(A) Show That $y _ { t } = \beta _ { 1 } y _ { t - 1 } + u _ { t }$

Essay

$$\text { Consider the standard AR(1) } Y _ { t } = \beta _ { 0 } + \beta _ { 1 } Y _ { t - 1 } + u _ { t } \text {, where the usual assumptions hold. }$$ (a) Show that $y _ { t } = \beta _ { 1 } y _ { t - 1 } + u _ { t }$ , where $y _ { t }$ is $Y _ { t }$ with the mean removed, i.e., $y _ { t } = Y _ { t } - E \left( Y _ { t } \right)$ . Show that $E \left( y _ { t } \right) = 0$ .