The most common seasonal models are $\begin{align*} &\mathrm{A} &&X_{t}= m_{t}+S_{t}+\epsilon_{t}\\ &\mathrm{B} &&X_{t}= m_{t} S_{t} + \epsilon_{t}\\ &\mathrm{C} &&X_{t}=m_{t}S_{t}\epsilon_{t} \end{align*}$where $m_{t}$ is the mean level without seasonality (modeled as a trend), and $S_{t}$ is the seasonal effect. - Model $\mathrm{A}$ uses **additive seasonality**, and models $\mathrm{B,C}$ have **multiplicative seasonality**; the error term in $\mathrm{C}$ is multiplicative as well. - In reality, the seasonal effect can be somewhere between multiplicative and additive, with its effect increasing with the mean but not as fast as the linear proportion implied in the multiplicative model. For isolating the trend, the most common filter is the **moving average**$\mathrm{Sm}(x_{t}):= \frac{1}{2s}\left( \frac{1}{2}x_{t-s}+ x_{t-s+1} + \dots + x_{t+s-1}+\frac{1}{2}x_{t+s} \right)$where $2s$ is the amount of time between seasons (e.g. $2s=12$ for monthly data). The $1 / 2$ factors on either end ensures the season/month corresponding to the start of the year (cycle) receives the same weight as the rest. - Given the trend, the stationary part can be computed as $x_{t}-\mathrm{Sm}(x_{t}),\,\, \frac{x_{t}}{\mathrm{Sm}(x_{t})}$for additive and multiplicative models respectively. For finding the seasonal effects in the stationary component, the easiest method is to compute the month/season's value averaged over all cycles: $S_{t}=\mathrm{mean}_{i \in \mathbb{Z}}(x_{t+2is})$and the residual after removing $S_{t}$ is the truly random variations. The differencing operator $\nabla$ can be generalized to take differences between non-neighbors: $\nabla_{r}x_{t}=x_{t}-x_{t-r}$where **seasonal differencing** use $r=12$ or $r=4$ for monthly and quarterly data.