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

A quarterly series follows the seasonal AR process y_t = 0.6 y_{t-4} + e_t, which is stationary. Which statement about its theoretical autocorrelation function is correct?

The ACF is nonzero only at multiples of four, decaying geometrically: 0.6 at lag 4, 0.36 at lag 8, about 0.216 at lag 12, with zero at other lags. The seasonal lag is the only channel of dependence.

  1. Arho(4) = 0.6, rho(8) = 0.36, and autocorrelations at lags not multiples of 4 are zeroCorrect
  2. Brho(1) = 0.6 and rho(4) = 0.36
  3. Crho(4) = 0.6, rho(8) = 0.6, rho(12) = 0.6
  4. Drho(4) = 0.6, rho(8) = 0.36, and rho(1), rho(2), rho(3) are 0.6, 0.36, 0.22

Explanation

The process links y_t only to y_{t-4}, so correlation propagates in steps of four: rho(4k) = 0.6^k. Lags 4, 8, 12 give 0.6, 0.36, 0.216, and other lags are zero. The distractor with constant 0.6 ignores geometric decay.

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