Estimating a non-parametric memory kernel for mutually-exciting point processes

Self- and cross-excitation in point processes are commonly captured in the financial econometrics literature using a multivariate exponential memory kernel. In this article, the exponential assumption is relaxed and the resultant non-parametric memory kernel is estimated by a method based on second-order cumulants. The estimator is shown to be consistent and asymptotically normally distributed and performs well under simulation. An empirical application based on 10 international stock indices is presented. Two different indices of contagion between markets are constructed from the point process models in order to examine interconnection over time. A conclusion which emerges from these results is the assumption that a parametric kernel may be too restrictive as the application reveals interesting features, and in some cases substantial differences, between the exponential and non-parametric kernels.