IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Core concepts of time series models
A random walk is defined by X_t = X_{t-1} + e_t with X_0 = 0, where e_t is white noise with variance 4. What is Var(X_25)?
The variance is 100. Starting at zero, X_25 is the sum of 25 independent shocks, each with variance 4, so variances add to 25 times 4. The variance rises linearly with time, which is why the random walk is non-stationary.
- A4
- B20
- C25
- D100Correct
- 625
Explanation
X_25 is the sum of 25 independent white noise terms, so its variance is 25 x 4 = 100. Choosing 25 forgets sigma squared; 625 squares the number of terms; 20 uses sigma times sqrt(25)... mixing steps incorrectly.
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
- X_t is a random walk with X_0 = 0 and shock variance sigma squared. For s < t, what is Corr(X_s, X_t) when s = 9 and t = 36?
- An analyst fits an ARMA model to monthly claim counts for an Indian insurer and tests residuals. Which finding would most clearly indicate t…
- Which condition makes the MA(1) process X_t = e_t + θ e_{t-1} invertible?
- 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 …
- An AR(2) process X_t = 0.5 X_{t-1} + 0.3 X_{t-2} + e_t is being checked for stationarity. Which condition must hold?
- A stationary AR(1) process X_t = 0.8 X_{t-1} + e_t has white noise variance 3.6. What is the variance of X_t?