ar1 recursive expansion

ar1.typ

$
Y_t = beta_0 + beta_1 Y_(t-1) + epsilon_t
$

Take note of:

- $abs(beta_1) < 1$
- $epsilon_t tilde.op "WN"(0, sigma^2)$
- $Y_(t-1) = beta_0 + beta_1 Y_(t-2) + epsilon_(t-1) $

#let sub_eq = [
  #set text(red)
  $Y_(t-2) = beta_0 + beta_1 Y_(t-3) + epsilon_(t-2)$
]
Expand $Y_t$ with recursive substitution:
$
Y_t &= beta_0 + beta_1 Y_(t-1) + epsilon_t \
&= beta_0 + beta_0 beta_1 + beta_1^2 Y_(t-2) + epsilon_t + beta_1 epsilon_(t+1) \
#sub_eq\
&= beta_0 + beta_0 beta_1 + beta_0 beta_1^2 + beta_1^3 Y_(t-3) + epsilon_(t-1) + beta_1^2 epsilon_(t-2) \
&#h(4cm) dots.v \
&= beta_0(1 + beta_1 + beta_1^2 + ... + beta_1^k) + beta_1^(k+1) Y_(t-k-1) + epsilon_t + beta_1 epsilon_(t-1) + beta_1^2 epsilon_(t+2) + ... + beta_1^k epsilon_(t-k) \
&#h(4cm) dots.v \
&= beta_0(1 + beta_1 + beta_1^2 + ... ) +lim_(k->infinity) beta_1^(k+1) Y_(t-k-1) + epsilon_t + beta_1 epsilon_(t-1) + beta_1^2 epsilon_(t+2) + ...
$

Apply geometric progression formula ( https://en.wikipedia.org/wiki/Geometric_series ):
$ sum_(k=0)^infinity a r^k = a/(1-r) "for" abs(r)<1 $

$
Y_t = beta_0/(1-beta_1) + epsilon_t + beta_1 epsilon_(t-1) + beta_1^2 epsilon_(t-2) + beta_1^3 epsilon_(t-3) + ...
$

Furthermore:

$ E[Y_t] = beta_0/(1-beta_1) $

$ v a r[Y_t] &= sigma^2 + beta_2^2 + beta_1^4 sigma^2 + ... \
&= sigma^2 (1 + beta_1^2 + beta_1^4 + beta_1^6 + ...) \
&= sigma^2 /(1 - beta_1^2) \
& "geometric progression formula again, just with" r=beta_1^2
$