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

A stationary ARMA(1,1) model is Y_t = 3 + 0.6 Y_{t-1} + e_t + 0.5 e_{t-1}. What is the long-run (unconditional) mean of Y, and what does the multi-step forecast converge to?

The unconditional mean is the intercept divided by one minus phi, which is 3 divided by 0.4, or 7.5. Forecasts mean-revert toward 7.5 as the horizon increases because the AR coefficient is below one in absolute value and the MA effect dies after one step.

  1. A7.5, and forecasts converge to 7.5 as the horizon growsCorrect
  2. B3.0, and forecasts converge to 3.0 as the horizon grows
  3. C5.0, and forecasts converge to 7.5 as the horizon grows
  4. D7.5, and forecasts diverge because of the MA term

Explanation

The mean mu = 3/(1-0.6)=7.5. The MA term affects only the first forecast step; beyond that future shocks have zero expectation and the forecast decays geometrically at rate 0.6 toward the mean. The model is stationary since |phi|<1.

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