Consider the standard AR(1)Yt = β0 + β1Yt-1 + ut,where the usual assumptions hold.
(a)Show that yt = β0Yt-1 + ut,where yt is Yt with the mean removed,i.e. ,yt = Yt - E(Yt).Show that E(Yt)= 0.
(b)Show that the r-period ahead forecast E( +r )= .If 0

Question

Consider the standard AR(1)Yt = β0 + β1Yt-1 + ut,where the usual assumptions hold.
(a)Show that yt = β0Yt-1 + ut,where yt is Yt with the mean removed,i.e. ,yt = Yt - E(Yt).Show that E(Yt)= 0.
(b)Show that the r-period ahead forecast E(  +r  )=   .If 0 < β1 < 1,how does the r-period ahead forecast behave as r becomes large? What is the forecast of  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(  +r  )= 0.5  .Show that in the present case this is given by r = -  .

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The Answer of Consider the standard AR(1)Yt = β0 + β1Yt-1 + ut,where the usual assumptions hold.
(a)Show that yt = β0Yt-1 + ut,where yt is Yt with the mean removed,i.e. ,yt = Yt - E(Yt).Show that E(Yt)= 0.
(b)Show that the r-period ahead forecast E(  +r  )=   .If 0 < β1 < 1,how does the r-period ahead forecast behave as r becomes large? What is the forecast of  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(  +r  )= 0.5  .Show that in the present case this is given by r = -  .

Consider the Standard AR(1)Yt = β0 + β1Yt-1 + Ut,where