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

An analyst fits Y_t = 0.2 + Y_{t-1} + e_t, where e_t is white noise with variance 1.00 and Y_0 = 50. What are the expected value and variance of Y_t at t = 40, from time 0?

The expected value is 58 and the variance is 40. Drift accumulates each period, giving 50 + 40 x 0.2 = 58, while shock variance also accumulates over 40 periods to 40 x 1 = 40. Drift does not change the variance.

  1. AMean 58; variance 40Correct
  2. BMean 50.2; variance 1
  3. CMean 58; variance 1
  4. DMean 50.2; variance 40

Explanation

With drift, E(Y_40) = 50 + 40 x 0.2 = 58. Variance accumulates shock variances: 40 x 1.00 = 40. The drift affects only the mean, not the variance, so options with variance 1 ignore accumulation, and the 50.2 option ignores accumulation of drift.

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