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

Y_t = 0.6 Y_{t-1} + ε_t, where ε_t is white noise with variance 1.28. What is the unconditional variance of Y_t, and what is the autocorrelation of Y_t at lag 2?

The variance is 2.00 and the lag-2 autocorrelation is 0.36. For a stationary AR(1) driven by white noise, variance equals the shock variance divided by one minus phi squared, so 1.28 divided by 0.64. Autocorrelation decays as phi to the power of the lag, 0.6 squared.

  1. AVariance 2.00; lag-2 autocorrelation 0.36Correct
  2. BVariance 2.00; lag-2 autocorrelation 0.60
  3. CVariance 1.28; lag-2 autocorrelation 0.36
  4. DVariance 3.56; lag-2 autocorrelation 0.36

Explanation

For a stationary AR(1), variance = σ²/(1 − φ²) = 1.28/(1 − 0.36) = 1.28/0.64 = 2.00. Autocorrelation at lag k is φ^k, so lag 2 gives 0.36. Option B uses lag-1 value; C ignores persistence.

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