# Does ARIMA(0,1,11)4 exist?

I have, without luck, searched to find what an ARIMA model that is differenced once, and has two moving average terms (one at lag 1 and one at lag 4) is called.

The equation for what I am asking for is the following: $(1-B)Y_t= (1-0.5B-0.2B^4)a_t$.

What I have come up with of possibilities are:
ARIMA(0,1,2)4
ARIMA(0,1,1,1)4
ARIMA(0,1,1 1)4

This a ARIMA(0,1,4) model with zero restriction on some of the parameters. Usually nobody fits such types of restricted models, since it is hard to justify the restrictions.

If you really want to give this model a name, you need to invent a new notation. This means stating unambiguously the model and corresponding notation. For example ARIMA(p,d,q) means that the model is written like this

$$(1-\phi_1B-...-\phi_pB^p)(1-B)^dY_t = (1+ \theta_1B+...+\theta_qB^q)Z_t$$

where $Z_t$ is the white noise, and lag polynomials do not have common roots. So you need to write down something similar. Then you need to make this notation known, since the point of the notation is that people understand it in unambiguous way.

No. The reason is because the pdq notation is too simple to define that model. More general solutions like SAS and AUTOBOX , SPSS and others allow these kinds of models to be formed and estimated. It may be time to put away simple approaches and start to do more comprehensive/correct modelling.

• Thank you for the answer IrishStat. I have identified that model as the most fitting model with SAS PROC ARIMA in SAS BASE 9.4. But i will just write the equation instead, then. Thank once again!
– Kir
Commented Jul 9, 2015 at 13:39
• The SAS Proc does not test for the existence of Pulses, Level Shifts , Seasonal Pulses or Local Time Trends AND the constancy of both parameters and error variance over time . These considerations and more warrant you downloading a 30 day free trial of AUTOBOX (which I have helped develop) . Commented Jul 9, 2015 at 15:19
• What i have done is the following: PROC ARIMA DATA=SASUSER.sales; *where intervention=0; IDENTIFY VAR=sales(1) crosscorr=(intervention(1)) nlag=10; estimate noint q=(1,4) input=(intervention) method=ml; forecast out=results lead=0; RUN; This is, from what i understand a BoxTiao intervention analysis where the test is that the intervention is abrupt and permanent (w). E.g. Woodfield does the same : sascommunity.org/sugi/SUGI87/Sugi-12-57%20Woodfield.pdf . Am i way off?
– Kir
Commented Jul 10, 2015 at 13:27
• I am sorry for the ugly formatting, i am not quite sure how to change that.
– Kir
Commented Jul 10, 2015 at 13:31
• You are doing Intervention Modelling which requires that you pre-specify the nature of the intervention and the form of the response.In your case you are specifying a permanent and instantaneous response. Intervention Detection actually Identifies the kind of intervention and the nature of the response. Commented Jul 10, 2015 at 13:37