FRM Part I · FRM Exam Part I · Nonstationary Time Series
A random walk without drift has Y_t = Y_{t-1} + e_t, with e_t white noise of variance 9 and Y_0 = 0. What is the variance of Y_16, and what is the standard deviation of the 16-step-ahead forecast error at time 0?
Variance is 144 and the standard deviation is 12. A random walk accumulates 16 independent shocks each of variance 9, so variance is 16 × 9 = 144, and its square root is 12. The forecast error equals that accumulated shock sum.
- AVariance 144, standard deviation 12Correct
- BVariance 9, standard deviation 3
- CVariance 144, standard deviation 144
- DVariance 48, standard deviation 12
Explanation
Y_16 is the sum of 16 independent shocks, so variance = 16 × 9 = 144. The standard deviation is √144 = 12. Option with variance 9 ignores accumulation, and standard deviation 144 confuses variance with standard deviation.
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