IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Core concepts of time series models
A stationary, zero-mean AR(2) process X_t = 0.5 X_{t-1} + 0.2 X_{t-2} + e_t is considered. What is the autocorrelation at lag 1, rho_1?
The lag-1 autocorrelation is 0.625. The Yule-Walker equation rho_1 = 0.5 + 0.2 rho_1 rearranges to rho_1 = 0.5 divided by 0.8. Simply using the first coefficient 0.5 or adding the coefficients ignores how the second lag feeds back through rho_1.
- A0.25
- B0.40
- C0.50
- D0.625Correct
- 0.70
Explanation
Yule-Walker at lag 1 gives rho_1 = a1 + a2 rho_1, so rho_1 = a1/(1 - a2) = 0.5/0.8 = 0.625. Answering 0.5 ignores the second lag's contribution. The option 0.70 adds the coefficients, which is not valid.
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