McConnell and Perez-Quiros (2000) and Sensier and van Dijk (2004), we test
for a break in volatility using the mean of the absolute value of the demeaned series. Let
WT(v) denote the Wald test of the null hypothesis H0 : i1 = i2 in the regression
r

2
jyit   ^ ij = i1(1   I(t > i)) + i2I(t > i) + "it; t = 1;:::;T (7)
where, in our context, ^ i is again the sample mean of yit, i is the specied break date and
I(A) is an indicator function for the event A5. We treat the break date i as unknown
5If yit is iid and follows a normal distribution with mean i and variance 2
i ,
p
2jyit ^ ij is a unbiased
5and use the sup-Wald statistic developed by Andrews (1993), given by6
SupW = sup
minimax
WT(i): (8)
Both pre- and post-break periods are required to contain at least 10 % of the available
observations, that is we set min = [T] and max = [(1   )T] + 1 with  = 0:10, where
[] denotes integer part. We use the method of Hansen (1997) to obtain approximate
asymptotic p-values.
This volatility break test is applied prior to the causality-in-variance test. If, using
(8), the null hypothesis of no volatility change in yit is rejected at a 5% signicance level,
we take the estimated volatility break at ^ i (the time period that minimizes the sum of
squared residuals in (7)) into account when standardizing the series. This is achieved by
replacing ^ hit in (3) by the sample variance before (after) ^ i for all t  (>) ^ i.
Results are shown in panels (c) and (d) of Figures 1-5. In all cases this pre-testing
procedure yields empirical rejection frequencies close to the nominal signicance level of
5%. For K = 1, the procedure tends to yield some under-sizing, with some evidence
of over-sizing for larger K. Nevertheless, these distortions are relative modest, with (for
example) the empirical size almost always being between 4% and 8% when T = 160.
Therefore, pre-testing for breaks in volatility provides an eective solution to the size
problem of the causality-in-variance tests.
5 Conclusions
In this paper we have examined the eects of volatility breaks on the tests for causality-
in-variance developed by Cheung and Ng (1996) and Hong (2001). Based on an extensive
set of Monte Carlo simulations, we demonstrate that these tests suer from severe size
estimator of i.
6Average and exponential statistics, computed as AveW = 1
max min+1
Pmax
i=min WT(i) and ExpW =
ln

1
max min+1
Pmax
i=min exp
 1
2WT(i)

, respectively, render qualitatively similar results. The same
holds for Lagrange Multiplier and Likelihood Ratio based tests. Details are available upon request.
6distortions when such breaks are neglected. Pre-testing for structural changes in volatility
is shown to remedy this problem. Therefore, we recommend that these causality-in-
variance tests should be applied only after such pre-testing for breaks in volatility.
References
Andreou, E. and E. Ghysels (2002), Detecting multiple breaks in nancial market volatility
dynamics, Journal of Applied Econometrics 17, 579{600.
Andrews, D.W.K. (1993), Tests for parameter instability and structural change with unknown
change point, Econometrica 61, 821{856.
Billio, M. and L. Pelizzon (2003), Volatility and shocks spillover before and after EMU in Euro-
pean stock markets, Journal of Multinational Financial Management 13, 323{340.
Cheung, Y.-W. and L.K. Ng (1996), A causality-in-variance test and its application to nancial
market prices, Journal of Econometrics 72, 33{48.
Hansen, B.E. (1997), Approximate asymptotic p values for structural-change tests, Journal of
Business and Economic Statistics 15, 60{67.
Hong, Y. (2001), A test for volatility spillover with application to exchange rates, Journal of
Econometrics 103, 183{224.
King, M. and S. Wadhwani (1990), Transmission of volatility between stock markets, Review of
Financial Studies 3, 5-33.
Lamoureux, C. and W. Lastrapes (1990), Persistence in variance, structural-change, and the
GARCH model, Journal of Business and Economic Statistics 8, 225{234.
Lin, W.-L., R.F. Engle and T. Ito (1994), Do bulls and bears move across borders? International
transmission stock returns and volatility, Review of Financial Studies 7, 507-538.
McConnell, M.M. and G. Perez-Quiros (2000), Output uctuations in the United States: What
has changed since the early 1980s?, American Economic Review 90, 1464{1476.
Morgenstern, O. (1959), International Financial Transactions and the Business Cycle, Princeton,
NJ: Princeton University Press.
Ng, A. (2000), Volatility spillover eects from Japan and the US to the Pacic-Basin, Journal
of International Money and Finance 19, 207{233.
Pantelidis, T. and N. Pittis (2004), Testing for Granger causality in variance in the presence of
causality in mean, Economics Letters 85, 201{207.
Sensier, M. and D. van Dijk (2004), Testing for volatility changes in US macroeconomic time
series, Review of Economics and Statistics 86, 833{839.
Stock, J.H. and M.W. Watson (2003), Has the business cycle changed and why?, in M. Gertler
and K.S. Rogo (eds.) NBER Macroeconomics Annual 2002, pp. 159{218.
Susmel, R. and R.F. Engle (1994), Hourly volatility spillovers between international equity mar-
kets, Journal of International Money and Finance 13, 3-25.
7(a) Raw series, T = 160 (b) Raw series, T = 480
(c) Pre-tested series, T = 160 (d) Pre-tested series, T = 480
Figure 1: Rejection frequencies of the Cheung-Ng test (5) at 5% nominal signicance level.
Positive (negative) values of K indicate the null hypothesis is no causality-in-variance from
y2t to y1t (from y1t to y2t). Experiment A: shift in volatility of y2t only.
8(a) Raw series, T = 160 (b) Raw series, T = 480
(c) Pre-tested series, T = 160 (d) Pre-tested series, T = 480
Figure 2: Rejection frequencies of the Cheung-Ng test (5) at 5% nominal signicance level.
Positive (negative) values of K indicate the null hypothesis is no causality-in-variance from
y2t to y1t (from y1t to y2t). Experiment B: simultaneous and identical shift in volatility of
y1t and y2t.
9(a) Raw series, T = 160 (b) Raw series, T = 480
(c) Pre-tested series, T = 160 (d) Pre-tested series, T = 480
Figure 3: Rejection frequencies of the Cheung-Ng test (5) at 5% nominal signicance level.
Positive (negative) values of K indicate the null hypothesis is no causality-in-variance from
y2t to y1t (from y1t to y2t). Experiment C: simultaneous but opposite shift in volatility of
y1t and y2t.
10(a) Raw series, T = 160 (b) Raw series, T = 480
(c) Pre-tested series, T = 160 (d) Pre-tested series, T = 480
Figure 4: Rejection frequencies of the Cheung-Ng test (5) at 5% nominal signicance level.
Positive (negative) values of K indicate the null hypothesis is no causality-in-variance from
y2t to y1t (from y1t to y2t). Experiment D: simultaneous and identical decline in volatility
of y1t and y2t.
11(a) Raw series, T = 160 (b) Raw series, T = 480
(c) Pre-tested series, T = 160 (d) Pre-tested series, T = 480
Figure 5: Rejection frequencies of the Cheung-Ng test (5) at 5% nominal signicance level.
Positive (negative) values of K indicate the null hypothesis is no causality-in-variance from
y2t to y1t (from y1t to y2t). Experiment F: identical decline in volatility of y1t and y2t with
dierent timing.
12

A causality-in-variance test and its application to  market prices,

A test for volatility spillover with application to exchange rates,

Approximate asymptotic p values for structural-change tests,

Detecting multiple breaks in  market volatility dynamics,

Do bulls and bears move across borders? International transmission stock returns and volatility,

Has the business cycle changed and why?,

Hourly volatility spillovers between international equity markets,

Output uctuations in the United States: What has changed since the early 1980s?,

Persistence in variance, structural-change, and the GARCH model,

Testing for Granger causality in variance in the presence of causality in mean,

Testing for volatility changes in US macroeconomic time series,

Tests for parameter instability and structural change with unknown change point,

Transmission of volatility between stock markets,

Volatility and shocks spillover before and after EMU in European stock markets,

Volatility spillover eects from Japan and the US to the Paci

Testing for causality in variance in the presence of breaks

Testing for causality in variance in the presence of breaks

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics