MA

253. MA

Moving average

MA (𝑞)

𝑥_{𝑡} = 𝑐 + 𝜃 (𝐵) 𝜀_{𝑡}

𝜃 (𝐵) = 1 + 𝜃_{1} 𝐵 + 𝜃_{2} 𝐵^{2} + \dots + 𝜃_{𝑞} 𝐵^{𝑞}

i.e.,

𝑥_{𝑡} = 𝑐 + 𝜀_{𝑡} + 𝜃_{1} 𝜀_{𝑡 - 1} + \dots + 𝜃_{𝑞} 𝜀_{𝑡 - 𝑞}

Always stationary. Invertible if roots of $𝜃 (𝐵) = 0$ lie outside the unit circle.

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

Example:

MA (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 $𝑞 = 1$ into the MA $(𝑞)$ recursion:

𝑥_{𝑡} = 𝑐 + 𝜀_{𝑡} + 𝜃_{1} 𝜀_{𝑡 - 1}

Forecast (set $𝜀_{𝑡} = 0$ ):

{\hat{𝑥}}_{𝑡} = 𝑐 + 𝜃_{1} 𝜀_{𝑡 - 1}

Innovation:

𝜀_{𝑡} = 𝑥_{𝑡} - {\hat{𝑥}}_{𝑡}

Step 2 — apply at $𝑡 = 1$

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

{\hat{𝑥}}_{1} = 0 + 0.5 \cdot 0 = 0

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

Step 3 — iterate

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

Like AR(1), MA(1) cannot capture the trend or seasonality in $𝑥$ .