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

A random walk without drift is defined as y_t = y_{t-1} + e_t, with y_0 = 0 and e_t i.i.d. with variance 0.25. What is the variance of y_t at t = 16 and the variance of the first difference at that date?

The variance of the random walk at t = 16 is 16 times 0.25, which equals 4.0. The first difference is just the white-noise shock, so its variance remains 0.25. Differencing therefore turns the nonstationary series into a stationary one.

  1. AVariance of y_16 is 4.0; variance of the first difference is 0.25Correct
  2. BVariance of y_16 is 0.25; variance of the first difference is 4.0
  3. CVariance of y_16 is 16.0; variance of the first difference is 0.25
  4. DVariance of y_16 is 4.0; variance of the first difference is 4.0

Explanation

Var(y_t) = t × sigma² = 16 × 0.25 = 4.0. The first difference equals e_t, so its variance stays at 0.25 and is constant. Using 16 as the variance (ignoring sigma²) is the key error, and 16 × 0.25 confirms 4.0.

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