IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Core concepts of time series models
The stationary AR(1) process X_t = 0.8 X_{t-1} + e_t has e_t white noise with variance 9. A stationary ARMA(1,1) process is not considered. What is the variance of X_t?
The variance of X_t is 25. For a stationary AR(1) process, the variance equals the noise variance divided by one minus the squared coefficient. That is 9 divided by 0.36, which equals 25.
- A9
- B11.25
- C25
- D45Correct
- 81
Explanation
For AR(1), Var(X) = sigma^2/(1-a^2) = 9/(1-0.64) = 9/0.36 = 25. Option 9 ignores the dependence. Option 11.25 uses 9/(1-0.8^2)... incorrectly as 9/0.8, and 45 is not derived from the formula.
Did you get it right without looking?
One question tells you little. A timed set on Core concepts of time series models shows your real accuracy, how long you take and where you lose marks.
More Core concepts of time series models questions
- A time series analyst inspects the sample ACF and PACF of a stationary series. The sample ACF decays gradually towards zero, while the sampl…
- A stationary AR(1) model is X_t - 50 = 0.6(X_{t-1} - 50) + e_t, with white noise variance 16. The latest observation is x_100 = 60. What is …
- Let e_t be white noise with variance 9. A process is defined by Y_t = e_t + 0.4 e_{t-1}. Using the autocorrelation at lag 1 of this process,…
- For the MA(1) process X_t = e_t + 0.5 e_{t-1}, where e_t is white noise with variance sigma^2, what is the autocorrelation at lag 1?
- A stationary, zero-mean AR(2) process X_t = 0.5 X_{t-1} + 0.2 X_{t-2} + e_t is considered. What is the autocorrelation at lag 1, rho_1?
- A stationary AR(1) process has X_t - mu = 0.5 (X_{t-1} - mu) + e_t with mu = 40. Given X_10 = 48, what is the best forecast of X_12 at time …