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In the book Forecasting with Exponential Smoothing by Rob J. Hyndman it is mentioned that there is a heuristic method to get the initial states for ets models, prior to the optimization of the likelihood function. Regarding the seasonal component, the method says that being $m$ the length of the seasonality, one must compute a $2 \ x \ m$ moving average of the first few years. What do they mean by moving average, a process $f_t = \frac{1}{2q+1}\sum_{j = -q}^{q}{y_{t-j}}$, being $\left\{ y_t \right\}_t$ the series? In this case, is $q = 2m$? It is also said that the moving average should be calculated for $t = m/2 + 1, \ m/2+2, \ \cdots$, but if $m$ is not even, would not $t$ be fractionary?

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A $2\times m$-MA is not the same as a $2m$-MA. See my introductory book: https://www.otexts.org/fpp2/moving-averages.html

If $m$ is not even, then a $m$-MA would be used. For most seasonal data, $m$ is even ($m=12$ for monthly, $m=4$ for quarterly).

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