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

An analyst fits y_t = 0.4 + 1.0 y_{t-1} + e_t with sigma² = 0.09 to a price series that starts at y_0 = 50. Using the model, what are the expected value and the standard deviation of the forecast error for y_{t+9}, made at time t, with y_t = 50?

The expected value nine steps ahead is 50 plus nine times the drift of 0.4, giving 53.6. Forecast error variance accumulates as nine times 0.09, which is 0.81, so the standard deviation is 0.90. Uncertainty grows with the square root of the horizon.

  1. AExpected value 53.6; standard deviation 0.90Correct
  2. BExpected value 53.6; standard deviation 0.27
  3. CExpected value 50.4; standard deviation 0.90
  4. DExpected value 53.6; standard deviation 2.70

Explanation

Expected value = 50 + 9×0.4 = 53.6. The forecast error is the sum of nine shocks, with variance 9×0.09 = 0.81, so the standard deviation is 0.9. Option 0.27 is wrong because it multiplies sigma (0.3) by 9 incorrectly as sigma²×3; 2.70 multiplies sigma by 9 instead of sqrt(9).

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