COGARCH (p,q) : Simulation and Inference with the yuima Package

Abstract

In this paper we show how to simulate and estimate a COGARCH(p, q) model in the R package yuima. Several routines for simulation and estimation are introduced. In particular, for the generation of a COGARCH(p, q) trajectory, the user can choose between two alternative schemes. The first is based on the Euler discretization of the stochastic differential equations that identify a COGARCH(p, q) model while the second considers the explicit solution of the equations defining the variance process.Estimation is based on the matching of the empirical with the theoretical autocorrelation function. Three different approaches are implemented: minimization of the mean squared error, minimization of the absolute mean error and the generalized method of moments where the weighting matrix is continuously updated. Numerical examples are given in order to explain methods and classes used in the yuima package.

Discrete‐Time Approximation of a Cogarch(p,q) Model and its Estimation

“…As shown in Chadraa () and recently computed using Teugels martingales in Iacus et al (), the conditional first moment and the conditional variance for the increments are respectively …”

“…The estimation procedure discussed in Section 4 can be performed using the qmle function in yuima, which controls, through the Diagnostic.Cogarch function, whether the model parameters satisfy the conditions used for the identification of set Θ in the optimization routine (see Iacus et al. , , for details).…”

“…We apply the estimation method developed in Section 4 to simulated data generated on a regular grid through the simulate function in yuima, where we choose the Euler–Maruyama discretization of the stochastic differential equation (SDE) in (see Iacus et al. , , for more details). To verify whether the pseudo maximum‐likelihood method gives plausible estimates, we perform a simulation‐estimation study for different values of orders p and q and for two different driving Lévy processes.…”

“…whether the model parameters satisfy the conditions used for the identification of set Θ in the optimization routine (see Iacus et al, 2017, for details).…”

“…The results derived for a COGARCH(p, q) model in Chadraa (2009) are used in this article for extending the estimation procedure based on the maximization of the pseudo log-likelihood function. This estimation method is then implemented in the yuima package available on CRAN (see Brouste et al, 2014;Iacus et al, 2017;Iacus and Yoshida, 2018, for more details on yuima package).…”