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

A analyst models a monthly log price series as Y_t = Y_{t-1} + e_t, where e_t is white noise with variance 0.0004. Y_0 = 4.60 is known. What is the variance of the forecast error for Y_{t+9}, given information at time t?

The forecast error variance is 0.0036. In a random walk, the nine-step error is the sum of nine independent shocks, so variance grows linearly with the horizon: 9 times 0.0004. It does not stay at the one-step value or grow with the square of the horizon.

  1. A0.0004
  2. B0.0036Correct
  3. C0.0060
  4. D0.0324

Explanation

For a pure random walk, the h-step forecast error is the sum of h independent shocks, so variance equals h times sigma squared. With h = 9 this is 9 x 0.0004 = 0.0036. 0.0324 wrongly squares the horizon (81 x 0.0004); 0.0004 ignores accumulation.

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