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

An AR(1) process Y_t = 0.9 Y_{t-1} + e_t and a random walk Y_t = Y_{t-1} + e_t both have shock variance 1. Which statement correctly contrasts them for long-horizon forecasting?

The stationary AR(1) forecast error variance converges to its unconditional variance, 1 divided by (1 minus 0.81), about 5.26, whereas the random walk's error variance grows linearly without bound. Mean reversion bounds uncertainty only when the coefficient is below one in absolute value.

  1. ABoth have forecast error variance that grows without bound
  2. BThe AR(1) has forecast error variance converging to 1/(1-0.81), about 5.26, while the random walk's grows without boundCorrect
  3. CThe AR(1) has forecast error variance converging to 1/(1-0.9), or 10
  4. DThe random walk has forecast error variance converging to 1

Explanation

For the stationary AR(1), the long-horizon forecast error variance approaches the unconditional variance sigma^2/(1-phi^2) = 1/0.19 ≈ 5.26. The random walk's grows linearly with horizon. Option 3 forgets to square phi.

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