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

A time series follows X_t = 0.6 X_{t-1} + e_t, where e_t is white noise. Which statement about this process is correct?

The process is stationary because the root of 1 - 0.6B is 1/0.6, about 1.67, which lies outside the unit circle. Stationarity of an AR(1) depends only on the absolute value of the coefficient being below one, not its sign or the noise variance.

  1. AIt is stationary because the root of 1 - 0.6B lies outside the unit circleCorrect
  2. BIt is non-stationary because the coefficient is positive
  3. CIt is stationary only if the white noise variance equals 1
  4. DIt is non-stationary because its autocorrelations decay geometrically
  5. It is stationary because its partial autocorrelation function decays geometrically

Explanation

The characteristic polynomial 1 - 0.6B has root B = 1/0.6 = 1.667, which has modulus greater than 1, so the AR(1) process is stationary. Geometric decay of the ACF is a feature of stationary AR(1), not a sign of non-stationarity. The PACF of an AR(1) cuts off after lag 1, so the last option is wrong.

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