Time-Series Modeling and Decomposition

 The simplest seasonal model for monthly seasonality can be written as

$S_{t}=\Sigma_{j=1}^{12} \alpha_{j} d_{j t}+u_{t}, d_{j t}=\left\{\begin{array}{cc} 1, & j=t \pm 12 k, k_{0}=0,1,2, \ldots, 11 \\ 0, & \text { otherwise } \end{array}\right.$                (14)

subject to $\Sigma_{j=1}^{12} \alpha_{j}=0,\left\{u_{t}\right\}$, $\left\{u_{t}\right\}$ is assumed white noise. The $\alpha_{j}$ are the seasonal effects and the $d_{j t}$'s are dummy variables.

Model (14) can be equivalently written by means of sines and cosines

$S_{t}=\Sigma_{j=1}^{6}\left[\alpha_{j} \cos \left(\lambda_{j} t\right)+\beta_{j} \sin \left(\lambda_{j} t\right)\right]$                                                                    (15)

where $\lambda_{j}=2 \pi j / 12, j=1,2, \ldots, 6 \text { and } \beta_{6}=0$. The $\lambda_{j}$s are known as the seasonal frequencies, with $j$ corresponding to cycles lasting 12, 6, 4, 3, 2.4 and 2 months respectively.

In order to represent stochastic seasonality, the $\alpha_{j}$ of model (14) are specified as random variables instead of constant coefficients. Such a model is

$S_{t}=S_{t-12}+\omega_{t}$,                                                                                                     (16.a)

or $\left(1-B^{12}\right) S_{t}=\omega_{t}$,                                                                                            (16.b)

subject to constraints $\sum_{j=0}^{11} S_{t-j}=\omega_{t}$ where $\omega_{t}$ is assumed white noise.

Model (16.a) specifies seasonality as a non-stationary random walk process. Since $\left(1-B^{s}\right) \equiv(1-B)\left(1+B+\ldots+B^{s-1}\right)$, model-based seasonal adjustment method assigns $(1-B)$ to the trend and $S(B)=\sum_{j=0}^{s-1} B^{j}$ to the seasonal component. Hence, the corresponding seasonal model is

$\sum_{j=0}^{s-1} S_{t-j}=\omega_{t}$,                                                                                                     (17)

which entails a volatile seasonal behaviour, because the sum is not constrained to 0 but to the value of $\omega_{t}$. Indeed, the spectrum of $\sum_{j=0}^{s-1} B^{j}$ (not shown here) displays broad bands at the high seasonal frequencies, i.e. corresponding to cycles of 4, 3, and 2.4 months.

Model (17) has been used in many structural time series models. A very important variant to model (17) was introduced by Hillmer and Tiao and largely discussed in Bell and Hillmer, that is

$\sum_{j=0}^{s-1} \quad S_{t-j}=\eta_{s}(B) b_{t}$                                                                                        (18)

where $\eta_{s}(B)$ is a moving average of $s-1$ minimum order and $b_{t} \sim W N\left(0, \sigma_{b}^{2}\right)$. The moving average component enables seasonality to evolve gradually. Indeed, the moving average eliminates the afore mentioned bands at the high seasonal frequencies.

Another stochastic seasonality model is based on trigonometric functions defined as

$S_{t}=\Sigma_{j=1}^{[s / 2]} \gamma_{j t}$                                                                                                        (19)

where $\gamma_{j t}$ denotes the seasonal effects generated by

$\left[\begin{array}{c} \gamma_{j t} \\ \gamma_{j t}^{*} \end{array}\right]=\left[\begin{array}{cc} \cos & \sin \\ -\sin \lambda_{j} & \cos \end{array}\right]\left[\begin{array}{l} \gamma_{j, t-1} \\ \gamma_{j, t-1}^{*} \end{array}\right]+\left[\begin{array}{c} \omega_{j t} \\ \omega_{j t}^{*} \end{array}\right]$                                               (20)

and $\lambda_{j}=2 \pi j / s, j=1, \ldots,[s / 2]$ and $t=1, \ldots, T$. The seasonal innovation $\omega_{j t}$ and $\omega_{j t}^{*}$ are mutually uncorrelated with zero means and common variance $\sigma_{\omega}^{2}$.