Recursive Computation of the Parameters of Periodic Autoregressive Moving‐average Processes

Abstract

An algorithm for recursive computation of the parameters of periodic autoregressive moving-average (ARMA) processes is given. It also provides recursions for stationary multivariate ARMA processes. A procedure for simultaneous estimation of the order and the parameters of a periodic ARMA process is outlined. computation of parameters; order determination.

Asymptotic results for Fourier-PARMA time series

“…1. Periodic time series models and their practical applications are discussed in Adams and Goodwin (1995), Anderson and Vecchia (1993), Meerschaert (1997, 1998), Anderson et al (1999), Basawa et al (2004), Boshnakov (1996), Gautier (2006), Jones and Brelsford (1967), Lund andBasawa (1999, 2000), Lund (2006), Nowicka-Zagrajek and Wyłomań ska (2006), Pagano (1978), Roy and Saidi (2008), Salas et al (1982Salas et al ( , 1985, Shao and Lund (2004), Tesfaye et al (2005), Tjøstheim and Paulsen (1982), Troutman (1979), Vecchia (1985aVecchia ( , 1985b, Vecchia and Ballerini (1991), Ula (1990Ula ( , 1993, Smadi (1997, 2003) and Wyłomań ska (2008). See also the recent book of Franses and Paap (2004) as well as Hipel and McLeod (1994).…”

Computation and Characterization of Autocorrelations and Partial Autocorrelations in Periodic ARMA Models

“…Second, Sakai (1982, 1983) and Boshnakov (1995) explore Durbin–Levinson type algorithms for periodic series. These algorithms can be used to extract a PARMA PACF and to recursively fit PARMA models in the orders p and q .…”

“…In general, multivariate and periodic time series differ structurally: the components of a multivariate time series may not be time-ordered, whereas the components of a periodic time series are necessarily time-ordered. Second, Sakai (1982Sakai ( , 1983 and Boshnakov (1995) explore Durbin-Levinson type algorithms for periodic series. These algorithms can be used to extract a PARMA PACF and to recursively fit PARMA models in the orders p and q.…”

“…The autoregressive and moving‐average model orders during season ν are p (ν) and q (ν) respectively, and φ 1 (ν),…,φ p (ν) and θ 1 (ν),…,θ p (ν) are the season ν autoregressive and moving‐average model coefficients, respectively. For mathematical purposes, p (ν) and q (ν) can be taken as constant in ν– merely set p = max 1≤ν≤ T p (ν), q = max 1≤ν≤ T q (ν), and take φ k (ν) = 0 for k > p (ν) and θ k (ν) = 0 for k > q (ν) (Boshnakov, 1995). Such simplification is inappropriate, however, when specific values of p (ν) and q (ν) are under investigation (as is the case in Section 4).…”

“…; h p ðmÞ are the season m autoregressive and moving-average model coefficients, respectively. For mathematical purposes, pðmÞ and qðmÞ can be taken as constant in m -merely set p ¼ max 1OmOT pðmÞ; q ¼ max 1OmOT qðmÞ, and take / k ðmÞ ¼ 0 for k > pðmÞ and h k ðmÞ ¼ 0 for k > qðmÞ (Boshnakov, 1995). Such simplification is inappropriate, however, when specific values of pðmÞ and qðmÞ are under investigation (as is the case in Section 4).…”