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

Abstract

In this article, we construct a sequence of discrete‐time stochastic processes that converges in the Skorokhod metric to a COGARCH(p,q) model. The result is useful for the estimation of the COGARCH(p,q) on irregularly spaced time series data. The proposed estimation procedure is based on the maximization of a pseudo log‐likelihood function and is implemented in the yuima package.

Portfolio Selection with Irregular Time Grids: an example using an ICA-COGARCH(1, 1) approach

“…As shown in Iacus et al (2018), the pair 5 (G i,n , V i,n ) n∈N converges in the Skorokhod distance to the solution (G t , V t ) t≥0 of the system in (1). We are particularly interested in the behavior of (G i,n ) n∈N and (V i,n ) n∈N for p = q = 1 where the random coefficients C i,n , D i,n and the variance process V i,n read:…”

“…The reparameterization in (9) allows to obtain the GARCH(1, 1) specification as a special case of the COGARCH(1, 1) in the case of equally spaced time grids. The discrete process in (5) has been used in Iacus et al (2018) for the construction of a pseudo-maximum likelihood estimation procedure for a COGARCH( p, q) model based on the assumption of normality for i,n . This procedure generalizes the approach proposed in Maller et al (2008) for a COGARCH(1, 1) model.…”

“…Using the same arguments in Iacus et al (2018) we construct a piecewise constant vector process X n t = AS n t defined on the partition of the interval [0, T ] introduced in (2). The piecewise vector process S n t has the following form:…”

“…As shown in Iacus et al (2018), each component of the vector S n t converges uniformly in probability to S t ; therefore, we have:…”

“…Differences emerge in the estimation of the two models. Recently, Iacus et al (2018) proposed a discrete time process that is shown to converge to the COGARCH( p, q) model. The innovations in the approximating process are constructed using the first jump approximation method (see Maller and Szimayer 2007), while model parameters are estimated through the pseudo-maximum likelihood method.…”