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FRM Part I · FRM Exam Part I · Nonstationary Time Series

A risk team analyzes monthly log values of a series that has a stochastic seasonal pattern. Seasonal differences, Y_t - Y_{t-12}, look stationary, whereas the series with only dummy variables retains a unit root at seasonal frequencies. Which conclusion is best supported?

Seasonal differencing is appropriate. When dummies leave a seasonal unit root, the seasonal pattern is stochastic and changes over time, so fixed monthly means cannot capture it. Differencing at lag 12 removes this seasonal nonstationarity and yields a stationary series.

  1. ADeterministic seasonal dummies are inadequate; seasonal differencing is appropriate because the seasonal pattern is stochastic and evolves over timeCorrect
  2. BSeasonal dummies are sufficient because seasonal patterns are always deterministic
  3. CFirst differencing alone is required, and seasonal differencing would add a unit root
  4. DThe series is stationary around fixed monthly means

Explanation

Dummies capture fixed seasonal means. If residuals retain a seasonal unit root, the seasonal pattern drifts, and seasonal differencing removes it. Seasonal differencing does not create a unit root here since the differenced series looks stationary.

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