On the Invertibility of Periodic Moving‐average Models

Abstract

A sufficient condition for the invertibility of univariate periodic movingaverage models has been given by Cipra and Ghysels and Hall. We show that this condition is not a necessary one, and provide a necessary and sufficient condition for the general rn-variate, d-periodical moving-average MA(q) case.Much attention has been given recently to periodical autoregressive movingaverage (ARMA) processes, mainly because of their applications as an alternative to traditional seasonal models, in econometrics (see, for example, Lutkepohl, 1991a,b, or Ghysels andHall, 1992a) as well as in hydrology and environmental studies (McLeod, 1993).Though the identification and estimation of periodic ARMA models has been studied in some detail (see, for example, Vecchia, 1985;Anderson and Vecchia, 1993), the available results remain somewhat incomplete, in the moving-average (MA) case, due to the lack of an appropriate characterization of invertibility. Without such a property, periodic ARMA processes, just as their non-periodic counterparts, remain unidentifiable. Moreover, non-invertible MA models, whether periodical or not, produce inefficient forecasts; see Hallin (1986).Let L denote, as usual, the lag operator, and consider the rn-dimensional linear difference operator of order q , @,(L) = 8 , ; o + 8,;IL + * * . + 8,;,L4 t E Z(1) where eti0, . . ., @, ; , are m x m matrices such that \8,;ol # 0, t E Z. If, for some integer d > 1,( 2 ) the operator Q),(L) is called periodical. If, moreover, d is the smallest integer such that (2) holds, d is called the period, and we say that 8 , ( L ) is d-periodical.Denote by { E , , t E Z} a second-order, m-dimensional white noise process: E(E,E;) = 6(s; t ) Z 0143-9782/94/03 263-268 JOURNAL OF TIME SERIES ANALYSIS Vol. 15, No. 3 0

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

“…Since invertibility conditions in terms of moving average (MA) parameters are analogous to stationarity conditions in terms of AR parameters, we adopt their approach to the determination of periodic stationarity conditions for a fixed p th-order multivariate PARw(p) process and call itp-span lumping. Although Bentarzi and Hallin [1994] claimed the superiority of their approach over w-span lumping in terms of the degrees of the determinantal equations involved, it will be shown here that their approach may yield the periodic stationarity conditions in an analytically simpler form only when p < w. …”

“…We will employ this result here for expressing the periodic stationarity conditions for any PARMA process in terms of eigenvalues. Bentarzi and Hallin [1994] studied the invertibility of periodic moving average (PMA) processes and obtained the invertibility conditions for a w period and fixed qth-order multivariate PMAw(q) process by considering a lumping of the process over a span of q rather than over a span of w, which leads to a PMAs(1) process, s being a function of w and q. Since invertibility conditions in terms of moving average (MA) parameters are analogous to stationarity conditions in terms of AR parameters, we adopt their approach to the determination of periodic stationarity conditions for a fixed p th-order multivariate PARw(p) process and call itp-span lumping.…”

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)…”

Model-Building Problem of Periodically Correlatedm-Variate Moving Average Processes

“…Consequently, much attention has been given recently to periodic linear models as a promising alternative to traditional seasonal models (see e.g., Box and Jenkins, 1976). Most of the existing periodic literature is concerned with identification, estimation, and testing problems; see Adams and Goodwin (1995), Bentarzi and Hallin (1994, 1996, and 1997, Anderson and Vecchia (1993), Li and Hui (1988), Salas et al (1982) Pagano (1978), C 8 ipra (1985), Vecchia (1985), Tiao and Grupe (1980), Parzen and Pagano (1979), Troutman (1979), Jones and Brelsford (1967), and many others. On the other hand, the invertibility property of periodic moving average models, which is of primary importance (for the model identifiability problem as well as for forecasting purposes), has been studied by C 8 ipra (1985) and Ghysels and Hall (1992).…”