(Requires Appendix Material)The Long-Run, Stationary State Solution of an AD(p,q)model $a _ { 0 }$

Essay

(Requires Appendix material)The long-run, stationary state solution of an AD(p,q)model, which can be written as A(L)Y_t = ?₀ + c(L)X_t-1 + u_t, where $a _ { 0 }$ = 1, and $a _ { j }$ = -?_j, c_j = ?_j, can be found by setting L=1 in the two lag polynomials. Explain. Derive the long-run solution for the estimated ADL(4,4)of the change in the inflation rate on unemployment:
$\widehat { \Delta I n f }$ _t = 1.32 - .36 ?Inf_t-1 - 0.34vInf_t-2 + 0.7?Inf_t-3 - 0.3?Inf_t-4
-2.68Unemp_t-1 + 3.43Unemp_{t-2 -} 1.04Unemp_t-3 + .07Unemp_t-4
Assume that the inflation rate is constant in the long-run and calculate the resulting unemployment rate. What does the solution represent? Is it reasonable to assume that this long-run solution is constant over the estimation period 1962-1999? If not, how could you detect the instability?