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

A stationary MA(1) process is Y_t = e_t + 0.5 e_{t-1}, with e_t white noise of variance 1. At time T the residual is e_T = 2 and the mean is zero. What are the one-step-ahead forecast of Y_{T+1} and the two-step-ahead forecast of Y_{T+2}, respectively?

The forecasts are 1.0 and 0.0. The one-step forecast equals theta times the latest residual, 0.5 times 2, because future shocks have zero expectation. Beyond one step an MA(1) forecast reverts to the unconditional mean, here zero.

  1. A1.0 and 0.0Correct
  2. B0.5 and 0.0
  3. C1.0 and 1.0
  4. D2.0 and 0.0

Explanation

One-step forecast = theta x e_T = 0.5 x 2 = 1.0, since E[e_{T+1}] = 0. Two steps ahead both shocks are in the future, so the forecast equals the mean, 0. Option 0.5 and 0.0 uses theta without e_T. Option 1.0 and 1.0 wrongly carries the shock forward. Option 2.0 and 0.0 ignores theta.

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