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In forecasting principles and practice, the update equations are given as:

$$l_t = \alpha\frac{y_t}{s_{t-m}} + (1-\alpha)(l_{t-1} + b_{t-1})$$ $$b_t = \beta^*(l_t-l_{t-1}) + (1-\beta^*)b_{t-1}$$ $$s_t = \gamma\frac{y_t}{l_{t-1}+b_{t-1}} + (1-\gamma)s_{t-m}$$

However, in other texts I've found, such as this one and this one, the last one is given as

$$s_t = \gamma\frac{y_t}{l_t} + (1-\gamma)s_{t-m}$$

So, which is it?

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  • $\begingroup$ I wish there was a way to tag users like @user:159 who is the author of that text (Rob Hyndman). I pinged him by email. $\endgroup$
    – Avraham
    Dec 28, 2021 at 19:55

1 Answer 1

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As Hyndman writes regarding additive model, seasonal component there is usually expressed in one of two ways, either:

$$ s_{t}=\gamma\left (y_{t} - l_{t-1} - b_{t-1} \right ) + \left ( 1-\gamma \right )s_{t-m} $$

or:

$$ s_{t}=\gamma^{*}\left (y_{t} - l_{t} \right ) + \left ( 1-\gamma^{*} \right )s_{t-m}$$

where

$$ \gamma = \gamma^{*}\left (1-\alpha \right ) $$

Similarly for multiplicative model we have two representations:

$$ s_{t}=\gamma \frac{y_{t}}{l_{t-1} + b_{t-1}} + \left ( 1-\gamma \right )s_{t-m} $$

and

$$ s_{t}=\gamma^{*} \frac{y_{t}}{l_{t}} + \left ( 1-\gamma^{*} \right )s_{t-m} $$

Apparently Hyndman just didn't bother providing the second one.

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  • $\begingroup$ Ah yes, I missed that in one case gamma is between 0 and 1, and in the other it's between 0 and 1-alpha. Thanks! $\endgroup$ Dec 29, 2021 at 9:40

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