Consider the following two dynamic regression models:
Model 1 yt = Î±ytâˆ’1 + Î²xt + t
Model 2 yt =Î³xt1âˆ’Ï†L +t1âˆ’Î¸L
where L is the lag operator and Î±, Î², Î³, Ï†, and Î¸ are parameters. Assume that t âˆ¼ N(0, Ïƒ2) (not necessarily the same Ïƒ for each model!).
Answer the following:
(a) Under what conditions are these the same dynamic model?
(b) What hypothesis test(s) would we need to use to determine whether these are the same or different model?
(c) If the parameters are different, how would you characterize each of the models and what different interpretations would exist for the impact of a one unit change or shock to xt in each model?