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Macro-Econometrics

Homework Assignment #2 (100 points)

1.  Consider the following difference equation:

(a) Derive the homogenous solution for this difference equation.  Please show your work.

(b) Derive the particular solution for this difference equation using the method of undetermined coefficients.  Please show your work.

(c) Suppose that ε₁ = +1.  Assuming that there are no further shocks to y(t), use the derived coefficients of the particular solution to trace out the time path of the effects of ε₁ on y₁, y₂, y₃,…..   Discuss the shape of this impulse response function and what it implies for the stability of the difference equation stated at the beginning of this problem (a similar exercise was done in Part B, slide #12, of the lecture).

(d)  Suppose that the initial conditions are as follows:  y₀ = 0 and ε_t = 0 for t≤ 0.  Impose the initial conditions in order to find the general solution.

2. Suppose that a research has estimated the following relationship for the US inflation rate:

(a) Derive the homogenous solution.  What does the homogeneous solution tell us about the long-run stability of this estimated equation?

(b) Derive the particular solution using the method of undetermined coefficients.  Please show your work.

(c) Suppose that ε₂ = +1.  Trace out the time path of the effects of ε₂ on π₂, π₃, π₄,…..  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?

(d) Suppose that the initial conditions are as follows:  π₀ = 10, π₁ = 11, and  ε_t = 0 for t≤ 1.  Impose the initial conditions in order to find the general solution.