Seasonal ARIMA

ARIMA($p$,$d$,$q$) models rely on the assumption that the underlying data is stationary or can be made stationary by differencing it $d$ number of times. However, some time series will not be stationary even after differencing. Seasonal series are a common example. Seasonality is a widespread phenomenon in time series analysis:

• People consume more during Christmas.

• Households consume less energy at night.

• Temperatures are higher in summer than in winter.

Fortunately for us, we can extend the logic of the ARIMA model to account for these seasonal (regular) variations. This is what the Seasonal Autoregressive Integrated Moving Average (SARIMA) model does.

Introducing seasonality in the formula

To understand the SARIMA model, let’s break it down into two blocks, the ARIMA block and the seasonal block.

The ARIMA block

We use this part to model the dependence of $y_t$ on its lagged values. We can think of this as a standard ARIMA model. It is composed of an AR part with $p$ lags, AR($p$), an MA part with $q$ lags, MA($q$), and an integration of order $d$, $I(d)$ ...