FRM Part I · FRM Exam Part I · Stationary Time Series
A zero-mean covariance stationary AR(1) process has φ = 0.8. The most recent observation is Y_T = 10. What is the best linear forecast of Y_(T+3), and what happens to long-horizon forecasts as the horizon grows?
The three-step forecast is 0.8 cubed times 10, which equals 5.12. Since the AR coefficient has absolute value below one, forecasts decay geometrically toward the unconditional mean, here zero, as the horizon lengthens.
- A5.12; they converge to the unconditional mean of zeroCorrect
- B8.00; they remain at the last observed value
- C5.12; they converge to 10
- D2.40; they converge to zero at a constant additive rate
Explanation
The h-step forecast is φ^h·Y_T = 0.8³×10 = 0.512×10 = 5.12. Because |φ|<1, forecasts decay geometrically toward the unconditional mean, which is zero here. Using 0.8×3 = 2.4 treats decay as additive, which is wrong.
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