Estimation of slowly decreasing Hawkes kernels: application to high-frequency order book dynamics

We present a modified version of the non parametric Hawkes kernel estimation procedure studied in Bacry and Muzy arXiv:1401.0903, 2014 that is adapted to slowly decreasing kernels. We show on numerical simulations involving a reasonable number of events that this method allows us to estimate faithfully a power-law decreasing kernel over at least six decades. We then propose a eight-dimensional Hawkes model for all events associated with the first level of some asset order book. Applying our estimation procedure to this model, allows us to uncover the main properties of the coupled dynamics of trade, limit and cancel orders in relationship with the mid-price variations.