Generalized Givens Rotations Applied to Complex Joint Eigenvalue Decomposition

This paper shows the different ways of using generalized Givens rotations in complex joint eigenvaluedecomposition (JEVD) problem. It presents the different schemes of generalized Givens rotation, justifies the introducedapproximations and focuses on the process of extending an algorithm developed for real JEVD to the complex JEVD.Several Joint Diagonalization problem use generalized Givens rotations to achieve the solution, many algorithmsdeveloped in the real case exist in the literature and are not generalized to the complex case. Hence, we show herein asimple and not trivial way to get the complex case from the real one. Simulation results are provided to highlight theeffectiveness and behaviour of the proposed techniques for different scenarios.

Combining sign and parametric restrictions in SVARs by utilising Givens rotations

This paper shows how to impose parametric restrictions in conjunction with sign restrictions to separate the shocks in SVARs. In sign restrictions, it is common to rotate an initial set of orthogonal shocks by utilising a Givens rotation matrix. In this paper, we show how to construct the Givens rotation matrix when parametric restrictions are part of the identification in sign restricted SVARs. The properties of Givens matrices are such that the parametric restrictions imply a system of equations which can be solved for the unknown parameters (or “angles”) in a rotation matrix, conditional on the values of the parameters which are drawn. The Givens rotation matrix formed in this manner is such that the parametric restrictions on the impulse responses are satisfied on each draw in sign restrictions. The method is demonstrated in an influential SVAR and is shown to generate results similar to those from a recent method which imposes the orthogonality and zero parametric restrictions on the columns of the rotation matrix in sign restrictions.