Problem 3 (Thinking of simulations). Suppose that you have been tasked to graphically
illustrate the AR(1) time series
$X_t = \phi X_{t-1} + \epsilon_t$.
Time Series (SS4861/9861)
Assignment 4
You decide to simulate i.i.d. standard normal errors $\epsilon_t$ and then recursively calculate $X_t$:
$X_1 = \phi X_0 + \epsilon_1$,
$X_2 = \phi X_1 + \epsilon_2$,
$X_3 = \phi X_2 + \epsilon_3$,
and so on.
But you need to set an initial value $X_0$ because without it you cannot obtain any subsequent
X. Hence, you choose a constant $c$ as the initial value, that is, you set $X_0 := c$. This leads
to the time series
$Y_1 = \phi c + \epsilon_1$,
$Y_2 = \phi Y_1 + \epsilon_2$,
$Y_3 = \phi Y_2 + \epsilon_3$,
and so on.
Estimate the difference $X_t - Y_t$ for $t \ge 1$ and establish its limit when $t \to \infty$.
