RDP 2014-12: A State-space Approach to Australian GDP Measurement Appendix A: Identification

We use the results contained in Appendix A of Aruoba et al (2013) and Section 4 of Komunjer and Ng (2011) to prove that the model presented in Section 2.2 is identified with a single parameter restriction. In particular, and ignoring the constants, the model can be written as

where s_t=Δy_t, , A = ρ, B = [1,0,0,0], C = [ρ,ρ,ρ]′ and

where ε_t =[ε_G,t, ε_E,t, ε_I,t, ε_P,t] ∼ N(0,Σ).

Assuming that 0 ≤ ρ < 1 and that Σ is positive definite, and noting that the rows of D are linearly independent, ensures that Assumptions 1, 2 and 4-NS of Komunjer and Ng (2011) are satisfied, while Appendix A of Aruoba et al (2013) shows that Assumption 5-NS of Komunjer and Ng is satisfied. Then by Proposition 1-NS of Komunjer and Ng, two models (with the second model indexed by a * subscript) are observationally equivalent if and only if ρ_* = ρ and

where p solves .

If p_* = p then the above equations imply that Σ_* = Σ and the models are identical. If p_* ≠ p we can write p_* as p_* = p + δ for some δ ≠ 0, in which case Equation (A3) becomes . From Equation (A2) we have

so that σ_GE* = σ_GE − δ, σ_GI* = σ_GI − δ and σ_GP* = σ_GP − δ. Finally,

so that from Equation (A1) we have

so that , σ_IP* = σ_IP + δ and . Hence the ‘star’ model is observationally equivalent to the ‘non-star’ model if and only if

for some δ. As such, we need to place at least one restriction on Σ to ensure an identified model.