Consider the Following Distributed Lag Model Is Serially Uncorrelated, and X Is Strictly Exogenous

Essay

Consider the following distributed lag model $$Y _ { t } = \beta _ { 0 } + \beta _ { 1 } X _ { t } + \beta _ { 2 } X _ { t - 1 } + u _ { t } \text {, where } u _ { t } = \phi _ { 1 } u _ { t - 1 } + \tilde { u } _ { t } , \tilde { u } _ { t }$$ is serially uncorrelated, and X is strictly exogenous.
(a)How many parameters are there to be estimated between the two equations?
(b)Using the two equations of the model above, derive the ADL form of the model.
(c)There are five regressors in the ADL model, namely Y_t-1, X_t, X_t-1, X_t-2 and the constant. Estimating the ADL model linearly will give you five coefficients. Can you derive the parameters of the original two equation model from these five estimates? Why or why not?
(d)What alternative method do you have to retrieve the parameters of the two equation model?