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

A trend-stationary series follows y_t = 10 + 0.5t + e_t, where e_t is white noise with variance 4. An analyst removes the trend by regressing y_t on t and keeps the residuals. Which statement best describes the residuals and the effect of a shock?

The residuals are stationary white noise, and shocks have only temporary effects. In a trend-stationary process the series reverts to its deterministic trend after a shock, so detrending removes nonstationarity. Permanent shock effects are characteristic of unit root processes, not deterministic trends.

  1. AThe residuals are stationary and a shock to e_t has only a temporary effect on the deviation from trendCorrect
  2. BThe residuals have a unit root and a shock has a permanent effect on the level
  3. CThe residuals are stationary but a shock permanently shifts the trend line
  4. DThe residuals have variance that grows with t, so they are nonstationary

Explanation

Detrending a trend-stationary series leaves e_t, which is white noise and therefore covariance stationary. Shocks do not accumulate, so the series returns to its deterministic trend. A unit root would be the case only for a stochastic trend such as a random walk with drift.

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