Gresnigt, F., Kole, E. and Franses, P.H. (2017). Specification Testing in Hawkes Models Journal of Financial Econometrics, 15(1):139--171.

We propose various specification tests for Hawkes models based on the Lagrange Multiplier (LM) principle. Hawkes models can be used to model the occurrence of extreme events in financial markets. Our specific testing focus is on extending a univariate model to a multivariate model, that is, we examine whether there is a conditional dependence between extreme events in markets. Simulations show that the test has good size and power, in particular for sample sizes that are typically encountered in practice. Applying the specification test for dependence to U.S. stocks, bonds, and exchange rate data, we find strong evidence for cross-excitation within segments as well as between segments, which cannot simply be explained by volatility spillovers. Therefore, we recommend that univariate Hawkes models be extended to account for the cross-triggering phenomenon.