als appear to be significantly autocorrelated. Moreover, the Lagrange Multiplier Ž
. LM test documents that ARCH is present in the standardized residuals. Overall,
our findings imply that the simple multivariate model does not address the issue of heteroscedasticity adequately, which points towards a heteroscedastic model, e.g. a
multivariate GARCH model. Another benefit of employing the more involved multivariate GARCH specification is that it will enable us to look further into the
persistency of the announcement shocks.
For illustrative purposes, the properties of the daily excess returns of the 3- and 10-year bonds are shown graphically in Fig. 1. The graphs also suggest that a
model including heteroscedasticity is required to describe the evolution of the bond excess returns as there are signs of volatility clustering. Setting up the
multivariate GARCH model is the subject of the following section.
3. The multivariate model
In the previous section, we have found evidence that any potential multivariate model of the announcement effects of the covariance structure of the government
bond returns should allow for heteroscedasticity. Consequently, we suggest to apply a multivariate GARCH model. No specific multivariate GARCH specifica-
tion is always preferred above all other specifications and, apart from Kroner and
Ž .
Ng 1998 , there are hardly any comparative studies of multivariate GARCH models. Hence, it remains to decide which particular multivariate GARCH model
to use, and to this end, we list our model choice criteria: Firstly, the total number of parameters must be kept at a minimum, which is even more important here
because we introduce additional parameters to account for the announcements. Secondly, the model must guarantee that the conditional covariance matrix is
positive definite. Thirdly, the model must be formulated such that we can make interesting conclusions as to the impacts of macroeconomic announcements on the
covariance structure.
The limitation that the number of parameters should be fairly small restricts the Ž
. available alternatives to the diagonal BEKK model, cf. Bollerslev et al. 1988 , the
Ž .
Ž .
Constant Conditional Correlations model CCORR , cf. Bollerslev 1990 and the Ž
. Factor-ARCH model, cf. Engle et al. 1990 . We assume that the reader is familiar
with these models. It is not a straightforward problem which factors to apply in the factor ARCH-model, and there is no consensus as to how to define these. Hence,
we remove this specification from our shortlist.
7
7
A previous version of this paper applied the factor-ARCH model, more hereon follows.
The main objection to the CCORR model is that large simultaneous shocks of opposite signs to two assets increase their conditional covariance. This objection is
weakened by the fact that it is not likely that large positive and large negative shocks occur simultaneously for bonds of different maturities. The disadvantage of
the diagonal BEKK model is that it still involves many parameters compared to
Ž Ž
. the CCORR model. In fact, in the simplest setup GARCH 1,1 and no announce-
. ment effects , there are 108 parameters in the diagonal BEKK model, compared to
33 in the CCORR model. Both the CCORR model and the diagonal BEKK model, Ž
. appropriately extended, meet the third criterion. Moreover, Jeffrey 1998 docu-
Ž .
ments that the forward rate volatility and, thus, the interest rate volatility varies with the maturity, this feature is also accommodated by both specifications.
The above indicates that we might prefer the CCORR model extended to accommodate various differences between announcement and non-announcement
days. In the CCORR model, the conditional variances evolve according to GARCH processes. The conditional correlations are assumed to be constant, which
explains the phrase CCORR. Furthermore, the conditional covariances are propor- tional to the product of the corresponding volatilities.
The details of how the CCORR model is extended to accommodate announce- ment effects is presented in Section 3.2, but first the specification of the condi-
tional mean equation is put forward in Section 3.1. 3.1. Conditional means
We consider the excess returns of N different government bonds, R for i s 1,
i t
. . . , N, and the conditional mean vector of R is denoted m :
t t
A
R s m q 1 q d I ´ , 3
Ž .
i t i t
i t i t
where ´ is the vector of error terms and I
A
is an indicator function for
t t
Ž announcement days we will return to a discussion of the parameter d as well as
i
. the distribution of ´ in the following section . When modeling m we have to
t t
ensure that the error terms are serially uncorrelated. Because our main interest lies in how the conditional covariance matrix evolves over time, we make the
simplifying assumption that R
evolves according to a vector autoregressive
t
Ž Ž ..
process of order one VAR 1 where a level effect for announcement days is
included, that is:
m F q F R
q F
A
I
A
. 4
Ž .
t 1
t I1 t
F and F
A
are N = 1 vectors, and F is an N = N matrix. The conditional mean
1
Ž .
Ž . specification is in line with Jones et al. 1998 who consider an AR 1 process
Ž . combined with a level effect for announcement days, i.e. Eq. 4 where F
is
1
assumed to be diagonal.
3.2. Conditional coÕariance matrix In this section, we describe how the conditional covariance matrix evolves over
time. Firstly, the conditional variance of R is assumed to be greater on
i t
Ž .
Ž . announcement days by 1 q d , cf. Eq. 3 . The conditional covariance matrix of
i
the vector of error terms, ´ , is denoted H and is described by an appropriately
t t
Ž
extended version of the CCORR model. The diagonal elements of H i.e. the
t
. Ž
. conditional variances are assumed to evolve according to extended GARCH 1,1
processes: h
s s q f q f
A
I
A
q f
y
I
A
I
y
´
2
q w h ,
5
Ž .
Ž .
i i , t i
i i
ty1 i
ty1 i , ty1 i , ty1
i i i , ty1
where I
-
s 1 if ´ - 0 and 0 else. To ensure that the conditional variance is
i t i t
strictly positive, the following constraints are imposed: s 0,
f ,f q f
A
,f q f
A
q f
y
,w ,G 0. 6
Ž .
i i
i i
i i
i i
Ž . Ž
. In Eq. 5 , we follow Li and Engle 1998 and allow positive and negative
announcement shocks to have different implications on the future conditional Ž .
volatility, in some sense including a leverage effect. The specification in Eq. 5 is, Ž
. in fact, a version of the Glosten et al. 1993 GARCH model, where we have
included announcement effects. There is not a unique way of measuring volatility persistence; however, one
simple and often used metric is the sum of the GARCH parameters. The Ž .
specification in Eq. 5 permits differences in the persistence on announcement and non-announcement days in that the sum of the GARCH parameters is greater
A
Ž
A y
. Ž
. on announcement days by f
f q f for positive negative announcement
i i
i
shocks. When the market incorporates the information related to the announce- Ž
.
A
ments faster slower
than other kinds of information, the parameter f is
i
Ž .
negative positive . Ž
.
The off-diagonal elements of H i.e. the conditional covariances are described
t
by:
A A
h s r
1 q r I h
h ,
7
Ž .
Ž .
i j, t i j
i j t i i , t
j j, t
where i j. The conditional covariances are proportional to the product of the applicable conditional volatilities. In accordance with the CCORR model, we
assume that the conditional correlations are constant. We stress, that the assump- tion of constant conditional correlations is nothing but a convenient working
preposition. Moreover, the conditional correlations change deterministically by the proportion r
A
on announcement days. The conditional covariance between bond i
i j
and j is also allowed to be different on announcement days, though, it is not quite as simple to state its magnitude: The conditional covariance between R
and R
i t jt
A A
equals 1 q d I
1 q d I h . If we disregard differences in h
between
i t j t
i j, t i j, t
announcement days and non-event days, we see that the conditional covariance is greater on announcement days by 1 q d
1 q d .
i j
Ž Ž ..
The specification that we apply for the diagonal elements of H Eq. 5
is
t
Ž .
similar to the univariate specification in Jones et al. 1998 . Their most general specification of the conditional variance of the bond excess return for a given
Ž . maturity is a Aregime switchingB model, which corresponds to Eq. 3 expanded
A
Ž by the multiplicative factor 1 q c I
i.e. the conditional variance deterministi-
i ty1
. Ž .
cally changes regime the day following an announcement , w in Eq. 5 is
i
allowed to differ on announcement days, and shocks are reduced to being symmetric, i.e. f
A
s 0. For all maturities examined, they find that the conditional
i
variance does not follow the regime switching process and that the GARCH parameter for the lagged conditional variance is identical on announcement and
non-announcement days. Their results lead us to analyze the GARCH specification Ž .
in Eq. 5 . 3.3. Estimation practicalities
The multivariate model set up above is estimated using a two-stage estimation Ž .
technique. In the first step, the conditional mean Eq. 4 , is estimated using Ž
. Ordinary Least Squares OLS and applying White’s heteroscedasticity consistent
standard errors. The differences between the returns and their conditional mean are termed the VAR residuals: y R y m . The VAR residuals are now the basis
t t
t
time series. The use of the residuals as observed data have been applied by Kroner Ž
. and Ng 1998 amongst others in a similar setting.
In the last step, we estimate the second moment equation. The estimation is Ž
. conducted using Quasi Maximum Likelihood QML with a Gaussian likelihood
function and applying robust standard errors, cf. Bollerslev and Wooldridge Ž
. 1992 . The log-likelihood function can be stated in an appealingly simple manner
because H can be partitioned as follows: H s D G D , where:
t t
t t
t
h
11 , t
. .
D s ,
t
. h
N N , t
and: 1
r 1 q r
A
I
A
Ž .
i j i j t
. .
G s .
t
.
A A
r 1 q r I
1
Ž .
i j i j t
Ž Ž
. Ž
. Hence, the log-likelihood for observation number t is y1r2 N ln 2p q ln G q
t X
Ž .
y1
.
8,9,10
2ln D q ´ D G D
´ .
t t
t t
t t
4. Empirical results from the treasury bond market
