Christian Francq

Periodically correlated time series are common in many scientific fields where the observed phenomena may have significant seasonal behavior in mean, variance and covariance structure, namely in hydrology, meteorology, finance and economy. An important class of stochastic models for describing such periodicity in mean and in covariances, are the periodic autoregressive moving average (PARMA) models. The main goal of this paper is to study the asymptotic properties of least squares (LS) estimation for invertible and causal PARMA models 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. 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. 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) on least squares estimation of PARMA models with independent errors (strong PARMA).

The paper is organized as follows. In Section 2, we provide examples of weak periodic noises and of nonlinear processes admitting a weak PARMA representation. The asymptotic results are described in Section 3. Arguing as in Francq *et al.* (2005), a consistent estimator of the asymptotic covariance matrix of LS estimators under the assumption of a weak noise is also proposed. In Section 4, we present two examples 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. Monte Carlo results are described in Section 5. 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. Our results are exploited in Section 6 to address the question of day-of-the-week seasonality of four European stock market indices.

**References:**

Basawa, I. V. and Lund, R. (2001). Large sample properties of parameter estimates for periodic ARMA models. *Journal of Time Series Analysis* 22, 651-663.

Francq, C., Roy, R. and Zakoïan, J. M. (2005). Diagnostic checking in ARMA models with uncorrelated errors. *Journal of the American Statistical Association* 100, 532-544.

**Keywords:** Seasonal time series models; Dependent errors; Weighted least squares estimation; Asymptotic theory

**Biography:** Roch Roy is currently an adjunct professor in the Department of Mathematics and Statistics of the Université de Montréal, after many years as full professor. His research interests are concerned with time series methods and their applications in various fields such as econometrics, epidemiology, environment.