# [R] [gently off topic] arima seasonal question

Rolf Turner rolf at math.unb.ca
Fri Jul 2 15:46:36 CEST 2004

```The seasonal aspect of arima models allows, essentially, for a
special realtionship between X_t and X_{t+s} where s is the
``seasonality'' of the model.  It (``the model'') couldn't care less
what the time ***units*** are --- they could be weeks, quarters,
days, hours, microseconds, 1.14135*microseconds, ....  What matters
is:  Do you have reason to believe that there is a special
relationship between X_t and X_{t+s}???  If so, go for it.  If not,
don't.

Such relationships are ***most likely*** to arise in quarterly and
monthly data --- with s = 4 in the quarterly data, s = 12 in the
monthly data.  You could conceiveably get seasonality with s = 7 in
daily data; at a stretch with s = 30 (pretending all months are 30
days long ... a bit dubious).  You might (ah, well, sort of ....)
also have s = 365 seasonality in daily data, but such a large s is
unlikely to ``work'' very well.  You might get seasonality with s =
52 in weekly data.  (Dubious.)  You might get seasonality with s = 24
in hourly data.  U.s.w.

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)
> f1
.
Coefficients:
sar1  intercept
0.4987    -0.0775
s.e.  0.0499     0.1051

sigma^2 estimated as 0.8536:  log likelihood = -402.51,  aic = 811.02

> f2
.
Coefficients:
ar1  ar2  ar3     ar4  intercept
0    0    0  0.4987    -0.0774
s.e.    0    0    0  0.0499     0.1051

sigma^2 estimated as 0.8536:  log likelihood = -402.51,  aic = 811.02
===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===

Hope this is a bit enlightening.

cheers,

Rolf Turner
rolf at math.unb.ca

> Hello R People:
>
> When using the arima function with the seasonal option, are the
> seasonal options only good for monthly and quarterly data, please?
>
> Also, I believe that weekly and daily data are not appropriate for
> seasonal parm estimation via arima.
>