DNB WORKING PAPER
A Copula-Based Autoregressive Conditional Dependence
Model of International Stock Markets
Rob van den Goorbergh
No. 22/December 2004
A Copula-Based Autoregressive Conditional Dependence Model of
International Stock Markets
Rob van den Goorbergh *
* Views expressed are those of the individual authors and do not neccessarily reflect official
positions of De Nederlandsche Bank.
Working Paper No. 022/2004
December 2004
De Nederlandsche Bank NV
P.O. Box 98
1000 AB AMSTERDAM
The Netherlands
A Copula-Based Autoregressive
Conditional Dependence Model of
International Stock Markets∗
Rob W. J. van den Goorbergh†
December 9, 2004
Abstract
This paper investigates the level and development of cross-country
stock market dependence using daily returns on stock indices. The
use of copulas allows us to build flexible models of the joint distribution of stock index returns. In particular, we apply univariate AR(p)GARCH(1,1) models to the margins with possibly skewed and fat tailed
return innovations, while modelling the dependence between markets
using parametric families of copulas which offer various alternatives to
the commonly assumed normal dependence structure. Moreover, the
dependence across stock markets is allowed to vary over time through
a GARCH-like autoregressive conditional copula model. Using synchronous daily returns on U.S., U.K., and French stock indices, we
find strong evidence that the conditional dependence between pairs of
each of these markets varies over time. All market pairs show high
levels of dependence persistence. The performance of the copula-based
approach is compared with Engle’s (2002) dynamic conditional correlation model and found to be superior.
JEL classification: G15, C51, C12, C13, C32.
Keywords: stock markets, dependence, copulas, synchronicity.
∗
This paper was written while the author was a post-doc research fellow at the Research
Department of De Nederlandsche Bank. The views expressed in this paper are the author’s
and do not necessarily reflect those of his current or previous employer. The author
would like to thank Carsten Folkertsma, Leo de Haan, Juan Carlos Rodriguez, Maarten
van Rooij, Peter Vlaar, Bas Werker, and seminar participants at the Tinbergen Institute
for helpful comments and suggestions. Remaining errors are the author’s alone.
†
Research Department, ABP Investments, PO Box 75753, 1118 ZX Schiphol, The
Netherlands. E-mail: r.van.den.goorbergh@abp.nl, Phone: +31 (0)20 405 5892, Fax:
+31 (0)20 405 9809.
1
1
Introduction
The interdependence between world stock markets is important for various
reasons. Economists argue that the comovement of world equity markets
is a sign of global economic and financial integration. Portfolio managers
are concerned with the level of cross-country stock market correlations as
they seek to exploit diversification benefits from international portfolios.
Regulatory requirements urge banks to build internal models for multiple,
interdependent market risks. Furthermore, stock market interdependence is
crucial to the valuation of financial derivatives whose payoff depends on two
or more underlying assets, such as better-of-two-markets options.
The modelling of the dependence between stock markets, or any kind
of (economic) variables for that matter, involves the description of the joint
distribution of the process of interest. Traditionally, multivariate extensions
of univariate models have been used for this purpose. In particular, the
successful univariate generalized autoregressive conditional heteroskedasticity (GARCH) models pioneered by Engle (1982) and Bollerslev (1986) and
their many variants and refinements have been extended to multivariate
versions typically by assuming a joint normal distribution for the return
innovations and some specification for the conditional variance-covariance
matrix.1
However, a stark contrast exists between the levels of sophistication that
have been achieved in univariate models and in multivariate models. In
univariate models, various distribution functions have been proposed for
the normalized return innovations beyond the normal distribution originally
used by Engle (1982). To accommodate for excess kurtosis or fat tails in
the error distribution, many authors, including Engle and Bollerslev (1986),
have used a Student’s t distribution. Nelson (1990) uses the generalized
exponential distribution. To capture the possibility of skewness in the error distribution, in addition to kurtosis, Hansen (1994) proposes a skewed t
density function. Moreover, Hansen also allows for time variation in these
shape parameters. He suggests modelling the skewness and kurtosis pa1
Two strands of multivariate GARCH models exist. One strand specifies the conditional covariances, in addition to the variances; the other specifies the conditional correlations, in addition to the variances. Examples of the former include the VECH model of
Bollerslev, Engle and Wooldridge (1988), the BEKK model of Engle and Kroner (1995),
the factor-GARCH model of Engle, Ng and Rothschild (1990), and the asymmetric dynamic covariance (ADC) model of Kroner and Ng (1998). Examples of the latter include
the constant conditional correlation (CCC) model of Bollerslev (1990) and the dynamic
conditional correlation (DCC) model of Engle (2002). See Bauwens, Laurent and Rombouts (2003) for an overview.
2
rameters, similarly to the mean and variance processes, as functions of the
conditioning information.
In the multivariate case such flexible modelling of the joint error distribution is generally much more difficult using conventional techniques. The
reason is that conventional techniques construct the joint error distribution
as a multivariate extension of a univariate error distribution. This extension is straightforward in case of normality, but it poses serious obstacles
for departures from normality. For example, one may use the multivariate t
distribution, which is a generalization of the univariate t distribution. However, as Patton (2003) points out, the multivariate t distribution imposes
that all the marginal distributions have the same degrees of freedom parameter, which implies that all assets have equally heavy tails. Flexible density
models from multivariate extensions of univariate distributions become even
more unwieldy if the possibility of skewness, or different degrees of skewness,
is to be allowed for. More generally still, the marginal distributions may be
very different from each other altogether, leaving the multivariate extension
approach in such circumstances of little use.
This paper uses copula theory to model the multivariate distribution of
daily stock returns. A copula links together two or more marginal distributions to form a multivariate distribution. We make use of Sklar’s (1959)
theorem, which states that any multivariate continuous distribution function
can be uniquely factored into its margins and a copula. Interestingly, all the
univariate information is contained in the margins, while the dependence is
fully captured by the copula. In contrast, the usual correlation coefficient
is not sufficient to describe the dependence structure unless the joint distribution of the variables of interest is normal or, more generally, elliptical.
See, for instance, Embrechts, McNeil and Straumann (2002) for a discussion
on the shortcomings of correlation. From a modelling point of view, the
advantage of the copula-based approach is that appropriate marginal distributions for the components of a multivariate system can be selected by
any desired method, and then linked through a suitably chosen copula, or
family of copulas, to form the joint distribution of the components. Hence,
in the case of international stock markets, we are able to use this result to
model the joint distribution of stock returns without sacrificing the level of
sophistication of the modelling of the individual markets. In fact, we argue
that if the marginal distributions are not specified adequately, this will blur
our perception of the interdependence between stock markets.
In this paper we consider various parametric copula families to capture
the dependence between stock returns in international markets. These copula families include the normal copula, which is commonly (and usually
3
implicitly) assumed, and several non-normal alternatives. These alternative
copulas have different features including asymmetric dependence and tail
dependence. Our interest is to find which copula fits the data best. In this
horse race the dependence structure is not treated as fixed, but as possibly varying over time. To capture the dynamics of the copula, we impose
an autoregressive model on a measure of association which translates into
a value for the copula parameter of each family. In the present paper we
use Kendall’s tau for this purpose. The proposed autoregressive model for
Kendall’s tau is similar in spirit to the GARCH model for the variance. An
autoregressive term captures the persistence in the dependence structure,
while the shocks to dependence are governed by a forcing variable. In particular, we propose a forcing variable that increases dependence if markets
move in the same direction and decreases dependence if they move in opposite directions. A distinct advantage of this parametric approach to modelling time variation in the copula, contrary to a nonparametric approach, is
that it does not suffer from a lack of precision when the copula model is wellspecified. On the other hand, a parametric copula model may of course be
misspecified. That is why we consider a large number of parametric copula
families and do elaborate diagnostic testing.
A similar dynamic-copula approach has been used in the foreign exchange market literature by Patton (2003), who found time variation to
be significant in a copula model for asymmetric dependence between two
exchange rates. Patton (2004) applies the same method to U.S. small-cap
and large-cap portfolios to study the effect of asymmetric dependence on
asset allocation. The present paper draws on Patton’s idea to consider time
variation in the conditional copula. However, the objective of this paper
is very different. The main focus here is to describe the time variation of
cross-country stock market dependence at a daily frequency. Moreover, we
use a different estimation procedure that links all parametric copulas under
consideration to the same measure of association, which makes it possible
to compare the implications of different copula models for the dynamics of
the dependence structure. We propose an evolution equation for the dependence structure that is parsimonious, easy to interpret, and close to the
well-known GARCH model for the variance. Another contribution of this
paper is that we allow for asymmetry in the response of the dependence
parameter to negative and positive innovations, which enables us to test the
frequently suggested hypothesis that asset prices have a greater tendency to
move together during market downturns than during market upswings; see,
for instance, Boyer, Gibson and Loretan (1999), Rodriguez (2003), Patton
(2003, 2004) and the references therein.
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An important practical problem in the modelling of international equity
prices is the time difference present in many data records. The use of closing
prices from markets in different time zones, such as the New York Stock Exchange, the London Stock Exchange, and the Paris Bourse, leads to biased
cross-market correlations. The reason is that news happening after trading
hours in Europe, but before closing time in the U.S., say, will be reflected
in U.S. prices but not in European prices until the next day. By using lowfrequency data one may alleviate the non-synchronicity problem albeit at
the cost of a reduced sample size as well as an inability to model short-term
return dynamics. Since the goal of this paper is to do just that, another
approach must be followed. Authors including Burns, Engle and Mezrich
(1998) have proposed synchronicity adjustment models which construct ‘synchronized’ correlations from close-to-close returns. However, Martens and
Poon (2001) show that these models do not remedy the non-synchronicity
problem. The corrected correlations are reported to be substantially different from their synchronous counterparts, which the authors compute from
series of daily international stock market prices recorded at 16:00 London
Time. Moreover, it is not clear how to make non-synchronicity adjustments
once one allows for dependence structures that are more general than the
Gaussian case. The current study uses the same synchronous series Martens
and Poon use in order to model at a daily frequency the dynamics of world
stock market dependence.
The proposed autoregressive conditional dependence model was applied
to the S&P 500, the FTSE 100, and the CAC 40 indices for the period
3 August 1990 until 10 March 2004. We find that the conditional dependence between these markets is not constant over time, but moving with
market shocks. Nevertheless, high levels of persistence are observed in the
conditional dependence at a daily frequency. For each pair of markets, the
dependence structure was best described by a Student’s t copula. Only for
the dependence between the S&P 500 and the CAC 40 do we find evidence
of an asymmetric response of dependence to joint positive and joint negative events. That is, U.S. and French stocks have a greater tendency to
move together in case of bad news than in case of good news. Furthermore,
the performance of the copula-based approach was compared with Engle’s
(2002) dynamic conditional correlation model and found to be superior.
The remainder of this paper is organized as follows. Section 2 presents
a brief review of copula theory and discusses the estimation of copula models. Section 3 describes a copula-based parametric model of international
stock returns. The estimation results are discussed in Section 4. Section 5
concludes.
5
2
Copula theory
As mentioned in the Introduction, a copula is a function that links together
two or more marginal distributions to form a joint distribution. The study of
copulas originates with Sklar (1959) and has had various applications in the
statistics literature. Examples include Clayton (1978), Schweizer and Wolff
(1981), Genest (1987), Oakes (1989), and Genest and Rivest (1993). Only in
the last five years or so have copulas been used in economics and finance. See,
for instance, the work of Rosenberg (1998, 1999, 2003), Coutant, Durrleman,
Rapuch and Roncalli (2001), Embrechts et al. (2002), Cherubini and Luciano
(2002), Patton (2003, 2004), and Fermanian and Scaillet (2003). In this
section we present a brief discussion of copula theory for the bivariate case,
which is also the focus of our application discussed in the remainder of the
paper. The discussion is largely taken from Nelsen (1999) who also treats
the multivariate case.
First we define copulas from the statistical viewpoint and mention some
well-known examples. We then state Sklar’s theorem and discuss its probabilistic interpretation. Finally, we discuss the relation between dependence
measures and copulas.
Definition 2.1 A two-dimensional copula is a function C : [0, 1]2 → [0, 1]
such that
1. C(u, 0) = C(0, v) = 0 for every u, v ∈ [0, 1]
2. C(u, 1) = u and C(1, v) = v for every u, v ∈ [0, 1]
3. C(u2 , v2 ) − C(u2 , v1 ) − C(u1 , v2 ) + C(u1 , v1 ) ≥ 0 for every (u1 , v1 ),
(u2 , v2 ) ∈ [0, 1] × [0, 1]
One can think of a copula as a function which assigns a number in the unit
interval [0, 1] to any point in the unit square [0, 1] × [0, 1]. The third defining
property is the definition of the two-dimensional analog of a nondecreasing
one-dimensional function. A function that fulfills this property is called
2-increasing or quasi-monotone.
Note from the above definition that a copula is a joint distribution
function whose margins are uniform on the unit interval. Hence, from a
probabilistic standpoint, a copula is also the joint distribution of a pair
of uniform [0, 1] random variables. Some elementary examples of copulas
include the product copula and the Fréchet-Hoeffding bounds. The product copula, Π(u, v) = uv, is the unique copula corresponding to independence. The Fréchet-Hoeffding bounds, M (u, v) = min(u, v) and W (u, v) =
6
max(u + v − 1, 0), correspond to perfect positive and perfect negative dependence, respectively. Any copula can be shown to be in between these
bounds, i.e., if C is a copula, then W (u, v) ≤ C(u, v) ≤ M (u, v) for every
(u, v) ∈ [0, 1]2 .
The most important result in copula theory is Sklar’s theorem, which is
stated below.
Theorem 2.1 Sklar’s theorem. Let H be a joint distribution function
with margins F and G. Then there exists a copula C with
H(x, y) = C(F (x), G(y))
(1)
¯ × IR.
¯ If F and G are continuous, then C is unique;
for every (x, y) ∈ IR
otherwise, C is uniquely determined on RanF ×RanG. Conversely, if C is a
copula and F and G are distribution functions, then the function H defined
by (1) is a joint distribution function with margins F and G.
See Nelsen (1999) for a proof. Sklar’s theorem says that any joint distribution can be factored into the marginal distributions of the components and
a copula describing the dependence between the components. The converse
of Sklar’s theorem implies that a joint distribution can be obtained from any
two marginal distributions and any copula. This result is very useful from a
modeler’s point of view; one can specify suitable marginal distributions for
the components of a multivariate system by any desired method, and then
link these through an appropriate copula to form the joint distribution of the
components. Sklar’s theorem thus dramatically increases the set of possible
parametric joint distributions compared with the conventional approach of
looking for multivariate extensions of univariate distributions.
Sklar’s theorem can be restated in terms of random variables and their
distributions functions.
Theorem 2.2 Let X and Y be random variables with distribution functions
F and G, respectively, and joint distribution H. Then there exists a copula
C such that (1) holds. If F and G are continuous, C is unique. Otherwise,
C is uniquely determined on RanF × RanG.
The copula in the above theorem can thus be referred to as the copula of
X and Y , or CX,Y . One implication of this theorem and the familiar result
that two random variables X and Y with continuous distribution functions
F and G are independent if and only if their joint distribution H is given
¯ 2 , is that the product copula
by H(x, y) = F (x)G(y) for every x, y ∈ IR
Π(u, v) = uv gives a unique characterization of independence. Furthermore,
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it is easily shown that copulas are invariant with respect to strictly monotone
transformations of the random variables, i.e., if α and β are strictly increasing functions on RanF and RanG respectively, then CX,Y = Cα(X),β(Y ) .
To go more deeply into this last issue and its relevance to finance, we
mention the example of two government bonds with different maturities.
Sklar’s theorem says that the joint distribution of the prices of these bonds at
any future point in time (prior to maturity) can be factored into the marginal
distributions of the bond prices and a copula describing the dependence
between the bond prices. Since the price of a bond is determined by its
yield to maturity as the discount rate which equates the present value of the
bond’s payments to its price, which is a strictly monotone transformation,
the copula of the bonds’ prices is identical to the copula of the bonds’ yields.
In other words, the prices of the bonds are ‘as dependent as’ their yields.
The respective marginal distributions, however, can (and will) of course be
very different. Note that Pearson’s correlation coefficient does not have this
property; the correlation between prices and the correlation between yields
are two distinct quantities, neither of which fully captures the dependence
between bonds across the term structure.
The example illustrates that any reasonable measure of dependence must
be a function of the copula and of the copula only. A popular dependence
measure that has this property is Kendall’s tau. The (population version
of) Kendall’s tau for two continuous random variables X and Y with copula
C is given by
ZZ
τC = 4
C(u, v)dC(u, v) − 1.
(2)
[0,1]2
Note that since Kendall’s tau depends on the copula only, it is invariant
with respect to strictly monotone transformations of the components X and
Y . We remark that the integral in Eq. (2) can be viewed as the expected
value of the function C(U, V ) of uniform [0, 1] random variables U and V
with joint distribution C. Kendall’s measure of dependence is used in the
remainder of the paper for the purpose of comparing different parametric
copula models.
2.1
Examples of copulas
In this paper we consider several parametric copula families to describe
the dependence structure of international equity markets. To illustrate the
many shapes that these families can accommodate, we present in Figures 1a
and 1b density contour plots of a number of bivariate distributions with
standard normal margins, each with a different parametric copula. Each
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plot shows contour curves for four different density levels, which are kept
constant across the plots. The functional forms of the copula families are
given in Appendix A. The copula parameters are chosen in such a way
that Kendall’s tau is equal to 0.5, implying a positive dependence structure.
This calibration is achieved by inverting relation (2). Appendix B provides
closed-form expressions—where available—of the relation between Kendall’s
tau and the parameter(s) of each copula family under consideration.
The top left plot in Figure 1a depicts the familiar concentric elliptical
contour curves of the bivariate normal distribution. A slightly different
picture emerges for the Student’s t copula (here depicted with degrees of
freedom parameter equal to 5), which allows for fatter tails. Both the normal
and Student’s t copulas are (radially) symmetric.2 A form of asymmetry
can be obtained with Clayton’s copula, whose contour curves are displayed
in the bottom left plot of Figure 1a. Note that the probability mass is
more concentrated in the negative than in the positive quadrant, indicating
a higher degree of dependence if both components are small than if both
components are large. The opposite can be achieved by rotating Clayton’s
copula. Alternative symmetric and asymmetric shapes are provided by the
families presented in Figure 1b.
2.2
Estimation
Under certain conditions the copula-based approach allows for a particularly
convenient way of estimating the parameters of a multivariate model using
maximum likelihood. Note from the definition of a copula in Eq. (1) that
the log of a multivariate density function is given by
log
∂2
C(F (x), G(y)) = log f (x) + log g(x) + log c(F (x), G(y)),
∂x∂y
(3)
where f and g denote the probability density functions of F and G, respectively, and c is the copula density, c(u, v) = ∂ 2 C(u, v)/∂u∂v. Now consider a
parametric model in which the marginal densities f and g depend on distinct
parameters θf and θg , and the copula density depends on parameters θc . Assuming the availability of a sample of size n, with observed random pairs
(xi , yi ), i = 1, . . . , n, we can write the log likelihood of the joint distribution
2
See Nelsen (1999) for a precise definition of different symmetry concepts.
9
as
L(θf , θg , θc ) :=
n
X
i=1
n
X
log f (xi ; θf ) +
n
X
log g(yi ; θg ) +
i=1
log c(F (xi ; θf ), G(yi ; θg ); θc ).
(4)
i=1
Hence, the log likelihood of the joint distribution is just the sum of the log
likelihoods of the margins and the log likelihood of the copula. Standard
maximum likelihood estimates may be obtained by maximizing the above
expression with respect to the parameters (θf0 , θg0 , θc0 )0 . In practice this can
involve a large numerical optimization problem with many parameters which
may be difficult to solve. However, given the partitioning of the parameter
vector into separate parameters for each margin and parameters for the copula, one may use Eq. (4) to break up the optimization problem into several
small optimizations, each with fewer parameters. First the log likelihoods of
the univariate margins are separately maximized to obtain estimates of θ f
and θg . Subsequently, these estimates are substituted into the log likelihood
of the copula which is then maximized over θc . This two-step procedure is
known as the method of inference functions for margins or IFM method. Xu
(1996) compares the efficiency of the IFM method relative to full maximum
likelihood for a number of multivariate models and finds the IFM method to
be highly efficient.3 Therefore we think it is safe to use the IFM method and
benefit from the huge reduction in complexity it implies for the numerical
optimization.
2.3
Conditioning information
While the previous section assumed iid observations, we now consider the
case of conditioning information. The use of conditional distributions allows
us to model the time variation of the joint process. Let (Xt , Yt ) our pair of
variables of interest at date t and let It−1 be all information available up
to date t − 1. This set includes all past values of X and Y . Sklar’s theorem is easily extended to continuous conditional distributions; see Patton
(2004). That is, the conditional distribution of (Xt , Yt )|It−1 can be uniquely
factored into the conditional margins of Xt |It−1 and Yt |It−1 , and a copula
which is conditional on It−1 . It is important to note that the information
set is the same for these conditional distributions. For example, the conditioning information of the conditional marginal distribution of X includes
3
See Joe (1997) for a summary of Xu’s results.
10
past values of X and but also past values of Y . In practice, this presents
a slight complication to the modeler in that he cannot just use any off-theshelf univariate model for any one variable. It must be tested empirically if
current or past values of the other variables can be excluded.
3
A model of international stock returns
This section discusses the set-up and the estimation of a model of the conditional distribution of international stock returns. We propose a copula-based
parametric model in which the parameter vector can be partitioned into distinct sets of parameters for each of the conditional marginal distributions,
and a set of parameters for the conditional copula. As explained in the
previous section, this partitioning facilitates the estimation as it allows one
to break up a large numerical optimization problem with many parameters
into several small optimizations, each with fewer parameters.
First we discuss the data on international stock returns used in the empirical analysis. Subsequently, we present models for the conditional marginal
distributions of the stock returns, followed by an exploration of several models for the conditional copula. We perform various diagnostic tests to check
the validity of our specifications.
3.1
Data
Most studies which model international stock markets use returns computed
from closing prices. However, for exchanges with different trading hours,
the use of daily close-to-close returns leads to an underestimation of crosscountry returns correlations. To circumvent this non-synchronicity problem,
authors have resorted to low frequency data, having to accept an efficiency
loss due to a reduction in sample size as well as an inability to model shortterm dynamics. Alternatively, various procedures have been proposed, for
instance by Burns et al. (1998), to adjust correlation measures from nonsynchronous returns to reflect the contemporaneous dependence structure
of markets in different time zones. However, Martens and Poon (2001) use a
sample of synchronous stock market prices to show that these synchronized
measures of dependence are not robust to the model used to produce them
and that they can be very misleading.
To avoid such mismeasurement, this paper uses the same daily synchronous stock market prices from Datastream that were used by Martens
and Poon (2001) to test the effectiveness of synchronicity adjustment models. The price data, which is sampled at 16:00 London time, is available for
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various markets including the U.S. (S&P 500 index), the U.K. (FTSE 100
index), and France (CAC 40 index) since 3 August 1990. Our sample extends to 10 March 2004. Figure 2 plots the prices of these stock market
indices over time. Clearly, prices tend to move together across countries,
suggesting substantial positive dependence between stock market returns.
As a crude measure of dependence consider the correlation coefficients of
the time series. The sample correlations between each of the index returns
are between .66 and .74, which is roughly in line with what Martens and
Poon (2001) report for their sample which ends in November 1998. For markets whose trading hours overlap only for a short time, such as France and
the U.S., these values are double the figures implied by close-to-close data.
This demonstrates the seriousness of the non-synchronicity problem.
3.2
Flexible models for the margins
We denote by Xt (Yt ) the log return over period t − 1 to t on market X
(Y ). Several pairs of markets are considered: the U.S. and the U.K., the
U.S. and France, and the U.K. and France. In the present paper we limit
ourselves to the dependence between two markets at a time. The analysis
of the multivariate case is left as a topic of future research.
The marginal distributions are modelled using an AR(p) specification
for the mean equation. For the variance equation, we use the celebrated
GARCH(1, 1) model.4 Moreover, we allow for possible asymmetry in the
response of the conditional variance to positive and to negative shocks.
TARCH, or threshold ARCH, was introduced independently by Zakoian
(1994) and Glosten, Jagannathan and Runkle (1993) to model this type of
asymmetry, which is sometimes referred to as the leverage effect. Furthermore, Hansen’s (1994) skewed Student’s t distribution is used to model the
density of the normalized return innovations. This distribution is sufficiently
flexible to capture both fat tails and skewness. Special cases include the Student’s t distribution and the normal distribution. The density function is
given in Appendix C. We present the AR(p)-TARCH(1, 1) specification here
4
Autoregressive conditional heteroskedasticity or ARCH models were introduced by
Engle (1982) and generalized as GARCH (generalized ARCH) by Bollerslev (1986). The
GARCH orders are fixed to unity in light of the overwhelming success of this specification
in financial time series analysis.
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for market X.
Xt = κ +
σt2
εt
σt
=ω+
Pp
i=1 φi Xt−i + εt
2
β σt−1
+ αε2t−1 +
(5a)
γ ε2t−1 1{εt−1
∼ G(νt , λ t ),
< 0}
(5b)
(5c)
where σt2 is the conditional variance of Xt given information at t − 1, and
G(ν, λ) denotes the skewed Student’s t distribution with degrees of freedom
ν and skewness λ. One retrieves the usual GARCH model with normal
innovations for G(∞, 0) and γ = 0. The leverage effect is captured by the
last term in Eq. (5b): if γ > 0, then negative shocks have a greater impact
on the conditional variance than positive shocks.
To capture possible time variation in the higher order conditional moments, we propose modelling the degrees of freedom and skewness parameters as functions of past innovations following Hansen (1994). Moreover, we
extend Hansen’s approach by including autoregressive terms as it is likely
that current tail thickness depends on past tail thickness, and that current
skewness depends on past skewness. In particular, we specify laws of motion
of the form
0
νt0 = a + bεt−1 + cε2t−1 + ψ νt−1
,
(6)
where νt0 is an appropriate logistic transformation of νt which allows νt0 to
vary over the entire real line; recall that the degrees of freedom νt is constrained to lie in the region (2, ∞). The parameter ψ measures the persistence of the degrees of freedom over time. We use similar expressions to
specify the law of motion of the skewness parameter.5
Recall from the previous section that the copula approach to multivariate modelling requires a common information set It−1 for the conditional
distributions of Xt and Yt . Up to now, we have only included these variables’
own lags in the respective univariate models. It may very well be, however,
that lags of one variable affect, say, the conditional mean of the other. To
allow for potential spillovers of this type, lags of Y are added to the right
5 0
x is the logistic transformation of x if
x=L+
U −L
.
1 + exp{−x0 }
Notice that if x0 is allowed to vary over the entire real line, x will be restricted to the
interval [U, L]. Following Hansen (1994), we use in practice a lower bound of 2.1 and an
upper bound of 30 for the degrees of freedom, and a lower bound of −.9 and an upper
bound of .9 for the skewness parameter.
13
hand side of Eq. (5a). This is easily done as it does not interfere with our
approach of separate estimation of the margins.6
The correct specification of the marginal distributions is essential as it
is an important step in the estimation of the copula. From the probabilistic
interpretation of Sklar’s theorem it is clear that the copula of the pair (X, Y )
depends on the probability integral transforms F (X) and G(Y ). Hence, misspecification of the margins F (·) and G(·) leads necessarily to an incorrect
assessment of the dependence between X and Y . Therefore, it is important
to verify any model for the marginal distribution before analyzing the copula. To this end, we perform various misspecification tests in the empirical
analysis, which is described in Section 4.
3.3
Models for the copula
After specifying the marginal distributions, we can study the dependence between the two markets. Under the assumption that the margins are correct,
the conditional probability integral transforms Ut and Vt of the normalized
return innovations are uniform [0, 1] random variables whose joint distribution is equal to the conditional copula of the return innovations.7 The
conditional copula is modelled using various parametric copula families and
estimated using the IFM method. The copula family attaining the highest
likelihood is selected as the best model. As mentioned earlier, each copula family is linked to Kendall’s tau. In order to be able to compare the
estimated copula models, we will report the estimation results in terms of
Kendall’s tau. Moreover, we allow for time-varying dependence by proposing
a model for the evolution of Kendall’s tau. Let τt be Kendall’s tau conditional on information up to date t − 1, and let τt0 be an appropriate logistic
transformation of τt .8 We propose the following autoregressive conditional
6
The conditional variance may also be affected by past values of Y . In particular, some
multivariate GARCH models include lags of the conditional variance of one variable in
the conditional variance equation of the other. However, if we allow for such volatility
spillovers, which in principle can be done, the estimations of the marginal distributions
get entangled, so that instead of two small optimization problems with few parameters,
we have a single large optimization with many parameters. Therefore, we do not pursue
this possibility here.
7
The conditional probability integral transform Ut of the normalized return innovation
εt /σt is equal to G(εt /σt ; νt , λt ), where G(· ; ν, λ) denotes the cumulative distribution
function of Hansen’s (1994) skewed Student’s t distribution.
8
See footnote 5. We use an upper bound of .99 and a lower bound of −.99 for the
copula families that accommodate both negative and positive dependence and a lower
bound of .01 for those that only accommodate positive dependence.
14
dependence model:
0
τt0 = const + χτt−1
+ δ (Ut−1 − 12 )(Vt−1 − 12 ).
(7)
The model is similar in spirit to the GARCH model for the conditional
variance. It contains an autoregressive term to capture the persistence in
dependence, and a forcing variable which is a cross-product that is positive when both probability integral transforms are on the same side of the
median, and negative when they are on opposite sides. We expect δ to be
positive as dependence is likely to go up in case of joint positive or joint
negative events, and go down in case of opposing events.
It is frequently suggested that stock prices have a greater tendency to
move together during market downturns than during market upswings; see,
for instance, Boyer et al. (1999) and Patton (2003, 2004) and the references
therein. We are able to test this hypothesis in the context of our model
through the inclusion of asymmetric parametric copulas such as Clayton’s
and Gumbel’s copula. We also explore another potential manifestation of
asymmetric dependence. Similar to the TARCH model for the conditional
variance, we investigate the possibility that there is an asymmetry in the
way dependence is affected by shocks by adding to Eq. (7) a cross-product
multiplied by a dummy for every (but one) quadrant formed by the point
( 21 , 21 ):
0
0
0
τt0 = const + χτt−1
+ δ Ut−1
Vt−1
0
0
0
0
+ γnn Ut−1
Vt−1
1{Ut−1
< 0, Vt−1
< 0}
0
0
0
0
+ γpn Ut−1
Vt−1
1{Ut−1
≥ 0, Vt−1
< 0}
0
0
0
0
+ γnp Ut−1
Vt−1
1{Ut−1
< 0, Vt−1
≥ 0},
(8)
0
0
where Ut−1
≡ Ut−1 − 12 and Vt−1
≡ Vt−1 − 12 . The positive quadrant, for
which no dummy is included, serves as a benchmark. Various tests can be
done with this set-up. For example, if γnn > 0, then joint negative events
have a greater impact on dependence than joint positive events.
4
Estimation results
In this section we describe the results of the estimation of the conditional
margins and the conditional copula. Results are reported for the U.S.
(S&P 500 index), the U.K. (FTSE 100 index), and France (CAC 40 index).
15
4.1
Estimation of the margins
A general-to-specific method was employed for the estimation of the conditional marginal distributions of the stock returns. A parsimonious model
was selected by successfully eliminating the variable with the smallest tstatistic. The only exception to this rule are the intercepts which are kept
throughout all models. Table I reports the final result of this selection process for the S&P 500 index. Maximum likelihood estimates are accompanied
by White (1982) standard errors which are robust to misspecification.
The mean equation appeared to require an autoregressive term of lag six,
and strong support is found for the leverage effect in the variance equation.
We find evidence for time variation in the degrees of freedom parameter.
Return shocks seem to have a negative influence on the degrees of freedom. This is consistent with Hansen’s (1994) results for bond yields and
exchange rates. Moreover, we find that the model benefits from the inclusion of an autoregressive component, a possibility not investigated by
Hansen.9 In contrast, Patton (2004) and Harvey and Siddique (1999) find
conditional kurtosis to be constant for U.S. small-cap and large-cap returns.
On the other hand, these authors do find significant time variation in conditional skewness, while our results suggest that skewness is constant for the
S&P 500. As is reported by many authors, we find a significant negative
skew in the stock return distribution.
Lagged returns on the FTSE 100 or the CAC 40 index did not turn
up significantly in the mean equation of the S&P 500. Hence, spillovers
from other markets do not appear to be present in the level of the S&P 500
returns. This gives further justification to our approach of separately estimating the marginal distributions.
The last column of Table I shows the test statistics from a parameter
stability test based on the cumulative score functions which was introduced
by Nyblom (1989) and modified by Hansen (1990). Parameter stability is
rejected if the test statistic exceeds a critical value. The asymptotic 5 percent
critical value for the individual statistics is .47, and the asymptotic 1 percent
critical value is .75. Some of our parameters appear to be individually
stable over the estimation period, but the statistics for the intercepts and
the parameters in the variance equation hover between the 1 and 5 percent
critical values, with the leverage effect being the least stable. The model
fails a joint parameter constancy test at the 5 percent level. Nevertheless,
a substantial improvement is made with respect to a model in which the
9
A slightly worse performing model (in terms of the information criteria) is achieved
by leaving out the autoregressive term and including a second lag of the return innovation.
16
degrees of freedom parameter is restricted to be constant. This becomes
clear from Table II which shows the estimation results of this restricted
model. The Nyblom-Hansen statistic for the degrees of freedom is huge.
Also note the poor performance of the restricted model in terms of the
information criteria and the log likelihood; a likelihood ratio test of the
unrestricted model against this model produces a statistic of 24.0 which has
an asymptotic p-value far below 1 percent.
Further diagnostic testing is done through a Kolmogorov-Smirnov test
for the adequacy of the distribution model. Our specification survives this
test easily. Note, however, that the same is true for the restricted model in
Table II. This is likely to be due to the well-known fact that the KolmogorovSmirnov test has low power in detecting detailed features of a distribution.
Tables III and IV hold the estimation results for the U.K.’s FTSE 100
index and France’s CAC 40 index, respectively. No spillovers of the type
described above were found. The FTSE 100 needed a third-order lag in the
mean equation, while the CAC 40 required a more elaborate autoregressive
structure with three lagged returns of orders 1, 7, and 13. (Although the first
lag is individually insignificant, we included it in our model, since it is jointly
significant with the other included lagged returns. The three-lag specification was also favored by the information criteria.) Neither the FTSE nor the
CAC turned out to have significant leverage effects in the variance equation.
Furthermore, no skew was found for either index. Apparently, the Student’s
t distribution with time-varying degrees of freedom describes the return innovations for these indices adequately, whereas a skewed t distribution, also
with time-varying degrees of freedom, is required for the S&P 500.
The specifications for the variation of the degrees of freedom that were
preferred by the data are strikingly similar across indices. In each case,
a first-order autoregressive term captures the persistence in tail thickness,
while a past return innovation serves as a forcing variable with negative
sign. Persistence does vary across markets, however, with the level being
particularly high for the U.K. and an unstable relation for France. Nevertheless, it is remarkable that the three markets should share the same model
that fits the variation in tail thickness best. However, the large NyblomHansen test statistics for France may be a sign of a nonstationary feature of
the conditional distribution that could not be incorporated by adding extra
lags.
17
4.2
Estimation of the copula
Having estimated the conditional margins of the index returns, we are now
in a position to estimate the conditional copula. As a first pass, we try a
constant copula model. Figure 3 shows the support set of the histogram
of the estimated probability integral transforms (U, V ) of the normalized
return innovations of the S&P 500 and the FTSE 100. Clearly, there is
positive dependence on average, and a concentration of mass in the top
right and bottom left corners of the graph, indicating that large positive
shocks often happen simultaneously, as do large negative shocks. Table V
displays the IFM estimates of Kendall’s tau for several parametric copula
models for the case of the S&P 500 and the FTSE 100. All copula models
under consideration yield estimates between .42 and .45, with the exception
of the Clayton copulas, which give somewhat lower values. The normal and
Student’s t copulas produce the highest log likelihoods, with the optimum
being attained for the Student’s t copula with 11 degrees of freedom. Hence,
we get the best fit using a symmetric, fat-tailed copula density, if dependence
is assumed to be constant over time.
However, the high values of the Nyblom-Hansen statistics are indicative
of time variation in the degree of dependence between the indices. To allow
for this possibility, we estimated the autoregressive conditional dependence
model proposed in Eq. (7). Table VI shows the IFM estimates of this model
for the Student’s t copula with 11 degrees of freedom. This copula turned
out to attain the highest log likelihood (again). We find close to unit persistence in the dependence between the indices, and a significant and positive
effect of the cross-product. Hence, dependence increases in case of aligned
market shocks and decreases in case of opposite market shocks, as was to
be expected. We get very similar estimates for other copulas.
Note that the log likelihood increased considerably with respect to the
constant copula model. A likelihood ratio test based on the log-likelihood
difference rejects the constant copula model overwhelmingly. Moreover, the
Nyblom-Hansen test statistics reveal that the parameter estimates of the
autoregressive conditional dependence model are much more stable.
To further test the adequacy of the specified copula model, we conducted
hit tests à la Engle and Manganelli (2004). For this purpose we split the
support set of the copula density up into a number of regions and compared
the relative frequency of occurrence (“hits”) per region with the theoretical
probability of hitting the region. We chose the seven rectangular regions
defined by Patton (2003), which are depicted in Figure 4. The fit of the
copula model was tested by checking for serial correlation in the hits. This
18
was done using a linear probability model, as suggested by Engle and Manganelli (2004), in which a standardized hit dummy is regressed on its own
lags.10 The results of these hit tests for the individual regions and for the
regions jointly, are in Table VI. Clearly, the Student’s t copula passes the
hit tests with ease.
The data seemed to favor a symmetric copula model for the dependence
structure of the S&P 500 and the FTSE 100, implying that the level of dependence is not higher (or lower) for joint negative than for joint positive
events. To investigate this issue further, we test whether the model gains
explanatory power by augmenting it with dummies as in Eq. (8). Table VII
displays the results. All additional variables are individually insignificant,
while the log likelihood increases only marginally. The corresponding likelihood ratio test has an asymptotic p-value of 22 percent. The information
criteria also favor the symmetric model. Hence, there does not appear to
be a threshold or leverage effect in the conditional dependence between the
indices.
As for the dependence structure of the S&P 500 and the CAC 40, and
that of the FTSE 100 and the CAC 40, we again find that an autoregressive dependence model with the Student’s t copula fits the data best. The
results are in Tables VIII and IX, respectively. Whereas no significant leverage effect was found in the conditional dependence between S&P 500 and
FTSE 100, there did appear to be such an asymmetry in the dependence
between the S&P and the CAC 40. The level of dependence goes up by
twice the amount for joint negative shocks compared with joint positive
shocks. Thus, there is evidence to support the hypothesis that U.S. and
French stock returns have a greater tendency to move together in case of
bad news. Moreover, we find that there is a significant negative effect of
individual shocks in the French market. This means that once we correct
for the joint effects of shocks in both markets, an upswing in the French
market decreases U.S.-French dependence regardless of the price movement
in the U.S. market. The Nyblom statistics indicate that this specification is
particularly stable for the period under scrutiny. Hit tests reveal a poor fit
of the model only for region six, which corresponds to the rare situation of
having a bottom 25 percent shock in the U.S. and a top 25 percent shock in
France.
For the U.K. and France we find no leverage effect, but we do find a
10
Lags of one day, one week, and one month were included in the hit tests. Following
Patton’s (2003) suggestion, standardization was done by demeaning the hit dummies by
their theoretical mean (under the null), and scaling them by their theoretical standard
deviation.
19
significant individual effect of the French market, again with a negative
sign. In this case the data seemed to favor a Student’s t copula with 7
rather than 11 degrees of freedom, implying slightly fatter tails for the copula
density. Unfortunately, the model parameters are not as stable as for the
other market pairs and the hit tests indicate a poorer model fit. Including
higher order autoregressive lags and other lagged variables did not improve
the model.
4.3
Comparison with the DCC model
In order to determine the usefulness of the copula-based approach relative
to multivariate GARCH models, we compare our model to the dynamic
conditional correlation (DCC) model that was recently proposed by Engle
(2002). The DCC model is a multivariate GARCH model with time-varying
correlations. It assumes a (conditionally) joint normal distribution for the
return innovations. Note that this assumption implies normal conditional
margins, and a normal conditional copula, which is fully captured by the
correlation coefficients. The assumption of joint normality allows for a convenient two-step procedure for the estimation of the parameters, which is
similar to the IFM method described earlier. In the first step, univariate
GARCH models are estimated for each market. In the second step, the
parameters of the conditional correlation equation are estimated using the
standardized residuals from the first step.
The DCC model is defined as follows. Let ηt be the vector of return innovations at time t, and Ht its covariance matrix conditional on information
up to time t − 1. We have
ηt |It−1 ∼ N (0, Ht )
1/2
1/2
Ht = DHt Rt DHt ,
(9a)
(9b)
where Rt is the conditional correlation matrix, and DHt is a diagonal matrix with the diagonal elements of Ht —the conditional variances—on the
diagonal. The conditional variances are modelled by univariate GARCH
models, which are estimated in the first step of the estimation procedure.
The conditional correlation matrix is modelled as follows.
0
Qt = (1 − a − b)Q̄ + azt−1 zt−1
+ bQt−1
Rt =
−1/2
−1/2
D Q t Qt DQ t ,
−1/2
(10a)
(10b)
where zt = DHt ηt are the standardized return innovations. These are
20
estimated from the standardized residuals of the first estimation step. For
more details we refer to Engle (2002) and Engle and Sheppard (2001).
We apply the bivariate version of the DCC model to our three market pairs and compare its performance to the copula-based autoregressive
conditional dependence model. Table X shows the log likelihoods for both
models broken up into the log-likelihood contributions of the margins and
the log-likelihood contribution of the copula. In every case, we find that
the total log likelihood is substantially higher for the copula-based model
than for the DCC model. The results show that these improvements are
mostly made in the margins. The table further shows the p-values of hit
tests. While the copula-based model passes the individual and joint hit tests
for the pair S&P 500–FTSE 100, the DCC model fails to give an adequate
description of the conditional probability of hitting regions 2 and 5 as well
as all regions jointly. The copula-based approach is also found to be superior for the pair S&P 500–CAC 40. For the pair FTSE 100–CAC 40, the
results are mixed. The copula-based model fails in region 7, but its overall
performance is slightly better than the DCC model.
5
Conclusions
In this paper we have proposed a copula-based autoregressive conditional
dependence model to describe the daily co-movement of international stock
markets. The copula approach allowed us to construct multivariate distributions with a great deal of flexibility. Contrary to traditional multivariate
GARCH models, we were able to accurately model the margins using distinct parametric models which allow for different degrees of skewness and
tail thickness. Conditional dependence structures of several important international stock markets were found to vary over time, showing high degrees of
persistence. Using our conditional dependence model, one may test various
hypotheses about the dependence between asset returns. For example, we
tested the hypothesis of a leverage effect in the dependence structure, and
found support for this claim in the case of the S&P 500 and the CAC 40.
Furthermore, the copula-based approach was found to perform well in describing the joint conditional distribution of the asset returns relative to a
multivariate GARCH model.
The Student t copula gave the best description of the conditional dependence structure for each pair of market indices under consideration. An
advantage of this copula family is that it can be easily generalized to the
case of more than two assets. A drawback, however, is that while each
21
margin is allowed distinct degrees of freedom, there is only one degrees of
freedom parameter for the copula. The different degrees of tail-thickness of
market pairs that were found in this paper show that this may be a serious
restriction. Nevertheless, it would be interesting to extend the bivariate application to larger portfolios. The grouped t copula proposed by Daul, De
Giorgi, Lindskog and McNeil (2003) is a promising new development in this
area.
Several other extensions could be made to the model presented in this
paper. For instance, one may include volatility spillovers which have been
found to be present in multiple stock returns; see Baele (2004) and the references therein. However, this extension comes at the cost of an increased
complexity as the parameters of the margins cannot be estimated separately
anymore. Nevertheless, solving the large numerical optimization could be
helped using the parameter estimates of models without volatility spillovers
as starting values. Another interesting extension would be to estimate the
degrees of freedom of the copula density, or even model it using an autoregressive model similar to the univariate case. We leave these extensions for
further research.
22
A
Copula families
Below we list several parametric copulas which are used in this paper. The
copula density is referred to as c, while the cumulative distribution function
is denoted C.
Normala
Z
xN
Z
yN
pN (s, t; ρ)dsdt
1
1
2 2
2 2
cN (u, v; ρ) = p
exp −
ρ xN − 2ρxN yN + ρ yN ,
2(1 − ρ2 )
1 − ρ2
CN (u, v; ρ) =
−∞
−∞
where xN = Φ−1 (u), yN = Φ−1 (v), and ρ ∈ (0, 1). Special cases are
CN (u, v; −1) = W (u, v), CN (u, v; 0) = Π(u, v), and CN (u, v; 1) = M (u, v).
Student’s t b
Ct (u, v; ρ, ν) =
Z
xt
−∞
Z
1
yt
pt (s, t; ρ, ν)dsdt
−∞
ct (u, v; ρ, ν) = p
1 − ρ2
Γ
ν+2
Γ ν2
2
2
Γ ν+1
2
h
h
yt2
ν
i− ν+1
1+
x2t
ν
1+
i ν+2
x2t −2ρxt yt +yt2 − 2
ν
1+
2
,
−1
where xt = t−1
ν (u), yt = tν (v), ρ ∈ (0, 1), and ν > 0. Special cases
are Ct (u, v; −1, ν) = W (u, v), Ct (u, v; 0, ν) = Π(u, v), and Ct (u, v; 1, ν) =
M (u, v). Furthermore, we have Ct (u, v; ρ, ∞) = CN (u, v; ρ).
Clayton
CC (u, v; α) = u−α + v −α − 1
−1/α
cC (u, v; α) = (1 + α)(uv)−α−1 CC (u, v; α)2α+1 ,
a
Φ(·) is the standard (univariate) normal distribution function; pN (·, ·; ρ) denotes the
bivariate standard normal density function with correlation coefficient ρ:
1
1
2
2
p
.
x
−
2ρxy
+
y
pN (x, y; ρ) =
exp −
2(1 − ρ2 )
2π 1 − ρ2
b
tν (·) is the (univariate) Student’s t distribution function; pt (·, ·; ρ, ν) denotes the bivariate Student’s t density function with correlation coefficient ρ and degrees of freedom
ν:
− ν+2
2
Γ ν+2
1
x2 − 2ρxy + y 2
2
p
pt (x, y; ρ, ν) =
.
1
+
ν
ν
νπ 1 − ρ2 Γ 2
23
where α ∈ [−1, ∞)\{0}. Special cases include CC (u, v; −1) = W (u, v),
CC (u, v; 0) = Π(u, v), and CC (u, v; ∞) = M (u, v).
Plackett
CP (u, v; α) =
[1 + (α − 1)(u + v)] −
cP (u, v; α) = q
[1 + (α − 1)(u + v)]2 − 4α(α − 1)uv
2(α − 1)
α [1 + (α − 1)(u + v − 2uv)]
2
[1 + (α − 1)(u + v)] − 4α(α − 1)uv
3/2 ,
where α ∈ [0, ∞)\{1}. Special cases are CP (u, v; 0) = W (u, v), CP (u, v; 1) =
Π(u, v), and CP (u, v; ∞) = M (u, v).
Frank
(eαu − 1) (eαv − 1)
1
CF (u, v; α) = log 1 +
α
eα − 1
cF (u, v; α) =
α
α
e −1
eα(u+v)
1+
(eαu −1)(eαv −1)
eα −1
2 ,
where α ∈ (−∞, ∞)\{0}. Special cases include CF (u, v; ∞) = W (u, v),
CF (u, v; 0) = Π(u, v), and CF (u, v; −∞) = M (u, v).
Gumbel
n
o
CG (u, v; α) = exp − ([− log u]α + [− log v]α )1/α
cG (u, v; α) =
(log u × log v)α−1 CG (u, v; α)
uv ([− log u]α + [− log v]α )2−1/α
(α − 1 − log CG (u, v; α)) ,
where α ∈ [1, ∞). Special cases are CG (u, v; 1) = Π(u, v) and CG (u, v; ∞) =
M (u, v).
All above copula families except the Gumbel family are comprehensive,
in that they include the Fréchet-Hoeffding bounds and the product copula as
special cases. All except the Clayton and Gumbel families are symmetric.
Following Patton (2004) we expand our set of parametric copula families
by considering rotated versions of the asymmetric families. For example,
one obtains the rotated Clayton copula by u + v − 1 + CC (1 − u, 1 − v; α).
Contrary to Clayton’s copula, this rotated copula displays a greater degree
of dependence for joint high values of the pair (u, v) than for joint low values.
24
B
Kendall’s tau
The table below provides expressions—closed-form if available—of the relation between Kendall’s tau and the parameter for the copula families considered in Appendix A.
Normal, Student’s t
τ (ρ) =
2
π
Clayton
τ (α) =
α
2+α
Plackett
τ (α) = 4
Frank
τ (α) = 1 − 4 {D1 (−α) − 1} /α
Gumbel
τ (α) = 1 − 1/α
arcsin ρ
R 1R 1
0 0
Cα (u, v)dCα (u, v) − 1
Note: D1 denote the first-order Debye function, D1 (−θ) =
C
1
θ
Rθ
t
dt
0 et −1
+ θ2 .
Skewed Student’s t density
The probability density function of Hansen’s (1994) skewed t distribution is
given by
 2 − ν+1
2


bz+a
1

, if z < −a/b
bc 1 + ν−2 1−λ
g(z; ν, λ) =
2 − ν+1

2

1
bz+a

bc 1 + ν−2 1+λ
, if z ≥ −a/b,
where the degrees of freedom parameter ν ∈ (2, ∞) and the skewness parameter λ ∈ (−1, 1). The constants a, b, and c are given by
ν−2
,
ν−1
b2 = 1 + 3λ2 − a2 ,
a = 4λc
Γ ν+1
2
c= p
.
π(ν − 2) Γ ν2
1
This density function has a zero mean and a unit variance. For λ = 0 one
retrieves the Student’s t density (with unit variance). As a consequence,
the skewed t distribution specializes to the standard normal distribution for
λ = 0 and ν = ∞.
25
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29
Table I: Maximum likelihood estimates of an AR(p)-TARCH(1, 1) specification of the return on the S&P 500 with skewed t innovations and possibly
time-varying degrees of freedom and skewness parameters. S.E. are White
robust standard errors. Nyblom-Hansen stability test statistics are reported
in the last column along with the p-value of a Kolmogorov-Smirnov test for
the adequacy of the distribution model.
Variables
Estimate
S.E.
t-stat.
Nybloma
Mean equation
Intercept
.031
.014
2.2
.53
Xt−6
−.044
.017
−2.6
.23
Variance equation
Intercept
.009
.003
2.5
.63
2
σt−1
.92
.014
67
.51
ε2t−1
.030
.009
3.2
.51
ε2t−1 1{εt−1 < 0}
.083
.019
4.4
.84
Degrees of freedom
Intercept
−.54
.15
−3.7
.66
0
νt−1
.56
.12
4.6
.44
εt−1
−.78
.20
−4.0
.26
Skew parameter
−.12
.025
−4.6
.30
Model
Log likelihood
−4258.7
Nybloma
4.86
AIC
8537.3
(2.54)
BIC
8598.1
KS test
.78
a
For individual parameters the asymptotic 5 percent critical value is .47. The appropriate
critical value for the entire parameter vector is reported in brackets. The null hypothesis
of parameter stability is rejected if the test statistic exceeds the critical value.
30
Table II: Maximum likelihood estimates of an AR(p)-TARCH(1, 1) specification of the return on the S&P 500 with skewed t innovations in which the
degrees of freedom and skewness are restricted to be constant.
Variables
Estimate
S.E.
t-stat.
Nyblom
Mean equation
Intercept
.035
.014
2.5
.51
Xt−6
−.045
.018
−2.5
.24
Variance equation
Intercept
.010
.004
2.4
.65
2
σt−1
.92
.015
62
.51
ε2t−1
.019
.008
2.5
.45
ε2t−1 1{εt−1 < 0}
.099
.021
4.8
.91
Degrees of freedom
8.5
1.2
7.2
2.03
Skew parameter
−.11
.025
−4.5
.29
Model
Log likelihood
−4270.7
Nyblom
4.33
AIC
8557.3
(2.11)
BIC
8606.0
KS test
.82
Table III: Maximum likelihood estimates
cation of the return on the FTSE 100.
Variables
Estimate
Mean equation
Intercept
.027
Xt−3
−.056
Variance equation
Intercept
.014
2
σt−1
.91
2
εt−1
.087
Degrees of freedom
Intercept
−.19
0
νt−1
.84
εt−1
−.58
Model
Log likelihood
−4371.7
AIC
8759.5
BIC
8808.1
31
of an AR(p)-TARCH(1, 1) specifiS.E.
t-stat.
.015
.018
1.8
−3.0
.20
.25
.004
.014
.019
3.2
64
6.8
.29
.57
.64
.058
.035
.11
−3.2
24
−5.3
.58
.27
.09
Nyblom
2.42
(2.11)
.45
KS test
Nyblom
Table IV: Maximum likelihood estimates
cation of the return on the CAC 40.
Variables
Estimate
Mean equation
Intercept
.028
Xt−1
.022
Xt−7
−.040
Xt−13
.047
Variance equation
Intercept
.030
2
σt−1
.90
2
εt−1
.10
Degrees of freedom
Intercept
−.31
0
νt−1
.64
εt−1
−.96
Model
Log likelihood
−5333.7
AIC
10687.3
BIC
10748.1
of an AR(p)-TARCH(1, 1) specifiS.E.
t-stat.
.021
.017
.017
.017
1.4
1.3
−2.3
2.8
.45
.57
.19
.09
.010
.018
.017
3.0
51
5.8
.20
.30
.20
.13
.09
.19
−2.3
7.2
−5.1
1.39
1.12
.58
Nyblom
3.26
(2.54)
.61
KS test
Nyblom
Table V: IFM estimates of Kendall’s tau in a constant copula model of the
S&P 500 and the FTSE 100 for various parametric copula families.
Family
Estimate
S.E.
Log likelihood
Nyblom
Normal
.447
.0083
866.7
1.91
Student’s t
ν=7
.445
.0079
882.1
2.64
ν=9
.448
.0079
884.8
2.57
ν = 11
.450
.0079
885.2
2.51
ν = 13
.451
.0079
884.8
2.46
ν = 15
.451
.0080
884.0
2.42
Clayton
Rotated Clayton
Plackett
Frank
Gumbel
Rotated Gumbel
.358
.347
.436
.449
.420
.423
.0087
.0086
.0078
.0031
.0085
.0084
32
711.0
640.5
824.7
812.9
793.0
833.9
.63
2.74
2.72
2.65
2.89
1.46
Table VI: IFM estimates and hit tests for an autoregressive model of the evolution of Kendall’s tau for the S&P 500 and the FTSE 100 using a Student’s
t copula with 11 degrees of freedom.
Variable
Estimate
S.E.
t-stat.
Nyblom
Dependence
Intercept
.014
.0047
3.0
.16
0
τt−1
.97
.0068
144
.22
1
1
(Ut−1 − 2 )(Vt−1 − 2 )
.27
.064
4.2
.24
Model
Log likelihood
907.6
Nyblom
1.16
AIC
−1809.3
(1.01)
BIC
−1791.0
Hit tests
p-value
Region 1
.89
Region 2
.31
Region 3
.24
Region 4
.71
Region 5
.72
Region 6
.59
Region 7
.86
All regions
.85
Table VII: IFM estimates of an autoregressive model of the evolution of
Kendall’s tau for the S&P 500 and the FTSE 100 using a Student’s t copula
with 11 degrees of freedom and allowing for asymmetries in the response of
dependence to shocks.
Variable
Estimate
S.E. t-stat. Nyblom
Dependence
Intercept
.021
.013
1.5
.15
0
τt−1
.97
.012
78
.21
0 V0
Ut−1
.22
.092
2.4
.22
t−1
0 V 0 1{U 0
0
Ut−1
.050
.11
.45
.15
t−1
t−1 < 0, Vt−1 < 0}
0
0
0
0
Ut−1 Vt−1 1{Ut−1 ≥ 0, Vt−1 < 0}
.61
.46
1.3
.03
0 V 0 1{U 0
0
Ut−1
<
0,
V
≥
0}
−.064
.36
−.18
.13
t−1
t−1
t−1
Model
Log likelihood
909.6
Nyblom
1.42
AIC
−1807.3
(1.68)
BIC
−1770.8
33
Table VIII: IFM estimates of an autoregressive model of the evolution of
Kendall’s tau for the S&P 500 and the CAC 40 using a Student’s t copula
with 11 degrees of freedom.
Variable
Estimate S.E.
t-stat.
Nyblom
Dependence
Intercept
.0096 .0041
2.3
.08
0
τt−1
.99
.0042 232
.12
0
Vt−1
−.12
.051
−2.4
.09
0 V0
Ut−1
.17
.061
2.7
.12
t−1
0
0
0
0
Ut−1 Vt−1 1{Ut−1 < 0, Vt−1 < 0}
.18
.071
2.5
.11
Model
Log likelihood
824.3
Nyblom
.49
AIC
−1638.6
(1.47)
BIC
−1608.2
Hit tests
p-value
Region 1
.74
Region 2
.17
Region 3
.86
Region 4
.80
Region 5
.82
Region 6
.005
Region 7
.46
All regions
.22
34
Table IX: IFM estimates of an autoregressive model of the evolution of
Kendall’s tau for the FTSE 100 and the CAC 40 using a Student’s t copula
with 7 degrees of freedom.
Variable
Estimate
S.E.
t-stat.
Nyblom
Dependence
Intercept
.025
.0073
3.4
.52
0
τt−1
.97
.0054
178
.79
0
Vt−1
−.29
.091
−3.2
.64
0 V0
Ut−1
.62
.10
6.1
.92
t−1
Model
Log likelihood
1240.7
Nyblom
2.17
AIC
−2473.4
(1.24)
BIC
−2449.1
Hit tests
p-value
Region 1
.65
Region 2
.41
Region 3
.13
Region 4
.25
Region 5
.035
Region 6
.84
Region 7
.010
All regions
.044
35
Table X: Comparison of the dynamic conditional correlation (DCC) model
with the copula-based autoregressive conditional dependence (ACD) model.
S&P–FTSE
S&P–CAC
FTSE–CAC
DCC
ACD
DCC
ACD
DCC
ACD
Log likelihood
Margin#1 −4335.9 −4258.7 −4335.9 −4258.7 −4426.1 −4371.7
Margin#2 −4426.1 −4371.7 −5406.8 −5333.7 −5406.8 −5333.7
Copula
899.8
907.6
841.7
824.3
1185.0 1240.7
Joint density −7862.2 −7722.8 −8901.0 −8768.1 −8647.9 −8464.7
Hit tests (p-values)
Region 1
.86
.89
.77
.74
.40
.65
Region 2
.01
.31
.14
.17
.07
.41
Region 3
.93
.24
.19
.86
.21
.13
Region 4
.21
.71
1.00
.80
.48
.25
Region 5
.00
.72
.00
.82
.00
.04
Region 6
.17
.59
.01
.01
.56
.84
Region 7
.93
.86
.05
.46
.11
.01
All regions
.01
.85
.00
.22
.00
.04
36
Figure 1a: Density contour plots of bivariate distributions implied by different parametric copulas, all with standard normal margins and Kendall’s tau
equal to 0.5. The degrees of freedom of the Student’s t copula is set equal
to 5. The functional form of the copula families is given in Appendix A.
Normal
2
1
1
0
0
-1
-1
-2
-2
PSfrag replacements
-1
0
1
-2
-2
2
Clayton
2
2
1
1
0
0
-1
-1
-2
-2
-1
0
Student’s t
2
1
-2
-2
2
37
-1
0
1
2
Rotated Clayton
-1
0
1
2
Figure 1b: Density contour plots of bivariate distributions implied by different parametric copulas, all with standard normal margins and Kendall’s
tau equal to 0.5. The functional form of the copula families is given in
Appendix A.
Plackett
2
1
1
0
0
-1
-1
-2
-2
PSfrag replacements
-1
0
1
-2
-2
2
Gumbel
2
2
1
1
0
0
-1
-1
-2
-2
-1
0
Frank
2
1
-2
-2
2
38
-1
0
1
2
Rotated Gumbel
-1
0
1
2
Figure 2: Evolution of international stock market indices over time (3 August 1990 = 100).
450
PSfrag replacements
400
FTSE 100
CAC 40
S&P 500
350
300
250
200
150
100
50
1990
1995
2000
39
2005
Figure 3: Support set of the histogram of estimated probability integral
transforms (U, V ) of the normalized return innovations of the S&P 500 and
the FTSE 100 index.
1
.9
.8
.7
V
PSfrag replacements
.6
.5
.4
.3
.2
.1
0
0
.1
.2
.3
.4
.5
U
40
.6
.7
.8
.9
1
Figure 4: Patton’s (2003) regions used in the hit tests.
1
2
.9
6
4
.75
5
V
.25
3
7
.1
1
0
0
.1
.25
.75
U
41
.9
1
Previous DNB Working Papers in 2004
No. 1
No. 2
No. 3
No. 4
No. 5
No. 6
No. 7
No. 8
No. 9
No. 10
No. 11
No. 12
No. 13
No. 14
No. 15
No. 16
No. 17
No. 18
No. 19
No. 20
No. 21
Jacob A. Bikker, Laura Spierdijk and Pieter Jelle van der Sluis, Market Impact Costs of
Institutional Equity Trades
J.W.B. Bos and C.J.M. Kool, Bank Efficiency: The Role of Bank Strategy and Local Market
conditions
Marco Hoeberichts, Mewael Tesfaselassie and Sylvester Eijffinger, Central Bank
Communication and Output
Olivier Roodenburg, On the predictability of GDP data revisions in the Netherlands
Iman van Lelyveld and Franka Liedorp, Interbank Contagion in the Dutch Banking Sector
Joke Mooij, Corporate Culture of Central Banks: Lessons from the Past
David-Jan Jansen and Jakob de Haan, Look Who’s Talking: ECB Communication During
the First Years of EMU
Jaap Bos and Mindel van de Laar, Explaining Foreign Direct Investment in Central and
Eastern Europe: an Extended Gravity Approach
Jaap Bikker and Paul Metzemakers, Is bank capital procyclical? A cross-country analysis
Philipp Maier, EMU enlargement, inflation and adjustment of tradable goods prices: What
to expect
Karel-Jan Alsem, Steven Brakman, Lex Hoogduin and Gerard Kuper, The Impact of
Newspapers on Consumer Confidence: Does Spin Bias Exist?
Riemer P. Faber and Ad C.J. Stokman, Price convergence in Europe from a macro
perspective: Trends and determinants (1960-2003)
Wilko Bolt and Alexander F. Tieman, Skewed Pricing in Two-Sided Markets: An IO
approach
Robert S. Chirinko, Leo de Haan and Elmer Sterken, Asset Price Shocks, Real
Expenditures, and Financial Structure: A Multi-Country Analysis
Massimo Giuliodori, Monetary Policy Shocks and the Role of House Prices Across
European Countries
Tijs de Bie and Leo de Haan, Does market timing drive capital structures? A panel data
study for Dutch firms
Maria Demertzis and Nicola Viegi, Inflation Targets as Focal Points
Jacob Bikker and Jaap Bos, Trends in Competition and Profitability in the Banking
Industry: A Basic Framework
Nicole Jonker, Carsten Folkertsma and Harry Blijenberg, An Empirical analysis of price
setting behaviour in the Netherlands in the period 1998-2003 using micro data
Olivier Pierrard and Henri Sneessens, Biased Technological Shocks, Wage Rigidities and
Low-Skilled Unemployment
Henriëtte Prast and Iman van Lelyveld, New Architectures in the Regulation and
Supervision of Financial Markets and Institutions: The Netherlands