Impulse-response function and 90% confidence interval for the effect of a one-unit increase in quantity demanded on the price k periods later under four different normalizations.
Contours of log likelihood for cointegrated system under three different normalizations.
Fruhwirth-Schnatter suggested plotting the posterior distributions under alternative normalizations to try to find one that best respects the geometry of the posterior.
Abstract: The issue of normalization arises whenever two different values for a vector of unknown parameters imply the identical economic model.
Key words: normalization, mixture distributions, vector autoregressions, cointegration, regime switching, numerical Bayesian methods, small sample distributions, weak identification
Less widely appreciated is the fact that normalization can also materially affect the conclusions one draws with likelihood-based methods.
To our knowledge, all previous discussions have dealt with normalization in terms of the specific issues arising in a particular class of models, with no unifying treatment of the general nature of the problem and its solution.
Section 6 summarizes our practical recommendations for applied research in any setting requiring normalization.
We can illustrate the key issues associated with normalization through the following example.
The obvious (and, we will argue, correct) normalization is to restrict [sigma] > 0.
If one had adopted this normalization, the question of whether [sigma] is large or small would not be of fundamental interest, and why a researcher would even want to calculate the posterior mean and standard deviation of [sigma] is not at all clear.
Again, if this is the normalization one had imposed, it is not clear why one would ever want to calculate an object such as [partial derivative][y.
In this example, these issues are sufficiently transparent that no researcher would ever choose such a poor normalization or fall into these pitfalls.
Our starting point is the observation that the normalization problem is fundamentally a question of identification.
In the absence of a normalization condition, the structure would typically be globally unidentified but locally identified.