Consider the Standard AR(1)Y_t = β₀ + β₁Y_t-1 + U_t $y _ { T }$

Essay

Consider the standard AR(1)Y_t = β₀ + β₁Y_t-1 + u_t, where the usual assumptions hold.
(a)Show that y_t = β₀Y_t-1 + u_t, where y_t is Y_t with the mean removed, i.e., y_t = Y_t - E(Y_t). Show that E(Y_t)= 0.
(b)Show that the r-period ahead forecast E( $y _ { T }$ _+r
$|T$ )= $\beta _ { 1 } ^ { r }$ $y _ { T }$ If 0 < β₁ < 1, how does the r-period ahead forecast behave as r becomes large? What is the forecast of ${ } ^ { Y _ { T + r } } { |} _ { T }$ for large r?
(c)The median lag is the number of periods it takes a time series with zero mean to halve its current value (in expectation), i.e., the solution r to E( $y _ { T }$ _+r
$|T$ )= 0.5 $y _ { T }$ Show that in the present case this is given by r = - $\frac { \log ( 2 ) } { \log ( \beta 1 ) }$