APPENDIX OF “TIME-VARYING INTERCEPTS AND EQUILIBRIUM ANALYSIS: AN EXTENSION OF THE DYNAMIC ALMOST IDEAL DEMAND MODEL”

The algorithm for simulating (4.3) requires the knowledge of the full conditional posterior den-sities of γ, (β0,β1), Φ, Ω, µ, and V. Before describing the algorithm proper, we will derive the forms of these six densities. Since only the first and third terms of the right-hand side of (4.3) depend on γ, we have: p∗(γ | β0,β1,Φ,Ω, µ, V) ∝ L(β1, γ,Φ,Ω | data)fNO[β0;w0 − (Ir ⊗Π)x0,Σ]. (A1) The first term of this product is the likelihood corresponding to equation (3.4): Φ(L)wt − Φ(1)αt = [(x0t d0t) ⊗ Φ(1)]S γ + t for t = r + 1,..., r + T. (A2) Similarly, the second term may be viewed as a likelihood for the first r observations, corresponding to the following regression equation: (wt − αt) = (Π B) µ xt Okq×1 + ut = [(x0t O1×kq) ⊗ In] vec (Π B) + ut = [(x0t O1×kq) ⊗ In]S γ + ut. (A3) In this case, the nr disturbances uit (for t = 1,..., r) are jointly distributed as N(0,Σ) and are independent of the it, for t = r + 1,..., r + T. Since (A2) and (A3) involve the same coefficient vector γ, we may combine them into: y ∗ = X∗γ + ψ (A4) where ψ ∼ N(0,Ψ), with: y ∗ = w1 − α1 wr − α