FRM Part I · FRM Exam Part I · Stationary Time Series
A monthly series with annual seasonality is modeled as y_t = 0.5 y_{t-12} + e_t, where e_t is white noise with variance 1.5 and the process is covariance stationary. What is the unconditional variance of y_t?
The variance is 2.0. For a stationary seasonal AR with coefficient 0.5, variance equals noise variance divided by one minus phi squared, 1.5 divided by 0.75. Ignoring the squaring or the seasonal feedback gives wrong values such as 1.5 or 3.0.
- A2.0
- B3.0Correct
- C6.0
- D1.5
Explanation
For a seasonal AR with coefficient phi at lag 12, variance = sigma^2/(1-phi^2) = 1.5/(1-0.25) = 1.5/0.75 = 2.0. Check: 0.25*2.0 + 1.5 = 2.0. So the correct value is 2.0, not 3.0.
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