FRM Part I · FRM Exam Part I · Stationary Time Series
An analyst fits a model to monthly data: Y_t = 2 + 0.5*Y_{t-1} + 0.4*Y_{t-12} + e_t, where e_t is white noise. Assuming covariance stationarity and ignoring any interaction term, what is the long-run (unconditional) mean of Y?
The unconditional mean is 20. Setting the mean equal across periods gives mu = 2 + 0.5mu + 0.4mu, so mu = 2/(1 - 0.9) = 20. Both the first lag and the seasonal lag coefficients must be included in the denominator.
- A4
- B20Correct
- C40
- D10
Explanation
For a stationary model the mean satisfies mu = 2 + 0.5mu + 0.4mu, so mu(1 - 0.9) = 2 and mu = 2/0.1 = 20. The value 4 results from dividing by only 1 - 0.5 (ignoring the seasonal lag), giving 2/0.5 = 4.
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