Goodness-of-Fit Test of Mismatched Models for Self-Exciting Processes

We develop a goodness-of-fit (GOF) test for generative models of self-exciting processes by making a new connection to this problem with the classical statistical theory of Quasi-maximum-likelihood estimator (QMLE). We present a non-parametric self-normalizing statistic for the GOF test: the Generalized Score (GS) statistics, and explicitly capture the model misspecification when establishing the asymptotic distribution of the GS statistic. Numerical experiments based on simulation and real-data validate our theory and demonstrate the proposed GS test's good performance.