Overview
ARIMA stands for, auto-regressive, integrated, moving average model. There are three pieces in this definition.
- Auto Regressive
- Moving Average
- Integrated An integrated series is a series that is - stationary after differencing.
An ARIMA model of a time series, {% Y_t, Y_{t-1}, ... %} is first differenced into a new time series, {% X_t, X_{t-1}, ... %} that is stationary.
Next, the model for {% X_t %} is designed to be an ARMA (autoregressive moving average) model.
{% X_t - \phi_1 X_{t-1} - ... - \phi_p X_{t-p} = Z_t + \theta_1 Z_{t-1} + ... + \theta_q Z_{t-q} %}
here, the source of randomness comes from {% Z_t %}, which is given by
{% Z_t \sim WN(0, \sigma^2) %}
and sometimes referred to as the innovation.