Equation 8.1

[Dmitrij548]

Hello, is it correct to write the Equation 8.1 like this: y ̂_(T+1|T)=αy_T+α(1-α) y_(T-1)+α〖(1-α)〗^2 y_(T-2)+⋯+α〖(1-α)〗^n y_(T-n)

[robjhyndman]

It is not clear what $n$ is here. The length of the series in our notation is $T$.

The equation is deliberately infinite. That is, there is no end. Of course, there are a finite number of observations, so once we run out of observations, we have no way of computing the remaining terms.

If you expand the equation to include all observations, it is

$$\hat{y}_{T+1|T} = \alpha y_T + \alpha(1-\alpha) y_{T-1} + \alpha(1-\alpha)^2 y_{T-2} + \dots + \alpha(1-\alpha)^{T-1} y_{1} + c$$

where $c = (1-\alpha)^T \sum_{j=0}^{-\infty} \alpha(1-\alpha)^j y_{j}$ contains the unobserved terms. We can rewrite that as $c = (1-\alpha)^T\ell_0$, giving the weighted average form.

[Dmitrij548]

Thank you for the answer. Can You explain, what does ℓ0 mean?

[robjhyndman]

That's the level component (or fitted value) at time 0. Please read the section I linked to.

[Dmitrij548]

Thank You