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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Core concepts of time series models

If X_t follows a random walk X_t = X_{t-1} + e_t with e_t white noise, which transformation produces a stationary series?

Taking the first difference works. For a random walk, X_t minus X_{t-1} equals the white noise term e_t, which is stationary. Subtracting a mean cannot remove the variance growth caused by the unit root.

  1. ATaking the first difference, X_t - X_{t-1}Correct
  2. BSquaring each observation
  3. CTaking a moving average over 100 terms of the levels
  4. DSubtracting the sample mean from every X_t
  5. Multiplying each X_t by 1/t

Explanation

The first difference of a random walk equals e_t, which is white noise and stationary. Subtracting the mean does not remove the growing variance, and the other transformations do not remove the unit root.

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