Suppose that a research has estimated the following relation

Suppose that a research has estimated the following relationship for the US unemployment rate:

a.) Suppose that ?₂ = +1. Trace out the time path of the effects of ?₂ on u₂, u₃, u₄,….. Discuss the shape of this impulse response function and what it implies for the stability of the estimated equation stated at the beginning of this problem. Do you believe that the researcher’s equation is poorly estimated? Why or why not?    

b.) Suppose that the initial conditions are as follows: u₀ = 5, u₁ = 6, and ?_t = 0 for t? 1. Impose the initial conditions in order to find the general solution.

Please show all work, thanks!

Solution

Put u = 2 in our given equation:

u(2) = -0.5 + 0.5u(1) + 0.7u(0) + 1

Given u(0) = 5, u(1) = 6

Put the above values in 1 above to get;

u(2) = -0.5 + 0.5*6 + 0.7*5 + 1

=> u(2) = 7 for t = 2. Remember that here we have  2 _{= 1 (Given)}