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I simulated a time series using expressions (3.10a), (3.10b) from (Hyndman et al., 2008). Next, I'd like to use a simple exponential smoothing method to forecast for the next period. For a given initial point $\hat{y}_1$ I obtained the smoothing parameter $0<\alpha<2$, etc.

My question is as follows: am I going to predict the next value of the local level $l_t$ or the next value of the observed time series $y_t$? Regretfully, I still do not understand how one can predict a quantity which includes a random component, and if I got it right, time series without trend consists of level and such a component.

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For simple exponential smoothing, \begin{align*} y_t &= \ell_{t-1} + e_t \\ \ell_t &= \ell_{t-1} + \alpha e_t, \end{align*} where $e_t$ is a white noise error. So the forecast of $y_{t+1|t}$ is also a forecast of $\ell_{t+1|t}$. That is, both conditional distributions have the same mean, and the forecast is an estimate of that mean.

Everything you want to predict has a random component, otherwise you would know its value and there would be no need to predict it. Forecasts are just estimates of the means of conditional distributions.

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