Free CAS MAS-II Time Series with Constant Variance Practice Questions (2026)

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 questions 55 easy 55 medium 26 hard 2026 syllabus

Sample Questions

Question 1 Easy

Which of the following statements BEST describes a moving-average MA(q) process?

Solution

B is correct. A moving-average MA(q) process expresses

Y_t

as a finite linear combination of the contemporaneous innovation

\epsilon_t

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

Y_t = c + \phi\,Y_{t-1} + \epsilon_t

with

|\phi| < 1

and mean-zero white-noise innovations?

Solution

D is correct. For a stationary AR(1) process, the autocorrelation at lag

k

\rho_k = \phi^k

. Because

|\phi|<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-\phi)

and variance

\sigma_\epsilon^2/(1-\phi^2)

are both finite constants that do not depend on time, consistent with weak stationarity.

Question 3 Hard

Consider the stationary ARMA(1,1) process

Y_t = 0.4\,Y_{t-1} + \varepsilon_t + 0.3\,\varepsilon_{t-1}

, where

\varepsilon_t

is white noise with variance

\sigma^2 = 9

. Calculate the unconditional variance of

Y_t

Solution

B is correct. For a stationary ARMA(1,1) process

Y_t = \phi Y_{t-1} + \varepsilon_t + \theta \varepsilon_{t-1}

, the unconditional variance is

\gamma(0) = \frac{1 + 2\phi\theta + \theta^2}{1 - \phi^2}\,\sigma^2.

The numerator combines the direct shock

\varepsilon_t

, the lagged MA shock contribution, and the AR-MA cross term. Substituting

\phi = 0.4

\theta = 0.3

, and

\sigma^2 = 9

1 + 2(0.4)(0.3) + (0.3)^2 = 1 + 0.24 + 0.09 = 1.33,

1 - (0.4)^2 = 0.84.

Therefore

\gamma(0) = \frac{1.33}{0.84} \times 9 \approx 1.583 \times 9 \approx 14.25.

The value 14.25 falls in the interval "At least 12.5, but less than 15".

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Free CAS MAS-II (Modern Actuarial Statistics II) Time Series with Constant Variance Practice Questions

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