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FRM Part I · FRM Exam Part I · Stationary Time Series

For the MA(1) process y_t = e_t + 0.5 e_{t-1}, the last shock was e_T = 1.2. Which is the best one-step-ahead forecast of y_{T+1}, and the forecast of y_{T+2}, given zero mean?

The one-step forecast is 0.6 and the two-step forecast is 0. The next value depends on the known current shock through the coefficient 0.5, giving 0.6. Beyond one step, only unknown future shocks remain, so the forecast reverts to the zero mean.

  1. A0.6 for T+1 and 0.6 for T+2
  2. B0.6 for T+1 and 0 for T+2Correct
  3. C1.2 for T+1 and 0 for T+2
  4. D0 for T+1 and 0 for T+2

Explanation

y_{T+1} = e_{T+1} + 0.5 e_T, so the forecast is 0.5(1.2) = 0.6. y_{T+2} depends only on future shocks, whose expectations are zero, so the forecast is the mean, 0.

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