IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Core concepts of time series models
Consider the process X_t = 0.5 X_{t-1} + e_t, where e_t are independent N(0, 4) white noise terms, and the process has been running for a long time. What is the stationary variance of X_t?
The stationary variance is sigma squared divided by one minus alpha squared, which is 4 divided by 0.75, giving 16/3. Dividing by one minus alpha instead would wrongly give 8.
- A4
- B8
- C16/3Correct
- D6
- 16
Explanation
For a stationary AR(1), Var = sigma^2/(1-a^2) = 4/(1-0.25) = 4/0.75 = 16/3. Option 4 (8) comes from dividing by 1-a = 0.5 instead of 1-a^2. Option 0 ignores the autoregressive term.
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