# How to Identify a multiplicative SARIMA model?

In «Introduction to Time Series and Forecasting» by Brockwell and Davis, an $\{X_t\}$ SARIMA $(p,d,q)\times(P,D,Q)_s$ process is of the form:

$$\phi(B)\Phi(B^s)Y_t=\theta(B)\Theta(B^s)Z_t$$

where $Y_t=(1-B)^d(1-B^s)^DX_t$, and $Z_t \sim WN(0,\sigma^2)$, and $\deg(\phi,\Phi, \theta, \Theta)=(p,P,q,Q)$.

The authors state that one possibility for identification of this SARIMA model, is to

1. Find $d$ and $D$, so as to make the observations $\{Y_t\}$ stationary in appearance. - But how are $s,d,D$ found? Is it just by 'common sense'/plots? Also, when would we need $D>1$?

2. Examine the ACF and PACF of $\{Y_t\}$ at lags multiples of seasonality $s$, and determine $P$ and $Q$ as if determining an ARMA$(P,Q)$ process.

3. Then looking for all lags from $1$ to $s-1$ of $\{Y_t\}$ ACF and PACF we fit an ARMA$(p,q)$ model.

4. Given $p,d,q,P,D,Q,s$, we can determine the coefficients of the SARIMA polynomials $\phi, \Phi, \theta, \Theta$, by looking at $\{Y_t\}$ as an ARMA$(p+sP,q+sQ)$ process, in which some coefficients are zero. - I understand that we could rewrite the initial difference equation as an ARMA$(p+sP,q+sQ)$ process, but I don't get why we would have to constrain some coefficients to be zero... And which ones should be zero?

Any help would be appreciated.

It is very difficult to choose the parameters for a SARIMA model in order to produce an accurate forecast. There's generally two approaches:

1). Use a cross-validation approach which is very time consuming and computationally expensive. Also requires a lot of data.

Cross-Validation of Time-Series

2). Take advantage of the auto.arima function in the 'forecast' package in R. More on how that works here:

auto.arima

• Users are beginning to flag your posts because many of those posts, like this one, appear primarily to be links to the same software. Nobody's complaining about the software--only about the repetition. Please consider your message sent and think about elaborating on your recommendations in the future, lest they begin to seem more like ads and less like answers.
– whuber
Jul 8, 2018 at 23:49
• Thanks for the note whuber, I understand. I have no affiliation with Rob Hyndman for what it's worth. However his work is robust and I've found it to be one of the best references for many time-series related questions. I had some free time today, so just browsing through some past questions that were not answered; hence a few posts in a short period. Jul 9, 2018 at 3:25