FRM Part I · FRM Exam Part I · Nonstationary Time Series
A price series follows the random walk with drift P_t = 0.5 + P_(t-1) + e_t, where e_t is white noise with variance 4. After taking first differences, what are the mean and variance of the differenced series?
The differenced series has mean 0.5 and variance 4. Differencing a random walk with drift leaves the constant drift plus the white noise shock, so the mean is the drift and the variance is the shock variance, and the series is stationary.
- AMean 0.5, variance 4Correct
- BMean 0, variance 4
- CMean 0.5, variance 2
- DMean 0.5, variance 8
Explanation
ΔP_t = P_t − P_(t-1) = 0.5 + e_t. The mean is the drift, 0.5, and the variance equals the variance of e_t, which is 4. The result is stationary. Variance 8 wrongly doubles the variance, as if the difference of two independent shocks were taken.
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