Consider the Following Model Is Strictly
Exogenous $\sum _ { i = 1 } ^ { 3 } \beta _ { i } = 1$

Essay

Consider the following model $Y _ { t } = \beta _ { 0 } + \beta _ { 1 } X _ { t } + \beta _ { 2 } X _ { t - 1 } + \beta _ { 3 } Y _ { t - 1 } + u _ { t } \text {, where } X _ { t }$ is strictly
exogenous. Show that by imposing the restriction $\sum _ { i = 1 } ^ { 3 } \beta _ { i } = 1$ you can derive the following so-called Error Correction Mechanism (ECM) model
$\Delta Y _ { t } = \beta _ { 0 } + \beta _ { 1 } \Delta X _ { t } - \theta ( Y - X ) _ { t - 1 } + u _ { t }$
where $\theta = \beta _ { 1 } + \beta _ { 2 }$
What is the short-run (impact) response of a unit increase in X ? What is the long-run solution? Why do you think the term in parenthesis in the above expression is called ECM?