On the estimation of periodic ARMA models with uncorrelated but dependent errors

The main goal of this paper is to study the asymptotic properties of least squares (LS) estimation for invertible and causal PARMAmodels with uncorrelated but dependent errors (weak PARMA). Four different LS estimators are considered: ordinary least squares (OLS), weighted least squares (WLS) for an arbitrary vector of weights, generalized least squares (GLS) in which the weights correspond to the theoretical seasonal variances and quasi-generalized least squares (QLS) where the weights are the estimated seasonal variances. The strong consistency and the asymptotic normality are established for each of them. Obviously, their asymptotic covariance matrices depend on the vector of weights. Our work extends a result of Basawa and Lund (2001) for least squares estimation of PARMA models with independent errors (strong PARMA). The paper is organized as follows. The definitions of strong and weak PARMA processes are given in Section 2 and situations in which weak PARMA representations naturally arise are presented. In Section 3, we first recall the usual multivariate ARMA representation of a PARMA model. The four LS estimators are described and their asymptotic properties are given in Theorems 1 and 2. The proofs can be found in Francq, Roy and Saidi (2009) (FRS in the sequel). It is seen that the GLS estimators are optimal in the class of WLS estimators when the noise sequence is in a particular class of martingale differences. Arguing as in Francq, Roy and Zakoan (2005), a consistent estimator of the asymptotic covariance matrix of LS estimators under the assumption of a weak noise is proposed. In Section 4, we present an example of weak PARMA models for which the asymptotic covariance matrix of the least squares estimators is given in a close form and is compared to the corresponding matrix under the assumption of a strong noise. The difference can be huge. In FRS, Monte Carlo results are presented. Two different PARMA models with strong and weak noises were used to investigate the size and power of a Wald test based on the proposed consistent estimator of the asymptotic covariance matrix, under the assumption of either a weak or strong noise. The rate of convergence of the estimated asymptotic standard errors is also analysed. Finally, our results were exploited to address the question of day-of-the-week seasonality of four European stock market indices. The space constraint does not allow us to report these numerical results here.