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

An AR(2) process X_t = 0.5 X_{t-1} + 0.3 X_{t-2} + e_t is being checked for stationarity. Which condition must hold?

The AR(2) process is stationary when all roots of the characteristic polynomial 1 - 0.5z - 0.3z^2 lie outside the unit circle, which ensures the process does not explode and has constant mean and autocovariances.

  1. AThe roots of 1 - 0.5z - 0.3z^2 = 0 all lie outside the unit circleCorrect
  2. BThe roots of 1 - 0.5z - 0.3z^2 = 0 all lie inside the unit circle
  3. CThe roots of 1 + 0.5z + 0.3z^2 = 0 must be real
  4. DThe sum of the two coefficients must exceed 1
  5. The errors e_t must be normally distributed

Explanation

Stationarity of an AR process requires all roots of the characteristic polynomial 1 - phi1 z - phi2 z^2 to lie outside the unit circle in modulus. Roots inside the circle give explosive behaviour. Normality is not required and a coefficient sum above 1 indicates non-stationarity.

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