Statistical inference for highly correlated stationary point processes and noisy bivariate Neyman-Scott processes

Motivated by estimating the lead-lag relationships in high-frequency financial data, we propose noisy bivariate Neyman-Scott point processes with gamma kernels (NBNSP-G). NBNSP-G tolerates noises that are not necessarily Poissonian and has an intuitive interpretation. Our experiments suggest that NBNSP-G can explain the correlation of orders of two stocks well. A composite-type quasi-likelihood is employed to estimate the parameters of the model. However, when one tries to prove consistency and asymptotic normality, NBNSP-G breaks the boundedness assumption on the moment density functions commonly assumed in the literature. Therefore, under more relaxed conditions, we show consistency and asymptotic normality for bivariate point process models, which include NBNSP-G. Our numerical simulations also show that the estimator is indeed likely to converge.