Re: [R] [gently off topic] arima seasonal question

From: Rolf Turner <rolf_at_math.unb.ca>
Date: Fri 02 Jul 2004 - 23:46:36 EST


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@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.
>
> Is that correct, please?
>
> Thanks,
> Sincerely,
> Laura Holt
> mailto: lauraholt_983@hotmail.com



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