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

Two models are fitted to the same series. Model A: y_t = a + bt + e_t with e_t white noise. Model B: y_t = b + y_{t-1} + e_t with e_t white noise. Which statement is correct?

Detrending by regression on time makes model A stationary but not model B. A is a deterministic trend plus noise, while B is a random walk with drift whose shocks accumulate permanently, leaving a stochastic trend after detrending.

  1. ABoth models are trend-stationary, since each has a drift term b
  2. BShocks have permanent effects on the level in A and transitory effects in B
  3. CDetrending by regression on time yields a stationary series for A but not BCorrect
  4. DBoth models have forecast error variance that is constant as the horizon grows

Explanation

Model A is trend-stationary: removing a + bt leaves white noise. Model B is a random walk with drift, a unit-root process, so regression detrending leaves a stochastic trend and nonstationary residuals. Shock effects are transitory in A and permanent in B, and B's forecast error variance grows with the horizon.

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