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

An analyst models the log of a stock index level as y_t = 0.002 + y_{t-1} + e_t, where e_t is white noise with variance 0.0004. Which description best fits y_t?

The series is a random walk with drift. The coefficient on the lagged value is one, which creates a unit root, so shocks are permanent and the variance rises over time. That makes the process nonstationary rather than mean-reverting or trend-stationary.

  1. AA covariance-stationary AR(1) process with a long-run mean of 0.002
  2. BA random walk with drift, which is nonstationary because it has a unit rootCorrect
  3. CA trend-stationary process whose deviations from a deterministic trend are stationary
  4. DA stationary process whose variance is constant over time

Explanation

The coefficient on y_{t-1} equals 1, so the process has a unit root. With a constant added, it is a random walk with drift. Its variance grows with t, so it is not covariance-stationary. The trend-stationary option is wrong because the shocks have permanent effects, not temporary ones.

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