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FRM Part I · FRM Exam Part I · Stationary Time Series

A forecaster has a quarterly series with a stable seasonal pattern and models it as Y_t = c + 0.3*Y_{t-1} + 0.5*Y_{t-4} + e_t. Which statement about the sample autocorrelation function (ACF) of the fitted series is most accurate?

The ACF shows non-zero autocorrelation at lags 1 and 4 and around multiples of 4, decaying gradually, because the model is autoregressive rather than moving average. A cutoff after lag 4 would indicate an MA(4). The series is stationary because the coefficients sum to 0.8.

  1. AIt shows spikes at lags 4, 8 and 12 with no other pattern, since the seasonal term alone drives the dependence
  2. BIt is exactly zero at all lags beyond 4 because the model has only lags up to 4
  3. CIt shows non-zero autocorrelations at lags 1 and 4 and around multiples of 4, decaying gradually since the model is autoregressiveCorrect
  4. DIt cannot be computed because seasonal series are nonstationary by definition

Explanation

An autoregressive model with lags 1 and 4 has an ACF that decays gradually rather than cutting off, with notable values at lag 1, lag 4 and their combinations (e.g., 3, 5, 8). A cutoff at lag 4 would characterize an MA(4) process. Seasonal series can be stationary if the lag coefficients sum below 1 in magnitude (0.8 here).

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