📂Statistical Analysis

Arima Model

Model ¹

For the given white noise $\left\{ e_{t} \right\}_{t \in \mathbb{N}}$, it is defined as $$\nabla^{d} Y_{t} := \sum_{i = 1}^{p} \phi_{i} \nabla^{d} Y_{t-i} + e_{t} - \sum_{i = 1}^{q} \theta_{i} e_{t-i}$$ and this form is referred to as the $(p,d,q)$th ARIMA process $ARIMA (p,d,q)$. Such a form of time series analysis model is called ARIMA model.

Explanation

$ARI(p,d) \iff ARIMA(p,d,0)$ is referred to as AR model, and $IMA(d,q) \iff ARIMA(0,d,q)$ as MA model, though these terms are not commonly used. Preferably, expressions like $ARIMA(p,d,0)$ or $ARIMA(0,d,q)$ are favored.

Although the formula looks complicated, it’s not as difficult as it seems, as it merely involves changing $Y_{t}$ to $\nabla^{d} Y_{t}$ in the ARMA model $$Y_{t} = \sum_{i = 1}^{p} \phi_{i} Y_{t-i} + e^{t} - \sum_{i = 1}^{q} \theta_{i} e_{t-i}$$. It’s about analyzing data that has obtained stationarity through $d$ times of differencing in the ARMA model framework.