Exchange Rate Volatility Comovements and Spillovers before and after the
Introduction of Euro: A Multivariate GARCH Approach
Draft June 2008
Nikolaos Antonakakis
1
Abstract
This paper examines exchange rate volatility comovements and spillovers
among major currencies including the euro, the British pound, the Japanese yen and
the Swiss franc before and after the introduction of Euro. Two flexible Multivariate
GARCH models, namely the DCC and the BEKK, are employed. These models allow
correlations to be time-varying, in addition to the conditional variances and
covariances. The results of the empirical analysis show that significant volatility
spillovers and comovements across the four exchange rates exist, but have declined
since the introduction of the euro. These findings suggest that the launch of the euro
itself may have brought about greater stability in the global financial markets.
Moreover, the results show that euro is the dominant currency in volatility
transmission, as its volatility significantly affects the volatility expectations of the
franc, pound and the yen. This fact has serious implications for portfolio
diversification and risk management. An additional finding is that sterling has become
the least volatile currency in the group since the launch of euro. On this basis,
replacing the British pound with the euro might be inadvisable.
Keywords: Exchange rate volatility, Volatility spillovers and comovements,
Multivariate GARCH
1
Department of Economics, Business School, University of Strathclyde, SWD, Room 4.27, 130
Rottenrow, Glasgow, G4 0GE, Scotland, UK, Tel: +44 (0) 141 548 4340, Email:
[email protected]
1. Introduction
Undoubtedly, one of the most important events for the international financial
markets has been the implementation of the Economic Monetary Union on the 1st of
January 1999, with the euro as the common currency. The euro has subsequently
become a serious competitor of the dollar in international usage. According to the
Bank for International Settlements (BIS) (2007), the euro rapidly became the second
most traded currency behind the US dollar in the international exchange markets. The
BIS’s Triennial Central Bank Survey on Foreign Exchange and Derivative Market
Activity in 2007 showed that, from April 2004 to April 2007, the average daily
turnover of the euro accounted for 37% of all transactions (spot and forward
transactions, FX swaps etc.), while the US dollar accounts for 86.3%. The Japanese
yen, the British Pound and the Swiss franc follow the euro with average daily
turnovers of 16.5%, 15% and 6.8%, respectively. 2 As the Euro area is enlarged with
new members that will subsequently adopt the euro, there exists the possibility that
the new currency might in the near future rival or surpass the US dollar as the world’s
leading international reserve currency (Chinn and Frankel, 2005). One of the factors
that constitute in such development is the stability of the euro/US dollar exchange rate
volatility. Besides, the purpose of the euro participation countries was to cushion
internal and external shocks (e.g. monetary shocks). That is, to cushion shocks
arriving from a particular market as well as to shocks arriving from different markets
(the later known as ‘spillover’ effect). Given the status of the Euro, as the anchor
currency in the EMU, it would be interesting to examine its volatility and its impact
on other currencies.
The most recent studies that have investigated the volatility of the Euro since
its introduction were based on univariate framework and have not reached a
conclusion [Cotter (2005) and Coppel et al. (2000) reported no significant increase in
the Euro/USD volatility; Hau et al. (2002) found a slight decrease in the Euro/USD
volatility as opposed to the DM/USD volatility. Malik (2005) and Wan and Kao
(2008) found significant evidence that the euro is much more volatile than the British
pound but the persistence of volatility has decreased for the euro while increasing for
the British pound volatility post launch of the euro.] Even though exchange rate
2
The reason % shares exceed 100% is because two currencies are involved in each transaction hence,
the sum of the % shares of individual currencies used in the BIS report totals 200%.
volatility has been the focal point of recent research, little is known about volatility
linkages among exchange rates or the transmission of volatility between currencies
since the introduction of the euro.
Studies that analysed exchange rate volatility transmission were initiated by
Engle et al., (1990) were they have formulated two possible hypotheses named the
‘heat waves’ and the ‘meteor shower’. The former refers to exchange rate volatility in
one particular market having only country specific effects while, the later refers to
volatility being transmitted to other countries. The authors found supporting evidence
of both effects. On this basis, studies were extended by Bollerslev (1990), Kearney
and Patton (2000), Speight and McMillan (2001), Melvin and Melvin (2003), Black
and McMillan (2004) and by Calvet, Fisher and Thompson (2006) among others. The
main feature of these studies is the application of GARCH modelling to assess
volatility dependencies across currencies. The main conclusion among these studies is
the support of volatility comovements and spillovers between exchange rates. Some
of the reasons of such volatility dependencies across currencies might be attributed to
news announcements about central bank interventions and fundamentals or just to
speculation events.
This paper investigates exchange rate volatility dependencies and spillovers
between the Euro and several other major currencies specifically the British pound,
the Japanese yen and the Swiss franc before and after the introduction of Euro.
Recognizing this feature of co-movement, within a multivariate framework, results in
more efficient empirical models than the alternative of working with separate
univariate models. In addition, the study of volatility spillovers can be helpful when
considering the importance and extend of financial contagion across countries.
Examples of crises in international financial systems such as the Mexican crisis of
1995, the Asian crisis of 1997 and the Russian crisis of 1998 are gruesome reminders
of the damage than can result from contagion in global financial markets. These
currency crises evidently indicate that movements in one currency may affect other
currencies beyond the impact of macroeconomic fundamentals. Accounting for such
volatility linkages across exchange rates opens the door to better decision tools in
various areas, including portfolio diversification, asset pricing, option pricing,
hedging, policy making and risk management. Indeed, unlike at the beginning of the
1990s, several institutions have now developed the necessary skills to apply
econometric theory to these kinds of financial data with ease. The most obvious
approach to modelling the relations between volatilities, co-volatilities and spillovers
of exchange rates is to employ multivariate GARCH-type (MGARCH) models.
This paper contributes to the literature of exchange rate volatility
comovements and spillovers since the introduction of the euro in several respects.
First, it explores whether there has been significant exchange rate volatility
transmission between the euro and the British pound, the Japanese yen and the Swiss
franc against the US dollar since the introduction of the euro. Secondly, it compares
these findings with the pre-euro period. Thirdly, the Dynamic Conditional Correlation
(DCC) multivariate GARCH of Engle (2002) is examined which, according to my
knowledge it has not been applied to modelling exchange rate volatility transmission
and especially to pre- and post- euro exchange rates. Hence, it would be of interest to
investigate the performance of the DCC model in terms of capturing and comparing
volatility spillovers in the pre- and post- euro period. This model has the flexibility of
the univariate GARCH models and is less complicated than the conventional
multivariate GARCH models. The DCC model allows for correlation to be timevarying rather than relying on the assumption of constant correlations through time
(e.g. as implied by the Constant Conditional Correlation (CCC) multivariate GARCH
of Bollerslev (1990)). Besides, such an assumption is a rather unrealistic in many
empirical applications (e.g. see Longin and Solnik (1995) and Sheady (1997)). In
addition, the BEKK model is also applied in order to check the robustness of the
results from the DCC model. Another contribution of this paper is the inclusion of
exchange rate data up to 2007, a rather more efficient sample that allows a much more
rigorous and deeper study of the volatility dynamics of euro.
The remainder of this paper is organised as follows. Section 2 presents the
literature review. Section 3 describes the methodology and the data. Section 4
presents the empirical results and section 5 concludes.
2. Literature Review
A large number of studies find evidence in support of significant
comovements in volatility and spillovers among asset prices (see, Bae and Karolyi,
1994, Kim and Rogers 1995, Kim, 2003, Gannon, 2005 and Christodoulakis, 2007
among others). However, few studies have so far investigated volatility comovements
and spillovers among exchange rates and especially since the introduction of the euro.
One can divide these few studies into two frameworks. Those based on
univariate frameworks, such as granger-causality tests and vector autoregressions
(VAR) and those based on multivariate GARCH methodologies.
On a univariate framework, Inagaki (2007) uses a residual cross-correlation
function (CCF) approach originally proposed by Cheung and Ng (1996). The
objective was to construct a Granger (1969) causality test to investigate volatility
spillover between the British pound/US dollar and the euro/US dollar spot exchange
rates (employing daily data for the period spanning 5 January 1999 to 30 December
2004. The author uses the sample cross-correlation function between the two squared
standardized residuals obtained from univariate GARCH models to apply the Granger
causality test. The author finds unidirectional volatility spillover from the euro to the
British pound. Specifically, he finds that the euro Granger-causes the British pound
but the British pound does not Granger-causes the euro.
On similar grounds, Nikkinen et al. (2005) provided additional support for
Inagaki’s result. They investigated daily implied volatility spillovers within a VAR
framework between the British pound, the euro and the Swiss franc from January
2001 to September 2003. The authors found that the highest correlation exist between
the euro and the franc. In addition, the euro is the dominant currency in volatility
transmission as the euro was found to granger cause the pound and the franc but not
the opposite.
In dealing with volatility spillovers, it is argued that multivariate frameworks
are more appropriate than univariate ones. However, Inagaki (2007 p.163) states that
“although a standard method for investigating volatility spillover is … in the
framework of multivariate GARCH models, the uncertainty in the potential
interaction between two time series may complicate the formulation of multivariate
GARCH models” . In addition, he argues that the residual cross-correlation test “is
easier to implement than the multivariate method as it does not involve simultaneous
modelling”. Nonetheless, with the development of powerful econometric software,
the application of multivariate GARCH models is now relatively easily achieved.
In common with a growing literature, we argue that Multivariate GARCH
models are more appropriate than the univariate methods for analysing exchange rate
volatility spillovers, not least because such spillovers involve the simultaneous
interaction of at least two currencies.
Bollerslev (1990) employed a multivariate GARCH framework to investigate
European exchange rate volatility spillovers. He applied the Constant Conditional
Correlation (CCC) MGARCH to a set of five weekly European spot exchange rates
against the US dollar, namely the Deutsche Mark, the French franc, the Italian lira, the
Swiss franc and the British pound. The author found significant comovements in
volatility of these five exchange rates, and by comparing the pre- and post European
Monetary System (EMS) periods he found that the comovements were significantly
greater in the later period.
Calvet, Fisher and Thompson (2006) employed a Markov-Switching
Multifractal (MSM) approach to assess volatility comovements of the Deutsch mark
(Euro), the British Pound and the Japanese Yen against the US dollar, contrasting the
period after the demise of the Bretton Woods fixed exchange rate system until the 30th
October 2003. They were able to employ an MSM approach because they had a
sample of 7635 daily observations (from June 1, 1973 to October 30, 2003). Over
such a period, there is a higher probability of structural break(s). 3 In addition to the
MSM model, the authors applied the Constant Conditional Correlation (CCC)
MGARCH of Bollerslev (1990) and compared the results from the two methods. They
found that the MSM model outperformed the CCC-MGARCH (based upon the loglikelihood values and the BIC). The CCC-MGARCH explicitly restricts conditional
correlations to be constant overtime, and as discussed earlier, this is a rather
unrealistic assumption in many empirical applications.
Kearney and Patton (2000) offered an application of the so called BEKK 4
MGARCH model, proposed by Engle & Kroner (1995), to exchange rate volatility
spillovers. This approach does not impose the restrictive assumption of constant
conditional correlations through time and offers a very attractive formulation of
volatility spillovers. In addition, it can distinguish between “direct” volatility
spillovers (which are manifest in conditional variances) or/and of “indirect” spillovers
(which are manifest in the conditional covariances). The currencies investigated by
Kearney and Patton included the most important currencies within the EMS, that is
the Deutsche mark, the French Franc, the Italian lira, the British pound as well as the
European Currency Unit all against the US dollar. Their analysis covered data from
3
Especially when the sample includes the first and the second oil price shocks that caused periods of
turbulent volatility in global financial markets (see Mussa, 2000).
4
The acronym comes from the conjoint work of Baba, Engle, Kraft and Kroner.
April 1979 through to March 1997. The authors found evidence of both direct and
indirect volatility transmission within the EMS (through the conditional variances and
covariances, respectively) in both daily and weekly data for the European Currency
Unit (ECU), German mark, French franc, Italian lira and the British pound. In
addition, they provided evidence that the German mark held a dominant position in
terms of volatility transmission. The mark was found to be relatively insulated from
outside shocks while transmitting more volatility than the other currencies. Their
results obtained using weekly data were less significant than those on daily data. This
decrease in significance, derived from the temporal aggregation of the data from daily
to weekly, is in line with the conjecture that markets are more likely to transmit
volatility in active phases rather than in calm ones (e.g. see Ghose and Kroner, 1996).
Another very attractive formulation of volatility spillovers is the Dynamic
Conditional Correlation (DCC) multivariate GARCH model proposed by Engle
(2002) 5 and which is a generalization of the CCC model. This model relaxes the
assumption of constant conditional correlations by making the conditional correlation
matrix time dependent. Even though the DCC model has been applied to examine
volatility spillovers in stock returns (e.g. see Chiang, Jeon and Li, 2007), it has not
been applied, according to my knowledge, to examine volatility spillovers in
exchange returns. Hence, it would be of interest to investigate the performance of the
DCC model in terms of capturing and comparing volatility spillovers in the pre- and
post- euro period.
The next section describes the data and the methodology used in this analysis.
3. Data and Methodology
The data consist of daily spot exchange rates of the European euro, the British
pound, the Japanese yen and the Swiss franc against the US dollar over the period of
January 5, 1990 to December 31, 2007. The series were obtained from the Bank of
England online database. 6 The four currencies chosen are among the “most traded
currencies” as defined by the daily trading volume and the size of the economy in the
BIS report (2007). The data were split into two sub-periods; the periods i) prior to and
5
A similar specification of the DCC model has been proposed by Tse & Tsui (2002).
The website is: http://www.bankofengland.co.uk/statistics/index.htm and was accessed on January
2008.
6
ii) after the introduction of euro. Specifically, the date of separation is the January 1,
1999, the date on which exchange rates were irrevocably fixed against the euro. The
reason the sample period starts on January 1990 and ends on December 2007 giving
equal numbers of observations in the pre- and the post- euro period (specifically 2274
and 2272 daily observations, respectively). For the pre-euro period analysis, the
DM/US dollar spot rate replaces the euro/US dollar rate. The choice of separation date
and the same approach of replacing the euro/dollar with the DM/dollar rate was also
used by Cotter (2005) 7 and Calvet, Fisher and Thompson (2006). 8
Following previous work on spot exchange rates data, where the spot rates ()
are generally non-stationary, I focus on daily exchange rate returns defined as:
Δ ln St = ln ( St ) − ln ( St −1 )
(3.1)
where St is the spot exchange rate and ln is the natural logarithm.
The validity of this first logarithmic difference transformations in rendering
the underlying series stationary is confirmed by the results of unit root tests for nonstationarity using the Augmented Dickey–Fuller (ADF) test statistic, which fails to
reject the null hypothesis of a unit root in the spot exchange rates against a stationary
alternative, but reject that null for first logarithmic differences. 9
In this paper, the main Multivariate GARCH models employed is the Dynamic
Conditional Correlation (DCC) model proposed by Engle (2002). The DCC model is
adopted because it uses a two-step procedure that overcomes certain numerical
difficulties often arising in estimating multivariate GARCH models and also because
it enables the estimation of time-varying volatilities, co-volatilities and correlations.
The DCC model of Engle (2002) is defined as:
yt = μt (θ ) + ε t
H t = Dt Rt Dt
7
(3.2)
Even though Cotter (2005) has the same pre-euro sample (1 January 1990 - 31 December 1998), the
post-euro sample is rather small (1 January 1999 – 31 December 2001) to account for the effects of the
introduction of the euro.
8
Moreover, having artificial data for the euro (obtained from the Bank of England online database)
over the first sub-period, it is found that the unconditional correlation between the euro and the
Deutsche mark is 0.98763. This fact also highly supports the idea of replacing the Euro with the
Deutsche mark for the first sub-period.
9
The results of the unit root test are omitted for the sake of brevity, but are available from author upon
request.
where yt = ( yit ... ynt ) ' is a nx1 vector of exchange returns (specifically the euro,
pound, yen and the franc returns, thus n=4), μt (θ ) = ( μit ,..., μnt ) ' is the conditional
nx1 mean vector of yt , which can be specified as an ARMA process
Ψ i ( L)( yit − μit ) = Θi ( L)ε t
(3.3)
n1
μit = μi + ∑ δ ji χ j ,t ,
j =1
n
s
j =1
j =1
where L is the lag operator, Ψ i ( L) = 1 − ∑ψ ij Lj and Θi ( L) = 1 − ∑ θij Lj .
1/2
Dt = diag (hiit1/2 ...hnnt
) ' is a diagonal matrix of square root conditional variances, where
⎛ n(n − 1) ⎞
hiit can be defined as any univariate GARCH-type model, and Rt is the t × ⎜
⎟
⎝ 2 ⎠
matrix containing the time-varying conditional correlations defined as:
−1/2
−1/2
−1/2
−1/2
Rt = diag (q11,
t ...qnn ,t )Qt diag ( q11,t ...qnn ,t ) .
(3.4)
where Qt = (qij ,t ) is a nxn symmetric positive definite matrix given by:
Qt = (1 − α − β )Q + α ut −1ut′−1 + β Qt −1
(3.5)
where ut = (u1t u2t ...unt )′ is the nx1 vector of standardized residuals, Q is the nxn
unconditional variance matrix of ut , and α and β are nonnegative scalar parameters
satisfying α + β < 1 .
In order to check whether conditional correlations are time varying or not, I
apply the Constant Conditional Correlation (CCC) of Bollerslev (1990) and compare
it with the DCC model. The CCC model is defined as:
yt = μt (θ ) + ε t
H t = Dt RDt = ( ρij hiit h jjt )
(3.6)
where μt (θ ) and Dt are as specified in equations (3.2) and (3.3), but
R = ( ρij )
(3.7)
is a symmetric positive definite matrix with ρii = 1, ∀i matrix that contains the
constant conditional correlations ρij .
The CCC model of Bollerslev (1990) has a GARCH(1,1) specification for each
conditional variance in Dt :
hiit = ωi + α iε i2,t −1 + β hii ,t −1 i = 1, 2,..., n.
(3.8)
This model contains n(n + 5) / 2 parameters. The positive definiteness of H t is
ensured if and only if all the n conditional variances are positive and R is positive
definite. The unconditional variances are easily obtained, as in the univariate case, but
the unconditional covariances are difficult to calculate because of the nonlinearity in
the second equation of (3.6).
Having estimated the CCC model of Bollerslev (1990), I apply two tests for
constant conditional correlations of Tse (2000) and Engle and Sheppard (2001) in the
full and the pre- and post- euro samples. In the former test, the null hypothesis, which
is that conditional correlations remain constant over time, is specified as:
H 0 : hijt = ρij hiit h jjt ,
(3.9)
where the conditional variances are GARCH-type models and the alternative
hypothesis is that conditional correlations are time-dependent and specified as:
H a : hijt = ρijt hiit h jjt
(3.10)
This test statistic is an LM statistic which under the null is asymptotically χ 2
distributed with n(n − 1) / 2 degrees of freedom.
In the constant correlation test of Engle and Sheppard (2001), the null hypothesis is
specified as:
H 0 : Rt = R ∀t
(3.11)
And the alternative is specified as:
H a : vech( Rt ) = vech( R ) + β1*vech( Rt −1 ) + ... + β p* vech( Rt − p ) (3.12)
The test is easy to employ as H 0 entails that the coefficients in the regression
X t = β 0* + β1* X t −1 + ...β p* X t − p + ut* are equal to zero, where X t = vechu ( zˆt zˆt' − I N ) ,
vechu is like the vech operator but it only selects the elements under the main
diagonal, zˆt = Rˆ −1/2 Dˆ t−1εˆt is the N x 1 vector of standardized residuals (under the null),
1/ 2
and Dt = diag (h111/t2 ...hnnt
).
In addition to these tests, the CCC and DCC models are evaluated by
diagnostic checking on the standardized residuals and squared standardized residuals.
Moreover, since these models are nested, four information criteria, namely the Akaike
Information Criteria (AIC), Schwartz Bayesian Criteria (SBC), Hannan-Quinn
Criteria (HQC) and the Shibata Criteria are applied to enhance the selection of the
most appropriate between these two models.
Having proved that conditional correlations do not remain constant over time,
the BEKK 10 model proposed by Engle and Kroner (1995) is also applied, to check the
robustness of the results obtained from the DCC model.
The Full-BEKK model of Engle and Kroner (1995) is defined as:
yt = μt (θ ) + ε t
q
H t = C ′C + ∑
i =1
K
p
∑ Ai′ε t −iε t′−i Ai + ∑
k =1
j =1
K
∑ B′H
k =1
i
t− j
Bj
(3.13)
where μt (θ ) is as specified in (3.3), C is an upper triangular matrix, and A and B
are nxn matrices. This Full-BEKK model is a bit general as it involves a summation
over K terms. Whenever K>1 an identification problem arises, as there are several
parameterizations that yield the same representation of the model. The Full-BEKK
contains ( p + q ) Kn 2 + n(n + 1) / 2 thus, obtaining convergence may therefore be
difficult because the previous equation is not linear in parameters. Since, numerical
difficulties are so common in the estimation of the Full-BEKK model, it is typically
assumed that p=q=K=1 in applications of the Full-BEKK. In addition, as the number
of variables increases, so does the number of the estimated parameters, but
exponentially. For n=2 variables the number of estimated parameters for a full BEKK
equals to 11 but for n=4 equals to 42. This makes things even more complicated in
terms of ensuring the positive definiteness of the conditional variance matrix.
Not surprisingly, new models more tractable have been developed that impose
additional structure, e.g. in the form of diagonal parameter matrices of factors, and
that ensure to positive definitiveness of the conditional variance matrix. There have
been various more flexible extensions of the MGARCH model 11 which are able to
ensure the positive definiteness of the conditional variance matrix such as the
Diagonal BEKK (D-BEKK) that assumes a diagonal conditional variance matrix
specification that reduces the number of parameters to n(n + 5) / 2 (e.g. for n=4 it is
equal to 18). In addition, the positivity constraint of the conditional variance matrix of
the D-BEKK model is guaranteed and easily imposed through the Cholesky
decomposition. There can be imposed several other restrictions, however they are out
10
11
The acronym comes from the conjoint work of Baba, Engle, Kraft and Kroner.
For a survey on multivariate GARCH models see Bauwens, Laurent and Rombouts (2006).
of scope of this paper at the moment. In the diagonal BEKK the A and B matrices
are diagonal. In this paper, the Diagonal-BEKK model is applied because it is more
tractable and the positive definiteness of the variance-covariance matrix is ensured.
The D-BEKK is specified as:
yt = μt (θ ) + ε t
q
p
i =1
j =1
(3.14)
H t = C ′C + ∑ Ai′ε t −iε t′−i Ai + ∑ Bi′H t − j B j
where μt (θ ) is as specified in (3.3), C is an upper triangular matrix, and A and B
are diagonal matrices. The coefficients of matrices A and B indicate the innovations
in each specific market and the persistence or the rate of the decay of news in each
specific market, respectively. 12 The D-BEKK is covariance stationary if and only if:
q
q
i =1
j =1
∑ ann2 ,i + ∑ bnn2 , j < 1, ∀n = 1, 2,..., N .
(3.15)
All the multivariate GARCH models employed in this paper are estimated
using the Quasi-Maximum Likelihood (QML) estimator under a multivariate Student
distribution (see Harvey, Ruiz, and Shephard, 1992 and Fiorentini, Sentana, and
Calzolari, 2003). The multivariate Student distribution is applied as it is well known
that the normality assumption of the innovations is rejected in most empirical
applications dealing with daily exchange rate data. This adds an extra parameter to the
estimation of each model, namely the degrees of freedom parameter, denoted by υ .
When υ tends to infinity, the Student distribution tends to the normal density. When
it tends to zero, the tails of the density become thicker and thicker. The parameter
value indicates the order of existence of the moments, e.g. if υ = 2 , the second
moments do not exist, but the first moments exist. For this reason it is convenient to
assume that υ > 2 , so that the conditional variance-covariance matrix H t is always
interpretable as a conditional covariance matrix. Under this assumption, the Student
density can be defined as:
⎛υ + n ⎞
Γ⎜
⎟
2 ⎠
⎝
g zt θ ,υ =
n
⎛n⎞
Γ ⎜ ⎟ ⎡⎣π (υ − 2 ) ⎤⎦ 2
⎝2⎠
12
⎡ zt ' zt ⎤
⎢1 + υ − 2 ⎥
⎣
⎦
−
n +υ
2
(3.16)
The coefficients a11, a22, a33 and a44 indicate the impact of news in the euro(mark), pound, yen and
the franc markets, respectively, whereas, the coefficients b11, b22, b33 and b44 indicate the persistence of
news in these markets, respectively.
Where Γ (.) is the Gamma function. The density function of yt (exchange returns) is
easily obtained by applying:
f yt θ , Ωt −1 = H t
Where H t
−1/2
−1/ 2
g H t−1/2 ( yt − μt υ
(3.17)
is the Jacobian that arises in the transformation from the innovations to
the observables.
4. Empirical results
4. 1 Descriptive statistics
Table 1 presents descriptive statistics of the mark(euro), the pound, the yen and
the franc returns series for the pre- and the post-euro period. The returns are calculated by
taking the first logarithmic differences in exchange rates as denoted in equation (3.1). The
means show all returns with small negative returns for each sample period. The daily
unconditional standard deviations and variances of the spot exchange rates show an
interesting feature. For each of the sample period, the pound’s return daily standard
deviation (variance) is the smallest. The euro, the yen and the franc rank in the second,
third and fourth place, respectively. In addition, the unconditional standard deviations
(and subsequently the unconditional variances) for all the returns have declined since the
introduction of the euro. Specifically, the euro-mark, pound, yen and the franc standard
deviations (variances) have declined from 0.6649 (0.4421), 0.6071 (0.3686), 0.7356
(0.5410) and 0.7353 (0.5406) in the pre-euro period to 0.5949 (0.3539), 0.5027 (0.2527),
0.6163 (0.3799) and 0.6373 (0.4061) in the post-euro period. 13 This indicates that the
launch of the euro itself might induced greater stability in these four currencies. The
excess kurtosis coefficient is highly significantly greater than zero for each of the four
currencies at each sample period indicating non-normality of returns. 14 However, the
excess kurtosis has significantly declined since the introduction of the euro. This fact also
13
An F-test on the null hypothesis that the standard deviation in the pre-euro period is greater than the
standard deviation in the post-euro period was not rejected for each return series. The results can be
provided from the author upon request.
4
E ⎡( y − μ ) ⎤
⎣
⎦ − 3 . A distribution with positive excess
14
The excess kurtosis is defined as: K =
4
σ
kurtosis is said to have heavy tails, implying that the distribution puts more mass on the tails of its
support than a normal distribution does. If returns are normally distributed, then the excess kurtosis
coefficient should be zero.
provides evidence that the launch of the euro coincided with greater stability, as extreme
events (such as currency crisis) are seem to have declined in frequency compared to the
pre-euro period. In addition, the Jarque-Bera statistic confirms that exchange returns are,
as expected, not normally distributed since the null hypothesis of normally distributed
returns is persuasively rejected and the data are clearly skewed.
The Ljung-Box Q statistic tests the null hypothesis of no serial correlation and is
calculated using up to 10 lags for both daily returns and squared returns series. A
significant Q statistic rejects the null hypothesis of no serial correlation in returns, while a
significant Q statistic for the squared returns series is rejecting the null hypothesis of
homoskedastic square returns. Table 1 reports the Q statistics to be insignificant at 10 lags
across each returns series, apart from the pound returns in the pre-euro period. This
indicates that all returns but the pound can be characterized as random walk processes.
However, the Q statistic in the squared returns is significant for each return series
indicating strong non-linear dependencies. This is also supported by Engle’s ARCH-LM
statistic. The last column of Table 1 clearly shows the presence of ARCH effects in
returns up to 5 lags. The null hypothesis of no ARCH effects is rejected for each series at
5% level of significance. 15
Figures 2 and 3, which plot returns in the left hand side and their autocorrelation
functions in the right hand side, support the phenomenon of no autocorrelation in returns
(apart from the pound returns for the 1 lag) and volatility clustering. The latter means that
large (small) changes tend to be followed by large (small) changes of either sign.
Thus, the returns series are characterised by non-randomness and the presence of
ARCH effects, and the square returns by the presence of higher order serial correlation
and non-linear dependency. The findings of higher order serial correlation, and non-linear
dependency support the decision of modelling exchange rate volatility through a
GARCH-type process.
Moreover, in Figure 1 which shows the evolution of the exchange rates 16 over
time, one can observe the presence of comovements and especially between the three
“European” currencies, namely the Euro (mark), the pound and the franc. These
comovements are evident in both the pre- and the post- euro period. Since the beginning
of 1990 until early in 1991, all four currencies appreciated against the dollar. Immediately
after that, a sharp depreciation took place till mid 1991 followed by an appreciation
period, with couple depreciation periods in the mid 1992 and 1993, until the first quarter
15
16
For the Swiss franc the null hypothesis of no ARCH effects is rejected at 10% level of significance.
In order to analyse comovements, in figure 1, exchange rates were scaled by their means.
of 1995. Since then, the mark (euro), the yen and the franc have depreciated against the
dollar until the introduction of the euro, and the pound remained relatively stable. Since
the launch of the euro, the presence of comovements in all four currencies is again highly
evident but in yen between mid 1999 until 2002 and mid 2005 until the end of 2007.
Hence, such evidence indicates modelling these series through a multivariate GARCH
framework.
4.2 Volatility comovements and spillovers
Before proceeding with the multivariate GARCH modelling, Table 3 presents the
unconditional sample correlations between these currencies. These correlations indicate
that the market expectations of future exchange rate volatilities are contemporaneously
and positively correlated across these four major currencies. They range from 0.3143 for
the yen-pound correlation to 0.94 for the euro-franc correlation. In addition, it can be
clearly seen that the unconditional correlation of the euro (mark)-franc is the greatest for
both the pre- and post- euro periods. The euro (mark)-pound, franc-pound, franc-yen,
euro (mark)-yen and the pound-yen correlations capture the second, third, fourth, fifth
and sixth place, respectively. The fact that the observed correlations are higher between
the European currencies is in line with the phenomenon of intra-regional currency
contagion rather inter-regional. A representative example of such intra-regional currency
contagion is the Asian currency crisis in 1997. Another interesting feature is that all the
pairwise unconditional correlations (apart from the euro (mark)-franc) have declined in
the post euro period. This means that volatility spillovers have declined since the launch
of the euro, suggesting relative stability since the launch of the euro.
Rather than looking on unconditional (constant) correlations it is more
appropriate to examine time-varying conditional correlations. In order to check
whether correlations are time-varying, I employ both the CCC and DCC multivariate
GARCH models and compare their performance. The estimation results of the CCC
and the DCC models, with a AR(2) filter for the conditional mean specification, for
the full sample are shown on Table 3. Even though both models are well specified, on
the basis that the univariate and multivariate tests on the standardized and squared
standardized residuals do not find evidence of significant serial correlation (apart
from the univariate Q statistic on the pound’s squared standardized residuals of the
DCC model), the two test statistics for constant conditional correlations of Tse (2000)
and Engle and Sheppard (2001) reject the null hypothesis of constant conditional
correlations at 5% level of significance. All the information criteria along with the log
likelihood value support the DCC specification of time-varying conditional
correlation. This model is also supported by Figure 4, in where the blue and the red
lines denote correlations of the CCC and DCC models, respectively. From this figure
it can be clearly seen that each six correlations are non constant throughout the whole
sample. 17 In addition, under the DCC model not all dynamic conditional correlations
are significant as opposed to the CCC model, in which all constant conditional
correlations are significant. According to the preferred DCC model only correlations
between European currencies are significantly positive. Correlations between the
euro-yen, pound-yen and franc-yen are insignificantly positive. This means that news
arriving from one of the euro, pound or the franc markets positively affects the other
two but it does not affect the yen market. This is in line with the literature that
currency contagions/spillovers are of intra-regional rather than inter-regional nature
(see Glick and Rose, 1999).
Now that it has been documented that correlations do not remain constant
throughout the whole sample the next step is to estimate the DCC of Engle (1992) for
the pre- and the post- euro period.
Table 4 presents the empirical results of the DCC model for the pre- and the
post- euro period. 18 An AR(3)-DCC-MGARCH(1,1) and a random walk DCCMGARCH(1,1) model were chosen in order to remove any serial correlation in
returns for the pre and the post-euro, respectively. 19 For both periods, the DCC model
seems to be well specified, as the diagnostic tests for serial correlation including the
multivariate ones, which are of more interest, report no evidence of serial correlation.
Hosking’s (1980) and Li and McLeod (1981) multivariate versions of the the LjungBox test statistics do not reject the null hypothesis of no serial correlation up to 30
lags. There is strong evidence of significant dynamic conditional correlations for the
both the pre- and the post-euro period. The correlation coefficients are reported, with
17
The sharp drop in correlations of euro (mark)-pound and euro (mark)-franc in the end of 1998 is due
to the fact of replacing the mark with the euro.
18
A CCC model was also applied for the pre- and the post- euro period but was rejected against the
DCC, as both tests for constant correlations rejected the null of no constant correlations at 5% level of
significance. For the pre-euro period the test statistics of Tse (2000) and Engle and Sheppard (2001)
were 74.1633 and 66.1108 with p-values of [0.0000] and [0.0000], respectively. For the post-euro
period were 71.5886 and 327.913 with [0.0000] and [0.0000] p-values, respectively. For the shake of
brevity, the results of the CCC model for the pre- and post- euro period are not presented but can be
provided from the author upon request.
19
The Akaike Information Criterion (AIC), the Schwarz Bayesian Criterion (SBC) and overfitting
approaches were used to determine the optimal lag structure of mean returns.
their significance becoming greater in the post-euro period, thus indicating that
volatility spillovers are more significant since the launch of the euro. As expected, the
largest volatility spillovers are between currencies belonging to countries
geographically closer to each other, namely the euro, the pound and the franc. The
strongest in magnitude volatility spillovers occur between the euro-franc, the europound and the franc-pound currencies for both periods. Specifically, the estimated
correlation coefficients between the franc-euro(mark), pound-euro(mark) and the
franc-pound are 0.8515, 0.8053 and 0.7490, respectively, in the pre-euro period and
0.9496, 0.6656 and 0.6632, respectively in the post-euro period. The lowest are
between the yen and the other three currencies. Specifically, the estimated correlation
coefficients between the yen-euro(mark), yen-pound and the yen-franc are 0.3190,
0.3030 and 0.2855, respectively, in the pre-euro period and 0.4224, 0.3342 and
0.4521, respectively in the post-euro period. This is in line with the literature that
currency contagions/spillovers are of intra-regional rather than inter-regional nature
(see Glick and Rose, 1999). These results show that the euro(mark) is the dominant
currency in volatility transmission, as the highest four correlation are between
currencies involving the euro (specifically. the euro-franc, euro-pound and the euroyen) for both the pre- and post- euro periods. This means that the euro volatility
significantly affects the volatility expectations of the franc, pound and the yen. This is
in line with the results of Nikkinen et al., (2006). The authors having daily data for the
period between 2 January 2001 and 29 September 2003, found that the implied
volatility of the euro significantly affects the volatility expectations of the pound and
the franc. In this paper, not only additional evidence of the dominant position of the
euro volatility transmission to the yen is provided, but it is also supported to a more
extended sample. This fact has serious implications for portfolio diversification and
risk management.
Another interesting feature of Table 4 is that on average volatility spillovers
have increased since the launch of euro. The fact that correlations increase over time
can be interpreted by the globalization of financial markets. Out of the six dynamic
conditional correlation coefficients, four have increased (namely the euro-yen, eurofranc, pound-yen and the franc-yen) and two have declined (the euro-pound and the
franc-pound) since the introduction of the euro. A standard Z-test statistic suggested
by Morrison (1983) was applied for statistic inference. 20 One time, to test the null
hypothesis of no increase in correlations, for each of the four previously mentioned
correlations that have increased, and another time, to test the null hypothesis of no
decrease in correlations 21 for each the two previously mentioned correlations that
have declined. The results of this test are presented on Table 5. The null hypothesis of
no increase in correlations since the launch of the euro is rejected for the euro-yen,
euro-franc and the franc-yen correlations but no for the pound-yen at 1% level of
significance. In the other case, the null hypothesis of no decrease in correlations since
the launch of the euro is rejected for both the euro-pound and the franc-pound at 1%
level of significance.
An interesting finding is that, a shock arriving in the euro market in the preeuro period is being transmitted to the pound market by 80.53% while, a shock of the
same magnitude in the post-euro period is being transmitted to the by 66.56% pound
market. The lower the volatility spillovers of the euro to the pound in the post-euro
period has obviously important implications on the highly debated issue of the UK
adopting the euro, but also for risk management, portfolio diversification etc. This
particular finding definitely requires more research along this dimension before
economists converge to any specific decision on this highly debated issue.
Figures 5 and 6 show the evolution of conditional correlations obtained from
the DCC model between these four returns for both sub-samples. 22 One can clearly
observe that correlations do not remain constant over time in both the pre- and the
post- euro period as, correlations range between -0.165 to 0.964 and -0.017 to 0.979,
respectively. Specifically, in the pre-euro period correlation coefficients of each two
return series were higher between 1991 and 1992, in 1995 and in 1997-1998. Coupled
20
The Z-test statistic suggested by Morrison (1983) for the null hypothesis of no increase in correlation
Z 0 − Z1
1 ⎛ 1 + ρ0 ⎞
1 ⎛ 1 + ρ1 ⎞
, where Z 0 = ln ⎜
⎟ , Z1 = ln ⎜
⎟ , ρ0 and
2 ⎝ 1 − ρ0 ⎠
2 ⎝ 1 − ρ1 ⎠
1
1
+
N 0 − 3 N1 − 3
ρ1 refer to pre-euro and post-euro correlations, respectively, N 0 = 2274 and N1 = 2272 . This test
is defined as:
T=
statistic is approximately normally distributed and is fairly robust to the non-normality of correlation
coefficients.
21
22
In the case of the null of no decrease in correlation, the null is defined as:
T=
Z1 − Z 0
1
1
+
N 0 − 3 N1 − 3
In addition, the blue lines in figures 5 and 6 show the constant conditional correlations obtained from
the CCC model.
with the fact that conditional variances and covariances were higher during the same
periods, as shown from Figures 7 and 8, is in agreement with empirical studies of
relationships between currency markets for which the correlation (or the spillover
effect) between two markets tend to increase during periods of extreme episodes such
as political and/or currency crisis. The large increases in conditional variances,
covariances and correlations particularly for the pre-euro period coincide, hence,
possibly associated with the cut of the discount rate from 6% to 5.5% by Federal
Reserve on the 14 April 1991, with the political turmoil in Russia (where news of a
coup in the Soviet Union against president Gorbachev arrived on 19 August 1991),
with the EMS crisis in 1992 (with the pound being suspended from the ERM on 16
September 1992), the Mexican-Peso crisis in 1994-1995 and the Asian crisis in 19971998 (see Lobo, 2002).
The same applies for the post-euro period. Correlations are greater during
2000 till the beginning of 2001, at the end of 2002 and in beginning of 2004 which
coincide with periods of greater conditional variances and covariances.
Another interesting feature of Figures 7 and 9 is that conditional variances of
each currency have declined since the launch of the euro. This is also supported by the
F-test of the hypothesis that each currency’s conditional variance 23 in the pre-euro
period is less than that in the post-euro period against the alternative that, each
conditional variance is less in the post-euro period. The F-test for the euro(mark),
pound, yen and franc is 2.817, 10.009, 18.834 and 4.009 with a p-value of [0.000],
[0.000], [0.000] and [0.000], respectively, clearly rejecting the null hypothesis that
each conditional variance in the pre-euro period is less than that in the post-euro
period. This means that each currency is less volatile in the post-euro period; hence
the possibility of extreme episodes is reduced. In other words, the introduction of the
euro coincided with greater stability in the global financial markets. The fact the
euro/dollar volatility is less than the mark/dollar volatility is in line with the empirical
results of Hau et al (2002).
Among the euro, pound, yen and the franc, the yen remains the most volatile
currency in the pre- and post- euro period. In the pre-euro period the franc, the pound
and the mark (euro) capture the second, third and fourth place, respectively. In the
post-euro period, the euro, franc and the pound rank as the second, third and fourth
23
The conditional variances of the AR(3)-DCC-MGARCH(1,1) and DCC-MGARCH(1,1) for the preand the post-euro period, respectively, where used to perform the F-test.
most volatile currencies, respectively. This is in line with the empirical results of
Malik (2005) that the euro is more volatility than the pound in the post-euro period.
This means that the pound became less volatile than the euro, since the latter’s
introduction, as compared to the pre-euro period. This partly contradicts with the
results of Wan and Kao (2008). The authors find that the euro is more volatile than
the pound but the volatility of the former has declined and of the latter has increased
since the launch of the euro. There are possible statistical explanations for their
findings that the volatility of the pound has increased since the launch of the euro.
One of which is that their estimation sample is up to March 31, 2006 and does not
include 2007 a period characterised by its lower volatility in the global financial
markets as shown in Figure 9. Another reason is the choice of the pre- and the posteuro periods of their sample, which might affect estimation results. The date of
separation of their data is the January 2002 (the date of the euro notes and coins began
circulating) and not January 1999 when the euro was introduced. In addition, their
pre-euro period ranges from January 1999 till the end of 2001, a rather small sample
to account for the dynamics of exchange rates. Last but not least, their analysis is
based on a univariate framework.
Looking at the evolution of the conditional covariances from Figures 8 and 10
one can observe another interesting feature. Each of the six conditional covariances
has declined since the launch of the euro, indicating that comovents and spillovers
among these currencies have declined in the post-euro period. The greatest range in
comovements between currency markets exist between the euro(mark)-franc for both
sub-periods. Before the introduction of euro the second, third, fourth, fifth and last in
magnitude comovements are between the yen-franc, euro(mark)-yen, euro(mark)pound, franc-pound and euro(mark)-franc, respectively. The equivalent ones for the
post-euro period are between the euro-pound, pound-franc, franc-yen, euro-yen and
pound-yen, respectively. Since the euro(mark) capture the first places in the
magnitude of comovements in volatility it clearly plays a dominant role in this
volatility transmission and especially after the introduction of the euro. It transmits
volatility among each of the four currencies and especially to those geographically
closer. The dominance of the mark in volatility spillovers was also reported in an
empirical study of Kearney and Patton (2000) within the EMS period. In this study we
provide evidence that the mark and the euro, the currency the former being replaced
by, still holds a dominant position in terms of volatility transmission even after the
introduction of the euro.
The second multivariate GARCH specification that is able to account for
volatility spillovers is the diagonal-BEKK model. The estimation results of the
diagonal-BEKK model for the pre- and the post- euro period are reported on Table 5.
Again an AR(3) and a random walk Diagonal-BEKK-MGARCH(1,1) for the pre- and
the post-euro periods, respectively, were sufficient to filter any remaining serial
correlation in the conditional mean specification.
The coefficients of matrices A and B indicate the innovations in each specific
market and the persistence or the rate of the decay of news, respectively. 24 All ann and
bnn estimated coefficients are significantly different from zero apart from the a11 for
the pre-euro period, which is insignificant at 5% level of significance. This means that
the impact of news in the euro(mark) do not affect that specific market in the pre-euro
period. However, not much attention has to be paid for the pre-euro period as the
diagnostic tests for the AR(3) Diagonal-BEKK-MGARCH(1,1) are not very
supportive. 25 The multivariate versions of the Ljung-Box statistic reject the null
hypothesis of no serial correlation in the standardized and squared standardized
residuals. Specifically, Hosking (1980) and Li and McLeod (1981) statistics reject the
null hypothesis of no serial correlation up to 30 lags at 5% level of significance. For
the post-euro period, the diagonal-BEKK seems to be well specified as there is no
evidence of multivariate serial correlation in the standardized and squared
standardized residuals according Hosking (1980) and Li and McLeod (1981)
statistics. Hence, the analysis of the diagonal BEKK model will be limited for the
post-euro period.
According to Table 6, all ann and bnn estimated coefficients in the post-euro
period are statistically significant at 1% level of significance. The significance of
estimated coefficients ann means that news/shocks in a specific market are of great
importance for future volatility in that specific market. The highest estimated
coefficient is a22, and a44, a11 and a33 rank in the second, third and fourth place
24
The coefficients a11, a22, a33 and a44 indicate the impact of news in the euro(mark), pound, yen and
the franc markets, respectively, whereas, the coefficients b11, b22, b33 and b44 indicate the persistence of
news.
25
Other specification of the diagonal-BEKK were applied but the AR(3)-Diagonal-BEKKMGARCH(1,1) was preferred by the AIC the BIC, SBC and Shibata information criteria and
overfitting approaches.
respectively. This means the British traders pay more attention to news arriving in the
pound market compared to Japanese traders pay to news arriving in the yen market.
The significantly high estimated coefficients bnn indicate the highly persistence
of news in these four currency markets. The highest estimated coefficient is b33, and
b11, b44 and b22 rank in the second, third and fourth place respectively. This means that
shocks in the yen market persist more than shocks in the pound market in terms of
future volatility. This is also in line with the empirical results obtained from the DCC
model where it was found that conditional variances are the greatest in the yen and the
lowest in the pound. 26 In addition, the lowest persistence is found in the pound market
(b22). This means that the pound become the most stable currency since the
introduction of the euro. This has several implications for its future development. For
instance, on these grounds alone, proponents of the UK adopting the euro as its
currency might need to revisit their opinion in the near future.
Figures 14, 15 and 16 which plot the conditional variances, covariances and
correlations, respectively, are almost identical with those obtained from the DCC
model (Figures 9, 10 and 6, respectively) hence, the analysis of these figures remains
the same for the BEKK specification.
5. Conclusion
The implementation of the European Union and especially the news of the
introduction of the euro and its adoption as a single currency brought many debates
about its future developments in terms of stability, spillovers etc. in the euro and the
global financial markets.
This paper examined exchange rate volatility comovements and spillovers of
four highly traded currencies, specifically the euro, the pound, the yen and the franc
against the dollar, for the period before and after the introduction of the euro. The
paper contributes to the literature of exchange rate volatility spillovers and
comovements in two ways. Firstly, it examines volatility comovements and spillovers
in both the pre- and the post-euro period and extends the post-euro period sample with
data up to 2007. Secondly, two flexible multivariate models were applied that allowed
conditional variances, covariances and correlations to be time-varying rather than
26
See Figure 9
relying on the restrictive assumption of constant correlations. That assumption was
rejected by the data.
The DCC model, based on diagnostic checking, was preferred against the
alternative BEKK model. The latter specification performed well in the post-euro
period but not equally well in the pre-euro period. According to the former, it was
found evidence of significant conditional variances, covariances and correlations for
each sub-sample which have declined since the introduction of the euro. In other
words, volatility comovements and spillovers between the euro, pound, yen and the
franc have a smaller impact in the post-euro period. This means that the launch of the
euro coincided with greater stability in the euro and the global financial markets,
which is in line with Mundell’s (1998) claim of a decrease in volatility in the
expansion of the euro due to the European’s Central Bank commission for price
stability. Whether the volatility of the four currencies reflects the underlying volatility
in the economic fundamentals of these regions is definitely an important avenue for
future research.
Moreover, the results showed that the euro(mark) is the dominant currency in
volatility transmission, as its volatility significantly affects the volatility expectations
of the franc, pound and the yen. This is in line with the results of Nikkinen et al.,
(2006). The authors having daily data for the period between 2 January 2001 and 29
September 2003, found that the implied volatility of the euro significantly affects the
volatility expectations of the pound and the franc. In this paper, not only additional
evidence of the dominant position of the euro volatility transmission to the yen is
provided, but it is also supported to a more extended sample. This fact has serious
implications for portfolio diversification and risk management.
In addition, it was found that the pound is the least volatile among these four
currencies since the launch of the euro. As already pointed out, exchange rate
volatility is just one aspect of the many, which need to be considered before making
any conclusive argument as to whether UK should replace pound with euro or not. On
these ground alone, this suggest that the UK may not be well advised to adopt the
euro in the near future. The higher volatility of euro than BP has obviously important
implications for many other financial markets and this particular finding definitely
requires more research along this dimension before economists converge to any
specific decision on this highly debated issue.
The analysis in this paper was based on the assumption that shocks or news in
one market affects that specific market and other markets symmetrically. This means
that the impact of negative and positive shocks of the same amplitude to exchange
rate volatility is the same. It would be of interest for further research to investigate
whether the impact of news could have an asymmetric impact on exchange rate
volatility.
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Mundell, R. A., 1998 “What the Euro Means for the Dollar and the International
Monetary System.” Atlantic Economic Journal, 26, pp. 227-237.
•
Mussa, M., 2000 “The Relationship between the Euro and the Dollar.” Journal of
Policy Modeling, 22, pp. 369-379.
•
Nikkinen, J., Sahlstrom P., and Vahamaa, S., 2006 “Implied Volatility Linkages
among Major European Currencies.” International Financial Markets, Institutions
and Money, 16, pp. 87-103.
•
Sheady, E., 1997. “Correlation in International Equity and Currency Markets: A
Risk Adjusted Perspective.” Working paper 17, Macquarie University.
•
Speight A. E. H., and McMillan, D. G., 2001 “Volatility spillovers in East
European Black-Market Exchange Rates.” Journal of International Money and
Finance, 20, pp. 367-378.
•
Tse, Y., 2000. “A Test for Constant Correlations in a Multivariate GARCH
Model.” Journal of Econometrics, 98, pp. 107-127.
•
Tse, Y., and Tsui, A., 2002. “A Multivariate GARCH Model with Time-Varying
Correlations.” Journal of Business and Economic Statistics, 20, pp. 351-362.
•
Wan, J.-Y., and Kao, C.-W. 2008 “The Euro and Pound Volatility Dynamics: An
Investigation from Conditional Jump Process.” Research in International Business
and Finance, 22, pp.193-207.
List of Tables
Table 1: Descriptive Statistics of Returns, Pre- & Post- Period
Pre-Euro period: 05/01/1990 – 31/12/1998
DM
GBP
JPY
CHF
(EURO)
Mean
Standard
Deviation
Variance
Skewnes
Excess
Kurtosis
Normality
Test (JB)
Q(10)
Q2(10)
ARCH(5)
Post-Euro period: 05/01/1990 – 31/12/1998
EURO
GBP
JPY
CHF
-5.8975e-006
-9.9118e-006
-0.00010766
-5.3067e-005
-9.4285e-005
-7.9984e-005
-3.8447e-006
-8.3327e-005
0.6649
0.6071
0.7356
0.7353
0.5949
0.5027
0.6163
0.6373
0.4421
0.0606
[0.2377]
2.2534
[0.0000]**
482.53
[0.0000]**
17.6864
[0.0605]
200.561
[0.0000]**
17.048
[0.0000]**
0.3686
0.2384
[0.0000]**
3.3374
[0.0000]**
1076.9
[0.0000]**
37.1312
[0.0001]**
286.728
[0.0000]**
25.917
[0.0000]**
0.5410
-0.7894
[0.0000]**
6.1570
[0.0000]**
3828.0
[0.0000]**
18.0518
[0.0541]
252.008
[0.0000]**
24.128
[0.0000]**
0.5406
-0.1553
[0.0025]**
1.8936
[0.0000]**
348.88
[0.0000]**
16.8692
[0.0773]
140.878
[0.0000]**
13.638
[0.0000]**
0.3539
-0.18675
[0.0003]**
1.2247
[0.0000]**
155.19
[0.0000]**
2.93054
[0.9830]
50.2921
[0.0000]**
4.2426
[0.0008]**
0.2527
-0.023314
[0.6498]
0.8476
[0.0000]**
68.219
[0.0000]**
10.6938
[0.3819]
64.9244
[0.0000]**
4.1899
[0.0009]**
0.3799
-0.28212
[0.0000]**
1.5772
[0.0000]**
265.62
[0.0000]**
5.00038
[0.8912]
40.3288
[0.0000]**
4.0884
[0.0011]**
0.4061
-0.19405
[0.0002]**
1.0460
[0.0000]**
117.83
[0.0000]**
8.22549
[0.6068]
23.3099
[0.0097]**
2.1723
[0.0546]
Notes: [] denote p-values. Q(10) and Q2(10) is the Ljung-Box statistic for serial correlation in raw
series and squared series, respectively. * 5% significant; ** 1% significant.
Table 2: Unconditional Correlations, Pre- & Post- Euro Period
Pre-Euro period: 05/01/1990 – 31/12/1998 (Observations: 2274)
DLEUR(DM)
DLGBP
DLJPY
1.0000
DLEUR(DM)
0.7188
1.0000
DLGBP
0.4992
0.3514
1.0000
DLJPY
0.9209
0.6825
0.5012
DLCHF
Post-Euro period: 05/01/1999 – 31/12/2007 (Observations: 2272)
DLEUR
DLGBP
DLJPY
1.0000
DLEUR
0.6960
1.0000
DLGBP
0.3440
0.3143
1.0000
DLJPY
0.9400
0.6804
0.3934
DLCHF
DLCHF
1.0000
DLCHF
1.0000
Table 3: Estimation Results of CCC and DCC models – Full sample (05/01/90-31/12/07)
CCC-MGARCH(1,1)
DCC-MGARCH(1,1)
Panel A: 1-step, univariate GARCH estimates and univariate diagnostic tests
EUR
GBP
JPY
CHF
EUR
GBP
JPY
-0.0001
-0.0001
-0.00001
-0.0001
-0.0001
-0.0001
-0.00001
Constant
(8.351e-005)
(7.79e-005) (9.24e-005) (9.69e-005) (8.351e-005)
(7.79e-005)
(9.24e-005)
(m)
CHF
[-0.7811]
[-1.640]
[-0.0511]
[-1.002]
[-0.7811]
[-1.640]
[-0.0511]
-0.0001
(9.69e-005)
[-1.002]
Constant (v)
0.0295
(0.012308)
[2.400]*
0.0026
(0.0008)
[3.145]**
0.0079
(0.0030)
[2.634]**
0.0048
(0.0022)
[2.170]*
0.0295
(0.012308)
[2.400]*
0.0026
(0.0008)
[3.145]**
0.0079
(0.0030)
[2.634]**
0.0048
(0.0022)
[2.170]*
α
0.0759
(0.010386)
[7.304]**
0.0379
(0.0059)
[6.405]**
0.0455
(0.0093)
[4.920]**
0.0275
(0.0053)
[5.215]**
0.0759
(0.010386)
[7.304]**
0.0379
(0.0059)
[6.405]**
0.0455
(0.0093)
[4.920]**
0.0275
(0.0053)
[5.215]**
β
0.8542
(0.033955)
[25.16]**
0.9535
(0.0067)
[143.4]**
0.9374
(0.0136)
[68.71]**
0.9624
(0.0084)
[114.5]**
0.8542
(0.033955)
[25.16]**
0.9535
(0.0067)
[143.4]**
0.9374
(0.0136)
[68.71]**
0.9624
(0.0084)
[114.5]**
Q(30)
42.8590
{0.0603}
39.3085
{0.1190}
25.2724
{0.7117}
18.0968
{0.9570}
40.3202
{0.0988}
40.2744
{0.0997}
27.0350
{0.6214}
19.8495
{0.9204}
Q2(30)
0.0589
{1.0000}
5.3959
{0.9999}
32.9728
{0.3237}
0.2337
{1.0000}
0.0731
[1.0000}
74.6051
{0.0000}**
29.7339
{0.4793}
0.3809
{1.0000}
Panel B: 2-step, correlation estimates and multivariate diagnostic tests
0.5985 (0.0118) [50.92]**
0.5349 (0.0397) [13.46]**
ρGBPEUR
0.3115 (0.0195) [15.98]**
0.0900 (0.0670) [1.343]
ρJPYEUR
0.8193 (0.0040) [203.7]**
0.7328 (0.0286) [25.65]**
ρCHFEUR
0.2960 (0.0140) [21.28]**
0.0826 (0.0666) [1.241]
ρJPYGBP
0.6553 (0.0086) [76.18]**
0.6105 (0.0357) [17.08]**
ρCHFGBP
0.3967 (0.0141) [28.24]**
0.1243 (0.0694) [1.791]
ρCHFJPY
0.0286 (0.0023) [12.57]**
α
0.9645 (0.0029) [334.5]**
β
4.2178 (0.1128) [37.38]**
4.3685 (0.1250) [34.96]**
df
70684.8
71035.9
Log-Lik
-31.0771
-31.2306
AIC
31.0333
-31.1840
SBC
-31.0617
-31.2142
HQC
-31.0772
-31.2307
Shibata
3874.93 {0.0000}**
LMC(6)
12.7051 {0.0480}*
E-S Test(5)
477.799 {0.4940}
500.722 {0.2281}
H(30)
496.682 {0.2683}
420.647 {0.9721}
H2(30)
477.894 {0.4928}
500.765 {0.2277}
Li-McL(30)
496.841 {0.2666}
421.074 {0.9712}
Li-McL2(30)
Notes: LMC and E-S stands for the statistic for constant correlations of Tse (2000) and Engle and
Sheppard (2001), respectively. Q() and Q2() are the Ljung-Box portmanteau tests statistic for serial
correlation in the univariate standardized and squared standardized residuals, respectively. H(), H2()
and Li-McL(), Li-McL2() are the multivariate versions of Ljung-Box statistic of Hosking (1980) and Li
and McLeod (1981), respectively. (), [] and {} denote standard errors, t-values and p-values,
respectively. * 5% significant; ** 1% significant.
Table 4: Estimation Results of DCC model – Pre- & Post- Euro Period
Pre-Euro Period (05/01/90-31/12/98) 2274 Obs.
Post-Euro Period (05/01/99-31/12/07) 2272 Obs.
Panel A: 1-step, univariate GARCH estimates and univariate diagnostic tests
EUR
GBP
JPY
CHF
EUR
GBP
JPY
CHF
-0.000017
-0.0001
-0.00003
-0.00005
-0.0002
-0.0001
0.00002
-0.0001
Constant
(0.00012)
(0.0001)
(0.00013)
(0.00014)
(0.0001)
(0.0001)
(0.00012)
(0.0001)
(m)
[-0.1361]
[-1.102]
[-0.2419]
[-0.3640]
[-1.651]
[-1.240]
[0.1772]
[-0.9591]
Constant (v)
0.0062
(0.0036)
[1.706]
0.0023
(0.0011)
[2.026]*
0.0110
(0.0062)
[1.773]
0.0135
(0.0064)
[2.123]*
0.0007
(0.0006)
[1.18]
0.0041
(0.0019)
[2.148]*
0.0085
(0.0035)
[2.446]*
0.0009
(0.0009)
[0.9724]
α
0.0426
(0.0113)
[3.776]**
0.0401
(0.0107)
[3.742]**
0.0675
(0.0206)
[3.272]**
0.0412
(0.0109)
[3.768]**
0.0211
(0.0054)
[3.918]**
0.0344
(0.0078)
[4.421]**
0.0305
(0.0085)
[3.603]**
0.0148
(0.0050)
[2.974]**
β
0.9433
(0.0177)
[53.43]**
0.9538
(0.0125)
[76.50]**
0.9137
(0.0285)
[32.03]**
0.9334
(0.0197)
[47.47]**
0.9769
(0.0059)
[166.0]**
0.9497
(0.0121)
[78.26]**
0.9469
(0.0136)
(69.74)**
0.9830
(0.0063)
[156.9]**
Q(30)
33.7972
{0.2890}
27.4653
{0.5988}
38.0391
{0.1488}
30.7953
{0.4255}
60.1044
{0.0009}**
38.7183
{0.1322}
17.7635
{0.9622}
26.9527
{0.6258}
Q2(30)
55.7948
{0.0029}**
17.0378
{0.9721}
28.7945
{0.5284}
25.4660
{0.7021}
21.6422
{0.8667}
25.3339
{0.7087}
18.7981
{0.9441}
84.8199
{0.000}**
Panel B: 2-step, correlation estimates and multivariate diagnostic tests
0.8053 (0.0577) [13.97]**
0.6656 (0.0407) [16.36]**
ρGBPEUR
0.3190 (0.1147) [2.781]**
0.4224 (0.0671) [6.294]**
ρJPYEUR
0.8515 (0.0389) [21.90]**
0.9496 (0.0073) [130.7]**
ρCHFEUR
0.3030 (0.1255) [2.413]*
0.3342 (0.0714) [4.679]**
ρJPYGBP
0.7490 (0.0613) [12.22]**
0.6632 (0.0416) [15.95]**
ρCHFGBP
0.2855 (0.1175) [2.429]*
0.4521 (0.0662) [6.827]**
ρCHFJPY
0.0217 (0.0022) [9.605]**
0.0188 (0.0027) [7.007]**
α
0.9762 (0.0027) [357.8]**
0.9758 (0.0042) [234.6]**
β
6.8754 (0.4196) [16.38]**
7.8911 (0.5327) [14.81]**
df
36944.3
37973.1
Log-Lik
-32.4603
-33.4050
AIC
-32.3671
-33.3420
SBC
-32.4263
-33.3820
HQC
-32.4608
-33.4052
Shibata
506.319 {0.1706}
520.023 {0.1005}
H(30)
501.884 {0.2173}
522.162 {0.0796}
H2(30)
506.081 {0.1725}
519.968 {0.1008}
Li-McL(30)
502.657 {0.2102}
522.419 {0.0784}
Li-McL2(30)
Notes: Q() and Q2() are the Ljung-Box portmanteau tests statistic for serial correlation in the univariate
standardized and squared standardized residuals, respectively. H(), H2() and Li-McL(), Li-McL2() are
the multivariate versions of Ljung-Box statistic of Hosking (1980) and Li and McLeod (1981),
respectively. (), [] and {} denote standard errors, t-values and p-values, respectively. * 5% significant;
** 1% significant.
Table 5: Test of significant increases or decreases in correlations
Pre-Euro
correlations
Post-Euro
correlations
Ho: no increase Ho: no decrease
in correlations in correlations
0.8053
0.6656
-10.4677**
ρGBPEUR
0.3190
0.4224
-4.0453**
ρJPYEUR
0.8515
0.9496
-19.0721**
ρCHFEUR
0.3030
0.3342
-1.1700
ρJPYGBP
0.7490
0.6632
-5.8003**
ρCHFGBP
0.2855
0.4521
-6.5249**
ρCHFJPY
Notes: Correlations are obtained from the DCC model of Table 4. The 1% and 5% critical values for a
one-sided test of the null are −2.32 and −1.64, respectively. ** and * indicate statistical significance at
the 1% and 5% levels, respectively.
Table 6: Estimation Results of BEKK model – Pre- & Post- Euro Period
Pre-Euro (05/01/90-31/12/98)
Post-Euro (05/01/99-31/12/07)
0.00001 (0.0001) [0.1429]
-0.0001 (0.0001)
[-1.107]
Constant11 (m)
-0.00001 (0.0001) [-0.0936]
-0.0002 (9.4513e-005) [-1.605]
Constant22 (m)
0.0002 (0.0001)
[1.421]
0.0001 (0.0001) [0.9979]
Constant33 (m)
-0.00003 (0.0001)
[-0.320]
0.0001 (0.0001) [0.4991]
Constant44 (m)
0.0004 (0.0009) [0.4990]
0.0005 (7.2781e-005) [7.081]**
c11
0.0003 (0.0006 ) [0.5561]
0.0004 (9.6308e-005) [3.858]**
c12
0.0002 (0.0003) [0.8210]
0.0002 (0.0002)
[1.187]
c13
0.0004 (0.0010) [0.4095]
0.0005 (7.1340e-005) [7.317]**
c14
0.0002 (0.0004) [0.5496]
0.0004 (8.3090e-005) [4.404]**
c22
0.00001 (9.0835e-005) [0.1199]
0.00004 (3.8244e-005) [1.013]
c23
0.0001 (6.7609e-005) [0.6601]
0.00002 (1.4202e-005) [1.651]
c24
0.0005 (0.00045)
[1.1450]
0.0004 (0.0006)
[0.656]
c33
0.000004 (3.1327e-005) [0.1378]
0.00003 (1.449e-005) [2.338]*
c34
0.0002
(0.0005)
[0.4169]
0.0002 (2.7486e-005) [5.886]**
c44
0.1726 (0.0923) [1.870]
0.1609 (0.0113) [14.30]**
a11
0.1711 (0.0834) [2.053]*
0.1643 (0.0150) [10.94]**
a22
0.1900 (0.0496) [3.835]**
0.1372 (0.0398) [3.447]**
a33
0.1746 (0.0847) [2.061]*
0.1627 (0.0108) [15.05]**
a44
0.9831 (0.0241) [40.80]**
0.9834 (0.0023) [426.8]**
b11
0.9832 (0.0214) [45.97]**
0.9820 (0.0040) [245.1]**
b22
0.9790 (0.0148) [65.95]**
0.9882 (0.0117) [84.19]**
b33
0.9836 (0.0238) [41.29]**
0.9833 (0.0022) [457.9]**
b44
6.4676 (0.5707) [11.33]**
7.7651 (0.5843) [13.29]**
df
36962.3
37985.4
Log-Lik
-32.4778
-33.4176
AIC
-32.3896
-33.3596
SBC
-32.4457
-33.3965
HQC
-32.4783
-33.4178
Shibata
25.6366 {0.6935}
61.4911 {0.0006}**
Q11(30)
44.3966 {0.0439}*
39.5251 {0.1144}
Q22(30)
40.4940 {0.0957}
18.9052 {0.9420}
Q33(30)
32.5123 {0.3441}
25.7848 {0.6861}
Q44(30)
75.9176 {0.00001}**
21.6559 {0.8663}
Q211(30)
22.4316 {0.8379}
Q222(30)
26.2461 {0.6625}
29.8950 {0.4710}
21.7755 {0.8621}
Q233(30)
26.4728 {0.6508}
Q244(30)
69.0051 {0.0001}**
529.242 {0.0489}*
524.188 {0.0799}
H(30)
535.180 {0.0358}*
507.338 {0.1707}
H2(30)
529.010 {0.0497}*
524.114 {0.0802}
Li-McL(30)
536.231 {0.0334}*
507.612 {0.1685}
Li-McL2(30)
Notes: Q() and Q2() are the Ljung-Box portmanteau tests statistic for serial correlation in the univariate
standardized and squared standardized residuals, respectively. H(), H2() and Li-McL(), Li-McL2() are
the multivariate versions of Ljung-Box statistic of Hosking (1980) and Li and McLeod (1981),
respectively. (), [] and {} denote standard errors, t-values and p-values, respectively. * 5% significant;
** 1% significant.
List of Figures
Figure 1: Pre- & Post Euro Exchange Rates
DMEURO
JPY
GBP
CHF
1.8
1.6
1.4
1990
1991
DMEURO
JPY
1.2
1992
1993
1994
1995
1996
1997
1998
2001
2002
2003
2004
2005
2006
2007
GBP
CHF
1.0
0.8
1999
2000
Figure 2: Returns & ACF – Pre-Euro Period
0.025
1
DLDMEURO
0.000
ACF-DLDMEURO
0
-0.025
1990
0.050
1992
1994
1996
1998
0
1
DLGBP
5
10
15
20
10
15
20
10
15
20
10
15
20
ACF-DLGBP
0.025
0
0.000
-0.025
1990
0.05
1992
1994
1996
1998
0
1
DLJPY
0.00
5
ACF-DLJPY
0
-0.05
1990
0.025
1992
1994
1996
1998
0
1
DLCHF
0.000
5
ACF-DLCHF
0
-0.025
1990
1992
1994
1996
1998
0
5
Figure 3: Returns & ACF of Retruns– Post-Euro Period
0.025
1
DLDMEURO
0.000
ACF-DLDMEURO
0
-0.025
2000
2002
2004
2006
1
DLGBP
0.02
0
0.00
5
10
15
20
10
15
20
10
15
20
10
15
20
ACF-DLGBP
0
-0.02
2000
2002
2004
2006
1
DLJPY
0.02
0
0.00
5
ACF-DLJPY
0
-0.02
2000
2002
2004
2006
DLCHF
0.02
0
1
0.00
5
ACF-DLCHF
0
-0.02
2000
2002
2004
2006
0
5
Figure 4: Conditional Correlations of CCC Vs DCC model – Whole Sample (05/01/90-31/12/07)
CORR_DLDMEURO_DLGBP-CCC
CORR_DLDMEURO_DLGBP-DCC
CORR_DLDMEURO_DLJPY-CCC
CORR_DLDMEURO_DLJPY-DCC
0.5
0.5
0.0
0.0
1990
1.00
1995
2000
CORR_DLDMEURO_DLCHF-CCC
2005
1990
CORR_DLDMEURO_DLCHF-DCC
1995
CORR_DLGBP_DLJPY-CCC
2000
2005
CORR_DLGBP_DLJPY-DCC
0.5
0.75
0.50
0.0
0.25
1990
1.00
1995
CORR_DLGBP_DLCHF-CCC
2000
2005
1990
CORR_DLGBP_DLCHF-DCC
1995
CORR_DLJPY_DLCHF-CCC
2000
2005
CORR_DLJPY_DLCHF-DCC
0.5
0.75
0.50
0.0
0.25
1990
1995
2000
2005
1990
1995
2000
2005
Figure 5: Conditional Correlations of CCC Vs DCC model – Pre-Euro (05/01/90-31/12/98)
1.00
CORR_DLDMEURO_DLGBP-DCC
CORR_DLDMEURO_DLGBP-CCC
0.75
CORR_DLDMEURO_DLJPY-DCC
CORR_DLDMEURO_DLJPY-CCC
0.75
0.50
0.50
0.25
0.25
1990
1992
1994
1996
CORR_DLDMEURO_DLCHF-DCC
1998
CORR_DLDMEURO_DLCHF-CCC
1990
0.75
1992
1994
CORR_DLGBP_DLJPY-DCC
1996
1998
CORR_DLGBP_DLJPY-CCC
0.50
0.9
0.25
0.00
0.8
1990
1.00
1992
1994
CORR_DLGBP_DLCHF-DCC
1996
1998
1990
CORR_DLGBP_DLCHF-CCC
1992
1994
CORR_DLJPY_DLCHF-DCC
1996
1998
CORR_DLJPY_DLCHF-CCC
0.75
0.75
0.50
0.50
0.25
0.25
1990
1992
1994
1996
1998
1990
1992
1994
1996
1998
Figure 6: Conditional Correlations of CCC Vs DCC model – Post-Euro (05/01/99-31/12/07)
CORR_DLDMEURO_DLGBP-DCC
CORR_DLDMEURO_DLGBP-CCC
0.75
CORR_DLDMEURO_DLJPY-DCC
CORR_DLDMEURO_DLJPY-CCC
0.8
0.50
0.25
0.6
0.00
2000
1.00
2002
2004
CORR_DLDMEURO_DLCHF-DCC
2006
CORR_DLDMEURO_DLCHF-CCC
2000
0.75
0.95
0.50
0.90
0.25
0.85
0.00
2000
2002
CORR_DLGBP_DLCHF-DCC
2004
2006
CORR_DLGBP_DLCHF-CCC
CORR_DLGBP_DLJPY-DCC
2000
0.75
2002
2002
CORR_DLJPY_DLCHF-DCC
2004
2006
CORR_DLGBP_DLJPY-CCC
2004
2006
CORR_DLJPY_DLCHF-CCC
0.8
0.50
0.25
0.6
0.00
2000
2002
2004
2006
2000
2002
2004
2006
Figure 7: Conditional Variances of DCC model – Pre-Euro (05/01/90-31/12/98)
0.00015
CondVar_DLDMEURO-DCC
CondVar_DLGBP-DCC
0.00015
0.0001
0.0001
5.00e-5
5.00e-5
1990
0.0005
1992
1994
1996
1998
1990
CondVar_DLJPY-DCC
1992
1994
1996
1998
1996
1998
CondVar_DLCHF-DCC
0.0004
0.00015
0.0003
0.0001
0.0002
0.0001
5.00e-5
1990
1992
1994
1996
1998
1990
1992
1994
Figure 8: Conditional Covariances of DCC model – Pre-Euro Period (05/01/90-31/12/98)
COV_DLDMEURO_DLGBP-DCC
COV_DLDMEURO_DLJPY-DCC
0.0001
0.0001
5.00e-5
1990
0.00015
5.00e-5
1992
1994
1996
1998
1990
7.5e-5
COV_DLDMEURO_DLCHF-DCC
1992
1994
1996
1998
1996
1998
1996
1998
COV_DLGBP_DLJPY-DCC
5.0e-5
0.0001
2.5e-5
5.00e-5
0.0
1990
0.0001
1992
1994
1996
1998
1990
0.00015
COV_DLGBP_DLCHF-DCC
1992
1994
COV_DLJPY_DLCHF-DCC
0.0001
5.00e-5
5.00e-5
1990
1992
1994
1996
1998
1990
1992
1994
Figure 9: Conditional Variances of DCC model – Post-Euro Period (05/01/99-31/12/07)
8e-5
CondVar_DLDMEURO-DCC
6e-5
CondVar_DLGBP-DCC
5e-5
6e-5
4e-5
4e-5
3e-5
2e-5
2e-5
2000
2002
2004
2006
2000
CondVar_DLJPY-DCC
2002
2004
2006
2004
2006
CondVar_DLCHF-DCC
8e-5
6e-5
6e-5
4e-5
4e-5
2e-5
2000
2002
2004
2006
2000
2002
Figure 10: Conditional Covariances of DCC model – Post-Euro Period (05/01/99-31/12/07)
4e-5
COV_DLDMEURO_DLGBP-DCC
COV_DLDMEURO_DLJPY-DCC
4e-5
2e-5
2e-5
0
2000
2002
2004
2006
2000
3e-5
COV_DLDMEURO_DLCHF-DCC
2002
2004
2006
2004
2006
2004
2006
COV_DLGBP_DLJPY-DCC
2e-5
5.0e-5
1e-5
2.5e-5
0
2000
2002
2004
2006
2000
4e-5
COV_DLGBP_DLCHF-DCC
2002
COV_DLJPY_DLCHF-DCC
4e-5
2e-5
2e-5
0
2000
2002
2004
2006
2000
2002
Figure 11: Conditional Variances of BEKK model – Pre-Euro (05/01/90-31/12/98)
0.00013
0.00015
CondVar_DLDMEURO-BEKK
0.0001
CondVar_DLGBP-BEKK
0.0001
7.5e-5
5.00e-5
5.0e-5
2.5e-5
1990
1992
1994
1996
1998
1990
CondVar_DLJPY-BEKK
1992
1994
1996
1998
1996
1998
CondVar_DLCHF-BEKK
0.00015
0.0003
0.0002
0.0001
0.0001
5.00e-5
1990
1992
1994
1996
1998
1990
1992
1994
Figure 12: Conditional Covariances of BEKK model – Pre-Euro (05/01/90-31/12/98)
0.0001
COV_DLDMEURO_DLGBP-BEKK
0.0001
5.00e-5
1990
COV_DLDMEURO_DLJPY-BEKK
5.00e-5
1992
1994
1996
1998
COV_DLDMEURO_DLCHF-BEKK
1990
5.0e-5
1992
1994
1996
1998
1996
1998
1996
1998
COV_DLGBP_DLJPY-BEKK
0.0001
2.5e-5
5.00e-5
0.0
1990
0.0001
1992
1994
1996
1998
1990
COV_DLGBP_DLCHF-BEKK
1992
1994
COV_DLJPY_DLCHF-BEKK
0.0001
7.5e-5
5.0e-5
5.00e-5
2.5e-5
1990
1992
1994
1996
1998
1990
1992
1994
Figure 13: Conditional Correlations of BEKK model – Pre-Euro (05/01/90-31/12/98)
1.00
CORR_DLDMEURO_DLGBP-BEKK
CORR_DLDMEURO_DLJPY-BEKK
0.75
0.75
0.50
0.50
0.25
0.25
1990
1.0
1992
1994
1996
1998
1990
0.75
CORR_DLDMEURO_DLCHF-BEKK
1992
1994
1996
1998
1996
1998
1996
1998
CORR_DLGBP_DLJPY-BEKK
0.9
0.25
0.8
1990
1.00
1992
1994
1996
1998
CORR_DLGBP_DLCHF-BEKK
1990
0.75
1992
1994
CORR_DLJPY_DLCHF-BEKK
0.75
0.50
0.50
0.25
0.25
1990
1992
1994
1996
1998
1990
1992
1994
Figure 14: Conditional Variances of BEKK model – Post-Euro (05/01/99-31/12/07)
8e-5
CondVar_DLDMEURO-BEKK
6e-5
CondVar_DLGBP-BEKK
5e-5
6e-5
4e-5
4e-5
3e-5
2e-5
2e-5
2000
8e-5
2002
2004
2006
2000
8e-5
CondVar_DLJPY-BEKK
6e-5
2002
2004
2006
2004
2006
CondVar_DLCHF-BEKK
6e-5
4e-5
4e-5
2e-5
2e-5
2000
2002
2004
2006
2000
2002
Figure 15: Conditional Covariances of BEKK model – Post-Euro (05/01/99-31/12/07)
4e-5
COV_DLDMEURO_DLGBP-BEKK
COV_DLDMEURO_DLJPY-BEKK
4e-5
2e-5
2e-5
0
2000
7.5e-5
2002
2004
2006
2000
3e-5
COV_DLDMEURO_DLCHF-BEKK
2002
2004
2006
2004
2006
2004
2006
COV_DLGBP_DLJPY-BEKK
2e-5
5.0e-5
1e-5
0
2.5e-5
2000
2002
2004
2006
COV_DLGBP_DLCHF-BEKK
2000
4e-5
2002
COV_DLJPY_DLCHF-BEKK
4e-5
2e-5
2e-5
0
2000
2002
2004
2006
2000
2002
Figure 16: Conditional Correlations of BEKK model – Post-Euro (05/01/99-31/12/07)
0.75
CORR_DLDMEURO_DLGBP-BEKK
CORR_DLDMEURO_DLJPY-BEKK
0.8
0.50
0.6
0.25
0.4
0.00
2000
1.0
2002
2004
2006
2000
0.75
CORR_DLDMEURO_DLCHF-BEKK
2002
2004
2006
2004
2006
2004
2006
CORR_DLGBP_DLJPY-BEKK
0.50
0.9
0.25
0.00
0.8
2000
2002
2004
2006
2000
0.75
CORR_DLGBP_DLCHF-BEKK
0.8
2002
CORR_DLJPY_DLCHF-BEKK
0.50
0.25
0.6
0.00
2000
2002
2004
2006
2000
2002