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A correlated bivariate Poisson jump model for foreign exchange

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A correlated bivariate Poisson jump model for foreign exchange Powered By Docstoc
					Empirical Economics (2003) 28:669–685
DOI 10.1007/s00181-003-0153-9




A correlated bivariate Poisson jump model
for foreign exchange
Wing H. Chan
Department of Economics, Wilfrid Laurier University, Waterloo, Ontario, Canada N2L 3C5
(e-mail: wchan@wlu.ca)

First Version Received: June 2001/Final Version Received: July 2002



Abstract. This paper develops a new bivariate jump model to study jump
dynamics in foreign exchange returns. The model extends a multivariate
GARCH parameterization to include a bivariate correlated jump process. The
conditional covariance matrix has the Baba, Engle, Kraft, and Kroner (1989)
structure, while the bivariate jumps are governed by a Correlated Bivariate
Poisson (CBP) function. Using daily data we find evidence of both indepen-
dent currency specific jumps, as well as jumps common to both exchange rates
of the Canadian dollar and Japanese Yen against the U.S. dollar. The paper
concludes by investigating a time-varying structure for the arrival of jumps
that relaxes the assumption of constant and bounded jump correlation
imposed by the CBP function.

Key words: Correlated Poisson jump, bivariate GARCH, time-varying jump
intensity


1. Introduction

Empirical research has so far failed to find a predictable component of
exchange rates using linear models and this failure has led to a shift of focus
towards nonlinear modeling. The Autoregressive Heteroskedasticity (ARCH)


I am indebted to two anonymous referees and the editor, Baldev Raj for helpful suggestions. I am
also grateful for helpful comments from Adolf Buse, Ramazan Gencay, Rehim Kilic, John
Maheu, Alex Maynard, Denis Pelletier, Denise Young, and seminar participants at the Tenth
Annual Symposium of the Society for Nonlinear Dynamics and Econometrics (SNDE), Federal
Reserve Bank of Atlanta 2002; the Midwest Econometrics Group (MEG) Meetings, Federal
Reserve Bank of Kansas City 2001; Canadian Economics Association (CEA) Meetings, McGill
University 2001.
670                                                                    W. H. Chan


model, proposed in the seminal paper by Engle (1982) and later generalized
(GARCH) by Bollerslev (1986), has been most influential. This parsimonious
structure implies serial correlation in the second moment and volatility clus-
tering, suggesting that periods of high (low) volatility are likely to be followed
by periods of high (low) volatility.
    Multivariate GARCH models (Bollerslev 1990, Baba et al. 1989, Boller-
slev et al. 1988, Diebold and Nerlove 1989, Engle et al. 1990) emerge as a
natural extension of the univariate model. The motivations behind the mul-
tivariate generalization are possible volatility spillover effects and a quest for
an understanding of how one market might influence another. For example, if
all currencies being studied are expressed in terms of a common denomination
(U.S. dollar), any shock to the U.S. market may easily be transmitted to all
currencies, producing a similar GARCH effect.
    Although multivariate GARCH models are adequate in terms of account-
ing for heteroskedasticity, these models do not fully capture another stylized
fact: leptokurtosis in the unconditional distribution, often observed in
financial data. Many solutions have been proposed in the literature. For
example, the normal density can be replaced by a fat tail distribution such as
the Student t distribution or Power Exponential distribution. Other alter-
natives include the Poisson jump model of Press (1967) which introduces an
independent jump process with the arrival of jumps governed by a Poisson
distribution. This model has been applied successfully to daily exchange rates
by Akgiray and Booth (1988), Tucker and Pond (1988) and Hsieh (1989). This
approach is attractive as more can be learned from modeling leptokurtosis as
some systematic pattern than by a fat tail distribution. Although jumps are
unobservable, an expost filter can always be constructed to infer the prob-
ability of jumps.
    The theoretical framework of the Poisson jump model has been extended
to permit a time-varying jump distribution in recent literature. For example,
Das (1998) uses dummy variables to capture day of the week effect on the jump
intensity. Chernov et al. (1999) allow the jump intensity to depend on the size
of previous jumps and a stochastic volaility factor. Chan and Maheu (2002)
model jump intensity as an approximate ARMA process.
    The presence of jumps can be explained by either news content entering
the market or more interestingly market microstructure – order flow recently
proposed by Evans and Lyons (2001). The former implies that market par-
ticipants may react to certain kind of unanticipated news systematically over
time. Modeling these pattern is no easy task, the Poisson distribution pro-
vides a simple entry point which has proven to be useful in empirical studies.
The market microstructure approach (Evans and Lyons 2001) relies on
portfolio shift not being common knowledge. Dealers observe interdealer
order flow to learn about these shift. As the market gradually aggregating
these informations, the transactions between dealers and non-dealer public
may create series of jumps in the exchange rate they are trading.
    This paper develops a new bivariate jump model to study jump dynamics
in foreign exchange returns. The model extends a multivariate GARCH
parameterization to include a correlated jump process. The conditional
covariance matrix has the Baba et al. (1989) structure, while the bivariate
jumps are governed by a Correlated Bivariate Poisson (CBP) function. This
function provides a bivariate discrete counting process which has been used to
solve problems in different context such as the relationship between voluntary
Correlated bivariate Poisson jumps                                         671


and involuntary job changes (Jung and Winkelmann 1993), and firm’s deci-
sion to enter or exit an industry (Mayer and Chappell 1992). Modeling two
discrete dependent variables using the CBP function is also discussed in
Gourieroux et al. (1984).
    There are several advantages of using this CBP-GARCH model: (i) It
mixes smooth volatility movement with abrupt changes in returns. The incor-
poration of jumps provides one possible solution to account for unconditional
leptokurtosis. (ii) It allows one to identify two types of systematic jumps:
jumps specific to one currency and jumps that occur to both currencies at
the same time. (iii) The frequency of jumps may change over time depending
on the market conditions. (iv) The interrelationship between currencies are
driven by two distinct sources: normal random noises and systematic cor-
related jumps. Jump dynamics may provide a better understanding of the
comovement between currencies, which has important implications in risk
management and hedging such as deriving the optimal hedging ratio (Baillie
and Myers 1991).
    The model is applied to ten years of daily spot exchange rates on the
Canadian Dollar (CD) and Japanese Yen (JY) against the US dollar. We find
systematic independent as well as correlated jumps with significant jump size
in both currencies. However, the Canadian dollar on average experiences
more positive jumps causing depreciation, whereas the Japanese Yen
encounters mostly negative jumps. We have also generalized the jump
frequency to be time varying and find that the arrival of independent jumps is
determined by the currency’s volatility, whereas the arrival of correlated
jumps is jointly determined by the volatilities in both currencies.
    The paper is organized as follow: Section 2 describes the Correlated
Bivariate Poisson (CBP) jump model. Section 3 provides a simple data descrip-
tion with summary statistics. Section 4 applies the CBP-GARCH model to the
foreign exchange rates in our data set. Section 5 offers conclusion.


2. Model

The model is a combination of the GARCH model (Bollerslev 1986) and
the Poisson Correlation function (M’Kendrick 1926, Campbell 1934). The
Correlated Bivariate Poisson (CBP-GARCH) model is defined as follow:
   Rt ¼ l þ et þ Jt                                                      ð2:1Þ
where Rt is a 2 Â 1 vector of returns consisting of a constant mean l, a
random disturbance et , and a jump component Jt . The random disturbance
follows a bivariate normal distribution with zero mean and variance covar-
              ~
iance matrix Ht .
    Within any single time period t, a currency may experience ‘‘n’’ number of
jumps depending on the news content entering the market. The jump com-
ponent therefore is constructed as a sum of a series of random variables Yi :
    Xn
       Yi ¼ Y1 þ Y2 þ Y3 þ Á Á Á þ Yn                                    ð2:2Þ
    i¼1

Each of these random variables can be interpreted as a jump size which is
governed by a normal distribution with constant mean h and constant
672                                                                                           W. H. Chan


variance d. We assume that these mean and variance parameters remain
the same across time, but differ across currencies. In other words, the jump
sizes for the two currencies can be characterized as
                        2                              2
      Y1t; i $ N ðh1 ; d1 Þ and      Y2t; j $ N ðh2 ; d2 Þ:                                        ð2:3Þ

    The jump component enters the mean equation with an expected value
of zero which is achieved by subtracting the expected values from the
series of random jumps. In a bivariate framework, the jump component is
defined as
         " Pn                     P n1t         #
              i¼1 Y1t; i À EtÀ1 ð  i¼1 Y1t; i Þ
               1t

    Jt ¼ P n2t                    P n2t           :               ð2:4Þ
             j¼1 Y2t; j À EtÀ1 ð   j¼1 Y2t; j Þ

Jt has a bivariate normal distribution with zero mean and variance covariance
matrix Dt . The normal disturbance and the jump components are assumed to
be independent.
     Two discrete counting variables n1t and n2t control the arrival of jumps
and they are constructed by three independent Poisson variables namely
  Ã    Ã         Ã
n1t ; n2t , and n3t . Each one of these variables has a probability density function
given by

          Ã                   eÀki kj
                                    i
      P ðnit ¼ j j UtÀ1 Þ ¼           :                                                            ð2:5Þ
                                j!

                                    Ã
The expected value and variance of nit are each equal to ki , which is also
referred to as the expected number of jumps or the jump intensity.
    The correlated jump intensity counters (M’Kendrick 1926, Campbell
1934) are defined as
             Ã     Ã                      Ã     Ã
      n1t ¼ n1t þ n3t       and    n2t ¼ n2t þ n3t :                                               ð2:6Þ

By construction, each of these counting variables, nit , is capable of gen-
erating independent jumps as well as correlated jumps. The independent
                           Ã       Ã
jumps are initiated by n1t and n2t in time period t. The correlated jumps
                                                      Ã
are produced by the additional Poisson variable n3t which contributes
jumps to both series.
                                                                Ã
    Using the change of variables method and integrating out n3t yields the
joint probability density for n1t and n2t as:

                                          minði; jÞ
                                           X                                k1 k2jÀk k3
                                                                             iÀk      k
      P ðn1t ¼ i; n2t ¼ j j UtÀ1 Þ ¼                  eÀðk1 þk2 þk3 Þ                              ð2:7Þ
                                            k¼0
                                                                        ði À kÞ!ð j À kÞ!k!
and the expected number of jumps is equal to

      Eðnit Þ ¼ ki þ k3 :                                                                          ð2:8Þ

Since both n1t and n2t are monotone functions of independent Poisson ran-
dom variables, their covariance is k3 and their correlation is always positive in
the form of
Correlated bivariate Poisson jumps                                                              673


                                      k3
    Corrðn1t ; n2t Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :                              ð2:9Þ
                        ðk1 þ k3 Þðk2 þ k3 Þ
The assumption of positive correlation can be relaxed by using other bivariate
distributions. However, in this model, there is built-in flexibility in terms of
jumps. Although the model assumes that the number of jumps in the two
series are positively related, the mean of the jump sizes can be different. This
implies that it is possible for both series to experience a jump at the same time,
but that one jump may have a positive effect on one series and a negative
effect on the other.
    Note that the bivariate Poisson function simplifies to two independent
Poisson process when k3 ¼ 0. When either one of the independent jump
intensities k1 or k2 is equal to zero, the distribution is called semi-Poisson. In
the situation of k1 ¼ 0, positive probability is only assigned to the situation
where n1t      n2t . In the case of k1 ¼ k2 ¼ 0, the only possible values that the
number of jumps can take is n1t ¼ n2t .
    A drawback of using the Poisson Correlated function is that the corre-
lation is limited to the range:
                                "                      #
                                   k1 þ k3 1=2 k2 þ k3 1=2
    0 Corrðn1t ; n2t Þ      min               ;               :             ð2:10Þ
                                   k2 þ k3       k1 þ k3
Given a constant value for each jump intensity parameter ki , the correlation
between n1t and n2t will have a theoretically imposed upper bound. To relax the
assumption of constant correlation and the implicitly imposed upper bound,
we propose a simple parametric structure for each of the ki , so that the like-
lihood of jumps may change over time. The time varying jump intensities are
defined as
                2 2
    k1t ¼ k1 þ g1 r1tÀ1 ;                                                                   ð2:11Þ
                2 2
    k2t ¼ k2 þ g2 r2tÀ1 ;                                                                   ð2:12Þ
                2 2        2 2
    k3t ¼ k3 þ g3 r1tÀ1 þ g4 r2tÀ1 ;                                                        ð2:13Þ

where ritÀ1 is the rate of return for currency i at time t À 1. The jump
intensities are assumed to be related to market conditions1 which are reflected
    2
in ritÀ1 as an approximation of last period’s volatility. Similarly, the covar-
iance k3t is governed by the variations in last period volatilities from both
series. This parametric structure not only introduces additional jump

1
  The jump intensities are positively related to the volatilities. This assumption can be relaxed by
using logarithms such as
                        2
  k1t ¼ exp½k1 þ g1 ln r1tÀ1 Š                                                               ð2:14Þ

                        2
  k2t ¼ exp½k2 þ g2 ln r2tÀ1 Š                                                               ð2:15Þ

                        2             2
  k3t ¼ exp½k3 þ g3 ln r1tÀ1 þ g4 ln r2tÀ1 Š                                                 ð2:16Þ

This specification allows for both positive and negative correlation between the jump intensities
and past volatilities.
674                                                                                             W. H. Chan


dynamics to the model, but also allows for a time varying correlation between
the counting variables n1t and n2t . Since kit is realized at time t À 1, one step
ahead forecasts of the average number of jumps is also possible.
    Combining the GARCH model with the CBP function, the probability
density function for Rt given i jumps in currency 1 and j jumps in currency 2
is defined by
                                             1
      f ðRt j n1t ¼ i; n2t ¼ j; UtÀ1 Þ ¼           jHij; t jÀ1=2 exp½Àuij; t Hij; t uij; t Š;
                                                                       0       À1
                                                                                                    ð2:17Þ
                                           2p N =2
where uij; t is the usual error term with the jump component Jij; t representing
the effect of i and j jumps:
                                                            !
                              r1t À l1 À ih1 þ ðk1 þ k3 Þh1
   uij; t ¼ Rt À l À Jij; t ¼                                 :           ð2:18Þ
                              r2t À l2 À jh2 þ ðk2 þ k3 Þh2

Hij; t is the variance covariance matrix of the returns given i jumps in currency
1 and j jumps in currency 2. Under the assumption that the normal dis-
turbance, et , is independent of the jump component, Hij; t can be separated
into two parts: the variance covariance matrix for the normal random dis-
             ~
turbance Ht and the variance covariance matrix for the jump component Dij; t .
                                       ~
     The variance covariance matrix Ht for the normal disturbance is assumed
to have the bivariate symmetric BEKK form (Baba et al. 1989) which is
defined by

                       e e0
      Ht ¼ C 0 C þ A 0 ~tÀ1 ~tÀ1 A þ B 0 HtÀ1 B
      ~                                  ~                                                          ð2:19Þ

where C is an upper triangular matrix, and A and B are symmetric matrices.
~tÀ1 refers to the sum of a disturbance and a jump component. The positive
e
definiteness of the variance covariance matrix is ensured by the quadratic
form.
    The variance covariance matrix for the jump component Dij; t is derived
from the assumption that the correlation between the jump sizes is constant
across contemporaneous equations and zero across time:
      CorrðY1t ; Y2t Þ ¼ q12   and     CorrðY1t ; Y2s Þ ¼ 0 t 6¼ s                                  ð2:20Þ
Therefore, conditional on i and j jumps, the variances of the jump compo-
        P           P           2         2
nentsffiffiffiffi i Y1t and j Y2t are id1 and jd2 , respectively. The covariance will be
   p
q12 ijd1 d2 which completes the specification of the variance covariance
matrix for the jump component as
                     2        pffiffiffiffi    !
                  id1      q12 ijd1 d2
   Dij; t ¼       pffiffiffiffi          2      :                                ð2:21Þ
               q12 ijd1 d2     jd2
   The variance covariance matrix for the CBP-GARCH model Hij; t will
                                       ~
always be positive definite as long as Ht is positive definite. By construction,
the variance matrix for the jump component Dij; t is well defined given i and j
jumps and therefore Hij; t as the sum of two positive definite matrix will also
be positive definite.
Correlated bivariate Poisson jumps                                                 675


   Introducing correlated jumps in the bivariate model has important
implications for the covariance between the currencies. Assuming that there
are no jumps in the series, the covariance will be measured by the off-diagonal
elements in the BEKK structure as h12; t . The time varying covariance will be
solely determined by the characteristics of the normal disturbances. In the
presence of jumps, the covariance between two currencies will be driven not
only by the covariance between the normal disturbances, but also by the
characteristics of the jumps.
   Finally, to complete the specification, the conditional density of returns is
defined by
                   XX
                   1 1
   P ðRt jUtÀ1 Þ ¼        f ðRt j n1t ¼ i; n2t ¼ j; UtÀ1 ÞP ðn1t ¼ i; n2t ¼ j j UtÀ1 Þ:
                    i¼0 j¼0

                                                                    ð2:22Þ
Although jumps are unobservable, an expost filter can be constructed as

                                  f ðRt j n1t ¼ i; n2t ¼ j; UtÀ1 Þ
   P ðn1t ¼ i; n2t ¼ j j Ut Þ ¼
                                            P ðRt jUtÀ1 Þ

                                  Â P ðn1t ¼ i; n2t ¼ j j UtÀ1 Þ                ð2:23Þ

to identify jumps in the series. The log likelihood function is simply the sum
of the log conditional densities:
           XN
    ln L ¼    ln P ðRt jUtÀ1 Þ:                                          ð2:24Þ
            t¼1
Since the information matrix is not block diagonal, evaluation of the full like-
lihood is required.
    To estimate the CBP-GARCH model, a truncation point must be selected
for the probability function in Eq. (2.22). We choose a truncation point which
is sufficiently large so that the likelihood function and parameter estimates
stabilize to a set of converged values. The models are estimated with a
Fortran 77 compiler using a variable metric function maximization (Koval
1997) subroutine.



3. Data

We use daily spot exchange rates for the Canadian Dollar (CD) and Japanese
Yen (JY) relative to the U.S. Dollar from January 4, 1989 to December 31,
1998. Table 1 provides summary statistics for the returns expressed as one
hundred times the first differenced logarithms.
   The summary statistics show that there is significant excess kurtosis in the
CD and the modified Ljung Box (West and Cho 1995) statistics indicate serial
correlation in the squared returns. Similar descriptive statistics are found for
the JY. Serial correlation exists in both the first and second moments. The
excess kurtosis coefficient is slightly larger in JY as compared to the one in
CD. The KPSS test statistics (Kwiatkowski et al. 1992) with five lags are 0.153
676                                                                                    W. H. Chan

Table 1. Summary statistics (1989–1998)

             Canadian Dollar

             Mean               Standard           Skewness            Excess             Q(5)
                                deviation                              kurtosis

Rt           0.0098             0.2833             0.0659              5.9336             7.34
             (0.0056)           (0.0062)           (0.1546)            (0.5250)           [0.06]
jRt j        0.2066             0.1940             2.1017              9.9956             280.00
             (0.0038)           (0.0057)           (0.1556)            (1.2766)           [0.00]
Rt2          0.0803             0.1785             6.5232              67.859             252.00
             (0.0035)           (0.0144)           (0.6781)            (12.1154)          [0.00]

             Japanese Yen

             Mean               Standard           Skewness            Excess             Q(5)
                                deviation                              kurtosis

Rt           À0.0040            0.7263             À0.6453             7.8111             13.60
             (0.0144)           (0.0188)           (0.2166)            (1.2252)           [0.01]
jRt j        0.5098             0.5174             2.5027              13.7631            266.00
             (0.0102)           (0.0183)           (0.2717)            (3.1018)           [0.00]
Rt2          0.5276             1.3783             9.4629              156.08             276.00
             (0.0273)           (0.1703)           (2.0221)            (54.58)            [0.00]

Summary statistics for daily Canadian Dollar and Japanese Yen spot returns from January 4,
1989 to Dec 31, 1998. Q(5) are modified Ljung-Box Statistics robust to heteroskedasticity for
serial correlation with 5 lags. Standard errors robust to heteroskedasticity are in parentheses, and
p-values are in square brackets.




for CD and 0.124 for JY which do not exceed the 95% critical value of 0.463.
Therefore, the null hypothesis of stationary process is not rejected for both
series.
    Figure 1 present plots of spot returns for the two currencies. A general
trend of depreciation of the Canadian dollar is apparent for the last decade,
while the Japanese Yen has experienced both periods of appreciation and
depreciation over the sample period. Volatility clustering is common in both
return series and therefore suggests the plausibility of the GARCH structure.
There is no close relationship between the two series as the simple correlation
coefficient between the returns is 0.08.


4. Results

The results for the univariate Poisson-GARCH model are presented in
Table 2. The first thing to note are the strong GARCH effects and the per-
sistence of the conditional variance, with parameters a þ b ¼ 0:9668 for CD
and 0.9701 for JY. The Poisson jump components are also very important in
modeling these exchange rates as shown by the significance of both the jump
intensity and size parameters. The estimated jump intensities are 0.1839 and
0.2661 for CD and JY, respectively. The jumps in CD are mostly positive with
a small variance, whereas the jumps in JY have a negative mean and a
relatively larger variance. The LB statistics show no serial correlation in the
Correlated bivariate Poisson jumps                                          677




Fig. 1. Returns: Canadian Dollar (A) and Japanese Yen (B)


standardized residuals. Overall, these results indicate that although the
(previously reported) simple correlation coefficient (0.08) shows no indication
of a relationship between the returns, both exchange rate return series exhibit
systematic jumps. Any attempt to model the two series jointly must take into
account these jump components.
    The bivariate models are reported in Table 3. BEKK, CBP-GARCH, and
CBP-GARCH-R2 refer to the plain bivariate GARCH model, the GARCH
model with Correlated Bivariate Poisson (CBP) jumps, and the CBP-
GARCH model with time varying jump intensities, respectively. All three
models show a similar GARCH effect. In comparison with the BEKK model,
the GARCH structures of the jump models are essentially the same. The
estimated cross equation parameters are all small (<0.01) and insignificant.
The parameters associated with the autoregressive conditional variance, b1
and b2 , and the squared past prediction errors, a1 and a2 , are of the same
order of magnitude. These results suggest that adding a jump component does
not qualitatively or quantitatively affect the conditional variance structure of
678                                                                                    W. H. Chan

Table 2. Estimates of the constant jump models
             qffiffiffiffi   X nt
   Rt ¼ l þ ht zt þ       Yt; k
                                k¼1

                       2                    2
      Yt; k $ NIDðh; d Þ;        ht ¼ x þ aetÀ1 þ bhtÀ1
                     expÀk k j
      P ðnt ¼ jÞ ¼                ;   zt $ NIDð0; 1Þ
                           j!

Parameter                                              Canada                          Japan

l                                                      0.0051                          À0.0045
                                                       (0.0050)                        (0.0131)
x                                                      0.0004                          0.0012
                                                       (0.0002)                        (0.0014)
a                                                      0.0690                          0.0335
                                                       (0.0102)                        (0.0063)
b                                                      0.8978                          0.9366
                                                       (0.0143)                        (0.0125)
d                                                      0.3316                          0.8531
                                                       (0.0424)                        (0.1034)
h                                                      0.0649                          À0.1980
                                                       (0.0273)                        (0.0727)
k                                                      0.1839                          0.2661
                                                       (0.0424)                        (0.0845)

Q                                                      8.19                            6.59
                                                       [0.14]                          [0.25]
Q2                                                     5.96                            10.36
                                                       [0.30]                          [0.06]

log likelihood                                         À154.44                         À2496.53

Standard errors are in parentheses. p-values are in square brackets. Q is the modified Ljung-Box
portmanteau test, robust to heteroskedasticity, for serial correlation in the standardized residuals
with 5 lags for the respective models. Q 2 is the same test for the squared standardized residuals.



the random disturbance. The CBP-GARCH model does, however, use ad-
ditional information ( jump dynamics) already available in the data series to
better understand the comovement between the returns.
    Turning to the jump component of the CBP-GARCH model, all param-
eters related to the jump dynamics are highly significant. The likelihood ratio
test statistic for the presence of jumps is 405.08 with p-value equal to 0.0.2
The mean h and variance d of the jump sizes have values closely resembling to
the results from the univariate Poisson-GARCH model. For CD, the mean
and variance are 0.0649 and 0.3316 for the univariate case and 0.0685 and
0.3185 for the CBP-GARCH model. For JY, the mean and variance are
À0.1980 and 0.8531 for the univariate case and À0.2175 and 0.8654 for the
bivariate case. Note that even after introducing the correlated jumps, the two


2
  Testing for jumps is complicated by unidentification of the nuisance parameters hi and di under
the null hypothesis. Drost, Nijman, and Werker (1998) propose a kurtosis-based test for the
presence of jumps. However, the power of this test is not clear in this bivariate setting.
Correlated bivariate Poisson jumps                                                               679

Table 3. Estimates of the CBP-GARCH models

              Mean & variance                                           Jump component

              BEKK          CBP-          CBP-                          CBP-          CBP-
                            GARCH         GARCH-R2                      GARCH         GARCH-R2

l1            0.0020        0.0049        0.0017              d1        0.3185        0.3031
              (0.0048)      (0.0050)      (0.0046)                      (0.0370)      (0.0328)
l2            0.0052        À0.0035       0.0082              h1        0.0685        0.0569
              (0.0126)      (0.0131)      (0.0123)                      (0.0239)      (0.0207)
c1            0.0366        0.0217        0.0208              d2        0.8654        0.8125
              (0.0053)      (0.0057)      (0.0052)                      (0.1191)      (0.0917)
c2            0.1040        0.0400        0.0396              h2        À0.2175       À0.1819
              (0.0133)      (0.0321)      (0.0159)                      (0.0782)      (0.0597)
c12           À0.0093       0.0030        0.0032              k1        0.1160        0.1021
              (0.0077)      (0.0201)      (0.0092)                      (0.0558)      (0.0540)
a1            0.2511        0.2532        0.2272              k2        0.1478        0.1431
              (0.0170)      (0.0195)      (0.0207)                      (0.0793)      (0.0717)
a2            0.2265        0.1664        0.1521              k3        0.1036        0.1145
              (0.0185)      (0.0177)      (0.0156)                      (0.0381)      (0.0474)
a12           À0.0086       À0.0053       À0.0009             g1                      0.8001
              (0.0073)      (0.0062)      (0.0062)                                    (0.2549)
b1            0.9594        0.9485        0.9550              g2                      0.2852
              (0.0060)      (0.0078)      (0.0073)                                    (0.1015)
b2            0.9637        0.9728        0.9750              g3                      0.4588
              (0.0061)      (0.0082)      (0.0052)                                    (0.2344)
b12           0.0034        0.0024        0.0015              g4                      0.1796
              (0.0027)      (0.0042)      (0.0024)                                    (0.1026)
                                                              q12                     À0.1754
                                                                        0.0883        (0.1099)
                                                                        (0.1168)
Q1            8.80          9.18          8.89
              [0.11]        [0.10]        [0.11]
 2
Q1            4.71          5.01          4.11
              [0.45]        [0.41]        [0.53]
Q2            7.53          7.64          7.69
              [0.18]        [0.17]        [0.17]
 2
Q2            2.74          6.45          8.32
              [0.73]        [0.26]        [0.13]
ln L          À2822.27      À2619.73      À2601.95

BEKK, CBP-GARCH, and CBP-GARCH-R2 refer to the plain bivariate GARCH model, the
GARCH model with Correlated Bivariate Poisson jumps, and the CBP-GARCH model with time
varying jump intensities, respectively. Standard errors are in parentheses. p-values are in square
brackets. Qi is the modified Ljung-Box portmanteau test, robust to heteroskedasticity, for serial
correlation in the standardized residuals with 5 lags for the respective models. Qi2 is the same test
for the squared standardized residuals.



currencies still appear to have opposite jump sizes with CD experienced mostly
positive jumps in the last decade and JY encountered mostly negative jumps,
with triple the size of the ones in CD on average.
   The jump intensities provide the keys to how the jumps can be broken
down into independent and correlated components. In the univariate
Poisson GARCH model, the jump intensity (the average number of jumps) in
CD and JY are reported as 0.1839 and 0.2661, respectively. The equivalent
680                                                                 W. H. Chan




Fig. 2. Conditional probabilities of jumps


                                                    ^
measure in the CBP-GARCH model is k1 þ k3 ¼ 0:2196 for CD^
      ^    ^
and k2 þ k3 ¼ 0:2514 for JY. Since independent jumps are initiated by k1
and k2 , the average numbers of independent jumps in CD and JY are
0.1160 and 0.1478, respectively. The average number of correlated jumps,
k3 , in both series is estimated as 0.1036, implying that almost half the
jumps occurring in both series are correlated, while the other half are
independent.
    There is significant correlation in the arrival of jumps between the two
currencies which is shown by Corrðn1t ; n2t Þ ¼ 0:4312. The joint probabilities
of jumps are depicted in the top panel of Fig. 2. Given the set of estimated
intensity parameters from the CBP-GARCH model, the joint probabilities of
jumps are centered at the origin and the probability of having over two jumps
in both currencies on the same day is highly unlikely. This is a reasonable
result given the expected number of jumps on average are 0.2196 in CD
and 0.2514 in JY which are not even close to one jump. The other possible
explanation is that this is due to the fact that we assume constant correlation
between the counting variables across time. The probability of jumps may
change depending on the market conditions which motivates the time varying
generalization of the jump intensities.
Correlated bivariate Poisson jumps                                             681




Fig. 3. Time varying jump intensities


    We next generalize the jump intensity parameters to be time varying and
positively related to previous volatilities as specified in Eqs. (2.11)–(2.13). The
jump components in the CBP-GARCH-R2 model are significant and the
                            ^
jump intensity parameters ki remain in the same range as before, with changes
of less than 0.01. The most interesting results relate to the time varying para-
meters which are on average very large in size and highly significant. Take for
example the case of CD, the effect of last period’s volatility on jump intensity
g1 is almost sevenfold the size of the constant term k1 ¼ 0:1021. In other
words, the likelihood of having jumps is directly related to the market con-
ditions, reflected by the change in volatilities.
    Figure 3 graphs the jump dynamics which govern the independent jumps.
       ^         ^
Both k1 and k2 exhibit high variations in the sample period, and the
assumption of constant intensities is clearly invalid. The jump intensities for
CD and JY vary around the range of 0.1 to 1.4 and 0.1 to 1.7, respectively.
Having more than 0.5 jumps in any of the two currencies in one single day is
not uncommon. The covariance and correlation between the arrival of jumps
682                                                                       W. H. Chan




Fig. 4. Correlated jump intensity (A) and jump counter correlations (B)


                                                                    ^
are depicted in Fig. 4. Surprisingly the correlated jump intensity k3 seems to
exhibit variations with higher magnitude compared to the other two
                                 ^
independent jump intensities. k3 has values as low as 0.1145 and as high as
2.350. The results imply that the correlated jump dynamics are very important
in understanding the comovement between the two currencies.
    The joint probabilities created by these time varying jump intensities are
depicted in the lower panel of Figure 2. We have chosen the values k1 ¼ 1:1738,
k2 ¼ 1:7166 and k3 ¼ 2:3961 on October 8, 1998 to depict the effect of relaxing
the assumption of constant intensities on the joint probabilities. The graph
clearly shows that the joint probabilities are centered around three jumps in
both currencies. In fact, if we look at a particular number, the probability of
having 2 jumps in CD and 3 jumps in JY is 0.05. It is not uncommon to have 2
or 3 jumps jointly in both currencies depending on the date. This indicates that
the correlated jump dynamics seem to be the source of the changing rela-
tionship between the two currencies.
Correlated bivariate Poisson jumps                                           683


    The same conclusion can be made by looking at the correlation
between n1t and n2t in Fig. 4. The correlation varies from 0.4710 to 0.6515
and the assumption of constant correlation is clearly rejected. Diagnostic
statistics show no serial correlation in standardized residuals and squared
standardized residuals. The likelihood ratio test statistic of no time varying
jump dynamics is 35.56 and the null hypothesis is rejected using the critical
value from the chi square distribution with four degrees of freedom.
    In summary, the BEKK structure is adequate in modeling the conditional
variance covariance structure between CD and JY. However, a jump com-
ponent must be added to the model in order to more fully capture all the
dynamics in the mean equation. With the Poisson Correlation function, we are
able to identify independent as well as correlated jumps in the two currencies.
A further generalization also discovers that the correlated jump dynamics may
evolve over time which may serve as a good indicator to the future movement
of the currencies.
    As a cautionary note, in general these models perform well in a variety of
situations and we have shown that the CBP-GARCH models work well with
daily exchange rate series. However, the jump component may change with
time aggregation, and full characterization of the underlying data generating
process requires further investigation. A promising approach is to generalize
                                                              ¸
model parameters as functions of frequency dynamics (Gencay et al. 2002 and
      ¸
Gencay et al. 2002). The jump model proposed in this paper is designed for
data series of relative high frequency. Jumps may disappear as we aggregate
data over time smoothing out most of the abrupt changes. In addition, a long
data series is required to support the Poisson structure with the possibility of
infinite number of jumps.


5. Conclusion

In this paper, we propose a bivariate GARCH model with jumps for the
foreign exchange market. The BEKK structure is adopted for the condi-
tional variance covariance matrix and the jump component is governed by the
Poisson Correlation function. The model is applied to ten years of daily
spot exchange rates for Canadian Dollar and Japanese Yen against the U.S.
dollar.
    The model provides several improvements over existing models. First, the
CBP-GARCH model combines the popular multivariate GARCH model with
a jump component so that it can capture smooth volatility movements as well
as abrupt changes in the rates of return. Second, using the Poisson Correlation
function to govern the jump component, the model is able to generate cor-
related jumps in both series in addition to the independent jumps. Third, we
generalize the model to have time varying jump intensities controlling the
arrival of jumps which helps to understand the relationship between the jump
dynamics and volatilities. Finally, allowing time varying jump intensities also
relax the assumption of constant and bounded jump correlation between
currencies. This CBP-GARCH model can be useful for asset pricing, modeling
risk premium in foreign currency futures, and modeling optimal commodity
hedge ratio.
    Our results show empirical properties of two foreign exchange rates:
Canadian dollar and Japanese Yen. There are significant independent as
684                                                                               W. H. Chan


well as correlated jumps in both series. The frequency of jumps depends on
the past volatilities in both series. The time varying jump intensities may
provide important information as how the correlation between the two
currencies may evolve over time. Future research should focus on scale
consistency and time aggregation to study the relationship of foreign
exchange rates across different time horizons


References
Akgiray V, Booth GG (1988) Mixed jump-diffusion process modeling of exchange rate
    movements. Review of Economics and Statistics 70:631–637
Baba T, Engle RF, Kraft D, Kroner KF (1989) Multivariate simultaneous generalized arch.
    UCSD, Department of Economics, manuscript
Baillie RT, Myers RJ (1991) Bivariate garch estimation of the optimal commodity futures hedge.
    Journal of Applied Econometrics 6:109–124
Bollerslev T (1986) Generalized autoregressive conditional heteroskedasticity. Journal of
    Econometrics 31:309–328
Bollerslev T (1990) Modelling the coherence in short-run nominal exchange rates: a multivariate
    generalized arch model. Review of Economics and Statistics 72:498–504
Bollerslev T, Engle RF, Wooldridge JM (1988) A capital asset pricing model with time-varying
    covariances. Journal of Political Economy 96:116–131
Campbell JT (1934) The Poisson correlation function. Proceedings of the Edinburgh Mathe-
    matical Society, Series 2 4:18–26
Chan WH, Maheu JM (2002) Conditional jump dynamics in stock market returns. Journal of
    Business & Economic Statistics 20:377–389
Chernov M, Gallant AR, Ghysels E, Tauchen G (1999) A new class of stochastic volatility models
    with jumps: theory and estimation. Working Paper, CIRANO
Das SR (1998) Poisson-gaussian processes and the bond market. NBER Working Paper 6631
Diebold FX, Nerlove M (1989) The dynamics of exchange rate volatility: a multivariate latent
    factor arch model. Journal of Applied Econometrics 4:1–21
Drost FC, Nijman TE, Werker BJM (1998) Estimation and testing in models containing both
    jumps and conditional heteroskedasticity. Journal of Business and Economic Statistics
    16:237–243
Engle RF (1982) Autoregressive conditional heteroskedasticity with estimates of the UK
    inflation. Econometrica 50:987–1008
Engle RF, Ng VK, Rothschild M (1990) Asset pricing with a factor-arch covariance structure:
    empirical estimates for treasury bills. Journal of Econometrics 45:213–237
Evans MDD, Lyons RK (2001) Order flow and exchange rate dynamics. Haas School of
    Business, UC Berkeley, Working Paper
     ¸
Gencay R, Ballocchi G, Daorogna M, Olsen R, Pictet O (2002) Real-time trading models and the
    statistical properties of foreign exchange rates. International Economic Review 43:463–491
       ¸
Gencay R, Daorogna M, Olsen R, Pictet O (2002) Real-time foreign trading models and market
    behavior. Journal of Economic Dynamics and Control forthcoming
Gourieroux C, Monfort A, Trognon A (1984) Pseudo maximum likelihood methods: applications
    to poisson models. Econometrica 52:701–720
Hsieh D (1989) Testing for nonlinearity in daily foreign exchange rate changes. Journal of
    Business 62:339–368
Jung RC, Winkelmann R (1993) Two aspects of labour mobility: a bivariate poisson regression
    approach. Empirical Economics 18:543–556
Koval JJ (1997) Algorithm as 319: variable metric function minimization. Applied Statistics
    46:515–521
Kwiatkowski D, Phillips PCB, Schmidt P, Shin Y (1992) Testing the null hypothesis of
    stationarity against the alternative of a unit root. Journal of Econometrics 54:159–178
Mayer WJ, Chappell WF (1992) Determinants of entry and exit: an application of the
    compounded bivariate poisson distribution to U.S. industries. Southern Economic Journal
    58:770–778
Correlated bivariate Poisson jumps                                                        685

M’Kendrick AG (1926) Applications of mathematics to medical problems. Proceedings of the
    Edinburgh Mathematical Society 44:98–130
Press SJ (1967) A compound events model for security prices. Journal of Business 40:317–335
Tucker AL, Pond L (1988) The probability distribution of foreign exchanges: tests of candidate
    processes. Review of Economics and Statistics 70:638–647
West KD, Cho D (1995) The predictive ability of several models of exchange rate volatility.
    Journal of Econometrics 69:367–391

				
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