Identifiability and Whittle Estimation of Periodic ARMA Models

The Periodic Autoregressive Moving Average (PARMA) models are generally assumed to be identifiable. However, this assumption becomes not true if some model conditions are not specified. This paper fills this gap by providing verifiable conditions for the identifiability of PARMA models and, in addition, the Whittle likelihood estimator (WLE) is proposed to estimate the model parameters. This estimator is strongly consistent and asymptotically normal. The Monte Carlo simulation investigation shows that the WLE is a very attractive alternative to the Gaussian maximum likelihood estimator (MLE) for large data sets. Although the estimators have similar accuracy, the computational cost of the MLE is much higher. The methods are applied to fit a PARMA model to the sulfur dioxide (SO \({ }_{2}\) ) daily average pollutant concentrations measured in the city of Vitória (ES), Brazil.