FRM Exam Part I · Nonstationary Time Series

Seasonality and Seasonal Adjustment in Time Series

Updated 11 October 2026 · Fact-checked

Seasonality is a pattern that repeats at a fixed calendar interval, such as every quarter or month. You model it with seasonal dummy variables, one per season minus one if you keep an intercept, or remove it with seasonal differencing, Y(t) − Y(t−s), where s is the season length.

Understand Seasonality and Seasonal Adjustment

Seasonality is a regular pattern that repeats every year, quarter or month. Retail sales rise in December. Natural gas demand peaks in winter. Bank lending can spike at quarter-end. If you ignore it, your forecasts are biased and your residuals show a repeating pattern.

There are two main ways to handle it. The first is to model the seasonal pattern directly with seasonal dummy variables. The second is to remove it by taking a seasonal difference, which compares each observation with the one from the same season a year earlier.

With dummies, you add one 0/1 variable for each season. For quarterly data you have four seasons. If your regression has an intercept, include only three dummies. The omitted season is the base season, and the intercept is its mean level. Each dummy coefficient is the average gap between that season and the base season. Including all four dummies plus an intercept creates perfect multicollinearity. This is the dummy variable trap.

An alternative is to drop the intercept and use all four dummies. Then each coefficient is simply the mean level of that season. Both forms fit identically and give the same forecasts.

Seasonal differencing subtracts the value s periods earlier, where s is 12 for monthly data and 4 for quarterly data. This removes a stable seasonal pattern and also any seasonal-length cycle in the level. You lose the first s observations. If the data also has a stochastic trend, you may need a regular first difference as well. Seasonality can also be captured inside ARMA models by adding seasonal lags, such as a lag-4 term for quarterly data.

Key formulas to remember

Seasonal dummy regression (quarterly, with intercept): Y(t) = β0 + γ2·D2(t) + γ3·D3(t) + γ4·D4(t) + ε(t); Q1 is the base season. D(i) = 1 in quarter i, else 0. β0 is the Q1 mean; γ(i) is the Q(i) minus Q1 difference.
Number of dummies: Dummies needed = s − 1 (with intercept); s is the number of seasons per year. Use s dummies only if you drop the intercept.
Seasonal means without intercept: Y(t) = α1·D1(t) + α2·D2(t) + α3·D3(t) + α4·D4(t) + ε(t); Each α(i) is the mean of season i. Here α(i) = β0 + γ(i) from the intercept form.
Seasonal difference: ΔsY(t) = Y(t) − Y(t−s); s = 4 for quarterly and 12 for monthly data. You lose s observations.
Seasonal AR term (idea): Y(t) = φ·Y(t−s) + ε(t); A significant autocorrelation at lag s, 2s and so on points to seasonality.

How to solve Seasonality and Seasonal Adjustment questions

Use this sequence for any question on seasonality, whether it asks you to interpret dummies, forecast, or choose a remedy.

1Identify the data frequency and set s: 4 for quarterly, 12 for monthly.
2Check whether the regression has an intercept. This decides how many dummies you need and what each coefficient means.
3Name the base season, which is the one with no dummy. If there is no intercept, there is no base season.
4For a fitted value, add the intercept and the dummy coefficient of the target season. For the base season, use the intercept alone.
5For a seasonal difference, subtract the observation s periods back from the current one. Do not subtract the previous period.
6If a question mentions a trend too, remember dummies do not remove a trend. Add a trend term or difference the data.
7Check the answer: the interpretation of a dummy coefficient is always relative to the base season.

Quickest way: Read the intercept, then add the dummy

When to use it: Use this for multiple-choice questions that give a fitted dummy regression and ask for a seasonal forecast or the meaning of a coefficient.

Look for the intercept. If present, count dummies and confirm there are s − 1.
Forecast for season i = intercept + coefficient of dummy i. Base season = intercept.
Gap between two non-base seasons = difference of their coefficients.
For seasonal differences, subtract the value from s periods earlier. A quick check is that the first s data points vanish.
If all s dummies and an intercept appear together, spot the trap: perfect multicollinearity.

Common mistakes in Seasonality and Seasonal Adjustment

Including s dummies and an intercept
It feels natural to give every season its own variable.
Fix: With an intercept use s − 1 dummies. The dummies would sum to the intercept column, so OLS cannot be estimated. Otherwise drop the intercept.
Reading a dummy coefficient as the season's mean
Students forget the base season.
Fix: With an intercept, the coefficient is the gap from the base season. The season mean is intercept plus coefficient.
Using a first difference to remove seasonality
Differencing is associated with trends, so students apply Y(t) − Y(t−1).
Fix: Seasonal patterns need a lag-s difference, Y(t) − Y(t−s). A first difference only addresses a stochastic trend.
Assuming dummies handle a trend
Seasonal dummies are mistaken for a general adjustment tool.
Fix: Dummies capture a fixed seasonal pattern around a level. Add a time trend or difference the series if there is a trend.
Forgetting the observations lost in seasonal differencing
Students remember only the one lost with a first difference.
Fix: Seasonal differencing loses s observations. With 48 monthly points and s = 12, you keep 36.

Worked examples

Example 1

A quarterly sales series (in USD millions) is regressed on seasonal dummies with an intercept: Y = 50 + 8·D2 + 15·D3 − 5·D4, where Q1 is the base season. What is the fitted value for Q3, and how much higher is Q3 than Q4?

Show the solution

Q1 is the base season, since it has no dummy. The intercept 50 is the Q1 mean.
Q3 fitted value = 50 + 15 = 65.
Q4 fitted value = 50 − 5 = 45.
Difference Q3 − Q4 = 65 − 45 = 20, which also equals 15 − (−5).

Answer: The Q3 fitted value is USD 65 million, and Q3 is USD 20 million higher than Q4.

Example 2

A monthly series has 60 observations. You take a seasonal difference to remove annual seasonality. The values for January of year 1 and January of year 2 are 120 and 132. What is the seasonal difference at January of year 2, and how many observations remain?

Show the solution

For monthly data, s = 12.
Seasonal difference = Y(t) − Y(t−12) = 132 − 120 = 12.
Observations lost = s = 12.
Observations remaining = 60 − 12 = 48.

Answer: The seasonal difference is 12, and 48 observations remain.

Exam tips

Check the intercept first. Many questions test whether you know the dummy trap.
Expect to compute a fitted value for one season. Add the intercept and the relevant coefficient, and nothing else.
Watch the wording: lag-s difference means seasonal; lag-1 difference does not remove seasonality.
If a question shows autocorrelation spikes at lags 4, 8, 12 on quarterly data, the intended diagnosis is seasonality.
Do the arithmetic slowly. Wrong options often use the coefficient alone instead of intercept plus coefficient.

Practice questions from Nonstationary Time Series

Seasonality and Seasonal Adjustment: frequently asked questions

Why can't I include a dummy for every season with an intercept?

The dummies add up to a column of ones, which is the same as the intercept column. This is perfect multicollinearity, so OLS has no unique solution. Drop one dummy or drop the intercept.

Does the choice of base season change the forecasts?

No. Changing the base season changes the intercept and the coefficients, but the fitted value for each season stays the same. Only the interpretation of the coefficients changes.

When should I use seasonal differencing instead of dummies?

Use dummies when the seasonal pattern is stable and you want to estimate its size. Use seasonal differencing when the seasonal pattern itself may drift over time. Differencing discards the seasonal level, so you cannot read seasonal effects from it.

How many observations do I lose with seasonal differencing?

You lose s observations, where s is the number of periods in a seasonal cycle. That is 12 for monthly data and 4 for quarterly data.