📂Statistical Analysis

Autoregressive Moving Average Model

Model ¹

White noise $\left\{ e_{t} \right\}_{t \in \mathbb{N}}$ is defined as $$ Y_{t} := \phi_{1} Y_{t-1} + \phi_{2} Y_{t-2} + \cdots + \phi_{p} Y_{t-p} +e_{t} - \theta_{1} e_{t-1} - \theta_{2} e_{t-2} - \cdots - \theta_{q} e_{t-q} $$ such a process is called a $(p,q)$th order autoregressive moving average process $ARMA(p,q)$.

Explanation

The ARMA model is simply a combination of the Moving Average Process and the Autoregressive Process. For instance, if it is of order $(1,1)$, then it is expressed as $$ ARMA(1,1) : Y_{t} = \phi Y_{t-1} + e_{t} - \theta e_{t-1} $$ However, since the ARMA model still has some limitations as a model, Differencing is used to improve it, leading to the more commonly used ARIMA model. Essentially, all are eventually considered as ARMA models.