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Describe how a covariance-stationary ARMA model can represent seasonality.

FRM® Part I · Quantitative Analysis · Stationary Time Series · FRM1.QA.10.n

The idea

Seasonality is a regular pattern at a fixed period s, for example s = 12 for monthly data or s = 4 for quarterly data. A covariance-stationary ARMA model captures it by adding AR or MA terms at the seasonal lag. An additive seasonal AR includes y(t−s) with its own coefficient, and a seasonal MA includes ε(t−s). In the multiplicative form the polynomials multiply. Model = (1 − φL)(1 − ΦLˢ)y(t) = ε(t). Expanded, this gives y(t) = φy(t−1) + Φy(t−s) − φΦy(t−s−1) + ε(t). So an extra lag s + 1 appears, with a coefficient equal to minus the product φΦ. A sample autocorrelation with spikes at lags s, 2s and so on points to seasonality. Stationarity needs the seasonal coefficient to be below one in absolute value.

Formulas

Expanded multiplicative seasonal AR model

y(t) = φy(t−1) + Φy(t−s) − φΦy(t−s−1) + ε(t)

y(t)
the series at time t
φ
the ordinary AR(1) coefficient
Φ
the seasonal AR coefficient
s
the seasonal period, 12 for monthly data
ε(t)
the white noise shock

From (1 − φL)(1 − ΦLˢ)y(t) = ε(t). The trap is missing the lag s + 1 term or its negative sign.

A practice question

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

Monthly data show strong sample autocorrelation spikes at lags 12, 24 and 36. How can a covariance-stationary ARMA model capture this seasonality?

  1. By adding a deterministic trend to remove the lagsWhy not: A trend shifts the mean over time, which violates stationarity and does not address spikes at specific lags.
  2. By including autoregressive or moving average terms at the seasonal lag of 12 - correct
  3. By adding a unit root so the spikes disappearWhy not: Unit roots create non-stationarity; covariance-stationary models capture seasonality with seasonal lags.
  4. By using only a lag-1 AR term with a large coefficientWhy not: A large lag-1 coefficient creates decaying autocorrelation at all lags, not spikes at multiples of 12.

Seasonal ARMA models add AR or MA terms at the seasonal period s (12 for monthly data). Spikes at s, 2s and 3s in the sample ACF point to such terms.

A harder one

A multiplicative seasonal AR model for monthly data is (1 - 0.5 L)(1 - 0.4 L^12) y_t = e_t. When expanded, the coefficient on y_(t-13) in the equation y_t = ... + e_t is:

  1. 0.20Why not: Gets the sign wrong; the product term in (1 - 0.5 L)(1 - 0.4 L^12) is +0.2 L^13 on the left, so it moves to the right as -0.2.
  2. -0.20 - correct
  3. 0.90Why not: Adds the two coefficients, 0.5 + 0.4, instead of multiplying them.
  4. 0.00Why not: Assumes an additive seasonal model with only lags 1 and 12; the multiplicative form generates a cross term at lag 13.

(1 - 0.5L)(1 - 0.4L^12) = 1 - 0.5L - 0.4L^12 + 0.2L^13, so y_t = 0.5 y_(t-1) + 0.4 y_(t-12) - 0.2 y_(t-13) + e_t.

The hardest tier

A seasonal AR process with monthly data is y_t = 0.9 y_(t-12) + e_t. The autocorrelation at lag 24 is closest to:

  1. 0.81 - correct
  2. 0.90Why not: Assumes the autocorrelation does not decay between seasonal lags; it falls by 0.9 at each additional year.
  3. 0.00Why not: Applies the MA cut-off pattern; an AR process has autocorrelations at all multiples of the seasonal lag.
  4. 1.80Why not: Adds the contributions of two seasonal lags instead of multiplying: 0.9 + 0.9.

The process is an AR(1) in steps of 12 months, so rho(12k) = 0.9^k. rho(24) = 0.81, and all other lags are zero. Its roots lie outside the unit circle since 0.9 < 1.

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