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

A stationary AR(1) process is X_t = 10 + 0.6(X_{t-1} - 10) + e_t, with e_t white noise with variance 4. The latest observation is X_100 = 15. What is the 3-step-ahead point forecast of X_103?

The 3-step forecast is 11.08. For a stationary AR(1) the forecast decays geometrically towards the mean: 10 plus 0.6 cubed times the current deviation of 5. Forecasts converge to the long-run mean of 10 as the horizon lengthens.

  1. A11.08Correct
  2. B12.00
  3. C11.00
  4. D13.00
  5. 10.00

Explanation

The forecast is mu + phi^k (x_n - mu) = 10 + 0.6^3 x 5 = 10 + 0.216 x 5 = 11.08. Using phi instead of phi^3 gives 13.00. Using 0.6^3 without the mean shift would give a value that ignores mu.

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