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I am learning time series analysis from Forecasting: Principles and Practice by Hyndman & Athanasopoulos. In chapter 8, the ETS (Error, Trend, Seasonality - a state space implementation of Exponential Smoothing) model is written as

$$\hat{y}_{T+1|T} = \alpha y_T + \alpha(1-\alpha) y_{T-1} + \alpha(1-\alpha)^2 y_{T-2}+ \cdots$$ where $\alpha$ is a smoothing parameter that takes the value between 0 and 1. When this $\alpha$ is 1 the ETS is equivalent to the NAIVE method and when this parameter takes on the value of 0, the ETS is equivalent to the MEAN method.

While I understand how ETS = NAIVE method when the value of $\alpha$ is 1, I fail to fully understand how this method becomes equivalent to the MEAN when $\alpha$ takes on the value of 0. There is explanation for why this is, in the given chapter but I find it a bit tricky to understand. I would appreciate it if someone could explain how come when $\alpha$=0, the ETS method is equivalent to the MEAN method.

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    $\begingroup$ Please explain the meaning of all the acronyms in the post to help readers who are not experts in the area. $\endgroup$
    – Zhanxiong
    Commented Feb 16, 2023 at 4:49
  • $\begingroup$ Thanks, will keep that in mind in the future :) $\endgroup$
    – ask9
    Commented Feb 17, 2023 at 2:58

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You've missed the last term. The full expression is $$\hat{y}_{T+1|T} = \sum_{j=0}^{T-1} \alpha(1-\alpha)^j y_{T-j} + (1-\alpha)^T \ell_{0},$$ so when $\alpha = 0$, $$\hat{y}_{T+1|T} = \ell_{0}.$$ If the parameter $\ell_0$ is estimated using least squares, we are minimizing $$\sum_{t=1}^T (y_t - \ell_0)^2$$ which has minimum at $$\hat{\ell}_0 = \frac{1}{T}\sum_{t=1}^T y_t.$$

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  • $\begingroup$ Keep reading, just another paragraph or two. $\endgroup$ Commented Feb 17, 2023 at 2:55
  • $\begingroup$ As given in the book: $\hat y_{4|3}= \alpha y_3 + (1- \alpha) \hat y_{3|2}$ equals the previous observation ($\alpha y_3) + (1- \alpha)$ times our best estimate of previous observation $\hat y_{3|2}$. When we go further back to the initial value, our one step ahead forecast = $(\alpha y_1) + (1-\alpha) \ell_0$ where $\ell_0$ is estimate of the first observation When $\alpha$=0 then all else cancels & we've $\hat y_{t+1|T}=\ell_0 and $ell_0$=\bar y$ because for $\ell_0$ to min.SSE the value we obtain is the mean of the data. Thanks for Fpp3 Prof.Hyndman; it's a great learning resource. $\endgroup$
    – ask9
    Commented Feb 17, 2023 at 2:56

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