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

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.

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

$$h$$ 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.