Quasi-Maximum Likelihood Estimation of Heteroskedastic Fractional Time Series Models

In a recent paper Hualde and Robinson (2011) establish consistency and asymptotic normality for conditional sum-of-squares estimators, which are equivalent to conditional quasi-maximum likelihood estimators, in parametric fractional time series models driven by conditionally homoskedastic shocks. In contrast to earlier results in the literature, their results apply over an arbitrarily large set of admissible parameter values for the (unknown) memory parameter covering both stationary and non-stationary processes and invertible and non-invertible processes. In this paper we extend their results to the case where the shocks can display conditional and unconditional heteroskedasticity of a quite general and unknown form. We establish that the consistency result presented in Hualde and Robinson (2011) continues to hold under heteroskedasticity as does asymptotic normality. However, we demonstrate that the covariance matrix of the limiting distribution of the estimator now depends on nuisance parameters which derive both from the weak dependence in the process (as is also the case for the corresponding result in Hualde and Robinson, 2011) but additionally from the heteroskedasticity present in the shocks. Asymptotically pivotal inference can be performed on the parameters of the heteroskedastic model, provided a conventional "sandwich" estimator of the variance is employed.