How to Identify a multiplicative SARIMA model?

Question

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.

Answer 1

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