Multivariate Hawkes processes (MHP) are a class of point processes in which events at different coordinates interact through mutual excitation. The weighted adjacency matrix of the MHP encodes the strength of the relations, and shares its support with the causal graph of interactions of the process. We consider the problem of testing for causal relationships across the dimensions of a marked MHP. The null hypothesis is that a joint group of adjacency coefficients are null, corresponding to the absence of interactions. The alternative is that they are positive, and the associated interactions do exist. To this end, we introduce a novel estimation procedure in the context of a large sample of independent event sequences. We construct the associated likelihood ratio test and derive the asymptotic distribution of the test statistic as a mixture of chi squared laws. We offer two applications on financial datasets to illustrate the performance of our method. In the first one, our test reveals a deviation from a static equilibrium in bidders' strategies on retail online auctions. In the second one, we uncover some factors at play in the dynamics of German intraday power prices.

翻译：多元霍克斯过程是一类点过程，其中不同坐标上的事件通过相互激励产生交互。其加权邻接矩阵编码了关系强度，且与过程交互的因果图共享支持集。本文考虑标记多元霍克斯过程中跨维度的因果关系检验问题。原假设为邻接系数的联合组为零，对应无交互存在；备择假设为这些系数为正，且相关交互确实存在。为此，我们针对大量独立事件序列的样本，提出了一种新的估计方法。我们构建了相应的似然比检验，并推导出检验统计量的渐近分布为卡方分布的混合。通过两个金融数据集的应用，我们展示了本方法的性能：第一个应用中，检验揭示了零售在线拍卖中竞标者策略偏离静态均衡的现象；第二个应用则揭示了德国日内电力价格动态中的若干影响因素。