Seasonality

Study guides: Exam MAS-II, Exam CP-321, Exam GH-201C, Exam GH-201U

Seasonality is a pattern in a Time Series that repeats at a fixed, known period s — 4 for quarterly data, 12 for monthly, 7 for daily. It is distinct from a cycle, whose length is not fixed.

Y_t = m_t + s_t + z_t \quad \text{with } s_t = s_{t+s} \text{ and } \sum_{j=1}^{s}s_j = 0

\text{seasonal ARIMA: } (p,d,q)\times(P,D,Q)_s

• Additive seasonality adds a fixed amount each period; multiplicative scales the level, so it grows with the trend. A log transform turns the second into the first
• Signature on the ACF: significant spikes at lags s, 2s, 3s, \dots that do not die away
• Handling it: seasonal Differencing \nabla_s Y_t = Y_t - Y_{t-s} removes a stochastic seasonal pattern; seasonal dummies or a sine/cosine pair in a regression remove a deterministic one
• A seasonal ARIMA multiplies the seasonal and non-seasonal polynomials, so it models within-period and across-period dependence at once
• Actuarial cases: winter freeze losses, quarterly reporting lags, the seasonal pattern in the reporting of accident-year claims

Additive or Multiplicative? Quarterly claim counts show a Q1 excess over the yearly average of +180 in year 1 (average 900), +310 in year 5 (average 1,550) and +420 in year 9 (average 2,100). Which form of seasonality, and what transform helps?

Answer As a share of the level: 180/900 = 20\%, 310/1550 = 20\%, 420/2100 = 20\% — constant in proportional terms, growing in absolute terms. Multiplicative. Fit on \log Y_t, where the Q1 effect becomes a constant \log(1.20) = 0.182 addition and an additive seasonal model applies. Modelling the raw series additively would under-fit the early years and over-fit the late ones.