Analysis of Financial Time Series, Third Edition

3.10 The CHARMA Model

Many other econometric models have been proposed in the literature to describe the evolution of the conditional variance in Eq. (3.2). We mention the conditional heteroscedastic ARMA (CHARMA) model that uses random coefficients to produce conditional heteroscedasticity; see Tsay (1987). The CHARMA model is not the same as the ARCH model, but the two models have similar second-order conditional properties. A CHARMA model is defined as

(3.36)

where {ηt} is a Gaussian white noise series with mean zero and variance , {δt} = {(δ1t, … , δmt)′} is a sequence of iid random vectors with mean zero and nonnegative definite covariance matrix Ω, and {δt} is independent of {ηt}. In this section, we use some basic properties of vector and matrix operations to simplify the presentation. Readers may consult Appendix A of Chapter 8 for a brief review of these properties. For m > 0, the model can be written as

where at−1 = (at−1, … , at−m)′ is a vector of lagged values of at and is available at time t − 1. The conditional variance of at of the CHARMA model in Eq. (3.36) is then