FRM Part I · FRM Exam Part I · Stationary Time Series
A stationary ARMA(1,1) process is Y_t = 2 + 0.6Y_{t-1} + ε_t + 0.3ε_{t-1}, with E[ε_t] = 0. What is the unconditional mean of Y_t?
The unconditional mean is 5.00. Taking expectations, the white-noise terms vanish, leaving μ = 2 + 0.6μ, so μ = 2/(1 − 0.6) = 5. The moving-average coefficient does not enter the mean of a stationary ARMA process.
- A1.25
- B2.00
- C5.00Correct
- D20.00
Explanation
Taking expectations gives μ = 2 + 0.6μ, since the MA term has zero mean. So μ = 2/(1 - 0.6) = 5.00. The value 1.25 comes from wrongly using 1 + 0.6 in the denominator. The value 20 comes from wrongly subtracting θ as well (2/0.1).
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