On the estimation and diagnostic checking of the ARFIMA–HYGARCH model

The estimation and diagnostic checking of the fractional autoregressive integrated moving average with hyperbolic generalized autoregressive conditional heteroscedasticity (ARFIMA–HYGARCH) model is considered. The ARFIMA–HYGARCH model is a long-memory model for the conditional mean that also allows for long memory in the conditional variance, the latter given by an HYGARCH model that nests both the GARCH and integrated GARCH models. It is therefore important to provide a thorough treatment of its statistical inference. Asymptotic properties of the maximum likelihood estimators under the Student’s t distribution are established, and the asymptotic normality of the Gaussian quasi-maximum likelihood estimation is also derived. Two portmanteau test statistics based on the residual autocorrelations and squared residual autocorrelations are defined and their asymptotic distributions are derived. These tests will be useful in model diagnostic checking. Simulation results show that the tests have reasonable empirical size and power.