IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Applications of time series models
A stationary AR(1) process X_t = 0.5 X_{t-1} + e_t has e_t white noise with variance 9. What is the variance of X_t?
The variance is 12. For a stationary AR(1) process the variance equals the noise variance divided by one minus the squared coefficient, so 9 divided by 0.75 gives 12.
- A9
- B12Correct
- C18
- D6.75
- 4.5
Explanation
Var(X) = sigma^2 / (1 - alpha^2) = 9 / (1 - 0.25) = 9 / 0.75 = 12. Check: 0.25 x 12 + 9 = 12. Option 9 ignores the autoregressive effect, and 6.75 multiplies by 0.75 instead of dividing.
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