Sliding spans
The sliding spans technique involves the comparison of the correlated seasonal adjustments of a given period obtained by applying the adjustment procedure to a sequence of two, three or four overlapping spans of data, all of which contain this period (month or quarter)1.
Each period that belongs to more than one span is examined to see if its seasonal adjustments vary more than a specified amount across the spans2. For the multiplicative decomposition a seasonal factor is regarded to be unreliable if the following condition is fulfilled:
\(S_{t}^{\max} = \frac{\max_{k \in N_{t}}S_{t}\left( k \right) - \min_{k \in N_{t}}S_{t}(k)}{\min_{k \in N_{t}}S_{t}(k)} > 0.03\), [1]
where:
$S_{t}(k)$ – the seasonal factor estimated from span $k$ for month (quarter) $t$;
$N_{t}$ – {$\text{k}$: month (quarter) $\text{t}$ is in the $k$-th span}.
For the additive decomposition JDemetra+ uses the rule in equation [2] for checking for the reliability of the seasonal factor.
\(S_{t}^{\max} = \frac{\max_{k \in N_{t}}S_{t}\left( k \right) - \min_{k \in N_{t}}S_{t}(k)}{\sqrt{\frac{\sum_{i}^{n}y_{i}^{2}}{n}}} > 0.03\), [2]
where:
$n$ – number of observations of the orginal time series $y_{i}$.
The month-to-month percentage change in the seasonally adjusted value from span $k$ for month $t$ is calculated as:
\(\text{MM}_{m}\left(k\right) = \frac{A_{m}\left(k\right) - A_{m - 1}\left(k\right)}{A_{m - 1}\left(k\right)}\), [3]
where:
$A_{m}\left( k \right)$ – the seasonally (and trading day) adjusted value from span $k$ for month $t$;
$\text{MM}_{m}\left( k \right)$ is considered unreliable if the statistics below is higher than 0.03.
\(\text{MM}_{m}^{\max} = \max_{k \in {N1}_{m}}\text{MM}_{m}\left( k \right) - \min_{k \in {N1}_{m}}\text{MM}_{m}\left( k \right) > 0.03\), [4]
where:
${N1}_{t}$ – {$\text{k}$: month $\text{t}$ and $t$-1 are in the $k$-th span}.
The respective formula for the quarter-to-quarter percentage change in the seasonally adjusted value from span $k$ for quarter $t$ is calculated as:
\(\text{QQ}_{q}\left( k \right) = \frac{A_{q}\left( k \right) - A_{q - 1}\left( k \right)}{A_{q - 1}\left( k \right)}\), [5]
where:
$A_{q}\left( k \right)$ – the seasonally (and trading day) adjusted value from span $k$ for quarter $q$.
$\text{QQ}_{q}\left( k \right)$ is considered unreliable if the statistics below is higher than 0.03.
\(\text{QQ}_{q}^{\max} = \max_{k \in {N1}_{q}}\text{QQ}_{q}\left( k \right) - \min_{k \in {N1}_{t}}\text{QQ}_{q}\left( k \right) > 0.03\), [6]
where:
${N1}_{q}$ – {$\text{k}$: quarter $\text{t}$ and $t$-1 are in the $k$-th span}.
The respective diagnostic can be also performed for the trading days/working days component.