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

A time series follows the process X_t = 0.6 X_{t-1} + e_t, where e_t is white noise with variance 4. 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 1, not on 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 is below 1
  4. DIt is non-stationary because the autocorrelations decay geometrically
  5. It is stationary because its autocorrelation function cuts off after lag 1

Explanation

The characteristic polynomial 1 - 0.6z has root z = 1/0.6 = 1.67, which is outside the unit circle, so the AR(1) is stationary. Geometric decay of the ACF is a feature of stationary AR(1), and a cutoff after lag 1 is a feature of MA(1), not AR(1). Stationarity does not depend on the noise variance.

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