Autoregressive integrated moving average

Remarks

For $d = 0$ this is a causal ARMA($p$, $q$) process. For $d \ge 1$ the process is not stationary in general. It is determined by $Y$ only up to $d$ starting values, since $\nabla^d$ kills polynomials in $t$ of degree less than $d$. A series whose $d$-th difference is stationary but whose $(d-1)$-th difference is not is called integrated of order $d$, written I($d$).

Example: the random walk $X_t = X_{t-1} + \varepsilon_t$ is ARIMA($0$, $1$, $0$), because $\nabla X_t = \varepsilon_t$ is a white noise, which is ARMA($0$, $0$). The model $X_t = X_{t-1} + 0.5 (X_{t-1} - X_{t-2}) + \varepsilon_t$ is ARIMA($1$, $1$, $0$), since $\nabla X_t = 0.5 \nabla X_{t-1} + \varepsilon_t$.

Non-example: the explosive series with $X_0 = 0$ and $X_t = 1.5 X_{t-1} + \varepsilon_t$ for $t \ge 1$ is not ARIMA for any $d$. Its autoregressive polynomial $1 - 1.5 z$ has the root $2/3$ inside the unit disk, so the variance of $X_t$ grows geometrically. Differencing does not remove this explosive root, because $\nabla^d$ only contributes the root $1$, so $\nabla^d X_t$ is never stationary.

Seasonal ARIMA models also apply seasonal differences $1 - B^s$ for a period $s$, such as $s = 12$ for monthly data.