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Atlanta Fed Working Papers

Normalization in Econometrics

James D. Hamilton, Daniel F. Waggoner, and Tao Zha
Working Paper 2004-13
June 2004

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The issue of normalization arises whenever two different values for a vector of unknown parameters imply the identical economic model. A normalization does not just imply a rule for selecting which point, among equivalent ones, to call the maximum likelihood estimator (MLE). It also governs the topography of the set of points that go into a small-sample confidence interval associated with that MLE. A poor normalization can lead to multimodal distributions, confidence intervals that are disjoint, and very misleading characterizations of the true statistical uncertainty. This paper introduces the identification principle as a framework upon which a normalization should be imposed, according to which the boundaries of the allowable parameter space should correspond to loci along which the model is locally unidentified. The authors illustrate these issues with examples taken from mixture models, structural VARs, and cointegration.

JEL classification: C1

Key words: normalization, mixture distributions, vector autoregressions, cointegration, regime switching, numerical Bayesian methods, small sample distributions, weak identification

This research was supported by the National Science Foundation under Grant No. SES-0215754. Computer code used in this study can be downloaded free of charge from 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 James D. Hamilton, Department of Economics, 0508, University of California, San Diego, La Jolla, CA 92093-0508,; Daniel F. Waggoner, Research Department, Federal Reserve Bank of Atlanta, 1000 Peachtree Street N.E., Atlanta, GA 30309-4470, 404-498-8278,; or Tao Zha, Research Department, Federal Reserve Bank of Atlanta, 1000 Peachtree Street N.E., Atlanta, GA 30309-4470, 404-498-8353,

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