Free CAS MAS-II (Modern Actuarial Statistics II) Time Series with Constant Variance Practice Questions
Time series with constant variance on CAS Exam MAS-II covers AR, MA, and ARIMA model framework and identification, ACF and PACF diagnostics, stationarity conditions, deterministic vs. stochastic trends, seasonality via regression and seasonal differencing, forecast construction, and prediction-interval interpretation (CAS).
136 questions55 easy55 medium26 hard2026 syllabus
Sample Questions
Question 1
Easy
Which of the following statements BEST describes a moving-average MA(q) process?
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Correct Answer: B
Solution
B is correct. A moving-average MA(q) process expresses Yt as a finite linear combination of the contemporaneous innovation ϵt and its q most recent lags. Because no lagged values of the series itself appear on the right-hand side, MA(q) processes are always weakly stationary for any choice of MA coefficients.
Question 2
Medium
Which of the following statements is a property of a stationary AR(1) process Yt=c+ϕYt−1+ϵt with ∣ϕ∣<1 and mean-zero white-noise innovations?
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Correct Answer: D
Solution
D is correct. For a stationary AR(1) process, the autocorrelation at lag k is ρk=ϕk. Because ∣ϕ∣<1, the absolute autocorrelations shrink geometrically as the lag k grows, which is the hallmark ACF signature of an AR(1). The mean c/(1−ϕ) and variance σϵ2/(1−ϕ2) are both finite constants that do not depend on time, consistent with weak stationarity.
Question 3
Hard
Consider the stationary ARMA(1,1) process Yt=0.4Yt−1+εt+0.3εt−1, where εt is white noise with variance σ2=9. Calculate the unconditional variance of Yt.
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Correct Answer: A
Solution
A is correct. For a stationary ARMA(1,1) process Yt=ϕYt−1+εt+θεt−1, the unconditional variance is γ(0)=1−ϕ21+2ϕθ+θ2σ2. The numerator combines the direct shock εt, the lagged MA shock contribution, and the AR-MA cross term. Substituting ϕ=0.4, θ=0.3, and σ2=9: 1+2(0.4)(0.3)+(0.3)2=1+0.24+0.09=1.33,1−(0.4)2=0.84. Therefore γ(0)=0.841.33×9≈1.583×9≈14.25. The value 14.25 falls in the interval "Less than 15".
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