Autoregressive moving-average model ($\mathrm{ARMA}(p,q)$)

An autoregressive–moving-average model satisfieswhere $B$ is the backshift operator, $\varphi$ and $\theta$ are finite polynomials, and $\epsilon$ is white noise.

The backshift operator acts on a time series by $B{X}_t=X_{t-1}$ and hence $B^j{X}_t=X_{t-j}$.

An autoregressive model expresses the current value as a linear combination of finitely many past values plus white noise.

An autoregressive process of order one satisfies $X_t=\varphi X_{t-1}+{\epsilon }_t$. It is causal and weakly stationary when $|\varphi |<1$.

A moving-average model expresses the current value as a finite linear combination of present and past white-noise innovations.

A moving-average process of order one has the form $X_t={\epsilon }_t+\theta {\epsilon }_{t-1}$ and zero autocovariance beyond lag one.