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FRM Part I · FRM Exam Part I · Stationary Time Series

A stationary AR(1) process is given by Y_t = 2.0 + 0.6 Y_{t-1} + e_t, where e_t is white noise. What is the long-run (unconditional) mean of Y_t?

The long-run mean is 5.0. For a stationary AR(1), the mean equals the intercept divided by one minus the autoregressive coefficient, so 2.0 divided by 0.4 gives 5.0. At that level the process satisfies its own equation on average.

  1. A3.2
  2. B5.0Correct
  3. C2.0
  4. D1.2

Explanation

For a stationary AR(1), the mean is mu = c/(1 - phi) = 2.0/(1 - 0.6) = 2.0/0.4 = 5.0. Check: 2.0 + 0.6*5.0 = 5.0. The value 3.2 comes from wrongly using 2.0/(1 - 0.6) with 0.6 added, 2.0*1.6, and 1.2 is simply c*phi.

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