22 ARIMA Models

ARIMA stands for AutoRegressive Integrated Moving Average. It is a generalization of the simpler AutoRegressive Moving Average and adds the notion of integration.

This acronym is descriptive, capturing the key aspects of the model itself. Briefly, they are:

• AR: Autoregression. A model that uses the dependent relationship between an observation and some number of lagged observations.
• I: Integrated. The use of differencing of raw observations (i.e. subtracting an observation from an observation at the previous time step) in order to make the time series stationary.
• MA: Moving Average. A model that uses the dependency between an observation and residual errors from a moving average model applied to lagged observations. Each of these components are explicitly specified in the model as a parameter.

A standard notation is used of ARIMA(p,d,q) where the parameters are substituted with integer values to quickly indicate the specific ARIMA model being used.

The parameters of the ARIMA model are defined as follows:

• \(p\): The number of lag observations included in the model, also called the lag order.
• \(d\): The number of times that the raw observations are differenced, also called the degree of differencing.
• \(q\): The size of the moving average window, also called the order of moving average.

Un proceso estocástico \(\left(Y_{t}\right)\) es integrado de orden \(d\left(d \geq 0\right.\) entero) si y sólo si \(\left(Y_{t}\right)\) sigue un modelo autorregresivo-integrado-media móvil de orden \((p, d, q)\), o \(\operatorname{ARIMA}(p, d, q)\) (del inglés AutoRegressive-Integrated-Moving Average), del tipo \[ \phi(B) \nabla^{d} Y_{t}=\mu+\theta(B) \omega_{t} \quad \forall t=0, \pm 1, \pm 2, \ldots \] donde las raíces de las ecuaciones \(\phi(x)=0\) y \(\theta(x)=0\) están fuera del círculo unitario.