Evolutive seasonality test (Moving seasonality test)

The evolutive seasonality test is based on a two-way analysis of variance model. The model uses the values from complete years only. Depending on the decomposition type for the Seasonal – Irregular component it uses [1] (in the case of a multiplicative model) or [2] (in the case of an additive model):

$\left|\text{SI}_{\text{ij}} - 1 \right| = X_{\text{ij}} = b_{i} + m_{j} + e_{\text{ij}}$, [1]

$\left| \text{SI}_{\text{ij}} \right| = X_{\text{ij}} = b_{i} + m_{j} + e_{\text{ij}}$, [2]

where:

$m_{j}$ – the monthly or quarterly effect for $j$-th period, $j = (1,\ldots,k)$, where $k = 12$ for a monthly series and $k = 4$ for a quarterly series;

$b_{j}$ – the annual effect $i$, $(i = 1,\ldots,N)$ where $N$ is the number of complete years;

$e_{\text{ij}}$ – the residual effect.

The test is based on the following decomposition:

$S^{2} = S_{A}^{2} + S_{B}^{2} + S_{R}^{2},$ [3]

where:

$S^{2} = \sum_{j = 1}^{k}{\sum_{i = 1}^{N}\left( {\overline{X}}_{\text{ij}} - {\overline{X}}_{\bullet \bullet} \right)^{2}}\$ –the total sum of squares;

$S_{A}^{2} = N\sum_{j = 1}^{k}\left( {\overline{X}}_{\bullet j} - {\overline{X}}_{\bullet \bullet} \right)^{2}$ – the inter-month (inter-quarter, respectively) sum of squares, which mainly measures the magnitude of the seasonality;

$S_{B}^{2} = k\sum_{i = 1}^{N}\left( {\overline{X}}_{i \bullet} - {\overline{X}}_{\bullet \bullet} \right)^{2}$ – the inter-year sum of squares, which mainly measures the year-to-year movement of seasonality;

$S_{R}^{2} = \sum_{i = 1}^{N}{\sum_{j = 1}^{k}\left( {\overline{X}}_{\text{ij}} - {\overline{X}}_{i \bullet} - {\overline{X}}_{\bullet j} - {\overline{X}}_{\bullet \bullet} \right)^{2}}$ – the residual sum of squares.

The null hypothesis $H_{0}\ $is that $b_{1} = b_{2} = … = b_{N}$ which means that there is no change in seasonality over the years. This hypothesis is verified by the following test statistic:

$F_{M} = \frac{\frac{S_{B}^{2}}{(n - 1)}}{\frac{S_{R}^{2}}{(n - 1)(k - 1)}}$, [4]

which follows an $F$-distribution with $k - 1$ and $n - k$ degrees of freedom.