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Testing for Co-jumps in Financial Markets

Testing for Co-jumps in Financial Markets Abstract In this paper, we introduce the notion of co-jumps within the co-features framework. We formulate a limiting theory of co-jumps and discuss their discrete sample properties. In the presence of idiosyncratic price jumps, we identify the notion of weak co-jumps. We illustrate the empirical relevance of the proposed framework via an empirical application using the components of the Dow Jones Industrial Average 30 index running from 1 January 2010 to 30 June 2012, sampled at a five-min frequency. This paper proposes a novel theoretical framework to assess common price jumps in a multivariate framework using the notion of co-features, that is, the existence of a linear combination of time series in which individual features are eliminated, as originally proposed by Engle and Kozicki (1993) and more recently reconsidered in the special issue of Journal of Business and Economic Statistics (2007). There is a huge body of literature on the identification of price jumps in the univariate context. Several procedures have been proposed to test for the presence of price jumps defined as discontinuity in the price process. See, for example, Aït-Sahalia and Jacod (2009); Aït-Sahalia and Jacod (2011); Aït-Sahalia and Jacod (2012); Andersen, Dobrev, and Schaumburg (2011, 2012); Barndorff-Nielsen and Shephard (2004b, 2006); Jiang and Oomen (2008); Lee and Mykland (2008); Lee and Hannig (2010); Huang and Tauchen (2005), and Mancini (2009). Dumitru and Urga (2012) evaluate the performance of alternative non-parametric price jump tests. In contrast, a multivariate framework allows one to identify common jumps between stochastic processes as highlighted in the seminal work by Barndorff-Nielsen and Shephard (2004a). Bollerslev, Law, and Tauchen (2008) test for the presence of portfolio-wide systemic price jumps and focus in particular on systemic common jumps without counterparts on the individual time series level. This framework is extended by Liao and Anderson (2011) using the range-based indicators proposed by Bannouh, Van Dijk, and Martens (2009). Jacod and Todorov (2009) propose a procedure to test for the joint occurrence of price jump arrivals at a pair of time series. In an empirical study, Lahaye, Laurent, and Neely (2011) estimate the joint probabilities of common price jump arrivals and also suggest a joint statistic for the estimation of common price jumps and map common jumps in response to specific macro-news for a broader range of assets such as USD exchange rates, US Treasury bonds futures, and US equity futures. Based on factor regressions techniques in Bollerslev, Todorov, and Li (2013), Bollerslev, Li, and Todorov (2016) relate the identification of co-jumps to estimating factors and loadings for the strict factor model and verify the method on the sensitivity of the stock price jumps of Microsoft to the market jumps. In the case of an unknown factor structure, Aït-Sahalia and Xiu (2015) and Pelger (2016) provide estimators based on principal component analysis. Li, Todorov, and Tauchen (2017) propose a framework to evaluate the dependency between jumps of two processes and to test for the relationship implied by the linear standard factor model. Caporin, Kolokolov, and Renò (2015) introduce a non-parametric test based on the smoothed estimators of integrated variance to provide evidence for statistically significant multivariate jumps in stock prices. Gilder, Shackleton, and Taylor (2014) analyze the contemporaneous co-jumps of US equities and link them to Federal Fund Target Rate announcements. Jiang, Lo, and Verdelhan (2011) conclude that surprises related to macroeconomic news announcements have limited power in explaining jumps for bonds. Aït-Sahalia, Cacho-Diaz, and Hurd (2009) use common price jumps for assets in the same sector to evaluate the optimal portfolio in the presence of jumps. Finally, in a recent paper, Bandi and Renò (2016) propose a novel identification strategy for price and volatility co-jumps to relate some significant price changes to volatility jumps. This paper contributes to the current literature on common price jumps as follows: we propose a novel notion of co-jumps identified within the co-feature framework. In particular, the notion of co-jumps is linked to the diversification of price jumps out of a basket of assets. Thus, co-jumps can be intuitively understood as a possibility to diversify the price jumps completely out of a portfolio. Bollerslev, Law, and Tauchen (2008) discuss the case of a portfolio of common jumps which cannot be diversified out, and as such it serves to identify common jumps. We further extend the notion of co-jumps to cases where each asset has idiosyncratic price jumps, implying the absence of co-jumps. We define weak co-jumps as a linear combination of assets with minimum contribution of price jumps to the quadratic covariance. This notion is further supported by the empirical results of Bollerslev, Law, and Tauchen (2008) and Lahaye, Laurent, and Neely (2011). We report an empirical illustration of the co-jump framework using the individual assets of the Dow Jones Industrial Average 30 (DJIA 30) index running from 1 January 2010 to 30 June 2012, sampled at a five-minute frequency. The paper is organized as follows: In Section 1, we provide the definition and the main properties of co-jumps, and specify the procedure to test for the presence of co-jumps. In Section 2, we report an empirical illustration of the co-jumps using constituents of the DJIA 30 index and provide a robustness check with respect to the multiple testing bias. Section 3 concludes. 1 Modeling Co-Jumps In this paper, we introduce the notion of co-jumps within the co-feature framework. Consider an N-dimensional vector of log-prices, log⁡P={log⁡Pt}0≤t≤T ⁠, defined on a filtered probability space (Ω,F,(Ft)t≥0,P) over the finite time interval [0,T] ⁠. The vector of log-prices is a semi-martingale Ft-adapted and its continuous-time dynamics can be specified by the following stochastic differential equation: dlog⁡Pt=μtdt+σtdBt+dJt , (1) where μt is N-dimensional vector of drift processes, σt is the (N×N)-dimensional covariance matrix, dBt is the N-dimensional vector of independent standard Brownian motions, and dJt is the N-dimensional vector of pure jump Lévy processes. The presence of price jumps in Equation (1) implies that a (N×N)-dimensional quadratic variation process Σt can be written as: Σt=Σt(c)+Σt(d) , (2) where Σt(c) represents the continuous part of the semi-martingale process, Σt(c)=∫0tσsσ′sds ,  with {Σt(c)}i,j<∞ , i,j=1,…,N ,<∞ , (3) and Σt(d) represents the discontinuous part of the semi-martingale process, Σt(d)=∑j=1Ntcjc′j ,  with {Σt(d)}i,j<∞ , i,j=1,…,N ,<∞ , (4) where cj is an N-dimensional vector for which there exists at least one i=1,…,N such that dlog⁡Ptj−(i)>0 ⁠, and Nt is the number of tj≤t ⁠. The decomposition of the quadratic variance allows us to map the presence of price jumps in terms of quadratic variation. 1.1 Co-jumps Consider the integrated counterpart of the N-dimensional process described in Equation (1) log⁡Pt=∫0tμsds+∫0tσsdBs+∑j=1Ntcj , (5) where each of the components has discontinuities in the interval [0,t] ⁠. The N-dimensional Brownian semi-martingale process with finite-activity jumps is closed with respect to the stochastic integration under a linear transformation given by a (p×N)-dimensional matrix Ω, where the matrix can in general be time-dependent (see Jacod and Shiryaev, 2003). The p-dimensional process given as a linear transformation of log⁡Pt can be written as Ωlog⁡Pt=∫0tΩμsds+∫0tΩσsdBs+∑j=1NtΩcj=∫0tμs(p)ds+∫0tσs(p)dBs+∑j=1Ntcj(p) , (6) which is a p-dimensional Brownian semi-martingale with finite activity jumps. If log⁡Pt is a Brownian semi-martingale, the product Ωlog⁡Pt is Brownian semi-martingale as well. Our aim is to find a Ω such that Ωlog⁡Pt is a Brownian semi-martingale with Σt(d)≡0 ⁠. If Ω exists, Ωlog⁡Pt does not have any price jumps despite the presence of price jumps in each component. This characteristic is in fact the notion of co-features, as introduced by Engle and Kozicki (1993), and in the special issue of the Journal of Business and Economic Statistics (2007). 1.1.1 Definition of co-jumps For the N-dimensional process log⁡Pt defined in Equation (5) with each of the components having a discontinuity in the interval [0,t] ⁠, co-jumps are defined as the existence of the N-dimensional constant vector Ω, different from the zero vector, such that for the process Ωlog⁡Pt the discontinuous part of the semi-martingale process in the covariance disappears Σt(d)=∑j=1NtΩ′cjc′jΩ=0 . The vector Ω is called the co-jump vector and the space of all co-jump vectors spans the co-jump space. The vector is an ex post computation that sheds some light on the commonality of equity price jumps, and the task of finding an ex ante co-feature vector is deferred to future research. 1.2 Identification of Jumps Following Barndorff-Nielsen and Shephard (2006), let us now consider the G^Ω-statistic defined as G^Ω=M1/2IV̂M−QV̂MIQ̂M , (7) where IV̂M is the estimator of the Integrated Variance (IV̂M→p∫0tσs2ds) ⁠, QV̂M is the estimator of the Quadratic Variance (QV̂M→p∫0tσs2ds+∑j=1Ntcj2) ⁠, IQ̂M is the estimator of the Integrated Quarticity (IQ̂M→p∫0tσs4ds) ⁠. For a univariate log-price process log⁡Pt generated by Equation (1), under the null hypothesis of no price jumps, G^Ω→DN(0,ϑ) with →D denoting a stable convergence in law and ϑ is some known constant depending on the particular choice of estimators used. Thus, for the N-dimensional process log⁡Pt in the interval [0,t] there is a co-jump if a vector Ω exists such that the G^Ω-statistic for the univariate process Ωlog⁡Pt does not reject the null hypothesis. The asymptotic properties of the G^Ω-statistic under the null hypothesis hold when there is no discontinuous part of the price process Ωlog⁡Pt ⁠. We identify co-jumps when the discontinuous part of the quadratic variance disappears, that is, Σt(d)=0 ⁠. In this paper, we consider a sparse sampling approach to deal with market micro-structure noise since it provides a reasonable trade-off between accuracy and numerical feasibility at chosen sampling frequency. However, our framework can be extended to employ alternative techniques such as the pre-averaging method by Podolskij and Vetter (2009), employed by Aït-Sahalia and Jacod (2009) and Aït-Sahalia, Jacod, and Li (2012), or the combination of different time scales by Zhang, Mykland, and Ait-Sahalia (2005) and Zhang (2011). 1.3 An Additional Co-jumps Feature: Weak Co-jumps The notion of co-jumps introduced above aims to find a linear combination which eliminates the jumps. When idiosyncratic price jumps are present for each component log⁡Pt (see, for instance, Jiang, Lo, and Verdelhan, 2011; Lahaye, Laurent, and Neely, 2011; Lee, 2012) co-jumps do not exist as they cannot be fully eliminated. To this purpose, we modify the notion of co-jumps such that we weaken the requirement for the elimination of the jump term in Equation (6). 1.3.1 Definition of weak co-jumps We define weak co-jumps as a linear combination which minimizes the presence of price jumps in Ωlog⁡Pt ⁠. The minimization of price jumps is done through the G^Ω-statistic. In the presence of price jumps, that is, when the null hypothesis does not hold, G^Ω→p∞ ⁠, with →p denoting convergence in probability. Thus, we define a weak co-jump as a linear combination(s), Ω, which maximizes the G^Ω-statistic. Then, the weak co-jump portfolio (vector) is Ω=arg⁡max⁡Ω*,|Ω*|=1 G^Ω*. Thus, for an N-dimensional process log⁡Pt we evaluate the difference between two vectors Ω(1) and Ω(2) via the G^Ω-statistic. 2 An Empirical Illustration In this section, we illustrate the empirical validity of the proposed theoretical framework by evaluating the presence of co-jumps in high-frequency data. 2.1 Data and Index Selection We use the individual assets of the DJIA 30 index running from 1 January 2010 to 30 June 2012 provided by the NYSE TAQ database. We use data on trades only and utilize the appropriate cleaning mechanism by Barndorff-Nielsen et al. (2009). As a result, the data are sampled at a five-minute frequency. Such a sampling frequency filters out the presence of the market micro-structure noise, while preserving the high-frequency features. The trading day starts at 9:30 a.m. and ends at 4:00 p.m., which yields 79 log-prices per day. Our sample contains 621 trading days in total. We split the DJIA 30 index into six indices, each with five companies, based on the capitalization at the beginning of the sample. We illustrate the notion of co-jumps using the high-cap index containing the five most capitalized companies and the low-cap index the five least capitalized companies in the DJIA 30. Table 1 presents the composition of each of the indices as well as the market capitalization of companies at the beginning of the sample. The results using the indices with the remaining DJIA 30 companies are available upon request. Table 1. Market capitalization and descriptive statistics for DJIA 30 ID Index selection Descriptive statistics of five-min log-returns (%) Market cap ($bn) No. σ S K Min Max XOM 360.98 High-cap 0.129 −0.089 10.970 −1.627 1.586 MSFT 266.46 0.147 −0.069 12.601 −2.177 2.190 WMT 211.16 0.102 0.101 12.735 −1.518 1.222 PG 183.81 0.096 −0.038 15.844 −1.529 1.630 JNJ 175.23 0.096 −0.208 14.461 −1.368 1.238 BA 39.03 Low-cap 0.160 −0.060 9.647 −1.737 1.658 CAT 37.16 0.191 −0.148 9.925 −2.282 1.949 DD 31.72 0.162 0.011 9.972 −2.035 1.924 TRV 28.74 0.129 0.077 12.942 −1.541 1.496 AA 12.47 0.223 −0.121 9.384 −2.722 2.178 ID Index selection Descriptive statistics of five-min log-returns (%) Market cap ($bn) No. σ S K Min Max XOM 360.98 High-cap 0.129 −0.089 10.970 −1.627 1.586 MSFT 266.46 0.147 −0.069 12.601 −2.177 2.190 WMT 211.16 0.102 0.101 12.735 −1.518 1.222 PG 183.81 0.096 −0.038 15.844 −1.529 1.630 JNJ 175.23 0.096 −0.208 14.461 −1.368 1.238 BA 39.03 Low-cap 0.160 −0.060 9.647 −1.737 1.658 CAT 37.16 0.191 −0.148 9.925 −2.282 1.949 DD 31.72 0.162 0.011 9.972 −2.035 1.924 TRV 28.74 0.129 0.077 12.942 −1.541 1.496 AA 12.47 0.223 −0.121 9.384 −2.722 2.178 Notes: The table contains market capitalization in $bn as the markets closed on 31 December 2009 as retrieved from Bloomberg and standard deviation (σ), skewness (S), kurtosis (K), and minimum (Min) and maximum (Max) log-return of the five most capitalized (XOM, Exxon Mobil Corp; MSFT, Microsoft Corp; WMT, Wal-Mart Stores Inc., PG, Procter & Gamble Co.; JNJ, Johnson & Johnson) and the five least capitalized (BA, Boeing Co.; CAT, Caterpillar Inc.; DD, E.I. DuPont de Nemours & Co.; TRV, Travelers Cos. Inc.; AA, Alcoa Corp.) members of the DJIA 30. Table 1. Market capitalization and descriptive statistics for DJIA 30 ID Index selection Descriptive statistics of five-min log-returns (%) Market cap ($bn) No. σ S K Min Max XOM 360.98 High-cap 0.129 −0.089 10.970 −1.627 1.586 MSFT 266.46 0.147 −0.069 12.601 −2.177 2.190 WMT 211.16 0.102 0.101 12.735 −1.518 1.222 PG 183.81 0.096 −0.038 15.844 −1.529 1.630 JNJ 175.23 0.096 −0.208 14.461 −1.368 1.238 BA 39.03 Low-cap 0.160 −0.060 9.647 −1.737 1.658 CAT 37.16 0.191 −0.148 9.925 −2.282 1.949 DD 31.72 0.162 0.011 9.972 −2.035 1.924 TRV 28.74 0.129 0.077 12.942 −1.541 1.496 AA 12.47 0.223 −0.121 9.384 −2.722 2.178 ID Index selection Descriptive statistics of five-min log-returns (%) Market cap ($bn) No. σ S K Min Max XOM 360.98 High-cap 0.129 −0.089 10.970 −1.627 1.586 MSFT 266.46 0.147 −0.069 12.601 −2.177 2.190 WMT 211.16 0.102 0.101 12.735 −1.518 1.222 PG 183.81 0.096 −0.038 15.844 −1.529 1.630 JNJ 175.23 0.096 −0.208 14.461 −1.368 1.238 BA 39.03 Low-cap 0.160 −0.060 9.647 −1.737 1.658 CAT 37.16 0.191 −0.148 9.925 −2.282 1.949 DD 31.72 0.162 0.011 9.972 −2.035 1.924 TRV 28.74 0.129 0.077 12.942 −1.541 1.496 AA 12.47 0.223 −0.121 9.384 −2.722 2.178 Notes: The table contains market capitalization in $bn as the markets closed on 31 December 2009 as retrieved from Bloomberg and standard deviation (σ), skewness (S), kurtosis (K), and minimum (Min) and maximum (Max) log-return of the five most capitalized (XOM, Exxon Mobil Corp; MSFT, Microsoft Corp; WMT, Wal-Mart Stores Inc., PG, Procter & Gamble Co.; JNJ, Johnson & Johnson) and the five least capitalized (BA, Boeing Co.; CAT, Caterpillar Inc.; DD, E.I. DuPont de Nemours & Co.; TRV, Travelers Cos. Inc.; AA, Alcoa Corp.) members of the DJIA 30. The descriptive statistics reported in Table 1 reveal the large kurtosis for each asset and support the deviation from normality at a five-minute frequency consistently across all equities. 2.2 Co-jumps We now employ the notion of co-jumps with the G^Ω-statistic calculated for each trading day. We use α=0.05 to test for the null hypothesis that there is no price jump(s) during the given trading day. Following Barndorff-Nielsen and Shephard (2006), we estimate the Quadratic Variance, (QV̂) ⁠, the Integrated Variance, (IV̂) ⁠, and the Integrated Quarticity, (IQ̂) as:edit QV̂D=∑i=1MDri,D2 , (8) IV̂D=MDMD−1μ1−2∑i=2MD|ri−1,D||ri,D| , (9) IQ̂D=MDMD−31MDμ1−4∑i=4MD|ri−3,D||ri−2,D||ri−1,D||ri,D| . (10) where ri,D is the i-th log-return on the day indexed by D, where each day is divided into MD = 78 equally sized five-minute buckets, and μ1=E[|z|]=2/π with z∼N(0,1) ⁠. In such a case, the G^-statistic converges as G^D→DN(0,ϑ) with ϑ=(π2/4)+π−5≅0.609 ⁠. The test for the presence of price jumps during the trading day D at α=0.05 has the form H0:G^Ω(D)≥ϑΦ−1(α) no jumpHA:G^Ω(D)<ϑΦ−1(α) jump(s) , where Φ−1 is the inverse cumulative function of the standard normal distribution giving ϑΦ−1(α)≅−1.284 ⁠. In Figure 1, panels (a) and (b) depict the results of the co-jumps exercise for the high-cap and low-cap indices, the most and the least capitalized set of assets in the DJIA 30, respectively. For every trading day, we find the co-jump vector Ω such that it maximizes the G^Ω-statistic (crosses). For every trading day and each index, we test for the presence of co-jumps and confirm the presence of co-jumps as G^Ω(D)≥−1.284 ⁠, which is captured by the black long-dash line. This means that at the given sampling frequency, a linear combination of assets exists in the index such that the price jumps diversify out. Figure 1. Open in new tabDownload slide Co-jumps properties. (a) Co-jumps G^Ω-statistic: high-cap index. (b) co-jumps G^Ω-statistic: low-cap index. (c) Co-jump vector magnitudes: high-cap index. (d) Co-jump vector magnitudes: low-cap index. Notes: Panels (a) and (b) depict the G^Ω-statistic for the co-jump vector (crosses), for the equally weighted index (triangles), and the gray shaded area captures the region in which lies the G^Ω-statistic for each individual asset in the index. The black long-dash line denotes the α=0.05 critical value to test for the presence of price jumps, ϑΦ−1(α)≅−1.284 ⁠. Panels (c) and (d) depict the minimum and maximum of the co-jump vectors. The solid black line corresponds to the value of the equally weighted index. The vectors are normalized as that ∑i=15Ω(i)2=1 ⁠. Figure 1. Open in new tabDownload slide Co-jumps properties. (a) Co-jumps G^Ω-statistic: high-cap index. (b) co-jumps G^Ω-statistic: low-cap index. (c) Co-jump vector magnitudes: high-cap index. (d) Co-jump vector magnitudes: low-cap index. Notes: Panels (a) and (b) depict the G^Ω-statistic for the co-jump vector (crosses), for the equally weighted index (triangles), and the gray shaded area captures the region in which lies the G^Ω-statistic for each individual asset in the index. The black long-dash line denotes the α=0.05 critical value to test for the presence of price jumps, ϑΦ−1(α)≅−1.284 ⁠. Panels (c) and (d) depict the minimum and maximum of the co-jump vectors. The solid black line corresponds to the value of the equally weighted index. The vectors are normalized as that ∑i=15Ω(i)2=1 ⁠. Further, each of the two figures depicts the range (gray shaded area) of the individual G^-statistics calculated for each asset in the index. The results show that, for the majority of the trading days, at least one asset exists in the index such that the null is rejected for both indices. At the same time, there is no case where the null would be rejected for every asset and, therefore, there is no co-jump for all five assets at the same time. In addition, the two figures report the G^Ω-statistic for equally weighted index (triangles). The results indicate that in the majority of cases, the G^Ω-statistic for the equally weighted index is in the range implied by the individual assets. However, a significant number of cases show that the equally weighted index may either amplify or suppress the presence of price jumps. Table 2 suggests that the popular “1/N” strategy, or employing the equally weighted index, is not optimal for dealing with price jumps. Table 2. Number of co-jumps vs. the individual assets Portfolio 1 2 3 4 5 6 (A) G^Ω(1/N)<min⁡G^Ω(i) 60 51 47 55 60 70 (B) min⁡G^Ω(i)≤G^Ω(1/N)≤max⁡G^Ω(i) 494 492 466 484 469 469 (C) GΩ^(1/N)>max⁡G^Ω(i) 67 78 108 82 92 82 Portfolio 1 2 3 4 5 6 (A) G^Ω(1/N)<min⁡G^Ω(i) 60 51 47 55 60 70 (B) min⁡G^Ω(i)≤G^Ω(1/N)≤max⁡G^Ω(i) 494 492 466 484 469 469 (C) GΩ^(1/N)>max⁡G^Ω(i) 67 78 108 82 92 82 Notes: The table evaluates the frequency of: (A) the equally weighted index amplifies price jumps, the G^Ω(1/N)-statistic for the equally weighted portfolio is smaller than any individual asset; (B) the price jumps for the equally weighted index are comparable with price jumps at individual assets, the G^Ω(1/N)-statistic is in the range implied by the individual assets; and (C) the equally weighted index suppresses price jumps, the G^Ω(1/N)-statistic is higher than any individual assets. Table 2. Number of co-jumps vs. the individual assets Portfolio 1 2 3 4 5 6 (A) G^Ω(1/N)<min⁡G^Ω(i) 60 51 47 55 60 70 (B) min⁡G^Ω(i)≤G^Ω(1/N)≤max⁡G^Ω(i) 494 492 466 484 469 469 (C) GΩ^(1/N)>max⁡G^Ω(i) 67 78 108 82 92 82 Portfolio 1 2 3 4 5 6 (A) G^Ω(1/N)<min⁡G^Ω(i) 60 51 47 55 60 70 (B) min⁡G^Ω(i)≤G^Ω(1/N)≤max⁡G^Ω(i) 494 492 466 484 469 469 (C) GΩ^(1/N)>max⁡G^Ω(i) 67 78 108 82 92 82 Notes: The table evaluates the frequency of: (A) the equally weighted index amplifies price jumps, the G^Ω(1/N)-statistic for the equally weighted portfolio is smaller than any individual asset; (B) the price jumps for the equally weighted index are comparable with price jumps at individual assets, the G^Ω(1/N)-statistic is in the range implied by the individual assets; and (C) the equally weighted index suppresses price jumps, the G^Ω(1/N)-statistic is higher than any individual assets. To assess how much the individual assets contribute to the co-jumps, Figure 1, panels (c) and (d) present the range of the components of each co-jump vector identified above for the high-cap and low-cap indices, respectively. We consider co-jump vectors, normalized such that ∑i=15Ω(i)2=1 ⁠. First, the figure depicts the minimum and maximum of the magnitude of the co-jump vectors. In particular, for high-cap index, the maximum magnitude oscillated around 0.75, while the minimum oscillated around 0.1 with the least magnitude taking the value of 2.02·10−5 and the largest one 9.87·10−1 ⁠, taken from all indices. Therefore, each asset significantly contributes to the co-jump vector and the diversification of price jumps is clearly not caused by picking up an asset with few or no price jumps. The low-cap index provides the same qualitative conclusion. The results show the presence of co-jump vectors. From the index perspective, the price jumps can be ex post diversified out at a five-minute frequency. Further, the equally weighted index is not in general sufficient to eliminate price jumps. In some cases, it amplifies price jumps and thus the deviation from Gaussianity. 3 Conclusions In this paper, we employed the co-feature framework to introduce the notion of co-jumps defined as a linear combination of assets which is free of price jumps. We extended the notion of co-jumps to assets with idiosyncratic price jumps to define the weak co-jumps as a linear combination which minimizes the price jumps. We then linked the concept to the optimization of an index of assets with price jumps. We evaluated the empirical validity of the proposed framework using assets from the DJIA 30 index from 1 January 2010 to 30 June 2012 sampled at a five-minute frequency. We considered two indices, the high-cap index and the low-cap index, based on the market capitalization and tested for co-jumps. The results showed the presence of co-jumps at five-minute frequency, meaning that price jumps could be diversified out. However, our analysis showed that such diversification in general could not be achieved by creating equally weighted indices. Thus, the optimization in terms of removing price jumps should be considered as independent criteria. The findings in this paper suggest some further developments. First, it will be interesting to extend the framework in this paper to the case of a more general price arrival process, for example, mutually correlated self-exciting price jumps. Second, the sensitivity of the proposed framework, and in particular of the measure of commonality, can be transformed in the proper testing procedure for asynchronicity among the price jumps. Finally, it will also be interesting to develop an extension of the notion of co-arrivals to define the information measures capturing the different features of the multivariate arrival process. This is part of our ongoing research agenda. Acknowledgements This paper is a substantially revised version of a manuscript previously circulated under the title “Co-features in Finance: Co-arrivals and Co-jumps”. We wish to thank participants in the Conference on Skewness, Heavy Tails, Market Crashes, and Dynamics (University of Cambridge, 28–29 April 2014), the 13th OxMetrics User Conference (Aarhus, 5–6 September 2013), the International Conference on Systemic Risk, Contagion and Jumps (Cass Business School, 25 January 2013), in particular Yacine Aït-Sahalia and Neil Shephard, the Finance Research Workshop (Cass Business School, 4 March 2013), in particular Richard Payne, and the Economics Division Workshop (Nottingham Trent University, 6 March 2013) for comments. We are grateful to Simona Boffelli for discussions and insightful comments on various versions of the paper. Special thanks to the Editor, George Tauchen, for his patience, and the useful comments and suggestions provided during the review process which have directed us toward the successful conclusion of this submission. We also wish to thank an Associate Editor and two anonymous referees for very constructive suggestions which greatly helped to improve the paper. The usual disclaimer applies. This work was supported by the European Community's Seventh Framework Program FP7-PEOPLE-2011-IEF [under grant agreement number PIEF-GA-2011-302098 to J.N.] (Price Jump Dynamics), the Centre for Econometric Analysis, and the GAČR [grant number 14-27047S to J.N.]. References Aït-Sahalia Y. , Cacho-Diaz J. , Hurd T. . 2009 . Portfolio Choice with Jumps: A Closed-Form Solution . The Annals of Applied Probability 19 ( 2 ): 556 – 584 . Google Scholar Crossref Search ADS WorldCat Aït-Sahalia Y. , Jacod J. . 2009 . Testing for Jumps in a Discretely Observed Process . 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Testing for Co-jumps in Financial Markets

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Abstract

Abstract In this paper, we introduce the notion of co-jumps within the co-features framework. We formulate a limiting theory of co-jumps and discuss their discrete sample properties. In the presence of idiosyncratic price jumps, we identify the notion of weak co-jumps. We illustrate the empirical relevance of the proposed framework via an empirical application using the components of the Dow Jones Industrial Average 30 index running from 1 January 2010 to 30 June 2012, sampled at a five-min frequency. This paper proposes a novel theoretical framework to assess common price jumps in a multivariate framework using the notion of co-features, that is, the existence of a linear combination of time series in which individual features are eliminated, as originally proposed by Engle and Kozicki (1993) and more recently reconsidered in the special issue of Journal of Business and Economic Statistics (2007). There is a huge body of literature on the identification of price jumps in the univariate context. Several procedures have been proposed to test for the presence of price jumps defined as discontinuity in the price process. See, for example, Aït-Sahalia and Jacod (2009); Aït-Sahalia and Jacod (2011); Aït-Sahalia and Jacod (2012); Andersen, Dobrev, and Schaumburg (2011, 2012); Barndorff-Nielsen and Shephard (2004b, 2006); Jiang and Oomen (2008); Lee and Mykland (2008); Lee and Hannig (2010); Huang and Tauchen (2005), and Mancini (2009). Dumitru and Urga (2012) evaluate the performance of alternative non-parametric price jump tests. In contrast, a multivariate framework allows one to identify common jumps between stochastic processes as highlighted in the seminal work by Barndorff-Nielsen and Shephard (2004a). Bollerslev, Law, and Tauchen (2008) test for the presence of portfolio-wide systemic price jumps and focus in particular on systemic common jumps without counterparts on the individual time series level. This framework is extended by Liao and Anderson (2011) using the range-based indicators proposed by Bannouh, Van Dijk, and Martens (2009). Jacod and Todorov (2009) propose a procedure to test for the joint occurrence of price jump arrivals at a pair of time series. In an empirical study, Lahaye, Laurent, and Neely (2011) estimate the joint probabilities of common price jump arrivals and also suggest a joint statistic for the estimation of common price jumps and map common jumps in response to specific macro-news for a broader range of assets such as USD exchange rates, US Treasury bonds futures, and US equity futures. Based on factor regressions techniques in Bollerslev, Todorov, and Li (2013), Bollerslev, Li, and Todorov (2016) relate the identification of co-jumps to estimating factors and loadings for the strict factor model and verify the method on the sensitivity of the stock price jumps of Microsoft to the market jumps. In the case of an unknown factor structure, Aït-Sahalia and Xiu (2015) and Pelger (2016) provide estimators based on principal component analysis. Li, Todorov, and Tauchen (2017) propose a framework to evaluate the dependency between jumps of two processes and to test for the relationship implied by the linear standard factor model. Caporin, Kolokolov, and Renò (2015) introduce a non-parametric test based on the smoothed estimators of integrated variance to provide evidence for statistically significant multivariate jumps in stock prices. Gilder, Shackleton, and Taylor (2014) analyze the contemporaneous co-jumps of US equities and link them to Federal Fund Target Rate announcements. Jiang, Lo, and Verdelhan (2011) conclude that surprises related to macroeconomic news announcements have limited power in explaining jumps for bonds. Aït-Sahalia, Cacho-Diaz, and Hurd (2009) use common price jumps for assets in the same sector to evaluate the optimal portfolio in the presence of jumps. Finally, in a recent paper, Bandi and Renò (2016) propose a novel identification strategy for price and volatility co-jumps to relate some significant price changes to volatility jumps. This paper contributes to the current literature on common price jumps as follows: we propose a novel notion of co-jumps identified within the co-feature framework. In particular, the notion of co-jumps is linked to the diversification of price jumps out of a basket of assets. Thus, co-jumps can be intuitively understood as a possibility to diversify the price jumps completely out of a portfolio. Bollerslev, Law, and Tauchen (2008) discuss the case of a portfolio of common jumps which cannot be diversified out, and as such it serves to identify common jumps. We further extend the notion of co-jumps to cases where each asset has idiosyncratic price jumps, implying the absence of co-jumps. We define weak co-jumps as a linear combination of assets with minimum contribution of price jumps to the quadratic covariance. This notion is further supported by the empirical results of Bollerslev, Law, and Tauchen (2008) and Lahaye, Laurent, and Neely (2011). We report an empirical illustration of the co-jump framework using the individual assets of the Dow Jones Industrial Average 30 (DJIA 30) index running from 1 January 2010 to 30 June 2012, sampled at a five-minute frequency. The paper is organized as follows: In Section 1, we provide the definition and the main properties of co-jumps, and specify the procedure to test for the presence of co-jumps. In Section 2, we report an empirical illustration of the co-jumps using constituents of the DJIA 30 index and provide a robustness check with respect to the multiple testing bias. Section 3 concludes. 1 Modeling Co-Jumps In this paper, we introduce the notion of co-jumps within the co-feature framework. Consider an N-dimensional vector of log-prices, log⁡P={log⁡Pt}0≤t≤T ⁠, defined on a filtered probability space (Ω,F,(Ft)t≥0,P) over the finite time interval [0,T] ⁠. The vector of log-prices is a semi-martingale Ft-adapted and its continuous-time dynamics can be specified by the following stochastic differential equation: dlog⁡Pt=μtdt+σtdBt+dJt , (1) where μt is N-dimensional vector of drift processes, σt is the (N×N)-dimensional covariance matrix, dBt is the N-dimensional vector of independent standard Brownian motions, and dJt is the N-dimensional vector of pure jump Lévy processes. The presence of price jumps in Equation (1) implies that a (N×N)-dimensional quadratic variation process Σt can be written as: Σt=Σt(c)+Σt(d) , (2) where Σt(c) represents the continuous part of the semi-martingale process, Σt(c)=∫0tσsσ′sds ,  with {Σt(c)}i,j<∞ , i,j=1,…,N ,<∞ , (3) and Σt(d) represents the discontinuous part of the semi-martingale process, Σt(d)=∑j=1Ntcjc′j ,  with {Σt(d)}i,j<∞ , i,j=1,…,N ,<∞ , (4) where cj is an N-dimensional vector for which there exists at least one i=1,…,N such that dlog⁡Ptj−(i)>0 ⁠, and Nt is the number of tj≤t ⁠. The decomposition of the quadratic variance allows us to map the presence of price jumps in terms of quadratic variation. 1.1 Co-jumps Consider the integrated counterpart of the N-dimensional process described in Equation (1) log⁡Pt=∫0tμsds+∫0tσsdBs+∑j=1Ntcj , (5) where each of the components has discontinuities in the interval [0,t] ⁠. The N-dimensional Brownian semi-martingale process with finite-activity jumps is closed with respect to the stochastic integration under a linear transformation given by a (p×N)-dimensional matrix Ω, where the matrix can in general be time-dependent (see Jacod and Shiryaev, 2003). The p-dimensional process given as a linear transformation of log⁡Pt can be written as Ωlog⁡Pt=∫0tΩμsds+∫0tΩσsdBs+∑j=1NtΩcj=∫0tμs(p)ds+∫0tσs(p)dBs+∑j=1Ntcj(p) , (6) which is a p-dimensional Brownian semi-martingale with finite activity jumps. If log⁡Pt is a Brownian semi-martingale, the product Ωlog⁡Pt is Brownian semi-martingale as well. Our aim is to find a Ω such that Ωlog⁡Pt is a Brownian semi-martingale with Σt(d)≡0 ⁠. If Ω exists, Ωlog⁡Pt does not have any price jumps despite the presence of price jumps in each component. This characteristic is in fact the notion of co-features, as introduced by Engle and Kozicki (1993), and in the special issue of the Journal of Business and Economic Statistics (2007). 1.1.1 Definition of co-jumps For the N-dimensional process log⁡Pt defined in Equation (5) with each of the components having a discontinuity in the interval [0,t] ⁠, co-jumps are defined as the existence of the N-dimensional constant vector Ω, different from the zero vector, such that for the process Ωlog⁡Pt the discontinuous part of the semi-martingale process in the covariance disappears Σt(d)=∑j=1NtΩ′cjc′jΩ=0 . The vector Ω is called the co-jump vector and the space of all co-jump vectors spans the co-jump space. The vector is an ex post computation that sheds some light on the commonality of equity price jumps, and the task of finding an ex ante co-feature vector is deferred to future research. 1.2 Identification of Jumps Following Barndorff-Nielsen and Shephard (2006), let us now consider the G^Ω-statistic defined as G^Ω=M1/2IV̂M−QV̂MIQ̂M , (7) where IV̂M is the estimator of the Integrated Variance (IV̂M→p∫0tσs2ds) ⁠, QV̂M is the estimator of the Quadratic Variance (QV̂M→p∫0tσs2ds+∑j=1Ntcj2) ⁠, IQ̂M is the estimator of the Integrated Quarticity (IQ̂M→p∫0tσs4ds) ⁠. For a univariate log-price process log⁡Pt generated by Equation (1), under the null hypothesis of no price jumps, G^Ω→DN(0,ϑ) with →D denoting a stable convergence in law and ϑ is some known constant depending on the particular choice of estimators used. Thus, for the N-dimensional process log⁡Pt in the interval [0,t] there is a co-jump if a vector Ω exists such that the G^Ω-statistic for the univariate process Ωlog⁡Pt does not reject the null hypothesis. The asymptotic properties of the G^Ω-statistic under the null hypothesis hold when there is no discontinuous part of the price process Ωlog⁡Pt ⁠. We identify co-jumps when the discontinuous part of the quadratic variance disappears, that is, Σt(d)=0 ⁠. In this paper, we consider a sparse sampling approach to deal with market micro-structure noise since it provides a reasonable trade-off between accuracy and numerical feasibility at chosen sampling frequency. However, our framework can be extended to employ alternative techniques such as the pre-averaging method by Podolskij and Vetter (2009), employed by Aït-Sahalia and Jacod (2009) and Aït-Sahalia, Jacod, and Li (2012), or the combination of different time scales by Zhang, Mykland, and Ait-Sahalia (2005) and Zhang (2011). 1.3 An Additional Co-jumps Feature: Weak Co-jumps The notion of co-jumps introduced above aims to find a linear combination which eliminates the jumps. When idiosyncratic price jumps are present for each component log⁡Pt (see, for instance, Jiang, Lo, and Verdelhan, 2011; Lahaye, Laurent, and Neely, 2011; Lee, 2012) co-jumps do not exist as they cannot be fully eliminated. To this purpose, we modify the notion of co-jumps such that we weaken the requirement for the elimination of the jump term in Equation (6). 1.3.1 Definition of weak co-jumps We define weak co-jumps as a linear combination which minimizes the presence of price jumps in Ωlog⁡Pt ⁠. The minimization of price jumps is done through the G^Ω-statistic. In the presence of price jumps, that is, when the null hypothesis does not hold, G^Ω→p∞ ⁠, with →p denoting convergence in probability. Thus, we define a weak co-jump as a linear combination(s), Ω, which maximizes the G^Ω-statistic. Then, the weak co-jump portfolio (vector) is Ω=arg⁡max⁡Ω*,|Ω*|=1 G^Ω*. Thus, for an N-dimensional process log⁡Pt we evaluate the difference between two vectors Ω(1) and Ω(2) via the G^Ω-statistic. 2 An Empirical Illustration In this section, we illustrate the empirical validity of the proposed theoretical framework by evaluating the presence of co-jumps in high-frequency data. 2.1 Data and Index Selection We use the individual assets of the DJIA 30 index running from 1 January 2010 to 30 June 2012 provided by the NYSE TAQ database. We use data on trades only and utilize the appropriate cleaning mechanism by Barndorff-Nielsen et al. (2009). As a result, the data are sampled at a five-minute frequency. Such a sampling frequency filters out the presence of the market micro-structure noise, while preserving the high-frequency features. The trading day starts at 9:30 a.m. and ends at 4:00 p.m., which yields 79 log-prices per day. Our sample contains 621 trading days in total. We split the DJIA 30 index into six indices, each with five companies, based on the capitalization at the beginning of the sample. We illustrate the notion of co-jumps using the high-cap index containing the five most capitalized companies and the low-cap index the five least capitalized companies in the DJIA 30. Table 1 presents the composition of each of the indices as well as the market capitalization of companies at the beginning of the sample. The results using the indices with the remaining DJIA 30 companies are available upon request. Table 1. Market capitalization and descriptive statistics for DJIA 30 ID Index selection Descriptive statistics of five-min log-returns (%) Market cap ($bn) No. σ S K Min Max XOM 360.98 High-cap 0.129 −0.089 10.970 −1.627 1.586 MSFT 266.46 0.147 −0.069 12.601 −2.177 2.190 WMT 211.16 0.102 0.101 12.735 −1.518 1.222 PG 183.81 0.096 −0.038 15.844 −1.529 1.630 JNJ 175.23 0.096 −0.208 14.461 −1.368 1.238 BA 39.03 Low-cap 0.160 −0.060 9.647 −1.737 1.658 CAT 37.16 0.191 −0.148 9.925 −2.282 1.949 DD 31.72 0.162 0.011 9.972 −2.035 1.924 TRV 28.74 0.129 0.077 12.942 −1.541 1.496 AA 12.47 0.223 −0.121 9.384 −2.722 2.178 ID Index selection Descriptive statistics of five-min log-returns (%) Market cap ($bn) No. σ S K Min Max XOM 360.98 High-cap 0.129 −0.089 10.970 −1.627 1.586 MSFT 266.46 0.147 −0.069 12.601 −2.177 2.190 WMT 211.16 0.102 0.101 12.735 −1.518 1.222 PG 183.81 0.096 −0.038 15.844 −1.529 1.630 JNJ 175.23 0.096 −0.208 14.461 −1.368 1.238 BA 39.03 Low-cap 0.160 −0.060 9.647 −1.737 1.658 CAT 37.16 0.191 −0.148 9.925 −2.282 1.949 DD 31.72 0.162 0.011 9.972 −2.035 1.924 TRV 28.74 0.129 0.077 12.942 −1.541 1.496 AA 12.47 0.223 −0.121 9.384 −2.722 2.178 Notes: The table contains market capitalization in $bn as the markets closed on 31 December 2009 as retrieved from Bloomberg and standard deviation (σ), skewness (S), kurtosis (K), and minimum (Min) and maximum (Max) log-return of the five most capitalized (XOM, Exxon Mobil Corp; MSFT, Microsoft Corp; WMT, Wal-Mart Stores Inc., PG, Procter & Gamble Co.; JNJ, Johnson & Johnson) and the five least capitalized (BA, Boeing Co.; CAT, Caterpillar Inc.; DD, E.I. DuPont de Nemours & Co.; TRV, Travelers Cos. Inc.; AA, Alcoa Corp.) members of the DJIA 30. Table 1. Market capitalization and descriptive statistics for DJIA 30 ID Index selection Descriptive statistics of five-min log-returns (%) Market cap ($bn) No. σ S K Min Max XOM 360.98 High-cap 0.129 −0.089 10.970 −1.627 1.586 MSFT 266.46 0.147 −0.069 12.601 −2.177 2.190 WMT 211.16 0.102 0.101 12.735 −1.518 1.222 PG 183.81 0.096 −0.038 15.844 −1.529 1.630 JNJ 175.23 0.096 −0.208 14.461 −1.368 1.238 BA 39.03 Low-cap 0.160 −0.060 9.647 −1.737 1.658 CAT 37.16 0.191 −0.148 9.925 −2.282 1.949 DD 31.72 0.162 0.011 9.972 −2.035 1.924 TRV 28.74 0.129 0.077 12.942 −1.541 1.496 AA 12.47 0.223 −0.121 9.384 −2.722 2.178 ID Index selection Descriptive statistics of five-min log-returns (%) Market cap ($bn) No. σ S K Min Max XOM 360.98 High-cap 0.129 −0.089 10.970 −1.627 1.586 MSFT 266.46 0.147 −0.069 12.601 −2.177 2.190 WMT 211.16 0.102 0.101 12.735 −1.518 1.222 PG 183.81 0.096 −0.038 15.844 −1.529 1.630 JNJ 175.23 0.096 −0.208 14.461 −1.368 1.238 BA 39.03 Low-cap 0.160 −0.060 9.647 −1.737 1.658 CAT 37.16 0.191 −0.148 9.925 −2.282 1.949 DD 31.72 0.162 0.011 9.972 −2.035 1.924 TRV 28.74 0.129 0.077 12.942 −1.541 1.496 AA 12.47 0.223 −0.121 9.384 −2.722 2.178 Notes: The table contains market capitalization in $bn as the markets closed on 31 December 2009 as retrieved from Bloomberg and standard deviation (σ), skewness (S), kurtosis (K), and minimum (Min) and maximum (Max) log-return of the five most capitalized (XOM, Exxon Mobil Corp; MSFT, Microsoft Corp; WMT, Wal-Mart Stores Inc., PG, Procter & Gamble Co.; JNJ, Johnson & Johnson) and the five least capitalized (BA, Boeing Co.; CAT, Caterpillar Inc.; DD, E.I. DuPont de Nemours & Co.; TRV, Travelers Cos. Inc.; AA, Alcoa Corp.) members of the DJIA 30. The descriptive statistics reported in Table 1 reveal the large kurtosis for each asset and support the deviation from normality at a five-minute frequency consistently across all equities. 2.2 Co-jumps We now employ the notion of co-jumps with the G^Ω-statistic calculated for each trading day. We use α=0.05 to test for the null hypothesis that there is no price jump(s) during the given trading day. Following Barndorff-Nielsen and Shephard (2006), we estimate the Quadratic Variance, (QV̂) ⁠, the Integrated Variance, (IV̂) ⁠, and the Integrated Quarticity, (IQ̂) as:edit QV̂D=∑i=1MDri,D2 , (8) IV̂D=MDMD−1μ1−2∑i=2MD|ri−1,D||ri,D| , (9) IQ̂D=MDMD−31MDμ1−4∑i=4MD|ri−3,D||ri−2,D||ri−1,D||ri,D| . (10) where ri,D is the i-th log-return on the day indexed by D, where each day is divided into MD = 78 equally sized five-minute buckets, and μ1=E[|z|]=2/π with z∼N(0,1) ⁠. In such a case, the G^-statistic converges as G^D→DN(0,ϑ) with ϑ=(π2/4)+π−5≅0.609 ⁠. The test for the presence of price jumps during the trading day D at α=0.05 has the form H0:G^Ω(D)≥ϑΦ−1(α) no jumpHA:G^Ω(D)<ϑΦ−1(α) jump(s) , where Φ−1 is the inverse cumulative function of the standard normal distribution giving ϑΦ−1(α)≅−1.284 ⁠. In Figure 1, panels (a) and (b) depict the results of the co-jumps exercise for the high-cap and low-cap indices, the most and the least capitalized set of assets in the DJIA 30, respectively. For every trading day, we find the co-jump vector Ω such that it maximizes the G^Ω-statistic (crosses). For every trading day and each index, we test for the presence of co-jumps and confirm the presence of co-jumps as G^Ω(D)≥−1.284 ⁠, which is captured by the black long-dash line. This means that at the given sampling frequency, a linear combination of assets exists in the index such that the price jumps diversify out. Figure 1. Open in new tabDownload slide Co-jumps properties. (a) Co-jumps G^Ω-statistic: high-cap index. (b) co-jumps G^Ω-statistic: low-cap index. (c) Co-jump vector magnitudes: high-cap index. (d) Co-jump vector magnitudes: low-cap index. Notes: Panels (a) and (b) depict the G^Ω-statistic for the co-jump vector (crosses), for the equally weighted index (triangles), and the gray shaded area captures the region in which lies the G^Ω-statistic for each individual asset in the index. The black long-dash line denotes the α=0.05 critical value to test for the presence of price jumps, ϑΦ−1(α)≅−1.284 ⁠. Panels (c) and (d) depict the minimum and maximum of the co-jump vectors. The solid black line corresponds to the value of the equally weighted index. The vectors are normalized as that ∑i=15Ω(i)2=1 ⁠. Figure 1. Open in new tabDownload slide Co-jumps properties. (a) Co-jumps G^Ω-statistic: high-cap index. (b) co-jumps G^Ω-statistic: low-cap index. (c) Co-jump vector magnitudes: high-cap index. (d) Co-jump vector magnitudes: low-cap index. Notes: Panels (a) and (b) depict the G^Ω-statistic for the co-jump vector (crosses), for the equally weighted index (triangles), and the gray shaded area captures the region in which lies the G^Ω-statistic for each individual asset in the index. The black long-dash line denotes the α=0.05 critical value to test for the presence of price jumps, ϑΦ−1(α)≅−1.284 ⁠. Panels (c) and (d) depict the minimum and maximum of the co-jump vectors. The solid black line corresponds to the value of the equally weighted index. The vectors are normalized as that ∑i=15Ω(i)2=1 ⁠. Further, each of the two figures depicts the range (gray shaded area) of the individual G^-statistics calculated for each asset in the index. The results show that, for the majority of the trading days, at least one asset exists in the index such that the null is rejected for both indices. At the same time, there is no case where the null would be rejected for every asset and, therefore, there is no co-jump for all five assets at the same time. In addition, the two figures report the G^Ω-statistic for equally weighted index (triangles). The results indicate that in the majority of cases, the G^Ω-statistic for the equally weighted index is in the range implied by the individual assets. However, a significant number of cases show that the equally weighted index may either amplify or suppress the presence of price jumps. Table 2 suggests that the popular “1/N” strategy, or employing the equally weighted index, is not optimal for dealing with price jumps. Table 2. Number of co-jumps vs. the individual assets Portfolio 1 2 3 4 5 6 (A) G^Ω(1/N)<min⁡G^Ω(i) 60 51 47 55 60 70 (B) min⁡G^Ω(i)≤G^Ω(1/N)≤max⁡G^Ω(i) 494 492 466 484 469 469 (C) GΩ^(1/N)>max⁡G^Ω(i) 67 78 108 82 92 82 Portfolio 1 2 3 4 5 6 (A) G^Ω(1/N)<min⁡G^Ω(i) 60 51 47 55 60 70 (B) min⁡G^Ω(i)≤G^Ω(1/N)≤max⁡G^Ω(i) 494 492 466 484 469 469 (C) GΩ^(1/N)>max⁡G^Ω(i) 67 78 108 82 92 82 Notes: The table evaluates the frequency of: (A) the equally weighted index amplifies price jumps, the G^Ω(1/N)-statistic for the equally weighted portfolio is smaller than any individual asset; (B) the price jumps for the equally weighted index are comparable with price jumps at individual assets, the G^Ω(1/N)-statistic is in the range implied by the individual assets; and (C) the equally weighted index suppresses price jumps, the G^Ω(1/N)-statistic is higher than any individual assets. Table 2. Number of co-jumps vs. the individual assets Portfolio 1 2 3 4 5 6 (A) G^Ω(1/N)<min⁡G^Ω(i) 60 51 47 55 60 70 (B) min⁡G^Ω(i)≤G^Ω(1/N)≤max⁡G^Ω(i) 494 492 466 484 469 469 (C) GΩ^(1/N)>max⁡G^Ω(i) 67 78 108 82 92 82 Portfolio 1 2 3 4 5 6 (A) G^Ω(1/N)<min⁡G^Ω(i) 60 51 47 55 60 70 (B) min⁡G^Ω(i)≤G^Ω(1/N)≤max⁡G^Ω(i) 494 492 466 484 469 469 (C) GΩ^(1/N)>max⁡G^Ω(i) 67 78 108 82 92 82 Notes: The table evaluates the frequency of: (A) the equally weighted index amplifies price jumps, the G^Ω(1/N)-statistic for the equally weighted portfolio is smaller than any individual asset; (B) the price jumps for the equally weighted index are comparable with price jumps at individual assets, the G^Ω(1/N)-statistic is in the range implied by the individual assets; and (C) the equally weighted index suppresses price jumps, the G^Ω(1/N)-statistic is higher than any individual assets. To assess how much the individual assets contribute to the co-jumps, Figure 1, panels (c) and (d) present the range of the components of each co-jump vector identified above for the high-cap and low-cap indices, respectively. We consider co-jump vectors, normalized such that ∑i=15Ω(i)2=1 ⁠. First, the figure depicts the minimum and maximum of the magnitude of the co-jump vectors. In particular, for high-cap index, the maximum magnitude oscillated around 0.75, while the minimum oscillated around 0.1 with the least magnitude taking the value of 2.02·10−5 and the largest one 9.87·10−1 ⁠, taken from all indices. Therefore, each asset significantly contributes to the co-jump vector and the diversification of price jumps is clearly not caused by picking up an asset with few or no price jumps. The low-cap index provides the same qualitative conclusion. The results show the presence of co-jump vectors. From the index perspective, the price jumps can be ex post diversified out at a five-minute frequency. Further, the equally weighted index is not in general sufficient to eliminate price jumps. In some cases, it amplifies price jumps and thus the deviation from Gaussianity. 3 Conclusions In this paper, we employed the co-feature framework to introduce the notion of co-jumps defined as a linear combination of assets which is free of price jumps. We extended the notion of co-jumps to assets with idiosyncratic price jumps to define the weak co-jumps as a linear combination which minimizes the price jumps. We then linked the concept to the optimization of an index of assets with price jumps. We evaluated the empirical validity of the proposed framework using assets from the DJIA 30 index from 1 January 2010 to 30 June 2012 sampled at a five-minute frequency. We considered two indices, the high-cap index and the low-cap index, based on the market capitalization and tested for co-jumps. The results showed the presence of co-jumps at five-minute frequency, meaning that price jumps could be diversified out. However, our analysis showed that such diversification in general could not be achieved by creating equally weighted indices. Thus, the optimization in terms of removing price jumps should be considered as independent criteria. The findings in this paper suggest some further developments. First, it will be interesting to extend the framework in this paper to the case of a more general price arrival process, for example, mutually correlated self-exciting price jumps. Second, the sensitivity of the proposed framework, and in particular of the measure of commonality, can be transformed in the proper testing procedure for asynchronicity among the price jumps. Finally, it will also be interesting to develop an extension of the notion of co-arrivals to define the information measures capturing the different features of the multivariate arrival process. This is part of our ongoing research agenda. Acknowledgements This paper is a substantially revised version of a manuscript previously circulated under the title “Co-features in Finance: Co-arrivals and Co-jumps”. We wish to thank participants in the Conference on Skewness, Heavy Tails, Market Crashes, and Dynamics (University of Cambridge, 28–29 April 2014), the 13th OxMetrics User Conference (Aarhus, 5–6 September 2013), the International Conference on Systemic Risk, Contagion and Jumps (Cass Business School, 25 January 2013), in particular Yacine Aït-Sahalia and Neil Shephard, the Finance Research Workshop (Cass Business School, 4 March 2013), in particular Richard Payne, and the Economics Division Workshop (Nottingham Trent University, 6 March 2013) for comments. We are grateful to Simona Boffelli for discussions and insightful comments on various versions of the paper. 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Journal of Financial Econometrics – Oxford University Press

Published: Jan 1, 2018

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