FRM Part I · FRM Exam Part I · Stationary Time Series
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 2?
The lag-2 autocovariance is 0.60. Only the shock e_{t-2} appears in both y_t and y_{t-2}, with coefficients 0.3 and 1, so the covariance is the shock variance 2 times 0.3. Leaving out the variance gives 0.30, which is wrong.
- A0.30
- B0.60Correct
- C0.84
- D0.00
Explanation
Cov(y_t, y_{t-2}) = sigma^2 * theta_2 = 2 * 0.3 = 0.60. Only the e_{t-2} term is shared between y_t and y_{t-2}, with coefficients 0.3 and 1. 0.30 omits the variance; 0.84 is the lag-1 autocovariance, 2(0.4 + 0.4*0.3) = 1.04, so not that either, but it ignores the cutoff logic.
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