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

An MA(1) process is Y_t = ε_t + 0.5·ε_(t-1), with ε_t white noise of variance σ². What is the first-order autocorrelation of Y_t, and what is the autocorrelation at lag 2?

For an MA(1) with θ = 0.5, the lag-1 autocorrelation is θ/(1+θ²) = 0.5/1.25 = 0.40. MA(1) autocorrelations cut off after lag 1, so the lag-2 autocorrelation is zero.

  1. A0.40 at lag 1; 0 at lag 2Correct
  2. B0.50 at lag 1; 0 at lag 2
  3. C0.40 at lag 1; 0.16 at lag 2
  4. D0.25 at lag 1; 0 at lag 2

Explanation

For MA(1), ρ1 = θ/(1+θ²) = 0.5/1.25 = 0.40. The autocorrelation cuts off after lag 1, so ρ2 = 0. Using θ directly (0.50) ignores the variance denominator; 0.16 wrongly applies geometric decay as in an AR process.

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