RDP 2013-11: Issues in Estimating New-Keynesian Phillips Curves in the Presence of Unknown Structural Change Appendix A: Relevant Instruments

Setting means to zero, the system used is:

where the shocks ε_jt are uncorrelated with one another and not autocorrelated. It is known that the solution to this system would have the form (for inflation and potential output):

Substituting Equation (A4) into Equation (A1) and taking the expectation we get:

Gathering terms in Equation (A6) produces:

where . Substituting for r_t from Equation (A3) we get:

where .

Leading π_t from Equation (A8) and substituting for x_t+1 from Equation (A5) gives

Substituting from Equation (A5) again we get:

where b₁ = a₁ + a₂ϕ₄, b₂ = a₂ϕ₅, b₃ = a₃ + a₂ϕ₆, ξ_t+1 = η_t+1a₂v_2t+1. Now using Equation (A3) for r_t in Equation (A9) we get:

where d₁ = b₁ +b₃λ₃, d₂ = b₂ + b₃λ₂, d₃ = b₃λ₁, ζ_t+1 = ξ_t+1 + b₃ε_3t. Now replace π_t by Equation (A8):

where f₁ = d₁a₁, f₂ = d₁a₂ + d₂, f₃ = d₁a₃ + d₃ and e_t+1 = d₁η_t + ζ_t+1. Hence π_t+1 does not depend on x_t−1 and so the latter is not an instrument.