An estimation procedure for the Hawkes process
An estimation procedure for the Hawkes process
复制标题
霍克斯过程的估计程序
DOI：
10.1080/14697688.2016.1211312
复制
发表时间：
2015
影响因子：
1.3
通讯作者：
Matthias Kirchner
中科院分区：
文献类型：
--
作者：
Matthias Kirchner
给出了多元Hawkes点过程的一种非参数估计方法。时间线被切成箱，并且--对于每个组件工艺--计算每个箱中的点数。由于早先在基什内尔[Stoch.流程。2016,162,2494-2525]，所得到的‘仓位计数序列’的分布可以由被称为(多变量)INAR(P)模型的整数值自回归模型来近似。我们将INAR(P)模型表示为具有白噪声新息的标准向量值线性自回归时间序列(VAR(P))。我们分别建立了VAR(P)和Inar(P)模型条件最小二乘估计的相合性和渐近正态。经过适当的缩放后，这些时间序列估计产生了潜在多变量Hawkes过程的估计以及相应的方差估计。估计依赖于面元大小和支持度S。我们讨论了这些参数的影响和选择。所有结果都是以这样一种方式呈现的，即计算机实现，例如在R中，是直接的。仿真研究证实了该估计方法的有效性。在文章的第二部分，我们给出了一个数据实例，将该方法应用于金融限价订单数据中的二元事件流。我们用一个双变量的Hawkes模型来描述限价和市场订单到达的联合过程。分析表明，这两个过程之间存在显着的不对称关系：入市指令对限价指令流有很大的激励作用，而市场指令流几乎不受限价指令的影响。对于估计的兴奋函数，我们观察了幂规律形状，对0.003 S以下滞后的抑制效应，第二周期，以及在0.01，0.1和0.5S处的局部极大值。
In this paper, we present a nonparametric estimation procedure for the multivariate Hawkes point process. The timeline is cut into bins and—for each component process—the number of points in each bin is counted. As a consequence of earlier results in Kirchner [Stoch. Process. Appl., 2016, 162, 2494–2525], the distribution of the resulting ‘bin-count sequences’ can be approximated by an integer-valued autoregressive model known as the (multivariate) INAR(p) model. We represent the INAR(p) model as a standard vector-valued linear autoregressive time series with white-noise innovations (VAR(p)). We establish consistency and asymptotic normality for conditional least-squares estimation of the VAR(p), respectively, the INAR(p) model. After appropriate scaling, these time-series estimates yield estimates for the underlying multivariate Hawkes process as well as corresponding variance estimates. The estimates depend on a bin-size and a support s. We discuss the impact and the choice of these parameters. All results are presented in such a way that computer implementation, e.g. in R, is straightforward. Simulation studies confirm the effectiveness of our estimation procedure. In the second part of the paper, we present a data example where the method is applied to bivariate event-streams in financial limit-order-book data. We fit a bivariate Hawkes model on the joint process of limit and market order arrivals. The analysis exhibits a remarkably asymmetric relation between the two component processes: incoming market orders excite the limit-order flow heavily whereas the market-order flow is hardly affected by incoming limit orders. For the estimated excitement functions, we observe power-law shapes, inhibitory effects for lags under 0.003 s, second periodicities and local maxima at 0.01, 0.1 and 0.5 s.