Estimation of Parameters in the ARMA Model When the Characteristic Polynomial of the MA Operator Has a Unit Zero

Abstract

We consider the estimation of parameters in the time series model $X(t) = \sum^q_{j = 1}a_jX(t - j) + \varepsilon(t) - \varepsilon(t - j) - \sum^p_{j = 1} c_j\{\varepsilon(t - j) - \varepsilon(t - j - 1)\}$ where the $\varepsilon(t)$ are independently identically distributed random variables with zero mean and variance $\sigma^2$. We compute the exact $\log$ likelihood of the model, propose and justify an asymptotic approximation of it. The latter will be used to derive estimates of the parameters which are shown to be asymptotically normal and efficient.