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

A risk analyst models monthly retail sales of a chain using an AR(1) model and notices that the residuals show large positive autocorrelation at lags 12, 24 and 36 but little autocorrelation elsewhere. Which interpretation and remedy is most appropriate?

The pattern of residual autocorrelation at lags 12, 24 and 36 in monthly data signals unmodeled seasonality. The remedy is to add seasonal components, such as monthly dummy variables or a seasonal lag. A unit root would show persistent autocorrelation at all lags, not just seasonal ones.

  1. AThe series has a unit root; it should be differenced once and the AR(1) specification retained
  2. BThe residuals are white noise because only a few lags are significant; no change is required
  3. CThe series has seasonality; seasonal terms such as monthly dummy variables or a seasonal lag should be addedCorrect
  4. DThe series is covariance stationary with a time-varying variance; a GARCH model is required

Explanation

Spikes in residual autocorrelation at multiples of 12 for monthly data indicate a yearly seasonal pattern left unmodeled. Adding seasonal dummies or a seasonal lag (e.g., lag 12) captures it. A unit root would show slowly decaying autocorrelation at all lags, not spikes at seasonal lags.

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