RDP 2013-06: Estimating and Identifying Empirical BVAR-DSGE Models for Small Open Economies Appendix A: Identifying the Large Economy

This appendix shows that the problem of selecting Q^L to match the impulse responses of the large and small economies to large economy shocks can be obtained in the same way as when only the impact on the large economy is considered, with some matrices redefined.

The problem is

The solution closely follows Schönemann (1966). In this appendix I adopt his notation, which is different to that throughout the rest of the paper. Let T ≡ Q, , and . I can then define two residual matrices E₁ = A₁T − B₁ and E₂ = A₂T − B₂. The problem can then be rewritten as:

where tr(.) denotes the trace.

The objective function is:

using that tr(A) = tr(A′).

The constraint can be rewritten as:

where ℒ is a matrix of Lagrange multipliers. The Lagrangian, g, then is:

Partially differentiating g with respect to the elements of T provides the first order conditions (apart from the constraint):

using the symmetry of and , and where 0 is a matrix.

Now P ≡ P₁ + P₂ and S ≡ S₁ + S₂. Note that as the sum of two symmetric matrices is symmetric, P is symmetric. Equation (A2) can then be rewritten as

and hence

Equation (A3) is the same as Equation (1.9) in Schönemann (1966). This shows that these first order conditions are the same as those to the original problem, with some of the matrices redefined. Consequently the solution is the same; the optimal T, T*, is obtained by a singular-value decomposition of S, which yields , and setting T* = WV′. In the notation used in this paper S corresponds to T^L, which equals . A singular value decomposition of T^L yields T^L = U^L W^L V^L′, and the solution for T* above corresponds to Q^L* = U^L V^L′.