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Atlanta Fed Working Papers


Likelihood-Preserving Normalization in Multiple Equation Models

Daniel F. Waggoner and Tao Zha
Federal Reserve Bank of Atlanta
Working Paper 2000-8
June 2000

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Causal analysis in multiple equation models often revolves around the evaluation of the effects of an exogenous shift in a structural equation. When taking into account the uncertainty implied by the shape of the likelihood, we argue that how normalization is implemented matters for inferential conclusions around the maximum likelihood (ML) estimates of such effects. We develop a general method that eliminates the distortion of finite-sample inferences about these ML estimates after normalization. We show that our likelihood-preserving normalization always maintains coherent economic interpretations while an arbitrary implementation of normalization can lead to ill-determined inferential results.

JEL classification: C32, E52

Key words: Bayesian methods, causal analysis, supply and demand, simultaneity, likelihood shape, equilibrium effects


The authors are indebted to John Geweke and Chris Sims, whose encouragement and suggestions have led to significant improvement of this paper. The authors have also benefited from discussions with Roberto Chang, Tom Cunningham, Joel Horowitz, Clive Granger, Eric Leeper, Adrian Pagan, Ellis Tallman, Chuck Whiteman, Arnold Zellner, and seminar participants at the University of Iowa, Indiana University, UCSD, and UCLA. 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 Daniel Waggoner, Economist, Research Department, Federal Reserve Bank of Atlanta, 104 Marietta Street, N.W., Atlanta, Georgia 30303-2713, 404/498-8278, daniel.f.waggoner@atl.frb.org, or Tao Zha, Senior Economist and Policy Adviser, Research Department, Federal Reserve Bank of Atlanta, 104 Marietta Street, N.W., Atlanta, Georgia 30303-2713, 404/498-8353, tao.zha@atl.frb.org.

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