# [R] Re: Seasonal ARMA model

Prof Brian Ripley ripley at stats.ox.ac.uk
Sun Jul 4 11:24:47 CEST 2004

```On Sun, 4 Jul 2004, Ajay Shah wrote:

> > It might clarify your thinking to note that a seasonal ARIMA model
> > is just an ``ordinary'' ARIMA model with some coefficients
> > constrained to be 0 in an efficient way.  E.g.  a seasonal AR(1) s =
> > 4 model is the same as an ordinary (nonseasonal) AR(4) model with
> > coefficients theta_1, theta_2, and theta_3 constrained to be 0.  You
> > can get the same answer as from a seasonal model by using the
> > ``fixed'' argument to arima.  E.g.:
>
>    set.seed(42)
>    x <- arima.sim(list(ar=c(0,0,0,0.5)),300)
>    f1 = arima(x,seasonal=list(order=c(1,0,0),period=4))
>    f2 = arima(x,order=c(4,0,0),fixed=c(0,0,0,NA,NA),transform.pars=FALSE)
>
> Is there a convenient URL which shows the mathematics of the seasonal
> ARMA model, as implemented by R?

No, but there is a book, MASS4 (see the FAQ).  Although the software is in
base R it was in fact written by me to support MASS4.

R follows S-PLUS in some of its choices of signs, which do differ between
accounts.

> I understand f2 fine. I understand that you are saying that f1 is just
> an AR(4) with the lags 1,2,3 constrained to 0. But I'm unable to
> generalise this. What would be the meaning of mixing up both order and
> seasonal? E.g. what would it mean to do something like:
>
>  arima(x,order=c(2,0,0),seasonal=list(order=c(2,0,0),period=12))

That is in MASS4 and most of the books referenced on the help page.

--
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595

```