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

Let X_t = 5 + 0.4t + e_t, where e_t are independent N(0, 4) variables. Which statement is correct?

The series is not stationary because its mean 5 + 0.4t changes with time. Its first difference equals 0.4 plus e_t minus e_{t-1}, which has constant mean and lag-dependent autocovariances only, so it is stationary. This is a trend-stationary series.

  1. AX_t is weakly stationary because the variance is constant
  2. BX_t is not stationary because its mean depends on t, but the first difference is stationaryCorrect
  3. CX_t is not stationary and no differencing can make it stationary
  4. DX_t is stationary because the autocovariances depend only on lag
  5. The first difference of X_t has mean 0.4t

Explanation

E[X_t] = 5 + 0.4t varies with t, so X_t is not stationary. Y_t = X_t - X_{t-1} = 0.4 + e_t - e_{t-1}, which has constant mean 0.4 and autocovariances depending only on lag, so it is stationary. Option 4 fails since the mean is not constant.

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