Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 218 / 6 / b / Solution

Fit a trend ${\hat{T}}_t$, form $D_t=X_t-{\hat{T}}_t$, and estimate the period-25 seasonal effect byThe residual is $R_t=X_t-{\hat{T}}_t-{\hat{S}}_{t\mathrm{mod}25}$. This additive decomposition is sensible when seasonal amplitude does not systematically change with the level or trend; multiplicative seasonality would require a logarithmic transform or ratio decomposition.