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

Monthly log sales of a firm show a seasonal pattern in which the seasonal fluctuation is the same size each year, while a stochastic trend also exists. Which transformation is most appropriate to remove both the seasonal pattern and the unit root in the log series?

Apply a seasonal difference at lag 12 to the log series, and if a unit root persists, also take a first difference. The seasonal difference removes the stable annual pattern, while the first difference removes the stochastic trend; lag-1 differencing alone leaves seasonality.

  1. ATake the seasonal difference at lag 12 and, if a unit root remains, first difference the resultCorrect
  2. BRegress on a constant and take residuals only
  3. CTake only a first difference at lag 1
  4. DTake a moving average of length 3

Explanation

A lag-12 seasonal difference removes a stable seasonal pattern, and a further first difference removes the remaining stochastic trend. A lag-1 difference alone leaves seasonality, demeaning removes neither, and a short moving average only smooths noise.

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