Authors: Stephen Crowley
Definitions from the theory of point processes are recalled. Models of intensity function parametrization and maximum likelihood estimation from data are explored. Closed-form log-likelihood expressions are given for the (exponential) Hawkes (univariate and multivariate) process, Autoregressive Conditional Duration(ACD), with both exponential and Weibull distributed errors, and a hybrid model combining the ACD and the exponential Hawkes models. Formulas are also derived, however without the elegant recursions of the exponential kernels, for kernels of the Weibull and Gamma type and comparison of the Weibullfit vs exponential kernel fits viaQQand probability plots are provided. The additional complexity of the Hawkes-Weibull or the ACD-Hawkes appears to not be worth the tradeoff. Diurnal, or daily, adjustment of the deterministic predictable part of the intensity variation via piecewise polynomial splines is discussed. Data from the symbol SPY on three different electronic markets is used to estimate model parameters and generate illustrative plots. The parameters were estimated without diurnal adjustments, a repeat of the analysis with adjustments is due in a future version of this article. The connection of the Hawkes process to quantum theory is briefly mentioned. Prediction of the next point of a Hawkes process is briefly discussed and a closed-form expression in terms of the Lambert W function for the standard exponential kernel with P=1 is calculated.
Comments: 41 Pages.
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