Can you show me an example of an ARIMA(1,1,1) model and its equations?

Conditions

• $p=1$ (one AR term)
• $d=1$ (one first difference)
• $q=1$ (one MA term)

Reasoning, step by step

1. Apply first-order differencing ($d=1$) to the original time series $Y_t$ to obtain the stationary series $W_t$.
2. Model the differenced series $W_t$ using an ARMA(1,1) process, incorporating one AR term ($p=1$) and one MA term ($q=1$).

Example

Slide displays ARIMA(1,1,1) with equations $Y_t - Y_{t-1} = W_t$ and $W_t = \omega + \phi_1 W_{t-1} + \theta_1 e_{t-1} + e_t$. Speaker explains that for an ARIMA(1,1,1), we first take a difference ($d=1$) to go from Y to W, then have a lag of $W (p=1)$, and a lag of error ($q=1$).

Common misconceptions

• Thinking the ARMA equation is applied directly to $Y_t$.
• Confusing the differenced series $W_t$ with the original series $Y_t$.

Watch the explanation

Explore next

• What do the parameters p, d, and q represent in an ARIMA model?
• How do you make time series data stationary?