250. Summary

251. ARIMA family (Box-Jenkins models)

■ AR $𝜑\left(𝐵\right)$ ■ MA $𝜃\left(𝐵\right)$ ■ differencing ${\left(1-𝐵\right)}^{𝑑}$

Each model is written using the backshift operator $𝐵$, where $𝐵{𝑥}_{𝑡}={𝑥}_{𝑡-1}$ and ${𝐵}^{𝑘}{𝑥}_{𝑡}={𝑥}_{𝑡-𝑘}$. The compact form

expresses ARIMA as a product of three operators acting on the series: the AR polynomial, the differencing operator, and the MA polynomial. ${𝜀}_{𝑡}$ is white noise with mean zero and variance ${𝜎}^2$.

For multivariate models, ${𝒙}_{𝑡}\in {ℝ}^{𝐾}$ is a vector of $𝐾$ time series, and operator coefficients become $𝐾\times 𝐾$ matrices.

251.1. Univariate models

251.2. Multivariate (vector) models

For all models below, ${𝒙}_{𝑡}={\left({𝑥}_{1,𝑡},\dots ,{𝑥}_{𝐾,𝑡}\right)}^{\top }\in {ℝ}^{𝐾}$ is a vector of $𝐾$ jointly modeled time series. Operator coefficients ${𝚽}_{𝑖}$ and ${𝚯}_{𝑗}$ are $𝐾\times 𝐾$ matrices, and ${𝜺}_{𝑡}$ is multivariate white noise with covariance matrix $𝚺\in {ℝ}^{𝐾\times 𝐾}$.