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

Which of the following processes, with e_t independent white noise of constant variance, is NOT weakly stationary?

The random walk X_t = X_{t-1} + e_t is not weakly stationary because its variance equals t times sigma squared and grows with time. The other processes have constant mean and lag-dependent covariance only.

  1. AX_t = 0.8 X_{t-1} + e_t
  2. BX_t = e_t - 0.5 e_{t-1}
  3. CX_t = 5 + e_t + 0.3 e_{t-1}
  4. DX_t = X_{t-1} + e_t, starting from X_0 = 0Correct
  5. X_t = 0.3 X_{t-1} - 0.2 X_{t-2} + e_t

Explanation

The random walk has Var(X_t) = t sigma^2, which grows with t, so the autocovariance is not a function of lag alone. The AR(1) with 0.8 has root outside the unit circle, the MAs are stationary, and the AR(2) satisfies the stationarity conditions.

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