ARMA

254. ARMA

Autoregressive moving average

ARMA (𝑝, 𝑞)

𝜑 (𝐵) 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}

Combines AR and MA structure. Requires stationary input series.

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

Example:

ARMA (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 $𝑝 = 1$ , $𝑞 = 1$ into the ARMA $(𝑝, 𝑞)$ recursion:

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

Forecast (set $𝜀_{𝑡} = 0$ ):

{\hat{𝑥}}_{𝑡} = 𝑐 + 𝜑_{1} 𝑥_{𝑡 - 1} + 𝜃_{1} 𝜀_{𝑡 - 1}

Innovation:

𝜀_{𝑡} = 𝑥_{𝑡} - {\hat{𝑥}}_{𝑡}

Step 2 — apply at $𝑡 = 1$

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

{\hat{𝑥}}_{1} = 0 + 0.5 \cdot 0 + 0.5 \cdot 0 = 0

𝜀_{1} = 𝑥_{1} - {\hat{𝑥}}_{1} = 12 - 0 = 12

Step 3 — iterate

Each row uses the previous observation $𝑥_{𝑡 - 1}$ and the previous innovation $𝜀_{𝑡 - 1}$ . Values rounded to 4 decimal places.