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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 ð2:14Þ 2 k2t ¼ exp½k2 þ g2 ln r2tÀ1 ð2:15Þ 2 2 k3t ¼ exp½k3 þ g3 ln r1tÀ1 þ g4 ln r2tÀ1 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 ; 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. 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