Often time series possess a seasonal component that repeats every s observations. For monthly observations s= 12 (12 in 1 year), for quarterly observations s= 4 (4 in 1 year). In order to deal with seasonality, ARIMA processes have been generalized: **SARIMA** models (Seasonal Autoregressive Integrated Moving Average Model) have then been formulated.

Φ (B) ∆d Xt = θ (B) αt

where αt is such that

sΦ (Bs) ∆Ds αt = sΘ (Bs) at

hence

Φ (B)s Φ (Bs) ∆Ds ∆d Xt = θ (B)s Θ (Bs) αt

and we write Xt ∼ ARIMA (p, d, q) × (P, D, Q)s. The idea is that **SARIMA** models are ARIMA (p, d, q) models whose residuals αt are ARIMA (P, D, Q). With ARIMA (P, D, Q) we intend ARIMA models whose operators are defined on Bs and successive powers.

Concepts of admissible regions SARIMA are analog to the admissible regions for ARIMA processes; they are just expressed in terms of Bs powers.

Now, consider some examples (special cases):

- Xt = at – s Θ1 at−12 is ARIMA (0, 0, 0) × (0, 0, 1)12 model. There is only one seasonal MA component, specified by s = 12, Q = 1. So, the ACF is characterized by finite extension and takes value only at lag k = 12. The PACF is infinite extended with exponential decay, visible at multiple of 12 lags, which is alternate or monotonic according to the sign of Θ
- Xt =s Φ1 Xt−12 + at is ARIMA (0, 0, 0) × (1, 0, 0)12 model. The seasonal AR component is specified by s = 12, P = 1. So, the ACF is characterized by infinite extension. The PACF is with finite extended and takes value only at lag k = 12.

## CHARACTERISTICS OF ARIMA PROCESSES

- d = 0 stationary process
- d = 1 no stationary process: the level changes in time, but the increase is constant → level is no stationary, but its increments are
- d = 2 no stationary process: both level and increments are stationary

When Xt is no stationary, its theoretical ACF is not defined (only the empirical ACF is). However, by observing the behavior of processes that are nearly stationary we can put in evidence the following regularities:

- The ACF decreases extremely slowly to zero, the decrease is not exponential by linear.
- The PACF takes value 1 for k = 1 and zero elsewhere.

These characteristics of ACF and PACF are motivated by the dominance of the trend on the other dynamics in the series. Unless the trend is removed, nothing else can be recognized from ACF and PACF (e.g. other MA or AR components).

One great and solid example can be find in Aleasoft