A, B, C’s (and D’s) for Understanding VARs

Jesús Fernández-Villaverde, Juan Francisco Rubio-Ramírez, and Thomas Sargent
Working Paper 2005-9
May 2005

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The dynamics of a linear (or linearized) dynamic stochastic economic model can be expressed in terms of matrices (A, B, C, D) that define a state-space system. An associated state space system (A, K, C, Σ) determines a vector autoregression (VAR) for observables available to an econometrician. We review circumstances in which the impulse response of the VAR resembles the impulse response associated with the economic model. We give four examples that illustrate a simple condition for checking whether the mapping from VAR shocks to economic shocks is invertible. The condition applies when there are equal numbers of VAR and economic shocks.

JEL classification: C32, E10

Key words: vector autoregression, economic shocks, innovations, invertibility

The authors thank James Nason and Mark Watson for very insightful criticisms of an earlier draft. The views expressed here are the authors’ and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. Any remaining errors are the authors’ responsibility. Please address questions regarding content to Jesús Fernández-Villaverde, University of Pennsylvania, 160 McNeil Building, 3718 Locust Walk, Philadelphia, PA 215-898-1504, jesusfv@econ.upenn.edu.; Juan Francisco Rubio-Ramírez, Federal Reserve Bank of Atlanta, 1000 Peachtree Street, NE, Atlanta, GA 30309, 404-498-8057, juan.rubio@atl.frb.org; and Thomas Sargent, New York University and Hoover Institution, 269 Mercer Street, New York, NY 10003, 212-998-3548, ts43@nyu.edu.