Daily Correlation and Volatility Dynamics between National Stock
Markets with Non-overlapping Trading Hours
Yiguo Sun 1
Department of Economics
University of Guelph
Guelph, ON N1G2W1
Canada
February 2008
(Preliminary Draft)
Abstract
This paper studies the daily correlation and volatility dynamics between two national markets
with non-overlapping trading hours via a dual synchronization procedure. The duality method is
motivated by the asynchronous problem: the use of daily close-to-close market returns results two
different within 24-hour return cross correlations between the two national markets according to
transaction time instead of calendar time. We take the asymmetric within 24-hour correlations as
the evidence that one national economy has stronger impact on the other one than vice versa. The
method is applied to Australian and the USA markets.
Key Words: non-synchronization; (reversed) echelon form of VARMA; BEKK model
JEL Classification: C32; C53; G15
1. Introduction
With the fast growing telecommunication technology and capital mobility, combined
with the increasing international involvement in trades of good and services and policy
coordination, it is stylized fact that an individual nation’s financial market is hardly
completely immune to shocks from foreign countries. In addition, a nation’s market index
is one of the important leading indicators used to predict the nation’s economic status in
the following several months. Therefore, understanding the inter-market relationships
1
Email address: yisun@uoguelph.ca. This draft is preliminary and revision is under way.
1
helps to predict how the changes of one nation’s economic situation influence another
nation’s economy. Take the impact of the recent turbulences originated from the USA
sub-primary credit markets for an example. The sharp drops of Asian and European
market prices in January 2008 have manifested the markets’ concerns of the potential
impact of the USA economic slowdown on global economy. 2 Although a nation’s
financial market can recover from extreme negative foreign shocks in the long-run
period, the short-run suffering may not be easily swallowed. It is thus pertinent to
understand the short-run interdependence of prices and price volatilities and correlations
across national markets.
Given the crucial role of inter-market dependency, numerous research works have
been produced in the last two decades. Few are named here: Burns et al. (1998), Hamo, et
al. (1990), Koch and Koch (1991), Koutmos and Booth (1995), Longin and Solnik
(1995), Martens and Poon’s (2001), among many others. This paper joins the literature
by studying the daily correlation and volatility dynamics between two national markets
with non-overlapping trading hours.
Specifically, we name the market opening earlier between two non-overlapping
markets to be market E and the other market W. 3 We then denote their respective market
returns by re ,t and rw,t on calendar date t, where the time refers to the time recorded in
market W. Figure 1 plots the time line of the market returns on two consecutive calendar
dates. Since rw,t −1 , re ,t , and rw,t are observed consecutively with less-than-24-hour time
increment, we can always construct two within 24-hour correlation measures:
ρ A = corr (re,t , rw,t ) and ρ B = corr (rw,t −1 , re,t )
(1)
If we interpret ρ A as a measure of market E’s influence on the next market W and ρ B as
a measure of market W’s influence on the next market E, ρ B > ρ A then implies that
market W’s economy has stronger influence on market E’s economy than vice versa.
Also, ρ A = ρ B implies symmetric influence between the two national economies and
2
A word of caution here: we will not over-emphasize the predicative power of stock index returns on
national economy, as it is hard to measure to what extent that stock price changes are attributed to changes
in fundamental values.
3
It is motivated by the fact that international markets open and close sequentially every day from the
eastern hemisphere to the western hemisphere.
2
none of the national economies lead another. The asynchronous problem mentioned in
the literature refers to the asymmetric case, i.e. ρ A ≠ ρ B such that none of the intra-day
correlations can be treated as the contemporaneous correlation between the two national
markets.
To overcome the asynchronous problem, Burns et al. (1998, BEM henceforth)
proposed to calculate the contemporaneous (conditional) covariance and correlation from
synchronized returns {(rˆe,t , rˆw,t )}, where the hypothetical return rˆe,t synchronizes with rw,t
on calendar date t = 1,2,
. Specifically, their synchronization procedure works like this:
splitting the time period from calendar date t-1 to calendar date t into two time intervals,
1 = t − (t − 1) = t1 + t2 , the synchronized return rˆe,t takes the sum of re ,t − Et −1 (re ,t ) , the
return occurring during time t1 , and Et re,t +1 , the expected return occurring during time t2 .
Here, t1 is the time length between market W’s closing time on calendar date t-1 and
market E’s closing time on calendar date t and t2 is the time length between market E’s
closing time on calendar date t and market W’s closing time on calendar date t.
Figure 1. Time line of asynchronous and synchronized returns
rˆe ,t
t-1
re ,t −1
rw,t −1
(r
e ,t
re ,t
t1
+
rw,t
t2
=1
− Et −1 (re ,t )) + Et re,t +1 = rˆe,t
BEM’s methodology takes the closing time of market W on each day as the reference
time. Note that, on calendar date t, re,t and rw,t are observed sequentially within 24 hours,
so are rw,t −1 and re ,t . We therefore question what additional information we can obtain if
we synchronize the returns from market W with those from market E closing within 24
hours; that is, on a calendar date t, taking the closing time of market E as the reference
time, we attempt to construct ~
r synchronizing with r .
w, t
e,t
3
Hence, a dual method is proposed to solve the asymmetric or asynchronous problem
defined by Eq. (1). What we do is this: to correct the biasedness of ρ A in measuring the
contemporaneous correlation between the two markets, we construct synchronized data
{(rˆ
AUS , t
, rˆUS ,t )} from
{(r
AUS ,t
, rUS ,t )}; to correct the biasedness of ρ B in measuring the
contemporaneous correlation between the two markets, we construct synchronized data
{(~rUS ,t , ~rAUS ,t )} from {(rUS ,t −1 , rAUS ,t )};. We call the first synchronization procedure method A
and the second one method B, and apparently, each method reflects one side story of a
mirror.
Intuitively, if the synchronization procedure is adequate, the (cross) serial return
correlations of {(rˆAUS ,t , rˆUS ,t )} should be very close to those of {(~
rUS ,t , ~
rAUS ,t )}. In addition, if
method A leads to larger corrections on the contemporaneous correlations between the
two markets than method B does, we explain that market W has stronger influence on
market E than vice versa. Further, the methods are complementary. In the opening of
market W on calendar date t, method B can be used to estimate the conditional
correlation of market W with market E with return data up to rw,t −1 and re,t , while method
A only uses return data up to re,t −1 and rw,t −1 ; apparently, method B provides more
accurate information than method A does. In contrast, in the opening of market E on
calendar date t, similar argument applies and method A is preferred to method B.
Therefore, both methods can be used rotationally, depending on which market’s risks are
to be hedged.
In the end, applying the dual approach to study the inter-market relationships
between Australian and the USA markets, we find that method A leads to large
corrections on the contemporaneous conditional correlations while method B gives
negligible corrections, and that the two methods give similar (cross) autocorrelation
functions for synchronized returns as expected. We interpret this as an evidence to
support that the USA market has stronger influence on Australian market than vice versa.
Of course, the conclusion of our empirical findings is not new, we just use this set of data
to illustrate the usefulness of the dual synchronization procedures and introduce this
method to readers as an alternative in learning the lead-lag relationship between two
national markets with non-overlapping trading hours.
4
The rest of the paper is organized as follows. Section 2 explains the identification
and estimation of dynamic conditional models of bivariate returns and gives the
synchronization procedure of Burns et al. (1998). We then illustrate the dual method to
the daily Australian and the USA market returns in Section 3. Section 4 concludes.
2. Dynamic modeling and return synchronization
In this section, we first give mathematical representation of the dynamic structures on
both conditional mean and conditional second moment equations of the original,
asynchronous return data. Following Burns et al.’s (1998) method, we then construct the
synchronized returns from the asynchronous return data. Moreover, we explain how the
synchronization procedure changes the dynamic structures of the original, asynchronous
returns data.
2.1. Dynamic structure model of conditional mean returns
Koch and Koch (1991) considered the pure vector autoregressive model VAR(p) and
Burns et al. (1998) assumed vector moving average of order one (VMA(1)) model to
capture the short-term serial return correlations across the markets. Theoretically, an
invertible VARMA(p,q) model can be well approximated by a VAR model including as
many lag terms as possible; however, the estimation efficiency and accuracy will be
reduced with the increase of the number of parameters to be estimated. On the other
hand, a misspecified VMA(1) model could give misleading inferences on the causal
relations between market index returns. Therefore, we start with estimating a general
bivariate VARMA model and let the data to select the parsimonious model best for the
data.
Denote Yt = (xt
yt ) and ε t = (ε x ,t
T
ε y ,t )T . When method A is applied, we have
(xt , yt ) = (re,t , rw,t ) . When method B is applied, we have (xt , yt ) = (rw,t −1 , re,t ) . In either case,
xt is observed within 24-hour earlier than yt is. We consider the following VARMA(p,q)
model
5
p
p
q
q
i =1
i =1
i =1
i =1
p
p
q
q
i =0
i =1
i =0
i =1
xt = ∑φ11,i xt − i + ∑ φ12,i yt − i + ε x ,t + ∑ψ 11,iε x ,t − i + ∑ψ 12,iε y ,t − i
(2)
yt = ∑ φ21,i xt − i + ∑ φ22,i yt − i + ε y ,t + ∑ψ 21,iε x ,t − i + ∑ψ 22,iε y ,t − i
which takes Koch and Koch’s (1991) and Burns et al.’s (1998) models as special cases
and assumes ψ 21,0 = −φ21,0 for the sake of identification. Also, E (ε t | I t −1 ) = 0 and
(
)
E ε t ε tT | I t −1 = H t , a positive definite matrix, for any calendar date t. The information set
available on calendar date t is given by I t −1 = {(xs , ys , ε x , s , ε y , s ) : s < t}. Apparently, other
factors being fixed, the parameters in front of xt and ε x,t in the equation of yt and those
in front of yt −1 and ε y ,t −1 in the equation of xt capture the within 24-hour inter-market
effects, and parameters φ12,i , ψ 12,i , φ21,i , and ψ 21,i (i>0) capture the inter-market relations
beyond 24 hours. As in Koch and Koch (1991), model (2) can be used to distinguish
intra-day effects from inter-day effects between the two markets. If the international
stock markets are efficient, finance theory predicts that inter-market adjustments should
be completed within 24 hours, not beyond a day.
Rewritting the models above in matrix format gives
p
q
i =1
i =1
= ∑ Φ iYt − i + Φ 0ε t + ∑ Ψiε t − i ,
Φ 0Yt
(3)
0⎤
⎡φ11,i φ12,i ⎤
⎡ψ 11,i ψ 12,i ⎤
and for i ≥ 1 , Φ i = ⎢
and Ψi = ⎢
⎥
⎥
⎥.
1⎦
⎣φ21,i φ22,i ⎦
⎣ψ 21,i ψ 22,i ⎦
⎡ 1
where Φ 0 = ⎢
⎣− φ21,0
The standard software such as Splus considers the model with parameter matrices
~
~
Φ i = Φ 0−1Φ i and Ψi = Φ 0−1Ψi ; that is, Φ 0−1 is multiplied to both sides of the model above:
Yt
p
q
~
~
= ∑ Φ iYt − i + ε t + ∑ Ψiε t − i , t = max( p, q ) + 1,
i =1
,T .
(4)
i =1
p
q
~
~
~
~
Define Φ(L ) = I − ∑ Φ i Li and Ψ (L ) = I − ∑ Ψi Li to be its characteristics functions of
i =1
i =1
{(
~ ~
the VARMA(p,q) model. The parameters Φ i , Ψ j
)} are identified if we assume that (a)
~
~
Φ (L ) and Ψ (L ) have no common left factors other than unimodular ones, and (b) with
6
([
])
~ ~
q as small as possible, and p as small as possible given q , rank Φ p , Ψq = 2 , the
dimension of Yt (see Reinsel, 1993, Sect. 2.3.4). In addition, the return vector, {Yt } , is
~
stationary if the roots of Φ( z ) = 0 all lie outside the unit circle, and it is invertible if the
~
roots of Ψ ( z ) = 0 all lie outside the unit circle.
However, when φ21,0 is also of our concern, we need to directly estimate model
(3), instead of model (4). The identification of model (3) has been studied Tsy (1991) and
Lutkepohl and Poskitt (1996), and references therein. Lutkepohl and Poskitt (1996)
suggest constructing parsimonious and identifiable VARMA models via the (reversed)
echelon form VARMA models. An echelon form VARMA model is determined by the
Kronecker indices (n1,n2), which gives the largest polynomial order of the first and
second row of the characteristic functions of the model, and this model has an equivalent
mathematical representation of VARMA(max(n1,n2), max(n1,n2)). In particular, if n1=n2,
the model becomes the standard VARMA(n1,n1) model with φ21,0 = 0 . When n2 + 1 ≤ n1
holds, φ21,0 is a free parameter to be estimated, and ( xt , ε x ,t ) observed earlier does
directly affect the prediction of yt . Therefore, starting from the echelon form VARMA
model and constructing the models in transaction sequence instead of calendar sequence,
we are able to identify the intra-day effects from both markets. Finally, the optimal
Kronecker index vector is selected via information criteria such as AIC, BIC and HQ
statistics, as explained in Lutkepohl and Poskitt (1996).
2.2. Constructing synchronized returns
Following BEM’s methodology, we construct the synchronized returns, Ŷt , from the
compounded close-to-close daily returns:
p
q
i =1
i =1
~
~
Yˆt = EtYt +1 + (Yt − Et −1Yt ) = ∑ Φ iYt +1− i + ε t + ∑ Ψiε t +1− i
(5)
where Et (⋅) = E (⋅ | I t ) . Apparently, Yˆt ≠ Yt if EtYt +1 ≠ Et −1Yt . In addition, if { Yt } is serially
uncorrelated, we have Yˆt = Yt and the asynchronous trading is of no importance. Given
7
that asynchronous returns follow a general VARMA(p,q) model, what will be the
stochastic property of the synchronized returns? Simple calculations give that
p
q
~
~
Yˆt = ∑ Φ iYt +1− i + ε t + ∑ Ψiε t +1− i
i =1
i =1
q
⎞ q ~
~
~ ⎛ ~
= ∑ Φ i ⎜⎜ ∑ Φ jYt +1− i − j + ε t +1− i + ∑ Ψ jε t +1− i − j ⎟⎟ + ∑ Ψiε t +1− i + ε t
i =1
j =1
⎝ j =1
⎠ i =1
p
p
(6)
p
p
q
~
~
~
= ∑ Φ iYˆt − i + ε t + ∑ Φ i (ε t +1− i − ε t − i ) + ∑ Ψiε t +1− i
i =1
i =1
(
i =1
)
i =1
p −1
(
)
q −1
~
~
~
~
~
~
~
= ∑ Φ iYˆt − i + I + Φ1 + Ψ1 ε t + ∑ Φ i +1 − Φ i ε t − i + ∑ Ψi +1ε t − i − Φ pε t − p
p
i =1
i =1
{}
The synchronized returns, Yˆt , thus follow VARMA( p, max(q − 1, p )) model, and the
VARMA model of the synchronized returns shares the same autoregressive parameter
matrices as that of the asynchronous returns {Yt } . In other words, the synchronization
procedure preserves the autoregressive structure of the asynchronous returns. This simple
proof is consistent with BEM’s (1998) idea that the synchronization procedure corrects
the short-term dynamic linkage between two non-overlapping national markets. In
addition, the last equation of Eq. (6) indicates that Et −1Yˆt ≠ Et −1Yt in general. Taking
()
~
VAR(1) for an example, we obtain Et −1 Yˆt = Φ1 Et −1 (Yt ) , and it means that the mean of the
synchronized returns will not be the same as that of the raw returns conditional on
information set I t −1 unless the raw return data is an I(1) sequence.
2.3. Dynamic conditional second moments
Denote the time-varying conditional covariance of the asynchronous returns, conditional
on the information set I t −1 by
⎡ hxx ,t
H t = Var (Yt | I t −1 ) = E ε tε tT | I t −1 = ⎢
⎣hyx ,t
(
)
hxy ,t ⎤
,
hyy ,t ⎥⎦
where we caution that we cannot interpret hxy ,t to be the contemporaneous conditional
covariance between the two market returns due to the non-synchronicity. If the
VARMA(1,1) model is the proper model to fit the raw data, the time-varying conditional
covariance of the synchronized returns, conditional on the information set I t −1 will be
8
(
) (
)
Hˆ t = Var Yˆt | I t −1 = E ξtξtT | I t −1 = ΛH t ΛT
(7)
~
~
2
where Λ = I + Φ1 + Ψ1 = (λij )i , j =1 and ξt = Λε t . If Λ is a lower triangular matrix,
2
hˆxx ,t = λ11
hxx ,t . If Λ is an upper triangular matrix, hˆyy , t = λ222 hyy ,t . Otherwise, the
synchronized conditional variance in each market is also affected by innovations from the
other market.
Throughout this paper, we assume E (Yt ) = 0 , since we can always remove the sample
means from the data, so the unconditional mean of the synchronized returns is
()
p
~
ˆ
E Yt = ∑ Φ i E (Yt +1− i ) = 0 , and its unconditional covariance can be estimated by its
i =1
sample covariance.
Next, to quantify the dynamic structure of the conditional second moments of the
asynchronous returns, we use Engle and Kroner’s (1995) BEKK(a,b) model to capture
the conditional heteroskedasticities of the data:
a
b
i =1
i =1
H t = W + ∑ AiT ε t −1ε tT−1 Ai + ∑ BiT H t −1 Bi
(8)
where W = (ωij ) = C T C with a non-singular lower triangular matrix C . The series {Yt } is
a
stationary if the eigenvalues of
b
∑ A ⊗ A + ∑ B ⊗ B are all less than one in modulus.
i =1
i
i
i =1
i
i
Of course, many variant of multivariate GARCH models can be used to capture time
varying volatilities and correlations. For example, the VECH model of Bollerslev, Engle,
and Wooldridge (1988), the constant conditional correlation (CCC) model of Bollerslev
(1990), the factor ARCH (FARCH) model of Engle, Ng, and Rothschild (1990), the
asymmetric dynamic covariance (ADC) matrix model of Kroner and Ng (1998), among
many others. The selection of the BEKK model is due to its popularity and good
performance documented in many empirical studies.
9
2.4. Consistent estimator via Quasi-maximum likelihood method
The quasi-MLE of Bollerslev and Wooldridge (1992) maximizes the following likelihood
T
T
T
t =1
t =1
t =1
function: L(θ ) = ∑ lt (θ ) = −T ln (2π ) − 12 ∑ ln H t − 12 ∑ ε tT H t−1ε t , where H t is given by
Eq. (8) and ε t is derived from Eq. (3) if the echelon form VARMA model is considered
and it is derived from Eq. (4) if the standard VARMA model is considered. To identify
the parameter vector θ , we need some parameter restrictions: the diagonal elements of C
cannot be zero for the assurance of a p.d.f. matrix W , and the upper left corner element
of A and B are positive as in Engle and Kroner (1995) otherwise –Aii and –Bii also satisfy
the multivariate GARCH models (8). Of course, we could impose other restrictions on A
and B, but the current assumptions can be easily imposed by using a112 ,i and b112 ,i as the
respective upper left corner element of A and B. Moreover, stationarity and invertibility
are enforced on the model using the reparameterization discussed in Ansley and Kohn
(1986).
In addition, the QMLE involves the calculation of the score function and the
Hessian matrix. Due to the complication of the objective function, it is common practice
to use numerical gradients to substitute analytical ones, which may cause estimation
accuracy loss, especially when the number of parameters is high. We therefore apply the
variance targeting method of Engle and Mezrich (1996) to reduce the number of
a
b
i =1
i =1
parameters by three; that is, we fix W = H 0 − ∑ AiT H 0 Ai − ∑ BiT H 0 Bi in Eq. (8) and H 0
is replaced with the sample covariance of {Yt }. To differentiate the models estimated, we
call a model without the variance target restriction to be full model and a model with the
restriction to be variance-targeting model.
3. Empirical application
In this section, we use the Australian and USA market indexes to illustrate the usefulness
of the dual method when making inferences on dynamic interdependence between two
national markets with non-overlapping trading hours. The Australian (AUS) and the USA
markets locate at the GMT+10 and GMT-5 time zones, respectively. On a given calendar
date, the AUS market opens within 24 hours after the USA market closed on the previous
10
date, and the USA market opens within 24 hours after the AUS market is closed. The
AUS and USA markets have no overlapping trading hours at all. The daily closing prices
are downloaded from DataStream and span from January 1, 1991 to December 31, 2006.
The merged data contain 4,016 observations recorded in New York time. In addition, all
prices are in the US dollar and we scaled up the daily compound returns by 100, before
removing the day-of-week effects from the raw data.
3.1 Summary statistics
Table 1 gives summary statistics for the close-to-close index returns, where the second
and fifth columns are for the asynchronous AUS and US daily market returns,
respectively. The US daily returns are more volatile than the AUS daily returns, as their
respective standard deviations are 1.0641 and .9947 basis point per day. We also
calculate the frequency that the returns are positive: 50.35% and 50.41% for the AUS and
the US markets, respectively. Unconditionally, we have the most uncertainty according to
entropy theory in predicting the signs of the market return from each of the two markets
on any given day, although our experience tells us that we can do better with conditional
prediction.
Next, we calculated the conventional measures of the coefficients of skewness
and kurtosis:
Skewness = E (rt − Ert )
3
[E(r − Er ) ]
2
t
t
3
2
and Kurtosis = E (rt − Ert )
4
[E (r − Er ) ] − 3
2 2
t
t
where the coefficient of skewness is zero for any symmetric distribution and the
coefficient of kurtosis is zero for normal distributions. They are consistently estimated by
their sample analogs
T
T
⎛r −r ⎞
⎛r −r ⎞
−1
SKˆ = T −1 ∑ ⎜ t
⎟ and KRˆ = T ∑ ⎜ t
⎟ − 3,
⎠
⎠
t =1 ⎝ σˆ
t =1 ⎝ σˆ
3
4
where r and σ̂ are the sample mean and sample deviation of
{rt },
respectively.
However, using Monte Carlo simulations, Kim and White (2004) show that the
conventional measures of the coefficients of skewness and kurtosis from sample
moments give seriously biased estimation in finite samples. We therefore apply
subsampling method to calculate the critical values when testing for zero skewness and
11
kurtosis. With 1000 subsampling replications, we reject neither symmetry nor zero
kurtosis at the significance level of 5%. However, it does not imply that the returns are
normally distributed, as Jarque-Bera tests strongly reject the normality assumption for
both return series at the level of 1%.
Table 1 also gives the summary statistics for the synchronized returns derived from
both methods A and B. We find that the summary statistics of the synchronized returns
are close to those of the asynchronous returns. In addition, for Australian market, the
summary statistics of the synchronized returns based on method B are closer to those of
the original returns than are those based on method A, and the opposite is true for the
USA market.
3.2 Autocorrelations and cross (serial) correlations
The LM statistics and the Ljung-Box statistics given in Table 1 indicate strong rejection
of no autoregressive conditional heteroskedascity and of no serial correlations at the 1%
level for both the AUS and the USA market returns.
In Table 2, we then present the estimated serial return correlations of each market
and the estimated cross serial return correlations between the two markets up to lag 5.
The columns below ‘asynchronous’ are calculated from the raw data, where method A
applies to the data set
{(r
AUS , t
, rUS ,t )} and method B applies to
{(r
US ,t −1
, rAUS ,t )}. Firstly,
reading the serial correlations of the AUS market returns, we find that only the
autocorrelation of lag two, -.0333 is weakly significant at the 5% level for the
asynchronous data, and that none of the first five autocorrelations for the synchronized
returns is significant at the 5% level. Secondly, reading the serial correlations of the USA
market returns, we find that its fifth autocorrelation is negative for both asynchronous and
synchronized data at the 5% level. In addition, the synchronized US market returns
derived from both methods A and B have significant negative autocorrelation of lag one
at the 1% level.
Finally, we look at the cross serial return correlations between the two markets.
At the 1% level, we find significant within 24-hour return cross correlations: the
correlation between {rUS ,t −1} and {rAUS ,t } is .4025, and the correlation between {rAUS ,t } and
12
{r } is .0727, but there is no significant return cross correlation beyond a day.
US , t
The
return correlation between the AUS market and the last USA market is stronger than that
between the AUS market and the next USA market. As explained in Section 1, the
asymmetry of the intra-day correlations shows that the USA market has a stronger impact
on Australian market than vice versa.
Using synchronized returns via method A, we find that the “contemporaneous
correlation” increases to .4884 from .0727 and that there is no significant cross serial
return correlation beyond a day. When method B is applied, the “contemporaneous
correlation” between synchronized data, {~
rUS ,t } and {~
rAUS ,t }, increases to .5027 from
.4026, and there is no significant cross serial return correlation beyond a day, either.
Hence, the synchronized returns absorb the intra-day influences between the two markets,
and leave no significant cross serial return correlations beyond a day.
The results in Table 2 indicate that the synchronized returns derived from method
A share the same (cross) serial correlation patterns as those derived from method B do,
however the two methods adjust the biased correlation relationships differently. First,
under method A, ρ A = corr (rAUS ,t , rUS ,t ) = .0727 severely understates the contemporaneous
correlation between the two markets, since the information from the USA market comes
after Australian market closes on any calendar date. Therefore, to correct the bias, the
synchronized AUS market returns have to take into account the potential impacts of the
innovations from the USA market. On the other hand, under method B,
ρ B = corr (rUS ,t −1 , rAUS ,t ) = .4026 is very close to the adjusted contemporaneous cross
correlation .5027, since Australian market has absorbed information from the last USA
market, while the USA market does not response strongly to the potential information
from the next Australian market. Consequently, method A takes more adjustment than
method B does, and more noises may be associated with method A than with method B
due to the heavy adjustment.
In the next subsection, we will provide more evidence to support the current
discussion, and illuminate the complementary nature of methods A and B when making
inferences on price and volatility spillovers between the two markets.
13
3.3 QMLE results and inferences on dynamic interdependency
We estimated a series of the (reversed) echelon form VARMA(p,q) models defined in
Eq. (3) plus the BEKK(a,b) model for non-negative integer p, q ≤ 2 and a, b ≤ 2 . Among
many models passing the diagnostic statistics calculated by Splus, we find that
VARMA(1,1) model plus the BEKK(1,1) model minimizes information criteria such as
BIC and HQ statistics under both method A and method B. Table 3 gives our estimation
results under both methods A and B for both full model and variance-targeting model. In
addition, the AIC, BIC and HQ statistics given at the end of Table 3 prefer the variancetargeting model to the full model for the chosen model, and the estimation results do not
change significantly with the restriction imposed.
Before explaining the estimation results, for convenience, we give the information
set associated with method A and method B here. When method A is applied, the data
under consideration is
{(x , y ) : x
t
t
t
= rAUS ,t , yt = rUS ,t , t = 1,2,
},
and the information
available on calendar date t is
I tA−1 = {(rAUS , s , rUS , s , ε AUS , s , ε US , s ) : s < t}
When
method
{(x , y ) : x
t
t
t
B
is
applied,
= rUS ,t −1 , yt = rAUS ,t , t = 2,3,
the
(9)
data
under
consideration
is
}, and the information available on calendar date t
is
I tB−1 = {(rUS , s −1 , rAUS , s , ε US , s −1 , ε AUS , s ) : s < t}
(10)
3.3.1 Price spillovers
Method A. In the equation of rAUS ,t , the autoregressive coefficient in front of rUS ,t −1 , φ12,1 ,
is insignificant, but the moving average coefficient in front of ε US ,t −1 , ψ 12,1 , is significant
at the 5% level, indicating intra-day price spillover from the USA market to Australian
market induces short-term return cross serial correlation between the two markets. In the
equation of rUS ,t , the coefficient in front of rAUS ,t −1 , φ21,1 ,
and that in front of
ε AUS ,t −1 ,ψ 21,1 , are both significant at the 5% level, which may not be interpreted as the
evidence of market inefficiency, as it could result from the omission of rAUS ,t and ε AUS ,t
from the conditional information set given by Eq. (9).
14
Method B. In the equation of rUS ,t −1 , the insignificance of the coefficients in front
of (rUS ,t − 2 , rAUS ,t −1 ) and (ε US ,t − 2 , ε AUS ,t −1 ) indicates no own lagged effect and no intra-day
price spillover from the AUS market to the US market. In the equation of rAUS ,t , the
insignificance of φ21,1 and ψ 21,1 implies no price spillover from the US market to the
AUS market beyond a day. 4
The first equations of both method A and method B depict the intra-day impact of
the first market on the next market. In particular, other factors being fixed, method A
indicated that 72.9% of a unit of noise from the last USA market is absorbed by
Australian market, and method B reveals no significant impact of Australian market on
the next USA market. These observations are consistent with our findings on ρ A << ρ B as
stated in Section 3.2.
3.3.2 Volatility spillovers
The mathematical representation of the BEKK(1,1) model is given by
H t = W + AT ε t −1ε tT−1 A + B T H t −1 B .
(11)
Under method A, the Wald statistic on a joint test of A12 = A21 = B12 = B21 = 0 equals
5.9979 with a p-value of .20, which supports the diagonal bivariate GARCH(1,1) model
with hxx ,t = w11 + A112 ε x2,t −1 + B112 hxx ,t −1 and hyy ,t = w22 + A222 ε y2,t −1 + B222 hyy ,t −1 . We draw the
same conclusion using method B, as the Wald statistic for the joint test equals 6.5893
with a p-value of .16. However, the bivariate VARMA(1,1) model plus diagonal
GARCH(1,1) model does not pass the diagnostic tests and has much larger AIC, BIC and
HQ values than the bivariate VARMA(1,1) model plus BEKK(1,1) model. Therefore,
the finding on volatility spillover between the two markets is inconclusive.
3.3.3 Conditional second moments of synchronized returns
Since {Yt } follows the stationary VARMA(1,1) model with mathematical representation
(
)
~
~
Yt = Φ1Yt −1 + ε t + Ψ1ε t −1 , E (ε t | I t −1 ) = 0 and E ε t ε tT | I t −1 = H t for all t.
And following the argument made in Section 2, we have for the synchronized returns
4
Throughout this paper,
εt
is used to denote the error on calendar date t; however, it has different
value/meaning when different methods are used.
15
(
)
(
)
~
~ −1
~
~
~
~
Yˆt = Φ1Yt + I + Ψ1 ε t = Φ1Yˆt −1 + ξ t − Φ1 I + Φ1 + Ψ1 ξ t −1 ,
(12)
~
~
2
where Λ = I + Φ1 + Ψ1 = (λij )i , j =1 , ξt = Λε t , and Hˆ t = ΛH t ΛT . Since {Yt } are serially
~
~
correlated, econometric theory implies that Φ1 + Ψ1 = 0 does not hold, nor does Hˆ t = H t .
To understand how the dual methods explained in Section 2 adjust the conditional
covariance and correlations, we start with testing for the significance of the elements in
Λ , see our discussion made in Section 2 below Eq. (6).
Applying
method
A,
we
have
λˆ11 = .9783 ,
λˆ12 = .4311 ,
λˆ21 = .0190 ,
and λˆ22 = 1.0057 . We first test for λ12 = 0 and λ21 = 0 , and the respective Wald statistic
equals 514.4397 and 9.725652. Consequently, both λ12 and λ21 are significant at the 5%
level. Secondly, the hypothesis of λ11 = λ22 = 1 is not rejected at the 5% level, as the
associated Wald statistic takes a value of 1.2611 with a p-value .5322. In addition, we
obtain strong rejection on λ12 = λ21 with the Wald statistic being equal to 406.2258; the
adjustment is asymmetric. Other factors being fixed, on calendar date t, the
synchronization procedure transfers 43% of ε US ,t to Australian market, while only 1.9%
of ε AUS ,t is transferred to the USA market. Moreover, the estimated conditional
covariance of the synchronized returns is given by
⎡.9571hxx ,t + .8434hxy ,t + .1858hyy ,t
Hˆ t = ⎢
⎣.0186hxx ,t + .9920hxy ,t + .4335hyy ,t
.0186hxx ,t + .9920hxy ,t + .43351hyy ,t ⎤
, (13)
.0004hxx ,t + .0382hxy ,t + 1.0114hyy ,t ⎥⎦
where x = AUS and y = US , and significant adjustments are made to the calculation of
hˆxx ,t and hˆxy ,t . Therefore, after synchronizing the two market returns, we find that the
inter-market relation has more impact on the volatilities of the AUS market returns than
the US market returns, and that the conditional contemporaneous covariance between the
two series of synchronized returns is strongly influenced by the conditional volatilities of
the USA market returns. Simply speaking, the USA market has stronger influence on
Australian market than vice versa.
Applying
method
B,
we
have
λˆ11 = .9740 ,
λˆ12 = .0716 ,
λˆ21 = .0495 ,
and λˆ22 = 1.0056 . The respective Wald statistic is 22.7074 and 11.5549 for testing for
16
λ12 = 0 and λ21 = 0 . We thus reject each of the hypotheses at the 5% level. Second, the
hypothesis of λ11 = λ22 = 1 is not rejected at the 5% level, as the associated Wald statistic
takes a value of 1.7246 with a p-value .4222. In addition, the hypothesis of λ12 = λ21 is
not rejected at the 5% level, as the Wald statistic equals .9584 with a p-value of .3276;
the adjustment is symmetric. The estimated conditional covariance of the synchronized
return is given by
⎡.9487hxx ,t −1 + .1395hxy ,t + .0051hyy ,t
Hˆ t = ⎢
⎣.0482hxx ,t −1 + .9831hxy ,t + .0720hyy ,t
.0482hxx ,t −1 + .9831hxy ,t + .0720hyy ,t ⎤
,
.0025hxx ,t −1 + .0996hxy ,t + 1.0113hyy ,t ⎥⎦
(14)
where x = US and y = AUS . It follows that the inter-market relation also has significant
impact on the volatilities of the AUS market and the USA market, although the impact is
much weaker than when method A is used.
Again, the results above indicate that method A makes heavier adjustment than
method B does when capturing the dynamic interdependency of conditional second
moments between the USA and Australian markets, due to the stronger influence of the
USA market on the next AUS market.
Next, Table 4 presents the summary statistics of the estimated conditional second
moments for asynchronous and synchronized returns under both method A and method
B. The results indicate that the distribution of the estimated synchronized conditional
standard deviations under method B shift to the right of that under method A. In table 5,
we notice that the linear regression of R̂tA on 1 and R̂tB has a very low R2, while the
A
A
B
ˆA
ˆB
ˆB
regressions of hˆAUS
, t on 1 and hAUS , t , of hUS , t on 1 and hUS , t , and of Ŝ t on 1 and Ŝ t all
have decent R2 values. As we have I tA−1 = I tB−1 ∪ {(rUS ,t −1 , ε US ,t −1 )} for all t by Eq. (9) and Eq.
(10), the high correlation between hˆ jA,t and hˆ Bj ,t (j=AUS or US) implies that the
estimations of conditional volatilities are relatively robust to the information change from
I tB−1 to I tA−1 , which is consistent with the high persistency of conditional volatilities as
shown in Table 3. Basing on the same logic, the low correlation between R̂tA and R̂tB
could result from the low persistency of the conditional correlations. On other words, the
conditional correlation is sensitive to the information set used, and we need to update the
17
estimation of conditional correlations frequently as new information comes in when
monitoring short-term value at risk and updating time-varying beta or hedge ratios in
practice. It thus supports the rotational usage of method A and method B empirically.
Specifically, in the opening of the USA market on calendar date t, we can
estimate Ĥ tB via method B basing on the information set I tB = I tA−1 ∪ {(rAUS ,t , ε AUS ,t )} ,
which is better than Hˆ tA−1 calculated via method A with information set I tA−1 . In contrast,
in the opening of Australian market on calendar date t, we can estimate Ĥ tA via method
A basing on the information set I tA−1 = I tB−1 ∪ {(rUS ,t −1 , ε US ,t −1 )}, which is preferred to Hˆ tB−1
calculated via method B with information set I tB−1 .
4
Conclusion
This paper investigates the short-term dynamics of two national stock markets with nonoverlap trading hours via a dual synchronization procedure. We explain that the dual
method is complementary and can be used rotationally when monitoring short-term value
at risk and updating time-varying beta or hedge ratios in practice. In addition, we show
that the synchronization procedure preserves the autoregressive structure of the
asynchronous returns and only corrects the short-term dynamic linkage between two nonoverlapping national markets.
Applying this idea to Australian and the USA daily market returns, we find strong
intra-day price spillover from the USA market to Australian market. The synchronized
returns derived from method A share the same (cross) serial correlation patterns as those
derived from method B, although method A requires heavier adjustment from the
asynchronous data than method B does.
Finally, the finance literature has widely documented that the prediction of
conditional volatilities and correlations are very sensitive to the choice of econometrics
models used to fit the conditional variances. However, the current research emphasizes
on the introduction of the dual synchronization procedure, not on finding the best model
to fit the conditional heteroskedasticity of the returns.
18
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19
13. Kroner, K.F., Ng, V.K. (1998) Modeling asymmetric comovements of asset returns.
The Review of Financial Studies 11, 817-844.
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Sinica 1, 247-269.
20
Table 1. Summary statistics of asynchronous and synchronized returns
Statistics
AUS
asynchronous synchronized
synchronized asynchronous
A
B
Min.
-6.9527
-9.1342
-7.2852
-7.1859
Mean
-6.17E-17
-5.58E-05
.000244
2.52E-17
Max.
7.2164
6.6114
7.3330
5.5230
Stdev.
1.0641
1.0776
1.0903
.9947
Skewness
-.1345
-.2293
-.1196
-.0988
Kurtosis
-.4281
-.0072
-.3977
1.1525
.5035
.5045
.5014
.5040
Pr(r>0)
JB
1118.97
1534.00
1142.43
2891.96
Arch(12)
141.51
127.93
161.37
489.38
LB(12)
24.53
14.57
22.13
29.00
US
synchronized
A
-7.2654
3.32E-06
5.5289
1.0024
-0.0918
1.1408
0.5027
2874.81
492.25
31.83
synchronized
B
-7.1157
.000117
5.6678
.9993
-.0681
1.0686
.5044
2772.43
485.78
30.96
JB test is the Jarque-Bera test for normality; the LB(12) tests for H0: ρ (1) =
= ρ (12 ) = 0 ; ARCH test
tests for the existence of conditional heteroskedasticity. All of the test statistics above reject the
corresponding null hypothesis at 1% level. ‘synchronized j’ refers to the synchronized data derived from
method j, j=A,B.
Table 2. (Cross) serial correlation matrices of asynchronous and synchronized returns
Order(i)
Asynchr Synchron Asynchr Synchron Asynchr Synchron Asynchr
onous
ized
onous
ized
onous
ized
onous
{
Method A: {(xt, yt): xt=rAUS,t and yt=rUS,t}; synchronized returns= xt = rˆAUS , t , yt = rˆUS , t
AUS.AU
S
0
1
2
3
4
5
1.0000
.0166
-.0333**
-.0009
-.0033
-.0304*
AUS.US
1.0000
-.0200
-.0187
.0099
-.0109
-.0278*
.0727***
.4025***
.0138
.0279*
-.0049
.0127
US.US
.4884***
.0234
.0237
.0118
.0011
-.0104
Method B: {(xt, yt): xt=rUS,t-1 and yt=rAUS,t}; synchronized returns=
AUS.AUS
AUS.US
1.0000
-.0108
-.0254
-.0288*
.0059
-.0391**
{x
t
}
US.AUS
1.0000
-.0409***
-.0158
-.0252
.0095
-.0377**
.0727***
-.0087
-.0029
.0123
-.0215
-.0177
=~
rUS ,t , yt = ~
rAUS ,t }
US.US
Synchron
ized
.4884***
-.0238
.0000
-.0013
-.0039
-.0262*
US.AUS
1.0000
1.0000
.4026***
.5027***
1.0000
1.0000
.4026***
.5027***
0
.0160
-.0112
.0138
-.0064
-.0108
-.0408***
.0724***
-.0005
1
-.0334***
-.0286*
.0279*
.0213
-.0253
-.0149
-.0087
-.0038
2
-.0009
-.0013
-.0049
-.0097
-.0287
-.0283
-.0033
-.0030
3
-.0034
-.0017
.0127
.0122
.0058
.0115
.0122
.0130
4
-.0304*
-.0301*
-.0113
-.0144
-.0391**
-.0374**
-.0217
-.0235
5
The column ‘x.x’ gives the serial correlation of series {xt}. The column ‘x.y’ gives the cross serial
correlation between {xt} and {Liyt}, where L is the lag operator. In addition, ‘Asynchronous’ refers to the
raw data and ‘Synchronized’ refers to the synchronized returns. ‘*’: significance at the 10% level; ‘**’:
significance at the 5% level; ‘***’: significance at the 1% level.
21
Table 3. Quasi-maximum likelihood estimation results
Parameter
Method A
Full
Variance Target
Estimate Std. Err. Estimate Std. Err.
.0998
.0863
.0958**
.0536
φ
11,1
Method B
Full
Variance Target
Estimate Std. Err. Estimate
Std. Err.
.1124
.1812
.1043
.2267
φ21,1
-.0618**
.0294
-.1016***
.0321
-.2837
.3484
-.2694
.4491
φ12,1
-1.4874*
.8672
-.2987
.4408
-.4816
.3868
-.4944
.5531
φ22,1
.5284***
.1861
.6691***
.148
.5792***
.1476
.5896***
.1573
ψ 11,1
-.1215
.0934
-.1213**
.0633
-.1383
.1773
-.13
.2222
ψ 21,1
.0808***
.0298
.1183***
.0331
.3332
.3495
.3184
.4552
ψ 12,1
1.9185**
.8656
.7290**
.4407
.5533
.3842
.5654
.5499
ψ 22,1
-.5228***
.1917
-.6633***
.1524
-.5735***
.1624
-.5856***
.1744
C11
C21
C22
A11
A21
A12
A22
B11
B21
B12
B22
.1248***
.0285
.0740***
.0178
.0415
.0273
.1236**
.0562
.0492*
.0284
.1090***
.0397
.1588***
.0205
.1619***
.0191
.2201***
.041
.2080***
.0332
-.0201
.0216
-.0228
.0192
.0019
.0308
.0041
.0268
.0436**
.0186
.0390***
.0168
.0416*
.0235
.0360*
.0219
.2269***
.0318
.2188***
.0272
.1772***
.0327
.1733***
.0285
.9785***
.0065
.9821***
.0046
.9750***
.0106
.9777***
.008
.0045
.0047
.0046
.0042
-.0056
.0086
-.0056
.0068
-.0112*
.0059
-.0083**
.0043
-.0063
.0073
-.0051
.0059
.9712***
.0084
.9727***
.0074
.9706***
.0122
.9729***
.0093
AIC
21171.86
21177.18
21116.62
21110.37
BIC
21291.53
21277.95
21236.28
21211.13
HQ
21214.27
21212.89
21159.03
21146.08
*: significance at the 10% level; **: significance at the 5% level; ***: significance at the 1% level
22
Table 4. Summary statistics of estimated conditional second moments
Variables
Min.
Q1
Median
mean
Q3
h
A
AUS , t
Max.
Stdev.
Uncond
0.7442
0.8569
0.9262
0.9597
1.0263
1.7764
0.1399
1.0641
A
AUS , t
0.7981
0.9292
1.0174
1.0616
1.1416
1.9830
0.1875
1.0776
B
AUS , t
0.7982
0.9284
1.0098
1.0488
1.1223
2.3079
0.1721
1.0641
B
hˆAUS
,t
hˆ
h
0.8147
0.9492
1.0340
1.0736
1.1486
2.4158
0.1787
1.0903
A
US , t
0.4426
0.6534
0.8188
0.9283
1.1165
2.4942
0.3752
0.9947
A
hˆUS
,t
0.4403
0.6555
0.8238
0.9345
1.1237
2.5051
0.3792
1.0024
B
US , t
h
h
0.4665
0.6479
0.8131
0.9244
1.1192
2.4475
0.3684
0.9948
B
US , t
0.7981
0.9292
1.0174
1.0616
1.1416
1.9830
0.3677
0.9993
S
A
t
-0.9517
0.0044
0.0652
0.0869
0.1333
1.4580
0.1717
0.0770
Ŝ
A
t
-0.0483
0.2493
0.3697
0.5377
0.6584
3.3249
0.4723
0.5275
S
B
t
-0.0439
0.2105
0.3247
0.4308
0.5212
3.9196
0.3827
0.4262
Ŝ
B
t
0.0934
0.2968
0.4269
0.5534
0.6534
4.5826
0.4383
0.5477
A
t
-0.4145
0.0056
0.0877
0.0908
0.1709
0.6204
0.1308
0.0727
A
t
-0.0706
0.3892
0.4685
0.4683
0.5489
0.8488
0.1276
0.4884
B
t
-0.0554
0.3083
0.4050
0.3978
0.4932
0.8108
0.1416
0.4026
B
t
0.1171
0.4279
0.5086
0.5036
0.5880
0.8520
0.1226
0.5027
hˆ
R
R̂
R
R̂
Notations: j = AUS and US and
i
j ,t
i
t
A
t
i =method A and method B; h , S and R
are the estimated
conditional standard deviation, conditional covariance and conditional correlation, respectively, on calendar
date t and ‘ x̂ ’ are calculated from the corresponding synchronized data. The last column gives the
unconditional values of the variables of interest.
23
Table 5. The OLS estimation results
Variables
Model
Conditional standard deviation: AUS
A
A
hˆAUS
, t ~ 1 + hAUS , t
1.1753***
.7692
~ 1+ h
-.0128***
1.0359***
.9959
~ 1+ h
B
AUS , t
.2316***
.6942***
.7291
A
ˆB
hˆAUS
, t ~ 1 + hAUS , t
A
AUS , t
.0777***
.9165***
.7625
A
US , t
-.0034***
1.0104***
.9992
B
B
hˆUS
, t ~ 1 + hUS , t
.0127***
.9971***
.9982
-.0005
1.0047*
.9731
-.0154***
1.0166***
.9712
.4020***
1.5623***
.3224
.0612***
1.1422***
.9947
.0639***
.0534***
.0142
.0584***
.8662***
.646
.4107***
.6344***
.4229
.1611***
.8611***
.9897
.1417***
-.1281***
.0192
.3158***
.3028***
.0846
hˆ
A
US , t
A
US , t
h
~ 1+ h
~ 1+ h
B
US , t
A
ˆB
hˆUS
, t ~ 1 + hUS , t
SˆtA ~ 1 + StA
SˆtB ~ 1 + StB
S ~ 1+ S
SˆtA ~ 1 + SˆtB
Rˆ A ~ 1 + R A
A
t
Conditional correlation
t
B
t
t
RˆtB ~ 1 + RtB
R ~ 1+ R
A
t
B
t
RˆtA ~ 1 + RˆtB
Notations:
R2
-.0663***
B
AUS , t
h
Conditional covariance
β
B
AUS , t
hˆ
Conditional standard deviation: US
α
j = AUS and US and i =method A and method B; h , S and R
i
j ,t
i
t
A
t
are the estimated
conditional standard deviation, conditional covariance and conditional correlation, respectively, on calendar
date t and ‘ x̂ ’ are calculated from the corresponding synchronized data. In addition, y ~ −1 + x means
that we run a liner regression model: y
= α + βx + error . The t-statistic is used to test whether α = 0 ,
β = 1.
24