# Using the data file usmacro, estimate the (operatorname{AR}(1)) model (G_{t}=alpha+phi G_{t-1}+v_{t}). From these estimates and those obtained

## Question:

Using the data file usmacro, estimate the $$\operatorname{AR}(1)$$ model $$G_{t}=\alpha+\phi G_{t-1}+v_{t}$$. From these estimates and those obtained in Exercise 9.16, use the results from Exercise 9.4 to find point and $$95 \%$$ interval forecasts for $$U_{2016 Q 2}, U_{2016 Q 3}$$, and $$U_{2016 Q 4}$$.

Data From Exercise 9.4:-

Consider the ARDL $$(2,1)$$ model

with auxiliary AR(1) model $$x_{t}=\alpha+\phi x_{t-1}+v_{t}$$, where $$I_{t}=\left\{y_{t}, y_{t-1}, \ldots, x_{t}, x_{t-1}, \ldots\right\}, E\left(e_{t} \mid I_{t-1}\right)=0$$, $$E\left(v_{t} \mid I_{t-1}\right)=0, \operatorname{var}\left(e_{t} \mid I_{t-1}\right)=\sigma_{e}^{2}, \operatorname{var}\left(v_{t} \mid I_{t-1}\right)=\sigma_{v}^{2}$$, and $$v_{t}$$ and $$e_{t}$$ are independent. Assume that sample observations are available for $$t=1,2, \ldots, T$$.

a. Show that the best forecasts for periods $$T+1, T+2$$ and $$T+3$$ are given by

b. Show that the variances of the forecast errors are given by

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