IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Applications of time series models
A random walk with drift is defined by X_t = X_{t-1} + 2 + e_t, where e_t is white noise with variance 9. Which statement is correct?
The first difference equals 2 + e_t, which is stationary with mean 2 and variance 9. The drift only shifts the mean of the differenced series and does not change the order of integration, so X_t is I(1) and non-stationary.
- AX_t is stationary with mean 2
- BThe first difference of X_t is white noise with mean 0
- CThe first difference of X_t is stationary with mean 2 and variance 9Correct
- DX_t is I(2) because of the drift
- The first difference of X_t has variance 18
Explanation
Differencing gives X_t - X_{t-1} = 2 + e_t. This is white noise shifted by a constant, so it is stationary with mean 2 and variance 9. The drift does not raise the order of integration; X_t is I(1).
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