A-A-N

238. A-A-N

Holt’s linear trend

ETS (𝐴, 𝐴, 𝑁)

𝑥_{𝑡} = 𝑙_{𝑡 - 1} + 𝑏_{𝑡 - 1} + 𝜀_{𝑡}

𝑙_{𝑡} = 𝑙_{𝑡 - 1} + 𝑏_{𝑡 - 1} + 𝛼 𝜀_{𝑡}

𝑏_{𝑡} = 𝑏_{𝑡 - 1} + 𝛽 𝜀_{𝑡}

{\hat{𝑥}}_{𝑡 + ℎ | 𝑡} = 𝑙_{𝑡} + ℎ 𝑏_{𝑡}

Example:

ETS (𝐴, 𝐴, 𝑁)

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

Observation:

𝑥_{𝑡} = 𝑙_{𝑡 - 1} + 𝑏_{𝑡 - 1} + 𝜀_{𝑡}

Conditional mean (one-step-ahead forecast ${\hat{𝑥}}_{𝑡 | 𝑡 - 1} = 𝜇_{𝑡}$ ):

𝜇_{𝑡} = 𝑙_{𝑡 - 1} + 𝑏_{𝑡 - 1}

Innovation:

𝜀_{𝑡} = 𝑥_{𝑡} - 𝜇_{𝑡}

State updates:

𝑙_{𝑡} = 𝑙_{𝑡 - 1} + 𝑏_{𝑡 - 1} + 𝛼 𝜀_{𝑡}

𝑏_{𝑡} = 𝑏_{𝑡 - 1} + 𝛽 𝜀_{𝑡}

Forecast $ℎ$ steps ahead from time $𝑡$ (using current-period states):

{\hat{𝑥}}_{𝑡 + ℎ | 𝑡} = 𝑙_{𝑡} + ℎ 𝑏_{𝑡}

where $ℎ \in {1, 2, 3, \dots}$ is the forecast horizon (how many steps ahead).

Step 2 — apply at $𝑡 = 1$

𝜇_{1} = 12 + 0.5 = 12.5

𝜀_{1} = 𝑥_{1} - 𝜇_{1} = 12 - 12.5 = - 0.5

𝑙_{1} = 12 + 0.5 + 0.5 \cdot (- 0.5) = 12.25

𝑏_{1} = 0.5 + 0.4 \cdot (- 0.5) = 0.3

Step 3 — iterate

Each column header is the equation that produced its values. Values rounded to 4 decimal places; arithmetic performed at full precision.