Charta is on Google Play: practise on the go. Get the app
Charta

Show how modelling seasonality can be done through regression analysis.

FRM® Part I · Quantitative Analysis · Non-Stationary Time Series · FRM1.QA.11.b

The idea

Seasonality is a pattern that repeats every s periods, such as four quarters or twelve months. It can be modelled by adding seasonal dummy variables to a regression. A dummy equals 1 in its own season and 0 otherwise. With an intercept in the model you include only s − 1 dummies. Including all s makes them add up to the intercept column and produces perfect multicollinearity, the dummy variable trap. The omitted season is the baseline. The intercept is its average, after allowing for any trend, and each dummy coefficient is that season's average difference from the baseline. Alternatively drop the intercept and use all s dummies, so each coefficient is the season's own mean. A joint F-test on the dummies tests for seasonality. The usual error is reading a coefficient as a level rather than a difference from the baseline.

A practice question

From the question bank, with the reason each wrong option is wrong.

Quarterly data are modelled with an intercept, a time trend and seasonal dummy variables. How many seasonal dummies should be included?

  1. Four, one for each quarterWhy not: Including all four with an intercept causes perfect multicollinearity, the dummy variable trap.
  2. One, indicating the peak quarter onlyWhy not: Cannot represent the separate effects of the other three quarters.
  3. Two, one for each half-yearWhy not: Confuses half-year seasonality with quarterly seasonality; four seasons need s - 1 = 3 dummies.
  4. Three, one for each quarter except a baseline quarter - correct

With s seasons and an intercept, s - 1 dummies are used. The omitted season is the baseline captured by the intercept.

A harder one

Quarterly revenue is modelled as y_t = 10 + 2 Q2_t + 5 Q3_t - 1 Q4_t + e_t, where Q2, Q3 and Q4 are dummies and Q1 is the omitted baseline. What is the expected revenue in a third quarter?

  1. 5Why not: Reads the Q3 coefficient as the level of revenue; it is only the difference from the baseline quarter.
  2. 15 - correct
  3. 17Why not: Adds the Q2 coefficient as well as the Q3 coefficient; only the dummy that is switched on contributes.
  4. 16Why not: Adds every dummy coefficient (2 + 5 - 1) to the intercept instead of only the Q3 coefficient.

In Q3 only the Q3 dummy equals 1, so expected revenue is 10 + 5 = 15. Each coefficient is a difference from the Q1 baseline of 10.

The hardest tier

Quarterly data are modelled as y_t = 20 + 3 Q2 + 8 Q3 + 5 Q4 + e_t with Q1 as the baseline. The model is re-estimated using Q4 as the baseline (intercept plus dummies for Q1, Q2 and Q3). What are the new intercept and the new coefficient on the Q1 dummy?

  1. Intercept 20 and Q1 coefficient 5Why not: Keeps the old intercept and flips only the sign; the intercept is now the Q4 mean.
  2. Intercept 25 and Q1 coefficient 5Why not: Gets the new baseline right but reverses the sign of the Q1 difference, which is now Q1 minus Q4.
  3. Intercept 20 and Q1 coefficient -5Why not: Leaves the intercept as the Q1 mean even though the baseline season has changed to Q4.
  4. Intercept 25 and Q1 coefficient -5 - correct

The fitted seasonal means are unchanged: Q1 = 20, Q2 = 23, Q3 = 28, Q4 = 25. With Q4 as baseline the intercept is 25 and Q1 is 20 - 25 = -5.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →

More in Non-Stationary Time Series