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# Normalization

BIBLIOGRAPHY

The issue of normalization arises when the nature of an economic model is unaffected by a vector of structural parameters (coefficients) that can be arbitrarily scaled. This scaling is formally defined as normalization. A primary example is the simultaneous-equation framework of money supply and money demand:

In equations (1) and (2), Mt stands for money stock, Rt for the nominal interest rate, yt for real output or real income, and Pt for the price level. All variables are expressed in log value. The money-supply and money-demand shocks, and are random variables with zero mean and are independent of each other in probability distribution.

It has been long recognized that the parameter αs and βs in equations (1) and (2) can be normalized (scaled) in any way, with no consequence on economic interpretations of this equation. The conventional rule is to normalize the supply equation (1) as

With and to normalize the demand equation (2) as

With and

The coefficient ᾱ M is often interpreted as an interest elasticity of the money supply, β̄ R as an elasticity of demand for money, β̄ y as a money-demand elasticity with respect to changes in output or income, and β̄ p as an elasticity of demand for money with respect to changes in the price level. The following analysis for this example applies to any supply-demand framework in which Mt is replaced by the quantity of a commodity under consideration, Rt replaced by the price of this commodity, and yt and Pt replaced by any variables that affect supply but not demand.

In the 1970s econometricians began to recognize that how the supply or demand equation is normalized affects the estimator of the supply or demand elasticity (ᾱ M or β̄ R ) when the two-stage least squares (2SLS) approach is employed. The quality of this estimator is sensitive to the strength of instruments used in the 2SLS estimation, which in turn depends on whether the price variable or the quantity variable is normalized to be on the left-hand side of the supply or demand equation, as in (3) or (4). There are other methods that one can use to estimate the supply and demand equations. One dominating alternative is the full-information maximum likelihood (ML) approach. This approach used to be computationally infeasible for many practical problems. As computing technology improves over time, the ML approach has become more feasible to implement. One advantage of the ML approach over the 2SLS approach is that the economic meaning of the ML estimates will not be affected by normalization.

Not until the 1990s, however, did it become known that normalization matters to small-sample statistical inference about the ML estimates. Likelihood-based small-sample inferences are affected because normalization governs the likelihood shape around the ML estimates. A poor normalization can lead to multimodal distribution, disjoint confidence intervals, and very misleading characterizations of the true statistical uncertainty.

Related to this discovery, in the Bayesian econometric literature there have been theoretical results showing that normalization can lead to ill-behaved posterior distributions when a flat or symmetric prior is used. The empirical and policy significance of these results has been largely unexplored until very recently. Daniel Waggoner and Tao Zha (2003) and James Hamilton, Waggoner, and Zha (2007) show that normalization can alter economic interpretations of dynamic responses of the variables Mt and Rt to a supply or demand shock or in the above example. They use this and other examples to demonstrate that inadequate normalization may confound statistical and economic interpretations.

There are a variety of economic applications in which normalization plays an important role in likelihood-based statistical inferences. Unfortunately, there is no mechanical way to implement the best normalization across different models. As a practical guide, therefore, it is essential to report the small-sample distributions of parameters of interest rather than the mean and standard deviation only. Bimodal and wide-spread distributions are the first clue that the chosen normalization may be inadequate. Carefully chosen normalization should follow the principle of preserving the likelihood shape around the ML estimate. A successful implementation of this principle for normalization is likely to maintain coherent economic interpretations when statistical uncertainty is summarized.

SEE ALSO Bayesian Econometrics; Demand; Econometrics; Matrix Algebra; Maximum Likelihood Regression; Regression Analysis; Simultaneous Equation Bias

## BIBLIOGRAPHY

Hamilton, James D., Daniel F. Waggoner, and Tao Zha. 2007. Normalization in Econometrics. Econometric Reviews 26 (24): 221252.

Waggoner, Daniel F., and Tao Zha. 2003. Likelihood Preserving Normalization in Multiple Equation Models. Journal of Econometrics 114: 329347.

Tao Zha

# normalization

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normalization (nor-mă-ly-zay-shŏn) n. (in psychiatry) the process of making the living conditions of people with learning disabilities as similar as possible to those of people who are not handicapped. This includes moves to living outside institutions and encouragement to cope with work, pay, social life, sexuality, and civil rights.

# normalization

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normalization The normal distribution is one of the key distributions providing the basis for probability statistics, so that when a particular distribution is not normal, some transformation of the data may be attempted so as to achieve a normal distribution—for example charting the logarithms of values instead of the values themselves. This is known as normalizing the distribution.