IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Applications of time series models
An MA(1) model is X_t = mu + e_t + 0.5 e_{t-1}, with mu = 20. At time n the last observation is X_n = 23 and the estimated residual is e_n = 2. What are the forecasts of X_{n+1} and X_{n+2}?
The forecasts are 21 and 20. The one-step forecast uses the known last residual: 20 plus 0.5 times 2. Beyond one step an MA(1) has no remaining known shocks, so the forecast reverts to the mean of 20.
- A21 and 20Correct
- B23 and 20
- C21 and 21
- D23 and 23
- 20 and 20
Explanation
X_{n+1} forecast = 20 + 0.5 x 2 = 21 since e_{n+1} has expectation zero. For X_{n+2} both error terms are future, so the forecast is mu = 20. Option 21 and 21 wrongly carries the MA term further.
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