INTERNATIONAL ECONOMIC REVIEW
Vol. 47, No. 2, May 2006
MODELLING ASYMMETRIC EXCHANGE RATE DEPENDENCE
∗
B
Y ANDREW J. PATTON
1
London School of Economics, U.K.
We test for asymmetry in a model of the dependence between the Deutsche
mark and the yen, in the sense that a different degree of correlation is exhibited
during joint appreciations against the U.S. dollar versus during joint deprecia-
tions. We consider an extension of the theory of copulas to allow for conditioning
variables, and employ it to construct flexible models of the conditional depen-
dence structure of these exchange rates. We find evidence that the mark–dollar
and yen–dollar exchange rates are more correlated when they are depreciating
against the dollar than when they are appreciating.
1. INTRODUCTION
Evidence that the univariate distributions of many common economic variables
are nonnormal has been widely reported, as far back as Mills (1927). Common
examples of deviations from normality include excess kurtosis (or fat tails) and
skewness in univariate distributions. Recent studies of equity returns have also
reported deviations from multivariate normality, in the form of asymmetric de-
pendence. One example of asymmetric dependence is where two returns exhibit
greater correlation during market downturns than market upturns, as reported in
Erb et al. (1994), Longin and Solnik (2001), and Ang and Chen (2002). Various
explanations for the presence of asymmetric dependence between equity returns
have been proffered. For example, Ribeiro and Veronesi (2002) suggest corre-
lations between international stock markets increase during market downturns
as a consequence of investors having greater uncertainty about the state of the
economy.
Much less attention has been paid to the possibility of asymmetric dependence
between exchange rates. Asymmetric responses of central banks to exchange rate
movements is a possible cause of asymmetric dependence. For example, a desire
∗
Manuscript received June 2003; revised January 2005.
1
This article is based on Chapter I of my Ph.D. dissertation, Patton (2002), and was previously cir-
culated under the title “Modelling Time-Varying Exchange Rate Dependence Using the Conditional
Copula.” I would like to thank Graham Elliott, Rob Engle, Raffaella Giacomini, Tony Hall, Joshua
Rosenberg, Kevin Sheppard, Allan Timmermann, four anonymous referees, and seminar participants
at the Econometric Society meetings in Maryland, Board of Governors of the Federal Reserve, Chicago
Graduate School of Business, Commonwealth Scientific and Industrial Research Organisation,
London School of Economics, Michigan State, Monash, Oxford, Pennsylvania, Princeton, Purdue,
Texas A&M, UCSD, UC-Riverside, University of Technology Sydney, and Yale for their comments
and suggestions. Financial support from the UCSD Project in Econometric Analysis Fellowship is
gratefully acknowledged. Please address correspondence to: A. J. Patton, Financial Markets Group,
London School of Economics,Houghton Street, London WC2A 2AE, U.K. E-mail: a.patton@lse.ac.uk.
527
528 PATTON
to maintain the competitiveness of Japanese exports to the United States. with
German exports to the United States. would lead the Bank of Japan to intervene
to ensure a matching depreciation of the yen against the dollar whenever the
Deutsche mark (DM) depreciated against the U.S. dollar. Such a scenario was
considered by Takagi (1999). On the other hand, a preference for price stability
could lead the Bank of Japan to intervene to ensure a matching appreciation of the
yen against the dollar whenever the DM appreciated against the U.S. dollar. An
imbalance in these two objectives could cause asymmetric dependence between
these exchange rates. If the competitiveness preference dominates the price sta-
bility preference, we would expect the DM and yen to be more dependent during
depreciations against the dollar than during appreciations. An alternative cause
could come from portfolio rebalancing: When the dollar strengthens there is of-
ten a shift of funds from other currencies into the dollar, whereas when the dollar
weakens much of these funds shift into the DM or euro instead of the yen, as the
former was/is the second most important currency.
2
Such rebalancing behavior
would also lead to greater dependence during depreciations of the DM and yen
against the dollar than during appreciations.
To investigate whether the dependence structure of these exchange rates is
asymmetric, we make use of a theorem due to Sklar (1959), which shows that any
n-dimensional joint distribution function may be decomposed into its n marginal
distributions, and a copula, which completely describes the dependence between
the n variables.
3
Thecopula isa moreinformative measureof dependencebetween
two (or more) variables than linear correlation, as when the joint distribution of
the variables of interest is nonelliptical the usual correlation coefficient is no longer
sufficient to describe the dependence structure.
By using an extension of Sklar’s theorem, we are able to exploit the success we
have had in the modeling of univariate densities by first specifying models for the
marginal distributions of a multivariate distribution of interest, and then specify-
ing a copula. For example, consider the modeling of the joint distribution of two
exchange rates: The Student’s t distribution has been found to provide a reason-
able fit to the conditional univariate distribution of daily exchange rate returns;
see Bollerslev (1987) among others. A natural starting point in the modeling of the
joint distribution of two exchange rates might then be a bivariate t distribution.
However, the standard bivariate Student’s t distribution has the restrictive prop-
erty that both marginal distributions have the same degrees of freedom parameter.
Studies such as Bollerslev (1987) have shown that different exchange rates have
different degrees of freedom parameters, and our empirical results confirm that
this is true for the Deutsche mark–U.S. dollar and yen–U.S. dollar exchange rates:
The restriction that both exchange rate returns have the same degrees of freedom
2
I thank a referee for providing these further suggestions on possible sources of asymmetric ex-
change rate dependence.
3
The word copula comes from Latin for a “link” or “bond,” and was coined by Sklar (1959), who
first proved the theorem that a collection of marginal distributions can be “coupled” together via
a copula to form a multivariate distribution. It has been given various names, such as dependence
function (Galambos, 1978, and Deheuvels, 1978), uniform representation (Kimeldorf and Sampson,
1975, and Hutchinson and Lai, 1990), or standard form (Cook and Johnson, 1981).
MODELING ASYMMETRIC DEPENDENCE 529
parameter is rejected by the data. Note also that this is possibly the most ideal
situation: where both assets turn out to have univariate distributions from the
same family, the Student’s t, and very similar degrees of freedom (6.2 for the mark
and 4.3 for the yen). We could imagine situations where the two variables of in-
terest have quite different marginal distributions, where no obvious choice for the
bivariate density exists. Further, the bivariate Student’s t distribution imposes a
symmetric dependence structure, ruling out the possibility that the exchange rates
may be more or less dependent during appreciations than during depreciations.
Decomposing the multivariate distribution into the marginal distributions and the
copula allows for the construction of better models of the individual variables than
would be possible if we constrained ourselves to look only at existing multivariate
distributions.
A useful parametric alternative to copula-based multivariate models is a multi-
variate regime switching model; see Ang and Bekaert (2002) for example. These
authors show that a mixture of two multivariate normal distributions can match
the asymmetric equity-return dependence found in Longin and Solnik (2001),
and thus may also be useful for studying asymmetric exchange rate dependence.
A detailed comparison of flexible copula-based models and flexible multivariate
regime switching models for exchange rates and/or equity returns would be an
interesting study, but we leave it for future research.
An alternative to parametric specifications of the multivariate distribution
would, of course, be a nonparametric estimate, as in Fermanian and Scaillet (2003)
for example, which can accommodate all possible distributional forms. One com-
mon drawback with nonparametric approaches is the lack of precision that occurs
when the dimension of the distribution of interest is moderately large (say over
four), or when we consider multivariate distributions conditioned on a state vector
(as is the case in this article). The trade-off for this lack of precision is the fact that
a parametric specification may be misspecified. It is for this reason that we devote
a great deal of attention to tests of goodness-of-fit of the proposed specifications.
This article makes two contributions. Our first contribution is to consider how
the theory of (unconditional) copulas may be extended to the conditional case,
thus allowing us to use copula theory in the analysis of time-varying conditional
dependence. Time variation in the conditional first and second moments of eco-
nomic time series has been widely reported, and so allowing for time variation
in the conditional dependence between economic time series seems natural. The
second and main contribution of the article is to show how we may use con-
ditional copulas for multivariate density modeling. We examine daily Deutsche
mark–U.S. dollar (DM–USD) and Japanese yen–U.S. dollar (Yen–USD) exchange
rates over the period January 1991 to December 2001, and propose a new copula
that allows for asymmetric dependence and includes symmetric dependence as a
special case. We find significant evidence that the dependence structure between
the DM–USD and Yen–USD exchange rates was asymmetric, consistent with the
asymmetric central bank behavior story presented above. We also find very strong
evidence of a structural break in the conditional copula following the introduc-
tion of the euro in January 1999: The level of dependence drops substantially,
the dynamics of conditional dependence change, and the dependence structure
530 PATTON
goes from significantly asymmetric in one direction to weakly asymmetric in the
opposite direction.
The modeling of the entire conditional joint distribution of these exchange
rates has a number of attractive features: Given the conditional joint distribution,
we can, of course, obtain conditional means, variances, and correlation, as well
as the time paths of any other dependence measure of interest, such as rank
correlation or tail dependence.
4
Further, there are economic situations where the
entire conditional joint density is required, such as the pricing of financial options
with multiple underlying assets (see Rosenberg, 2003) or in the calculation of
portfolio Value-at-Risk (VaR) (see Hull and White, 1998) or in a forecast situation
where the loss function of the forecast’s end user is unknown.
Despite the fact that copulas were introduced as a means of isolating the depen-
dence structure of a multivariate distribution over 40 years ago, it is only recently
that they attracted the attention of economists. In the last few years, numerous
papers have appeared, using copulas in such applications as multivariate option
pricing,asset allocation, models of default risk, integrated risk management, selec-
tivity bias, nonlinear autoregressive dependence, and contagion.
5
To our knowl-
edge, this article is one of the first to consider copulas for time-varying condi-
tional distributions, emphasize the importance of formal goodness-of-fit testing
for copulas and marginal distributions, and to employ statistical tests comparing
the goodness-of-fit of competing nonnested copulas.
The structure of the remainder of this article is as follows. In Section 2, we
present the theory of the conditional copula. In Section 3, we apply the theory of
conditional copulas to a study of the dependence structure of the Deutsche mark–
U.S. dollar and yen–U.S. dollar exchange rates. In that section, we discuss the
construction and evaluation of time-varying conditional copula models. We sum-
marize our results in Section 4. Details on the goodness-of-fit tests are presented
in the Appendix.
2. THE CONDITIONAL COPULA
In this section we review the theory of copulas and discuss the extension to
handle conditioning variables. Though in this article we focus on bivariate distri-
butions, it should be noted that the theory of copulas is applicable to the more
general multivariate case. We must first define the notation: The variables of in-
terest are X and Y and the conditioning variable is W, which may be a vector.
Let the joint distribution of (X, Y, W)beF
XYW
, denote the conditional distribu-
tion of (X, Y) given W,asF
XY|W
, and let the conditional marginal distributions
of X |W and Y |W be denoted F
X|W
and F
Y|W
, respectively. Recall that
4
This measure will be discussed in more detail in Section 3. Dependence during extreme events
has been the subject of much analysis in the financial contagion literature; see Hartmann et al. (2004)
among others.
5
See Frees et al. (1996), Bouy´e et al. (2000a, 2000b), Cherubini and Luciano (2001, 2002), Costinot
et al. (2000), Li (2000), Fermanian and Scaillet (2003), Embrechts et al. (2001), Granger et al. (forth-
coming), Frey and McNeil (2001), Rockinger and Jondeau (2001), Sancetta and Satchell (2001), Smith
(2003), Rodriguez (2003), Rosenberg (2003), Cherubini et al. (2004), Patton (2004a), Rosenberg and
Schuermann (2004), and Chen and Fan (2006).
MODELING ASYMMETRIC DEPENDENCE 531
F
X|W
(x|w) = F
XY|W
(x, ∞|w) and F
Y|W
(y |w) = F
XY|W
(∞, y |w). We will as-
sume in this article that the distribution function F
XYW
is sufficiently smooth for
all required derivatives to exist, and that F
X|W
, F
Y|W
, and F
XY|W
are continuous.
The latter assumptions are not necessary, but making them simplifies the presenta-
tion. Throughout this article,we will denote the distribution (or c.d.f.) of a random
variable using an uppercase letter, and the corresponding density (or p.d.f.) using
the lowercase letter. We will denote the extended real line as
¯
R ≡ R ∪ {±∞}.We
adopt the usual convention of denoting random variables in upper case, X
t
, and
realizations of random variables in lower case, x
t
.
A thorough review of (unconditional) copulas may be found in Nelsen (1999)
and Joe (1997). Briefly, copula theory enables us to decompose a joint distribution
into its marginal distributions and its dependence function, or copula:
F
XY
(x, y) = C(F
X
(x), F
Y
(y)), or(1)
f
xy
(x, y) = f
x
(x) · f
y
(y) · c(F
X
(x), F
Y
(y))(2)
where Equation (1) above decomposes a bivariate cdf, and Equation (2) decom-
poses a bivariate density. To provide some idea as to the flexibility that copula
theory gives us, we now consider various bivariate distributions, all with standard
normal marginal distributions and all implying a linear correlation coefficient, ρ,
of 0.5. The contour plots of these distributions are presented in Figure 1. In the
upper left corner of this figure is the standard bivariate normal distribution with
ρ = 0.5. The other elements of this figure show the dependence structures implied
by other copulas, with each copula calibrated so as to also yield ρ = 0.5. It is quite
clear that knowing the marginal distributions and linear correlation is not suffi-
cient to describe a joint distribution: Clayton’s copula, for example, has contours
that are quite peaked in the negative quadrant, implying greater dependence for
joint negative events than for joint positive events. Gumbel’s copula implies the
opposite. The functional form of the symmetrized Joe–Clayton will be given in
Section 3; the remaining copula functional forms may be found in Joe (1997) or
Patton (2004).
Now let us focus on the modifications required for the extension to conditional
distributions. Assume below that the dimension of the conditioning variable, W,
is 1. Then the conditional bivariate distribution of (X, Y) |W can be derived from
the unconditional joint distribution of (X, Y, W) as follows:
F
XY|W
(x, y |w) = f
w
(w)
−1
·
∂ F
XYW
(x, y,w)
∂w
, for w ∈ W
where f
w
is the unconditional density of W, and W is the support of W. However,
the conditional copula of (X, Y) |W cannot be derived from the unconditional
copula of (X, Y, W); further information is required.
6
One definition of the con-
ditional copula of (X, Y) |W is given below.
6
We thank a referee for pointing out that the conditional copula can be obtained given just the
unconditional copula of (X, Y, W) and the marginal density of W.
