IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Applications of time series models
For an AR(2) process X_t = 0.5 X_{t-1} + 0.2 X_{t-2} + e_t, the lag-1 autocorrelation rho_1 is obtained from the Yule-Walker equations. What is rho_1?
The lag-1 autocorrelation is 0.625. From the Yule-Walker equation rho_1 = 0.5 + 0.2 rho_1, so rho_1 = 0.5 divided by 0.8, which gives 0.625.
- A0.50
- B0.625Correct
- C0.40
- D0.70
- 0.25
Explanation
Yule-Walker: rho_1 = a1 + a2 rho_1, so rho_1 = a1/(1-a2) = 0.5/0.8 = 0.625. Check: 0.5 + 0.2*0.625 = 0.625, which holds. Choosing 0.5 ignores the second-lag contribution, and 0.7 simply adds the coefficients.
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