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Explain the construction of a point forecast h steps ahead when the series is seasonal.

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

The idea

To forecast a seasonal series h steps ahead from a trend-plus-seasonal-dummy regression, first work out which season period T + h falls in. Count h steps forward from the last observation, wrapping around after s periods. For monthly data, a forecast made after a December observation has h = 1 in January. Then evaluate the fitted equation at time T + h. Point forecast = b0 + b1 × (T + h) + the coefficient of the dummy for that season. The seasonal term is zero if it is the omitted season. Beware the two usual mistakes: using the season of the last observation instead of the season of the target date, and plugging in h rather than T + h for the trend term. Forecasts for the same season in different years differ only by the trend. For the exam, always identify the season and the time index first.

Formulas

Point forecast from a trend and seasonal-dummy model

ŷ(T + h) = b₀ + b₁ × (T + h) + season coefficient

b₀
the fitted intercept
b₁
the fitted trend slope
T
the time index of the last observation
h
the number of steps ahead
season coefficient
the dummy coefficient for the season of T + h, zero for the omitted season

Use the season of the target date, not of the last observation, and put T + h, not h, into the trend term.

A practice question

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

A model with a time trend and seasonal dummies is used to forecast y_(T+h). Which seasonal dummy is switched on?

  1. The dummy for the season of the last observation TWhy not: Uses the wrong season: the forecast date, not the last data point, determines the seasonal term.
  2. All dummies are set to zero because seasonal effects average outWhy not: Zero dummies correspond to the baseline season only, not to an average of all seasons.
  3. The dummy for the season in which period T + h falls - correct
  4. The dummy for season number h, counting from the first season of the yearWhy not: Treats h as a calendar position; the position depends on where T falls in the cycle.

The seasonal effect belongs to the date being forecast. Count h periods forward from T, wrapping around the cycle of length s.

A harder one

Quarterly sales are fitted as y_t = 100 + 2 t + 6 Q2 - 4 Q3 + 3 Q4, with Q1 the omitted season. The last observation is t = 40, which is a fourth quarter. What is the point forecast 3 quarters ahead?

  1. 192Why not: Uses the Q2 coefficient: counts the seasons one step off, from T+2.
  2. 176Why not: Applies the right seasonal term but leaves the trend at t = 40 rather than 43.
  3. 189Why not: Uses the season of the last observation (Q4) rather than the season of the forecast date.
  4. 182 - correct

Period 43 is 3 quarters after a Q4, so it is a Q3 (Q1, Q2, Q3). Forecast = 100 + 2 x 43 - 4 = 182.

The hardest tier

A quarterly model is y_t = b0 + 0.8 t + 5.0 Q2 + other dummies. The point forecasts for T + 2 and T + 10 are both second quarters. By how much do they differ?

  1. 11.4Why not: Adds the Q2 coefficient to the difference, though the same seasonal term is in both forecasts and cancels.
  2. 6.4 - correct
  3. 3.2Why not: Counts only four periods (one cycle) instead of the full eight-period gap.
  4. 8.0Why not: Counts the eight periods but forgets to multiply by the trend slope of 0.8.

Both forecasts include the same seasonal term, so the difference is only the trend change: 8 periods x 0.8 = 6.4.

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