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Atlanta Fed Working Papers

Normalization, Probability Distribution, and Impulse Responses

Daniel F. Waggoner and Tao Zha
Federal Reserve Bank of Atlanta
Working Paper 97-11
November 1997

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When impulse responses in dynamic multivariate models such as identified VARs are given economic interpretations, it is important that reliable statistical inferences be provided. Before probability assessments are provided, however, the model must be normalized. Contrary to the conventional wisdom, this paper argues that normalization, a rule of reversing signs of coefficients in equations in a particular way, could considerably affect the shape of the likelihood and thus probability bands for impulse responses. A new concept called ML distance normalization is introduced to avoid distorting the shape of the likelihood. Moreover, this paper develops a Monte Carlo simulation technique for implementing ML distance normalization.

JEL classification: C32, E52

Key words: ML distance normalization, likelihood, Monte Carlo method, posterior, impulse responses

The authors have benefited from discussions with John Geweke, Lars Hansen, Chuck Whiteman, and especially Chris Sims. The views expressed here are those of 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 F. Waggoner, Senior Quantitative Analyst, Federal Reserve Bank of Atlanta, 104 Marietta Street, N.W., Atlanta, Georgia 30303-2713, 404/498-8278, 404/498-8956 (fax), daniel.f., or Tao A. Zha, Economist, Federal Reserve Bank of Atlanta, 104 Marietta Street, N.W., Atlanta, Georgia 30303-2713, 404/498-8353, 404/498-8956 (fax),

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