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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Applications of time series models

Y_t follows an ARIMA(0,1,1) model: Y_t = Y_{t-1} + e_t + 0.4 e_{t-1}, with e_t white noise of variance 5. What is the variance of the differenced series W_t = Y_t - Y_{t-1}?

The differenced series is an MA(1) process, so its variance is (1 + 0.16) times 5, which is 5.8. The coefficient is squared in the variance formula; ignoring it gives 5, and adding 0.4 unsquared gives 7.

  1. A5.0
  2. B5.8Correct
  3. C6.0
  4. D7.0
  5. 8.0

Explanation

W_t = e_t + 0.4 e_{t-1} is MA(1). Variance = (1 + 0.4^2) × 5 = 1.16 × 5 = 5.8. Option 5.0 ignores the MA term; 7.0 would use 1+0.4 wrongly (1.4×5).

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