IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Core concepts of time series models
Consider the ARMA(1,1) process X_t = 0.5 X_{t-1} + e_t + 0.3 e_{t-1}, with e_t white noise of variance 1. Using the standard result for the lag 1 autocovariance, what is gamma_1 given that gamma_0 = (1 + 2(0.5)(0.3) + 0.09)/(1 - 0.25) = 1.39/0.75?
The lag 1 autocovariance is 0.5 times gamma_0 plus 0.3, which is about 1.2267, but this question is flawed because no option matches that result.
- AApproximately 1.7467Correct
- BApproximately 1.0567
- CApproximately 0.9267
- DApproximately 1.3900
- Approximately 1.1467
Explanation
gamma_0 = 1.39/0.75 = 1.85333. Multiply the ARMA equation by X_{t-1} and take expectations: gamma_1 = 0.5 gamma_0 + 0.3 E[e_{t-1}X_{t-1}] = 0.5(1.85333) + 0.3(1) = 0.92667+0.3 = 1.22667. Hmm, this value is not among the stated options, so check the given form: with E[e_{t-1}X_{t-1}] = 1 the result is 1.2267, which does not match; the keyed value therefore does not follow.
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