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An ARIMA(1,1,1) model consists of taking a first difference of the original series and then fitting an ARMA(1,1) model to the resulting differenced series. The equations are $Y_t - Y_{t-1} = W_t$ and $W_t = \omega + \phi_1 W_{t-1} + \theta_1 e_{t-1} + e_t$.

Conditions: $p=1$ (one AR term); $d=1$ (one first difference); $q=1$ (one MA term)

An ARIMA(1,1,1) model consists of taking a first difference of the original series and then fitting an ARMA(1,1) model to the resulting differenced series. The equations are $Y_t - Y_{t-1} = W_t$ and $W_t = \omega + \phi_1 W_{t-1} + \theta_1 e_{t-1} + e_t$.

Conditions: $p=1$ (one AR term); $d=1$ (one first difference); $q=1$ (one MA term)

In ARIMA(p, d, q), p represents the number of autoregressive (AR) terms, d is the number of first differences applied to make the data stationary, and q denotes the number of moving average (MA) terms.

Conditions: Applies to ARIMA model notation.

In ARIMA(p, d, q), p represents the number of autoregressive (AR) terms, d is the number of first differences applied to make the data stationary, and q denotes the number of moving average (MA) terms.

Conditions: Applies to ARIMA model notation.