(Solved): (a) Derive the forecasting formula for an AR(1) model. (6 marks) (b) A certain time series cons ...

(a) Derive the forecasting formula for an $AR(1)$ model. (6 marks) (b) A certain time series consisting of 100 observations was fitted by the AR(1) model: $(1?0.5B)\left(Z_t?7\right)=a_t$ The last four observations of the series were $Z_{97}=9.6,Z_{98}=9,Z_{99}=8.9$ and $Z_{100}=9.5$. Calculate the forecasts for the next three observations of the series, i.e., forecast $Z_{101},Z_{102}$, and $Z_{103}$. (9 marks) (c) Consider the AR(1) model with $Y_t=15,?=?0.5$, and $?=12$ : (i) Find the forecast values ${\stackrel{?}{Y}}_t(l)$ for $l=1,2$ and 8 . (9 marks) (ii) Assume that the AR(1) model has independently and identically distributed random variables $e_t$ with mean 0 and variance ${?}_2^2=0.4$. Calculate the $95\%$ confidence limits for the forecasts ${\hat{Y}}_i(1)$ and ${\hat{Y}}_i(2)$ calculated in c(i) above. (6 marks)