Seasonal effects and the deseasonalised series

What a seasonal effect is

• A seasonal effect is the average amount by which a particular quarter or month sits above or below the trend.
• In the additive model:

$$\text{seasonal effect} = \text{data} - \text{trend}$$

• Because the effect is an amount added to or subtracted from the trend, it is in the same units as the data — dollars, arrivals, kilowatt hours.
• A positive effect means that period is typically above trend; a negative effect means below.

How they are calculated

• The procedure the software follows:
  • Subtract the trend from each data value, leaving the seasonal plus irregular components.
  • Average these differences for each quarter across all the years — so all the Q1 differences are averaged to give the Q1 effect.
  • Adjust so the effects sum to zero.
• Averaging across years is what removes the irregular component, leaving the systematic seasonal pattern.

The effects sum to zero

• In an additive model, the seasonal effects must sum to zero across one complete cycle.
• This is your check. If your four quarterly effects are $+8\,200$, $-3\,100$, $-9\,400$ and $+4\,300$, they sum to zero ✓
• The reason is that a seasonal effect describes redistribution within the year, not a change in the annual total. What one quarter gains, others lose.

Interpreting them in context

• Always state the effect with its units and its meaning:
  • "The Quarter 1 seasonal effect is $+8\,200$, meaning that on average Quarter 1 sales are about $8 200 above the trend for that time."
• Then explain why, from the context. This pairing is where the Merit evidence comes from.
• Comparing effects is often revealing:
  • "The gap between the Q1 effect ($+8\,200$) and the Q3 effect ($-9\,400$) is about $17 600, so seasonality moves sales by more than $17 000 between the best and worst quarters — a substantial planning consideration for a business of this size."

The deseasonalised series

• Deseasonalising removes the seasonal pattern so that periods can be compared fairly:

$$\text{deseasonalised value} = \text{data} - \text{seasonal effect}$$

• What it is for: answering the question "is this quarter genuinely better, or is it just that Q1 is always good?"
• Official statistics are almost always published as seasonally adjusted figures for exactly this reason — Stats NZ reports seasonally adjusted employment and GDP so that quarter-to-quarter comparisons are meaningful.
• Plot the deseasonalised series. It should look like the trend plus random scatter, with no remaining zigzag. If a zigzag remains, the seasonal effects are wrong.

What the deseasonalised series is good for

Use	Why
Comparing consecutive quarters	Removes the automatic seasonal advantage
Spotting a turning point early	A trend change shows up before the moving average catches it
Identifying unusual periods	A point far from the trend after deseasonalising is genuinely unusual
Checking the model	Remaining pattern means the seasonal effects are wrong