First-order seasonal autoregressive processes with periodically varying parameters

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).…”

Estimation and model adequacy checking for multivariate seasonal autoregressive time series models with periodically varying parameters

“…They studied the stationarity conditions (in the periodic sense) and the limit distributions of the least squares estimators. Here, we extend the work of Basawa et al (2004) in three directions. First, we generalize the first-order process to a seasonal autoregressive stochastic process with periodically varying parameters of orders p 1 and p 2 , where p 1 and p 2 denote the orders in the seasonal VAR and VAR polynomials respectively.…”

“…., s, then a PVAR stochastic process of order p 2 in each season is obtained (see Lütkepohl 2005, amongst others). Letting p 1 = p 2 = 1 and d = 1, the SPVAR model defined by equation (1) reduces to the univariate SPAR(1,1) stochastic process introduced by Basawa et al (2004). Subsequent sections study the stationarity conditions (in the periodic sense) and the theoretical autocovariance function of the SPVAR(p 1 , p 2 ) time series model.…”

“…Here, we introduce a multivariate seasonal autoregressive model with periodically varying parameters, abbreviated by the acronym SPVAR, which includes the multivariate seasonal autoregressive time series model and also the multivariate periodic time series model as special cases. Combining periodic and seasonal time series models has been considered by Basawa et al (2004) in the particular situation of a first-order seasonal autoregressive process with periodically varying parameters. They studied the stationarity conditions (in the periodic sense) and the limit distributions of the least squares estimators.…”

“…Third, we consider diagnostic checking SPVAR time series models. Diagnostic checking was not developed in Basawa et al (2004). However, from a model building point of view, it is well recognized that checking the adequacy of models appears to be a fundamental step in the time series methodology.…”