Periodic stationarity conditions for periodic autoregressive moving average processes as eigenvalue problems

Abstract

Abstract. The determination of periodic stationarity conditions for periodic autoregressive moving average (PARMA) processes is a prerequisite to their analysis. Means of obtaining these conditions in analytically simple forms are sought. It is shown that periodic stationarity conditions for univariate and multivariate PARMA processes can always be reduced to eigenvalue problems, which are computationally and analytically easier to deal with. Two different lumpings of the periodic process are considered along this line. The first is the common w-span lumping over all w periods. The second lumping considered is the p-span lumping of the p th order periodic autoregressive process over p periods, which is based on a recently introduced lumping technique. It is shown that p-span lumping may yield the periodic stationarity conditions in an analytically simpler form as compared to w-span lumping when p < w.

Adaptive Estimation of Causal Periodic Autoregressive Model

“…Using the so-called "order span lumping" approach, which considers a lumping of the periodic process over a span of the order p (cf., Bentarzi, 1998;Bentarzi and Hallin, 1994;Ula and Smadi, 1997), the necessary and sufficient condition for a periodic P − AR model to be causal is that the roots of the determinantal equation (of degree p)…”

“…The considerable interest given to the class of linear and nonlinear periodic models is explained by its usefulness and appropriateness for modeling stationary periodically correlated processes (Gladyshev, 1961), which are very often met in many fields particularly in economics (e.g., Ghysels et al, 1996;Osborn and Smith, 1989; and others) as well as in hydrology and environmental studies (Bloomfield et al, 1994;Salas et al, 1982;Ula and Smadi, 1997;Vecchia et al, 1983;others).…”

Innovations algorithm asymptotics for periodically stationary time series with heavy tails

“…An important class of stochastic models for describing periodically stationary time series are the periodic ARMA models, in which the model parameters are allowed to vary with the season. Periodic ARMA models are developed by many authors including [1,2,[4][5][6][7]20,[22][23][24]26,28,30,31,[33][34][35][36][37][38][39][40][41].…”

On Least Squares Estimation in a Simple Linear Regression Model with Periodically Correlated Errors: A Cautionary Note

“…They are rather examined for a weaker type of stationarity named as periodic stationarity. This means that the mean and the variance of the time series is constant for each season and periodic with period ω and the autocovariance function depends on the time lag and season only (Ula and Smadi, 1997). For example, the PAR 4 (1) model above is periodic stationary if | ∏ 4 ν=1 ϕ 1 (ν)| < 1 (Obeysekera and Salas, 1986).…”