ARIMA

255. ARIMA

Autoregressive integrated moving average

ARIMA (𝑝, 𝑑, 𝑞)

𝜑 (𝐵) {(1 - 𝐵)}^{𝑑} 𝑥_{𝑡} = 𝑐 + 𝜃 (𝐵) 𝜀_{𝑡}

𝜑 (𝐵) = 1 - 𝜑_{1} 𝐵 - \dots - 𝜑_{𝑝} 𝐵^{𝑝}

𝜃 (𝐵) = 1 + 𝜃_{1} 𝐵 + \dots + 𝜃_{𝑞} 𝐵^{𝑞}

Differencing operator $(1 - 𝐵) 𝑦_{𝑡} = 𝑦_{𝑡} - 𝑦_{𝑡 - 1}$ removes trend.
Apply $𝑑$ times to make non-stationary series stationary.

Parameters: $𝜑_{1}, \dots, 𝜑_{𝑝}$ , $𝜃_{1}, \dots, 𝜃_{𝑞}$ , $𝑐$ , $𝜎^{2}$
Orders: $𝑝$ , $𝑑$ , $𝑞$

Example:

ARIMA (1, 1, 1)

Given

$𝑡$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
$x_{𝑡}$	12	10	8	11	14	12	9	13	16	14	11	15	18	16	13	17

Step 1 — formula

Substitute $𝑝 = 1$ , $𝑑 = 1$ , $𝑞 = 1$ into the ARIMA recursion:

(1 - 𝜑_{1} 𝐵) (1 - 𝐵) 𝑥_{𝑡} = 𝑐 + (1 + 𝜃_{1} 𝐵) 𝜀_{𝑡}

Difference — apply $(1 - 𝐵)$ to convert $𝑥_{𝑡}$ to a stationary series:

𝑑_{𝑡} ≔ (1 - 𝐵) 𝑥_{𝑡} = 𝑥_{𝑡} - 𝑥_{𝑡 - 1}

In the differenced space, the model is ARMA $(1, 1)$ :

𝑑_{𝑡} = 𝑐 + 𝜑_{1} 𝑑_{𝑡 - 1} + 𝜀_{𝑡} + 𝜃_{1} 𝜀_{𝑡 - 1}

Forecast the difference (set $𝜀_{𝑡} = 0$ ):

{\hat{𝑑}}_{𝑡} = 𝑐 + 𝜑_{1} 𝑑_{𝑡 - 1} + 𝜃_{1} 𝜀_{𝑡 - 1}

Undifference — convert the difference forecast back to a forecast for $𝑥_{𝑡}$ :

{\hat{𝑥}}_{𝑡} = 𝑥_{𝑡 - 1} + {\hat{𝑑}}_{𝑡}

Innovation:

𝜀_{𝑡} = 𝑑_{𝑡} - {\hat{𝑑}}_{𝑡} = 𝑥_{𝑡} - {\hat{𝑥}}_{𝑡}

Step 2 — apply at $𝑡 = 2$

First difference: $𝑑_{2} = 𝑥_{2} - 𝑥_{1} = 10 - 12 = - 2$ .

Plug in $𝜑_{1} = 0.5$ , $𝜃_{1} = 0.5$ , $𝑑_{1} = 0$ , $𝜀_{1} = 0$ :

{\hat{𝑑}}_{2} = 0 + 0.5 \cdot 0 + 0.5 \cdot 0 = 0

{\hat{𝑥}}_{2} = 𝑥_{1} + {\hat{𝑑}}_{2} = 12 + 0 = 12

𝜀_{2} = 𝑥_{2} - {\hat{𝑥}}_{2} = 10 - 12 = - 2

Step 3 — iterate

Two-stage pipeline at each $𝑡$ : difference $\to$ ARMA-forecast the difference $\to$ undifference. Values rounded to 4 decimal places.