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

Consider the process Y_t = 1.5 Y_{t-1} - 0.5 Y_{t-2} + e_t, with e_t white noise. Which statement is correct?

The polynomial factorises as (1-B)(1-0.5B), giving a unit root. So the process is ARIMA(1,1,0), and its first difference satisfies W_t = 0.5 W_{t-1} + e_t, a stationary AR(1). The sign of the coefficient is positive, not negative.

  1. AIt is a stationary AR(2) process because both coefficients are below 2 in size
  2. BIt is ARIMA(1,1,0), with first difference W_t = 0.5 W_{t-1} + e_tCorrect
  3. CIt is ARIMA(1,1,0), with first difference W_t = -0.5 W_{t-1} + e_t
  4. DIt is ARIMA(2,1,0), with first difference W_t = 0.5 W_{t-1} + e_t
  5. It is ARIMA(0,2,0)

Explanation

Characteristic polynomial 1 - 1.5B + 0.5B^2 = (1-B)(1-0.5B). One root is B=1 (unit root), so the process is integrated. Then (1-B)(1-0.5B)Y_t = e_t, so W_t = (1-B)Y_t satisfies W_t = 0.5W_{t-1} + e_t, a stationary AR(1). Hence ARIMA(1,1,0).

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