Generalized method of moment and indirect estimation of the ARasMA model (Q1297873)

An ARasMA\((p,q)\) model can be written in the form \[ y_{t}=\phi_{1}y_{t-1}+\ldots+ \phi_{p}y_{t-p}+u_{t}+\beta_{1}^{+}u_{t-1}^{+}+\ldots+ \beta_{q}^{+}u_{t-q}^{+}+\beta_{1}^{-}u_{t-1}^{-}+\ldots+ \beta_{q}^{-}u_{t-q}^{-}= \] \[ =\phi_{1}y_{t-1}+\ldots+ \phi_{p}y_{t-p}+u_{t}+\beta_{1}^{-}u_{t-1}+\ldots+ \beta_{q}^{-}u_{t-q}+\delta_{1}u_{t-1}^{+}+\ldots+ \delta_{q}u_{t-q}^{+}, \] where \(u^{+}_{i}=\max\{0,u_{i}\}\), \(u^{-}_{i}=\min\{0,u_{i}\}\), \(\delta_{i}=\beta_{i}^{+}-\beta_{i}^{-}\), and \(\{u_{t}\}\) is a sequence of independent and identically distributed random variables with zero mean and variance \(\sigma^{2}\). The authors propose and study the performance of the generalized method of moments (GMM) and indirect estimators for ARasMA\((p,q)\) models. For asMA\((1)\) models the Monte Carlo study indicates that the performances of the GMM estimator and its associated Wald test statistic are very similar to those of the maximum likelihood.