FRM Part I · FRM Exam Part I · Stationary Time Series
An analyst fits Y_t = 2 + 0.6Y_{t-1} + ε_t + 0.3ε_{t-1}. At time T, Y_T = 6.0 and the estimated residual is ε_T = 1.0. What is the two-step-ahead forecast of Y_{T+2}?
The two-step forecast is 5.54. The one-step forecast is 2 + 0.6×6 + 0.3×1 = 5.90. For the second step the future shock has zero expected value, so the moving-average term disappears and the forecast is 2 + 0.6×5.90 = 5.54.
- A5.00
- B5.54Correct
- C5.84
- D5.90
Explanation
One-step forecast: 2 + 0.6(6.0) + 0.3(1.0) = 5.90. For two steps, E[ε_{T+1}] = 0, so the MA term drops out: 2 + 0.6(5.90) = 5.54. Option 5.84 wrongly keeps the MA term again, 5.90 is the one-step forecast, and 5.00 is the long-run mean.
Did you get it right without looking?
One question tells you little. A timed set on Stationary Time Series shows your real accuracy, how long you take and where you lose marks.
More Stationary Time Series questions
- A risk analyst fits an ARMA(1,1) model to a stationary series: Y_t = 0.6 Y_{t-1} + e_t + 0.3 e_{t-1}, where e_t is white noise. Which statem…
- An analyst fits an MA(2) model y_t = e_t + 0.4 e_{t-1} + 0.3 e_{t-2}, with white noise variance 2. What is the autocovariance of y_t at lag …
- Using 100 residuals, an analyst computes sample autocorrelations of 0.20 at lag 1, -0.10 at lag 2, and 0.10 at lag 3. The Box-Pierce statist…
- A forecaster has a quarterly series with a stable seasonal pattern and models it as Y_t = c + 0.3*Y_{t-1} + 0.5*Y_{t-4} + e_t. Which stateme…
- An AR(1) model is Y_t = 1.0 + 0.5 Y_{t-1} + e_t, with sigma^2 = 4. The latest observation is Y_T = 6. What are the 2-step-ahead forecast and…
- 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_…