Time Series • Introduction to Autoregressive Models and Forecasting
Time Series / Seasonal Data

Seasonal Data

Notes 4 Introduction to Autoregressive Models and Forecasting
Quick concept

Seasonal data show a repeating pattern at a known calendar interval. That repeating pattern is seasonality. The seasonal period m is the length of one full seasonal cycle, measured in the same time units as the series.

Notes

Seasonal Data

Definition

Seasonal data show a repeating pattern at a known calendar interval. That repeating pattern is seasonality. The seasonal period m is the length of one full seasonal cycle, measured in the same time units as the series.

Example

Ice-cream sales that peak every summer have yearly seasonality. For monthly data, m = 12. For quarterly data with a yearly cycle, m = 4.

Seasonal Period m

Data frequency Typical yearly seasonality m
Monthly Same month each year 12
Quarterly Same quarter each year 4
Daily (weekly pattern) Same weekday each week 7

Not every daily series is seasonal at 7, and not every monthly series is seasonal at 12. m is chosen from knowledge of the calendar and from the plot/ACF, not copied blindly.

How Seasonality Appears on Plots

A time plot of seasonal data often shows a wave that returns at the same time of year (or week). An ACF plot may show a spike at lag m (for example lag 12 for months). A rising trend can sit underneath the seasonal wave; both can appear together.

Trend vs Seasonality vs Cyclical Movement

Feature Meaning Interval
Trend Long-term general rise, fall or stable level Not a fixed short calendar cycle
Seasonal pattern Repeating shape at a known seasonal period m Known (month, quarter, weekday, …)
Cyclical movement Longer swings not tied to a fixed short calendar period Irregular length (for example business-cycle style)

Why Ordinary ARIMA May Not Be Enough

Non-seasonal ARIMA(p, d, q) can handle trend-like wandering through d and short-lag AR/MA terms. Strong seasonality at lag m is a repeating calendar structure. If that structure is clear, a non-seasonal ARIMA may leave seasonal leftover in the residuals. Seasonal ARIMA (next topic) adds seasonal orders. This topic only states the need; it does not introduce SARIMA formulas.

Exam Note

Seasonality is a repeating pattern at a known period m. It is not the same as trend, and it is not the same as a long irregular cycle.

Exam-Oriented Key Points

  1. Seasonal data repeat at a known period m.
  2. Monthly yearly seasonality uses m = 12; quarterly yearly seasonality uses m = 4.
  3. Trend, seasonality and cyclical movement must be distinguished.
  4. Ordinary ARIMA may miss a strong seasonal lag; SARIMA is the seasonal extension.