15.1 Seasonal differencing

In practice many time series contain a seasonal periodic component which repeats every \(s\) observations: with monthly observations \(s = 12\), with quarterly observations \(s = 4\). Then \(X_t\) depends on terms such as \(X_{t-12}\), and perhaps \(X_{t-24}\), as well as on \(X_{t-1}, X_{t-2}, \dots \). A seasonal pattern can be removed by the seasonal difference \[ \nabla _sX_t = X_t - X_{t-s} = (1 - B^s)X_t , \] which compares each value with the value one season earlier (Proposition 2.4.4(ii)). A trend is removed by the ordinary difference \(\nabla = 1 - B\), and both are often needed. If \(d = D = 1\) and \(s = 12\), \begin {align*} W_t &= \nabla \nabla _{12}X_t \\ &= \nabla _{12}X_t - \nabla _{12}X_{t-1} \\ &= (X_t - X_{t-12}) - (X_{t-1} - X_{t-13}) , \end {align*}

the change from last month in the year-on-year change.

Questions on this section

Stuck on something here? Ask below and it stays attached to this topic.