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

An analyst models the log of a stock index as Y_t = Y_{t-1} + e_t, where e_t is white noise with variance 0.0004. Which statement correctly describes this process?

The process is a random walk without drift because the current value equals the prior value plus white noise. Its first difference is white noise, so differencing makes it stationary. The level series itself is not covariance stationary because its variance increases over time.

  1. AIt is covariance stationary with a constant mean equal to Y_0
  2. BIt is a random walk without drift, and its first difference is white noiseCorrect
  3. CIt is a trend-stationary process that reverts to a deterministic linear trend
  4. DIt is an AR(1) process with a coefficient of 0.5

Explanation

Y_t = Y_{t-1} + e_t is a random walk with no drift (AR(1) with coefficient 1). Differencing gives Y_t - Y_{t-1} = e_t, which is white noise. It is not covariance stationary because its variance grows with t, and it has no deterministic trend to revert to.

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