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):

, [1]

, [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:

[3]

where:

–the total sum of squares;

– the inter-month (inter-quarter, respectively) sum of squares, which mainly measures the magnitude of the seasonality;

– the inter-year sum of squares, which mainly measures the year-to-year movement of seasonality;

– 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:

, [4]

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