What is $h$ in the Holt-Winters model as denoted in Hyndman, R.J., & Athanasopoulos, G. (2018)?

Question

The Holt-Winters additive method model is defined to be

\begin{align*} \hat{y}_{t+h|t} &= \ell_{t} + hb_{t} + s_{t+h-m(k+1)} \\ \ell_{t} &= \alpha(y_{t} - s_{t-m}) + (1 - \alpha)(\ell_{t-1} + b_{t-1})\\ b_{t} &= \beta^*(\ell_{t} - \ell_{t-1}) + (1 - \beta^*)b_{t-1}\\ s_{t} &= \gamma (y_{t}-\ell_{t-1}-b_{t-1}) + (1-\gamma)s_{t-m}, \end{align*}

but the text I am reading doesn't mention on the relevant page what precisely $h$ represents. I see that it scales $b_t$ and shifts $s_{t+h+m(k+1)}$, but it isn't obvious to me what the motivation or meaning of this value is. The passage mentions that $k$ is the integer part of some ratio $\frac{h-1}{m}$, but this did not sufficiently clarify what the quantity $h$ is. What is this $h$ representing?

Reference

Hyndman, R.J., & Athanasopoulos, G. (2018) Forecasting: principles and practice, 2nd edition, OTexts: Melbourne, Australia. OTexts.com/fpp3. Accessed on 2022-07-19.

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Is $h$ the prediction horizon? If so, why does $h$ scale $b_t$ and shift $s_{t-m(k+1)}$?

Answer 1

$h$ is indeed the prediction horizon. $\hat{y}_{t+h|t}$ is the $h$-step ahead forecast made at time $t$, so it is for period $t+h$ and conditional on information up to period $t$; therefore the subscript is $t+h|t$.

Why does $h$ scale $b_t$? The term $b_t$ is the trend component, specifically its value at period $t$. If we forecast out $h$ periods, we expect $h$ increments due to the trend, the best estimate of which currently is $b_t$. Thus, the contribution of the trend to our point forecast is simply $hb_t$. (If the trend is dampened, you will see a dampening factor in this term, but this is an undampened trend model.)

Why does $h$ shift $s_{t+h-m(k+1)}$? We have a season of length $m$ periods. $k$ represents how many full seasonal cycles we forecast out. When we forecast out $h$ periods, it could be that we forecast less than one full seasonal cycle, e.g., six months in a yearly seasonal model. In this case, $h=6$ and $k=0$. Then $s_{t+h-m(k+1)}$ will "look back" to the last entry of the season component that corresponds to the period we are forecasting for. We have July; if we forecast out $h=6$ months, then $s_{t+h-m(k+1)}=s_{t+6-12(0+1)}=s_{t-6}$ will look back and pick the season component entry corresponding to January. If we forecast out more than one full cycle, $k>0$, and the index here will make sure we look at a component within the last cycle.