Seasonality for Time Series

A time-series y_i with no trend has seasonality of period c if E[y_i] = E[y_i+c].

If we have a stationary time series y_i and a deterministic time series s_i such that s_i = s_i+c for all i (and so s_i = s_i+kc for all integers k), then z_i = y_i + s_i would be a seasonal time series with period c. As shown in Regression with Seasonality, the seasonality of such time series can be modeled by using c–1 dummy variables.

A second way to model seasonality is to assume that s_i = μ_m(i) + ε_i where ε_i is a purely random time series and μ₀, …, μ_c-1 are constants where m(i) = MOD(i,c).

A third approach is to model seasonality as a sort of random walk, i.e. s_i = μ_m(i) + s_i-c + ε_i . If μ₀ = … = μ_c-1 = 0 then there is no drift; otherwise μ₀, …, μ_c-1  capture the seasonal drift.

Of course, seasonality can be modeled in many other ways.

Recall that for the lag function L^c(y_i) = y_i-c, and so (1–L^c)y_i = y_i – y_i-c. This is the principal way of expressing seasonality for SARIMA models.

Note too that if s_i is deterministic, then (1–L^c)s_i = s_i – s_i-c= 0.