Matrix Polynomial Factorization for Solving Vector Linear Rational Expectations Models and Computing Matrix Spectral Factorizations

• Activity Number: 310
• Type: Contributed
• Date/Time: Tuesday, August 11, 2015 : 8:30 AM to 10:20 AM
• Sponsor: Business and Economic Statistics Section
• Author(s): Peter Zadrozny*
• Companies: Bureau of Labor Statistics
• Keywords: theoretical and numerical factorization of matrix polynomials

Abstract

In economics, the key step in computing the solution of a multivariate or vector linear rational-expectations model (VLREM) is computing a dynamically-stable right factor of a left-right factorization of the lag-generating matrix polynomial of the model. The paper states conditions on the coefficients of a lag-generating matrix polynomial and proves that the conditions are necessary and sufficient for the polynomial to have a left-right factorization with a unique and dynamically-stable right factor. The result, thus, establishes necessary and sufficient conditions on finite-lag VLREMs under which they have unique, finite-lag, and dynamically-stable vector autoregressive (VAR) solutions. In statistical time-series analysis, the analogous problem is computing the left-right factorization of the finite-term matrix spectrum of a finite-lag vector moving-average (VMA) process such that the finite-lag right factor is the unique and miniphase representation of the process. Matrix spectral factorization (MSF) is a more specialized problem because the matrix polynomial to be factored has positive-definiteness and symmetry properties inherited from its data-autocovariance coefficient matric