Seasonal effects and forecasting

The seasonal effect

• The seasonal effect of a season is how much it is typically above or below the trend.
• For each time point, compute $\text{data} - \text{trend}$ (the trend from the centred moving average); then average these differences for each season across the years.
• In the additive model these are amounts, e.g. "Q1 averages $+5$ above trend; Q3 averages $-3$ below".

Seasonally adjusted data

• The seasonally adjusted value removes the season:

$\text{seasonally adjusted} = \text{data} - \text{seasonal effect}$

• This strips out the regular seasonal pattern so you can compare the underlying level — is this winter genuinely worse, or just the usual winter dip?

Forecasting

• To forecast a future value:
  1. Project the trend forward to that time (extend the trend / moving-average line).
  2. Add the seasonal effect for that season.

$\text{forecast} = \text{projected trend} + \text{seasonal effect}$

The limits of a forecast

• Projecting the trend assumes it continues — this is an extrapolation, so the further ahead, the less reliable the forecast.
• Irregular events (a shock, a one-off promotion) cannot be predicted from the model.