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

A stationary AR(1) process X_t = 0.8 X_{t-1} + e_t has white noise variance 3.6. What is the variance of X_t?

The variance is 10. For a stationary AR(1), the variance equals the noise variance divided by one minus the squared coefficient, so 3.6 divided by 0.36 gives 10. Dividing by one minus the coefficient instead of its square is a common error.

  1. A3.6
  2. B4.5
  3. C10.0
  4. D18.0Correct
  5. 7.2

Explanation

For an AR(1), Var(X) = sigma^2/(1 - a^2) = 3.6/(1 - 0.64) = 3.6/0.36 = 10. Check: 0.64 x 10 + 3.6 = 10. The option 4.5 comes from dividing by (1 - a) = 0.8 instead, and 18 is not obtained by any correct step.

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