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

A risk analyst fits a linear trend to a price series that actually follows a random walk with drift. Which is the most likely consequence?

Residuals stay highly persistent because a deterministic trend cannot remove a unit root. Variance of the leftover stochastic component keeps growing, so inference and R-squared are misleading. Differencing, not detrending, is the correct fix for a random walk with drift.

  1. AResiduals will be white noise, so the fit is valid and forecasts are reliable
  2. BResiduals will remain highly persistent because the stochastic trend is not removed, and standard errors and R-squared will be misleadingCorrect
  3. CThe estimated slope will be exactly zero in large samples
  4. DDetrending by the linear trend makes the series strictly stationary with constant variance

Explanation

A random walk with drift contains a unit root, so removing a deterministic trend leaves a stochastic component whose variance grows with time. The residuals stay persistent, giving spurious-regression problems such as inflated R-squared and unreliable t-statistics. The correct remedy is first differencing, not detrending.

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