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

For which of the following AR(2) processes, X_t = a1 X_{t-1} + a2 X_{t-2} + e_t, is the process stationary?

The pair a1 = 0.4, a2 = 0.3 is stationary because a1 + a2 = 0.7 is below 1, a2 - a1 = -0.1 is below 1 and |a2| is below 1. Every other option breaks at least one of these conditions.

  1. Aa1 = 0.5, a2 = 0.6
  2. Ba1 = 1.2, a2 = -0.2
  3. Ca1 = 0.4, a2 = 0.3Correct
  4. Da1 = -0.3, a2 = 0.8
  5. a1 = 1.5, a2 = -0.4

Explanation

Stationarity needs a2 + a1 < 1, a2 - a1 < 1 and |a2| < 1. Option 2: 0.7<1, -0.1<1, 0.3<1, so it is stationary. Option 0: a1+a2=1.1 fails. Option 1: sum is 1.0, a unit root. Option 3: a2 - a1 = 1.1 fails. Option 4: sum is 1.1 fails.

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