  A point process for event arrivals in high frequency trading is presented.
The intensity is the product of a Hawkes process and high dimensional functions
of covariates derived from the order book. Conditions for stationarity of the
process are stated. An algorithm is presented to estimate the model even in the
presence of billions of data points, possibly mapping covariates into a high
dimensional space. The large sample size can be common for high frequency data
applications using multiple liquid instruments. Convergence of the algorithm is
shown, consistency results under weak conditions is established, and a test
statistic to assess out of sample performance of different model specifications
is suggested. The methodology is applied to the study of four stocks that trade
on the New York Stock Exchange (NYSE). The out of sample testing procedure
suggests that capturing the nonlinearity of the order book information adds
value to the self exciting nature of high frequency trading events.
