Multiplicative seasonal ARIMA

[jensk]

Hello,

I have quarterly data and I want to estimate an ARIMA (1,1,0)X(2,1,0)4.
I know the model has the the following form:
(1- a1 B) (1- a4 B^4 - a8 B^8) (1-B)(1-B^4) Yt= u+Et

I also know that I need to use something like
d(Y,1,4) c ar(1) *representation for (1- a4 B^4 - a8 B^8)*

Could you please tell me how to represent the polynomial (1- a4 B^4 - a8 B^8) (i.e. AR(2)4) using the Eviews syntax?

Thank you very much.

[EViews Glenn]

From the EViews manual...

A SAR( ) term can be included in your equation specification for a seasonal autoregressive term with lag . The lag polynomial used in estimation is the product of the one specified by the AR terms and the one specified by the SAR terms. The purpose of the SAR is to allow you to form the product of lag polynomials.

I'm not sure from your notation how you're getting the L^8 term from the polynomial product, but that aside, you should look at the documentation on Seasonal ARMA for discussion.

[jensk]

From the EViews manual...

A SAR( ) term can be included in your equation specification for a seasonal autoregressive term with lag . The lag polynomial used in estimation is the product of the one specified by the AR terms and the one specified by the SAR terms. The purpose of the SAR is to allow you to form the product of lag polynomials.

I'm not sure from your notation how you're getting the L^8 term from the polynomial product, but that aside, you should look at the documentation on Seasonal ARMA for discussion.

Thanks for the reply. I'm using standard notation. I am dealing with quarterly data, so the seasonal polynomial of order 2 takes the following form: 1- a1s B^1*s - a2s B^2*s

From the manual, it is clear that a polynomial of the form (1- a4 B^4) can be represented as a sar(4). However, It's still not clear to me how to represent the polynomial 1- a4 B^4 - a8 B^8. Any thoughts would be appreciated.

[EViews Glenn]

I may be missing something, but couldn't you just do

Code: Select all

ar(4) ar(8)

[jensk]

I may be missing something, but couldn't you just do

Code: Select all

ar(4) ar(8)

Thank you for your observation. However, since this is a multiplicative model, (1- a1 B) (1- a4 B^4 - a8 B^8), I would also need to include an ar(5) and an ar(9) term. The issue with such an approach is that I would need to estimate 5 ar parameters instead of just 3, since the representation of the second order polynomial does not seem to be possible in Eviews.

[startz]

If this is really an ARIMA model no RHS variables and no MA terms, you can expand the polynomial in y lags by hand and estimate by nonlinear least squares. For example (1-a1B)(1-a2B)y = e would be

Code: Select all

ls y (c(1)+c(2))*y(-1) c(2)*c(3)*y(-2)

[EViews Glenn]

Sorry, I missed the part about multiplication of the latter polynomial by the former since I was just focusing on the last part of the post where you asked the specific question, not the first part where you talked about the specification. That said, doesn't

Code: Select all

ar(1) sar(4) sar(8)

give you

Code: Select all

(1 - rho(1)*L)(1 - theta(1)L^4 - theta(2)*L^8) u

which expands to

Code: Select all

u(t) = 1 - rho(1)*u(t-1) - theta(1)*u(t-4) + rho(1)*theta(1)*u(t-5) - theta(2)*u(t-8) + rho(1)*theta(2)*u(t-9))

which, again if I'm not misunderstanding (and I probably am), gives you the restricted AR(9) that you want. I'm not sure if these are the restrictions on the polynomial that you want, but you should be able to adjust the spec accordingly.

[runawayyy]

Excuse me, Could somebody tell me how to determine the parameter p used in SAR(p) ??
Is it true that p equals 4 in case of quarterly data and 12 in case of monthly data?
Thank you.