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

For the MA(1) process X_t = e_t + 0.5 e_{t-1}, with e_t white noise of variance 4, what is the lag-1 autocorrelation?

The lag-1 autocorrelation is 0.40. It equals θ/(1+θ²) = 0.5/1.25. The noise variance cancels between the autocovariance and variance, so it does not affect the autocorrelation.

  1. A0.25
  2. B0.40Correct
  3. C0.50
  4. D0.80
  5. 0.20

Explanation

ρ1 = θ/(1+θ²) = 0.5/(1+0.25) = 0.5/1.25 = 0.40. Check: γ0 = 4×1.25 = 5, γ1 = 4×0.5 = 2, and 2/5 = 0.4. Using 0.5 directly forgets the denominator, and the variance cancels out.

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