I am analyzing some data by applying 1d-Hawkes processes. To evaluate the fit of the model itself (and particularly the chosen kernel, i.e. exponential), I am performing a KS test.
However, it is not entirely clear to me on which data the KS has to be performed. My understanding, at this point, is that the KS test has to be run on the interarrival times (first-order differences) of the model residuals. Is this correct?
Thank you in advance!
This is correct, this residual analysis is the classic goodness of fit test for the estimation of Hawkes processes. Formally, consider a univariate Hawkes process $\mathcal{T}:=\{t_i: i\in \mathbb{N}^*\}$. Denote by $\lambda$ the conditional intensity of this Hawkes process, and by $\Lambda$ the compensator of this Hawkes process i.e.$\Lambda(t)=\int_0^t \lambda(t)dt$.
Now define for all $i\in \mathbb{N}^*$, $s_{i}=\Lambda(t_i)$ and consider the point process $\mathcal{S}:=\{s_i: i\in \mathbb{N}^*\}$. Then $\mathcal{S}$ is a standard Poisson process. One way to test that $\mathcal{S}$ is a standard Poisson process is indeed by performing a KS test on the residuals $(s_{i+1}-s_{i})$ since under the null hypothesis they should be exponentially distributed with rate $1$.
This classic result can be found in Theorem 1.20 in the thesis of Linigier (2009), Theorem 7.4.1 in the book of Daley and Vere-Jones (2003), or earlier in section 3.3 in Ogata (1988).
Note that the computation of the residuals of a Hawkes process is expensive in the general case: if $N_T$ jumps are observed, this computation is roughly in $O(N_T^2)$. The generalization of this GOF test to the case of a multidimensonal Hawkes process is straight forward: the time transforms for each dimension of the Hawkes are independent standard Poisson processes.
