A series is generated by
Xt=Xt−1+εt,X0=0X_t = X_{t-1} + \varepsilon_t, \qquad X_0 = 0Xt=Xt−1+εt,X0=0
where the shocks ε1,ε2,…\varepsilon_1, \varepsilon_2, \ldotsε1,ε2,… are independent, each with mean 000 and variance 444.
Which statement about X5X_5X5 is correct?
Select all that apply.
Var(X5)=4\text{Var}(X_5) = 4Var(X5)=4; each shock has variance 4 and the mean stays at 0, so the series is stationary.
Var(X5)=100\text{Var}(X_5) = 100Var(X5)=100; five steps multiply the spread by 5, so the variance scales by 525^252.
Var(X5)≈4.47\text{Var}(X_5) \approx 4.47Var(X5)≈4.47; that is the typical distance from the starting point after five steps.
Var(X5)=20\text{Var}(X_5) = 20Var(X5)=20; the variance grows with ttt, so the series is non-stationary.