Generalized Method of Moments (GMM) Estimator: A Robust Statistical Estimation Technique

A generalization of the method of moments estimator applicable when the number of moment conditions exceeds the number of parameters to be estimated, computed by minimizing the sum of squared differences between the population moments and the sample moments.

Historical Context

The Generalized Method of Moments (GMM) estimator was introduced by Lars Peter Hansen in 1982, significantly advancing econometrics and statistical analysis. It became a popular tool in empirical research, particularly in economics, finance, and related fields, due to its flexibility and robustness in parameter estimation when multiple moment conditions are available.

Types/Categories

• Linear GMM: Involves linear moment conditions and typically used in simpler models.
• Nonlinear GMM: Applies to models with nonlinear moment conditions, often used in more complex economic models.
• Dynamic GMM: Suitable for panel data with lagged dependent variables as instruments.

Key Events

• 1982: Introduction of GMM by Lars Peter Hansen.
• 1990s: Widespread adoption in empirical research, particularly in macroeconomics and finance.
• 2000s: Expansion of GMM applications to various fields, including dynamic panel data analysis.

Detailed Explanation

GMM is a generalization of the Method of Moments estimator. It is especially useful when the number of moment conditions (equations) exceeds the number of parameters to be estimated.

Mathematical Formulation

Given a vector of parameters \(\theta\) and moment conditions \(E[m_i(X, \theta)] = 0\), the GMM estimator \(\hat{\theta}\) is defined as:

$$ \hat{\theta} = \arg\min_{\theta} \left( \frac{1}{N} \sum_{i=1}^N m_i(X, \theta) \right)' W \left( \frac{1}{N} \sum_{i=1}^N m_i(X, \theta) \right) $$

where \(W\) is a positive semi-definite weighting matrix, and \(X\) represents the sample data.

Importance and Applicability

GMM is vital due to its flexibility:

• Applicable in both time series and cross-sectional data.
• Useful when traditional methods (like MLE) are difficult to apply or are inconsistent.
• Allows for robust parameter estimation even with heteroskedasticity or autocorrelation in error terms.

Examples

1. Economic Growth Models: Estimating parameters in a Solow growth model using national income data.
2. Financial Economics: Estimating risk premia in asset pricing models using return data.

Considerations

• Choice of Instruments: The validity and relevance of instruments critically affect GMM estimations.
• Over-identification Test: Use the Hansen (J) test to check for over-identifying restrictions.

Related Terms

• Method of Moments (MM): A simpler case of GMM with an equal number of moments and parameters.
• Maximum Likelihood Estimation (MLE): Another parameter estimation method, often compared with GMM.

Comparisons

• GMM vs. MLE: GMM is more flexible with fewer assumptions about the distribution of errors, but MLE is typically more efficient when the model assumptions hold.

Interesting Facts

• Lars Peter Hansen won the 2013 Nobel Prize in Economic Sciences for his development of GMM.

Inspirational Story

Lars Peter Hansen’s development of GMM was motivated by the need for more flexible econometric tools that could handle complex, real-world data. His groundbreaking work has inspired numerous advancements in economics and finance.

Famous Quotes

“Econometrics may be defined as the quantitative analysis of actual economic phenomena based on the concurrent development of theory and observation, related by appropriate methods of inference.” – Ragnar Frisch

Proverbs and Clichés

“Numbers never lie, but they don’t always tell the whole truth.”

Expressions, Jargon, and Slang

• Moment Conditions: Functions of data and parameters that represent theoretical relationships.
• Over-Identification: More moment conditions than parameters.
• Instrumental Variables: Variables used to achieve exogeneity in estimation.

FAQs

Q: What is the primary advantage of GMM over traditional estimation methods? A: GMM is flexible and can handle complex models with multiple moment conditions and endogenous variables.

Q: How is the weighting matrix chosen in GMM? A: The weighting matrix is often chosen to minimize the variance of the estimator, usually starting with an identity matrix and iteratively updated.

References

1. Hansen, L. P. (1982). Large Sample Properties of Generalized Method of Moments Estimators. Econometrica.
2. Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data. MIT Press.

Final Summary

The Generalized Method of Moments (GMM) Estimator is a powerful statistical tool essential in econometrics and finance for estimating model parameters when numerous moment conditions are present. Its flexibility, robustness to various data issues, and minimal assumptions make it a preferred choice for researchers dealing with complex empirical models.