# Shouldn't the seasonal ARIMA specification (11,0,0)x(1,0,0,12) be equivalent to non-seasonal (12,0,0)?

I know I'm not understanding something, but I'm working with monthly data that needs a seasonal component to it.

I was just testing different models and noticed that (11,0,0)x(1,0,0,12) wasn't giving the same evaluation as (12,0,0). Why is that? Aren't they essentially including the same terms, and with the same data set should nearly approximate one another?

I think I might have a misunderstanding though, could anyone shed some light?

EDIT: To clarify, my understanding is that we can formulate the AR(12) as:

$y_t=\beta_0+\beta_1y_{t-1}+...+\beta_{12}y_{t-12}$

Isn't this the same has having 11 AR terms and 1 seasonal AR term of 12-period frequency?

I think it has something to do with the polynomial backshift construction in the seasonal arima model, but I have no proof of that.

• why are there four numbers after the "x"? Dec 7 '17 at 21:17
• Sorry, I've seen that in text but i'm not sure if that's standard notation. the first tuple is the non-seasonal, the second is the seasonal p,d,q,m with m being the seasonal frequency. So (1,0,0,12) is the 12th backshift. Dec 7 '17 at 21:27

An Arima$(11,0,0)(1,0,0)_{12}$ is equivalent to a constrained ARIMA(23,0,0) because $$(1 - \phi_1B - \cdots \phi_{11}B^{11})(1 - \Phi_1 B^{12})Y_t = Z_t$$ is the same as $$(1 - \phi_1B - \cdots \phi_{12}B^{12} - \Phi_1 B^{12} + \phi_1 \Phi_1 B^{13} + \cdots + \phi_{11}\Phi_1 B^{23})Y_t = Z_t.$$