Homework 3

Homework 3
Winter 2011
Due Tuesday 2/1/2011

1. Use the Shapiro and Watson type model discussed in class to estimate the three variable system involving the federal funds rate, real GDP, and the CPI less food and energy. Impose the restrictions that shocks to GDP and shocks to the CPI have no effect on the federal funds rate in the long-run, and that shocks to the federal funds rate and the CPI have no effect on real GDP  in the long-run. Find the IRFs and VDCs for both the levels and differences representations. (The model is Δff, ΔlnGDP, and ΔlnCPI. The Δff is used simply to illustrate the technique. There are instances of articles in places like the AER where Δff has been used in VAR models in the past, but the level of the ff rate would normally be used today.)

2. Given the system of equations

          R_t = d_R + ∑ [A_RjR_t-j + B_RjΠ_t-j + C_RjY_t-j ]+ U_t

          Π_t = d_Π + ∑ [A_ΠjR_t-j + B_ΠjΠ_t-j + C_ΠjY_t-j ]+ V_t

          Y_t = d_Y + ∑ [A_YjR_t-j + B_YjΠ_t-j + C_YjY_t-j ]+ W_t

where the sum runs from j=1 to j=3, and U, V, and W are white noise errors, and given the definition of the real interest rate

          r_t = R_t - E[Π_t+1 | Ω_t],

where  Ω_t  = all information dated t and earlier and Π_t+1 is the inflation rate between t and t+1, (a) how many restrictions does the hypothesis r_t = λ₀ + λ₁r_t-1 + e_t impose on the model? (b) Derive the restricted set of equations. (c) Suppose the hypothesis is changed to r_t = λ₀ + λ₁r_t-1 + λ₂r_t-2 + e_t. What problems, if any, do you encounter in trying determine the number of restrictions this hypothesis imposes on the unrestricted VAR model?