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

Consider the process X_t = 1.5 X_{t-1} - 0.5 X_{t-2} + e_t, where e_t is white noise. Which description is correct?

It is an ARIMA(1,1,0) process. The polynomial 1 - 1.5B + 0.5B^2 factorises as (1-B)(1-0.5B), so it has a unit root. The first difference follows a stationary AR(1) with coefficient 0.5, so the process itself is not stationary.

  1. AA stationary AR(2) process, since both coefficients are less than 2 in magnitude
  2. BAn ARIMA(1,1,0) process, since the characteristic polynomial has a root equal to 1Correct
  3. CA stationary ARMA(2,1) process
  4. DAn ARIMA(0,1,1) process
  5. A non-invertible MA(2) process

Explanation

The characteristic polynomial is 1 - 1.5B + 0.5B^2 = (1-B)(1-0.5B). It has a unit root at B=1 and a root at B=2. Writing Y_t = X_t - X_{t-1} gives Y_t = 0.5 Y_{t-1} + e_t, a stationary AR(1), so X is ARIMA(1,1,0). Calling it stationary ignores the unit root.

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