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The equation for ARIMA(1,1,1) is $Z_t = \phi_1 Z_{t-1} + \theta_1 \epsilon_{t-1} + \epsilon_t$, where $Z_t$ is the differenced series, $\phi_1$ is the autoregressive coefficient, $\theta_1$ is the moving-average coefficient, and $\epsilon_t$ is the error term.

Conditions: The model is ARIMA(1,1,1).; $Z_t$ is the first difference of the original series $q_t$.; $p=1, d=1, q=1$.

The equation for ARIMA(1,1,1) is $Z_t = \phi_1 Z_{t-1} + \theta_1 \epsilon_{t-1} + \epsilon_t$, where $Z_t$ is the differenced series, $\phi_1$ is the autoregressive coefficient, $\theta_1$ is the moving-average coefficient, and $\epsilon_t$ is the error term.

Conditions: The model is ARIMA(1,1,1).; $Z_t$ is the first difference of the original series $q_t$.; $p=1, d=1, q=1$.

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)

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.

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.

The I in ARIMA stands for Integrated. It is implemented by modeling the differences of the original time series rather than the raw series itself.

Conditions: The context is the ARIMA time series model.; The explanation refers to the specific ARIMA(1,1,1) example shown on the board.

The I in ARIMA stands for Integrated. It is implemented by modeling the differences of the original time series rather than the raw series itself.

Conditions: The context is the ARIMA time series model.; The explanation refers to the specific ARIMA(1,1,1) example shown on the board.

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.