What is the equation for ARIMA(1,1,1)?

The $d$ in ARIMA(p,d,q) represents the order of differencing, also known as the integrated component. It indicates how many times the original series has been differenced to achieve stationarity.

Conditions: The model is denoted as ARIMA(p,d,q).; $d$ is a non-negative integer.

You would use a second difference if the first differenced series is still not stationary. The goal is to choose the lowest differencing order $d$ that makes the series stationary.

Conditions: The original series is non-stationary.; First differencing ($d=1$) does not result in a stationary series.; The modeler can inspect the stationarity of the differenced plots.

You recover the original series $q_k$ by summing the predicted differences $Z_t$ and adding them to the last known actual value $q_l$ (or $q_1$ depending on the starting index). The formula is $q_k = \sum_{i=1}^{k-l} Z_{k-i} + q_l$.

Conditions: You have forecasts for the differenced series $Z_t$.; You know the last observed value of the original series $q_l$.; The relationship $Z_t = q_{t+1} - q_t$ holds.

ARMA models require the time series to be stationary. A series with an upward trend has a moving mean, which violates the stationarity condition of having a constant mean over time.

Conditions: The model class is AR, MA, or ARMA.; The series exhibits a non-constant mean (trend).

Differencing is a technique used to transform a non-stationary time series into a stationary one by computing the differences between consecutive observations.

Conditions: Used when data exhibits non-stationarity.