IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Applications of time series models
An MA(1) process is X_t = e_t + 0.5 e_{t-1}, where e_t is white noise with variance 2. What is the autocorrelation at lag 1 and at lag 2?
Lag 1 autocorrelation is 0.4 and lag 2 is zero. Lag 1 equals theta divided by (1 + theta squared) = 0.5/1.25 = 0.4, and an MA(1) process has a cut-off, so all autocorrelations beyond lag 1 are zero.
- A0.5 and 0.25
- B0.4 and 0Correct
- C0.4 and 0.16
- D0.5 and 0
- 0.8 and 0
Explanation
Gamma0 = sigma^2(1+0.25) = 2.5; gamma1 = 0.5*2 = 1. So rho1 = 1/2.5 = 0.4. For an MA(1), autocorrelations are zero beyond lag 1, so rho2 = 0. The option 0.5 and 0 ignores the denominator 1 + theta^2.
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